Inverse Spectra (North-Holland Mathematical Library)

Preface

The notion of an inverse sequence and its limit first appeared in the well-known memoir by Alexandrov [3], where a special case of inverse spectra - the so called projective spectra - was considered. The concept of an inverse spectrum in its present form was first introduced by Lefschetz in [209] and later, in full generality, in [210]. In the meantime Freudenthal [152] introduced the notion of a morphism of inverse spectra. The foundations of the entire method of inverse spectra were laid down in these basic works. The book of Eilenberg and Steenrod [143] contains a detailed discussion of (algebraic) inverse spectra and must be mentioned as well. Subsequently, inverse spectra began to be widely studied and applied not only in the various major branches of Topology, but also in Functional Analysis and Algebra. This is not surprising because of the categorical nature of inverse spectra and of the extraordinary power of the related techniques. In Topology inverse spectra have been used both for the construction of objects with special properties (synthetic applications) and for investigation of complicated objects by means of their spectral approximation by simpler ones (analytic applications). For instance, using projective spectra Alexandrov approximated compacta by complexes and this method allowed him to extend the basic results of the combinatorial topology of polyhedra to arbitrary compacta. The next significant success of analytic applications of inverse spectra was the investigation by Pontrjagin and A.Weil of the structure of compact groups using approximation by Lie groups. Pontrjagin also considered continuous transfinite s p e c t r a - the so called Lie series. Since their creation, inverse spectra have been extensively used for constructing various exotic examples of spaces and maps. We mention here the example of the open map of a one-dimensional compactum onto a two-dimensional one constructed by Kolmogorov [196]. Several counterexamples in dimension theory, as well as in the theory of cardinal invariants, have been produced using this approach (see, for example, [147]). It should be emphasized that all the special dimension-raising maps presented in Chapter 4 are also constructed using the techniques of inverse spectra. Marde~id made the next step in analytic applications of inverse spectra in

vi

PREFACE

[213]. His principal result states that every (non-metrizable) compactum admits a representation as the limit space of an inverse spectrum, consisting of metrizable compacta of the same dimension. The significance of this statement becomes clear after remarking that there exist [213] one-dimensional non-metrizable compacta which do not have such representations, consisting of polyhedra. Further work in this direction made possible the development of the dimension theory (including both Lebesgue and cohomological) of non-metrizable compact spaces. The work of Haydon [164] was a landmark in the development of the method of inverse spectra. Haydon proved that the class of compact absolute extensors in dimension 0 coincides with the class of Dugundji compacta. But undoubtedly the most significant ingredient of his work was the discovery of what are now called Haydon spectra. It should be emphasized especially that in the work of Alexandrov on the homological theory of compacta and in the works of Pontrjagin and A.Weil on compact groups all of the results are natural extensions of the results in the metric case. Marde~id's theorem, mentioned above, also has, to some extent, a "countable" counterpart - the theorem of Freudenthal [152] which states that every metrizable compactum is the limit of an inverse sequence, consisting of polyhedra of the same dimension. In this sense Haydon's method has an "uncountable" nature. It is based on a special spectral construction whose roots are in a simple set-theoretical principle known as "the effect of uncountability." This principle, extracted in its present form by S~epin (see [278]), serves as the foundation for the Spectral Theorem. This theorem is perhaps the most powerful tool of the whole theory. It states the following: if the limit spaces of two uncountable inverse spectra (with some additional non-restrictive properties) are homeomorphic, then these spectra contain isomorphic cofinal subspectra. The Spectral Theorem has no analog for inverse sequences: for example, two inverse sequences, consisting of even- and odd-dimensional cubes, produce the same limit space - the Hilbert cube - but, of course, they contain no isomorphic subspectra whatsoever. Theorems stating that "limit spaces of spectra, consisting of "good" spaces are "good"" were typical in spectral applications. The Spectral Theorem allows us to obtain theorems stating that "limit spaces of spectra, consisting of "bad" spaces or "bad" projections are "bad"." Detailed discussion of various versions (including compact and non-compact cases) of the Spectral Theorem is presented in Chapter 1. The remainder of the book illustrates the extraordinary strength of the spectral method. It shows exactly how this method works on a range of examples and what kinds of difficulties it helps to overcome. We present various problems from different areas of Topology where the Spectral Theorem is effective. The list of applications presented in Chapter 8 includes: topological characterizations of uncountable powers of the Cantor cube and of the separable Baire space; spectral representations of topological groups (including a simple proof of the existence of Lie series); topological characterization of locally convex linear topological spaces that are homeomorphic to the powers of the real line (generalization of

PREFACE

vii

the classical result of Anderson-Kadec); the Spectral Theorem in Shape Category; actions of non-metrizable topological groups and the structure of fixed point sets in non-metrizable manifolds; the Spectral Theorem for Baire maps (isomorphisms); connections with direct spectra and E-products. The reader can find several other particular applications of the spectral technique in other parts of the book as well. Manifold theory, or, more generally, the theory of absolute extensors, is one of the major areas of modern topology where the spectral approach has already shown its full strength. We discuss this theory in detail. We start with metrizable infinite-dimensional manifolds. This subject is almost completely covered in three excellent books written by C. Bessaga and A. Pelczyr~ski [32], by T . n . C h a p m a n [69] and by J. van Mill [228]. For this reason, in Chapter 2 we present an updated survey of this theory. The reader can find here proofs of several statements of Q-manifold and /2-manifold theories. We begin with the introduction of the concept of (strong) Z-sets and discuss all major results of these theories including Torur~czyk's characterization theorems of the model spaces: countable infinite powers of the closed unit interval [298] and the real line [299]. Chapter 3 contains a solution of the well-known and long standing problem concerning coincidence of Lebesgue and integral cohomological dimensions. Chapter 4 can be considered as an independent introduction to Menger manifold theory. We present a detailed discussion of this theory including the basic geometric constructions. We also present examples of n-soft maps of (n + 1)dimensional Menger compacta onto the Hilbert cube. These maps, constructions of which, as was mentioned above, are performed using the spectral techniques, are used both in Menger manifold theory and later in the general theory of absolute extensors. Unlike Menger compacta very little is known about NSbeling universal spaces. In Chapter 5 we illustrate the close ties between Menger and NSbeling spaces by showing how NSbeling spaces can be identified with the so-called pseudo-interiors of Menger compacta. Using the spectral approach we construct an n-soft map of n-dimensional NSbeling space onto the separable Hilbert space. In Chapters 6 and 7 we develop the general theory of absolute extensors in dimension n, n E w. The Spectral Theorem plays a crucial role in almost every statement here. As the culmination of this theory we present its two major elements - the topological characterizations of uncountable powers of the closed unit interval and of the real line (originally done by S~epin [277] and the author [84] respectively). Coupled with the corresponding results of Torur~czyk we therefore obtain topological characterizations of all infinite products of the closed unit interval and the real line. Each section is ended with Historical and Bibliographical notes. Throughout the text the reader can find discussions of several unsolved problems. This book significantly extends and updates the previously published (in Rus-

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PREFACE

sian) book written by the author [102] (with the help of V. V. Fedorchuk). Responding to a suggestion to translate [102] I instead decided to rewrite book completely in order to include several new topics and recent developments. The result of this work is before the reader. I wish to express my thanks to Dr. J. van Mill for his support. I also thank Dr. R. B. Sher who read the entire manuscript and made many valuable suggestions. Alex Chigogidze Saskatoon, CANADA

The author was supported in part by NSERC Grant # OGP0155552.

CHAPTER

1

Inverse Spectra

In most cases all necessary notions are defined in the text. Nevertheless we assume that the reader is familiar with basics of general topology as it is presented in [145] or in any other standard textbook. All spaces under consideration are assumed to be completely regular (i.e. Tychonov) and maps are continuous. D, N, I and R stand for the two-point (discrete) space, natural numbers, the closed unit segment and the real line respectively, w and wl denote the first infinite and the first uncountable cardinals respectively.

1.1. P r e l i m i n a r y

information

1.1.1. C o m p l e t e n e s s a n d c o m p a c t n e s s . We begin with the key notion of completeness of a metrizable space. A sequence {xn" n E w} of points in a metric space (X, d) is called a Cauchy sequence if for every e > 0, there exists a natural number k = k(e), such that d(xn, Xm) < e whenever n, m _ k. As can be easily seen every convergent sequence of points in any metric space is a Cauchy sequence, but not vice versa. DEFINITION 1.1.1. A metric space ( X , d ) is complete if every Cauchy sequence in (X, d) is convergent to a point of X . Not every metric, generating the topology of a given complete metrizable space, is complete. However, the very property of existence of at least one complete metric generating the given topology of a metrizable space is a topological invariant. In what follows, we call separable completely metrizable spaces the

Polish spaces. It is well-known that the product of countably many Polish spaces is a Polish space.

2

1. INVERSE SPECTRA PROPOSITION 1.1.2. The following conditions are equivalent f o r any subspace

Y of any Polish space X :

(i) Y is a Polish space. (ii) Y is a G6-subspace of X . Since the real line R is a Polish space we conclude by P r o p o s i t i o n 1.1.2 t h a t the space of rational n u m b e r s (considered as a subspace of R) is not Polish. At the same time irrational n u m b e r s form a Polish space. T h e validity of the last fact can also be seen as follows. Since the space of irrationals is h o m e o m o r p h i c to the c o u n t a b l e infinite power N W of the space N of n a t u r a l numbers, it suffices to note t h a t N itself, as a closed subset of R, is a Polish space and to observe once again t h a t the o p e r a t i o n of taking c o u n t a b l e p r o d u c t s m a i n t a i n s the p r o p e r t y of being a Polish space. Similarly, we see t h a t the countable infinite power R W of the real line R is also a Polish space. T h e spaces N W and R W will play key roles below. T h e following s t a t e m e n t is one of the most convenient definitions of a Polish space. PROPOSITION 1.1.3. The following conditions are equivalent f o r any space X : (i) X is a Polish space. (ii) X is h o m e o m o r p h i c to a closed subspace of the space R W . A space X has the Baire property if an intersection of any countable collection of dense open subsets of X is dense. This p r o p e r t y of the real line R was shown by Baire (which is why the p r o p e r t y bears his name). A much more general s t a t e m e n t is true. PROPOSITION 1.1.4. E v e r y Polish space has the Baire property. Let us discuss the properties of the space N W in more detail (by the way, this space a p p e a r s in the l i t e r a t u r e under the n a m e of the separable Baire space as well). First of all, we present a classical result of A l e x a n d r o v - U r y s o h n , which gives a topological c h a r a c t e r i z a t i o n of this space. THEOREM 1.1.5. The following conditions are equivalent f o r any space X : (i) X is h o m e o m o r p h i c to N W . (ii) X is a z e r o - d i m e n s i o n a l Polish space w i t h o u t compact open subspaces. It is very interesting t h a t the space N ~ plays the same role with respect to the class of zero-dimensional Polish spaces, as the space R W does with respect to the class of all Polish spaces. T h e following s t a t e m e n t can be viewed as an i l l u s t r a t i o n of this fact. PROPOSITION 1.1.6. The following conditions are equivalent f o r any space X : (i) X is a z e r o - d i m e n s i o n a l Polish space. (ii) X is h o m e o m o r p h i c to a closed subspace of the space N W .

1.1. PRELIMINARY INFORMATION

3

It is worth noting t h a t all Polish spaces can be described by means of the space N ~ . Indeed, let us observe, first of all, t h a t the real line R can be presented as an open image of N ~ (the reader can easily check this fact by analyzing the proof of L e m m a 6.4.1). Further, taking the countable infinite power of the such an open m a p p i n g we see t h a t there exists an open m a p p i n g f : N "~ ---, R W. Consider now an a r b i t r a r y Polish space X, which, by P r o p o s i t i o n 1.1.3, can be identified with a closed subspace of R W . Clearly, the inverse image f - l ( X ) is a zero-dimensional Polish space and consequently, the p r o d u c t f - l ( X ) x N ~ is h o m e o m o r p h i c to N ~ (apply T h e o r e m 1.1.5). Clearly, the composition of the projection f - l ( x ) x Y ~ --, f - l ( X ) and of the open m a p p i n g f - l ( X ) --, Z is open. Therefore X is an open image of N ~ . T h e converse s t a t e m e n t is also well-known. T h u s we have proved the following theorem. THEOREM 1.1.7. The following conditions are equivalent f o r any space X : (i) X is a Polish space. (ii) X is an open image o f the space N ~ . An i m p o r t a n t subclass of the class of Polish spaces is formed by the c o m p a c t (metrizable) spaces. The theory of these spaces is one of the most developed areas of general topology, and for this reason we restrict ourselves to a very brief discussion. The C a n t o r cube D ~ and the Hilbert cube I ~ are basic objects for the entire class of (zero-dimensional) metrizable compacta. Their roles are emphasized by the following s t a t e m e n t s (compare with Propositions 1.1.3, 1.1.6). PROPOSITION 1.1.8. The following conditions are equivalent f o r any space X : (i) X is a metrizable c o m p a c t u m . (ii) X is h o m e o m o r p h i c to a closed subspace o f the Hilbert cube I ~ . PROPOSITION 1.1.9. The following conditions are equivalent f o r any space X : (i) X is a z e r o - d i m e n s i o n a l metrizable c o m p a c t u m . (ii) X is h o m e o m o r p h i c to a closed subspace of the C a n t o r cube D ~ . Here is a topological characterization of D ~ . THEOREM 1.1.10. The following conditions are equivalent f o r any space X : (i) X is h o m e o m o r p h i c to D ~ . (ii) X is a z e r o - d i m e n s i o n a l metrizable c o m p a c t u m w i t h o u t isolated (i.e. open) points.

We finish our brief discussion with the following surjective characterization of metrizable compacta. THEOREM 1.1.11. The following conditions are equivalent f o r any space X : (i) X is a metrizable c o m p a c t u m . (ii) X is a surjective image of D ~ .

4 1.1.2. R e a l c o m p a c t nition.

1. INVERSE SPECTRA s p a c e s . We begin with the following important defi-

DEFINITION 1.1.12. A space X is called realcompact if it is homeomorphic to a closed subspace of some power of the real line. Realcompact spaces have been introduced by Hewitt [170] (under a somewhat different name). The same class of spaces has been independently defined in terms of uniformities by Nachbin [239]. It is necessary to note that realcompact spaces appear in the literature under different names, such as: realcomplete spaces, Hewitt spaces, Q-spaces, R-spaces. This class of spaces plays a very i m p o r t a n t role in the sequel. Therefore, for the reader's convenience, we discuss some of their properties here. The following s t a t e m e n t is the straightforward consequence of the above definition. PROPOSITION 1.1.13. Every closed subspace of a realcompact space is realcompact, as is the product of an arbitrary family of realcompact spaces. COROLLARY 1.1.14. Let {Xt : t E T} be a family of realcompact subspaces of a space X . Then the intersection M{Xt :t E T} is realcompact. PROOF. Obviously, the intersection M{Xt : t E T} is homeomorphic to the intersection of the subspaces Y I { x t : t E T} and A x (the diagonal of the product x T ) . Consequently, the indicated intersection is closed in the product I-I{Xt : t E T}. It only remains to apply Proposition 1.1.13 ['-! COROLLARY 1.1.15. A functionally open subspace of a realcompact space is realcompact. PROOF. Let G be a functionally open subspaces of a realcompact space X. Consider the embedding i - A { ~ : ~ E C ( X ) } : X ~ R c ( z ) and identify the space X with its homeomorphic image i ( X ) in R e(X). It follows from the definition of the embedding i that X is C-embedded in R C(x). Consequently, there exists a functionally open subset V of R C(X) such that G -- V MX. Further, by Corollary 1.1.14, it only remains to show that any functionally open subspace V of an arbitrary power R A of the real line is realcompact. If I A I_< w, then V, as an open subspace of a Polish space, is realcompact. Suppose now that ]A ]> w. Since any continuous real-valued function defined on R A depends on countably many coordinates (see, for example, [145]), we can easily conclude that V = VB • R A-B, where B is a countable subset of A and VB is an open subspace of the product R B. It only remains to apply Proposition 1.1.13. [:] PROPOSITION 1.1.16. A realcompact and C-embedded subspace of a space is closed.

1.1. PRELIMINARY INFORMATION

5

PROOF. Let X be a r e a l c o m p a c t a n d C - e m b e d d e d subspace of a space Y. By Definition 1.1.12, there exist a set A and a h o m e o m o r p h i s m f : X ~ X ~ onto t h e closed subspace X ~ of the p r o d u c t R A. D e n o t e by ~ra : R A ---, Tla -- R, a E A, the projection onto the a - t h coordinate. Since X is C - e m b e d d e d in Y, for each a E A, t h e r e is a function ~ a E C ( Y ) such t h a t ~ I X = ~ a " f . Let = A { ~ a :c~ E A}. Clearly ~p m a p s t h e space Y into R A and ~ I X = f . It follows from t h e c o n t i n u i t y of ~ t h a t

~ ( c I y X ) C_ c I R , ~ ( X ) = c l n , f ( X )

= clR, X' = X'.

In o t h e r words, t h e restriction g - ~ I c l y X m a p s t h e closure c I y X onto X ~. In this s i t u a t i o n t h e c o m p o s i t i o n r = . f - l . g . c l y X ----, X is a r e t r a c t i o n . Hence, c l y X = X . [7 DEFINITION 1.1.17. A subspace X of a space Y is said to be z - e m b e d d e d in Y if every functionally open (closed) subset G of X can be written as G = X M G, where G is a f u n c t i o n a l l y open (closed) subset of Y . C - e m b e d d e d subspaces (closed subspaces of n o r m a l spaces, for e x a m p l e ) are z - e m b e d d e d . In a perfectly n o r m a l space every s u b s p a c e is z - e m b e d d e d . LEMMA 1.1.18. A functionally open subspace of any space is z-embedded. PROOF. Let X be a functionally o p e n subspace of a space Y. F i x a nonnegative function ~ E C ( Y ) such t h a t Y - X = {y E Y ' ~ ( y ) = 0} a n d consider a functionally closed subspace Z in X . O u r goal is to find a f u n c t i o n a l l y closed s u b s p a c e Z of Y such t h a t Z = X M ~5. Fix a n o n - n e g a t i v e function r E C ( X ) such t h a t Z = {x E X " r

= 0}. Define the function h " Y ~ R as follows:

ifyEY--X,

0, h(y)=

min{~(y)

,r

ifyEX.

Let Z = {y E Y 9 h(y) - 0}. Obviously, Z -- X M Z. Therefore, it only r e m a i n s to show c o n t i n u i t y of t h e function h. T h e c o n t i n u i t y of h at points of X follows from our definitions. If y E Y - X a n d e > 0, t h e n

(y 9 y . h(y) < ~} = (y e y . v(y) < ~} u (~ e X . r

< ~}.

T h e c o n t i n u i t y of ~ implies t h e o p e n n e s s of the first t e r m on t h e right side of this equation. T h e second t e r m is also o p e n in Y. This i m m e d i a t e l y follows from t h e c o n t i n u i t y of r a n d t h e o p e n n e s s of X in Y a n d c o m p l e t e s t h e verification of t h e c o n t i n u i t y of h. K1 LEMMA 1.1.19. A LindelSf subspace of an arbitrary space is z-embedded.

6

1. INVERSE SPECTRA

PROOF. Let X be a LindelSf subspace of a space Y and G be a functionally open subspace of X. For each point x E G take a functionally open subspace V~ such t h a t x E V x M X c_ G. From the open cover { V x M X 9 x E G} of the LindelSf space G select a countable subcover {Vn M X 9 n E w}. Clearly, G = X M (~, where (~ = M{Vn 9 n E w}. It only remains to note t h a t the set (~, as a countable union of functionally open subspaces, is itself functionally open in Y. [::] It should be noted t h a t z-embeddability of certain subspaces completely determines the class of so-called perfectly a-normal spaces. First, we give the corresponding definition. DEFINITION 1.1.20. A space is called perfectly a-normal if the closure of each open subset is f u n c t i o n a l l y closed. Obviously, perfectly normal spaces are perfectly a-normal. It can be shown [274] t h a t the p r o d u c t of an a r b i t r a r y family of metrizable spaces is also perfectly a-normal. PROPOSITION 1.1.21. The following conditions are equivalent for any space X:

(i) X is perfectly a - n o r m a l . (ii) All dense subsets of X are z-embedded. (ii) All open subsets of X are z-embedded. PROOF. (i) =~ (ii). Let Y be a dense subset of X. It is enough to show t h a t for every pair (Z1,Z2) of disjoint functionally closed subsets of Y there are functionally closed subsets F1 and F2 of X such t h a t F1 M F2 M Y -- 0 and Zi C Fi, i = 1,2. Consider open (in Y) neighborhoods G1 and G2, of Z1 and Z2 respectively, with disjoint closures. Since Y is dense in X we conclude t h a t the sets Fi -- c l x G i , i = 1,2 are canonically closed in X. Therefore, by (i), these sets are functionally closed in X. Clearly, F1 M F2 M Y -- 0 and Zi C Fi, i -- 1,2. (ii) ~ (ii). Let Y be an open subset of X. Clearly, the subset A -- Y U ( Y - c l x Y ) is dense in X. It can be easily seen t h a t each functionally closed subset Z of Y is functionally closed in A. By (ii), there is a functionally closed subset F in X such t h a t F M A -- Z. But then F M Y -- Z. (ii) =~ (i). Let Y be an open subset of X. As above, consider the set A -- Y U ( Y - c l x Y ) . A is open in X and consequently, by(//), is z-embedded in X. T h e set Y is functionally closed in A. Therefore, there is a functionally closed subset Z of X such t h a t Y - Z M A. Clearly, Z - c l x Y . F3 PROPOSITION 1.1.22. Let Y

be a z-embedded subset of a space X .

Then,

f o r each f u n c t i o n ~o e C ( Y ) there exist a countable f a m i l y { G n " n E w} of f u n c t i o n a l l y open subsets in Z and a f u n c t i o n r E C ( M { G n " n E w}) such that Y C_ n { G n " n 6 w} and O l Y - ~o.

1.1. PRELIMINARY INFORMATION

7

PROOF. Since every space can be identified with a C - e m b e d d e d subspace of R r for some T > w (see the proof of Corollary 1.1.15) we can assume w i t h o u t loss of generality t h a t X - R r. First we consider the case of a b o u n d e d function ~, i.e. ] ~ ]_ a, where a is a non-negative real number. We follow the proof of the well-known t h e o r e m of Brouwer-Tietze-Urysohn (see, for example, [7]). Let ~0 -- ~ and consider functionally closed subsets A0 -- {y E Y" ~0(Y) ~_ - ~ } and B0 -- {y E Y" ~0(y) >__ } of Y. Since, by our assumption, Y is z - e m b e d d e d in X, there exist functionally closed subsets A~ and B~ of X such t h a t A0 - A~ M Y and B0 -- B~ M Y. Let Go = X - (A~ M B~). Obviously, Go is functionally open in X and Y _C Go. Note also t h a t the sets fi-o = A~ M Go a n d / ~ 0 = B~ M Go are disjoint a n d functionally closed in Go. Consequently, there exists a function go E C ( G o ) such a a Ao = g 0 1 ( - ~ ) and /~0 = go 1 (~). Let ~1 = ~ o - g o / Y . t h a t I go I-< 3, It can be easily seen t h a t I ~1 [_< 23~. Continuing this process we o b t a i n a sequence of functions ~n E C ( Y ) satisfying the inequalities ] ~ n ]_< (5)2ha, n E w. We also have a sequence gn E C ( G n ) , n E w, (where Gn is a functionally open subset of X containing Y and Gn C Gn+l for each n E w) of functions satisfying the following conditions: I gn i_ A2 if and only if A1 D A2. W i t h this partial order expA becomes a directed set. If we consider only those subsets of the set A which have cardinality _ r, t h e n the c o r r e s p o n d i n g subcollection of expA, d e n o t e d by exprA, serves as a basic e x a m p l e of a T-complete set. Let us consider a second way of c o n s t r u c t i n g T-complete sets. Let A be an a r b i t r a r y directed set. We call two c o u n t a b l e chains in A equivalent if each of t h e m m a j o r a t e s the other. T h e collection of all equivalence classes of c o u n t a b l e chains in A is d e n o t e d by [A]~ and is called the w-completion of A. T h e order relation on [A]~ is defined by the relation of "majorantness"" one class majorates another if one of the chains in the first class majorates a chain belonging to the second class. It is easy to see that with respect to the described order the set [A]w is w-complete. Assigning to each element a E A the countable stationary chain (consisting of one element (a)) we obtain a natural embedding of the set A into its w-completion. If, for all A < T, the A-completions [A]), of A have already been defined, then the T-completion [A]r can be defined as the set of equivalence classes of chains of cardinality _< T from the union 0([A]~" A < T).An order on [A]r, as above, is generated by the relation of majorantness. Straightforward verification shows that the T-completion of an arbitrary directed set is T-complete for each infinite cardinal T.

10

1. INVERSE SPECTRA

PROPOSITION 1.1.27. Let { A t " t E T } be a collection of T-closed and cofinal subsets of a T-complete set A . T h e n the intersection M{At " t E T} is also cofinal and T-closed in A . PROOF. W i t h o u t loss of generality we can assume t h a t the set T is wellordered. Consider the p r o d u c t w x T endowed with the lexicographic order. Let a be an a r b i t r a r y element in A. In an obvious way, by transfinite induction, we define a chain B -- {a(n,t)" (n, t) E w x T} in A so t h a t the following conditions are satisfied: (i) a(n,t) >_ a for each (n, t) E w x T. (ii) a(n,t) _b},bEA.

PROPOSITION 1.1.29. Let A be a T-complete set, L C A 2, and suppose the following three conditions are satisfied: E x i s t e n c e : For each a E A there exists b E A such that (a,b) E L . Majorantness: I f (a, b) e L and c >_ b, then (a, c) E n . T - c l o s e d n e s s : Let {at " t e T } be a chain in A with a -- sup{at " t E T}. I f (at, c) E n f o r s o m e c e A and each t E T , then (a,c) E L . T h e n the set of all L-reflexive elements of A (an e l e m e n t a E A is said to be L-reflexive if (a, a) E L ) is cofinal and T-closed in A .

PROOF. First of all let us show t h a t the set of all L-reflexive elements is cofinal in A. Let a E A. We are going to c o n s t r u c t a chain B -- {ba" a < r}, indexed by all ordinals strictly less t h a n T and satisfying the following conditions: (a) b0 -- a. (b) (b~,ba) E L, whenever ~ < a < T. (C) b~ _< ba, whenever fl < a < T. Suppose t h a t the elements bz have already been c o n s t r u c t e d for all/~ smaller t h a n some a, where a < T. Let us c o n s t r u c t the element ba. If a is a limit ordinal t h e n let ba = sup{bz" ~ < a}. Clearly, ba > bt~ for e a c h ~ < a. Note t h a t ba >_ b~+l _> b~ and (b~,b~+l) E L. Therefore, by the m a j o r a n t n e s s condition, (b~, ba) E L for each ~ < a. If a is non-limit ordinal, t h e n a - ~ + 1. By the existence condition, there exists an element c E A such t h a t (b~, c) E L. Since A is directed we can find

1.2. INVERSE SPECTRA

11

an element b~ which majorates both bE and c, i.e. ba > bE and ba > c. The majorantness condition implies that (bE, b~) E L. This finishes the construction of the chain B. Let b = sup{ba: c~ < T). Let us show that b is a L-reflexive element. Indeed, by the majorantness condition, (b~,b) E L for each c~ < T. Therefore, by the closeness condition, (b, b) E L. This shows cofinality of the set of all L-reflexive elements in A. If C = {at: t E T ) is a chain, consisting of L-reflexive elements, I T I< T and a -- s u p C , then, by the majorantness condition, we have (at, a) E L whenever t E T. Finally, by the closeness condition, (a, a) E L. I--1

Historical and bibliographical notes 1.1. The results of subsection 1.1.1 are widely known. For example, Proposition 1.1.2 has been proven by Alexandrov in [2] (see also [208]). The characterization theorems 1.1.5 and 1.1.10 have been obtained by Alexandrov and Urysohn [8] and Brouwer [56] respectively. Theorems 1.1.7 and 1.1.11 belong respectively to Hausdorff and Alexandrov. Propositions 1.1.13, 1.1.16, as well as Corollary 1.1.14, have been proved by Shirota [281] (see also [185]). z-embedded subspaces were systematically studied in [40~ 41~ 42~ 10]. In particular, these references contain proofs of Lemmas 1.1.18 and 1.1.19. The property included in the definition of perfectly R-normal spaces was apparently known to Bokshtein (see also [144, 261]). The concept of perfect R-normality itself was independently introduced (under different names) by Blair [40] and Terada [292]. Shortly after the same notion was also defined by SEepin [274]. Propositions 1.1.21 and 1.1.22 appear in [40] and [42] respectively. Proposition 1.1.24 was in fact proved in [236]. Corollary 1.1.25 belongs to Hewitt [170]. Originally Proposition 1.1.26 was obtained in [40] and later rediscovered in [74]. Earlier, in the presence of normality of the given space coupled with closeness of summands, Proposition 1.1.26 was proved in [236]. It should be noted that the assumption of z-embeddedness of the summands in Proposition 1.1.26 is essential [237, 238]. The results of Subsection 1.1.3 were originally obtained in [278]; see also [107].

1.2. D e f i n i t i o n s a n d e l e m e n t a r y p r o p e r t i e s of i n v e r s e s p e c t r a If I ] { x a 9 c~ E A} is a Cartesian product of topological spaces Xa, c~ E A, and B _C A then ~ r B ' I - ] { X ~ ' a E A } ~ I ] { X ~ ' ~ E B } denotes the standard projection onto the corresponding subproduct. The same meaning is given to the projection 7rg 9 l l { X a 9 c~ E B} ~ l']{Xa 9 c~ E C}, C C B C A. For simplicity, sometimes we also use the following notation: X B = I ' ] { X ~ " ~ E B } , where B C_ A. We will maintain these notations throughout the whole text.

12

1. INVERSE SPECTRA

1 . 2 . 1 . D e f i n i t i o n o f i n v e r s e s p e c t r a . Let A be a directed set and suppose t h a t to each c~ E A there corresponds a topological space X a in such a way t h a t whenever/~ ~- c~, a continuous map P~a " X~ ---+ X a is also given. Suppose further that: 9 p ~ - p ~ - p~ for each triple (c~, ~, ~) of indexes with c~ ___ ~ ~ ~,.

9 pg - idx,~ for each c~ E A. In this s i t u a t i o n we say t h a t the inverse spectrum (inverse system) or simply spectrum S x = { X a , p ~ , A } is given. We call the spaces X a elements of the s p e c t r u m . T h e m a p p i n g s p~ 9 X~ ---+ X a are called projections (or bonding mappings) of the spectrum. An inverse sequence is a s p e c t r u m whose indexing set coincides wii;h w (the first infinite cardinal directed by its n a t u r a l order). A point {x~ 9 a E A} of the C a r t e s i a n p r o d u c t YI{x~ 9 c~ E A} of spaces X a is called a thread of the s p e c t r u m S x = { X a , p ~ , A } if p~(x~) = xa for any a , ~ E A such t h a t a ~ ~. T h e subspace of the p r o d u c t r I { x a 9 a E A} consisting of all t h r e a d s is called the inverse limit (or, simply, limit) of the s p e c t r u m S x = { X a , p ~ , A } and is d e n o t e d by l i m S z or, more explicitly, by lira { X a , p ~ , A } . By pc" l i m S x ~ X a , c~ E A, we denote the restriction of the projection r{~}" l-[{Xa 9 a E A} ---. X~, and we call this the a - t h limit projection of the s p e c t r u m S x . By definition of the p r o d u c t topology, the sets of the form lim S x M {~r~l(G~)" E B}, where B is a finite subset of the indexing set A and G~ is an open set in X~, c o n s t i t u t e a base of the limit space l i m S x . Let us show t h a t in fact we can be more specific. PROPOSITION 1.2.1. The collection of all sets p ~ l ( G a ) , where Ga is open in X a and ~ E A, forms a base for the topology of the limit space of the spectrum lim S x . PROOF. Let G = l i m S x Cl {~r~l(G~)" ~ E B}, where B is a finite subset of the indexing set A and G~ is open in X~. Take an index c~ E A such t h a t a ~ ~ for each ~ E B. T h e n the set G~ = M { ( p ~ ) - I ( G ~ ) 9 /3 E B} is open in Z a , and it easy to see t h a t G = p-~l(Ga).

K1

PROPOSITION 1.2.2. The limit space of a spectrum 8 x

-- {Xa,p~a,A} is a

closed subspace of the product l-I{x~ " a E A}. PROOF. Let x = {xa} E Y I { x a : a E A} - l i m S x . T h e n there exist indexes c~,~ E A such t h a t ~ > c~ and x~' - p~(xr ~ xa. Consider disjoint open subsets V and V ~ in X a such t h a t x~ E V ~ and xa E V. Clearly, the set U - ( p ~ ) - l ( Y ' ) is an open n e i g h b o r h o o d of the point x~ in X~. Therefore, the set W = 7r~-l(Y) M 7r~l(U) is an open n e i g h b o r h o o d of x in 1-I{xa 9 c~ E A}. It only remains to note t h a t W M l i m S x - 0.

K]

1.2. INVERSE SPECTRA

13

COROLLARY 1.2.3. The limit space of an inverse spectrum consisting of completely regular spaces is completely regular. Consider now a s p e c t r u m S x = { X a , p ~ , A} and a subspace Y of its limit space lira S x . T h e subspace Y defines the inverse s p e c t r u m S y = {Ya, q~, A } , where Ya = pa(Y), a E A, and q~ denotes the restriction p ~ / X ~ , ~ ~ a. In this s i t u a t i o n we say t h a t the s p e c t r u m S y = {Ya, q~, A} is induced by Y. Obviously, the p r o d u c t 1-I{Y~" a E A} is a subspace of the p r o d u c t Y I { x ~ . a E A} a n d the t h r e a d s of the s p e c t r u m S y are t h r e a d s of the s p e c t r u m S x . Therefore the limit space Y = lim S y is n a t u r a l l y e m b e d d e d in the limit space X -- lim S x . We claim t h a t = M{p~l(Va) 9 a E A}. Indeed, if x = { x a ' a E A} E M{p~I(Ya)" a E A}, t h e n x a E Ya for each a E A. This shows t h a t the t h r e a d x of the s p e c t r u m S x is in fact a t h r e a d of the s p e c t r u m By. Therefore, M{p~l(Ya) 9 a E A} C 9 . Conversely, if x E 9 , t h e n pa(x) E Ya for each a E A. Therefore, 9 C M{p~l(Ya) 9 a E A} and we have proved the following s t a t e m e n t . PROPOSITION 1.2.4. Let Y be a subspace of the limit space of a spectrum S x = { X a , p ~ , A } . Then the limit space lim S y of the induced (by Y ) spectrum SY = {Ya, q~, A} contains Y. The equality Y = lim S y holds if and only if Y = M{p~ 1 (Ya)" a E A}.

COROLLARY 1.2.5. Let Y be a subspace of the limit space X of a spectrum S x = {Xa,p~a,A} 9 Then the limit space of the spectrum {clx~(pa(Y), p~a/cIx~(pz(Y),A} coincides w~th c I x Y . PROOF. A p p l y P r o p o s i t i o n s 1.2.1 and 1.2.4

0

COROLLARY 1.2.6. Let Y be a closed subspace of the limit space of a spectrum S x = {Xa, p~, A } . Then Y = lim { p a ( Y ) , p ~ / p ~ ( Y ) , A }

and Y = lim {clx~ (pa(Y)),p~a/clxa(p~(Y)),A}. Suppose now t h a t a s p e c t r u m S z = {X~,p~a, A} is given and A' is a cofinal subset of the indexing set A. In this case we say t h a t the s p e c t r u m S~c = {Xa, p~, A'} is a cofinal subspectrum of S x . PROPOSITION 1.2.7. Let S~x = {Xa,p~a,A '} be a cofinal subspectrum of the spectrum S x = {Xa, p~, A} . Then the map .f " lim S x --~ lim S~: consisting of restricting all threads from X = lim S z to those from X ~ = lira S~: is a homeomorphism of X onto X ~.

14

1. INVERSE SPECTRA

PROOF. We c o n s t r u c t a m a p f~" X ~ --, X inverse to f. Let x ~ = {xa" For each a E A let x,~ = p ~ ( x / 3 ) , wheref~ ~ c~ a n d f ~ E A'. We claim t h a t the point x a does not d e p e n d on the choice of index f~ E A ~. Indeed, consider a t h i r d index -y E A ~ such t h a t ~/ ~ a. Since A is directed and A ~ is cofinal in A, t h e r e is a n o t h e r index 5 E A ~ such t h a t 5 __ f~ and 5 ~_ ~. T h e fact t h a t x ~ is a t h r e a d of the s p e c t r u m S ~ implies t h a t

E A'} E X'.

=

=

=

=

T h i s shows t h a t x = { x a ' a E A} is a well-defined point of the p r o d u c t 11 {Xa" a E A}. In fact, x is a t h r e a d of the s p e c t r u m 8 x . Indeed,

p/3p~ Thus, to each point x ~ E X ~ we have assigned a point x = f~(x ~) E X . It follows i m m e d i a t e l y from t h e above definition t h a t f f ~ ( x ~) -- x ~ for each x ~ E X ~ a n d f ~ f ( x ) = x for each x E X . Therefore, f is a bijection and f~ is the inverse off. Next we show t h a t f is an open m a p (the continuity of f is obvious). Let G a be an open subset of Xa, a E A, and f~ _ c~, where f~ E A ~. T h e n =

Therefore, the sets of the form p~l(Gf~), where f~ E A ~ and Gf~ is open in Xf~, c o n s t i t u t e a base for the t o p o l o g y of X (compare with P r o p o s i t i o n 1.2.1). Denote by qf~ the ~ - t h limit projection of the s p e c t r u m 8 ~ . T h e n

p ~ ( G / ~ ) - {x = { x , " a E A}" x/~ E G/~} and

q~l(a/3) - { x ' -

{xa" ~ E A'}" x/3 E a ~ } .

Consequently,

f p ~ l ( G / 3 ) -- q~l(Gf~). This shows t h a t f is a h o m e o m o r p h i s m .

[::]

COROLLARY 1.2.8. Let S x = { X a , p ~ , A }

be an inverse spectrum whose indexing set A has a maximal element ~. Then the inverse limit lira S x of S x is homeomorphic to the space X a . EXAMPLE 1.2.9. Suppose we are given a family {Xt" t E T } of spaces with I T I> - w. Obviously the set exp 0, ~1 < e and l)n E coy(X) be an open cover of X such t h a t f - l ( ] 2 n ) refines bin. Take an element V E Vn c o n t a i n i n g t h e point x. Since lim Xk -- x, t h e r e is an integer m such t h a t Xk E V for each k > m. Let U E bin such t h a t f - l ( V ) C U. Obviously, Yk E U for each k _> m. B u t dicing(U) < e. Therefore d(yk, Yk,) < e whenever k , k ' >_ m. This shows t h a t t h e sequence {yk" k E N } is a C a u c h y sequence. Let y = lim Yk. By the c o n t i n u i t y of f , x = f ( y ) E f ( Y ) . Therefore t h e set f ( Y ) is closed in X . T h e r e m a i n i n g p a r t of our s t a t e m e n t follows from L e m m a 2.1.1. Z] T h e proof of t h e following s t a t e m e n t is trivial. LEMMA 2.1.3. Let A be a closed subset of a Polish space Y , bt E cov(Y) and f E C ( Y , X ) . If the restriction f / A : A ---. X is a b i / A - m a p ( h e r e / d / A = {U M A: U E bi} E coy(A)), then there exists an open neighborhood G of A in Y such that the restriction f / G : G ~ X is a U / G - m a p .

2.1. ABSOLUTE EXTENSORS AND ABSOLUTE RETRACTS

35

In order to obtain other, not so trivial, properties of the space C(Y, X ) we describe the topology introduced above in a somewhat different way. It is easy to see that every bounded metric d on the space X generates the sup-metric d on C(Y, X). The definition is as follows:

d(f,g) = sup{d(f(y),g(y)): y E Y } , f , g e C(Y,X). The topology generated by the metric d on the set C (Y, X) is called the topology

of d-uniform convergence. By M etr(X) we denote the collection of all bounded metrics on a space X generating the topology of X. The collection {d: d E M e t r ( X ) } generates a topology on C(Y, X) which is precisely the topology of uniform convergence with respect to all metrics. Obviously, this topology is stronger than the topology of d-uniform convergence for each particular metric d. LEMMA 2.1.4. The limitation topology coincides with the topology of uniform convergence with respect to all metrics. PROOF. First, suppose that G is a neighborhood of f E C(Y, X) with respect to the topology of uniform convergence. This means that there exist a number e > 0 and a bounded metric d E M etr(X) such that

S ~ {g e c(Y, x ) : ~(f, g) < ~} c_ a. It is easy to check that B(f, bl) C_ {g E C ( Y , X ) : d ( f , g ) < e}, where U = {Bd(x, ~" x E X } (here Bd(x, r) denotes the open d-ball of radius r with center at x). For the converse, consider a neighborhood B, f, L/) of the map f in the limitation topology. Take a metric d E M e t r ( X ) such t h a t {Sd(x, 1): x E X } refines L/ (see, for example, [32, Theorem 4.1]). Obviously, {g E C(Y, X ) : d(f, g)
a. (c) Let V1 and V2 be open subsets of y N, B be a closed subset of V1 and B C V2 C_ 1/'1. Then f o r any two f u n c t i o n s a l E C(Vl,(0, c~)) and ~2 e C(V2, (0, oo)), there exists a stearing f u n c t i o n Ak on Vk, k = 1,2, such that A 1 / B = A 2 / B and Ak > ak, k = 1,2. (d) Let A be a stearing f u n c t i o n on an open subset V C y g . Suppose that h" y g x Y x [1, oo) ~ y N is a displacement isotopy for Y . Suppose also that on y g we have the metric oo

d({yn}, {y~}) = E

min{l'

P(Yn, Ytn)},

n=l

where p is a metric on Y . I f 2 -x(x) < 89 y N - V) for each x E V, then the map f " V x Y --~ y N, defined by letting

f ( ~ , y) = h(~, y, ~(~)),

(~, y) e v • Y

is a h o m e o m o r p h i s m of the product V x Y onto V. In addition, d(f(x,y),x)

< 2 -x(x)+l for each ( x , y ) E V x Y.

PROOF. Conditions (a)-(c) are obvious. Let us verify condition (d). For a point (x, y) e Y x Y take a number n such that A(x) e [n, n + 1]. Then, by the definition of displacement isotopy, we have the following two conditions:

(~) ~ s ( ~ , y) = ~ ( x ) . (ii) A f ( x , y ) = A(x).

2.3. R ~- AND I "~-MANIFOLDS

65

T h e first condition implies t h a t g ( f (x, y), x) < 2 - n < 2-~(z) +1 for all points ( x , y ) 9 Y • Y. In particular, f ( Y • Y ) c Y . T h e second condition implies t h a t the m a p g ( x ) - ( h / ( Y g x Y x {A(x)})) -1, x 9 V, sends the set Y into the p r o d u c t V x Y. Moreover, one can see t h a t g is an inverse of f. W1 PROPOSITION 2.3.8. I f a space Y a d m i t s a d i s p l a c e m e n t isotopy, then every open subspace of Y N is strictly Y - s t a b l e . PROOF. Let U be an open subspace of y N and A and B be disjoint closed subsets of U. Also let a E C ( U , (0, c~)). Our goal is to c o n s t r u c t a homeom o r p h i s m f : V • Y ~ (U • Y ) A such t h a t f ( z ) = z for each z E B • Y, and d ( p u f ( x , y ) , x ) < a ( x ) for each ( x , y ) e Y x Y (here d denotes the metric from condition (d) of L e m m a 2.3.7). Let U1 -- U, U2 -- U - A a n d let i l k ( X ) - - - l o g 2 a k ( x ) for each x e Uk, where a k ( X ) = ~1 m i n { a ( x ) , d ( x , y g

-- Uk)},

k = 1, 2.

By condition (c) of L e m m a 2.3.7, there exist two stearing functions Ak on Uk such t h a t )~I/B = )~2/B and Aa > ilk, k = 1,2. We define m a p s f k : Uk • Y --+ y y by letting

fk(x, y) = h(~, y, ~k(~)),

k = 1, 2

(here h denotes the displacement isotopy for Y). It follows from condition (d) of L e m m a 2.3.7 t h a t f k is a h o m e o m o r p h i s m of the p r o d u c t Uk • Y onto Uk and d ( f k ( x , y ) , x ) < 2 a k ( x ) for each ( x , y ) e Vk • Y, k = 1,2.. Now consider the m a p f3: (U • Y ) A ---+ U coinciding w i t h f2 on U2 • Y and with t h e i d e n t i t y m a p on A. Obviously, f3 is a h o m e o m o r p h i s m . Finally, let f - f 3 1 f l . It is not difficult to see t h a t f sends the p r o d u c t U • Y h o m e o m o r p h i c a l l y onto t h e space (U • Y ) A . It only remains to r e m a r k t h a t f ( z ) = z for each z 9 B • Y a n d t h a t d ( p u f ( x , y ) , x ) < 2 a l ( x ) + 2a2(x) _< c~(x) for each (x, y) 9 U • Y. W1 T h e s t a t e m e n t s proved so far in this section imply the following two i m p o r t a n t stability t h e o r e m s for R "~- and I "~ -manifolds. THEOREM 2.3.9. I f X is an R ~ -manifold, then X • R W ~ X . projection ~ x : X x R W ~ X is a n e a r - h o m e o m o r p h i s m .

Moreover, the

THEOREM 2.3.10. I f X is an R ~ -manifold, then X x I ~ ,~ X . Moreover, the proj e c t i o n ~ x : X • I ~ ~ X is a n e a r - h o m e o m o r p h i s m .

COROLLARY 2.3.11. I f X

is an R W - or I W -manifold,

then the proj ect i on

7r : X • I ~ X is a n e a r - h o m e o m o r p h i s m .

PROOF. Consider only the case of R ~ -manifolds (the r e m a i n i n g case is completely analogous to this one). D e n o t e by ~ x : X • R ~ --~ X and lr1: R W • I ---, R W the projections onto the first coordinates. By T h e o r e m 2.3.9, r x is a nearh o m e o m o r p h i s m . T h e projection 7rl is also a n e a r - h o m e o m o r p h i s m (see [25] for details). Now consider the following c o m m u t a t i v e diagram:

66

2. INFINITE-DIMENSIONAL MANIFOLDS

7rx x id X •

r xI

,.-X •

id x 7rl

X xR"

,,

7rx

,~ X

As observed above, b o t h horizontal arrows, as well as t h e left vertical arrow, are n e a r - h o m e o m o r p h i s m s . In this s i t u a t i o n t h e projection 7r is also a near-hom e o m o r p h i s m . [--1

2 . 3 . 2 . The strong universality o f I ~ - a n d R ~ - m a n i f o l d s . In Subsection 1.1.1 we have already seen ( P r o p o s i t i o n s 1.1.3 and 1.1.8) t h a t t h e spaces R ~ and I W are inuversal with respect to the classes of all Polish spaces a n d all c o m p a c t m e t r i z a b l e spaces respectively. F u r t h e r m o r e , in the first case, it is essential t h a t each Polish space a d m i t s a closed e m b e d d i n g into R ~ . This p r o p e r t y of the space R ~ is non-trivial, because it is easy to see every Polish (and, in general, every separable metrizable) space can be e m b e d d e d into t h e Hilbert cube I W as well. Below, we d e n o t e by Jt~,oo and B~,oo t h e class of all Polish spaces and the class of all m e t r i z a b l e c o m p a c t a respectively. T h e sense of these notations will become clear in Section 6.5. DEFINITION 2.3.12. Let 7~ E {Jtw,oo, Bw,oo}. A Polish space X is said to be strongly P - u n i v e r s a l if f o r each space Y f r o m the class 7~, the set of all closed embeddings of Y in X is dense in the space C (Y, X ) . PROPOSITION 2.3.13. E a c h I ~ - m a n i f o l d is strongly Bw,oo-universat. PROOF. We consider only t h e c o m p a c t case. Let X be a c o m p a c t I~-manifold. By T h e o r e m 2.3.10, it suffices to establish t h e s t r o n g B~,oo-universality of the p r o d u c t X x I ~ . Let f " Y --~ X x I ~ , where Y is a m e t r i z a b l e c o m p a c t u m and Lt E c o v ( X • I ~ ). W i t h o u t loss of generality we can a s s u m e t h a t the cover L/ has the form ]) • W , where • E c o y ( X ) and ]4? E c o v ( I ~ ). Take a finite subset A C w and an open cover ~)A E COV(IA) such t h a t the cover 7rAI()4)A) refines (here 7rA" I ~ --, I A d e n o t e s t h e projection onto the c o r r e s p o n d i n g s u b p r o d u c t ) . Since the space I w - A is h o m e o m o r p h i c to I ~ , we can see, by P r o p o s i t i o n 1.1.8, t h a t there is an (closed) e m b e d d i n g g" Y ~ I w - A . It only r e m a i n s to note t h a t t h e diagonal p r o d u c t 7rlfATrATr2Ag" Y ~ X • I W

2.3. R w- AND I w-MANIFOLDS

67

is an e m b e d d i n g / g - c l o s e to f (here lrl a n d r 2 d e n o t e t h e p r o j e c t i o n of X • I ~ o n t o t h e first a n d t h e second c o o r d i n a t e r e s p e c t i v e l y ) ,

i-1

PROPOSITION 2.3.14. Each R ~ -manifold is strongly ,4~,oo-universal. PROOF. B y T h e o r e m 2.3.9, t h e g e n e r a l case can be r e d u c e d to t h e case X = R ~ . T h e space R ~ is identified w i t h its c o u n t a b l e power X = 1-I{Rn~ : n E w}, w h e r e R ~ = R ~ for each n E w. Let lrn" X + R ~ d e n o t e t h e p r o j e c t i o n o n t o t h e c o r r e s p o n d i n g c o o r d i n a t e . Take a m e t r i c dn on t h e space R ~ b o u n d e d by 1 a n d consider t h e following m e t r i c on t h e p r o d u c t X :

d(x, (x') = m a x { d n ( r n ( x ) , r n ( x ' ) ) " n e w}, x , x ' e X . Let Y be a Polish space. S u p p o s e also t h a t m a p s f : Y - . X a n d a : X --, (0, 1) are given. For our p u r p o s e s it suffices to c o n s t r u c t a closed e m b e d d i n g g: Y - . X such t h a t d ( f ( y ) , g ( y ) ) 0 such that the d-diameter of f - l ( U ) is greater than e for each neighborhood U of y. It is easily seen that different choices of complete metrics for X yield the same non degenerate values. Indeed, if f ( X ) is dense in Y, a purely topological (in other words metric independent) description of a nondegenerate value y E Y is that either 9 f-l(y)

= 0,

9 f - 1 contains at least two points, or

9 f-l(y)_ {x} but f - l ( B ) is not a neighborhood basis for x, where B is a neighborhood basis for y. The set of all non degenerate values of f is denoted by N / . A straightforward verification shows that N / i s an Fa-subset of Y. Observe also that the restriction of f is a homeomorphism from f - 1 ( y _ N / ) to Y - N / . Of course, if f is a proper map, then N / {y E Y : I f - l ( y ) l > 1}. In general, fine homotopy equivalences are not surjective, though they have dense images, and points not in the image are necessarily non degenerate values. We begin with the following lemma.

2.4. TOPOLOGY OF R ~ - AND I "~-MANIFOLDS

73

LEMMA 2.4.2. Let X be a strongly .Aw,oo-universal Polish A N R-space. Then the space C (I W , X ) contains a dense set consisting of Z-embeddings with pairwise disjoint images. PROOF. Let d be a c o m p l e t e m e t r i c on X , and let /dk E c o y ( X ) d e n o t e an open cover of X whose e l e m e n t s have d i a m e t e r less t h a n ~, k E N . Let {an" n E N } be a c o u n t a b l e dense subset of t h e space C ( I W , X ) . Consider the map a" g x I W ~ X defined by letting a (n, a) = an(a) for each (n,a) e g x I W. By L e m m a 2.3.16, each c o m p a c t subset of X is a s t r o n g Z - s e t in X . T h e r e f o r e there exists a m a p #1" N x I W --~ X such t h a t beta l is /dl-close to a a n d # l ( g x I W ) M a ( N x I W ) = O. If m a p s #i" N x I W --~ X , each/d~-close to a a n d satisfying #i(N

x I ~ ) M (a(N

x I ~ ) U Uj e(i + 1)(m) for each m e M so t h a t every point a E N I has a neighborhood Wa such t h a t either

(iii)l H-lh~l(.f-l(Wa)) C_Fc(i+2),~(i) from some i >_ 0 and 7rH-lh~l(f-l(wa)) is contained in B~(m) for each m E M , or (iii)2 H - l h T l ( f - l ( W a ) ) C_ r~(1),~(0 ). Choosing e(1). Consider the set Z0 consisting of those points a E X t h a t do not have a n e i g h b o r h o o d Wa such t h a t 7rH-lhll(f-l(Wa)) is contained in B~(m)(m) for some m E M. This set is closed in X and is contained in N / . Thus, there is a closed f - s a t u r a t e d neighborhood No of f-l(Zo) disjoint from h l H ( U x {0}). Choose e(1) so t h a t F(0, el) is disjoint from H-lh~l(intNo) 3. Observe t h a t points in Z0 have neighborhoods satisfying (iii)2. Choosing e(i + 1). Consider the set Zi consisting of those points a E X t h a t do not have a neighborhood Wa such that H - l h ~ l ( f - l ( W a ) ) C_F(0, e(i)). This set is closed in X, and f - l ( Z i ) is disjoint from hlH(U x {0}). Choose any neighborhood N of h l H ( U x {0}) missing an f - s a t u r a t e d neighborhood of f - l ( z i ) , and any e(i + 1) with e(i + 1)(m) < ~(i)(m) for each m E U and with F(0, e(i + 1)) C_ H-lh~l(intN). Making the further restriction t h a t e(i)(m) < ~, i = 1 , 2 , . . . , we specify a h o m e o m o r p h i s m h0" M • I --~ M • I t h a t is the identity on M • {0, 1} and sends the graph of e(i) onto the graph of ( 1 - 5) ~, i = 1 , 2 , . . . 4. Setting ha -- hoH -1 and h = h2h3h~ 1, we claim t h a t h is the desired shrinking h o m e o m o r p h i s m . For, if a E N / and (iii)l is satisfied, then

hoU-lhll(.f-l(wa)) C

r((1

3here "0" denotes the constant map M --~ (0}. 4(1 - 5)i(m) = (1 - 5(m)) i.

-

~)~+2,(1

-

-

5) i)

2.4. TOPOLOGY OF R ~- AND I ~-MANIFOLDS

77

and so h o H - 1 h l l ( f - 1 ( W a ) ) is contained in B25(m)(m) • B2~(m)(t) for some (re.t) E i • I and condition (ii) implies that h ( f - l ( W a ) ) is sta02)-small. If a E N / and (iii)2 is satisfied, then

hoH-lh~l(f-l(Wa))

C_ F(1 - 5, 1)

and condition (i) implies that h ( f - l ( W a ) ) is st4(lZ)-small.

D

PROPOSITION 2.4.6. If f : M ~ X is a fine homotopy equivalence from an R W -manifold to a strongly Jtw,cc-universal Polish A N R - s p a c e and N / is a Zaset in X , then f is a near-homeomorphism. PROOF. The proof parallels closely that of Proposition 2.4.4. An approximating fine homotopy equivalence g is produced using Proposition 2.4.5 in place of Proposition 2.4.3. [-1 PROPOSITION 2.4.7. A fine homotopy equivalence f : M ---+ X from an R Wmanifold to a strongly .Aw,cc-universal Polish A N R-space is a near-homeomorphism. PROOF. By lemma 2.4.2, there exists a countable dense subset {en : n E N } C_ C ( I W , X ) , consisting of Z-embeddings with pairwise disjoint images. The map f : M --, X factors through the adjunction space as f -- plq, where the maps q: M --* M U / e l ( I w ) and P l : M U f e I ( I w ) --~ X are described in Subsection 2.3.1 on page 61. Obviously, Nq C_ e l ( I w ) and e l ( I W ) is a strong Z-set in M U I e l ( I W ). By Proposition 2.4.4, the map q, being clearly a fine homotopy equivalence, is a near-homeomorphism. Take a homeomorphism h approximating q and let f l = plh: M -~ X . Observe t h a t f l approximates f and is one-to-one over the set e l ( I W). Continuing this process, we obtain a sequence f = f0, f l , f 2 , . . , consisting of fine homotopy equivalences (from M to X) such t h a t f n + l a p p r o x i m a t e s fn and is one-to one over the set un+l k=lek(I W ). If f,~+l is sufficiently close to fn (for each n E N), then the map g - lim fn is also a fine homotopy equivalence approximating f. It is not hard oo 1e n(I W ). This fact, coupled with to see that g is o n e - t o - o n e over the set Un= the choice of the collection {en}, guarantees that Ng is a Zo-set in X. It only remains to apply Proposition 2.4.6. E] We are now ready to prove Theorem 2.4.1. Proof of Theorem 2.4.1. One part of the s t a t e m e n t follows from Proposition 2.3.14. Consider now a strongly jtw,~-universal Polish A N R - s p a c e X . By Theorem 2.3.22, there exists a fine homotopy equivalence f : M --, X where M is an R ~ -manifold. Proposition 2.4.7 implies t h a t f is a near-homeomorphism and, consequently, t h a t X is an R~-manifold. If, in addition, X is an absolute retract, then M is contractible (recall that f is a fine homotopy equivalence). It only remains to note t h a t in this case, according to Theorem 2.3.27, M (as well as X ) is homeomorphic to R ~ . This completes the proof of Theorem 2.4.1.

78

2. INFINITE-DIMENSIONAL MANIFOLDS Sometimes it is much more convenient to use the following version of T h e o r e m

2.4.1. PROPOSITION 2.4.8. Let X be a Polish A ( N ) R - s p a c e . Then the following conditions are equivalent: (i) Z is homeomorphic to R W (respectively, Z is an R W -manifold). (ii) The set

{ f e C ( N x I ~ , X ) " the collection { f ( { n } x I r176 ) ' n E N } is discrete} is dense in the space C ( N x I ~ , X ) . (iii) The set { f e C ( $ { I n" n e w } , X ) " the collection { f ( I n ) " n e w} is discrete} is dense in the space C ( ~ { I n " n E w } , X ) . In order to prove this s t a t e m e n t we need some preliminary results. The following l e m m a is geometrically obvious. LEMMA 2.4.9. Let K be an n-dimensional countable locally finite simplicial complex and let G be an open neighborhood of IK(n-1)l in IKI. Then the convex halls Co of the sets Iol- a , o ~ K , forms a discrete collection in IKI, PROOF. Assuming the contrary, there is a point x E c l C - C, where C = U{Co" a E K } . T h e n x E I~1 for some T E K (n-l). By an obvious geometrical argument, dist(la I - G , I~1)- dist(Co, I~1) for all a E s t ( r , K ) . Since I~1 c_ a, we have dist(C Cl Ist(r, K ) , I~1) > 0, contrary to the fact t h a t x e el C fq Irl. v1 LEMMA 2.4.10. Let X be a Polish A N R - s p a c e satisfying condition (iii) of Proposition 2.4.8, K be a finite-dimensional countable locally finite simplicial complex, and Lt e cov(IKI) be the cover of IKI by open stars of the vertices of K . Then the set of all U-maps of the polyhedron IKI into the product X x I ~~ is open and dense in the space C ( I K I , X x I ~ ). PROOF. T h a t the set of all U-maps from IKI to x x I ~ is open follows from L e m m a 2.1.1. Let us show t h a t this set is dense. If dim K = 0, then this follows immediately from condition (iii) of Proposition 2.4.8. Suppose that dim K = n and t h a t the l e m m a has already been proved for complexes of dimension < n - 1. Suppose also t h a t a map f" [K[ ~ X • I ~~ and an open cover 1) E cov(X x I ~ ) are given. Our goal is to show the existence of a / 4 - m a p g" [KI --, X x I "~ t h a t is V-close to f. By our inductive assumption and Corollary 2.1.23, we may assume without loss of generality t h a t the restriction f / l g ( n - 1 ) [ is an U-map. By L e m m a 2.1.3, there is an open neighborhood G of [K(n-1)[ in Igl such t h a t the restriction I / G is still a U-map. Denote by )4; E cov(X x I ~ ) a double star-refinement of 1) such t h a t for each map g" [K[--. X x I ~ t h a t is W-close to f , the restriction g i g is a n / , / - m a p (here we use L e m m a 2.1.1 and Corollary 2.1.23). By L e m m a 2.4.9, the

2.4. TOPOLOGY OF R ~- AND I ~~

79

c o m p l e m e n t I K I - G can be covered by a discrete (in IKI)collection {Ca" a e K } such t h a t Co ~ I n and Ca C_ i n t l a l for each a E g . By condition (iii) of Proposition 2.4.8 and Corollary 2.1.23, there exists a m a p go" IKI --+ X x I ~ such t h a t go is W-close to f and the collection {g0(Ca" a E K } is discrete in X x I W . Let {Da" a E K } be a discrete collection of open subsets of IKI such t h a t Ca C_ Da C_ i n t l a l for e a c h a E g . Take a f u n c t i o n ~ o " I g l - - + [0,1] such that ~o(U{Ca" a e g } ) = l

and ~ o ( [ g [ - U { D a "

aeK})=0.

A straightforward verification shows t h a t the diagonal p r o d u c t g I =g0/kgo" IKI---+X x I ~ x [0,1] is a / g - m a p . Since the projection rr" I " x [0, 1] --+ I ~~ is a n e a r - h o m e o m o r p h i s m (Corollary 2.3.11), there exists a h o m e o m o r p h i s m h ' X x I

~

x[O, 1] ---+XxI'"

t h a t is V-close to id x ft. T h e n the desired m a p g can be defined as the composition g - hg ~. F1 LEMMA 2.4.11. For each n E w and for each n - d i m e n s i o n a l Polish space X , there exists an (n + 1)-dimensional Polish absolute retract containing X as a closed subspace. PROOF. Represent X as the limit space of some inverse sequence S x = { X i ,Pi _i+1 } consisting of n-dimensional locally c o m p a c t p o l y h e d r a and surjective limit projections (see [179]). We construct a new inverse sequence s r = {Yi, q~+1} as follows. As the first element II1 of this s p e c t r u m we take c o n ( X 1 ) the cone over t h e p o l y h e d r o n X1. Obviously, Y1 is an (n + 1)-dimensional Polish absolute retract. We identify X1 with its n a t u r a l l y e m b e d d e d copy in Y1. Suppose that for each i, with i _< k, an (n + 1)-dimensional Polish absolute retract Yi, containing Xi as a closed subspace, and a fine h o m o t o p y equivalence qi-1 i . Yi '"+ Yi-1 , extending Pi-1, have already been constructed. As the i space Yk+l we take the m a p p i n g cylinder M a p ( p ~ +1) of the m a p Pkk+l , considered as a m a p from X k + l into Yk (recall t h a t p k + l ( X k + l ) = X k is closed in Yk). Obviously, Yk+l is an (n + 1)-dimensional Polish absolute retract (see [161] for details). T h e closed e m b e d d i n g of the p o l y h e d r o n X k+l into Yk+l and the k " Yk+l --+ Yk are defined canonically. It is also fine h o m o t o p y equivalence qk+l clear t h a t qkk+l(Xk+ 1) -- X k and q lk + lk/ x k + = Pkk+l . Let Y denote the limit space of the inverse sequence S y . It follows from the construction t h a t Y is an (n + 1)-dimensional Polish space containing X as a closed subspace. Since all projections of this s p e c t r u m are fine h o m o t o p y equivalences and since Y1 is a Polish absolute r e t r a c t we easily conclude t h a t Y is also an absolute r e t r a c t (see [112] for details). El

80

2. INFINITE-DIMENSIONAL MANIFOLDS

LEMMA 2.4.12. Let X be a Polish space satisfying condition (iii) of Proposition 2.4.8. Then for each finite-dimensional Polish space Y , the set of closed embeddings of Y into X x I ~ is dense in the space C ( Y , X x I "~). PROOF. By L e m m a s 2.1.1, 2.1.2 and Proposition 2.1.7, it suffices to show t h a t for each open cover IX E c o y ( Y ) , the set of all IX-maps is dense in the space C ( Y , X x I ~ ). By L e m m a 2.4.11 we may additionally assume t h a t Y is a finite-dimensional Polish absolute retract. But then, by T h e o r e m 2.1.19, we may assume, w i t h o u t loss of generality, t h a t Y = IK], where K is a finite-dimensional countable locally finite simplicial complex and IX is the cover of the polyhedron ]K I by open stars of its vertices. In this situation L e m m a 2.4.10 completes the proof, l1 LEMMA 2.4.13. Let X be a Polish A N R - s p a c e satisfying condition (iii) of Proposition 2.4.8, K be a countable locally finite simplicial complex, and IX be is the cover of the polyhedron ]K] by open stars of its vertices. Then the set of maps f : ]K I ---, X x I ~ such that the collection of fibers of the composition 7 r z f , where lrz : X x I"; ---. X is the projection, refines IX and is a dense and G h-subset of the ~par162c ( I g l , x x I ~ ). PROOF. Given integers m and n, consider for each a E K (n) - K (n-l) the image of ]hi under the (1 - G1 ) - h o m o t h e r y with respect to the barycenter of la]. Obviously, the collection .An,m of all such images is discrete in IK]. By condition (iii) of Proposition 2.4.8 and the compactness of I ~ , the set Fn,m = { f E C ( I K I , X

x I ~ ): the collection

~zf(A~.m)

is discrete in X }

is open and dense in C(IKI, X x I "~ ). It only remains to remark that the intersection MFn,m is the desired set. [Z] LEMMA 2.4.14. Let X be a Polish A N R - s p a c e satisfying condition (iii) of Proposition 2.4.8. Then for each Polish space Y , the set of closed embeddings of X into the p r o d u c t X • is dense in the s p a c e C ( Y , X x I "~). PROOF. As in the proof of L e m m a 2.4.12 it suffices to show t h a t if K is a locally c o m p a c t polyhedron and IX E c o y ( K ) , then the set of all/,/-maps is dense in the space C(Y, X x I ~ ). For simplicity we assume t h a t X is a Polish absolute retract (this simplification has a purely technical n a t u r e and does not affect the general argument). Represent K as the union K = U { K n : n E w} of its finite-dimensional s u b c o m p a c t a in such a way t h a t (a) K n C int K n + l , n E w, and (b) T h e collections { K 2 n + l - i n t K2n : n E w} and { K 2 n - i n t K2n-1 : n E N } are discrete in K. Next, take an arbitrary map f : K --, X x I ~ and introduce the notation A = U{K2n+I - int K2r, : n E w} and B = U { K 2 n - int K2n-1 : n E N}. Obviously, the sets A and B are closed in K. By Proposition 2.1.7, Corollary 2.1.23 and L e m m a 2.4.12, we may assume w i t h o u t loss of generality t h a t the restrictions

2.4. TOPOLOGY OF R w- AND I w -MANIFOLDS

81

S / K n are closed e m b e d d i n g s (we use the finite-dimensionality of K n also). Next, identifying each of the "rings" K2n +1 - i n t K2n, n 9 w, with a subspace of a finitedimensional c u b e I rim, we may also assume t h a t the subspace A is identified with a closed s u b s p a c e of the discrete sum @{In'~: m 9 w}. T h e restriction f / A can be e x t e n d e d to a m a p fl:

@ {in,,,: m 9 w} ---+ X x I ~ .

By condition (iii) of P r o p o s i t i o n 2.4.8, we m a y assume w i t h o u t loss of g e n e r a l i t y t h a t the collection {S#(Xnm): m 9 w} is discrete in t h e p r o d u c t X x I ~ . Therefore, the collection

{f#(K2n+l - int K 2 n ) : n e w} is also discrete in X x I W . As above, we can now conclude t h a t the restriction f / A is a closed e m b e d d i n g of A into X x I ~ . A c o m p l e t e l y analogous a r g u m e n t shows t h a t the restriction f / B is also a closed e m b e d d i n g of B into X x I ~ . Since every closed e m b e d d i n g is a U - m a p , there exists an open n e i g h b o r h o o d G of A in K such t h a t the m a p f / G is still a / g - m a p (here we a p p l y L e m m a 2.1.3; observe also that, by L e m m a 2.1.1 and Corollary 2.1.23, the restriction g / G o f a m a p g: K ~ X x I w t h a t is sufficiently close to f is also a / g - m a p ) . In addition, we assume t h a t G n B -- 0. Now take a function ~ : K ---+ [0, 1] such t h a t ~(A) = 0 and ~ ( K - G) = 1. A s t r a i g h t f o r w a r d verification shows t h a t the diagonal p r o d u c t

g'=fA~:K---+X

•

~ x [0,1]

is a H-map. Take a h o m e o m o r p h i s m h: I ~ • [0, 1] ~ I ~ sufficiently close to t h e projection 7r: I ~ x [0, 1] ~ I ~ . T h e n the c o m p o s i t i o n

( i d x x h)g~: K ---+ X x I ~ is a / g - m a p , sufficiently close to f.

F-1

We are now r e a d y to prove P r o p o s i t i o n 2.4.8. Proof of Proposition 2.4.8. T h e implications (i)===~(ii) and (ii)==~(iii) are obvious. Let us prove the validity of the implication ( i i i ) ~ ( i ) . By T h e o r e m 2.4.1, it suffices to show t h a t for each Polish space Y, the set of closed e m b e d d i n g of Y into X is dense in the space C(Y, X ) . As above, in order to show this last fact, it suffices to prove t h a t for each c o u n t a b l e locally finite simplicial c o m p l e x K the set of all U - m a p s of [g[ into X , w h e r e / d is the cover of [g[ by open stars (with respect to the t r i a n g u l a t i o n K ) of its vertices, is dense in the space C([K[, X ) . By L e m m a s 2.4.13, 2.4.14 and P r o p o s i t i o n 2.1.7, the space CIK[, X x I ~ ) contains a dense set F , consisting of closed e m b e d d i n g s such t h a t for each f E F t h e collection of fibers of the c o m p o s i t i o n ~ z f , where 7rx: X x I ~ ~ X d e n o t e s the projection, refines b/. Each such c o m p o s i t i o n is a p r o p e r map. Therefore, as can be seen in the proof of Corollary 2.1.9, each such c o m p o s i t i o n is a U-map. It only

82

2. INFINITE-DIMENSIONAL MANIFOLDS

remains to remark t h a t the set of maps g: IKI ~ X, t h a t can be represented as a composition g = r x f , f E F is dense in the space C ( I K I , X ). The proof of Proposition 2.4.8 is complete.

2.4.2. T o p o l o g y o f I ~ - m a n i f o l d s . The strategy of the proof of the main result of this Subsection (Theorem 2.4.18 and Proposition 2.4.19) is the same as that of Subsection 2.4.1. As a very useful (and sufficiently difficult, but manageable) exercise we recommend that the reader supply the omitted pieces of the corresponding proofs. PROPOSITION 2.4.15. Let f : M ~ X be a proper C E - m a p of an I ~-manifold onto a strongly Bw,oo-universal locally compact A N R - s p a c e X . If the set c l N i is a Z - s e t in X , then f is a near-homeomorphism. PROPOSITION 2.4.16. Let f : M --~ X be a proper C E - m a p of an I ~ - m a n i f o l d onto a strongly Bw,oo-universal locally compact A N R-space X . If the set N I is a Za-se$ in X , then f is a near-homeomorphism. As in Subsection 2.4.1 these two propositions imply the following statement. PROPOSITION 2.4.17. A proper C E - m a p of an I W -manifold onto a strongly Bw,c~-universal locally compact A N R-space is a near-homeomorphism. As in the case of R "~ -manifolds, the problem of the existence of a resolution for a given locally compact A N R - s p a c e X (i.e. the existence of a proper C E map f : M ~ X, where M is an I W-manifold) arises here as well. The existence of such a resolution follows, for example, from Theorem 2.3.21. Alternatively we can apply Miller's approach (see, for instance, [231] or [139]). Therefore, Proposition 2.4.17 implies the following characterization theorem. THEOREM 2.4.18. The following conditions are equivalent for each (locally) compact A ( N ) R - s p a c e X : (i) z i~ hom~omo~p~i~ to I ~ ( ~ p ~ t i ~ t y , x i~ ~ ~ - m ~ n i / o l d ) . (ii) X is strongly B~,oo-universal. Sometimes it is more convenient to use the following version of Theorem 2.4.18. PROPOSITION 2.4.19. The following conditions are equivalent for each locally compact A N R-space X : (i) X is an I ~ -manifold. (ii) The set {f EC(D •

~,x):

y({0} x I ~ ) A f ( { 1 }

is dense in the space C ( D • I "~ , X ) . 5Recall that D = {0,1) denotes the two-point discrete space

xI W)=0} 5

2.4. TOPOLOGY OF R ~- AND I ~-MANIFOLDS

83

(iii) For each n E w, the set

{f e

C ( D x In, X ) : f({0} x I n ) f3 f({1} • I n ) = O}

is dense in the space C ( D x I n, X ) . PROOF. The implications (i)==>(ii) and (ii)==~(iii) are obvious. Let us prove the validity of the implication ( i i i ) ~ ( i ) . According to Theorem 2.4.18 we only have to show the strong Bw,oo-universality of X. First of all let us establish the following fact: 9 For metrizable compacta T1 and T2 the set

{f c C(T1 9 T2, X):.Y(T1)n :(T~) = O} is dense in the space C (T1 (9 T2, X). Indeed, by Theorem 2.1.19, we may assume without loss of generality that T1 -]Ki], where Ki is a finite simplicial complex, i - 1, 2. For each pair of simplexes a l E K1 and a2 E g 2 , let

C(~1, ~2) : {f e C(IKll 9 IK21,X): f ( l ~ l l ) n f(l~2l) - 0}. By condition (iii) and Corollary 2.1.23, the set C ( a l , a2) is dense (and obviously open) in the space C ( I K l l (9 I g 2 ] , X ) . Consequently, according to Proposition 2.1.7, we conclude that the intersection N { C ( a l , a 2 ) : al C K l , a 2 C K2} is also a dense subset of the space C(IK~J 9 JK21,X). It only remains to observe that for each map f from this intersection, we obviously have f ( I g l ] ) N . f ( ] g 2 1 ) =

0. We proceed now to the direct proof of the strong B~,oo-universality of the space X. Consider a metrizable c o m p a c t u m Y and for each pair of disjoint s u b c o m p a c t a T1 and T2 of Y denote by C (T1, T2) the set

{S e c ( Y , X ) : S(T~)n S(T2) = 0}. By the fact established above, the set C(T1, T2) is dense (and obviously open) in the space C (Y, X). Take now a countable open basis 1) for the space Y. T h e n for each pair (V1,1:2) of elements of V, with cl V1 gl cl 1:2 - 0, the set C (cl V1, cl 1/2) is an open and dense subset of C(Y, X ) . Consequently, by Proposition 2.1.7, the intersection of all sets of the form C (cl V~, cl 1/2) is dense in the space C (Y, X). It only remains to observe that each map from this intersection embeds the c o m p a c t u m Y into the space X. I-1

84

2. INFINITE-DIMENSIONAL MANIFOLDS

2.4.3. A p p l i c a t i o n s o f c h a r a c t e r i z a t i o n t h e o r e m s . Theorems 2.4.1 and 2.4.18 (as well as their a l t e r n a t e versions Propositions 2.4.8 and 2.4.19) can be successfully used in m a n y different situations. In this Subsection we present some of them. PROPOSITION 2.4.20. Let f be a map between R W -manifolds. lowing conditions are equivalent: (i) f is approximately soft. (ii) f is a fine h o m o t o p y equivalence. (iii) f is a near-homeomorphism.

Then the fol-

PROOF. T h e implications (i)==~(ii) and ( i i ) - ~ ( i i i ) follow from Propositions 2.1.27 and 2.1.28 respectively. The implication ( i i i ) ~ ( i ) follows from Propositions 2.3.14 and 2.4.7. D PROPOSITION 2.4.21. Let f be a proper map between I ~ -manifolds. Then the following conditions are equivalent: (i) f is approximately soft. (ii) f is a fine h o m o t o p y equivalence. (iii) f is a C E - m a p . (iv) f is a near-homeomorphism. Corollary 2.2.35 states t h a t if A is a Za-set in an R ~ -manifold X, then the complement X - A is also an RW-manifold. Moreover, since the inclusion X - A r X is a (fine) h o m o t o p y equivalence, we can, by virtue of T h e o r e m 2.3.27, conclude t h a t the R"~-manifolds X - A and X are homeomorphic. Proposition 2.4.7 implies an even stronger result. PROPOSITION 2.4.22. Let A be a Z a - s e t in an R w -manifold X . inclusion X - A ~ X is a n e a r - h o m e o m o r p h i s m .

Then the

PROPOSITION 2.4.23. A countable infinite product of non-compact Polish absolute retracts is homeomorphic to R "~ . PROOF. Let X = 1-ln~__lXn, where each X n is a non-compact Polish absolute retract. Obviously, X itself is a Polish absolute retract. Consequently, by Theorem 2.4.1, it suffices to show the strong .A~,~-universality of X. It follows from the proof of Proposition 2.3.14 t h a t we will be done if each of the spaces X n , n C N, contains a closed copy of R "J . Represent the set N of natural numbers as a countable infinite disjoint union of infinite subsets Nk, k E N . T h e n oo

X---- I " [ Y k '

where Y k - - N ( X k ' n E N k } , k E N .

k--1

Therefore it only remains to show t h a t each of the spaces Yk, k E N , contains a closed copy of R ~ . This last condition will be satisfied if we show t h a t the

2.4. TOPOLOGY OF R ~

AND I w-MANIFOLDS

85

real line R admits a closed e m b e d d i n g into the p r o d u c t A x B of any two nonc o m p a c t Polish absolute retracts. Take c o u n t a b l e infinite and discrete subsets {an: n E Z} and {bn: n E Z} of the spaces A and B respectively (use nonc o m p a c t n e s s of these spaces). Let [an, an+I] denote the copy of the closed unit s e g m e n t in A with end points coinciding with a n and an+l. T h e same m e a n i n g shall be given to the symbol [bn, bn+l], n E Z. Let cn = (an, bn) E A x B and Cn+l = (an+I, bn+l) E A x B. Obviously the segment

[Cn, Cn+l] = ([an, an+l] X {bn}) U ({an+l x [bn, bn+l]) connects the points cn and Cn+l in A x B. Moreover, different s e g m e n t s of the form [cn, cn+l] are either disjoint or have exactly one c o m m o n point. It only r e m a i n s to observe t h a t the union U{[cn, Cn+l]: n E Z} is closed in A x B and is h o m e o m o r p h i c to the real line R. D PROPOSITION 2.4.24. A proper retract of an R ~

is an R ~

PROOF. Let r" M ~ X be a p r o p e r r e t r a c t i o n of an R ~ -manifold M into a s u b s p a c e X . First of all we show t h a t each c o m p a c t subset K C X is a Z - s e t i n X . Let 3 / E c o y ( X ) be an open cover of X and suppose a m a p f " I ~ ~ X is given. Define a m a p g" g x I ~ ~ M by letting g ( n , a ) - f ( a ) for each ( n , a ) E N x I ~ . T h e strong Jtoj,oo-universality of the R ~ -manifold M guarantees the existence of a map h" N x I ~ ~ M which is r - l ( U ) - c l o s e to g and such t h a t the collection {h({n} x I ~ )" n E N } is discrete in M . Consider t h e collection { r h ( { n } x I ~ E N}. P r o p e r n e s s of the r e t r a c t i o n r implies t h a t this collection is locally finite in X. Consequently, for each point x E K t h e r e is an open n e i g h b o r h o o d G x of x in X intersecting only finitely m a n y sets of the form r h ( { n } x I ~ ). Selecting a finite subcover of the cover {Gx" x E K } of K , we obtain an o p e n n e i g h b o r h o o d G of the c o m p a c t u m K in X which intersects only finitely m a n y sets having t h e indicated form. T h e r e f o r e t h e r e exists an index n E N such t h a t r h ( { n } x I ~ ) M K -- 0. Obviously, t h e m a p f ' " I ~~ -+ X, defined by letting f ' ( a ) = r h ( n , a) for each a E I ~~, is U-close to f a n d f ' ( I "~ ) M K = 0. Thus, every c o m p a c t subset of X is a Z-set in X. This fact, coupled with P r o p o s i t i o n 2.1.7 and Corollary 2.1.23, shows t h a t t h e set

Al={f

E C(N xI',X)"

f({n} xI')Mf({m}

xI ~)=0;

n#m}

is a dense G~-subset of the space C ( N x I ~ , X ) . On the o t h e r hand, the set

A2 = { f E C ( N x I ~ , X ) "

the collection { f ( { n } x I ~ ) ' n

E N } is locally finite}

is open and dense in the space C ( N x I ~ , X ) . By P r o p o s i t i o n 2.1.7, the intersection A1 M A2 is also dense in the space C ( N x I ~ , X ) . Finally, observe t h a t if f E A1MA2, t h e n the collection { f ( { n } x I ~ ): n E N } is discrete in X. P r o p o s i t i o n 2.4.8 completes the proof. [:]

86

2. INFINITE-DIMENSIONAL MANIFOLDS

The following result (compare with Proposition 2.4.23) follows directly from Proposition 2.4.19. PROPOSITION 2.4.25. A countable infinite product of n o n - t r i v i a ~ compact absolute retracts is h o m e o m o r p h i c to I "~ .

Let X be a Polish space. As usual, by e x p X we denote the hyperspace of all non-empty compact subsets of X endowed with the Hausdorff metric (or, equivalently, having the Vietoris topology; see, for example, [145]). The correspondence X ~ e x p X can be naturally extended to maps, i.e. for each map f : X - , Y we can assign a map exp f : exp X --, exp Y. Moreover, it is easy to see that this correspondence is functorial. Obviously the functor exp preserves the class of compacta. The following statement shows that exp significantly improves the properties of spaces. PROPOSITION 2.4.26. I f X is a P e a n o c o n t i n u u m , exp X is h o m e o m o r p h i c to I "~ .

i.e.

X

9 AE(1),

then

PROOF. Take a convex m e t r i c p on X (see [361). This means that each pair of points in X is contained in a subspace isometric (with respect to metric p) to a closed segment on the real line. Consider the following metric on exp X: d ( K 1 , K2) = max{p(x, Ki): i = 1,2, x 9 K1 U K2}.

By the result of Wojdyslawski [321], exp X is an AR-compactum. For each e > 0, the map fe defined by letting re(K) = {x 9 X : p ( x , K )

~ for each t E [a,b] and e E E.

PROPOSITION 2.4.30. Each infinite-dimensional separable Frgchet space is hom e o m o r p h i c to R ~ . PROOF. By T h e o r e m 2.1.17, X is a Polish absolute retract. Therefore, it suffices to verify condition (iii) of Proposition 2.4.8. If X contains a linear subspace X0 isomorphic to R ~ , then , by T h e o r e m 2.1.18, X is h o m e o m o r p h i c to the product R ~ • Y, where Y is a separable Frgchet space (namely, the factor space X / X o ) . By T h e o r e m s 2.1.17 and 2.3.22, X is h o m e o m o r p h i c to R W . Next, consider the case when X does not contain a linear subspace isomorphic to R ~ . In this case, there exists a n u m b e r E > 0 such t h a t no finite-dimensional linear subspace of X is an e-net in (X, d). Let us verify t h a t condition (iii) of Proposition 2.4.8 holds. Take two maps f : (9 ( I n : n E N } -+ X and (~: X --, (0, c~). W i t h o u t loss of generality we may assume t h a t c~ < c. In addition, by L e m m a 2.4.28, we may assume t h a t for each n E N, the c o m p a c t u m f ( I n) is contained in a finite-dimensional linear subspace of X. We construct a m a p g : ~ ( I n : n E N } --, X by induction. Let g / I 1 = f / i 1 and assume t h a t g has already been defined on the union @ ( I k : k < n } in such a way t h a t g ( I k ) , k < n, is contained in a finite-dimensional linear subspace E of X and the following condition is satisfied:

(,) lAg(q)- f(q)II < 5 ~ f ( q )

and lAg(q)- g(ql)ll > -~af(q)

for e a c h q , ql E I k , w h e r e k < m < n. Apply L e m m a 2.4.29 to the segment [a, b] -- a f ( I n) and proper (recall t h a t X is infinite-dimensional) linear subspace E. Let u: [a, b] --~ X be the m a p whose existence is g u a r a n t e e d by t h a t L e m m a . Let g(q) = f (q) + uc~ f (q), q E I n . This completes the inductive step of our construction. Let us show t h a t the map g: @ { I n : n E N } -~ X constructed above sends the collection {In: n E N } into a discrete (in X ) collection. Assuming the contrary, we can find sequences of indices kl < k2 < . . . and points qm E I k'~ such t h a t { g ( q m ) } converges in X. By (.), lira a f ( q m ) = 0 and the sequence { f ( q m ) } also converges in X . This, in turn, implies the false equality c~(lim f ( q m ) ) -- O. D PROPOSITION 2.4.31. Each infinite-dimensional compact convex subset of a separable Fr~chet space is h o m e o m o r p h i c to I ~ .

88

2. INFINITE-DIMENSIONAL MANIFOLDS

PROOF. Let K be an infinite-dimensional c o m p a c t convex subset of a separable Frgchet space. By T h e o r e m 2.1.17, K is an a b s o l u t e r e t r a c t . Let us verify t h a t condition (iii) of P r o p o s i t i o n 2.4.19 holds. Let e > 0 and let f , g" I n --, K be maps. By L e m m a 2.4.28, we m a y a s s u m e t h a t f ( I n) U g ( I n) is contained in t h e convex hull of a finite set {kl, k2, "., kin} C_ K . Since dim K ---- oc, there exists a point k E K which does not belong to t h e indicated convex hull. T h e comp a c t n e s s of K g u a r a n t e e s t h a t t h e r e is a 5 > 0 such t h a t 5 ( K - K ) is contained in the ball of radius e with c e n t e r at the origin. Let re(q) = (1 - 5 ) f ( q ) + 5k, q E I n. T h e n d(f~(q), f ( q ) ) = d ( 5 ( k - f ( q ) ) , O) < e, q E I n. Let gE = g. It only remains to n o t e t h a t fE(I n) M ge(I n) = 0.

[:]

2 . 4 . 4 . T r i v i a l B u n d l e s . In this Subsection we present p a r a m e t e r i z e d versions of T h e o r e m s 2.4.1 and 2.4.18. We also establish some corollaries which will be essential in C h a p t e r 6. First of all let us i n t r o d u c e some notations. Let a m a p f : X ~ Y be given 9 For any space Z a n d any m a p g: Z ---, X , C g ( Z , X ) shall d e n o t e the subspace of the space C ( Z , X ) consisting of all m a p s h: Z ~ X satisfying t h e equality f h -- f g . T h e following two t h e o r e m s characterize trivial R ~ - and I W-bundles. THEOREM 2.4.32 9 Let f : X ~ Y be a soft m a p between Polish A N R-spaces. T h e n the following conditions are equivalent: (i) f is a trivial bundle with fiber R "~ . (ii) For each Polish space Z and any m a p g: Z ~ embeddings is dense in the space C g ( Z , X ) .

X,

the set of closed

THEOREM 2.4.33. Let f : X --~ Y be a proper soft m a p between locally compact A N R-spaces. T h e n the following conditions are equivalent: (i) f is a trivial bundle with fiber I W . (ii) For each locally compact space Z and any proper m a p g: Z ~ set of closed embeddings is dense in the space c g ( z , x ) .

X,

the

T h e following s t a t e m e n t s are direct consequences of T h e o r e m s 2.4.32 and 2.4.33. PROPOSITION 2.4.34. Let f " X ~ Y be a soft m a p between Polish A N R spaces. T h e n the composition f i r x , where 1rx " X • R ~ ---. X is the projection, is a trivial bundle with fiber R ~ . PROPOSITION 2.4.35. Let f " X --~ Y be a proper soft m a p between locally compact A N R - s p a c e s . T h e n the composition f ~ r x , where ~ x " X • I w --. X is the projection, is a trivial bundle with fiber I • .

2.4. TOPOLOGY OF R ~- AND I ~-MANIFOLDS

89

PROPOSITION 2.4.36. Let f : X --, Y be a proper soft map between locally compact A N R - s p a e e s . I f f o r each open cover Lt E c o y ( X ) there are m a p s f l , f2: X --+ X , U-close to i d x , such that f l ( X ) n 12(X) -- q} and f f i i = 1, 2, then S is a trivial bundle with fiber I ~ .

-- f ,

PROOF. By T h e o r e m 2.4.33, it suffices to show t h a t if g: Z ~ X is a proper m a p of a locally compact space Z into X, then the set of closed e m b e d d i n g s of Z into X is dense in the space c g ( z , X ) . Take any two disjoint compact subsets K1 and K2 of Z. Obviously the set L ( K 1 , K 2 ) - {h e c g ( z , x ) :

h ( K 1 ) N h ( K 2 ) -- q}}

is open in the space c g ( z , x ) . Let us show t h a t these sets are dense in c g ( z , x ) . Take any p E C g ( Z , X ) and any b / E c o v ( X ) . Softness of the map f guarantees the existence of a refinement )2 E c o y ( X ) of b / s u c h t h a t the following condition is satisfied: (.) If a m a p h': K1 U K2 ~ X is ]2-close to the restriction p / ( K 1 U K2) and f h ~ = f p / ( K 1 U K2), then there exists a map h: Z --~ X t h a t is U-close to p and satisfies h i ( K 1 U K2) -- h ~ and f h -- f p . By our assumption, there are maps f l , f 2 : X ---, X, )2-close to i d x , such t h a t f l ( X ) N . f 2 ( Z ) = 0 a n d . f . f i = f , i = 1,2. Define the map h': K l U K 2 --* X by letting it be equal to f i p / K i on K i , i -- 1,2. Condition (.) guarantees the existence of a m a p h: Z ~ X such t h a t h is U-close to p, h / ( K 1 U K2) -- h l, and f h -- f p . It only remains to r e m a r k t h a t h C L ( K 1 , K2). Therefore, the set L ( K I , K 2 ) is open and dense in the space C g ( Z , X ) . Next, consider (in the space Z) a countable open basis B = {Vi: i E g } , the elements of which have compact closures. Let L denote the intersection of all sets of the form L(cl Yi, cl Yj), where Vi, Vj E B and cl Vi N cl Vj = 0. By Proposition 2.1.7, L is dense in the space C g ( Z , X ) . It only remains to r e m a r k t h a t each map h E L is a closed embedding. F-1 T h e proof of the following s t a t e m e n t is similar to t h a t of Proposition 2.4.19. PROPOSITION 2.4.37. Let f : X ~ Y be a proper soft map between locally compact A N R-spaces. Then the following conditions are equivalent: (i) f is a trivial bundle with fiber I • . (ii) For each map g: D • I ~ --~ X , the set {hecg(n

xI W,X): h({0}xI W)nh({0} xI ~)=0}

is dense in the space C g ( D x I ~ , X ) .

As an i m m e d i a t e consequence of Proposition 2.4.37 we have the following statement. PROPOSITION 2.4.38 " Let 3 = { X n ,~n ~n-F1 , w } be an inverse sequence consisting of locally compact A N R-spaces and proper soft short projections, each

90

2. INFINITE-DIMENSIONAL MANIFOLDS

of which has two sections with disjoint images. lira S ~ X o is a trivial bundle with fiber I ~ .

Then the limit projection p0:

T h e reader can provide the proof of the following useful s t a t e m e n t by analyzing the proof of P r o p o s i t i o n 2.4.8 (keeping in m i n d T h e o r e m 2.4.32). PROPOSITION 2.4.39. Let f : X --~ Y be a soft map between Polish A N R spaces. Then the following conditions are equivalent: (i) f is a trivial bundle with fiber R W . (ii) For each map g: N • I "~ ~ X , the set {h: C g ( N • I ~ , X ) : collection {h({n} • I ~ ): n E N } is discrete in X } is dense in the space C g ( N • I ~ , X ) .

PROPOSITION 2.4.40. Let S = { X n , p n n + l , w } be an inverse sequence consisting of Polish A N R - s p a c e s and soft short projections, each of which has a countable (infinite) f a m i l y of sections, the images of which f o r m discrete collections. Then the limit projection p0: lim S ---, X o is a trivial bundle with fiber R ~ .

PROOF. Obviously the limit projection p0 is a soft m a p and the limit space X - lim S is a Polish A N R-space. Consequently, we only have to verify condition (ii) of P r o p o s i t i o n 2.4.39. Take a metric dn on the space X n , n E w, b o u n d e d by ~ . On X we consider the metric d, defined as follows: d({xn}, {x~}) -- m a x { d g ( x n , x'n)" n e w}. Consider two m a p s f : N • I ~ --, X and a : X ~ (0, 1). Our goal is to construct a m a p g: N x I ~ ~ X satisfying the following conditions: (a) POg -- POf . (b) d ( f ( y ) , g(y)) 0 so small that any two e-close maps of any c o m p a c t u m O A have homotopic compositions with f. Take an index k such that for each i _> k the projection pi moves points not more than e and, additionally, the e-neighborhood Oe(pi(A)) of the set pi(A) in I ~ is contained in O A. By (ii), there is an index i(k) such t h a t a~+l(pk (k) < ~. Take a triangulation of mesh < ~ of the polyhedron Xi(k) and denote by Ai(k) the union of all those simplexes of this triangulation that intersect Pi(k)(A). Obviously, _

into

Pi(k)(A) C_ Ai(k) C OA. X i(k) ( n + l ) --, X k o f t h e ( n + l ) Bythechoiceoftheindexi(k) , thereexistsamapr dimensional skeleton of the polyhedron Xi(k) (with respect to the above indicated

triangulation) with at most n-dimensional image such t h a t dist(r Obviously, r rA(n+l)) i(k) C OA and the restrictions of the maps f r

and f onto

(n+l) (i.e. n-dimensional skeleton of Ai(k) are homotopic. ~(k) ( y (n-t-l)) Since dim Cv'i(k) < n , the map f admits an extension defined on the union (y(n+l)~ OA U C w , i(k) J" -

Consequently, the map f r

--- (n+l)

.y. ( n + l )

can be extended to ~'i(k)

. By the remark

made in the beginning, f liA(n+ i(k) 1) also has an extension to X(n+l)i(k). Now recall t h a t there are no obstructions to extending a map from the m-dimensional skeleton onto the (m + 1)-dimensional one if the range has trivial m-dimensional homotopy group. Therefore there is a map g" Xi(k) K ( Z , n ) extending Jl~i(k) 7,A(n+I) 9 Since the maps g/Ai(k) and f/Ai(k) coincide on the (n + 1)-skeleton, they are homotopic. This shows that f/Ai(k) has an extension to the space Xi(k). It only remains to remark t h a t f is homotopic to fpi(k)/A and, consequently, has an extension to X. By Theorem 3.1.2, dim Z X _ n. (i) ~ (ii). It is not hard to see t h a t the inequality dim Z X < 1 implies the inequality dim X _ 2. Consider a finite simplicial complex L. We now present a description of a connected C14~-complex L associated with L. First, represent L as the union L = L (n) tJ al LJ ... [_J~s, where the simplexes ai are indexed so t h a t n + 1 _ d i m a l < d i m a 2 _< --- _< as. We construct the complex L by induction on s. In the meantime the following conditions will be satisfied: (a) /~ = L (n) U g ( a l ) U - - - U K(as), where g ( a i ) is an Eilenberg-MacLane complex of type g ( r n ( a } n), n). (b) L (n) ~(n+l) L (n) and L (n) N K(ai) _(n) for each i < s. _~_

-~

.~

o

i

102

3. COHOMOLOGICAL DIMENSION

(c) aiNaj, g(aiNaj),

g(ai) Ng(aj)--

if d i m ( a ~ N a j ) < n if d i m ( a i N a j ) > _ n + l .

If s = 0, then we let f, = l (n). If a CW-complex L (n) U g ( a l ) U ... U g ( a k ) , associated to the complex L (n) Uaz U . . . U a k and satisfying the above conditions, has already been constructed, then the complex K(ak+l) can be obtained by killing all homotopy groups in dimensions larger than n of the complex

.~(n) g ( o a k + l ) = "k+l

U

u { g ( a i ) " ai C 0ak+l}.

Since the embedding Oak+ ~ (n)1 ~ K(Oak+l) induces isomorphisms of the homotopy groups of dimension _ n, we conclude that condition (a) is satisfied. The remaining conditions are obvious. We now proceed to the direct proof of the implication (i) ~ (ii). Let e > 0 and an index j be given. Consider a triangulation of the polyhedron Xj of mesh < 5" Consider the complex

f(j

=

X~ n)

U

g (61)

U""

U

g (as)

associated with this triangulation. Define a map i6j" X ~ )Cj so that the restriations of pj and i6j onto p-~Z(X~n)) coincide and such that for each k < s we have

p~(p~ (~k))c g(~k). ^

--1

We construct the desired map 16j by induction, defining it on the sets of the form

pj Z(x~n)U a z a . " U ak), where 0 < k _< s. If the map 16j has already been defined on the set pj Z(X n) U a z a . . . U ak), then, by the inductive assumption,

pj(pyl(Oak+l)) ~

g(ak+l).

Since dim Z X __ 0) if for every compactum C C_ Y there exists a compactum K C_ Y such that for each point x E X - f - l ( K ) and every map ~ " (S i, ,) --, (Y - K, f ( x ) ) there exists a map ~ " (S i, ,) --, ( X - f - l ( C ) , x ) and a homotopy f ~ "~ a(rel ,) in Y - C . We say that f 9 X - , Y induces a m o n o m o r p h i s m of i-th homotopy groups of ends if for every compactum C C_ Y there exists a compactum K C_ Y such that for every map ~ 9 S i --, X - f - l ( K ) with the property that f ~ is null-homotopic in Y - K it follows that a is null-homotopic in X - f - l ( C ) . As usual, f is said to induce an isomorphism of i-th homotopy groups of ends if it simultaneously induces an epimorphism and a monomorphism. PROPOSITION 4.1.11. A proper map f 9 X - , Y between at most (n ~- 1)dimensional locally compact LCn-spaces is a proper n - h o m o t o p y equivalence if and only if it induces isomorphisms of homotopy groups of dimension ~_ n and isomorphisms of homotopy groups of ends of dimension ~_ n. Note that proper n-homotopies have also been studied from the categorical point of view [169]. The following proposition will be used below and indicates a difference between the n-homotopy and usual homotopy theories (compare with [303]). PROPOSITION 4.1.12. Let M be an at m o s t (n -t- 1)-dimensional locally finite polyhedron. Suppose that there exists an at m o s t (n + 1)-dimensional finite polyn

hedron K and two maps f " M --~ K and g " K -+ M such that g f ~_ idM (i.e.

4.1. GENERAL THEORY

133

g is an n - h o m o t o p y domination). Then there exist an (n + 1)-dimensional finite polyhedron T, containing K as a subpolyhedron, and an n-homotopy equivalence h 9 T ~ M extending g such that f is a n-homotopy inverse of h. PROOF. Obviously, it suffices to consider only connected polyhedra. Consequently, the case n -- 0 is trivial. If n -- I, then, by the assumption, lrl(g)" l r l ( g ) --* ~rl(M) is an epimorphism and g e r ( l r l ( g ) ) is a finitely generated group. Select finitely m a n y generators of Ker(~rl(g)) and use t h e m to a t t a c h 2-cells to K and to e x t e n d g over these cells. In this way we obtain a 2-dimensional finite polyhedron T, containing K as a subpolyhedron, and a map h" T --, M , extending g, which induces an isomorphism of f u n d a m e n t a l groups. By Proposition 4.1.9, h is a 1-homotopy equivalence. Assume, by way of induction, t h a t the proposition is already proved in the cases n ~_ m, m _ 1, and consider the case n - m + 1. W i t h o u t loss of generality we can suppose t h a t f ( M (i)) C_ K(~) and g ( K (~)) C_ M (~) for each i _< m + 1. Since g f m+l idM it follows easily t h a t g f / M (re+l) m idM(m+l)" By the inductive hypothesis, there are an (m + 1)-dimensional finite polyhedron R, containing K (re+l) as a subpolyhedron, and an m - h o m o t o p y equivalence r" R - . M (m+l) e x t e n d i n g g / K (re+l). Sewing together the polyhedra K and R along naturally e m b e d d e d copies of K (re+l), we obtain the (m + 2)-dimensional finite polyhedron L, containing K and R as subpolyhedra, and the map s" L --. M which coincides with g on K and with r on R, whence s f = g f

m-l-1

~_ idM and f s / L (re+l) =

m

f r ~_ idL(m+~). By these conditions, we conclude t h a t ~i(s)" ~ri(L) --, ~ i ( i ) is an isomorphism for each i ___ m and an epimorphism for i -- m + 1. One can easily verify t h a t in this situation Ker(lrm+l(S)) is a finitely g e n e r a t e d Z(lrl(L))module. Select Z ( ~ l ( L ) ) - g e n e r a t o r s for Ker(lrm+l(S)) and use t h e m to a t t a c h (m + 2)-cells to L and to extend s over these cells. Let T denote the resulting (m + 2)-dimensional finite polyhedron, containing L as a subpolyhedron, and h" T --~ M the corresponding extension of s. T h e n lri(h) is an isomorphism for each i _< m + 1. Again, by Proposition 4.1.9, h is an (m + 1)-homotopy equivalence. This completes the inductive step and finishes the proof. [-1 T h e analogous s t a t e m e n t for proper n - h o m o t o p y dominations (near c~) will be discussed in Subsection 4.4.4

4.1.3. Z-set unknotting

and topological homogeneity.

PROPOSITION 4.1.13. Let A be a closed subset of a Polish A N E ( n ) - s p a c e X . Then the following conditions are equivalent: (i) A is a Z n - s e t . (ii) For each at most n-dimensional locally finite polyhedron P, the set { f E C ( P , X ) : f ( P ) A A - - 0 } is dense in C ( P , X ) . (iii) For each at most n-dimensional Polish space Y , the set { f e C (Y, X ) : f (Y) N A = 0} is dense in C (Y, X ) .

134

4. MENGER MANIFOLDS

Note that each Z n - s e t in any at m o s t n - d i m e n s i o n a l L C n - l - s p a c e is a Z-set. If, in Proposition 4.1.13, X is locally compact, then the listed conditions are equivalent to the following: (iv) For each at most n - d i m e n s i o n a l Polish space Y , the set { f 9 C ( Y , X ) : c l ( f ( Y ) ) N A - 0} is dense in C ( Y , X ) . Closed subsets satisfying t h e p r o p e r t y (4), as in the case of infinite-dimensional manifolds, are called strong Zn-sets. These sets are especially i m p o r t a n t in the non-locally c o m p a c t setting. PROPOSITION 4.1.14. One-point subsets of M e n g e r manifolds are Z-sets. T h e following s t a t e m e n t s are versions of the powerful Z-set u n k n o t t i n g theorem. THEOREM 4.1.15. Let Z1 and Z2 be two Z - s e t s in a ~ n + l - m a n i f o l d M , and let h : Z1 ---* Z2 be a h o m e o m o r p h i s m . Denote by ij : Z j --, M the inclusion map (j = 1,2). I f il and i2h are properly n-homotopic, then h extends to a h o m e o m o r p h i s m H : M ---, M which is properly n - h o m o t o p i c to idM.

COROLLARY 4.1.16. Every h o m e o m o r p h i s m between Z - s e t s of #n can be extended to an a u t o h o m e o m o r p h i s m of #n. For n - 0 this result is well-known. A closed subset of ~1 is a Z-set if and only if it does not locally s e p a r a t e ~1. T h e result for n - 1 originally a p p e a r e d

in [235]. A c o m p a c t u m X is called strongly locally homogeneous if for each point x E X and each n e i g h b o r h o o d U of x , there is a n e i g h b o r h o o d V of x contained in U such t h a t the following condition holds: for each point y E V, there is a h o m e o m o r p h i s m h : X ---, X such t h a t h(x) -- y and h / ( X - U) -- id.

COROLLARY 4.1.17. #n is topologically homogeneous. Moreover, it is strongly locally homogeneous. THEOREM 4.1.18. Let M be a #n-manifold. For each open c o v e r U E c o y ( M ) , there is an open cover )2 E c o y ( M ) with the following property: (,) if a h o m e o m o r p h i s m h : Z1 ---* Z2 between two Z - s e t s of M is )?-close to il (see notations in 2.3.5), then h can be extended to a h o m e o m o r p h i s m H : M ~ M which is l~-close to idM.

4.1. GENERAL THEORY

135

4.1.4. Topological characterization. The following characterization theorem [33, T h e o r e m 5.2.1] is central to the whole theory. We recall t h a t a space X has D D n P (Disjoint n-Disks P r o p e r t y ) if for each open cover L / 9 c o y ( X ) and any two m a p s a : I n ---. X and fl : I n ---+ X , there are maps a l : I n ---* X and ~1 : I n --. X such t h a t a l is U-close to a, ~1 is U-close to ~ and a l (I n) Aft1 (I n) = 0.

THEOREM 4.1.19. The following conditions are equivalent for any n - d i m e n sional locally compact A N E ( n ) - s p a c e X : (i) X is a #n-manifold. (ii) X is strongly B•,n-universal, i.e. has D D n p . (iii) Each map of the discrete union I n @ I n into X can be approximated arbitrarily closely by embeddings. (iv) Each proper map of any at m o s t n - d i m e n s i o n a l locally compact space into X can be approximated arbitrarily closely by closed embeddings. (v) Each proper map of any at m o s t n - d i m e n s i o n a l locally compact space into X can be approximated arbitrarily closely by Z-embeddings. (vi) Each proper map f : Y --. X of any at m o s t n - d i m e n s i o n a l locally compact space Y into X such that the restriction f /Yo onto a closed subset Iio is a Z-embedding can be arbitrarily closely approximated by Z-embeddings coinciding with f on Yo. Additionally, if X is c o m p a c t and ( n - 1)-connected (i.e. X 9 A E ( n ) ) , then conditions (ii)-(vi) give a topological characterization of the c o m p a c t u m #n. Note t h a t the t h e o r e m remains true even in the case n - c~ (see C h a p t e r 2). If n -- 0 and X is compact, condition (ii) trivially implies t h a t X has no isolated points. Therefore, in this case, X is h o m e o m o r p h i c to the C a n t o r disc o n t i n u u m as has already been noted above. A p p l y i n g the above characterization in the case k -- 1, we see t h a t a comp a c t u m is h o m e o m o r p h i c to #1 if and only if it is a 1-dimensional, locally connected c o n t i n u u m with D D 1 P . It is known (see [15] or [235]) t h a t a locally connected c o n t i n u u m has D D 1 P if and only if it has no local s e p a r a t i n g points and has no open subspaces e m b e d d a b l e in the plane. In this sense, Bestvina's characterization of #1 reduces to Anderson's. In Subsection 4.1.1, three m a j o r geometric constructions of the universal Menger c o m p a c t u m have been presented. Let us indicate another, spectral construction, given in [248]. Let {G~} be a basis of open sets of S 1, such t h a t G~ is an open cell with the p r o p e r t y t h a t diamG~ --. O. Let us construct an inverse sequence {Xi,p~ +1} as follows. We set X0 = S 1, and we get X~+I from X~ by "bubbling over G~" , i.e. X i + l is the quotient space o b t a i n e d from the disjoint union Xi ~ X~ by identifying the two copies of x 9 Xi precisely when p~(x) ~ Gi (here p*o" Z i ~ X o denotes the corresponding projection). T h e projection ~i ~ + 1 " X i + l ~ Xi is defined in the obvious way. Note t h a t p~+ 1is a retraction. One can check directly t h a t X = lim{Xi, Pi -i+1 } is a 1-dimensional 9

136

4. MENGER MANIFOLDS

A E ( 1 ) - c o m p a c t u m with D D 1 P and hence, by 4.1.19, X is a copy of #1. More careful consideration shows [33] that we get #n if we start with S '~ = X0 (and proceed as above). There are several other constructions of #n. To the best of our knowledge, all of them are defined as the limit spaces of inverse sequences(see, for example,

[155], [ls6]). 4.1.5. Approximation by Homeomorphisms. THEOREM 4.1.20. Proper U V n - m a p s between #n+l-manifolds are near-homeomorphisms. Another important result of I ~-manifold theory states that an infinite simple homotopy equivalence between/W-manifolds is homotopic to a homeomorphism. Let us note that this is not the case for homotopy equivalences (i.e. there exist non-homeomorphic but homotopy equivalent compact I ~ -manifolds). In #n-manifold theory we do not have simple homotopy obstructions and this significantly simplifies the corresponding result. THEOREM 4.1.21. Each proper n-homotopy equivalence between #n+l-manifolds is properly n-homotopic to a homeomorphism. The following result, due to Ferry [150, Proposition 1.7] and improved slightly in [188], also illustrates this situation. PROPOSITION 4.1.22. Let f 9 P ~ L be a map between compact polyhedra which induces an isomorphism between the i-th homotopy groups for each i < n. Then there is a compact polyhedron Z and U V n - m a p s ~ 9 Z ---. P and ~ 9 Z ~ L such that f a n ~.

Historical and bibliographical notes 4.1. Most of the results of this Section were obtained by Bestvina in his fundamental work [33]. The notion of nhomotopy is due to the author [87]. Proposition 4.1.12 was proved by the author

[100]. 4.2. n-soft m a p p i n g s of c o m p a c t a , raising d i m e n s i o n The following statement is the first resolution theorem for locally compact L C n - l - s p a c e s , in particular for #n-manifolds. THEOREM 4.2.1. Every locally compact L C n - l - s p a c e is an (n-invertibte) proper u v n - l - i m a g e of a #n-manifold. We also will be using the following result, which states that the u v n - r e s o l u tions from Theorem 4.2.1 can be improved over Z-sets.

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION PROPOSITION 4.2.2. Let f " M ~

X

be a proper U V n - s u r j e c t i o n of a #n+l_

m a n i f o l d onto a locally c o m p a c t L C n - s p a c e and let Z be a Z - s e t in M . X

is any proper m a p properly n - h o m o t o p i c to f / Z ,

uvn-surjection

137

I f g" Z ---.

then there exists a proper

h" M ---. X such that h / Z -- g.

Below, in this subsection, we prove much s t r o n g e r results. We begin by fixing a n a t u r a l n u m b e r n. Let B n+l -- { x E R n ' i i x i l < 1} d e n o t e t h e unit closed ball in R n+l. Consider a m a n y - v a l u e d r e t r a c t i o n l:ln" B n+l "--+ S n of the ball B n+l onto its boundary, the sphere S n - OB n+l, defined as follows" R n ( x ) - {y E S n" ( x , y ) >_ 4[[xI[ 2 - 3[[x[[} for each point x E B n+l.

Here (x, y) denotes the s t a n d a r d scalar p r o d u c t in the E u c l i d e a n space R n+l and Ilxll = v / ( x , x ) denotes the induced norm. T h e above to each point of the ball B~'+1 = {x E R n+l" IIxlI < If x E B n+~ - (B~ +1 U Sn), t h e n R n ( x ) is a ball on x E S n, then R n ( x ) = x, i.e. R n is a retraction. It continuous. Consider the g r a p h F Rn of this map, i.e.

m a n y - v a l u e d m a p assigns 89 the whole sphere S n. the sphere S n. Finally, if is easy to see t h a t R n is

FRo = { ( ( x , y) e B ~+1 • S ~ : y e R ~ ( ~ ) } . By Pn: FRn ---+ B n+l and qn: FRn --'+ S n we d e n o t e the restrictions to FR., of the n a t u r a l projections of the p r o d u c t B n+l x S n onto its coordinates. LEMMA 4.2.3. T h e m a p Pn" FRn -'+ B n+l is ( n -

1)-soft.

PROOF. C o n t i n u i t y of the r e t r a c t i o n R n implies openness of Pn. Therefore, by T h e o r e m 2.1.15, it suffices to show t h a t the collection { R n ( x ) " x E B n + l } is c o n n e c t e d and uniformly locally c o n n e c t e d in all dimensions less t h a n n - 1. As was m e n t i o n e d above, topologically t h e r e are only three t y p e s of elements of the collection { R n ( x ) " x e B n + l } . Indeed, R n ( x ) is the sphere S n (for x E B~'+I), X a closed ball B n with center at i[~-][ in the sphere S n (for x E B n+l - B~'+I), or a point (for x E Sn). C o n s e q u e n t l y the i n d i c a t e d collection is c o n n e c t e d in all dimensions not exceeding n - 1. Let us show the uniform local c o n n e c t i v i t y of this collection in all dimensions less t h a n n - 1. Let k _ n - 2, e > 0 and e Consider a point x E B n+l and an a r b i t r a r y m a p a" S k ---, R n ( x ) 5 - - ~. such t h a t the d i a m e t e r of its image a ( S k) is less t h a n 5. E v i d e n t l y t h e r e is a ball B n on the sphere S n, containing this image a ( S k ) . Let &" B k+l ---+ B n be an extension of a from the b o u n d a r y S k = OB k+l onto the whole B k+l. Additionally, we can assume t h a t if R n ( x ) = x (which occurs when x E S n ) , t h e n a = a. Obviously, in this case as well as in the case w h e n R n ( x ) = S n, ~ is the desired extension. Suppose now t h a t x E B n+l - (B n+l U s n ) . T h e n R n ( x ) is a ball on S n of non-zero diameter. If S '~ - R , ( x ) ~= B n, t h e n the intersection B n M R n ( x ) is h o m e o m o r p h i c to the n - d i m e n s i o n a l ball and, consequently, t h e r e exists a r e t r a c t i o n r" B n ~ B n M R n ( x ) . Therefore, in this case, the c o m p o s i t i o n a r is the desired extension of a with d i a m e t e r of the image less t h a n e. In the r e m a i n i n g case, when S ~ - R n ( x ) C B n, we first observe t h a t the b o u n d a r y

138

4. MENGER

MANIFOLDS

O R n ( x ) of the ball n n ( x ) is contained in B n. Since k 4- 1 < n, there is a h o m o t o p y (in B n) connecting the m a p 5 with a m a p / 5 " B k+l ~ B n N R n ( x ) fixed on ~ - l ( R n ( x ) ) . Then fl does not move points of OB k+l -- S k as well. Clearly, ~ is the required extension of c~. [7

LEMMA 4.2.4. The map Pn" FRn ~ B n + l is n-invertible. PROOF. Let us consider a lower semi-continuous many-valued retraction Fn" B n+l ---* S n defined as follows:

= ( S

I1 11 > ifll ll
m is naturally isomorphic to the diagram C~,j

(K). In

kn k-re,n-re(K). particular, Ei,'j (K) ~ Ei_m, j 9 Let K1 be a subcomplex of K.

Then the diagram Ek?(K1) can be k,n

naturally embedded into the diagram Ei, j (K) in the sense that for all indices i , j there exist embeddings ~oi,j" Ei,k,n j (K1) ~ E ki,j, n (K) such that the maps f.k'.n(K1)~,~ and gik,?(gl) coincide with the restrictions of the k,n

kn

maps fi,j (K) and gi,~ ( g ) respectively 9 Observe also that for each i > 0 and j > 0, we have

E ki,j, n ( K 1 ) - (fi,~n(K))-l(Eik,?(K1)) kn

LEMMA 4.2.11. If K is a finite simplicial complex, then Ei, ~ (K) is an A N R -

compactum. PROOF. We prove our statement by induction on k. First consider the case 1,n k = 1. Represent the space E0,1 ( g ) =1 g(n) I as the union a l U a2 U . . . U ar of

144

4. MENGER MANIFOLDS

its simplexes. Then 1 (al) U - - - U f -1,1 1 (fir). El:I n1 (K) = f -1,1

Since the map f1,1 is a projection of the graph of the many-valued retraction RK(,,_~), it can easily be seen that for each i _~ r we have Sl, l ( a i ) n Uj_ m0). Observe t h a t [ a l + l I - - M l - I n t l a l + l I. Therefore g-1

1

g-1

-1

1

k+l,k+lfl (M/+I) ---- k + l , k + l f l l ( I n t [ a / + l l) U gk+l,k+lfl (Ml) --- f o l ( c l ( g ~ , ~ + l ( X n t

[ hi+ 1 [)) U g-1 k+l,k+l f 1 1 ( M l ) .

Notice also t h a t

f o l ( d ( g l-1 , k + l ( I n t lal+l I)) ~ f l l ( I al+l [) • VX(al+l)B n. In this situation it suffices to show t h a t

fol(cl(g~,lk+l(IntIal+l I)) n g-lk+l,k+lfll (Ml) is an A N R-space. Note t h a t f o 1 ( c l ( g l-1 ,k+l(Int

f o l ( C l ( g l-1 ,k+l(Int

l al+l I)) N g k-1+ l , k + l f l l ( M l ) l al+l I)) N g l-1 ,k+lfll(Ml)

--

--

--1 --1 f o l ( c l ( g 1,k+1 -1 ( I n t [ hi_t_1 [)) Pl g l , k + l f I (0 ] al+ 1 [) Consider the projection

onto fll(Olaz+~l) which, as shown above, is an ANR-space. Let

r

fll(Olal+l[) • VXal+,)B n --+ E l , k + l

be the composition of f l x idvB n and the trivialization --1 (0lal+ll) 7I" OIal+ll • VB n ---+ gl,k+l

-1 (0 ] az+l I) --1 1( I n t ] al+l ]))I~l gl,k+l of the bundle gl,k+l/cl(gl,k+ flTr = g l , k + l r Consequently there exists a m a p 4: f11(Olal+11) • VX(a,+I) Bn --* Ek+l,k+l,

Notice t h a t

146

4. MENGER MANIFOLDS

lifting b o t h lr and r Since the collection {Tr, r separates points of the compactum f~l(Olal+ll) x VX(a~+I)B n, we conclude that r is injective. On the other hand,

r

x vx(o,+,)B ~)

-

fol(r

n g l-1 , k + l f -1l ( M t ) "

( g l-1 ,k+l(Intlal+ll))

Therefore

f o l ( d (o~,k+~(x~tl"z+~l)) -1 n g-1 k+~,k+~f~-1 (M~) is an A N R-space. --1 1(MI) is an ANR-space for each I. It only suffices to note Thus gk+l,k+lfl that Ek+1,k+1 = g-1 k+1,k+l f -11 (El,k). This finishes the proof.

D

LEMMA 4.2.12. For each finite simplicial complex K and each natural number n, there exist an ANR-compactum X and two maps f : X --*1 K I and g: X --~ IK(n)l satisfying the following conditions: (a) The map f is n-invertible, polyhedrally n-soft and ( n - 1)-soft.

(b) g / f - l ( [ g ( n ) l ) - f / f - l ( [ g ( n ) [ ) . (c) gf-l(lal)c_ Io(")1. n n+k

PROOF. Let d i m K = n + k. Consider the diagram ~r

X

(K) and let

~k,n+k(K )

= ~~k,k

k n+k

k n+k

f = f l , k " " fk,k: Ek' k (g)---* E0'~r (g) = Igl, k:.+k k:.+k g = gl,k gk,k: Zk, k ( g ) ---, Ek, o ( g ) - - I g ( n ) l

for each a e g .

By L e m m a 4.2.11, X is an ANR-compactum. By the properties of the di1)agrams s i , j ' the map f is n-invertible, polyhedrally n-soft and ( n soft. These properties also imply that g f - 1 = RK . . . . RK(,+k-1), and consequently f and g coincide on the inverse image f-l([K(n)[). For the same reason,

g.f-l(lal) c_ Io(n)l. D Remark 4.2.13. All statements proved in this section so far are valid for countable locally finite simplicial complexes as well. In such cases, the resulting spaces are locally compact A N R-spaces and the resulting maps are proper. Remark 4.2.14. More careful consideration shows that the compactum X from L e m m a 4.2.12 is a finite polyhedron. Respectively, if K is countable and locally finite then the polyhedron X is also countable and locally finite. We are ready to prove the main result of this section. THEOREM 4.2.15. For each natural number n, there exists a map fn: #n I ~ of the universal n-dimensional Menger compactum onto the Hilbert cube satisfying the following properties: (i) The map fn is n-invertible, polyhedrally n-soft and ( n - 1)-soft.

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION

147

(ii) For each at most n-dimensional compactum Y, each map f" Y --+ #n and each open cover lg E cov(#n), there is an embedding g" Y --~ #n that is lg-close to f and such that f ng = f n f .

PROOF. We construct the map fn as the limit projection of an quence 8 - {M{, c~ +1} consisting of the Hilbert cube manifolds (in M1 -- Io2 ). The construction is carried out by induction. The limit verse sequence is contained in the product I~' • I~' • --. • I~ • -.the metric

~(~, y) = r ~

inverse separticular, of this inwhich has

d~(x~, y~) 2(~+1)

where di is a (bounded by 1) metric on the Hilbert cube I.~ = ( y l , . . . , y i , . . . ) and xi, yi E I~ for each i. Let ai E Io2 for each i. Then each of the products I~' x --- x I m can be naturally identified with the subspace oo I~' x - - . x I m x {am+l} x - - - x {ai} x - - - of the product l I i = l i ~ . We construct an Io2-manifold as a subspace of the product I~ x --- x I~n and consider the O2 restriction (to Mi+l) of the projection ~ + 1 . 1-Ij=l I~ x Ii+1 ~ 1-I~=11~ as a projection a~+l. Mi+l ---+ Mi of the spectrum 8. In this situation, as can easily be seen, the limit space X = lim 8 is naturally embedded into Io2 - 1--Ii=l~176 i~'. Let M1 = I~ and suppose that the Io2-manifold Mj C YIi=lJI~ c Io2 has already been constructed. By Theorem 2.3.28, the Io2-manifold Mj is homeomorphic to the product of the Hilbert cube jIo2 and a finite polyhedron Kj. Obviously, we may assume that the composition of the projection wj" Kj x j I W ---+ Kj with a certain section sj" Kj" K j x j Io2 is a 2-J-move, i.e. for each point x E K j • Io2, we have d(x, sjwj(x)) < 2 - j . We also assume t h a t Kj is given together with a triangulation such t h a t diamsj(a) < 2 - j for each simplex a of this triangulation. By L e m m a 4.2.12, there exist an A N Rc o m p a c t u m X j + I and maps f j + l " X j + I ~ K j and gj+!" Xj+I ---+ K~ n) such t h a t f j + l is n-invertible, polyhedrally n-soft, ( n - 1)-soft and, in addition, -1 (n) for each simplex a of the triangulation given on Kj. Let Mj+I : Xj+I • Io2. By Theorem 2.3.21, Mj+I is a Io2-manifold. Next, represent the cube j+lIo2 as the product jIo2 x j + 1 I~' of two Hilbert cubes, and denote by yj+l" X j + I • Io2 • I~ --+ Xj+I x jIo2 the product of the identity map idx~+l and the projection ~ " jIo2 • I~ --+j Io2. Define the map ~+~ j 9 Mj+I ~ Mj by letting ~j+l

= (f3+~ • i d ~ ) ~ j + l .

In this situation we have the following diagram:

148

4. MENGER MANIFOLDS

Mj+I

= Xj+l

~ j + l -- i d x

x j I ~~ x j + 1 I~'

7r31

Xj+I xj I ~

Xj+I

f j+l x id

fj+l

,~ K j x j I ~~ = M j

,- K j

Let A" M j + I ~ IS+1 be an embedding. T h e n the desired e m b e d d i n g of the /"a-manifold M j + I into the p r o d u c t I i a x - - - x I~a+l can be defined as the diagonal p r o d u c t of the m a p ~ j + 1 and the e m b e d d i n g A. Let a jj + l ._ 7rJ+lj / M j + I , and j+l observe t h a t this map a j " M j + I ~ M j is n-invertible, polyhedrally n-soft and ( n - 1)-soft (because b o t h f j + l and 7r~ have the c o r r e s p o n d i n g properties). T h u s the c o n s t r u c t i o n of s p e c t r u m 8 = {Mi, a~ +1} is complete. Let ai" X = lira,5' --, M i d e n o t e the i-th limit projection of this s p e c t r u m . As the desired m a p fn" #n __~ i ~ we take the first limit projection a l " X ---+ M1 = I W 9 Of course, we still have to show t h a t X ~ #n. Let us investigate the properties of the c o m p a c t u m X and the m a p fn. Since each of the short projections a~ +1 of the s p e c t r u m S is n-invertible, p o l y h e d r a l l y n-soft and ( n - 1)-soft, we easily see t h a t fn (as well as all other limit projections of the s p e c t r u m S, see L e m m a 6.2.6) has the same properties. This proves one p a r t of the theorem. Let us now show t h a t dim X = n. Since M1 -- I w , M1 contains a topological copy of the n - d i m e n s i o n a l cube I n. T h e n-invertibility of fn then g u a r a n t e e s t h a t X also contains a copy of I n. Therefore dim X _> n. In order to show t h a t dim X _< n we proceed as follows. Obviously, all we need is an existence, for each i, of a ~ - m o v e of X into an n - d i m e n s i o n a l polyhedron. Such a move is i n d i c a t e d below" siwi(gi+l x idii~)~ii+lai+ 1 9 X ---+ K} n). Consequently, dim X - n . Next we show t h a t fn has the second p r o p e r t y formulated in the theorem. Let Y be an at most n - d i m e n s i o n a l c o m p a c t u m , f" Y -4 X be a m a p and

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION

149

bl E c o v ( X ) . Choose an index i and an open cover 12 E cov(Mi) such t h a t a~-l(Y) refines/~/. It follows from the construction of the m a p a~ +1 t h a t a~ +1 can represented as the composition 7i+lr/i+l, where ~i+l is a trivial bundle with fiber the Hilbert cube. Consequently, there exists an e m b e d d i n g hi+l" Y ~ M i + l such t h a t a~+lhi+l -- a i f . Since d i m Y < n, and since the limit projection a i + l " X ---+ M i + l in n-invertible, we can find a map h" Y ---+ X such t h a t a i + l h -- hi+l. It is not hard to see t h a t h is an embedding. Moreover, h is U-close to f and f n h = f n f . Finally, let us show t h a t X is homeomorphic to #n. By the property of f n established above, X satisfies the condition from T h e o r e m 4.1.19., i.e. every map of an at most n-dimensional c o m p a c t u m into X can be arbitrarily closely a p p r o x i m a t e d by embeddings. T h e equality dim X -- n was also established above. Therefore, by T h e o r e m 4.1.19, it suffices to show t h a t X E L C n - I N C n - 1 . First we show t h a t X E L C n-1. Let x be an a r b i t r a r y point of X and U be a neighborhood of x. Take an index i and a n e i g h b o r h o o d Ui of a i ( x ) in Mi such t h a t a~-l(Ui) C U. Since, by our construction, Mi is an A N R - c o m p a c t u m (even an I ~ manifold), M i E L C n-1. Consequently, there exists a neighborhood Vi of a i ( x ) in Mi such t h a t Vi C Ui and the following condition is satisfied: 9 for each k < n - 1 and each map ~i" S k -+ V~, there is a m a p r B k+l --+ U~ such t h a t ~i = r k. Let VaT, l ( v i ) . Clearly, V is a neighborhood of x contained in U. Take any map ~o" S k ---+ V, k _< n - 1. By the choice of the n e i g h b o r h o o d Vi, there is a map r B k+l --+ Ui such t h a t ai~o = r k. Since (Sk, B k+l) is a polyhedral pair and the map a i is polyhedrally n-soft (recall t h a t k < n - 1), we conclude t h a t there is a map r B k+l --+ X such t h a t the following d i a g r a m commutes:

X

Sk t

.

~Y

.Bk+ 1

In other words, r k = ~ and a i r = r This obviously implies t h a t k+l) C_C_U. Thus, X E L C n-1. A similar (but simpler) a r g u m e n t shows t h a t X C C n-1. Therefore, X ~ #n. T h e proof is finished. [2] r

In order to o b t a i n other i m p o r t a n t properties of the m a p fn we need some preliminary statements. T h e proof of the following one is, in fact, contained in

150

4. MENGER MANIFOLDS

the proof of L e m m a 4.2.6. LEMMA 4.2.16. For each e > O, there is a 5 > 0 such that for any map ~o: S n-1 --~ F R , , with diam(im(~o)) < 5, and any 5-homotopy H : S n-1 x[0, 1]---, B n+l, with Ho = pn~o, there exists a h o m o t o p y G: S n-1 x [0, 1] ~ FR, such that Go = ~o, pnG = H and d i a m G ( S n-1 x [0, 1]) < e. LEMMA 4.2.17. Let Y be an L C n - l - c o m p a c t u m , K be a finite simplicial complex and ~o: Y --~ Ig(n+l)l be a map. Then the fibered product (pullback) X of the spaces FRK(,~) = 1-'n+l and Y with respect to the maps Pn+l -- PRK(n ) : Pn+l --+

IK(r'+l)l and ~o: Y --, Ig(n+l)l is an L C n - l - c o m p a c t u m . i i c~ l:~ n PROOF. Let c~ (~,~=1~.i ) be the one-point compactification of a discrete collection of n-dimensional disks B.n and let x be the compactifying point. Let S~'-1 denote the boundary of the disk B~'. Assume the contrary. Then there exists a map r (U~IS~ '-1) ---, X such that for each i and for any extension of r -1 (to the disk B~) the diameter of its image is more than some positive number a. Let a be an (n + 1)-dimensional simplex in K (n+l) containing the point y = pn+l~O~r Here ~o~:X --+ Fr,+l denotes the canonical projection (parallel to ~o) of the fibered product X. The map q: X ~ Y has similar meaning (see the following diagram):

a (u~,S~ '-I)

r

, X

,- Fn+l

Pn+l

y

r

,._

IK(n+l) I

We identify the simplex a with the unit (n + 1)-dimensional ball B n+l. Also, we identify the restriction of the many-valued retraction R g ( n ) : l K ( n + l ) l --~ Ig(n)l to lal with the many-valued retraction R n : B n+l ~ OB n+l (see the beginning of this section). We consider two cases. C a s e 1. Assume first that y f~ OBg +1, and take a neighborhood V of y in IK(n+l)l disjoint from OB~ +1. The restriction of Pn+l to this neighborhood is n-soft by construction. But then the restriction of q to q-l(~o-l(Y)) is also nsoft. It only remains to observe that the latter fact, coupled with the condition Y E L C n - l , contradicts the assumption made in the beginning. C a s e 2. Now assume that y E OB~ +1. Since Y c L C n-1 we can conclude that there is an extension r (U~__lBn) ~ Y of the composition qr Take

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION

151

an e > 0 such that for each set F C X, the inequalities diam((t'(F)) < e and diam(q(F)) < e imply the inequality d i a m F < a. Next, choose 5 as in Lemma 4.2.16. Let b~ be the center of the ball B~. Connect the points 4(b~) and qr by paths ~i" [0, 1] ~ Y so that limi--.oo diam~i([O, 1]) = 0. Then there is a number k such that

max{diam(rlk([O, 1]) U r

diam((trlk([O , 1]) U (t(~(B~))} < 5.

Let Ht" S~ -1 x [0,1] --~ Y be a homotopy which c o n n e c t s r -1 with the constant map to the point 4(bk) inside of the image 4(B~), and then sends this constant map along the path ~k to another constant map to the point qr By L e m m a 4.2.16, the homotopy (tilt can be lifted to a homotopy Gt . S kn - 1 x [0, 11 --. Fn+l so that Go = (t'r G I ( S ~ -1) C_ Pn~I(Y) and d i a m ( G t ( S ~ -1 x [0, 1]) < e. Since Rn+I(Y) -1 "~ S n and the convex hall of G I ( S ~ -1) in the sphere S n has diameter less than e, we may conclude that, also shrinking c o n v ( G l ( S ~ -1) to a point, the homotopy Gt can be extended to an e-homotopy Lt" S~ -1 x [0, 1] --* Fn+l. Extending the homotopy Ht by adding the identity map, we obtain a homotopy Nt" S~ -1 • [0, 1] ~ Y such that (tiNt -- Pn+lLt. But then (recall that X is the fibered product in the above indicated diagram) we get a well-defined a-homotopy Mr" S~ -1 x [0,1] ~ X such that M0 = r -1 and M I ( S ~ -1) = r This contradiction finishes the proof. V1 LEMMA 4.2.18. Let K be an m-dimensional finite simplicial complex and f" X --+ [K[ be a map, constructed in Lemma 4.2.12. Then for each L C n - l - c o m p a c t u m Y , m > n, and for each map (t" Y ~ [K[, the fibered product Z of X and Y with respect to f and (t is an L C n - l - c o m p a c t u m . PROOF. Denote by X1 the inverse limit of the following diagram Fm

Fm-1

IKI

IK(m-1) [

Fn+2

...

[K(n+2) [

IK(n+l)[

and assume that p~" X1 ~ IK(i)l and q~" X1 ~ F~ denote the corresponding projections, n-softness of the projection p~ implies t h a t the fibered product y I of X1 and Y with respect to Pm I and (t is an L C n - l - c o m p a c t u m . Let (t~ Y~ ~ X1 be the projection of this fibered product, parallel to (t. It is easy to see that X is also the fibered product of spaces X1 and Fn+l with respect to maps ! . X l '"+ IK ( n + l ) [ and pn+l . Fn+l ---+ [ K ( n + l ) [. Then Z itself is the fibered Pn+l product of spaces Fn+l and Y~ with respect to Pn+l: Fn+l --~ IK(n+I)I and

152

Pn+l

4. MENGER MANIFOLDS

~ . y~

"-+ Ig

lemma is proved.

(n+l)

I" By L e m m a 4.2.17, Z is an L C n - l - c o m p a c t u m .

The

F'I

The following statement expresses one of the most important properties of the map fn: I~n --+ I~ constructed above. THEOREM 4.2.19. For each LCn-l-compactum Y contained in the Hilbert cube I ~, the inverse image f ~ l ( y ) is a #n-manifold. Additionally, i f Y E C n-l, then f ~ - l ( y ) is homeomorphic to #n. PROOF. We use the same notations as in the proof of Theorem 4.2.15. Recall t h a t the map f n : # n --~ I • was constructed as the limit projection of the inverse sequence S = {Mi, c~ +1} consisting of I ~ -manifolds (M1 -- I W) and n-invertible, polyhedrally n-soft and ( n - 1)-soft short projections. If Y is an LCn-l-compactum contained in I 0~ , then the inverse image fnZ(Y) is the limit space of the induced inverse sequence S ' - {Yi, r~+l}, where Y1 = Y, Yi+l = (c~+l)-l(Yi) and r~+1 = ~+l/Yi+l. Since the restriction of an ninvertible, polyhedrally n-soft and ( n - 1)-soft map onto the inverse image still has all these properties, we see that all short projections of the spectrum ,~' = {Yi, r~ +1} are n-invertible, polyhedrally n-soft and (n - 1)-soft. While proving Theorem 4.2.15 we have already seen t h a t the inverse limit of an inverse sequence consisting of L C n - l - c o m p a c t a and polyhedrally n-soft short projections is an LCn-l-compactum. Therefore it suffices to show that Yi E LC n-1 for each i. We prove this fact by induction. By assumption, Y1 E LC n-1. Assume t h a t Yi E LC n-1. The projection a~ +1 can be represented as the composition ")'i+1~i+1, where ~i+1 is a trivial bundle with fiber the Hilbert cube and "yi+l is homeomorphic to the map f~+l x i d i ~ . Here f~+l: X~+I ~ Ki is the map from Lemma 4.2.12 (see the diagram in the proof of Lemma 4.2.12). Consequently, by Lemma 4.2.18, ~i+l(Yi+l) is an LCn-l-compactum. But then Yi+l ~ ~i+l(Yi+l) x I ~ is also an LCn-l-compactum. Thus f ~ l ( y ) c LC n-1. Polyhedral n-softness of the short projections guarantees that if, in addition, Y E C n - l , then f n l ( Y ) e C n-1. Obviously, dim f ~ l ( y ) < dim #n = n. Theorem 4.2.15 shows that dim f n l ( Y ) ~n. Theorem 4.2.15 also guarantees that any map of an at most n-dimensional c o m p a c t u m into f~-I ( y ) can be arbitrarily closely approximated by embeddings. Therefore, by Theorem 4.1.19, f ~ Z ( y ) is a #n-manifold (and is homeomorphic to #n i f Y E Cn-1). [:]

Remark 4.2.20. In fact a stronger result can be proved: for any LC n-1c o m p a c t u m Y and for any map ~: Y ~ I W (not only for embeddings, as in Theorem 4.2.19) the fibered product of Y and #n with respect to ~ and fn is an L C n - 1_co m pact um. Summarizing the results proved above, we have.

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION

153

THEOREM 4.2.21. There are maps fn: #n ~ i W and gn: ttn ---* #n satisfying the following properties: (i) The maps fn and gn are n-invertible, ( n - 1 ) - s o f t and polyhedraUy n-soft. (ii) All the fibers of the maps fn and gn are homeomorphic to #n. (iii) The inverse images of L C n - l - c o m p a c t a under the maps fn and gn are #n-manifolds. (iv) The maps fn and gn both satisfy the parametric version of D D n P , that is, any two maps ~, ~ : I n ~ #n can be arbitrarily closely approximated by maps a I, ~l: I n ~ #n such that f n a l = f n ~ , f n ~ I = f n ~ , g n d = gnu, g n f l ' - - gnZ and i m ( a ' ) A im(Z') = O. Using the map fn as a guide, additional considerations allow us to obtain the following result. THEOREM 4.2.22. Any metrizable A ( N ) E ( n + 1)-compactum is an U V n - I image of: (i) An (n + 1)-dimensional A ( N ) E - c o m p a c t u m . (ii) The Hilbert cube I ~ (an I W -manifold, respectively). (iii) The (2n + 1)-dimensional cell ( a (2n + 1)-dimensional topological manifold, respectively). A simple comparison of the major ingredients of the Hilbert cube manifold theory (see Chapter 2) with the corresponding results of Menger manifold theory presented so far, shows that from a certain point of view the n-dimensional analog of the Hilbert cube Q should be considered to be, not the usual ndimensional cube I n, but the n-dimensional universal Menger compactum #n (moreover, the Hilbert cube itself may be viewed as the "infinite-dimensional Menger compactum"). In addition, one can observe a fairly deep analogy between the theories of #n-manifolds and Q-manifolds themselves. On the other hand, at first glance it is not clear what is the analog of the operation of "taking the product by Q" in #n-manifold theory - the operation which is involved in the formulations of triangulation (Theorem 2.3.28) and stability (Theorem 2.3.10) theorems for Q-manifolds. A decisive step in finding a "full" analog of this operation in #n-manifold theory is based on Theorems 4.2.15 and 4.2.19. First observe that taking the product X • Q of a space X and the Hilbert cube Q may be interpreted as taking the inverse image lr~-l(X) of a space X c Q, where r l " Q • Q ~ Q denotes the natural projection onto the first coordinate. It turns out that the map gn : f n / . f n l ( # n ) " #n ___+ttn in Theorem 4.2.15 may be thought of as the analog of the projection ~1 in the theory of #'~-manifolds. If this is agreed, everything then falls in place. The following statement is a triangulation theorem in #n-manifold theory. THEOREM 4.2.23. For any #n-manifold M , there is an n-dimensional polyhedron P such that for any embedding of P into #n the inverse image g~-l(p) is homeomorphic to M .

154

4. MENGER MANIFOLDS

PROOF. We consider the compact case. The locally compact case can be proved similarly. Take an n-dimensional finite polyhedron P and a map ~" P --~ M which induces isomorphism of t h e / - d i m e n s i o n a l homotopy groups for each i _< n - 1. E m b e d P into #n. It is easy to see that the composition

~gn/g~l(P) 9gnl(P)~

P ~ M

also induces isomorphism of t h e / - d i m e n s i o n a l homotopy groups for each i < n 1. Observe also t h a t the inverse image g ~ l ( p ) is a #n-manifold (we use Theorem 4.2.19). Therefore, by Theorem 4.1.21, M and g ~ l ( p ) are homeomorphic. [7 Here is the promised stability theorem for #"-manifolds. THEOREM 4.2.24. For any #n-manifold M in tt n, the inverse image g~-l(M) is homeomorphic to M . PROOF. If M is compact, then, by Theorem 4.2.19, the inverse image g n l ( M ) is a tin-manifold. Since the restriction g n / g n l ( M ) " g n l ( M ) ~ M is polyhedrally n-soft, we see t h a t it induces isomorphism of t h e / - d i m e n s i o n a l homotopy groups for each i < n - 1. By Theorem 4.1.21, g ~ l ( M ) is homeomorphic to M. The proof of the non-compact case is similar. Kl We conclude this section with the following statement which shows, in particular, that the map fn cannot be made n-soft. THEOREM 4.2.25. There is no n-soft map of an n-dimensional compactum onto a higher dimensional cube (the Hilbert cube in particular). PROOF. If n -- 0, the validity of our statement is obvious: the open image of a zero-dimensional c o m p a c t u m is zero-dimensional. Thus we may assume that n _> 1. If such a map g" Z --* I m existed, then Z must be connected (notice that 1-soft maps are monotone). Recall that cell-like maps cannot raise cohomological dimension (see Remark 3.1.11) and that the cohomological dimension of any cube coincides with its Lebesgue dimension: Therefore, g cannot be cell-like. It is clear now t h a t it suffices to prove the following claim. C l a i m . A n y non-constant n-soft map of a connected n-dimensional compactum is cell-like. Proof of Claim. Let f" X ---, Y be an n-soft map, where X is connected, d i m X = n and I Y I> 1. Since the fiber f - l ( y ) 6 L C n - I N C n - 1 for each y e Y we see t h a t g k ( f - l ( y ) ) = [-Ik(fkl(y)) = 0 for all k < n. Here g k denotes the singular homology with respect to the group of integers a n d / ~ k the (;'ech-homology. Then it follows that I ~ k ( f - l ( y ) ) = 0 for all k < n. Let us show that [-In(f-~(y)) = 0. If so, using the standard criterion that a finite-dimensional Peano continuum with trivial cohomology has trivial shape, we obtain the desired conclusion. Assume the contrary. Namely, t h a t / ~ n ( f - l ( y ) ) __/=0 for some y E Y. Take a map ~" f - l ( y ) ~ g ( Z , n ) (the Eilenberg-Maclane complex) which is not

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION

155

homotopic to a constant map. Consider a point x E Y different from y. W i t h o u t loss of generality we may assume t h a t I f - l ( x ) I 1 (otherwise shrink the fiber f - l ( x ) into a point). Since dimX - n, there is an extension ~: Z ~ K(Z,n). Next dengte by Y0 the set of all those points z E Y for which the restriction ~ / f - l ( z ) is homotopic to a constant map. Obviously, Yo is not empty. Indeed, x E Yo. Let us show t h a t Y0 is an open set in Y. Take a point z E Yo, and let h: f - l ( z ) x [0,1] ~ g ( Z , n ) be a h o m o t o p y connecting ~ / f - l ( z ) with a constant map to some point c E K ( Z , n), i.e.

h / ( f - l ( z ) x {0}) = Cp/f-l(z) and h / ( f - l ( z ) x {1}) = c. Consider the closed subset A -- (X x {0, 1} U ( f - l ( z ) x [0, 1]) of the product X x [0, 1] and define the m a p hi" A ~ K ( Z , n) by letting

hl(X

X

{ 1 } ) = c, h l / ( X x { 0 } ) = ~ a n d

hl/(f-l(z)

x

[0, 1 ] ) = h.

Since the CW-complex K (Z, n) is an absolute extensor with respect to the class of compact spaces, we can extend hi to a map h" U ---, K(Z, n), where U is a neighborhood of A in the p r o d u c t X x [0, 1]. T h e n U contains an open set of the form G x [0, 1], where G is a neighborhood of the point z in Y. Clearly G is contained in Y0 (see the definition of Y0). Since z was an arbitrarily chosen point of Y0, we conclude t h a t Y0 is open in Y. Next observe t h a t the complement Y - Y0 is also a n o n - e m p t y set, since it contains y. Let us now show t h a t this complement is also open in Y. Assume the contrary. T h e n there exists a sequence {zk} of points of Y0 such t h a t z = limzk E Y - ]I0. We construct a sequence {gk" f - l ( z ) --+ f--l(zk)} of maps, which converges to the identity map idy-~(z) in the space C ( f - l ( z ) , X ) . Since the space F -- f - l ( z ) x ({zk" k e N } U { z } ) is at most n-dimensional, n-softness of the m a p f guarantees the existence of a map r F --. X such t h a t

r

x

{z}) = idy-l(z)

and r

x

{zk}) C f - l ( z k )

for each k E g .

Let gk -- e l ( f - l ( z ) X {Zk}). T h u s we have a sequence {~gk}of maps each of which is homotopic to a constant map and which converges to the m a p ~ / f - 1 (z). This contradicts the fact t h a t sufficiently close maps into A N R-space are homotopic. Consequently, Y - Y0 is open in Y. Connectedness of Y shows t h a t this is impossible F-1

Historical and bibliographical notes 4.2. T h e o r e m 4.2.1 and Proposition 4.2.2 (a weaker version of it - w i t h o u t stating an n-invertibility) were proved in [33]. T h e construction of n-invertible, ( n - 1)-soft maps presented in this Section, as well as T h e o r e m s 4.2.23 and 4.2.24 in the compact case, are taken from [127](see [90] for the non-compact case). This result completes a circle of works of various

156

4. MENGER MANIFOLDS

authors concerning the existence of dimension raising maps. Historically, the first example of an open map (of the one-dimensional compactum onto the twodimensional "Pontryagin surface") of this sort was constructed by Kolmogorov in [196] in 1937. Further examples, with some additional properties, have been constructed in [11], [12], [15], [319], [320], [189], [190], [191], [200], [211], [245], [248], [304] etc. Theorems 4.2.22 and 4.2.25 are taken from [129] and [125]. Theorem 4.2.22 extends an earlier result from [71].

4.3. n-soft m a p p i n g s of P o l i s h spaces~ r a i s i n g d i m e n s i o n It has already been remarked in Section 4.2 that the maps fn and gn, constructed in that section, cannot be made n-soft. Also, both of them fail to satisfy the property of preservation of Z-sets in the inverse direction. In this section we construct a map with the last property which is "almost" n-soft. We begin with the following technical statement. LEMMA 4.3.1. If f" X ---. Y is an n-soft map, then the inverse image of each Zn-set in Y is a Zn-set in X . For each simplex a, denote the first and second barycentric subdivisions of a by fla and ~2a respectively. Ma denotes the closed star of the barycenter va of a in the triangulation ~2a" Ma = St(Va,~2a). We put No = Int(Ma). Finally let ra "(I a I - { v a } ) ~ ] Oa ]denote the canonical deformation retraction. LEMMA 4.3.2. For each n > 0 and each countable locally-finite simplicial complex K , there exist a countable locally finite simplicial complex B ~ and proper simplicial maps f~" IBm( I--*1 g I and g~" IBm( I---*1 g ( n + l ) I satisfying the following conditions: (i) f ~ is an (n + 1)-invertible u y n - m a p . (ii) If a is a simplex of K, then g ~ ( ( f ~ : ) - l ( ] a ])) C_I a (n+l) ]. (iii) f ~ c / ( f ~ ) - l ( [ K (n+l) I ) = g ~ / ( f ~ : ) - l ( ] K(n+l)I). (iv) There exists a subspace Ang of ] B ~ l such that the restriction f ~ / A ~ " A ~ --*1K[ is an (n + 1)-soft map and the complement l B ~ l - A ~ is a a Z-set in ] B ~ ]. (v) If Z is a Zn+l-set in I g I, then ( f ~ c ) - l ( z ) is a Zn+l-set in ] B ~ ]. PROOF. Let us consider an arbitrary simplex a and define two compact-valued retractions ~ , r l a I---*1 a(n+l) I. Definitions are given by induction on the (n + 1 +/)-dimensional skeleta of a. If x el a (n+l) I, then we put ~ ( x ) = x = r If T is an (n + 2)-dimensional face of a, then

n(x) __ { 107"1, ~~

rr(x),

if x e Nr, i f x e ITI--NT

4.3. n-SOFT MAPPINGS OF POLISH SPACES, RAISING DIMENSION

157

and

[OT[, r

--

r~-(x),

i f x 9 M~, if x 9 t~-1- M~

If T is an (n + 3)-dimensional face of a, then

~(~)

=

f I(0~)(-+1) I, t ~2(~(~)),

if x 9 Nr, if x 9 [ v [ - - N r

{ I(0T)(~+I)[, Can(rr(x)),

if x 9 Mr, if x 9 I~1- Mr.

and Can(x) =

Continuing this process, we obtain the desired retractions of [a] onto [a(n+l)[. Finally, if K is an arbitrary locally-finite simplicial complex, then the retractions ~ : , ~b~-IK} ~ ]K(n+l) 1 are defined as the unions of ~ and ~pa n, a 9 g . Now denote by B ~ the standard triangulation (induced by K) of the polyhedron {(x, y) 9 [g[ x ]g(n+l)[ 9 r ~ y}. We put also f~: = 7rl/[B~;[ and g~: = ~2/[B~I, where lr1" IKI • [K(n+l)l--* IKI and 7r2" IKI • IK("+~)I--' IK(n+l)[ denote the natural projections. Straightforward verification shows that: (a) the compact-valued retraction ~ : " [K[ ~ [K("+I)[ is lower semi-continuous. (b) the compact-valued retraction r IKI ---, IK(n+I)I is upper semi-continuous. (c) ~ - ( x ) C_ r for each point x 9 IKI. (d) the collection {r : x 9 IKI} is connected and uniformly locally connected in all dimensions less than n + 1 (we consider the standard metric

on

IKI).

Therefore, properties (i)-(iii) of the lemma are satisfied. Let us verify condition (iv). First of all, consider the subspace A~r = {(x,y) e ] K I • tK(n+l)l" y 9 qo~:(x)}

of IB~I and note that the complement I B ~ I - A~: is an Fa-subset of IB~I. By conditions (a), (c) and Theorem 2.1.15, the restriction f ~ / A ~ is an (n + 1)-soft map. Let us show that the complement IB~:I- A~. is a aZ-set in IB~I. Clearly it is sufficient to show that the last fact is true for the (n + 2)-dimensional simplex a. It follows from the construction that IB~:I- A~c = OMo • Io(-+a)l- T, where Z = {(~, y) e Iol • Io(~+1)1" ~ ( ~ ) = y}. Since OMa is a Z-set in Ma, we can conclude that OMo x [a(n+l)[ is a Z-set in M a x [a(n+l)[. Consequently, [B'~[-A'~ is a aZ-set in M a x [a(n+l)[. Consider now an open subspace U = [ B ~ [ - T of the polyhedron [B~[ Evidently, [ B ~ [ - A n C U C Mo x [a(n+l)[ Then the complement tBant- Aan is a aZ-set in U. Finally, for the same reason, we can conclude that [ B ~ ] - A~ is a aZ-set in [Ban[.

158

4. MENGER MANIFOLDS

The last condition is an easy consequence of condition (iv). Indeed, let Z be a Zn+~-set in [g[. Then, by (iv) and Lemma 4.3.1, the set

C 1 -=

{h e C ( I n + I , A ~ ) " ira(h) n ((S~7)-l(z) n A~:) -- 0}

is a dense G~-subset of C ( I n + I , A ~ ) . At the same time, by (iv), the set

6 2 - - {h e C ( I nq-1, [ B~z [)" ira(h) C Ang}

is also a dense G6-subset of C ( I n+l, IB~[). A Baire category argument finishes the proof, fl

THEOREM 4.3.3. Let n >_ 0 and let X be the limit space of an inverse sequence ~qx ---- {[ Xi [, p~+l} all spaces of which are locally finite polyhedra and all bonding maps of which are proper, simplicial and (n-b 1)-soft. Then there exists an (n-b 1)invertible proper UVn-surjection f ~ " M x ---* X of some #n+l-manifold M x onto X satisfying the following conditions: (i) For each (n -b 1)-dimensional locally compact space Z, closed subset Zo of Z, open cover Lt of M x , and proper map h" Z ~ M x , such that h/Zo is a fibered Z-embedding (with respect to f ~ ) , there exists a fibered Z-embedding (with respect to f ~ ) g" Z ---. M x which is U-close to h and such that f ~ g - f ~ h and g / Z o - - h/Zo. (ii) There exists a subspace A x of M x such that the restriction f ~ / A z " A x ---* X is an ( n + 1)-soft map and the complement M x - A x is a a Z-set in M x . (iii) If Z is a Zn+l-set in X , then ( f ~ ) - l ( Z ) is a Z-set in M x .

PROOF. For simplicity we consider only the compact case. The general case can be handled similarly. A standard Baire category argument reduces the proof to the case when Z0 - 0. After making these assumptions, we proceed as follows. We construct another polyhedral inverse sequences SM -- {[Ki[ , qi _i+1~~, all bonding maps of which are simplicial UVn-surjections. Further, we shall construct a family of ( n + 1)-invertible simplicial UYn-surjections fi: [Ki[ ---* [Xi[ which forms a strictly commutative (n+l)-invertible UYn-morphism {f~: [K~] [Xi[}: SM ~ 8 x . This means that if we fix an index i and consider the naturally arising diagram

4.3. n-SOFT MAPPINGS OF POLISH SPACES, RAISING DIMENSION

159

IKi+ll

i+1

r~

iKil

p~+l

........

.fi

~ [Xi[

~ , i + l A .e then the characteristic map li = ~i ~ J i + l of the diagram is an ( n + 1)-invertible UVn-surjection (here ITil denotes the fibered product of IKi{ and IXi+ll with _i+1 respect to maps fi and ~'i ; si and ri denote the corresponding projections of this fibered product). Then we shall obtain the space M x as the limit of t.~M and the map f ~ as the limit map of the morphism {f i}. Let K1 = Bnxl and f l = f ~ l " IKll --~ IXll (see L e m m a 4.3.2). We can suppose, of course, that mesh(Kx) < 1 (if not, then we consider a sufficiently fine subdivision of K1 and denote it again by K1). Suppose now that we have already 1 and simplicial constructed finite simplicial complexes Km with m e s h ( K m ) < ~-~ n UV -surjections fm" [Kml ~ IXml and qm-lm . IKml __~ IKm_ll, m _< i, in such a way that the characteristic maps of all the naturally arising rectangular diagrams are (n + 1)-invertible UVn-surjections. In order to perform the inductive step, consider the fibered product ITil of IKil and IX~+ll with respect to the maps f~ and p~+l and denote by ri" ITil--* Igil and si" ITi[--~ IXi+ll the corresponding projections. Now consider the map f ~ , ' l B L,I n ~ ILil where Li denotes the natural triangulation of the polyhedron ITil x i2n+3 (we again use L e m m a 4.3.2). 1 Let K i + l -- BnL~, assuming at the same time t h a t mesh(K{+l) < 2~--Tf" Also _i+l define f~+l = sili and qi = ril +i, where li = ~lf~, and ~1" ITil x i2n+3 __~ iTil denotes the projection onto the first coordinate. One can easily verify that all our requirements are satisfied and consequently the inductive step is complete. As already remarked, we let M x = limSM and f ~ = lim{fi}. Since all mentioned rectangular diagrams strictly commute (because, by the construction, their characteristic m a p s - li's - are surjective) and all fi's are UVn-surjections, we can conclude that their limit map f } " M x --~ X is also a VVn-surjection. The standard argument (see C h a p t e r 6) shows that (n + 1)invertibility of all fi's and a l l / i ' s implies (n + 1)-invertibility of f ~ . Let us now verify condition (i). Fix an index m and an open cover L/m of IKml such that qml(Llm) refines b/(recall that qi: M x --~ Igil and pi: X ~ IX~I denote the limit projections of the spectra SM and S x respectively). We are going to

160

4. MENGER MANIFOLDS

construct maps gi" Z ~ [Ki[ in such a way t h a t the following conditions are satisfied: (a) If i > m, then gi is an embedding. (b) q i~+1g i + l - - g i .

(c) f~g~ = p ~ / ] h . (d) If i > m and c~i" i n + l __~ IK~I is an arbitrary map, then there exists a m a p 13i . I n+l ~ IKil such t h a t f i ~ __. f i a i , q i - l ~ i ~ q ii_ l a i and im(Z~) n i~(g~) = O. Let gi = qih for each i _< m. We now indicate how the map gi+l can be constructed. First of all fix an e m b e d d i n g u" Z ~ I2n+3 (we use the inequality d i m Z _< n + 1). Now consider the diagonal p r o d u c t g m A p m + l f ~ h , which maps Z into ITm] 9 T h e n the diagonal product (gin • P m + l f xnh ) A u will be an embedding of Z into the product ILml = ITml • I2n+3. By L e m m a 4.3.2, the map f~,," [Kin+l[ ~ [Lml is (n + 1)-invertible. Consequently, there exists an embedding gm+l" Z ---* I g m + l l such t h a t f~,.,,gm+l = ( g m A P m + l f ~ h ) A u . A simple verification shows t h a t conditions (b) and (c) are satisfied. Now consider any m a p a m + l " i n + l ~ IKm+ll" Since u is an e m b e d d i n g and dim Z _ n + 1, there is a point a E i2n+3 such t h a t a qf ira(u). Let the same letter a denote the constant map which sends the whole cube I n+l to the point a. Consider the m a p (lmo~m+l/ka) . I n+l ----* ILml and observe t h a t the image of I n+l under this m a p does not intersect the image of Z under the composition f L m g m + l . Consider now any lifting ~m+l" I n+l ~ IKm+l[ of the product (lmoLm+lAa). Again, straightforward verification shows t h a t condition (d) is also satisfied. Therefore, continuing this process we obtain the maps g~ for each i. By (b), the diagonal p r o d u c t g of all g~'s maps Z into M x . By (a), g is an embedding. By (c), the desired equality f ~ g - f ~ h also holds. T h e choice of an index m and the equalities qmg = gm "-- qmh (which are true by our construction) show t h a t g and h are/g-close. Finally, let us show t h a t ira(g) is a fibered Z-set with respect to the map f ~ . Fix an open cover l; of M x and any map c~" I n+l --+ M x . Clearly we can assume t h a t there exist an index j _> m and an open cover l)j of ]Kj[ such t h a t q~-l(1)j) refines "g. As above we shall inductively construct maps ~i" I n+l ---* [Ki[ in such a way t h a t the following conditions are satisfied" - i + l ~ i + 1 = ~i. (e) qi (f) f ij3i = pif~ca.

(g) im ( ~ + ~) n im (g~+ ~) = r (h) Zj = q ~ . We let ~i ---- qia for each i < j (consequently, the last condition is automatically satisfied). Let us construct the m a p ~3j+1. By (d) (assuming t h a t i = j + 1), we obtain a map/3j+1" I n+l -~[ g y + l [ such t h a t i m ( Z j + l ) ~ i m ( g j + l ) = 0 (i.e. j+l condition (g) is satisfied), qj j3j+l = ~j and fj+lJ3j+l = p j + l f ~ a . For i > j + 2 , we can construct maps ~i" I n+l ~ [ Ki [ in a similar way to the construction of /~y+l. It only remains to r e m a r k t h a t if ~ is the diagonal p r o d u c t of all ~i's, then one can easily verify t h a t ira(Z) ~ ira(g) = O, f ~ Z = f ~ a and t h a t fl and a are

4.3. n-SOFT MAPPINGS OF POLISH SPACES, RAISING DIMENSION

161

])-close. This finishes the verification of condition (i). Let us now show that M x is an #n+l-manifold. Since mesh(Ki) ~ 0 we can conclude (conditions (ii) and (iii) of Lemma 4.3.2) that M x admits small maps into (n + 1)-dimensional polyhedra. Consequently, d i m M x _< n + 1. The inverse inequality is obvious, because, by the above verified condition (i), M x contains a copy of every (n + 1)-dimensional compactum. Since M x is, by the construction, the limit space of the polyhedral inverse sequence SM all bonding maps of which are UVn-surjections we conclude that M x is an LCn-compactum. Again, by (i), it follows immediately that M x satisfies the disjoint (n + 1)-disk property and hence, by Theorem 4.1.19, M x is an #n+l-manifold. Next, we verify condition (ii). By Lemma 4.3.2, there exists a subspace A1 of [KI[ such that the restriction f l / n l : A1 ~ [XI[ is an (n + 1)-soft map and the complement [ K I [ - A1 is a aZn+l-set of ]KI[. It follows from our construction that the fibered product R1 of A1 and [ X2 [ w i t h respect to the maps f l / A 1 and p2 is a subspace of [TI[. Clearly the natural projections of R1 onto A1 and IX2] coincide with the restrictions rl/R1 and Sl/R1 respectively. Since f l / A 1 is (n + 1)-soft, we conclude (see Lemma 6.2.5) that sl/R1 is also (n + 1)-soft. Consider the set R1 x i2n+3 and denote by A2 its inverse image under the (n + 1)-soft map f~l (see Lemma 4.3.2). Then the restriction f 2 / A 2 : A 2 ~ IX2[ is (n + 1)-soft. Since the bonding map p2 is (n + 1)-soft, using again Lemma 6.2.5, we see that the map rl is also ( n + l ) - s o f t . Consequently, by Lemma 4.3.1, the set ]TI[-R1 is a aZn+l-set in ]T1]. In this situation one can easily observe that, again by Lemma 4.3.1, the complement [ K 2 [ - A2 is a a Z n + l - s e t in [K2[. Continuing i + l / A i+ 1 ) and an in such a manner we obtain an inverse sequence SA = {Ai, ~i (n + 1)-soft morphism (in the sense of Chapter 6) morphism { f i / A i } of Sn to S x . Clearly, the limit space A x of the spectrum $A is a subspace of M x and the restriction f ~ / A x " A x ---* X , which coincides with the limit map of the morphism {f~/Ai}, is an (n + 1)-soft map (here we use L e m m a 6.2.7). It only remains to remark that the complement M x - A x is a aZ-set in M x . Condition (iii) is a direct consequence of condition (ii) and Lemma 4.3.1 (compare with the proof pf L e m m a 4.3.2). The proof is complete. [::] We also need the following statement. PROPOSITION 4.3.4. Each #n+lmanifold M can be represented as the limit space of an inverse sequence SM -- {Mi ,Pi i+1 } consisting of locally compact poly-

h~d~a a~d p~op~ ~imptic~al (n + 1)-~~t~bt~, n-~oft a~d polyh~d~aUy (,~ + 1)-~oft bonding maps. PROOF. By Proposition 4.1.10, there exists a proper n-homotopy equivalence a : ] K ]---~ M, where g is an at most (n + 1)-dimensional countable locally finite simplex. Let K0 = K. Suppose that the countable locally finite simplicial simplexes K~ and proper simplicial maps Pi-1 ~ 9 ]K~[ ~ [K~-I[, satisfying conditions

162

4. MENGER MANIFOLDS

(i) - (iii) from L e m m a 4.3.2, have already been c o n s t r u c t e d for each i _< m. We m a y also assume t h a t the m a p Pi-1 i 9 IKil - . IKi_ll has the following property: 9 For any m a p a" I n+l --, IKi_ll there exist two maps ill, f~2" I n+l --~ IKil such t h a t P~-lflj i ---- a, j -- 1,2, and im(f~l) N im(fl2) -- 0. In order to construct these objects for the next step, we apply L e m m a 4.3.2 to a sufficiently fine triangulation of the p o l y h e d r o n IKml x [0, 1]. In this way we get the next simplicial complex g m + l ---- B gmx[0,1] n m+l -- 7rlfg,,~ n x [0,1], 9 We let pm where ~rl"lgml x [0, 1] ~ Igml is the projection. As in the proof of Theorem 4.3.3, we can verify t h a t the limit space M ~ of the inverse sequence 8M -{Mi, Pi i + 1 } is a #n+l-manifold. Clearly, the limit projection p0" M ' --, IKI, being a p r o p e r U V n - m a p between at most (n + 1)-dimensional locally finite L C nspaces, is a proper n - h o m o t o p y equivalence. Therefore M and M ~ are properly n - h o m o t o p y equivalent. T h e o r e m 4.1.21 finishes the proof. V-1 Applying T h e o r e m 4.3.3 and P r o p o s i t i o n 4.3.4 we get the following s t a t e m e n t . THEOREM 4.3.5. Let n > 0 and X E ( # k . k >_ n + 2} U {I ~ }. Then there exists an (n + 1)-invertible UYn-surjection f ~ " #n+l __~ X , satisfying conditions ( i ) - (iii) of Theorem 4.3.3. We conclude this section with the following s t a t e m e n t . THEOREM 4.3.6. Let n ~_ O. For each locally compact polyhedron K , there exists a proper (n + 1)-invertible UYn-surjection h~" M ~ +1 ---. g of some #n+l_ manifold M ~ +1 onto K satisfying the following conditions: (i) xf L i~ ~ ~lo~d ~bpoly~d~o~ of K , t ~ it~ i ~ ~ im~g~ (h~)-I(L) is a #n+l-manifold. (ii) If L is a closed subpolyhedron of K and Z is a Z-set in L, then the

i~,~

im~g~ (h~)-l(z)

i~ ~ z - ~ t

i~

(h~)-~(L).

PROOF. R e p e a t the proof of T h e o r e m 4.3.3, first observing that if L is a s u b c o m p l e x of a countable locally finite simplicial complex K, then the simplicial complex B~ from L e m m a 4.3.2 is a s u b c o m p l e x of the complex B ~ and the map f ~ coincides with the restriction f ~ / I B ~ ]. []

Historical and bibliographical notes 4.3. L e m m a 4.3.1 was proved in [279]. All o t h e r results of this Section were o b t a i n e d by the a u t h o r [79], [84], [88], [97], [98]. The non-separable case was considered in [108].

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS

163

4.4. Further properties of M e n g e r manifolds Using the existence of the dimension raising maps constructed in Sections 4.2 and 4.3, we now discuss other major ingredients of Menger manifold theory. 4.4.1. n - h o m o t o p y kernel and Open E m b e d d i n g T h e o r e m . The open embedding theorem for I ~ -manifolds states (see C h a p t e r 2) that for each I ~ manifold X, the product X x [0, 1) can be embedded into I ~ as an open subspace. Observe that identifying X with X x [0, 1] (stability of I"~-manifolds), the product X x [0, 1) may be viewed as the complement of the image of an appropriately chosen Z-embedding of X into itself. Using this remark as a guide, we introduce the following notion. Consider a #n+l-manifold M and two Z-embeddings f , g : M --~ M each of which is properly n-homotopic to the identity map idM. Then the homeomorphism g f - 1 . f ( M ) --. g(M) is properly n-homotopic to idf(M) and, consequently, by Theorem 4.1.15, there exists a homeomorphism h: M ~ M extending g f - 1 . Clearly the restriction h / ( M - f ( M ) ) is a homeomorphism between the complements M - f ( M ) and M - g(M). This shows that the following definition does not depend on the choice of a Z-embedding.

DEFINITION 4.4.1. The n-homotopy kernel K e r n ( M ) of a #n+l-manifold M is defined to be the complement M - f ( M ) , where f : M ~ M is an arbitrary Z-embedding properly n-homotopic to idM.

PROPOSITION 4.4.2. Let M and N be #n+l-manifolds. Then the following conditions are equivalent: (i) N admits a proper UVn-surjection onto the product M x [0, 1). (ii) N is homeomorphic to K e r n ( M ) . PROOF. It suffices to show that K e r n ( M ) also admits a proper UVn-surjecti on onto the product M x [0, 1). Take a proper v v n - s u r j e c t i o n f : M1 : M x [0, 1], where M1 is also a #n+l-manifold (see Theorem 4.2.1). Consider the quotient space M2 of M1 with respect to the partition whose nontrivial elements are fibers f - l ( m , 1) over the Z-set M x {1} of the product M • [0, 1]. Clearly, by Theorem 4.1.19, M2 is a #n+l-manifold. Moreover, if we consider the naturally induced proper UVn-surjection of M1 onto M2, then we conclude, by Theorem 4.1.21, that M1 and M2 are even homeomorphic. Next, it is easy to see that the set g - l ( M • {1}) is a Z-set in M2 and the restriction g / g - l ( M • {1}) is a homeomorphism. By Theorem 4.1.20, the composition 7rMg, where ~rM: M x [0, 1] ~ M is the natural projection, can be arbitrarily closely approximated by homeomorphisms. In particular, there exists a homeomorphism h: M2 ---+ M that is properly n-homotopic to 7rMg. T h e n the map r -- h g - l i , where

164

4. MENGER MANIFOLDS

i: M --* M x {1} is a n a t u r a l h o m e o m o r p h i s m , is a Z - e m b e d d i n g properly nh o m o t o p i c to idM. Indeed, r - h g - l i ~n --p 7rMgg-1 i -- zrMi -- idM. Thus, by Definition 4.4.1, we conclude t h a t the c o m p l e m e n t M - r ( M ) is h o m e o m o r p h i c to K e r n ( M ) . Consequently, the space M 2 - g - l ( M • { 1 } ) - h - l ( M - r ( M ) ) i s also h o m e o m o r p h i c to K e r n ( M ) . It only remains to note t h a t the space M 2 g - l ( M x {1}) a d m i t s a p r o p e r UVn-surjection onto the p r o d u c t M x [0, 1). V1

Now we are ready to prove the open e m b e d d i n g t h e o r e m for Itn+l-manifolds.

THEOREM 4.4.3. The n-homotopy kernel of each #n+l-manifold admits an open embedding into Itn+l.

PROOF. First of all let us show t h a t every Itn+l-manifold M admits a p r o p e r U V n - m a p onto a certain I w -manifold X. For this we take a proper UVn-map ~" M ~ P , where P is a locally c o m p a c t polyhedron (see T h e o r e m 4.2.23). Clearly the p r o d u c t P • I w --- X is a IW-manifold (see C h a p t e r 2). By T h e o r e m 4.2.1, there is a p r o p e r UVn-surjection r M ' ~ M • I w of some #n+l-manifold onto the locally c o m p a c t L C n - s p a c e M x I w . Since the composition r M r M ' M is also a proper U V n - m a p , we conclude, by T h e o r e m 4.1.20, that M ' and M are homeomorphic. It only remains to observe t h a t the required proper UV nsurjection h" M --* X is given by the composition (~o • idlw )r Now we proceed with the direct proof of our theorem. Take a proper UV nsurjection h" M ---+ X, where X is a I w -manifold. By the open e m b e d d i n g t h e o r e m f o r / W - m a n i f o l d s (see T h e o r e m 2.3.25), one m a y suppose that the product Z • [0, 1) lies in I w as an open subspace. Consider a p r o p e r (n + 1)-invertible UVn-map g" M1 ---* M x [0, 1], where M1 is a # n + l - m a n i f o l d (we use T h e o r e m 4.2.1). Since the m a p g is (n q- 1)-invertible and dim M -- n q- 1, we may assume, w i t h o u t loss of generality, t h a t the restriction g / g - l ( M x {1}) is a homeomorphism and A = g - l ( M • {1}) is a Z-set in M1. As above, we conclude, using T h e o r e m 4.1.20, t h a t the # n + l - m a n i f o l d s M and M1 are homeomorphic. Therefore, it only remains to show t h a t the c o m p l e m e n t M1 - A (which is obviously h o m e o m o r p h i c to K e r n ( M ) ) admits an open e m b e d d i n g into #n+l. Consider the m a p f" # n + l ___, i w of T h e o r e m 4.3.5 and let M2 denote the # n + l - m a n i f o l d f - l ( X x [0, 1)), which is open in # n + l . All t h a t remains to be shown is t h a t the # n + l - m a n i f o l d s M2 and M1 - A are homeomorphic. The last fact can be observed in the following way: b o t h # n + l - m a n i f o l d s MI - A and M2 admit proper U Y n - m a p s onto X • [0, 1) (consider the maps (h • id)g and f ) . Therefore, by T h e o r e m 4.1.21, they are homeomorphic. T h e following d i a g r a m helps to reconstruct the complete argument.

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS M2

f

M1-A

M1

g

~ M x [0, 1)

g

~M

x

[0,1]

~t n + l

f

hxid

•

h

x [0,1]

•

165

,~ I w

71"

M This finishes the proof.

~X

K]

4.4.2. n - h o m o t o p y C l a s s i f i c a t i o n T h e o r e m . Theorem 4.1.21 completely describes the proper n-homotopy classification of #n+l-manifolds. In particular, n-homotopy equivalent compact #n+l-manifolds are homeomorphic. Obviously, the last fact is incorrect in the non-compact case: compare #n+l and # n + l - { p t } . The main result of this subsection solves the n-homotopy classification problem of arbitrary ttn+lmanifolds (compare with Theorem 2.3.26). PROPOSITION 4.4.4. For each #n+l-manifold M, the spaces K e r n ( M ) and Kern(Kern(M)) are homeomorphic. PROOF. As above, take a proper UVn-surjection g: M --~ X of M onto a I w -manifold X. By Proposition 4.4.2, there exists a proper UVn-surjection f : Kern(M) ~ M x [0, 1). For the same reason, there exists a proper UV nsurjection h: K e r n ( K e r n ( M ) ) --~ Kern(M) x [0, 1). Consequently, we have two proper UVn-surjections:

p = (g x id)f: Kern(M) ~ M x [0, 1) and

q = (p x id)h: K e r n ( K e r n ( M ) ) ~ K e r n ( M ) x [0, 1),

166

4. MENGER MANIFOLDS

where id denotes the identity map of [0, 1). Since X is a Hilbert cube manifold, the product X x [0, 1] is homeomorphic to X (see Chapter 2). Remarking that the spaces [0, 1) x [0, 1) and [0, 1] x [0, 1) are homeomorphic, we have X x [0,1) x [ 0 , 1 ) ~ X x [0,1] x [0,1) ..~ X x [0, 1). Consequently, the #n+l-manifolds K e r n ( M ) and K e r n ( K e r n ( M ) ) admit proper UVn-surjections onto the same I ~ -manifold X x [0, 1). Therefore they are properly n-homotopy equivalent. Theorem 4.1.21 finishes the proof. V1 PROPOSITION 4.4.5. Let M be a #n+l-manifold. I r A is a Z-set in K e r n ( M ) , then the spaces K e r n ( M ) and K e r n ( K e r n ( M ) - A) are homeomorphic. PROOF. As in the proof of Proposition 4.4.4 consider three proper UV nsurjections:

g: M ~ X, f : K e r n ( M ) - - * M x[0, 1 ) a n d p =

(gxid)f: Kern(M)~

Xx[0,1),

where X is a I ~ -manifold and id denotes the identity map of [0, 1). Now we redefine the map p in such a way that the set p(A) will be a Z-set in X x [0, 1). For this, consider any Z-embedding r: A ~ X x [0, 1) properly n-homotopic to the restriction p/A. By Proposition 4.2.2, there exists a proper UVn-surjection q: K e r n ( M ) ---* X x [0,1) such that q/A = r. Moreover, as in the proof of Proposition 4.4.2, we can additionally suppose that A = q-lq(A). Consequently, the restriction

q / ( K e r n ( M ) - A): K e r n ( M ) -

A ---. X x [0, 1 ) - q(A)

is a proper UVn-surjection. As above (compare with the proof of Proposition 4.4.4) this implies that K e r n ( K e r n ( M ) - A) admits a proper UYn-surjection onto the product (X x [0, 1 ) - q(A)) x [0, 1). By the same argument, there exists a proper VYn-surjection of K e r n ( K e r n ( M ) ) onto the product X x [0, 1) x [0, 1). By Proposition 4.4.4, K e r n ( M ) and K e r n ( K e r n ( M ) ) are homeomorphic. Therefore, by Theorem 4.1.21, it suffices to show that the spaces (X • [ 0 , 1 ) - q ( A ) )

x [0,1) and X x [ 0 , 1 ) x [0,1)

are homeomorphic. Indeed, since q(A) is a Z-set in X x [0, 1), we conclude that the I~-manifolds X x [0, 1) and X x [0, 1 ) - q ( A ) are homotopy equivalent. Then, by the homotopy classification theorem for I W-manifolds (Theorem 2.3.26), the products X x [0, 1) x [0, 1) and (X x [0, 1 ) - q(A)) x [0, 1) are homeomorphic. This finishes the proof. E] PROPOSITION 4.4.6. Let a #n+l-manifold M be a Z-set of a #n+l-manifold N , and suppose the inclusion i: M --+ N is an n-homotopy equivalence. Then there exists a Z-set A in N such that the complement N - A is homeomorphic to K e r n ( M ) .

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS

167

PROOF9 By Theorem 4.2.23, there exist (n + 1)-invertible proper UVn-sur jections ~: M ---. L and r N ---. K1, where L and K1 are (n + 1)-dimensional locally compact polyhedra. The (n + 1)-invertibility of ~ implies the existence of a proper map s: L ~ M with ~s - idL. Then the composition r L --. K1 is properly homotopic to some proper piecewise linear (PL) map p: L ~ K1. Form the mapping cylinder M(p) -- K of the map p. Recall that this is the space obtained from the disjoint union (L • [0, 1])@ K1, by identifying (/, 1) with p(l) for each l E L. At the same time we identify L with L x {0}. Clearly L x [0, 1) can be considered as an open subspace of K. Since p is a proper PLmap, L • {0} and K1 are subpolyhedra of the polyhedron K. Let c: K ~ K1 be the collapse to the base, i.e. the natural retraction defined by sending (/, t) to p(l). Clearly, c is a proper CE-surjection that is a homotopy equivalence. Now we consider an (n + 1)-invertible proper UYn-surjection f~(" M~(+1 --. K , satisfying the conditions of Theorem 4.3.6. Clearly, the composition cf~" M~(+1 ---* K1 is a proper UVn-surjection, and hence, by Theorem 4.1.21, the #n+l-manifolds M ~ +1 and N are homeomorphic. By Theorem 4.3.6, the inverse image ( f ~ ) - l ( L x {0}) is a #n+l-manifold that, again by Theorem 4.1.21, is homeomorphic to M. One can easily verify, using the assumption and the specifics of the above construction, that the natural inclusion of ( f ~ ) - l ( L x {0}) into M ~ +1 is an n-homotopy equivalence. Moreover, by Theorem 4.3.6, the above inverse image is a Z-set in M ~ +1 (since L • {0} is a Z-set in K). Now redefining the above objects for simplicity, we have the following situation. A proper UVn-surjection f : N ~ K, satisfying the conditions of Theorem 4.3.6, is given, M -- f - l ( L • {0}) is a Z-set in N and the inclusion M ~ N is an n-homotopy equivalence. Clearly, K - K1 -- L x [0, 1), and hence the inverse i m a g e / - I ( L • [0, 1)) admits a proper UYn-surjection onto L • [0, 1). On the other hand, K e r n ( M ) admits a proper vVn-surjection onto M • [0, 1), and hence onto the product L • [0, 1) as well. Thus, by Theorem 4.1.21, the inverse image f - l ( L • [0, 1)) and K e r n ( M ) are homeomorphic. Consequently, to finish the proof it remains to construct an open embedding h: f - l ( L x [0, 1)) ---. N such t h a t the complement A -- g - h ( f - l ( L • [0, 1)) is a Z-set in N. Since N is a #n+l-manifold, by Theorem 4.1.19, there is a countable dense subset {~k: k -- 1 , 2 , . . . } of c ( I n + l , g ) consisting of Z-embeddings. As observed above, f - l ( L • {0}) is a Z-set in N and the inclusion f - l ( L x {0}) ~ N is an n-homotopy equivalence. It easily follows from Proposition 4.2.2 that in this case there exists a retraction rl: N --. f - l ( L x {0}) that is n-homotopic to idN. Consider the restriction 81 = r l / 9

9 f-l(n

x {0}-)U~I(/n+l)

---~ f - l ( n

1

x [0, ~)).

Clearly 81 is a proper map. By Theorem 4.1.19, Sl is properly n-homotopic to a Z-embedding

gl 9 I - I ( L x { 0 } . ) U ~ I

(/n+l

)-"> .f --1 (L x [0, 1 )) Z

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4. MENGER MANIFOLDS

that coincides with the identity map on f - l ( L x {0}). By Theorem 4.1.15, there exists a homeomorphism GI" N --~ N extending gl. Put hi = G~-1. Then hi is a homeomorphism such t h a t (a) h l / f - l ( L x { 0 } ) - id. (b) (ill(/n+l) C_ h l f - l ( L x [0, 1)). 1 Consider now the polyhedron K - (L x [0, ~)). Since the set L x { 89 is a Z-set in K (L x [0, ~)), 1 we can conclude, by Theorem 4.3.6, that the set h l f - l ( L x { 1}) is a Z-set in a #n+l-manifold N - h l f - l ( L • [0, 89 Moreover, since the inclusion -

1 L x {~}~K-L

1 x [0,~))

is a homotopy equivalence, we conclude that the inclusion

hlf-l(L x { })~

N-hl.f-l(L

x [0,~))

is an n-homotopy equivalence. Again, using the above construction, we see that there is a homeomorphism

h ,2. N - h l f

-1

1

(L x

))---+ N

which is the identity on h l f - l ( L x {89 ~o2(I n+l) C'I ( N -

hlf-l(L

x [0,

-

h 1 f - l ( L x [0, 1

and for which ))) C_ h ~ h l f - l ( L x [~,2-

)).

Extend h~ to a homeomorphism h2 defined on N by defining h2 = id on

h l f - l ( L x [0, 89 Then we have 1

~o1(In+l) U ~o2(I~+1) C_ h 2 h l f - l ( L x [0,2 9 g)). Inductively continuing this process, we construct homeomorphisms hk" N ~ N in such a way that hk+l = id on h k f - l ( L x [O,k. k--~]) and 1

V~l(In+l)IJ...iJ(Pk(In+l ) C_ h k h k _ l . . . h l f - l ( L x [O,k. k + 1) ). Define an open embedding h" f - l ( L x [0, 1)) ~ N by h(x) = limk--.oo h k ' " hi(x) for each x E f - l ( L x [0, 1)). Clearly U{~k ( i n + l )" k = 1 , 2 , . . . } _C h f - l ( L x [0, 1)) and consequently, by the choice of the family {~k" k = 1, 2 , . . . }, the complement [-1

N - h f - l ( L x [0, 1)) is a Z-set in N.

Now we are ready to prove the n-homotopy classification theorem for #n+l_ manifolds. THEOREM 4.4.7. #n+l-manifolds are n-homotopy equivalent if and only if their n-homotopy kernels are homeomorphic.

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS

169

PROOF. Let M and N be n - h o m o t o p y equivalent tin+l-manifolds. Take maps n n a" M ~ N and/3" N ~ M such t h a t ~ a " ~ i d M and a ~ " ~ i d N . As above, we can find (n + 1)-invertible proper UYn-surjections f " K e r n ( M ) ~ M • [0, 1) and g" K e r n ( N ) ~ N • [0, 1). Let ~" M x [0, 1) ~ N x [0, 1) and r g x [0, 1) --, M • [0, 1) be a proper maps such t h a t ~ is homotopic to a x id and r is homotopic to fl x id, where id denotes the identity map of [0, 1) (compare with [69, L e m m a 21.1]) . Since g is (n + 1)-invertible and dim K e r n ( M ) = n + 1, there exists a proper m a p r" K e r n ( M ) ~ K e r n ( N ) such t h a t gr = ~o.f. Similarly we have a proper map s" K e r n ( N ) ~ K e r n ( M ) such t h a t f s = Cg. In this situation one n

can verify directly t h a t sr "~ idKer,~(g). Moreover, by T h e o r e m 4.1.19, we can additionally suppose t h a t r and s are Z-embeddings. Now consider the Z-set r ( g e r n ( M ) ) in a # n + l - m a n i f o l d K e r n ( N ) . It follows immediately from the above construction t h a t the inclusion map

r(Kern(M)) ~ Kern(N) is an n - h o m o t o p y equivalence. By Proposition 4.4.6, there is a Z-set A in K e r n ( N ) such t h a t K e r n ( N ) - A is h o m e o m o r p h i c to K e r n ( r ( g e r n ( M ) ) ) -K e r n ( K e r n ( M ) ) (recall t h a t r is an embedding). Then, by Propositions 4.4.4 and 4.4.5,

K e r n ( M ) .~ K e r n ( K e r n ( K e r n ( M ) ) ) ..~ K e r n ( K e r n ( N ) as desired. T h e second part of the t h e o r e m is trivial.

A) ..~ K e r n ( N )

V1

4.4.3. n - s h a p e a n d t h e C o m p l e m e n t T h e o r e m . T h e famous Complement T h e o r e m for I • -manifolds [69] states t h a t if X and Y are Z-sets in I ~ , then their complements I ~ - X and I ~ - Y are h o m e o m o r p h i c if and only if the shapes of X and Y coincide, i.e. S h ( Z ) = S h ( Y ) . T h e obvious form of T h e C o m p l e m e n t T h e o r e m fails for #n+l. T h e equality of shapes of two Z-sets X and Y in # n + l is sufficient, but far from necessary, for the c o m p l e m e n t s # n + l - X and # n + l _ y to be homeomorphic. Indeed, it can be easily seen t h a t if the (n + 1)dimensional sphere S n+l is e m b e d d e d into # n + l as a Z-set, t h e n # n + l _ s n + l is h o m e o m o r p h i c to tt n+l - {pt}. At the same time S h ( S n+l) ~ Sh(pt). T h e problem was solved in [87] (see also [90], [92]) where the notion of n - s h a p e was introduced. T h e relation between n - S H A P E and n - H O M O T O P Y categories is of the same n a t u r e as t h a t between the categories of S H A P E and H O M O T O P Y . Roughly, n - S H A P E is a "spectral completion" of n - H O M O T O P Y . T h e main result in this direction is the following. THEOREM 4.4.8. Let X and Y be Z-sets in #n+l. The complements tt n+l - X and # n + l _ y are homeomorphic if and only if n - S h ( X ) -- n - S h ( Y ) .

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4. MENGER

MANIFOLDS

We would like to mention some corollaries of this theorem and the definition of n-shape itself. COROLLARY 4.4.9. I f S h ( X ) -- S h ( Y ) , then n -

S h ( X ) -- n - S h ( Y ) .

COROLLARY 4.4.10. I f X and Y are at most n-dimensional, then S h ( X ) S h ( Y ) if and only if n - S h ( X ) -- n - S h ( Y ) .

=

COROLLARY 4.4.11. I f Z - s e t s X and Y in #n+l are #n+l-manifolds, then the complements #n+l _ X and #n+l _ y are homeomorphic if and only if the compacta X and Y are homeomorphic. Let us emphasize that the notion of n-equivalence, introduced by Ferry [150] as a generalization of Whitehead's notion of n-type, coincides in several important cases with the notion of n-shape. Relations between these two concepts have been studied in [97].We conclude this section by noting that Theorem 4.4.8 was extended [280] to a larger class of subspaces than Z-sets. These are the so-called weak Z-sets.

4.4.4. M e n d e r m a n i f o l d s w i t h b o u n d a r i e s . The problem of putting a boundary on various types of manifolds were considered in [57] (PL manifolds), [282] (smooth manifolds) and [70] (I ~ - manifolds). It was proved in [70] that if an I W-manifold M satisfies certain minimal necessary homotopy-theoretical conditions (finite type and tameness at oo), then there are two obstructions a ~ ( M ) and Tc~(M) to M having a boundary. The first one is an element of the group li+___m{/C0rl(M - A ) " A C M A is compact}, where ]C0~rl is the projective class group functor. If c o o ( M ) -- O, then the second obstruction can be defined as an element of the first derived limit of the inverse system li.___m{YVhrl( M - A ) " A C M A is compact}, where ~Vh~rl is the Whitehead group functor. It was shown in [70] t h a t the different boundaries that can be put on M constitute a whole shape class and that a classification of all possible ways of putting boundaries on M can be done in terms o f t h e group l i m { Y ~ h ~ l ( M - A ) " A C M A is compact}. It should be emphasized that the above mentioned obstructions essentially involve the Wall's finiteness obstruction [303]. The natural analog of Wall's obstruction vanishes in the n-homotopy category. This is exactly what was stated in 4.1.12. We will see that this observation significantly simplifies the situation for #n+l-manifolds. First of all we need the following corollary of Proposition 4.1.12. PROPOSITION 4.4.12. I f a # n + l - m a n i f o l d M is n - h o m o t o p y dominated by an at m o s t (n + 1)-dimensional L C n - c o m p a c t u m , then M is n - h o m o t o p y equivalent to a compact #n+l-manifold.

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS

171

We say t h a t a #'~+l-manifold M admits a boundary if there exists a compact # n + l manifold N such t h a t M = N - Z, where Z is a Z-set in N. In this case we shall say t h a t N is a compactification of M corresponding to the b o u n d a r y Z, and conversely, Z is a b o u n d a r y of M corresponding to the compactification N. We also need the following definition [100]. DEFINITION 4.4.13. A space X is said to be n - t a m e at c~ if for each compactum A C X there exists a larger compactum B C X such that the inclusion X - B ~-~ X - A factors up to n-homotopy through an at most (n + 1)-dimensional finite polyhedron. PROPOSITION 4.4.14. I f a #n+l-manifold M is n-tame at c~, then M is nhomotopy equivalent to a compact #n+l-manifold. PROOF. Take a proper U V n - r e t r a c t i o n r: M ~ P of the given # n + l - m a n i f o l d M onto some (n + 1)-dimensional locally compact polyhedron P. It follows from elementary properties of proper U V n - m a p s t h a t P is n - t a m e at c~ as well. Using Proposition 4.1.8, one can easily see t h a t P is n - h o m o t o p y d o m i n a t e d by an at most (n + 1)-dimensional compact polyhedron. Proposition 4.4.12 finishes the proof. [::] Let us recall t h a t an I ~ -manifold M lying in a larger I ~ -manifold N is said to be clean if M is closed in N and the topological frontier of M in N is collared b o t h in M and N - I n t M . For obvious dimensional reasons we cannot directly define the corresponding notion for #n+l-manifolds. Nevertheless, the following notion is sufficient for us. DEFINITION 4.4.15. A #n+l-manifold M lying in a #n+l-manifold N is said to be n-clean in N provided that M is closed in N and there exists a closed subspace 5 ( M ) of M such that the following conditions are satisfied:

(i) 6(M) /~ ~ (ii) (iii) (iv) (v)

,~+~-m~ifold.

( N - M ) t2 5 ( M ) is a #n+l-manifold. 5 ( M ) is a Z - s e t in M . 5(M) is a Z - s e t in ( N - M ) U S(M). M - 5 ( M ) is open in g .

Sometimes we say t h a t M is n-clean with respect to 5 ( M ) . Let us indicate the s t a n d a r d situation in which n-clean submanifolds arise naturally. Suppose t h a t L is submanifold of a combinatorial PL-manifold P . Consider a proper UVn-surjection f : N ~ P of a # n + l - m a n i f o l d N from Theorem 4.3.6. Using the properties of f , it is easy to see t h a t M = f - l ( L ) is an n-clean submanifold of N with 5 ( M ) = f - l ( O L ) . Generally speaking, .f is not an open map and consequently 5 ( M ) does not necessarily coincide with the topological frontier of M in N.

172

4. MENGER

MANIFOLDS

LEMMA 4.4.16. Let N be a #n+l-manifold which is n - t a m e at oo. Suppose that M is a compact and n-clean submanifold of N . Then the #n+l-manifold (N - M ) U 5 ( M ) is n-homotopy equivalent to a compact #n+l-manifold. PROOF. By Proposition 4.4.14, it suffices to show t h a t the #n+l-manifold ( N - M ) U S ( M ) is n - t a m e at 00. Let A be a compact subspace of ( N - M ) U 6 ( M ) . Clearly, K1 = A U M is compact. Since N is n - t a m e at 00, there exists a c o m p a c t u m K2 such t h a t Kz C_ K2 C_ N and the inclusion N - K 2 ~ NK1 factors up to n - h o m o t o p y t h r o u g h an at most (n + 1)-dimensional compact polyhedron. Let B = ((N - M ) U 6 ( M ) ) gl K2. Clearly B is a c o m p a c t u m and A c B. Note t h a t D

((N-M)U6(M))-B=N-K2

and N - K I C _

((N-M)U6(M))-A.

Consequently, the inclusion ((N-M)U6(M))-B

~

((N-M)U6(M))-A

factors up to n - h o m o t o p y t h r o u g h an at most (n + 1)-dimensional compact polyhedron. Hence ( g - M ) U 6 ( M ) is n - t a m e at c~. FI LEMMA 4.4.17. A n y #n+l-manifold M can be written as a union M = Ui=IM~ such that all M i ' s are compact and n-clean and M~ C M i + l - 6 ( M i + 1 ) for each i= 1,2,.... oo

PROOF. It suffices to show t h a t for each c o m p a c t u m K C_ M, there exists a compact and n-clean M1 C_ M such t h a t K C_ M1 - 6 ( M 1 ) . As before, take a proper UVn-surjection g" M ~ X, where X is a I ~ -manifold. There is a compact and clean Y _c X such t h a t g ( K ) C_ I n t ( x ( Y ) (see [70]). By the relative triangulation theorem for I~-manifolds (see T h e o r e m 2.3.31), there exists a polyhedron P which can be w r i t t e n as a union of two subpolyhedra P1 and /)2 such t h a t X = P x I ~ , Y = P1 x I ~ , X - I n t x ( Y ) = P2 x I W and B d x ( Y ) = (P1 CI P2) x I W . Note also t h a t the subpolyhedron P1 F1P2 is a Z-set b o t h in P1 and P2. Consider now a proper UVn-surjection f" N ~ P of a #n+l-manifold N onto the polyhedron P satisfying the conditions of T h e o r e m 4.3.6. Consequently, we have two proper u v n - s u r j e c t i o n s f" N ~ P and 7rpg" M --+ P (here ~p" P x I ~ --, P denotes the n a t u r a l projection) of two #n+l-manifolds onto the polyhedron P. Consider an open cover b / - {P - 7 r g g ( K ) , I n t p ( P 1 ) } of P. By T h e o r e m 4.1.20, there exists a h o m e o m o r p h i s m h" M --+ N such t h a t the compositions 7rgg and f h are b/-close. Let M1 -- h - l f - l ( p 1 ) and 6(M1) = h - l f - l ( p 1 rl P2). By the properties of the map f, M1 is compact and n-clean. It only remains to note t h a t K C_ M1 - 6 ( M 1 ) . This finishes the proof. F1 We also need the following s t a t e m e n t , which is a direct consequence of the characterization T h e o r e m 4.1.19.

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS

173

PROPOSITION 4.4.18. Let a space M be the union of two closed subspaces M1 and M2. If M1, M2 and Mo = M1 A M2 are #n+l-manifolds and Mo is a Z-set both in M1 and M2, then M is a #n+l-manifold. PROOF. It suffices to show t h a t for any m a p f : X ~ M of an at most (n + 1)dimensional c o m p a c t u m X into M , and any open cover U E cov(M), t h e r e exists an e m b e d d i n g g: X ~ M , U-close to f . Let us consider the case when f ( X ) N Mi 7~ 0 for each i = 0, 1, 2. All o t h e r cases are trivial. By P r o p o s i t i o n 4.1.7, t h e r e exists an open cover 12 E coy(M) refining b / s u c h t h a t the following condition is satisfied: (.)~ for any at most (n + 1)-dimensional c o m p a c t u m B, closed s u b s p a c e A of B, and any two P-close m a p s c~1,c~2: A --. M such t h a t c~1 has an extension r B --. M , it follows t h a t C~2 also a d m i t s an extension r B ~ M which is/,/-close to ~1. Let Xi = f - l ( M i ) , i = 0, 1,2. Since M0 is a # n + l - m a n i f o l d , there is a Ze m b e d d i n g go: X0 --~ M0 such t h a t go and f / X o are ]2-close. By (*)n, t h e r e is an extension h: X --. M of G - ) such t h a t h and f are V-close. Since Mo is a Z-set in b o t h M1 and M2 we conclude t h a t go(Xo) is a Z - s e t b o t h in M1 and M2. Consequently, by T h e o r e m 4.1.19, for each i = 1, 2 there is a Z - e m b e d d i n g gi: Xi ~ Mi such t h a t gi/Xo = go and gi is U-close to h / X i . At the same time, w i t h o u t loss of generality we can assume t h a t one of these maps, say gl, has the following p r o p e r t y : g l ( X 1 - X0) N M0 = 0 (we once again use the fact t h a t M0 is a Z-set in M1). T h e n the map g, coinciding with gi on Xi (i = 1,2), is an embedding. It only remains to note t h a t g and .f are N-close. [] LEMMA 4.4.19. If a #n+l-manifold M is n-tame at oo, then we can write M = U~__IMi such that all Mi's are compact and n-clean, Mi C M i + l - ~ ( M i + l ) and the inclusion 5(Mi) r ( M i + l - Mi) U 5(Mi) is n-homotopy equivalence for each i = 1 , 2 , . . . . PROOF. Choose any c o m p a c t and n-clean submanifold A of M . By L e m m a 4.4.17, it suffices to find a c o m p a c t and n-clean submanifold B of M such t h a t A C_ B 5(B) and the inclusion 8(B) ~-~ ( M - B ) U 5(B) is an nh o m o t o p y equivalence. By L e m m a 4.4.16, the # n + l - m a n i f o l d ( M - A)8(A) is n - h o m o t o p y equivalent to some c o m p a c t # n + l - m a n i f o l d X. Fix the corresponding n - h o m o t o p y equivalence r ( M - a ) U S ( A ) + X and its n - h o m o t o p y inverse ~o1" X ~ (M - A ) U 5(A). Obviously there is a m a p ~92" (M - A ) 5 ( A ) ~ X such t h a t r 5(A) --~ X is a Z - e m b e d d i n g a n d r is as close to r as we wish. Similarly, there is a Z - e m b e d d i n g ~o2" X ~ ( M - A ) U 5(A) which is as close to ~Ol as we wish. In particular, we can assume t h a t r and ~o2 are n - h o m o t o p y equivalences. If r and ~o2 were chosen sufficiently close to r and ~ol respectively, then, by T h e o r e m 4.1.18, t h e r e exists a h o m e o m o r phism h" ( M - A ) U 5(A) ~ ( M - A ) U 5(A) which e x t e n d s the h o m e o m o r phism qo2r

5(A) ~

qo2r

and which is sufficiently close to the

174

4. MENGER MANIFOLDS

identity map of (M - A ) U 5(A). In particular, we can assume that h is nhomotopic to id(M_A)oh(A ). T h e n the n - h o m o t o p y equivalence ~p - h -1~2" X --~ (iA ) U 6(A) is a Z - e m b e d d i n g and 6(A) C ~ ( X ) -- Y . Since Y is a compact #n+l-manifold, there exists a U V n - r e t r a c t i o n s" Y ~ K onto a finite (n + 1)-dimensional polyhedron K (see T h e o r e m 4.2.23). Similarly, take a proper V Y n - r e t r a c t i o n r" ( M - A) U 6 ( A ) --, T, where T is a polyhedron. Let i" Y "--. ( M - A ) U 6 ( A ) denote the inclusion map and j" K ~ Y be a section of s (i.e. s j - i d g ) . Note t h a t i is an n - h o m o t o p y equivalence. Let p" K --, T be a P L - m a p homotopic to the composition r i j . Form the mapping cylinder M ( p ) -- P of the map p. For the reader's convenience, we again recall t h a t P is the space obtained from the disjoint union ( g x [0, 1]) @ T, by identifying (k, 1) with p ( k ) , k E g . At the same time we identify g with g x {0}. Since p is a P L - m a p , K x {0} and T are s u b p o l y h e d r a of the polyhedron P. Let c" P --. T be the collapse to the base, i.e. the natural retraction defined by sending (k, t) to p ( k ) for each (k, t) E K • [0, 1]. Obviously, c is a proper cell-like map t h a t is a proper h o m o t o p y equivalence. Now consider a proper U V n - s u r j e c t i o n f" N --. P of some # n + l - m a n i f o l d N onto P, satisfying the conditions of Theorem 4.3.6. T h e compact #n+l-manifolds Y and N1 = f - l ( K • {0}) admit U V n - s u r j e c t i o n s s" Y ~ K x {0} and f / N l " N1 ---* K • {0} onto the same polyhedron. Consequently, by T h e o r e m 4.1.21, there exists a homeomorphism n gl" Y ---* N1 such t h a t f g l ~ - s . Similarly, we have two proper U V n - s u r j e c t i o n s r" ( M - A ) U 5 ( A ) --. T and c f " N --~ T. As above, there is a homeomorphism g2" ( M - A ) U 5 ( A ) --. N such t h a t c f g2 ~ p r. By the construction and the corresponding properties of proper U V n - s u r j e c t i o n s , we have n

n

n

c f gl ~-- cs -- ps ~ r i j s ~_ ri ~ c f g2i.

Since c f is a proper n - h o m o t o p y equivalence, we conclude t h a t gl" Y ~ N and g 2 / Y " Y ~ N are n-homotopic. Consider the h o m e o m o r p h i s m

= ~g2--1 /g2(Y)" g 2 ( Y ) ~ N~. Clearly n

--1

~-- g2g2 / g 2 ( Y ) -- idg2(y ).

By the properties of the map f, N1 is a Z-set in N. Note also that, by our construction, g2(Y) is a Z-set in N as well. By T h e o r e m 4.1.15, we can find a h o m e o m o r p h i s m G: N ~ N extending a. Let H -- G g2. Note that H ( Y ) -G g 2 ( Y ) = a g 2 ( Y ) = g l . Finally, let B -- A U H - I ( f - I ( K

• [0, ~])) and 5(B) -- H - I ( f - I ( K

• { })).

It follows from the properties of the map f and Proposition 4.4.18 t h a t B is a compact and n-clean submanifold of M , A C B - 5(B), and the inclusion 5(B) ~ ( M - B ) U S ( B ) is an n - h o m o t o p y equivalence. To see this, observe t h a t

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS 1 the m a p p, and consequently the inclusion K x {~} ~ n - h o m o t o p y equivalence. [-1

175

P - ( K x [0, ~]), 1 is an

LEMMA 4.4.20. Let a #n+l-manifold M be a Z - s e t in a compact # n + l - m a n i fold N . I f the inclusion i" M --~ N is an n-homotopy equivalence, then there exists a UVn-retraction of N onto M . PROOF. Let j" N --~ M be an n - h o m o t o p y inverse of i. By T h e o r e m 4.1.21, ?1 there is a h o m e o m o r p h i s m h" N ~ M such t h a t h n J. T h e n hi rn~ j i ~_ idM. Consequently, by P r o p o s i t i o n 4.2.2, there is a UV'~-surjection r" N ~ M such t h a t ri = idM. U] T h e following result gives us a c h a r a c t e r i z a t i o n of t i n + l - m a n i f o l d s with b o u n d aries. THEOREM 4.4.21. A #n+l-manifold admits a boundary if and only if it is n-tame at oo. PROOF. Let M be a # n + l - m a n i f o l d which is n - t a m e at cx~. By L e m m a 4.4.19, we can represent M as a union M -- U ~ I M i such t h a t all t h e M i ' s are c o m p a c t a n d n-clean, Mi C M i + l - 5 ( M i + I ) and the inclusion 5(Mi) ~ ( M i + l - Mi) U 5(Mi) is an n - h o m o t o p y equivalence for each i = 1 , 2 , . . . . By L e m m a 4.4.20, for each i there exists a u y n - r e t r a c t i o n fi" ( M i + l - Mi) U 5(Mi) ~ 3(Mi). Let the u v n - r e t r a c t i o n ri" M i + l --* Mi coincide with fi on M i + l - Mi and with t h e identity on Mi. T h e n we have an inverse sequence S = {Mi, ri} consisting of c o m p a c t # n + l - m a n i f o l d s and U V n -retractions. By T h e o r e m 4.1.20, ri is a nearh o m e o m o r p h i s m for each i. By [58], each limit p r o j e c t i o n of the s p e c t r u m S is a n e a r - h o m e o m o r p h i s m as well. Consequently, N = lim S, being h o m e o m o r p h i c to M1, is a c o m p a c t # n + l - m a n i f o l d . Since 5(Mi) is a Z-set in Mi for each i, we conclude t h a t the subset Z = l i m { 5 ( M i + l ) , r i / 5 ( M i + l ) } is a Z - s e t in N. It only remains to note t h a t N - Z is n a t u r a l l y h o m e o m o r p h i c to M . Conversely, suppose t h a t the # n + l - m a n i f o l d M a d m i t s a b o u n d a r y . This m e a n s t h a t there are a c o m p a c t # n + l - m a n i f o l d N and a Z - s e t Z in N such that M = N-Z. Let us show t h a t M is n - t a m e at cx~. Let A be a c o m p a c t subspace of M . As in the proof of L e m m a 4.4.17, t h e r e exists a c o m p a c t and n-clean submanifold B of M such t h a t A C_ B - 5 ( B ) . It suffices to show t h a t ( M - B ) U 5(B) is n - h o m o t o p y equivalent to an at m o s t (n + 1)-dimensional finite p o l y h e d r o n . Indeed, it is easy to see t h a t ( M - B ) U 5 ( B ) is n - h o m o t o p y equivalent to a c o m p a c t # n + l - m a n i f o l d ( N - B ) U S ( B ) . It only remains to a p p l y P r o p o s i t i o n 4.1.10. El Not all # n + l - m a n i f o l d s a d m i t b o u n d a r i e s in the above sense. To see this, consider the 3-dimensional (topological) manifold W ( c o n s t r u c t e d by W h i t e h e a d ) which is defined as the c o m p l e m e n t in S 3 of a c o n t i n u u m W h which, in t u r n , is the intersection of a nested sequence of tori in S 3. T h e manifold W has an

176

4. MENGER MANIFOLDS

infinitely generated f u n d a m e n t a l group at c~. Let n >_ 1, and consider a #n+l_ manifold M and a proper U V n - s u r j e c t i o n f " M ---. W . Since n + 1 >_ 2, we see t h a t f induces an isomorphism of f u n d a m e n t a l groups of ends. T h e n it is easy to see t h a t M is not 1-tame at c~ and, therefore, cannot have a boundary. On the other hand, it can shown t h a t the Freudenthal compactification of any connected #l-manifold contains its end as a Z-set. Consequently, the Freudenthal compactification of any connected #1manifold is homeomorphic to #1. In other words, any connected # l - m a n i f o l d has a boundary. We also mention two related results. PROPOSITION 4.4.22. I f the compactum X is a boundary for a #n+l-manifold M , then the compactum Y is also a boundary for M if and only if dim Y < n + 1 andn-Sh(Y)=n-Sh(X). Two compactifications N and T of the same space M are said to be equivalent if for every c o m p a c t u m A C M there is a h o m e o m o r p h i s m of N onto T fixing A point wise. Of course, if the #n+l-manifolds N and T are compactifications of a #n+l_ manifold M , then the inclusions M ~ N and M --~ T are n-homotopy equivalences (because, N - M and T - M are Z-sets in N and T respectively). Consequently, N and T are h o m e o m o r p h i c as n - h o m o t o p y equivalent compact #n+l-manifolds (Theorem 4.1.21). A stronger result can be obtained. PROPOSITION 4.4.23. Every two # n + l - m a n i f o l d compactifications of a given # n + l - m a n i f o l d are equivalent in the above sense. T h e problem of w h e t h e r a # n + l - m a n i f o l d has a b o u n d a r y which is itself a # n + l - m a n i f o l d was also considered in [104]. DEFINITION 4.4.24. A proper map f " Y ---, X between at most (n + 1)-dimensional locally compact spaces is an n - d o m i n a t i o n near c~ provided that there exists a cofinite subspace X1 of X ( i.e. X1 is closed and X - X1 has compact closure) and a proper map g" X1 ---* Y such that f g is properly n-homotopic to the inclusion map X1 ~-* X . If, in addition, for some cofinite subspace Y1 of Y the composition g f /Y1 is properly n - h o m o t o p i c to the inclusion map ]I1 ~ Y , then we say that f is an equivalence near c 0 is a sufficiently small number. Let U be a n e i g h b o r h o o d of i d x in A u t h ( X ) with d i a m U < e. Let x , y E X with d(x, y) -- e, and let A be a small d i a m e t e r arc from x to y in X. One can define, using local setwise homogeneity, a convergent sequence {h~} in A u t h ( X ) "sliding x towards y along A" and such t h a t h -limh~ E A u t h ( X ) and h E Bd(U). Thus, each e - n e i g h b o r h o o d of i d x has none m p t y b o u n d a r y , and so dim A u t h ( X ) >_ 1. Therefore we have the following statement. THEOREM 4.5.2. Let X

be a locally setwise homogeneous continuum. A u t h ( X ) is at least 1-dimensional.

Then

S t r o n g local h o m o g e n e i t y of # " - m a n i f o l d s (Corollary 4.1.17), implies their local setwise homogeneity. Consequently, we have the following. COROLLARY 4.5.3. If M is a compact #"-manifold, then dim A u t h ( M ) >_ 1. Recall t h a t a space X is almost O-dimensional if it has a basis B of open sets such t h a t for each B E B, X - c l B -- U(Ui 9 i E N } where each Ui is b o t h open and closed. Clearly, each 0-dimensional space is almost 0-dimensional. Also, it is easy to see t h a t every almost 0-dimensional space is totally disconnected. T h e c o m p l e t e Erdhs space ~ = (x E 12" xi is irrational for each i} is a 1-dimensional space which is almost 0-dimensional. PROPOSITION 4.5.4. Each almost O-dimensional space is at most 1-dimensi-

onal. SKETCH OF PROOF. Let X be an almost 0-dimensional space and let B be a c o u n t a b l e basis witnessing this fact. Let ~ = ( f i " i E N } be a collection of continuous functions f i " X ~ (0, 1} such t h a t if B , B ' E 13 with cl(B) Mcl(B') -0, there is a f i E ~ ' w i t h f~(B) = 0 and f i ( B ' ) = 1. Let p be the metric o n X given by p(x, y) = ~-~i 2 - i I f i ( x ) - f i ( y ) ]. Let d be a t o t a l l y bounded metric on X. It suffices to show t h a t the metric dimension # d i m ( X , d ) < 1. Now, d ~ -- d + p is also a t o t a l l y b o u n d e d metric on X. Let Y be the completion of X with respect to d ~. It suffices to show t h a t for each t > 0, there is an open set U of Y containing X such t h a t each c o n t i n u u m in U has d i a m e t e r less t h a n t. Let 34 - {U open in Y 9 diam U < t / 3 and U MX E B}. Let C be any continu u m in W - U/d. I f B , B ' E B with B M C ~ 0 ~ B ' M C , then cl(B) N c l ( B ' ) =fi 0, for otherwise there is fi E 9r with f~(B) = 0 and f i ( B ' ) = 1. But f~(X) = {0, 1} and so, by the definition of d', Y = c l y ( f ~ - l ( O ) ) U c l y ( f ~ - l ( 1 ) ) , where cly(f/--l(0)) and c l y ( f ~ - l ( 1 ) ) are disjoint closed sets. Hence C, being a continuum, c a n n o t meet b o t h c l y ( f / - l ( 0 ) ) and c l y ( f ~ - l ( 1 ) ) which is a contradiction. It follows t h a t d i a m C < t. V1 PROPOSITION 4.5.5. If M is an Mkn-manifold with 0 ~_ n < k < oe, then A u t h ( M ) is almost O-dimensional.

4.5. HOMEOMORPHISM GROUPS

179

SKETCH OF PROOF. Let g E A u t h ( M ) and ~ > 0. Let h E A u t h ( M ) with d(g,h) > e. We shall show that there is an open and closed set U containing h such that d ( g , j ) > e for each j E U. Now, d(g, h) = c + 45 for some 5 > 0, and there is x E M so that d(g, h) = d(g(x), h(x)). Choose an n-sphere S in M such that g(S) c_ N ( g ( x ) , 5 ) and h(S) C_ N ( h ( x ) , 5). Since dim M = n, there is a retraction r : M ~ h(S) such that r ( M - N ( h ( x ) , 2 5 ) ) is constant. Let U -- {f E A u t h ( M ) : r f / S ~ .}. Then U is both open and closed in A u t h ( M ) because close maps into S are homotopic. Also h E U. Let f E N(g, e). T h e n f ( S ) C N ( g ( x ) , c + 5). Hence, r . f ( S ) is a point and f ~ U. D COROLLARY 4.5.6. Let 1 < n < cx~. If M is a compact #n-manifold, then d i m A u t h ( M ) - - 1. PROOF. Apply Corollary 4.5.3 and Propositions 4.5.4 and 4.5.5.

[:]

4.5.2. S i m p l i c i t y . Anderson [16], [17] originated a technique for identifying minimal, non-trivial normal subgroups of A u t h ( X ) for spaces with certain dilation and homogeneity properties. DEFINITION 4.5.7. Let X be a space. A subset A of X is deformable if for every non-empty open set U in X , there is h E A u t h ( X ) with h(A) C_ U. Let V be an open set. A collection ({Bi : i E N } , h) is called a dilation system in U if {B~} is a sequence of disjoint non-empty open sets in U with lim Bi -- {p} for some p E U and h E A u t h ( X ) supported on U such that h(B~+l) ---- B~ for each i. PROPOSITION 4.5.8. Let X be a metrizable space in which each non-empty open set contains a dilation system. Let G be a subgroup of A u t h ( X ) generated by all homeomorphisms which are supported by deformable subsets of X . I f G :fi {e}, then G is the smallest non-trivial normal subgroup of A u t h ( X ) . If X is a finite-dimensional manifold without boundary, then A u t h 0 ( X ) , the subgroup of homeomorphisms isotopic to the identity, is simple (see [151] and [141]). It is also known (see [220] and [323]) that Auth(/2) and A u t h ( I ~ ) are simple. DEFINITION 4.5.9. Let M be a #n+l-manifold. A pair (W, 5 ( W ) ) is an nclean pair if W is n-clean with respect to 5 ( W ) in the sense of Definition 4.4.15 and if, in addition, both W and 5 ( W ) are homeomorphic to #n+l. By Theorem 4.1.15 and the existence of n-clean pairs in #n+l, (compare with [184]), it follows that every open set in #n+l has a dilation system. Also, every proper closed set in #n+l is deformable. Since every element of A u t h ( # n+l) is stable (see Theorem 4.5.13 below) we have the following. THEOREM 4.5.10. A u t h ( # n+l) is simple.

180

4. MENGER MANIFOLDS

4.5.3. Stability of homeomorphisms. An a u t o h o m e o m o r p h i s m of a space X is said to be stable [32] if it can be expressed as the composition of finitely m a n y a u t o h o m e o m o r p h i s m s each of which is the identity on some n o n - e m p t y open subspace of X. It is well-known t h a t all a u t o h o m e o m o r p h i s m s of the Hilbert cube I ~ and the Hilbert space 12 are stable (see, for example, [32]). Every orientation-preserving h o m e o m o r p h i s m of R n is stable [194]. LEMMA 4.5.11. For each #n+l-manifold M , there is a Z-embedding a" M ---+ M which is properly n-homotopic to idM and which satisfies the following condition: (.) If F e A u t h ( M ) and F / a ( M ) = id, then F can be expressed as the composition of two autohomeomorphisms of M each of which is the identity on some open subspace of M . PROOF. Take a proper UVnsurjection g" M ---+ K , where K is an at most (n + 1)-dimensional locally compact p o l y h e d r o n ( T h e o r e m 4.2.23). Consider also a p r o p e r UVn-surjection f" MI" K x [-1, 2], satisfying conditions of T h e o r e m 4.3.6. It can easily be checked t h a t the inverse image M -- f - l ( g x [0,2]) is also a copy of the # n + l - m a n i f o l d M. Moreover, since the composition 7 r l f / M --* g x {0}, where ~rl" g x [0, 2] --. g x {0} denotes the projection, is a proper U V nsurjection, we conclude t h a t there exists a h o m e o m o r p h i s m a" M --, f - l ( K x {0}) such t h a t f a ..~n 7 r l f / M o T h e n a is properly n - h o m o t o p i c to idM. By the mp properties of f , we see t h a t a" M --~ M is a Z-embedding. Consequently, it only remains to show t h a t if F E A u t h ( M ) and F / f - l ( g x {0}) -- id, then F can be expressed as the composition of two a u t o h o m e o m o r p h i s m s of M each of which is the identity on some open subspace of M . Let U - F - l ( f - l ( g x (1,2])). Since F / a ( M ) - id we conclude t h a t U n f - l ( K x {0}) = 0. Consequently, there exists a sufficiently small tl > 0 such t h a t f - l ( K x [O, t l ] ) N ( U U F ( U ) ) = O. Let V : f - l ( K x [0, t0)), where to = l t l l + t1l . Consider the h o m e o m o r p h i s m ~" [0, tl] --* [ - 1 , t l ] defined as v~(x) = t-LL--x--1 1+tl Consider also an open cover ZX=

{ [ K x

--1,--

,K x

--~,tl

]}

of the p r o d u c t K x [-1, tl]. Again, by the properties of f , the inverse images N1 = f - l ( K x [0, tl]) and N2 = f - l ( K x [-1, tl]) are #n+l-manifolds. Clearly these a d m i t proper u v n - s u r j e c t i o n s f l = (id x t p ) f / N l " N1 ---+ K x [-1, tl] and f2 = f / N 2 " K x [-1, tl] onto the polyhedron K x [-1, tl]. T h e n we see t h a t the inverse image f - l ( K x {tl}) is a Z-set b o t h in N1 and N2. Moreover, since, v~(tl) = tl, we conclude t h a t the identity h o m e o m o r p h i s m h o f . f - l ( K x {tl}) satisfies the equality f2h = fl/f-l(K x {tl}). Consequently, there exists a h o m e o m o r p h i s m T" N1 ---+ N2 such t h a t T / f - I ( K x {tl}) = id and the composition f 2 T is U-close to f l . Now

4.5. HOMEOMORPHISM GROUPS

181

consider the homeomorphism H" M ~ M1 such that H / f - I ( K x [0, tl]) = T and H / f - I ( K x [tl,2]) = id. It is easy to verify that H ( V ) N M = 0 and H/(UUF(U))=id. Let f

a(x) = ~F(x),

[ x,

ifx eM ifxEM1-M.

Since F / f - I ( K x {0}) = id, we conclude t h a t G E Auth(M1). Now let F2 -H - 1 G H e A u t h ( M ) . If x e U, then H ( x ) - x. Hence, G H ( x ) - F ( x ) e F ( V ) . Consequently, H - 1 G H ( x ) -- F ( x ) . In other words, F2/U -- F l U . Similarly, if x e V, then H ( x ) e M1 - M . Hence G H ( x ) = H ( x ) and H - 1 G H ( x ) = H - 1 H ( x ) = x. This means that F 2 / V = id. Obviously, F -- F2F1, where F1 = F 2 1 F . It only remains to note t h a t F 1 / U -- id and F 2 / V -- id. 0 LEMMA 4.5.12. Let M be an #n+l-manifold and ~" M ~ M be a Z-embedding properly n-homotopic to idM. I f G E A u t h ( M ) and G / ~ ( M ) -- id, then G can be expressed as the composition of two autohomeomorphisms each of which is the identity on some open subspace of M . PROOF. Take a Z-embedding a satisfying condition (.) of L e m m a 4.5.11. Then the homeomorphism / 3 a - l : a ( M ) ~ ~ ( M ) is properly n-homotopic to the inclusion a ( M ) ~ M. Consequently, by Theorem 4.1.15, there exists a homeomorphism h: M ~ M extending ~ a -1. Consider the homeomorphism h - l G h E A u t h ( M ) . Evidently, h - l G h / a ( M ) -- id. By the choice of a, there exist F1, F2 E A u t h ( M ) and open subspaces 1/1 and 1/2 of M such that F i / V i = id, i -- 1, 2, and h - l G h -- F2F1. Then G -- G2G1, where Gi -- hFih -1, i -- 1, 2. It only remains to note t h a t G i / h ( V i ) -- id, i -- 1, 2. V] THEOREM 4.5.13. Let M be a #n+l-manifold and F E A u t h ( M ) . properly n-homotopic to idM, then F is stable.

I f F is

PROOF. Let ~" M --~ M be a Z-embedding properly n-homotopic to idM. Let a" M ---+ M be another Z-embedding properly n-homotopic to idM and such that a ( M ) N ( ~ ( M ) U F ( ~ ( M ) ) ) = 0. Consider the homeomorphism S" a ( M ) U f l ( M ) ~ a ( M ) U F ( ~ ( M ) )

which coincides with the identity on a ( M ) and with F on /~(M). Clearly, f is properly n-homotopic to the inclusion a ( M ) U ~ ( M ) ~ M . Therefore there exists an extension F2 E A u t h ( M ) of f . Let F1 = F 2 1 F . By L e m m a 4.5.12, F = F2F1 can be expressed as the composition of four a u t o h o m e o m o r p h i s m s each of which is the identity on some open subspace of M. V1 Since each map of #n+l into itself is (properly) n-homotopic to the identity, we obtain the following statement.

COROLLARY 4.5.14. Every autohomeomorphism of#n+1 is stable.

182

4. MENGER MANIFOLDS

4 . 5 . 4 . G r o u p a c t i o n s o n M e n d e r m a n i f o l d s . T h e well-known HilbertSmith conjecture asks w h e t h e r every c o m p a c t group acting effectively on a manifold is a Lie group. This is equivalent to asking w h e t h e r the group A n of p-adic integers acts effectively on a manifold. This long s t a n d i n g problem is still open. T h e situation is r a t h e r different for Mender manifolds. For instance, it is known [13] t h a t any c o m p a c t metrizable zero-dimensional topological group G acts freely on #1 so t h a t the orbit space # I / G is h o m e o m o r p h i c to tt 1. There are several constructions of group actions on #n-manifolds. THEOREM 4.5.15. Let M be a # " - m a n i f o l d . T h e n : (i) E v e r y c o m p a c t z e r o - d i m e n s i o n a l metrizable group G acts on M so that the orbit space M / G

is h o m e o m o r p h i c to M

(ii) Ap acts freely on M so that d i m M / A p -- n + 1 (iii) Ap acts on M so that dim M / A p = n + 2 T h e r e are u n e x p e c t e d ties between group actions on Mender c o m p a c t a and the H i l b e r t - S m i t h problem mentioned above. Namely, a positive solution to the following conjecture would prove t h a t there is no free Ap-action on a connected (topological) manifold M with dim M / A p < 0r

CONJECTURE 4.5.16. Let m and n be positive integers and G be a zero-dim e n s i o n a l c o m p a c t m e t r i c group. I f #mWn and #n are free G-spaces, then there is no equivariant map #mTn ~ #n.

Historical and bibliographical notes 4.5. T h e o r e m 4.5.2 was proved in [54]. Corollary 4.5.6 a p p e a r s in [244]. Corollary 4.5.14 is due to the author [94]. It was shown later t h a t all a u t o h o m e o m o r p h i s m s of all connected Mender manifolds are stable [266]. Item (i) of T h e o r e m 4.5.15 was proved in [131] (see also [265]). Items (ii) and (iii) a p p e a r in [219] (the last s t a t e m e n t is based on the work [257] ). Conjecture 4.5.16 appears in [1].

4.6. w-soft map

of a onto

E

In this Section, using T h e o r e m 4.2.21, we show t h a t there exists an "almost soft" m a p from a onto E. DEFINITION 4.6.1. A m a p f : X c o u n t a b l e - d i m e n s i o n a l I space B , g: A ---, X

~

Y

is called w - s o f t if f o r each strongly

closed subspace A

and h: B ---, Y with f g -- h / A ,

of B

and any two m a p s

there exists a m a p k ~

B ---, X

such that k / A -- g and f k -- h, i.e. if the following diagram

1Recall that a space is strongly countable-dimensional if it can be represented as the countable union of finite-dimensional closed subspaces.

4.6. w-SOFT MAP OF a ONTO E

X

A

183

~Y

~

,.-B

commutes.

Obviously, e v e r y w-soft m a p is n-soft for each n E w. In p a r t i c u l a r , e v e r y wsoft m a p is s u r j e c t i v e a n d o p e n ( c o m p a r e w i t h t h e p r o o f of i m p l i c a t i o n (ii) ~

(i)

in P r o p o s i t i o n 2.1.34).

LEMMA 4.6.2. Let

T

,-X

i

,~ X I

,~Y

be a c o m m u t a t i v e diagram, consisting of compact spaces, where i is an embedding and d i m T < n. T h e n there exists a c o m m u t a t i v e n - s o f t diagram 2

2This means that the diagonal product k~Ah ~, considered as the map of T ~ onto its image, is n-soft (see page 159).

184

4. M E N G E R M A N I F O L D S

TI

\

h!

~

T

,

~

XI

h

f

Z

g

~Y

where j is an embedding and d i m T ~ n - k } and Li+l = fl2Li[Mi+l.

130

4. MENGER MANIFOLDS

If M is the n-simplex with the standard simplicial complex structure, the resulting compactum DMi is denoted by T~. In particular, T 2k+1 is denoted by ~k. (Again this is justified by the Characterization Theorem). Observe that Mi+l is regarded as a regular neighborhood of the dual k-skeleton of Mi (with respect to Li). It might be worth noting the differences among these constructions. Consider the properties of the partitions which are naturally induced by each of the above constructions. For simplicity, we formulate these properties only for M~, L~ and

PROPOSITION 4.1.1. There are sequences {Pi}, { Q i } and { R i } of partitions of M ~ , L~ and T ~ , respectively, satisfying the following conditions: (a) Pi+l, Qi+l and Ri+l are refinements of Pi, Qi and Ri, respectively. (b) lim mesh Pi = lim mesh Qi = lim mesh Ri -- 0. (c) ord Pi = n ~- 1,1imordQi = c~ and ordRi = k + 1. (d) For each p l , p 2 , . . . , p t E Pi, D{pj : j -- 1 , . . . , t } is an at most k-dimensional L C k-1 D C k - l - c o m p a c t u m o r (e) For each q l , q 2 , . . . , a t E Qi, D{qj : j sional L C k-1 gl C k - l - c o m p a c t u m o r (f) For each r l , r 2 , . . . , r t 9 Ri, D{rj : j sional L C k - t CI C k - t -compactum.

an at m o s t -- 1 , . . . , t } an at m o s t -- 1 , . . . , t }

k - d i m e n s i o n a l cell. is an at most k-dimenk - d i m e n s i o n a l simplex. is a ( k - t + 1)-dimen-

PROOF. The partitions defined below satisfy the desired conditions: Pi-- {eD M~ : e E Li},Qi--

{ s D L~ : s E L i } a n d R i -

{ s D T~ : s E Li}.

[:] R e m a r k 4.1.2. (1) In the last case, if ( k , n ) -- (1,3) we have a partition of #1 with 0-dimensional intersections of all adjacent elements. In this sense, the partition determined by Bestvina's construction can be regarded as a generalization of the partition of the Menger curve considered in [14, 15] and [235].

(2) We may obtain characterizations of (compact) Menger manifolds as well as (compact) Q-manifolds in terms of the existence of certain types of partitions

[18s]. 4.1.2. n - h o m o t o p y . We are going to describe an adequate homotopy language for #n+l-manifold theory. This is the so called n-homotopy theory. The related notion of #n+l-homotopy was first exploited in [33]. DEFINITION 4.1.3. Two maps f , g : X - , Y are said to be n - h o m o t o p i c (writn

ten f "~ g) if the compositions f a and ga are homotopic in the usual sense f o r any map ~ : Z --, X of an at m o s t n - d i m e n s i o n a l space Z into X .

4.1. GENERAL THEORY

131

It can easily be seen [87, Proposition 2.3] that if dim X _ n + 1 and Y E L C n, then maps f , g 9 X - , Y are #n+l-homotopic in the sense of Bestvina [33, Definition 2.1.7] if and only if they are n-homotopic. Note also that if, in the above definition, we consider, instead of compact, only polyhedral Z, then we get Fox's definition of n-homotopy [154]. In practice it is convenient to use the following statement. PROPOSITION 4.1.4. M a p s f , g 9 X ~ Y are n - h o m o t o p i c if and only if f o r s o m e (or, equivalently, any) n - i n v e r t i b l e m a p ~ 9 Z ~ X with dim Z ~_ n, the c o m p o s i t i o n s f (~ and g(~ are h o m o t o p i c .

DEFINITION 4.1.5. A m a p ~ 9 A --. X is said to be n - i n v e r t i b l e if f o r any m a p 9 B --. X with d i m B ~_ n, there is a m a p ~/9 B --~ A such that ~/~ - ~.

Note that 0-invertible maps between metrizable compacta are precisely surjections with a regular averaging operator (see Definition 6.1.24). Note also that each compactum is an n-invertible image of an n-dimensional compactum. Of course, homotopic maps are n-homotopic for each n _ 0, but not conversely. Indeed, consider the identity map and the constant map of an arbitrary non-contractible L C ~ 1 7 6 C~176 Nevertheless, n-homotopic maps have several useful properties. PROPOSITION 4.1.6. For each Y

E L C n, there exists an open cover Lt E

c o y ( Y ) such that any two U - c l o s e m a p s of any space into Y are n - h o m o t o p i c .

PROPOSITION 4.1.7. ( n - H o m o t o p y E x t e n s i o n T h e o r e m ) . Let Y E L C n. T h e n f o r each 34 E c o y ( Y ) , there exists 1) E c o y ( Y ) refining lg such that the following condition holds: (*)n For any at m o s t (n -t- 1 ) - d i m e n s i o n a l space B , any closed subspace A of B , and any two V - c l o s e m a p s f , g : A --. Y such that f has an e x t e n s i o n F : B --+ Y , it follows that g also has an e x t e n s i o n G : B --~ Y which is U-close to F .

PROPOSITION 4.1.8. Let Y E L C n. Suppose that A is closed in B and dim B n + 1.

If maps f,g

9 A --, Y

are n - h o m o t o p i c and f

a d m i t s an e x t e n s i o n

F 9 B - . Y , then g also a d m i t s an e x t e n s i o n G 9 B --+ Y , and it m a y be a s s u m e d that F ~ G.

A map f : X -~ Y is an n - h o m o t o p y equivalence if there is a map g : Y --~ X such that g f ~ i d x and f g ~ i d y [87]. The spaces X and Y in this case are said to be n - h o m o t o p y equivalent. For example, any map between arcwise connected spaces is a 0-homotopy equivalence. Note also t h a t the (n + 1)-dimensional sphere S n+l is n-homotopy equivalent to the one-point space. In general, we have the following algebraic characterization of n-homotopy equivalences [316, Theorem 2].

132

4. MENGER MANIFOLDS

PROPOSITION 4.1.9. A map f " X -~ Y between at most (n + 1)-dimensional locally finite polyhedra is an n - h o m o t o p y equivalence if and only if it induces isomorphisms of homotopy groups of dimension ~_ n, i.e., f induces a bijection between the components of X and Y and the h o m o m o r p h i s m ~ k ( f ' ) " ~rk(Cx) --~ ~ k ( C y ) is an isomorphism for each k ~_ n and each pair of components C x C_ X and C y C_ Y with f ( C x ) C_ C y , where f l " C X - , C y denotes the restriction of

f. Recall that each A N R - c o m p a c t u m is homotopy equivalent to a finite polyhedron (Corollary 2.3.30). The following statement is an "n-homotopy version" of West's result. PROPOSITION 4.1.10. Every at most ( n T 1 ) - d i m e n s i o n a l locally compact L C nspace is properly n - h o m o t o p y equivalent to an at most (n + 1)-dimensional locally finite polyhedron. Therefore, 4.1.9 holds even for maps between at most (n + 1)-dimensional locally compact LCn-spaces. Proper n-homotopies, and all associated notions, are defined in the natural way and we do not repeat them here. In order to state an algebraic characterization of proper n-homotopy equivalences similar to Proposition 4.1.9, we need some preliminary definitions. We say that a proper map f 9 X --, Y between locally compact spaces induces an epimorphism of i-th homotopy groups of ends (i >__ 0) if for every compactum C C_ Y there exists a compactum K C_ Y such that for each point x E X - f - l ( K ) and every map ~ " (S i, ,) --, (Y - K, f ( x ) ) there exists a map ~ " (S i, ,) --, ( X - f - l ( C ) , x ) and a homotopy f ~ "~ a(rel ,) in Y - C . We say that f 9 X - , Y induces a m o n o m o r p h i s m of i-th homotopy groups of ends if for every compactum C C_ Y there exists a compactum K C_ Y such that for every map ~ 9 S i --, X - f - l ( K ) with the property that f ~ is null-homotopic in Y - K it follows that a is null-homotopic in X - f - l ( C ) . As usual, f is said to induce an isomorphism of i-th homotopy groups of ends if it simultaneously induces an epimorphism and a monomorphism. PROPOSITION 4.1.11. A proper map f 9 X - , Y between at most (n ~- 1)dimensional locally compact LCn-spaces is a proper n - h o m o t o p y equivalence if and only if it induces isomorphisms of homotopy groups of dimension ~_ n and isomorphisms of homotopy groups of ends of dimension ~_ n. Note that proper n-homotopies have also been studied from the categorical point of view [169]. The following proposition will be used below and indicates a difference between the n-homotopy and usual homotopy theories (compare with [303]). PROPOSITION 4.1.12. Let M be an at m o s t (n -t- 1)-dimensional locally finite polyhedron. Suppose that there exists an at m o s t (n + 1)-dimensional finite polyn

hedron K and two maps f " M --~ K and g " K -+ M such that g f ~_ idM (i.e.

4.1. GENERAL THEORY

133

g is an n - h o m o t o p y domination). Then there exist an (n + 1)-dimensional finite polyhedron T, containing K as a subpolyhedron, and an n-homotopy equivalence h 9 T ~ M extending g such that f is a n-homotopy inverse of h. PROOF. Obviously, it suffices to consider only connected polyhedra. Consequently, the case n -- 0 is trivial. If n -- I, then, by the assumption, lrl(g)" l r l ( g ) --* ~rl(M) is an epimorphism and g e r ( l r l ( g ) ) is a finitely generated group. Select finitely m a n y generators of Ker(~rl(g)) and use t h e m to a t t a c h 2-cells to K and to e x t e n d g over these cells. In this way we obtain a 2-dimensional finite polyhedron T, containing K as a subpolyhedron, and a map h" T --, M , extending g, which induces an isomorphism of f u n d a m e n t a l groups. By Proposition 4.1.9, h is a 1-homotopy equivalence. Assume, by way of induction, t h a t the proposition is already proved in the cases n ~_ m, m _ 1, and consider the case n - m + 1. W i t h o u t loss of generality we can suppose t h a t f ( M (i)) C_ K(~) and g ( K (~)) C_ M (~) for each i _< m + 1. Since g f m+l idM it follows easily t h a t g f / M (re+l) m idM(m+l)" By the inductive hypothesis, there are an (m + 1)-dimensional finite polyhedron R, containing K (re+l) as a subpolyhedron, and an m - h o m o t o p y equivalence r" R - . M (m+l) e x t e n d i n g g / K (re+l). Sewing together the polyhedra K and R along naturally e m b e d d e d copies of K (re+l), we obtain the (m + 2)-dimensional finite polyhedron L, containing K and R as subpolyhedra, and the map s" L --. M which coincides with g on K and with r on R, whence s f = g f

m-l-1

~_ idM and f s / L (re+l) =

m

f r ~_ idL(m+~). By these conditions, we conclude t h a t ~i(s)" ~ri(L) --, ~ i ( i ) is an isomorphism for each i ___ m and an epimorphism for i -- m + 1. One can easily verify t h a t in this situation Ker(lrm+l(S)) is a finitely g e n e r a t e d Z(lrl(L))module. Select Z ( ~ l ( L ) ) - g e n e r a t o r s for Ker(lrm+l(S)) and use t h e m to a t t a c h (m + 2)-cells to L and to extend s over these cells. Let T denote the resulting (m + 2)-dimensional finite polyhedron, containing L as a subpolyhedron, and h" T --~ M the corresponding extension of s. T h e n lri(h) is an isomorphism for each i _< m + 1. Again, by Proposition 4.1.9, h is an (m + 1)-homotopy equivalence. This completes the inductive step and finishes the proof. [-1 T h e analogous s t a t e m e n t for proper n - h o m o t o p y dominations (near c~) will be discussed in Subsection 4.4.4

4.1.3. Z-set unknotting

and topological homogeneity.

PROPOSITION 4.1.13. Let A be a closed subset of a Polish A N E ( n ) - s p a c e X . Then the following conditions are equivalent: (i) A is a Z n - s e t . (ii) For each at most n-dimensional locally finite polyhedron P, the set { f E C ( P , X ) : f ( P ) A A - - 0 } is dense in C ( P , X ) . (iii) For each at most n-dimensional Polish space Y , the set { f e C (Y, X ) : f (Y) N A = 0} is dense in C (Y, X ) .

134

4. MENGER MANIFOLDS

Note that each Z n - s e t in any at m o s t n - d i m e n s i o n a l L C n - l - s p a c e is a Z-set. If, in Proposition 4.1.13, X is locally compact, then the listed conditions are equivalent to the following: (iv) For each at most n - d i m e n s i o n a l Polish space Y , the set { f 9 C ( Y , X ) : c l ( f ( Y ) ) N A - 0} is dense in C ( Y , X ) . Closed subsets satisfying t h e p r o p e r t y (4), as in the case of infinite-dimensional manifolds, are called strong Zn-sets. These sets are especially i m p o r t a n t in the non-locally c o m p a c t setting. PROPOSITION 4.1.14. One-point subsets of M e n g e r manifolds are Z-sets. T h e following s t a t e m e n t s are versions of the powerful Z-set u n k n o t t i n g theorem. THEOREM 4.1.15. Let Z1 and Z2 be two Z - s e t s in a ~ n + l - m a n i f o l d M , and let h : Z1 ---* Z2 be a h o m e o m o r p h i s m . Denote by ij : Z j --, M the inclusion map (j = 1,2). I f il and i2h are properly n-homotopic, then h extends to a h o m e o m o r p h i s m H : M ---, M which is properly n - h o m o t o p i c to idM.

COROLLARY 4.1.16. Every h o m e o m o r p h i s m between Z - s e t s of #n can be extended to an a u t o h o m e o m o r p h i s m of #n. For n - 0 this result is well-known. A closed subset of ~1 is a Z-set if and only if it does not locally s e p a r a t e ~1. T h e result for n - 1 originally a p p e a r e d

in [235]. A c o m p a c t u m X is called strongly locally homogeneous if for each point x E X and each n e i g h b o r h o o d U of x , there is a n e i g h b o r h o o d V of x contained in U such t h a t the following condition holds: for each point y E V, there is a h o m e o m o r p h i s m h : X ---, X such t h a t h(x) -- y and h / ( X - U) -- id.

COROLLARY 4.1.17. #n is topologically homogeneous. Moreover, it is strongly locally homogeneous. THEOREM 4.1.18. Let M be a #n-manifold. For each open c o v e r U E c o y ( M ) , there is an open cover )2 E c o y ( M ) with the following property: (,) if a h o m e o m o r p h i s m h : Z1 ---* Z2 between two Z - s e t s of M is )?-close to il (see notations in 2.3.5), then h can be extended to a h o m e o m o r p h i s m H : M ~ M which is l~-close to idM.

4.1. GENERAL THEORY

135

4.1.4. Topological characterization. The following characterization theorem [33, T h e o r e m 5.2.1] is central to the whole theory. We recall t h a t a space X has D D n P (Disjoint n-Disks P r o p e r t y ) if for each open cover L / 9 c o y ( X ) and any two m a p s a : I n ---. X and fl : I n ---+ X , there are maps a l : I n ---* X and ~1 : I n --. X such t h a t a l is U-close to a, ~1 is U-close to ~ and a l (I n) Aft1 (I n) = 0.

THEOREM 4.1.19. The following conditions are equivalent for any n - d i m e n sional locally compact A N E ( n ) - s p a c e X : (i) X is a #n-manifold. (ii) X is strongly B•,n-universal, i.e. has D D n p . (iii) Each map of the discrete union I n @ I n into X can be approximated arbitrarily closely by embeddings. (iv) Each proper map of any at m o s t n - d i m e n s i o n a l locally compact space into X can be approximated arbitrarily closely by closed embeddings. (v) Each proper map of any at m o s t n - d i m e n s i o n a l locally compact space into X can be approximated arbitrarily closely by Z-embeddings. (vi) Each proper map f : Y --. X of any at m o s t n - d i m e n s i o n a l locally compact space Y into X such that the restriction f /Yo onto a closed subset Iio is a Z-embedding can be arbitrarily closely approximated by Z-embeddings coinciding with f on Yo. Additionally, if X is c o m p a c t and ( n - 1)-connected (i.e. X 9 A E ( n ) ) , then conditions (ii)-(vi) give a topological characterization of the c o m p a c t u m #n. Note t h a t the t h e o r e m remains true even in the case n - c~ (see C h a p t e r 2). If n -- 0 and X is compact, condition (ii) trivially implies t h a t X has no isolated points. Therefore, in this case, X is h o m e o m o r p h i c to the C a n t o r disc o n t i n u u m as has already been noted above. A p p l y i n g the above characterization in the case k -- 1, we see t h a t a comp a c t u m is h o m e o m o r p h i c to #1 if and only if it is a 1-dimensional, locally connected c o n t i n u u m with D D 1 P . It is known (see [15] or [235]) t h a t a locally connected c o n t i n u u m has D D 1 P if and only if it has no local s e p a r a t i n g points and has no open subspaces e m b e d d a b l e in the plane. In this sense, Bestvina's characterization of #1 reduces to Anderson's. In Subsection 4.1.1, three m a j o r geometric constructions of the universal Menger c o m p a c t u m have been presented. Let us indicate another, spectral construction, given in [248]. Let {G~} be a basis of open sets of S 1, such t h a t G~ is an open cell with the p r o p e r t y t h a t diamG~ --. O. Let us construct an inverse sequence {Xi,p~ +1} as follows. We set X0 = S 1, and we get X~+I from X~ by "bubbling over G~" , i.e. X i + l is the quotient space o b t a i n e d from the disjoint union Xi ~ X~ by identifying the two copies of x 9 Xi precisely when p~(x) ~ Gi (here p*o" Z i ~ X o denotes the corresponding projection). T h e projection ~i ~ + 1 " X i + l ~ Xi is defined in the obvious way. Note t h a t p~+ 1is a retraction. One can check directly t h a t X = lim{Xi, Pi -i+1 } is a 1-dimensional 9

136

4. MENGER MANIFOLDS

A E ( 1 ) - c o m p a c t u m with D D 1 P and hence, by 4.1.19, X is a copy of #1. More careful consideration shows [33] that we get #n if we start with S '~ = X0 (and proceed as above). There are several other constructions of #n. To the best of our knowledge, all of them are defined as the limit spaces of inverse sequences(see, for example,

[155], [ls6]). 4.1.5. Approximation by Homeomorphisms. THEOREM 4.1.20. Proper U V n - m a p s between #n+l-manifolds are near-homeomorphisms. Another important result of I ~-manifold theory states that an infinite simple homotopy equivalence between/W-manifolds is homotopic to a homeomorphism. Let us note that this is not the case for homotopy equivalences (i.e. there exist non-homeomorphic but homotopy equivalent compact I ~ -manifolds). In #n-manifold theory we do not have simple homotopy obstructions and this significantly simplifies the corresponding result. THEOREM 4.1.21. Each proper n-homotopy equivalence between #n+l-manifolds is properly n-homotopic to a homeomorphism. The following result, due to Ferry [150, Proposition 1.7] and improved slightly in [188], also illustrates this situation. PROPOSITION 4.1.22. Let f 9 P ~ L be a map between compact polyhedra which induces an isomorphism between the i-th homotopy groups for each i < n. Then there is a compact polyhedron Z and U V n - m a p s ~ 9 Z ---. P and ~ 9 Z ~ L such that f a n ~.

Historical and bibliographical notes 4.1. Most of the results of this Section were obtained by Bestvina in his fundamental work [33]. The notion of nhomotopy is due to the author [87]. Proposition 4.1.12 was proved by the author

[100]. 4.2. n-soft m a p p i n g s of c o m p a c t a , raising d i m e n s i o n The following statement is the first resolution theorem for locally compact L C n - l - s p a c e s , in particular for #n-manifolds. THEOREM 4.2.1. Every locally compact L C n - l - s p a c e is an (n-invertibte) proper u v n - l - i m a g e of a #n-manifold. We also will be using the following result, which states that the u v n - r e s o l u tions from Theorem 4.2.1 can be improved over Z-sets.

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION PROPOSITION 4.2.2. Let f " M ~

X

be a proper U V n - s u r j e c t i o n of a #n+l_

m a n i f o l d onto a locally c o m p a c t L C n - s p a c e and let Z be a Z - s e t in M . X

is any proper m a p properly n - h o m o t o p i c to f / Z ,

uvn-surjection

137

I f g" Z ---.

then there exists a proper

h" M ---. X such that h / Z -- g.

Below, in this subsection, we prove much s t r o n g e r results. We begin by fixing a n a t u r a l n u m b e r n. Let B n+l -- { x E R n ' i i x i l < 1} d e n o t e t h e unit closed ball in R n+l. Consider a m a n y - v a l u e d r e t r a c t i o n l:ln" B n+l "--+ S n of the ball B n+l onto its boundary, the sphere S n - OB n+l, defined as follows" R n ( x ) - {y E S n" ( x , y ) >_ 4[[xI[ 2 - 3[[x[[} for each point x E B n+l.

Here (x, y) denotes the s t a n d a r d scalar p r o d u c t in the E u c l i d e a n space R n+l and Ilxll = v / ( x , x ) denotes the induced norm. T h e above to each point of the ball B~'+1 = {x E R n+l" IIxlI < If x E B n+~ - (B~ +1 U Sn), t h e n R n ( x ) is a ball on x E S n, then R n ( x ) = x, i.e. R n is a retraction. It continuous. Consider the g r a p h F Rn of this map, i.e.

m a n y - v a l u e d m a p assigns 89 the whole sphere S n. the sphere S n. Finally, if is easy to see t h a t R n is

FRo = { ( ( x , y) e B ~+1 • S ~ : y e R ~ ( ~ ) } . By Pn: FRn ---+ B n+l and qn: FRn --'+ S n we d e n o t e the restrictions to FR., of the n a t u r a l projections of the p r o d u c t B n+l x S n onto its coordinates. LEMMA 4.2.3. T h e m a p Pn" FRn -'+ B n+l is ( n -

1)-soft.

PROOF. C o n t i n u i t y of the r e t r a c t i o n R n implies openness of Pn. Therefore, by T h e o r e m 2.1.15, it suffices to show t h a t the collection { R n ( x ) " x E B n + l } is c o n n e c t e d and uniformly locally c o n n e c t e d in all dimensions less t h a n n - 1. As was m e n t i o n e d above, topologically t h e r e are only three t y p e s of elements of the collection { R n ( x ) " x e B n + l } . Indeed, R n ( x ) is the sphere S n (for x E B~'+I), X a closed ball B n with center at i[~-][ in the sphere S n (for x E B n+l - B~'+I), or a point (for x E Sn). C o n s e q u e n t l y the i n d i c a t e d collection is c o n n e c t e d in all dimensions not exceeding n - 1. Let us show the uniform local c o n n e c t i v i t y of this collection in all dimensions less t h a n n - 1. Let k _ n - 2, e > 0 and e Consider a point x E B n+l and an a r b i t r a r y m a p a" S k ---, R n ( x ) 5 - - ~. such t h a t the d i a m e t e r of its image a ( S k) is less t h a n 5. E v i d e n t l y t h e r e is a ball B n on the sphere S n, containing this image a ( S k ) . Let &" B k+l ---+ B n be an extension of a from the b o u n d a r y S k = OB k+l onto the whole B k+l. Additionally, we can assume t h a t if R n ( x ) = x (which occurs when x E S n ) , t h e n a = a. Obviously, in this case as well as in the case w h e n R n ( x ) = S n, ~ is the desired extension. Suppose now t h a t x E B n+l - (B n+l U s n ) . T h e n R n ( x ) is a ball on S n of non-zero diameter. If S '~ - R , ( x ) ~= B n, t h e n the intersection B n M R n ( x ) is h o m e o m o r p h i c to the n - d i m e n s i o n a l ball and, consequently, t h e r e exists a r e t r a c t i o n r" B n ~ B n M R n ( x ) . Therefore, in this case, the c o m p o s i t i o n a r is the desired extension of a with d i a m e t e r of the image less t h a n e. In the r e m a i n i n g case, when S ~ - R n ( x ) C B n, we first observe t h a t the b o u n d a r y

138

4. MENGER

MANIFOLDS

O R n ( x ) of the ball n n ( x ) is contained in B n. Since k 4- 1 < n, there is a h o m o t o p y (in B n) connecting the m a p 5 with a m a p / 5 " B k+l ~ B n N R n ( x ) fixed on ~ - l ( R n ( x ) ) . Then fl does not move points of OB k+l -- S k as well. Clearly, ~ is the required extension of c~. [7

LEMMA 4.2.4. The map Pn" FRn ~ B n + l is n-invertible. PROOF. Let us consider a lower semi-continuous many-valued retraction Fn" B n+l ---* S n defined as follows:

= ( S

I1 11 > ifll ll
m is naturally isomorphic to the diagram C~,j

(K). In

kn k-re,n-re(K). particular, Ei,'j (K) ~ Ei_m, j 9 Let K1 be a subcomplex of K.

Then the diagram Ek?(K1) can be k,n

naturally embedded into the diagram Ei, j (K) in the sense that for all indices i , j there exist embeddings ~oi,j" Ei,k,n j (K1) ~ E ki,j, n (K) such that the maps f.k'.n(K1)~,~ and gik,?(gl) coincide with the restrictions of the k,n

kn

maps fi,j (K) and gi,~ ( g ) respectively 9 Observe also that for each i > 0 and j > 0, we have

E ki,j, n ( K 1 ) - (fi,~n(K))-l(Eik,?(K1)) kn

LEMMA 4.2.11. If K is a finite simplicial complex, then Ei, ~ (K) is an A N R -

compactum. PROOF. We prove our statement by induction on k. First consider the case 1,n k = 1. Represent the space E0,1 ( g ) =1 g(n) I as the union a l U a2 U . . . U ar of

144

4. MENGER MANIFOLDS

its simplexes. Then 1 (al) U - - - U f -1,1 1 (fir). El:I n1 (K) = f -1,1

Since the map f1,1 is a projection of the graph of the many-valued retraction RK(,,_~), it can easily be seen that for each i _~ r we have Sl, l ( a i ) n Uj_ m0). Observe t h a t [ a l + l I - - M l - I n t l a l + l I. Therefore g-1

1

g-1

-1

1

k+l,k+lfl (M/+I) ---- k + l , k + l f l l ( I n t [ a / + l l) U gk+l,k+lfl (Ml) --- f o l ( c l ( g ~ , ~ + l ( X n t

[ hi+ 1 [)) U g-1 k+l,k+l f 1 1 ( M l ) .

Notice also t h a t

f o l ( d ( g l-1 , k + l ( I n t lal+l I)) ~ f l l ( I al+l [) • VX(al+l)B n. In this situation it suffices to show t h a t

fol(cl(g~,lk+l(IntIal+l I)) n g-lk+l,k+lfll (Ml) is an A N R-space. Note t h a t f o 1 ( c l ( g l-1 ,k+l(Int

f o l ( C l ( g l-1 ,k+l(Int

l al+l I)) N g k-1+ l , k + l f l l ( M l ) l al+l I)) N g l-1 ,k+lfll(Ml)

--

--

--1 --1 f o l ( c l ( g 1,k+1 -1 ( I n t [ hi_t_1 [)) Pl g l , k + l f I (0 ] al+ 1 [) Consider the projection

onto fll(Olaz+~l) which, as shown above, is an ANR-space. Let

r

fll(Olal+l[) • VXal+,)B n --+ E l , k + l

be the composition of f l x idvB n and the trivialization --1 (0lal+ll) 7I" OIal+ll • VB n ---+ gl,k+l

-1 (0 ] az+l I) --1 1( I n t ] al+l ]))I~l gl,k+l of the bundle gl,k+l/cl(gl,k+ flTr = g l , k + l r Consequently there exists a m a p 4: f11(Olal+11) • VX(a,+I) Bn --* Ek+l,k+l,

Notice t h a t

146

4. MENGER MANIFOLDS

lifting b o t h lr and r Since the collection {Tr, r separates points of the compactum f~l(Olal+ll) x VX(a~+I)B n, we conclude that r is injective. On the other hand,

r

x vx(o,+,)B ~)

-

fol(r

n g l-1 , k + l f -1l ( M t ) "

( g l-1 ,k+l(Intlal+ll))

Therefore

f o l ( d (o~,k+~(x~tl"z+~l)) -1 n g-1 k+~,k+~f~-1 (M~) is an A N R-space. --1 1(MI) is an ANR-space for each I. It only suffices to note Thus gk+l,k+lfl that Ek+1,k+1 = g-1 k+1,k+l f -11 (El,k). This finishes the proof.

D

LEMMA 4.2.12. For each finite simplicial complex K and each natural number n, there exist an ANR-compactum X and two maps f : X --*1 K I and g: X --~ IK(n)l satisfying the following conditions: (a) The map f is n-invertible, polyhedrally n-soft and ( n - 1)-soft.

(b) g / f - l ( [ g ( n ) l ) - f / f - l ( [ g ( n ) [ ) . (c) gf-l(lal)c_ Io(")1. n n+k

PROOF. Let d i m K = n + k. Consider the diagram ~r

X

(K) and let

~k,n+k(K )

= ~~k,k

k n+k

k n+k

f = f l , k " " fk,k: Ek' k (g)---* E0'~r (g) = Igl, k:.+k k:.+k g = gl,k gk,k: Zk, k ( g ) ---, Ek, o ( g ) - - I g ( n ) l

for each a e g .

By L e m m a 4.2.11, X is an ANR-compactum. By the properties of the di1)agrams s i , j ' the map f is n-invertible, polyhedrally n-soft and ( n soft. These properties also imply that g f - 1 = RK . . . . RK(,+k-1), and consequently f and g coincide on the inverse image f-l([K(n)[). For the same reason,

g.f-l(lal) c_ Io(n)l. D Remark 4.2.13. All statements proved in this section so far are valid for countable locally finite simplicial complexes as well. In such cases, the resulting spaces are locally compact A N R-spaces and the resulting maps are proper. Remark 4.2.14. More careful consideration shows that the compactum X from L e m m a 4.2.12 is a finite polyhedron. Respectively, if K is countable and locally finite then the polyhedron X is also countable and locally finite. We are ready to prove the main result of this section. THEOREM 4.2.15. For each natural number n, there exists a map fn: #n I ~ of the universal n-dimensional Menger compactum onto the Hilbert cube satisfying the following properties: (i) The map fn is n-invertible, polyhedrally n-soft and ( n - 1)-soft.

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION

147

(ii) For each at most n-dimensional compactum Y, each map f" Y --+ #n and each open cover lg E cov(#n), there is an embedding g" Y --~ #n that is lg-close to f and such that f ng = f n f .

PROOF. We construct the map fn as the limit projection of an quence 8 - {M{, c~ +1} consisting of the Hilbert cube manifolds (in M1 -- Io2 ). The construction is carried out by induction. The limit verse sequence is contained in the product I~' • I~' • --. • I~ • -.the metric

~(~, y) = r ~

inverse separticular, of this inwhich has

d~(x~, y~) 2(~+1)

where di is a (bounded by 1) metric on the Hilbert cube I.~ = ( y l , . . . , y i , . . . ) and xi, yi E I~ for each i. Let ai E Io2 for each i. Then each of the products I~' x --- x I m can be naturally identified with the subspace oo I~' x - - . x I m x {am+l} x - - - x {ai} x - - - of the product l I i = l i ~ . We construct an Io2-manifold as a subspace of the product I~ x --- x I~n and consider the O2 restriction (to Mi+l) of the projection ~ + 1 . 1-Ij=l I~ x Ii+1 ~ 1-I~=11~ as a projection a~+l. Mi+l ---+ Mi of the spectrum 8. In this situation, as can easily be seen, the limit space X = lim 8 is naturally embedded into Io2 - 1--Ii=l~176 i~'. Let M1 = I~ and suppose that the Io2-manifold Mj C YIi=lJI~ c Io2 has already been constructed. By Theorem 2.3.28, the Io2-manifold Mj is homeomorphic to the product of the Hilbert cube jIo2 and a finite polyhedron Kj. Obviously, we may assume that the composition of the projection wj" Kj x j I W ---+ Kj with a certain section sj" Kj" K j x j Io2 is a 2-J-move, i.e. for each point x E K j • Io2, we have d(x, sjwj(x)) < 2 - j . We also assume t h a t Kj is given together with a triangulation such t h a t diamsj(a) < 2 - j for each simplex a of this triangulation. By L e m m a 4.2.12, there exist an A N Rc o m p a c t u m X j + I and maps f j + l " X j + I ~ K j and gj+!" Xj+I ---+ K~ n) such t h a t f j + l is n-invertible, polyhedrally n-soft, ( n - 1)-soft and, in addition, -1 (n) for each simplex a of the triangulation given on Kj. Let Mj+I : Xj+I • Io2. By Theorem 2.3.21, Mj+I is a Io2-manifold. Next, represent the cube j+lIo2 as the product jIo2 x j + 1 I~' of two Hilbert cubes, and denote by yj+l" X j + I • Io2 • I~ --+ Xj+I x jIo2 the product of the identity map idx~+l and the projection ~ " jIo2 • I~ --+j Io2. Define the map ~+~ j 9 Mj+I ~ Mj by letting ~j+l

= (f3+~ • i d ~ ) ~ j + l .

In this situation we have the following diagram:

148

4. MENGER MANIFOLDS

Mj+I

= Xj+l

~ j + l -- i d x

x j I ~~ x j + 1 I~'

7r31

Xj+I xj I ~

Xj+I

f j+l x id

fj+l

,~ K j x j I ~~ = M j

,- K j

Let A" M j + I ~ IS+1 be an embedding. T h e n the desired e m b e d d i n g of the /"a-manifold M j + I into the p r o d u c t I i a x - - - x I~a+l can be defined as the diagonal p r o d u c t of the m a p ~ j + 1 and the e m b e d d i n g A. Let a jj + l ._ 7rJ+lj / M j + I , and j+l observe t h a t this map a j " M j + I ~ M j is n-invertible, polyhedrally n-soft and ( n - 1)-soft (because b o t h f j + l and 7r~ have the c o r r e s p o n d i n g properties). T h u s the c o n s t r u c t i o n of s p e c t r u m 8 = {Mi, a~ +1} is complete. Let ai" X = lira,5' --, M i d e n o t e the i-th limit projection of this s p e c t r u m . As the desired m a p fn" #n __~ i ~ we take the first limit projection a l " X ---+ M1 = I W 9 Of course, we still have to show t h a t X ~ #n. Let us investigate the properties of the c o m p a c t u m X and the m a p fn. Since each of the short projections a~ +1 of the s p e c t r u m S is n-invertible, p o l y h e d r a l l y n-soft and ( n - 1)-soft, we easily see t h a t fn (as well as all other limit projections of the s p e c t r u m S, see L e m m a 6.2.6) has the same properties. This proves one p a r t of the theorem. Let us now show t h a t dim X = n. Since M1 -- I w , M1 contains a topological copy of the n - d i m e n s i o n a l cube I n. T h e n-invertibility of fn then g u a r a n t e e s t h a t X also contains a copy of I n. Therefore dim X _> n. In order to show t h a t dim X _< n we proceed as follows. Obviously, all we need is an existence, for each i, of a ~ - m o v e of X into an n - d i m e n s i o n a l polyhedron. Such a move is i n d i c a t e d below" siwi(gi+l x idii~)~ii+lai+ 1 9 X ---+ K} n). Consequently, dim X - n . Next we show t h a t fn has the second p r o p e r t y formulated in the theorem. Let Y be an at most n - d i m e n s i o n a l c o m p a c t u m , f" Y -4 X be a m a p and

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION

149

bl E c o v ( X ) . Choose an index i and an open cover 12 E cov(Mi) such t h a t a~-l(Y) refines/~/. It follows from the construction of the m a p a~ +1 t h a t a~ +1 can represented as the composition 7i+lr/i+l, where ~i+l is a trivial bundle with fiber the Hilbert cube. Consequently, there exists an e m b e d d i n g hi+l" Y ~ M i + l such t h a t a~+lhi+l -- a i f . Since d i m Y < n, and since the limit projection a i + l " X ---+ M i + l in n-invertible, we can find a map h" Y ---+ X such t h a t a i + l h -- hi+l. It is not hard to see t h a t h is an embedding. Moreover, h is U-close to f and f n h = f n f . Finally, let us show t h a t X is homeomorphic to #n. By the property of f n established above, X satisfies the condition from T h e o r e m 4.1.19., i.e. every map of an at most n-dimensional c o m p a c t u m into X can be arbitrarily closely a p p r o x i m a t e d by embeddings. T h e equality dim X -- n was also established above. Therefore, by T h e o r e m 4.1.19, it suffices to show t h a t X E L C n - I N C n - 1 . First we show t h a t X E L C n-1. Let x be an a r b i t r a r y point of X and U be a neighborhood of x. Take an index i and a n e i g h b o r h o o d Ui of a i ( x ) in Mi such t h a t a~-l(Ui) C U. Since, by our construction, Mi is an A N R - c o m p a c t u m (even an I ~ manifold), M i E L C n-1. Consequently, there exists a neighborhood Vi of a i ( x ) in Mi such t h a t Vi C Ui and the following condition is satisfied: 9 for each k < n - 1 and each map ~i" S k -+ V~, there is a m a p r B k+l --+ U~ such t h a t ~i = r k. Let VaT, l ( v i ) . Clearly, V is a neighborhood of x contained in U. Take any map ~o" S k ---+ V, k _< n - 1. By the choice of the n e i g h b o r h o o d Vi, there is a map r B k+l --+ Ui such t h a t ai~o = r k. Since (Sk, B k+l) is a polyhedral pair and the map a i is polyhedrally n-soft (recall t h a t k < n - 1), we conclude t h a t there is a map r B k+l --+ X such t h a t the following d i a g r a m commutes:

X

Sk t

.

~Y

.Bk+ 1

In other words, r k = ~ and a i r = r This obviously implies t h a t k+l) C_C_U. Thus, X E L C n-1. A similar (but simpler) a r g u m e n t shows t h a t X C C n-1. Therefore, X ~ #n. T h e proof is finished. [2] r

In order to o b t a i n other i m p o r t a n t properties of the m a p fn we need some preliminary statements. T h e proof of the following one is, in fact, contained in

150

4. MENGER MANIFOLDS

the proof of L e m m a 4.2.6. LEMMA 4.2.16. For each e > O, there is a 5 > 0 such that for any map ~o: S n-1 --~ F R , , with diam(im(~o)) < 5, and any 5-homotopy H : S n-1 x[0, 1]---, B n+l, with Ho = pn~o, there exists a h o m o t o p y G: S n-1 x [0, 1] ~ FR, such that Go = ~o, pnG = H and d i a m G ( S n-1 x [0, 1]) < e. LEMMA 4.2.17. Let Y be an L C n - l - c o m p a c t u m , K be a finite simplicial complex and ~o: Y --~ Ig(n+l)l be a map. Then the fibered product (pullback) X of the spaces FRK(,~) = 1-'n+l and Y with respect to the maps Pn+l -- PRK(n ) : Pn+l --+

IK(r'+l)l and ~o: Y --, Ig(n+l)l is an L C n - l - c o m p a c t u m . i i c~ l:~ n PROOF. Let c~ (~,~=1~.i ) be the one-point compactification of a discrete collection of n-dimensional disks B.n and let x be the compactifying point. Let S~'-1 denote the boundary of the disk B~'. Assume the contrary. Then there exists a map r (U~IS~ '-1) ---, X such that for each i and for any extension of r -1 (to the disk B~) the diameter of its image is more than some positive number a. Let a be an (n + 1)-dimensional simplex in K (n+l) containing the point y = pn+l~O~r Here ~o~:X --+ Fr,+l denotes the canonical projection (parallel to ~o) of the fibered product X. The map q: X ~ Y has similar meaning (see the following diagram):

a (u~,S~ '-I)

r

, X

,- Fn+l

Pn+l

y

r

,._

IK(n+l) I

We identify the simplex a with the unit (n + 1)-dimensional ball B n+l. Also, we identify the restriction of the many-valued retraction R g ( n ) : l K ( n + l ) l --~ Ig(n)l to lal with the many-valued retraction R n : B n+l ~ OB n+l (see the beginning of this section). We consider two cases. C a s e 1. Assume first that y f~ OBg +1, and take a neighborhood V of y in IK(n+l)l disjoint from OB~ +1. The restriction of Pn+l to this neighborhood is n-soft by construction. But then the restriction of q to q-l(~o-l(Y)) is also nsoft. It only remains to observe that the latter fact, coupled with the condition Y E L C n - l , contradicts the assumption made in the beginning. C a s e 2. Now assume that y E OB~ +1. Since Y c L C n-1 we can conclude that there is an extension r (U~__lBn) ~ Y of the composition qr Take

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION

151

an e > 0 such that for each set F C X, the inequalities diam((t'(F)) < e and diam(q(F)) < e imply the inequality d i a m F < a. Next, choose 5 as in Lemma 4.2.16. Let b~ be the center of the ball B~. Connect the points 4(b~) and qr by paths ~i" [0, 1] ~ Y so that limi--.oo diam~i([O, 1]) = 0. Then there is a number k such that

max{diam(rlk([O, 1]) U r

diam((trlk([O , 1]) U (t(~(B~))} < 5.

Let Ht" S~ -1 x [0,1] --~ Y be a homotopy which c o n n e c t s r -1 with the constant map to the point 4(bk) inside of the image 4(B~), and then sends this constant map along the path ~k to another constant map to the point qr By L e m m a 4.2.16, the homotopy (tilt can be lifted to a homotopy Gt . S kn - 1 x [0, 11 --. Fn+l so that Go = (t'r G I ( S ~ -1) C_ Pn~I(Y) and d i a m ( G t ( S ~ -1 x [0, 1]) < e. Since Rn+I(Y) -1 "~ S n and the convex hall of G I ( S ~ -1) in the sphere S n has diameter less than e, we may conclude that, also shrinking c o n v ( G l ( S ~ -1) to a point, the homotopy Gt can be extended to an e-homotopy Lt" S~ -1 x [0, 1] --* Fn+l. Extending the homotopy Ht by adding the identity map, we obtain a homotopy Nt" S~ -1 • [0, 1] ~ Y such that (tiNt -- Pn+lLt. But then (recall that X is the fibered product in the above indicated diagram) we get a well-defined a-homotopy Mr" S~ -1 x [0,1] ~ X such that M0 = r -1 and M I ( S ~ -1) = r This contradiction finishes the proof. V1 LEMMA 4.2.18. Let K be an m-dimensional finite simplicial complex and f" X --+ [K[ be a map, constructed in Lemma 4.2.12. Then for each L C n - l - c o m p a c t u m Y , m > n, and for each map (t" Y ~ [K[, the fibered product Z of X and Y with respect to f and (t is an L C n - l - c o m p a c t u m . PROOF. Denote by X1 the inverse limit of the following diagram Fm

Fm-1

IKI

IK(m-1) [

Fn+2

...

[K(n+2) [

IK(n+l)[

and assume that p~" X1 ~ IK(i)l and q~" X1 ~ F~ denote the corresponding projections, n-softness of the projection p~ implies t h a t the fibered product y I of X1 and Y with respect to Pm I and (t is an L C n - l - c o m p a c t u m . Let (t~ Y~ ~ X1 be the projection of this fibered product, parallel to (t. It is easy to see that X is also the fibered product of spaces X1 and Fn+l with respect to maps ! . X l '"+ IK ( n + l ) [ and pn+l . Fn+l ---+ [ K ( n + l ) [. Then Z itself is the fibered Pn+l product of spaces Fn+l and Y~ with respect to Pn+l: Fn+l --~ IK(n+I)I and

152

Pn+l

4. MENGER MANIFOLDS

~ . y~

"-+ Ig

lemma is proved.

(n+l)

I" By L e m m a 4.2.17, Z is an L C n - l - c o m p a c t u m .

The

F'I

The following statement expresses one of the most important properties of the map fn: I~n --+ I~ constructed above. THEOREM 4.2.19. For each LCn-l-compactum Y contained in the Hilbert cube I ~, the inverse image f ~ l ( y ) is a #n-manifold. Additionally, i f Y E C n-l, then f ~ - l ( y ) is homeomorphic to #n. PROOF. We use the same notations as in the proof of Theorem 4.2.15. Recall t h a t the map f n : # n --~ I • was constructed as the limit projection of the inverse sequence S = {Mi, c~ +1} consisting of I ~ -manifolds (M1 -- I W) and n-invertible, polyhedrally n-soft and ( n - 1)-soft short projections. If Y is an LCn-l-compactum contained in I 0~ , then the inverse image fnZ(Y) is the limit space of the induced inverse sequence S ' - {Yi, r~+l}, where Y1 = Y, Yi+l = (c~+l)-l(Yi) and r~+1 = ~+l/Yi+l. Since the restriction of an ninvertible, polyhedrally n-soft and ( n - 1)-soft map onto the inverse image still has all these properties, we see that all short projections of the spectrum ,~' = {Yi, r~ +1} are n-invertible, polyhedrally n-soft and (n - 1)-soft. While proving Theorem 4.2.15 we have already seen t h a t the inverse limit of an inverse sequence consisting of L C n - l - c o m p a c t a and polyhedrally n-soft short projections is an LCn-l-compactum. Therefore it suffices to show that Yi E LC n-1 for each i. We prove this fact by induction. By assumption, Y1 E LC n-1. Assume t h a t Yi E LC n-1. The projection a~ +1 can be represented as the composition ")'i+1~i+1, where ~i+1 is a trivial bundle with fiber the Hilbert cube and "yi+l is homeomorphic to the map f~+l x i d i ~ . Here f~+l: X~+I ~ Ki is the map from Lemma 4.2.12 (see the diagram in the proof of Lemma 4.2.12). Consequently, by Lemma 4.2.18, ~i+l(Yi+l) is an LCn-l-compactum. But then Yi+l ~ ~i+l(Yi+l) x I ~ is also an LCn-l-compactum. Thus f ~ l ( y ) c LC n-1. Polyhedral n-softness of the short projections guarantees that if, in addition, Y E C n - l , then f n l ( Y ) e C n-1. Obviously, dim f ~ l ( y ) < dim #n = n. Theorem 4.2.15 shows that dim f n l ( Y ) ~n. Theorem 4.2.15 also guarantees that any map of an at most n-dimensional c o m p a c t u m into f~-I ( y ) can be arbitrarily closely approximated by embeddings. Therefore, by Theorem 4.1.19, f ~ Z ( y ) is a #n-manifold (and is homeomorphic to #n i f Y E Cn-1). [:]

Remark 4.2.20. In fact a stronger result can be proved: for any LC n-1c o m p a c t u m Y and for any map ~: Y ~ I W (not only for embeddings, as in Theorem 4.2.19) the fibered product of Y and #n with respect to ~ and fn is an L C n - 1_co m pact um. Summarizing the results proved above, we have.

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153

THEOREM 4.2.21. There are maps fn: #n ~ i W and gn: ttn ---* #n satisfying the following properties: (i) The maps fn and gn are n-invertible, ( n - 1 ) - s o f t and polyhedraUy n-soft. (ii) All the fibers of the maps fn and gn are homeomorphic to #n. (iii) The inverse images of L C n - l - c o m p a c t a under the maps fn and gn are #n-manifolds. (iv) The maps fn and gn both satisfy the parametric version of D D n P , that is, any two maps ~, ~ : I n ~ #n can be arbitrarily closely approximated by maps a I, ~l: I n ~ #n such that f n a l = f n ~ , f n ~ I = f n ~ , g n d = gnu, g n f l ' - - gnZ and i m ( a ' ) A im(Z') = O. Using the map fn as a guide, additional considerations allow us to obtain the following result. THEOREM 4.2.22. Any metrizable A ( N ) E ( n + 1)-compactum is an U V n - I image of: (i) An (n + 1)-dimensional A ( N ) E - c o m p a c t u m . (ii) The Hilbert cube I ~ (an I W -manifold, respectively). (iii) The (2n + 1)-dimensional cell ( a (2n + 1)-dimensional topological manifold, respectively). A simple comparison of the major ingredients of the Hilbert cube manifold theory (see Chapter 2) with the corresponding results of Menger manifold theory presented so far, shows that from a certain point of view the n-dimensional analog of the Hilbert cube Q should be considered to be, not the usual ndimensional cube I n, but the n-dimensional universal Menger compactum #n (moreover, the Hilbert cube itself may be viewed as the "infinite-dimensional Menger compactum"). In addition, one can observe a fairly deep analogy between the theories of #n-manifolds and Q-manifolds themselves. On the other hand, at first glance it is not clear what is the analog of the operation of "taking the product by Q" in #n-manifold theory - the operation which is involved in the formulations of triangulation (Theorem 2.3.28) and stability (Theorem 2.3.10) theorems for Q-manifolds. A decisive step in finding a "full" analog of this operation in #n-manifold theory is based on Theorems 4.2.15 and 4.2.19. First observe that taking the product X • Q of a space X and the Hilbert cube Q may be interpreted as taking the inverse image lr~-l(X) of a space X c Q, where r l " Q • Q ~ Q denotes the natural projection onto the first coordinate. It turns out that the map gn : f n / . f n l ( # n ) " #n ___+ttn in Theorem 4.2.15 may be thought of as the analog of the projection ~1 in the theory of #'~-manifolds. If this is agreed, everything then falls in place. The following statement is a triangulation theorem in #n-manifold theory. THEOREM 4.2.23. For any #n-manifold M , there is an n-dimensional polyhedron P such that for any embedding of P into #n the inverse image g~-l(p) is homeomorphic to M .

154

4. MENGER MANIFOLDS

PROOF. We consider the compact case. The locally compact case can be proved similarly. Take an n-dimensional finite polyhedron P and a map ~" P --~ M which induces isomorphism of t h e / - d i m e n s i o n a l homotopy groups for each i _< n - 1. E m b e d P into #n. It is easy to see that the composition

~gn/g~l(P) 9gnl(P)~

P ~ M

also induces isomorphism of t h e / - d i m e n s i o n a l homotopy groups for each i < n 1. Observe also t h a t the inverse image g ~ l ( p ) is a #n-manifold (we use Theorem 4.2.19). Therefore, by Theorem 4.1.21, M and g ~ l ( p ) are homeomorphic. [7 Here is the promised stability theorem for #"-manifolds. THEOREM 4.2.24. For any #n-manifold M in tt n, the inverse image g~-l(M) is homeomorphic to M . PROOF. If M is compact, then, by Theorem 4.2.19, the inverse image g n l ( M ) is a tin-manifold. Since the restriction g n / g n l ( M ) " g n l ( M ) ~ M is polyhedrally n-soft, we see t h a t it induces isomorphism of t h e / - d i m e n s i o n a l homotopy groups for each i < n - 1. By Theorem 4.1.21, g ~ l ( M ) is homeomorphic to M. The proof of the non-compact case is similar. Kl We conclude this section with the following statement which shows, in particular, that the map fn cannot be made n-soft. THEOREM 4.2.25. There is no n-soft map of an n-dimensional compactum onto a higher dimensional cube (the Hilbert cube in particular). PROOF. If n -- 0, the validity of our statement is obvious: the open image of a zero-dimensional c o m p a c t u m is zero-dimensional. Thus we may assume that n _> 1. If such a map g" Z --* I m existed, then Z must be connected (notice that 1-soft maps are monotone). Recall that cell-like maps cannot raise cohomological dimension (see Remark 3.1.11) and that the cohomological dimension of any cube coincides with its Lebesgue dimension: Therefore, g cannot be cell-like. It is clear now t h a t it suffices to prove the following claim. C l a i m . A n y non-constant n-soft map of a connected n-dimensional compactum is cell-like. Proof of Claim. Let f" X ---, Y be an n-soft map, where X is connected, d i m X = n and I Y I> 1. Since the fiber f - l ( y ) 6 L C n - I N C n - 1 for each y e Y we see t h a t g k ( f - l ( y ) ) = [-Ik(fkl(y)) = 0 for all k < n. Here g k denotes the singular homology with respect to the group of integers a n d / ~ k the (;'ech-homology. Then it follows that I ~ k ( f - l ( y ) ) = 0 for all k < n. Let us show that [-In(f-~(y)) = 0. If so, using the standard criterion that a finite-dimensional Peano continuum with trivial cohomology has trivial shape, we obtain the desired conclusion. Assume the contrary. Namely, t h a t / ~ n ( f - l ( y ) ) __/=0 for some y E Y. Take a map ~" f - l ( y ) ~ g ( Z , n ) (the Eilenberg-Maclane complex) which is not

4.2. n-SOFT MAPPINGS OF COMPACTA, RAISING DIMENSION

155

homotopic to a constant map. Consider a point x E Y different from y. W i t h o u t loss of generality we may assume t h a t I f - l ( x ) I 1 (otherwise shrink the fiber f - l ( x ) into a point). Since dimX - n, there is an extension ~: Z ~ K(Z,n). Next dengte by Y0 the set of all those points z E Y for which the restriction ~ / f - l ( z ) is homotopic to a constant map. Obviously, Yo is not empty. Indeed, x E Yo. Let us show t h a t Y0 is an open set in Y. Take a point z E Yo, and let h: f - l ( z ) x [0,1] ~ g ( Z , n ) be a h o m o t o p y connecting ~ / f - l ( z ) with a constant map to some point c E K ( Z , n), i.e.

h / ( f - l ( z ) x {0}) = Cp/f-l(z) and h / ( f - l ( z ) x {1}) = c. Consider the closed subset A -- (X x {0, 1} U ( f - l ( z ) x [0, 1]) of the product X x [0, 1] and define the m a p hi" A ~ K ( Z , n) by letting

hl(X

X

{ 1 } ) = c, h l / ( X x { 0 } ) = ~ a n d

hl/(f-l(z)

x

[0, 1 ] ) = h.

Since the CW-complex K (Z, n) is an absolute extensor with respect to the class of compact spaces, we can extend hi to a map h" U ---, K(Z, n), where U is a neighborhood of A in the p r o d u c t X x [0, 1]. T h e n U contains an open set of the form G x [0, 1], where G is a neighborhood of the point z in Y. Clearly G is contained in Y0 (see the definition of Y0). Since z was an arbitrarily chosen point of Y0, we conclude t h a t Y0 is open in Y. Next observe t h a t the complement Y - Y0 is also a n o n - e m p t y set, since it contains y. Let us now show t h a t this complement is also open in Y. Assume the contrary. T h e n there exists a sequence {zk} of points of Y0 such t h a t z = limzk E Y - ]I0. We construct a sequence {gk" f - l ( z ) --+ f--l(zk)} of maps, which converges to the identity map idy-~(z) in the space C ( f - l ( z ) , X ) . Since the space F -- f - l ( z ) x ({zk" k e N } U { z } ) is at most n-dimensional, n-softness of the m a p f guarantees the existence of a map r F --. X such t h a t

r

x

{z}) = idy-l(z)

and r

x

{zk}) C f - l ( z k )

for each k E g .

Let gk -- e l ( f - l ( z ) X {Zk}). T h u s we have a sequence {~gk}of maps each of which is homotopic to a constant map and which converges to the m a p ~ / f - 1 (z). This contradicts the fact t h a t sufficiently close maps into A N R-space are homotopic. Consequently, Y - Y0 is open in Y. Connectedness of Y shows t h a t this is impossible F-1

Historical and bibliographical notes 4.2. T h e o r e m 4.2.1 and Proposition 4.2.2 (a weaker version of it - w i t h o u t stating an n-invertibility) were proved in [33]. T h e construction of n-invertible, ( n - 1)-soft maps presented in this Section, as well as T h e o r e m s 4.2.23 and 4.2.24 in the compact case, are taken from [127](see [90] for the non-compact case). This result completes a circle of works of various

156

4. MENGER MANIFOLDS

authors concerning the existence of dimension raising maps. Historically, the first example of an open map (of the one-dimensional compactum onto the twodimensional "Pontryagin surface") of this sort was constructed by Kolmogorov in [196] in 1937. Further examples, with some additional properties, have been constructed in [11], [12], [15], [319], [320], [189], [190], [191], [200], [211], [245], [248], [304] etc. Theorems 4.2.22 and 4.2.25 are taken from [129] and [125]. Theorem 4.2.22 extends an earlier result from [71].

4.3. n-soft m a p p i n g s of P o l i s h spaces~ r a i s i n g d i m e n s i o n It has already been remarked in Section 4.2 that the maps fn and gn, constructed in that section, cannot be made n-soft. Also, both of them fail to satisfy the property of preservation of Z-sets in the inverse direction. In this section we construct a map with the last property which is "almost" n-soft. We begin with the following technical statement. LEMMA 4.3.1. If f" X ---. Y is an n-soft map, then the inverse image of each Zn-set in Y is a Zn-set in X . For each simplex a, denote the first and second barycentric subdivisions of a by fla and ~2a respectively. Ma denotes the closed star of the barycenter va of a in the triangulation ~2a" Ma = St(Va,~2a). We put No = Int(Ma). Finally let ra "(I a I - { v a } ) ~ ] Oa ]denote the canonical deformation retraction. LEMMA 4.3.2. For each n > 0 and each countable locally-finite simplicial complex K , there exist a countable locally finite simplicial complex B ~ and proper simplicial maps f~" IBm( I--*1 g I and g~" IBm( I---*1 g ( n + l ) I satisfying the following conditions: (i) f ~ is an (n + 1)-invertible u y n - m a p . (ii) If a is a simplex of K, then g ~ ( ( f ~ : ) - l ( ] a ])) C_I a (n+l) ]. (iii) f ~ c / ( f ~ ) - l ( [ K (n+l) I ) = g ~ / ( f ~ : ) - l ( ] K(n+l)I). (iv) There exists a subspace Ang of ] B ~ l such that the restriction f ~ / A ~ " A ~ --*1K[ is an (n + 1)-soft map and the complement l B ~ l - A ~ is a a Z-set in ] B ~ ]. (v) If Z is a Zn+l-set in I g I, then ( f ~ c ) - l ( z ) is a Zn+l-set in ] B ~ ]. PROOF. Let us consider an arbitrary simplex a and define two compact-valued retractions ~ , r l a I---*1 a(n+l) I. Definitions are given by induction on the (n + 1 +/)-dimensional skeleta of a. If x el a (n+l) I, then we put ~ ( x ) = x = r If T is an (n + 2)-dimensional face of a, then

n(x) __ { 107"1, ~~

rr(x),

if x e Nr, i f x e ITI--NT

4.3. n-SOFT MAPPINGS OF POLISH SPACES, RAISING DIMENSION

157

and

[OT[, r

--

r~-(x),

i f x 9 M~, if x 9 t~-1- M~

If T is an (n + 3)-dimensional face of a, then

~(~)

=

f I(0~)(-+1) I, t ~2(~(~)),

if x 9 Nr, if x 9 [ v [ - - N r

{ I(0T)(~+I)[, Can(rr(x)),

if x 9 Mr, if x 9 I~1- Mr.

and Can(x) =

Continuing this process, we obtain the desired retractions of [a] onto [a(n+l)[. Finally, if K is an arbitrary locally-finite simplicial complex, then the retractions ~ : , ~b~-IK} ~ ]K(n+l) 1 are defined as the unions of ~ and ~pa n, a 9 g . Now denote by B ~ the standard triangulation (induced by K) of the polyhedron {(x, y) 9 [g[ x ]g(n+l)[ 9 r ~ y}. We put also f~: = 7rl/[B~;[ and g~: = ~2/[B~I, where lr1" IKI • [K(n+l)l--* IKI and 7r2" IKI • IK("+~)I--' IK(n+l)[ denote the natural projections. Straightforward verification shows that: (a) the compact-valued retraction ~ : " [K[ ~ [K("+I)[ is lower semi-continuous. (b) the compact-valued retraction r IKI ---, IK(n+I)I is upper semi-continuous. (c) ~ - ( x ) C_ r for each point x 9 IKI. (d) the collection {r : x 9 IKI} is connected and uniformly locally connected in all dimensions less than n + 1 (we consider the standard metric

on

IKI).

Therefore, properties (i)-(iii) of the lemma are satisfied. Let us verify condition (iv). First of all, consider the subspace A~r = {(x,y) e ] K I • tK(n+l)l" y 9 qo~:(x)}

of IB~I and note that the complement I B ~ I - A~: is an Fa-subset of IB~I. By conditions (a), (c) and Theorem 2.1.15, the restriction f ~ / A ~ is an (n + 1)-soft map. Let us show that the complement IB~:I- A~. is a aZ-set in IB~I. Clearly it is sufficient to show that the last fact is true for the (n + 2)-dimensional simplex a. It follows from the construction that IB~:I- A~c = OMo • Io(-+a)l- T, where Z = {(~, y) e Iol • Io(~+1)1" ~ ( ~ ) = y}. Since OMa is a Z-set in Ma, we can conclude that OMo x [a(n+l)[ is a Z-set in M a x [a(n+l)[. Consequently, [B'~[-A'~ is a aZ-set in M a x [a(n+l)[. Consider now an open subspace U = [ B ~ [ - T of the polyhedron [B~[ Evidently, [ B ~ [ - A n C U C Mo x [a(n+l)[ Then the complement tBant- Aan is a aZ-set in U. Finally, for the same reason, we can conclude that [ B ~ ] - A~ is a aZ-set in [Ban[.

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4. MENGER MANIFOLDS

The last condition is an easy consequence of condition (iv). Indeed, let Z be a Zn+~-set in [g[. Then, by (iv) and Lemma 4.3.1, the set

C 1 -=

{h e C ( I n + I , A ~ ) " ira(h) n ((S~7)-l(z) n A~:) -- 0}

is a dense G~-subset of C ( I n + I , A ~ ) . At the same time, by (iv), the set

6 2 - - {h e C ( I nq-1, [ B~z [)" ira(h) C Ang}

is also a dense G6-subset of C ( I n+l, IB~[). A Baire category argument finishes the proof, fl

THEOREM 4.3.3. Let n >_ 0 and let X be the limit space of an inverse sequence ~qx ---- {[ Xi [, p~+l} all spaces of which are locally finite polyhedra and all bonding maps of which are proper, simplicial and (n-b 1)-soft. Then there exists an (n-b 1)invertible proper UVn-surjection f ~ " M x ---* X of some #n+l-manifold M x onto X satisfying the following conditions: (i) For each (n -b 1)-dimensional locally compact space Z, closed subset Zo of Z, open cover Lt of M x , and proper map h" Z ~ M x , such that h/Zo is a fibered Z-embedding (with respect to f ~ ) , there exists a fibered Z-embedding (with respect to f ~ ) g" Z ---. M x which is U-close to h and such that f ~ g - f ~ h and g / Z o - - h/Zo. (ii) There exists a subspace A x of M x such that the restriction f ~ / A z " A x ---* X is an ( n + 1)-soft map and the complement M x - A x is a a Z-set in M x . (iii) If Z is a Zn+l-set in X , then ( f ~ ) - l ( Z ) is a Z-set in M x .

PROOF. For simplicity we consider only the compact case. The general case can be handled similarly. A standard Baire category argument reduces the proof to the case when Z0 - 0. After making these assumptions, we proceed as follows. We construct another polyhedral inverse sequences SM -- {[Ki[ , qi _i+1~~, all bonding maps of which are simplicial UVn-surjections. Further, we shall construct a family of ( n + 1)-invertible simplicial UYn-surjections fi: [Ki[ ---* [Xi[ which forms a strictly commutative (n+l)-invertible UYn-morphism {f~: [K~] [Xi[}: SM ~ 8 x . This means that if we fix an index i and consider the naturally arising diagram

4.3. n-SOFT MAPPINGS OF POLISH SPACES, RAISING DIMENSION

159

IKi+ll

i+1

r~

iKil

p~+l

........

.fi

~ [Xi[

~ , i + l A .e then the characteristic map li = ~i ~ J i + l of the diagram is an ( n + 1)-invertible UVn-surjection (here ITil denotes the fibered product of IKi{ and IXi+ll with _i+1 respect to maps fi and ~'i ; si and ri denote the corresponding projections of this fibered product). Then we shall obtain the space M x as the limit of t.~M and the map f ~ as the limit map of the morphism {f i}. Let K1 = Bnxl and f l = f ~ l " IKll --~ IXll (see L e m m a 4.3.2). We can suppose, of course, that mesh(Kx) < 1 (if not, then we consider a sufficiently fine subdivision of K1 and denote it again by K1). Suppose now that we have already 1 and simplicial constructed finite simplicial complexes Km with m e s h ( K m ) < ~-~ n UV -surjections fm" [Kml ~ IXml and qm-lm . IKml __~ IKm_ll, m _< i, in such a way that the characteristic maps of all the naturally arising rectangular diagrams are (n + 1)-invertible UVn-surjections. In order to perform the inductive step, consider the fibered product ITil of IKil and IX~+ll with respect to the maps f~ and p~+l and denote by ri" ITil--* Igil and si" ITi[--~ IXi+ll the corresponding projections. Now consider the map f ~ , ' l B L,I n ~ ILil where Li denotes the natural triangulation of the polyhedron ITil x i2n+3 (we again use L e m m a 4.3.2). 1 Let K i + l -- BnL~, assuming at the same time t h a t mesh(K{+l) < 2~--Tf" Also _i+l define f~+l = sili and qi = ril +i, where li = ~lf~, and ~1" ITil x i2n+3 __~ iTil denotes the projection onto the first coordinate. One can easily verify that all our requirements are satisfied and consequently the inductive step is complete. As already remarked, we let M x = limSM and f ~ = lim{fi}. Since all mentioned rectangular diagrams strictly commute (because, by the construction, their characteristic m a p s - li's - are surjective) and all fi's are UVn-surjections, we can conclude that their limit map f } " M x --~ X is also a VVn-surjection. The standard argument (see C h a p t e r 6) shows that (n + 1)invertibility of all fi's and a l l / i ' s implies (n + 1)-invertibility of f ~ . Let us now verify condition (i). Fix an index m and an open cover L/m of IKml such that qml(Llm) refines b/(recall that qi: M x --~ Igil and pi: X ~ IX~I denote the limit projections of the spectra SM and S x respectively). We are going to

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4. MENGER MANIFOLDS

construct maps gi" Z ~ [Ki[ in such a way t h a t the following conditions are satisfied: (a) If i > m, then gi is an embedding. (b) q i~+1g i + l - - g i .

(c) f~g~ = p ~ / ] h . (d) If i > m and c~i" i n + l __~ IK~I is an arbitrary map, then there exists a m a p 13i . I n+l ~ IKil such t h a t f i ~ __. f i a i , q i - l ~ i ~ q ii_ l a i and im(Z~) n i~(g~) = O. Let gi = qih for each i _< m. We now indicate how the map gi+l can be constructed. First of all fix an e m b e d d i n g u" Z ~ I2n+3 (we use the inequality d i m Z _< n + 1). Now consider the diagonal p r o d u c t g m A p m + l f ~ h , which maps Z into ITm] 9 T h e n the diagonal product (gin • P m + l f xnh ) A u will be an embedding of Z into the product ILml = ITml • I2n+3. By L e m m a 4.3.2, the map f~,," [Kin+l[ ~ [Lml is (n + 1)-invertible. Consequently, there exists an embedding gm+l" Z ---* I g m + l l such t h a t f~,.,,gm+l = ( g m A P m + l f ~ h ) A u . A simple verification shows t h a t conditions (b) and (c) are satisfied. Now consider any m a p a m + l " i n + l ~ IKm+ll" Since u is an e m b e d d i n g and dim Z _ n + 1, there is a point a E i2n+3 such t h a t a qf ira(u). Let the same letter a denote the constant map which sends the whole cube I n+l to the point a. Consider the m a p (lmo~m+l/ka) . I n+l ----* ILml and observe t h a t the image of I n+l under this m a p does not intersect the image of Z under the composition f L m g m + l . Consider now any lifting ~m+l" I n+l ~ IKm+l[ of the product (lmoLm+lAa). Again, straightforward verification shows t h a t condition (d) is also satisfied. Therefore, continuing this process we obtain the maps g~ for each i. By (b), the diagonal p r o d u c t g of all g~'s maps Z into M x . By (a), g is an embedding. By (c), the desired equality f ~ g - f ~ h also holds. T h e choice of an index m and the equalities qmg = gm "-- qmh (which are true by our construction) show t h a t g and h are/g-close. Finally, let us show t h a t ira(g) is a fibered Z-set with respect to the map f ~ . Fix an open cover l; of M x and any map c~" I n+l --+ M x . Clearly we can assume t h a t there exist an index j _> m and an open cover l)j of ]Kj[ such t h a t q~-l(1)j) refines "g. As above we shall inductively construct maps ~i" I n+l ---* [Ki[ in such a way t h a t the following conditions are satisfied" - i + l ~ i + 1 = ~i. (e) qi (f) f ij3i = pif~ca.

(g) im ( ~ + ~) n im (g~+ ~) = r (h) Zj = q ~ . We let ~i ---- qia for each i < j (consequently, the last condition is automatically satisfied). Let us construct the m a p ~3j+1. By (d) (assuming t h a t i = j + 1), we obtain a map/3j+1" I n+l -~[ g y + l [ such t h a t i m ( Z j + l ) ~ i m ( g j + l ) = 0 (i.e. j+l condition (g) is satisfied), qj j3j+l = ~j and fj+lJ3j+l = p j + l f ~ a . For i > j + 2 , we can construct maps ~i" I n+l ~ [ Ki [ in a similar way to the construction of /~y+l. It only remains to r e m a r k t h a t if ~ is the diagonal p r o d u c t of all ~i's, then one can easily verify t h a t ira(Z) ~ ira(g) = O, f ~ Z = f ~ a and t h a t fl and a are

4.3. n-SOFT MAPPINGS OF POLISH SPACES, RAISING DIMENSION

161

])-close. This finishes the verification of condition (i). Let us now show that M x is an #n+l-manifold. Since mesh(Ki) ~ 0 we can conclude (conditions (ii) and (iii) of Lemma 4.3.2) that M x admits small maps into (n + 1)-dimensional polyhedra. Consequently, d i m M x _< n + 1. The inverse inequality is obvious, because, by the above verified condition (i), M x contains a copy of every (n + 1)-dimensional compactum. Since M x is, by the construction, the limit space of the polyhedral inverse sequence SM all bonding maps of which are UVn-surjections we conclude that M x is an LCn-compactum. Again, by (i), it follows immediately that M x satisfies the disjoint (n + 1)-disk property and hence, by Theorem 4.1.19, M x is an #n+l-manifold. Next, we verify condition (ii). By Lemma 4.3.2, there exists a subspace A1 of [KI[ such that the restriction f l / n l : A1 ~ [XI[ is an (n + 1)-soft map and the complement [ K I [ - A1 is a aZn+l-set of ]KI[. It follows from our construction that the fibered product R1 of A1 and [ X2 [ w i t h respect to the maps f l / A 1 and p2 is a subspace of [TI[. Clearly the natural projections of R1 onto A1 and IX2] coincide with the restrictions rl/R1 and Sl/R1 respectively. Since f l / A 1 is (n + 1)-soft, we conclude (see Lemma 6.2.5) that sl/R1 is also (n + 1)-soft. Consider the set R1 x i2n+3 and denote by A2 its inverse image under the (n + 1)-soft map f~l (see Lemma 4.3.2). Then the restriction f 2 / A 2 : A 2 ~ IX2[ is (n + 1)-soft. Since the bonding map p2 is (n + 1)-soft, using again Lemma 6.2.5, we see that the map rl is also ( n + l ) - s o f t . Consequently, by Lemma 4.3.1, the set ]TI[-R1 is a aZn+l-set in ]T1]. In this situation one can easily observe that, again by Lemma 4.3.1, the complement [ K 2 [ - A2 is a a Z n + l - s e t in [K2[. Continuing i + l / A i+ 1 ) and an in such a manner we obtain an inverse sequence SA = {Ai, ~i (n + 1)-soft morphism (in the sense of Chapter 6) morphism { f i / A i } of Sn to S x . Clearly, the limit space A x of the spectrum $A is a subspace of M x and the restriction f ~ / A x " A x ---* X , which coincides with the limit map of the morphism {f~/Ai}, is an (n + 1)-soft map (here we use L e m m a 6.2.7). It only remains to remark that the complement M x - A x is a aZ-set in M x . Condition (iii) is a direct consequence of condition (ii) and Lemma 4.3.1 (compare with the proof pf L e m m a 4.3.2). The proof is complete. [::] We also need the following statement. PROPOSITION 4.3.4. Each #n+lmanifold M can be represented as the limit space of an inverse sequence SM -- {Mi ,Pi i+1 } consisting of locally compact poly-

h~d~a a~d p~op~ ~imptic~al (n + 1)-~~t~bt~, n-~oft a~d polyh~d~aUy (,~ + 1)-~oft bonding maps. PROOF. By Proposition 4.1.10, there exists a proper n-homotopy equivalence a : ] K ]---~ M, where g is an at most (n + 1)-dimensional countable locally finite simplex. Let K0 = K. Suppose that the countable locally finite simplicial simplexes K~ and proper simplicial maps Pi-1 ~ 9 ]K~[ ~ [K~-I[, satisfying conditions

162

4. MENGER MANIFOLDS

(i) - (iii) from L e m m a 4.3.2, have already been c o n s t r u c t e d for each i _< m. We m a y also assume t h a t the m a p Pi-1 i 9 IKil - . IKi_ll has the following property: 9 For any m a p a" I n+l --, IKi_ll there exist two maps ill, f~2" I n+l --~ IKil such t h a t P~-lflj i ---- a, j -- 1,2, and im(f~l) N im(fl2) -- 0. In order to construct these objects for the next step, we apply L e m m a 4.3.2 to a sufficiently fine triangulation of the p o l y h e d r o n IKml x [0, 1]. In this way we get the next simplicial complex g m + l ---- B gmx[0,1] n m+l -- 7rlfg,,~ n x [0,1], 9 We let pm where ~rl"lgml x [0, 1] ~ Igml is the projection. As in the proof of Theorem 4.3.3, we can verify t h a t the limit space M ~ of the inverse sequence 8M -{Mi, Pi i + 1 } is a #n+l-manifold. Clearly, the limit projection p0" M ' --, IKI, being a p r o p e r U V n - m a p between at most (n + 1)-dimensional locally finite L C nspaces, is a proper n - h o m o t o p y equivalence. Therefore M and M ~ are properly n - h o m o t o p y equivalent. T h e o r e m 4.1.21 finishes the proof. V-1 Applying T h e o r e m 4.3.3 and P r o p o s i t i o n 4.3.4 we get the following s t a t e m e n t . THEOREM 4.3.5. Let n > 0 and X E ( # k . k >_ n + 2} U {I ~ }. Then there exists an (n + 1)-invertible UYn-surjection f ~ " #n+l __~ X , satisfying conditions ( i ) - (iii) of Theorem 4.3.3. We conclude this section with the following s t a t e m e n t . THEOREM 4.3.6. Let n ~_ O. For each locally compact polyhedron K , there exists a proper (n + 1)-invertible UYn-surjection h~" M ~ +1 ---. g of some #n+l_ manifold M ~ +1 onto K satisfying the following conditions: (i) xf L i~ ~ ~lo~d ~bpoly~d~o~ of K , t ~ it~ i ~ ~ im~g~ (h~)-I(L) is a #n+l-manifold. (ii) If L is a closed subpolyhedron of K and Z is a Z-set in L, then the

i~,~

im~g~ (h~)-l(z)

i~ ~ z - ~ t

i~

(h~)-~(L).

PROOF. R e p e a t the proof of T h e o r e m 4.3.3, first observing that if L is a s u b c o m p l e x of a countable locally finite simplicial complex K, then the simplicial complex B~ from L e m m a 4.3.2 is a s u b c o m p l e x of the complex B ~ and the map f ~ coincides with the restriction f ~ / I B ~ ]. []

Historical and bibliographical notes 4.3. L e m m a 4.3.1 was proved in [279]. All o t h e r results of this Section were o b t a i n e d by the a u t h o r [79], [84], [88], [97], [98]. The non-separable case was considered in [108].

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS

163

4.4. Further properties of M e n g e r manifolds Using the existence of the dimension raising maps constructed in Sections 4.2 and 4.3, we now discuss other major ingredients of Menger manifold theory. 4.4.1. n - h o m o t o p y kernel and Open E m b e d d i n g T h e o r e m . The open embedding theorem for I ~ -manifolds states (see C h a p t e r 2) that for each I ~ manifold X, the product X x [0, 1) can be embedded into I ~ as an open subspace. Observe that identifying X with X x [0, 1] (stability of I"~-manifolds), the product X x [0, 1) may be viewed as the complement of the image of an appropriately chosen Z-embedding of X into itself. Using this remark as a guide, we introduce the following notion. Consider a #n+l-manifold M and two Z-embeddings f , g : M --~ M each of which is properly n-homotopic to the identity map idM. Then the homeomorphism g f - 1 . f ( M ) --. g(M) is properly n-homotopic to idf(M) and, consequently, by Theorem 4.1.15, there exists a homeomorphism h: M ~ M extending g f - 1 . Clearly the restriction h / ( M - f ( M ) ) is a homeomorphism between the complements M - f ( M ) and M - g(M). This shows that the following definition does not depend on the choice of a Z-embedding.

DEFINITION 4.4.1. The n-homotopy kernel K e r n ( M ) of a #n+l-manifold M is defined to be the complement M - f ( M ) , where f : M ~ M is an arbitrary Z-embedding properly n-homotopic to idM.

PROPOSITION 4.4.2. Let M and N be #n+l-manifolds. Then the following conditions are equivalent: (i) N admits a proper UVn-surjection onto the product M x [0, 1). (ii) N is homeomorphic to K e r n ( M ) . PROOF. It suffices to show that K e r n ( M ) also admits a proper UVn-surjecti on onto the product M x [0, 1). Take a proper v v n - s u r j e c t i o n f : M1 : M x [0, 1], where M1 is also a #n+l-manifold (see Theorem 4.2.1). Consider the quotient space M2 of M1 with respect to the partition whose nontrivial elements are fibers f - l ( m , 1) over the Z-set M x {1} of the product M • [0, 1]. Clearly, by Theorem 4.1.19, M2 is a #n+l-manifold. Moreover, if we consider the naturally induced proper UVn-surjection of M1 onto M2, then we conclude, by Theorem 4.1.21, that M1 and M2 are even homeomorphic. Next, it is easy to see that the set g - l ( M • {1}) is a Z-set in M2 and the restriction g / g - l ( M • {1}) is a homeomorphism. By Theorem 4.1.20, the composition 7rMg, where ~rM: M x [0, 1] ~ M is the natural projection, can be arbitrarily closely approximated by homeomorphisms. In particular, there exists a homeomorphism h: M2 ---+ M that is properly n-homotopic to 7rMg. T h e n the map r -- h g - l i , where

164

4. MENGER MANIFOLDS

i: M --* M x {1} is a n a t u r a l h o m e o m o r p h i s m , is a Z - e m b e d d i n g properly nh o m o t o p i c to idM. Indeed, r - h g - l i ~n --p 7rMgg-1 i -- zrMi -- idM. Thus, by Definition 4.4.1, we conclude t h a t the c o m p l e m e n t M - r ( M ) is h o m e o m o r p h i c to K e r n ( M ) . Consequently, the space M 2 - g - l ( M • { 1 } ) - h - l ( M - r ( M ) ) i s also h o m e o m o r p h i c to K e r n ( M ) . It only remains to note t h a t the space M 2 g - l ( M x {1}) a d m i t s a p r o p e r UVn-surjection onto the p r o d u c t M x [0, 1). V1

Now we are ready to prove the open e m b e d d i n g t h e o r e m for Itn+l-manifolds.

THEOREM 4.4.3. The n-homotopy kernel of each #n+l-manifold admits an open embedding into Itn+l.

PROOF. First of all let us show t h a t every Itn+l-manifold M admits a p r o p e r U V n - m a p onto a certain I w -manifold X. For this we take a proper UVn-map ~" M ~ P , where P is a locally c o m p a c t polyhedron (see T h e o r e m 4.2.23). Clearly the p r o d u c t P • I w --- X is a IW-manifold (see C h a p t e r 2). By T h e o r e m 4.2.1, there is a p r o p e r UVn-surjection r M ' ~ M • I w of some #n+l-manifold onto the locally c o m p a c t L C n - s p a c e M x I w . Since the composition r M r M ' M is also a proper U V n - m a p , we conclude, by T h e o r e m 4.1.20, that M ' and M are homeomorphic. It only remains to observe t h a t the required proper UV nsurjection h" M --* X is given by the composition (~o • idlw )r Now we proceed with the direct proof of our theorem. Take a proper UV nsurjection h" M ---+ X, where X is a I w -manifold. By the open e m b e d d i n g t h e o r e m f o r / W - m a n i f o l d s (see T h e o r e m 2.3.25), one m a y suppose that the product Z • [0, 1) lies in I w as an open subspace. Consider a p r o p e r (n + 1)-invertible UVn-map g" M1 ---* M x [0, 1], where M1 is a # n + l - m a n i f o l d (we use T h e o r e m 4.2.1). Since the m a p g is (n q- 1)-invertible and dim M -- n q- 1, we may assume, w i t h o u t loss of generality, t h a t the restriction g / g - l ( M x {1}) is a homeomorphism and A = g - l ( M • {1}) is a Z-set in M1. As above, we conclude, using T h e o r e m 4.1.20, t h a t the # n + l - m a n i f o l d s M and M1 are homeomorphic. Therefore, it only remains to show t h a t the c o m p l e m e n t M1 - A (which is obviously h o m e o m o r p h i c to K e r n ( M ) ) admits an open e m b e d d i n g into #n+l. Consider the m a p f" # n + l ___, i w of T h e o r e m 4.3.5 and let M2 denote the # n + l - m a n i f o l d f - l ( X x [0, 1)), which is open in # n + l . All t h a t remains to be shown is t h a t the # n + l - m a n i f o l d s M2 and M1 - A are homeomorphic. The last fact can be observed in the following way: b o t h # n + l - m a n i f o l d s MI - A and M2 admit proper U Y n - m a p s onto X • [0, 1) (consider the maps (h • id)g and f ) . Therefore, by T h e o r e m 4.1.21, they are homeomorphic. T h e following d i a g r a m helps to reconstruct the complete argument.

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS M2

f

M1-A

M1

g

~ M x [0, 1)

g

~M

x

[0,1]

~t n + l

f

hxid

•

h

x [0,1]

•

165

,~ I w

71"

M This finishes the proof.

~X

K]

4.4.2. n - h o m o t o p y C l a s s i f i c a t i o n T h e o r e m . Theorem 4.1.21 completely describes the proper n-homotopy classification of #n+l-manifolds. In particular, n-homotopy equivalent compact #n+l-manifolds are homeomorphic. Obviously, the last fact is incorrect in the non-compact case: compare #n+l and # n + l - { p t } . The main result of this subsection solves the n-homotopy classification problem of arbitrary ttn+lmanifolds (compare with Theorem 2.3.26). PROPOSITION 4.4.4. For each #n+l-manifold M, the spaces K e r n ( M ) and Kern(Kern(M)) are homeomorphic. PROOF. As above, take a proper UVn-surjection g: M --~ X of M onto a I w -manifold X. By Proposition 4.4.2, there exists a proper UVn-surjection f : Kern(M) ~ M x [0, 1). For the same reason, there exists a proper UV nsurjection h: K e r n ( K e r n ( M ) ) --~ Kern(M) x [0, 1). Consequently, we have two proper UVn-surjections:

p = (g x id)f: Kern(M) ~ M x [0, 1) and

q = (p x id)h: K e r n ( K e r n ( M ) ) ~ K e r n ( M ) x [0, 1),

166

4. MENGER MANIFOLDS

where id denotes the identity map of [0, 1). Since X is a Hilbert cube manifold, the product X x [0, 1] is homeomorphic to X (see Chapter 2). Remarking that the spaces [0, 1) x [0, 1) and [0, 1] x [0, 1) are homeomorphic, we have X x [0,1) x [ 0 , 1 ) ~ X x [0,1] x [0,1) ..~ X x [0, 1). Consequently, the #n+l-manifolds K e r n ( M ) and K e r n ( K e r n ( M ) ) admit proper UVn-surjections onto the same I ~ -manifold X x [0, 1). Therefore they are properly n-homotopy equivalent. Theorem 4.1.21 finishes the proof. V1 PROPOSITION 4.4.5. Let M be a #n+l-manifold. I r A is a Z-set in K e r n ( M ) , then the spaces K e r n ( M ) and K e r n ( K e r n ( M ) - A) are homeomorphic. PROOF. As in the proof of Proposition 4.4.4 consider three proper UV nsurjections:

g: M ~ X, f : K e r n ( M ) - - * M x[0, 1 ) a n d p =

(gxid)f: Kern(M)~

Xx[0,1),

where X is a I ~ -manifold and id denotes the identity map of [0, 1). Now we redefine the map p in such a way that the set p(A) will be a Z-set in X x [0, 1). For this, consider any Z-embedding r: A ~ X x [0, 1) properly n-homotopic to the restriction p/A. By Proposition 4.2.2, there exists a proper UVn-surjection q: K e r n ( M ) ---* X x [0,1) such that q/A = r. Moreover, as in the proof of Proposition 4.4.2, we can additionally suppose that A = q-lq(A). Consequently, the restriction

q / ( K e r n ( M ) - A): K e r n ( M ) -

A ---. X x [0, 1 ) - q(A)

is a proper UVn-surjection. As above (compare with the proof of Proposition 4.4.4) this implies that K e r n ( K e r n ( M ) - A) admits a proper UYn-surjection onto the product (X x [0, 1 ) - q(A)) x [0, 1). By the same argument, there exists a proper VYn-surjection of K e r n ( K e r n ( M ) ) onto the product X x [0, 1) x [0, 1). By Proposition 4.4.4, K e r n ( M ) and K e r n ( K e r n ( M ) ) are homeomorphic. Therefore, by Theorem 4.1.21, it suffices to show that the spaces (X • [ 0 , 1 ) - q ( A ) )

x [0,1) and X x [ 0 , 1 ) x [0,1)

are homeomorphic. Indeed, since q(A) is a Z-set in X x [0, 1), we conclude that the I~-manifolds X x [0, 1) and X x [0, 1 ) - q ( A ) are homotopy equivalent. Then, by the homotopy classification theorem for I W-manifolds (Theorem 2.3.26), the products X x [0, 1) x [0, 1) and (X x [0, 1 ) - q(A)) x [0, 1) are homeomorphic. This finishes the proof. E] PROPOSITION 4.4.6. Let a #n+l-manifold M be a Z-set of a #n+l-manifold N , and suppose the inclusion i: M --+ N is an n-homotopy equivalence. Then there exists a Z-set A in N such that the complement N - A is homeomorphic to K e r n ( M ) .

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS

167

PROOF9 By Theorem 4.2.23, there exist (n + 1)-invertible proper UVn-sur jections ~: M ---. L and r N ---. K1, where L and K1 are (n + 1)-dimensional locally compact polyhedra. The (n + 1)-invertibility of ~ implies the existence of a proper map s: L ~ M with ~s - idL. Then the composition r L --. K1 is properly homotopic to some proper piecewise linear (PL) map p: L ~ K1. Form the mapping cylinder M(p) -- K of the map p. Recall that this is the space obtained from the disjoint union (L • [0, 1])@ K1, by identifying (/, 1) with p(l) for each l E L. At the same time we identify L with L x {0}. Clearly L x [0, 1) can be considered as an open subspace of K. Since p is a proper PLmap, L • {0} and K1 are subpolyhedra of the polyhedron K. Let c: K ~ K1 be the collapse to the base, i.e. the natural retraction defined by sending (/, t) to p(l). Clearly, c is a proper CE-surjection that is a homotopy equivalence. Now we consider an (n + 1)-invertible proper UYn-surjection f~(" M~(+1 --. K , satisfying the conditions of Theorem 4.3.6. Clearly, the composition cf~" M~(+1 ---* K1 is a proper UVn-surjection, and hence, by Theorem 4.1.21, the #n+l-manifolds M ~ +1 and N are homeomorphic. By Theorem 4.3.6, the inverse image ( f ~ ) - l ( L x {0}) is a #n+l-manifold that, again by Theorem 4.1.21, is homeomorphic to M. One can easily verify, using the assumption and the specifics of the above construction, that the natural inclusion of ( f ~ ) - l ( L x {0}) into M ~ +1 is an n-homotopy equivalence. Moreover, by Theorem 4.3.6, the above inverse image is a Z-set in M ~ +1 (since L • {0} is a Z-set in K). Now redefining the above objects for simplicity, we have the following situation. A proper UVn-surjection f : N ~ K, satisfying the conditions of Theorem 4.3.6, is given, M -- f - l ( L • {0}) is a Z-set in N and the inclusion M ~ N is an n-homotopy equivalence. Clearly, K - K1 -- L x [0, 1), and hence the inverse i m a g e / - I ( L • [0, 1)) admits a proper UYn-surjection onto L • [0, 1). On the other hand, K e r n ( M ) admits a proper vVn-surjection onto M • [0, 1), and hence onto the product L • [0, 1) as well. Thus, by Theorem 4.1.21, the inverse image f - l ( L • [0, 1)) and K e r n ( M ) are homeomorphic. Consequently, to finish the proof it remains to construct an open embedding h: f - l ( L x [0, 1)) ---. N such t h a t the complement A -- g - h ( f - l ( L • [0, 1)) is a Z-set in N. Since N is a #n+l-manifold, by Theorem 4.1.19, there is a countable dense subset {~k: k -- 1 , 2 , . . . } of c ( I n + l , g ) consisting of Z-embeddings. As observed above, f - l ( L • {0}) is a Z-set in N and the inclusion f - l ( L x {0}) ~ N is an n-homotopy equivalence. It easily follows from Proposition 4.2.2 that in this case there exists a retraction rl: N --. f - l ( L x {0}) that is n-homotopic to idN. Consider the restriction 81 = r l / 9

9 f-l(n

x {0}-)U~I(/n+l)

---~ f - l ( n

1

x [0, ~)).

Clearly 81 is a proper map. By Theorem 4.1.19, Sl is properly n-homotopic to a Z-embedding

gl 9 I - I ( L x { 0 } . ) U ~ I

(/n+l

)-"> .f --1 (L x [0, 1 )) Z

168

4. MENGER MANIFOLDS

that coincides with the identity map on f - l ( L x {0}). By Theorem 4.1.15, there exists a homeomorphism GI" N --~ N extending gl. Put hi = G~-1. Then hi is a homeomorphism such t h a t (a) h l / f - l ( L x { 0 } ) - id. (b) (ill(/n+l) C_ h l f - l ( L x [0, 1)). 1 Consider now the polyhedron K - (L x [0, ~)). Since the set L x { 89 is a Z-set in K (L x [0, ~)), 1 we can conclude, by Theorem 4.3.6, that the set h l f - l ( L x { 1}) is a Z-set in a #n+l-manifold N - h l f - l ( L • [0, 89 Moreover, since the inclusion -

1 L x {~}~K-L

1 x [0,~))

is a homotopy equivalence, we conclude that the inclusion

hlf-l(L x { })~

N-hl.f-l(L

x [0,~))

is an n-homotopy equivalence. Again, using the above construction, we see that there is a homeomorphism

h ,2. N - h l f

-1

1

(L x

))---+ N

which is the identity on h l f - l ( L x {89 ~o2(I n+l) C'I ( N -

hlf-l(L

x [0,

-

h 1 f - l ( L x [0, 1

and for which ))) C_ h ~ h l f - l ( L x [~,2-

)).

Extend h~ to a homeomorphism h2 defined on N by defining h2 = id on

h l f - l ( L x [0, 89 Then we have 1

~o1(In+l) U ~o2(I~+1) C_ h 2 h l f - l ( L x [0,2 9 g)). Inductively continuing this process, we construct homeomorphisms hk" N ~ N in such a way that hk+l = id on h k f - l ( L x [O,k. k--~]) and 1

V~l(In+l)IJ...iJ(Pk(In+l ) C_ h k h k _ l . . . h l f - l ( L x [O,k. k + 1) ). Define an open embedding h" f - l ( L x [0, 1)) ~ N by h(x) = limk--.oo h k ' " hi(x) for each x E f - l ( L x [0, 1)). Clearly U{~k ( i n + l )" k = 1 , 2 , . . . } _C h f - l ( L x [0, 1)) and consequently, by the choice of the family {~k" k = 1, 2 , . . . }, the complement [-1

N - h f - l ( L x [0, 1)) is a Z-set in N.

Now we are ready to prove the n-homotopy classification theorem for #n+l_ manifolds. THEOREM 4.4.7. #n+l-manifolds are n-homotopy equivalent if and only if their n-homotopy kernels are homeomorphic.

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS

169

PROOF. Let M and N be n - h o m o t o p y equivalent tin+l-manifolds. Take maps n n a" M ~ N and/3" N ~ M such t h a t ~ a " ~ i d M and a ~ " ~ i d N . As above, we can find (n + 1)-invertible proper UYn-surjections f " K e r n ( M ) ~ M • [0, 1) and g" K e r n ( N ) ~ N • [0, 1). Let ~" M x [0, 1) ~ N x [0, 1) and r g x [0, 1) --, M • [0, 1) be a proper maps such t h a t ~ is homotopic to a x id and r is homotopic to fl x id, where id denotes the identity map of [0, 1) (compare with [69, L e m m a 21.1]) . Since g is (n + 1)-invertible and dim K e r n ( M ) = n + 1, there exists a proper m a p r" K e r n ( M ) ~ K e r n ( N ) such t h a t gr = ~o.f. Similarly we have a proper map s" K e r n ( N ) ~ K e r n ( M ) such t h a t f s = Cg. In this situation one n

can verify directly t h a t sr "~ idKer,~(g). Moreover, by T h e o r e m 4.1.19, we can additionally suppose t h a t r and s are Z-embeddings. Now consider the Z-set r ( g e r n ( M ) ) in a # n + l - m a n i f o l d K e r n ( N ) . It follows immediately from the above construction t h a t the inclusion map

r(Kern(M)) ~ Kern(N) is an n - h o m o t o p y equivalence. By Proposition 4.4.6, there is a Z-set A in K e r n ( N ) such t h a t K e r n ( N ) - A is h o m e o m o r p h i c to K e r n ( r ( g e r n ( M ) ) ) -K e r n ( K e r n ( M ) ) (recall t h a t r is an embedding). Then, by Propositions 4.4.4 and 4.4.5,

K e r n ( M ) .~ K e r n ( K e r n ( K e r n ( M ) ) ) ..~ K e r n ( K e r n ( N ) as desired. T h e second part of the t h e o r e m is trivial.

A) ..~ K e r n ( N )

V1

4.4.3. n - s h a p e a n d t h e C o m p l e m e n t T h e o r e m . T h e famous Complement T h e o r e m for I • -manifolds [69] states t h a t if X and Y are Z-sets in I ~ , then their complements I ~ - X and I ~ - Y are h o m e o m o r p h i c if and only if the shapes of X and Y coincide, i.e. S h ( Z ) = S h ( Y ) . T h e obvious form of T h e C o m p l e m e n t T h e o r e m fails for #n+l. T h e equality of shapes of two Z-sets X and Y in # n + l is sufficient, but far from necessary, for the c o m p l e m e n t s # n + l - X and # n + l _ y to be homeomorphic. Indeed, it can be easily seen t h a t if the (n + 1)dimensional sphere S n+l is e m b e d d e d into # n + l as a Z-set, t h e n # n + l _ s n + l is h o m e o m o r p h i c to tt n+l - {pt}. At the same time S h ( S n+l) ~ Sh(pt). T h e problem was solved in [87] (see also [90], [92]) where the notion of n - s h a p e was introduced. T h e relation between n - S H A P E and n - H O M O T O P Y categories is of the same n a t u r e as t h a t between the categories of S H A P E and H O M O T O P Y . Roughly, n - S H A P E is a "spectral completion" of n - H O M O T O P Y . T h e main result in this direction is the following. THEOREM 4.4.8. Let X and Y be Z-sets in #n+l. The complements tt n+l - X and # n + l _ y are homeomorphic if and only if n - S h ( X ) -- n - S h ( Y ) .

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We would like to mention some corollaries of this theorem and the definition of n-shape itself. COROLLARY 4.4.9. I f S h ( X ) -- S h ( Y ) , then n -

S h ( X ) -- n - S h ( Y ) .

COROLLARY 4.4.10. I f X and Y are at most n-dimensional, then S h ( X ) S h ( Y ) if and only if n - S h ( X ) -- n - S h ( Y ) .

=

COROLLARY 4.4.11. I f Z - s e t s X and Y in #n+l are #n+l-manifolds, then the complements #n+l _ X and #n+l _ y are homeomorphic if and only if the compacta X and Y are homeomorphic. Let us emphasize that the notion of n-equivalence, introduced by Ferry [150] as a generalization of Whitehead's notion of n-type, coincides in several important cases with the notion of n-shape. Relations between these two concepts have been studied in [97].We conclude this section by noting that Theorem 4.4.8 was extended [280] to a larger class of subspaces than Z-sets. These are the so-called weak Z-sets.

4.4.4. M e n d e r m a n i f o l d s w i t h b o u n d a r i e s . The problem of putting a boundary on various types of manifolds were considered in [57] (PL manifolds), [282] (smooth manifolds) and [70] (I ~ - manifolds). It was proved in [70] that if an I W-manifold M satisfies certain minimal necessary homotopy-theoretical conditions (finite type and tameness at oo), then there are two obstructions a ~ ( M ) and Tc~(M) to M having a boundary. The first one is an element of the group li+___m{/C0rl(M - A ) " A C M A is compact}, where ]C0~rl is the projective class group functor. If c o o ( M ) -- O, then the second obstruction can be defined as an element of the first derived limit of the inverse system li.___m{YVhrl( M - A ) " A C M A is compact}, where ~Vh~rl is the Whitehead group functor. It was shown in [70] t h a t the different boundaries that can be put on M constitute a whole shape class and that a classification of all possible ways of putting boundaries on M can be done in terms o f t h e group l i m { Y ~ h ~ l ( M - A ) " A C M A is compact}. It should be emphasized that the above mentioned obstructions essentially involve the Wall's finiteness obstruction [303]. The natural analog of Wall's obstruction vanishes in the n-homotopy category. This is exactly what was stated in 4.1.12. We will see that this observation significantly simplifies the situation for #n+l-manifolds. First of all we need the following corollary of Proposition 4.1.12. PROPOSITION 4.4.12. I f a # n + l - m a n i f o l d M is n - h o m o t o p y dominated by an at m o s t (n + 1)-dimensional L C n - c o m p a c t u m , then M is n - h o m o t o p y equivalent to a compact #n+l-manifold.

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS

171

We say t h a t a #'~+l-manifold M admits a boundary if there exists a compact # n + l manifold N such t h a t M = N - Z, where Z is a Z-set in N. In this case we shall say t h a t N is a compactification of M corresponding to the b o u n d a r y Z, and conversely, Z is a b o u n d a r y of M corresponding to the compactification N. We also need the following definition [100]. DEFINITION 4.4.13. A space X is said to be n - t a m e at c~ if for each compactum A C X there exists a larger compactum B C X such that the inclusion X - B ~-~ X - A factors up to n-homotopy through an at most (n + 1)-dimensional finite polyhedron. PROPOSITION 4.4.14. I f a #n+l-manifold M is n-tame at c~, then M is nhomotopy equivalent to a compact #n+l-manifold. PROOF. Take a proper U V n - r e t r a c t i o n r: M ~ P of the given # n + l - m a n i f o l d M onto some (n + 1)-dimensional locally compact polyhedron P. It follows from elementary properties of proper U V n - m a p s t h a t P is n - t a m e at c~ as well. Using Proposition 4.1.8, one can easily see t h a t P is n - h o m o t o p y d o m i n a t e d by an at most (n + 1)-dimensional compact polyhedron. Proposition 4.4.12 finishes the proof. [::] Let us recall t h a t an I ~ -manifold M lying in a larger I ~ -manifold N is said to be clean if M is closed in N and the topological frontier of M in N is collared b o t h in M and N - I n t M . For obvious dimensional reasons we cannot directly define the corresponding notion for #n+l-manifolds. Nevertheless, the following notion is sufficient for us. DEFINITION 4.4.15. A #n+l-manifold M lying in a #n+l-manifold N is said to be n-clean in N provided that M is closed in N and there exists a closed subspace 5 ( M ) of M such that the following conditions are satisfied:

(i) 6(M) /~ ~ (ii) (iii) (iv) (v)

,~+~-m~ifold.

( N - M ) t2 5 ( M ) is a #n+l-manifold. 5 ( M ) is a Z - s e t in M . 5(M) is a Z - s e t in ( N - M ) U S(M). M - 5 ( M ) is open in g .

Sometimes we say t h a t M is n-clean with respect to 5 ( M ) . Let us indicate the s t a n d a r d situation in which n-clean submanifolds arise naturally. Suppose t h a t L is submanifold of a combinatorial PL-manifold P . Consider a proper UVn-surjection f : N ~ P of a # n + l - m a n i f o l d N from Theorem 4.3.6. Using the properties of f , it is easy to see t h a t M = f - l ( L ) is an n-clean submanifold of N with 5 ( M ) = f - l ( O L ) . Generally speaking, .f is not an open map and consequently 5 ( M ) does not necessarily coincide with the topological frontier of M in N.

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LEMMA 4.4.16. Let N be a #n+l-manifold which is n - t a m e at oo. Suppose that M is a compact and n-clean submanifold of N . Then the #n+l-manifold (N - M ) U 5 ( M ) is n-homotopy equivalent to a compact #n+l-manifold. PROOF. By Proposition 4.4.14, it suffices to show t h a t the #n+l-manifold ( N - M ) U S ( M ) is n - t a m e at 00. Let A be a compact subspace of ( N - M ) U 6 ( M ) . Clearly, K1 = A U M is compact. Since N is n - t a m e at 00, there exists a c o m p a c t u m K2 such t h a t Kz C_ K2 C_ N and the inclusion N - K 2 ~ NK1 factors up to n - h o m o t o p y t h r o u g h an at most (n + 1)-dimensional compact polyhedron. Let B = ((N - M ) U 6 ( M ) ) gl K2. Clearly B is a c o m p a c t u m and A c B. Note t h a t D

((N-M)U6(M))-B=N-K2

and N - K I C _

((N-M)U6(M))-A.

Consequently, the inclusion ((N-M)U6(M))-B

~

((N-M)U6(M))-A

factors up to n - h o m o t o p y t h r o u g h an at most (n + 1)-dimensional compact polyhedron. Hence ( g - M ) U 6 ( M ) is n - t a m e at c~. FI LEMMA 4.4.17. A n y #n+l-manifold M can be written as a union M = Ui=IM~ such that all M i ' s are compact and n-clean and M~ C M i + l - 6 ( M i + 1 ) for each i= 1,2,.... oo

PROOF. It suffices to show t h a t for each c o m p a c t u m K C_ M, there exists a compact and n-clean M1 C_ M such t h a t K C_ M1 - 6 ( M 1 ) . As before, take a proper UVn-surjection g" M ~ X, where X is a I ~ -manifold. There is a compact and clean Y _c X such t h a t g ( K ) C_ I n t ( x ( Y ) (see [70]). By the relative triangulation theorem for I~-manifolds (see T h e o r e m 2.3.31), there exists a polyhedron P which can be w r i t t e n as a union of two subpolyhedra P1 and /)2 such t h a t X = P x I ~ , Y = P1 x I ~ , X - I n t x ( Y ) = P2 x I W and B d x ( Y ) = (P1 CI P2) x I W . Note also t h a t the subpolyhedron P1 F1P2 is a Z-set b o t h in P1 and P2. Consider now a proper UVn-surjection f" N ~ P of a #n+l-manifold N onto the polyhedron P satisfying the conditions of T h e o r e m 4.3.6. Consequently, we have two proper u v n - s u r j e c t i o n s f" N ~ P and 7rpg" M --+ P (here ~p" P x I ~ --, P denotes the n a t u r a l projection) of two #n+l-manifolds onto the polyhedron P. Consider an open cover b / - {P - 7 r g g ( K ) , I n t p ( P 1 ) } of P. By T h e o r e m 4.1.20, there exists a h o m e o m o r p h i s m h" M --+ N such t h a t the compositions 7rgg and f h are b/-close. Let M1 -- h - l f - l ( p 1 ) and 6(M1) = h - l f - l ( p 1 rl P2). By the properties of the map f, M1 is compact and n-clean. It only remains to note t h a t K C_ M1 - 6 ( M 1 ) . This finishes the proof. F1 We also need the following s t a t e m e n t , which is a direct consequence of the characterization T h e o r e m 4.1.19.

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS

173

PROPOSITION 4.4.18. Let a space M be the union of two closed subspaces M1 and M2. If M1, M2 and Mo = M1 A M2 are #n+l-manifolds and Mo is a Z-set both in M1 and M2, then M is a #n+l-manifold. PROOF. It suffices to show t h a t for any m a p f : X ~ M of an at most (n + 1)dimensional c o m p a c t u m X into M , and any open cover U E cov(M), t h e r e exists an e m b e d d i n g g: X ~ M , U-close to f . Let us consider the case when f ( X ) N Mi 7~ 0 for each i = 0, 1, 2. All o t h e r cases are trivial. By P r o p o s i t i o n 4.1.7, t h e r e exists an open cover 12 E coy(M) refining b / s u c h t h a t the following condition is satisfied: (.)~ for any at most (n + 1)-dimensional c o m p a c t u m B, closed s u b s p a c e A of B, and any two P-close m a p s c~1,c~2: A --. M such t h a t c~1 has an extension r B --. M , it follows t h a t C~2 also a d m i t s an extension r B ~ M which is/,/-close to ~1. Let Xi = f - l ( M i ) , i = 0, 1,2. Since M0 is a # n + l - m a n i f o l d , there is a Ze m b e d d i n g go: X0 --~ M0 such t h a t go and f / X o are ]2-close. By (*)n, t h e r e is an extension h: X --. M of G - ) such t h a t h and f are V-close. Since Mo is a Z-set in b o t h M1 and M2 we conclude t h a t go(Xo) is a Z - s e t b o t h in M1 and M2. Consequently, by T h e o r e m 4.1.19, for each i = 1, 2 there is a Z - e m b e d d i n g gi: Xi ~ Mi such t h a t gi/Xo = go and gi is U-close to h / X i . At the same time, w i t h o u t loss of generality we can assume t h a t one of these maps, say gl, has the following p r o p e r t y : g l ( X 1 - X0) N M0 = 0 (we once again use the fact t h a t M0 is a Z-set in M1). T h e n the map g, coinciding with gi on Xi (i = 1,2), is an embedding. It only remains to note t h a t g and .f are N-close. [] LEMMA 4.4.19. If a #n+l-manifold M is n-tame at oo, then we can write M = U~__IMi such that all Mi's are compact and n-clean, Mi C M i + l - ~ ( M i + l ) and the inclusion 5(Mi) r ( M i + l - Mi) U 5(Mi) is n-homotopy equivalence for each i = 1 , 2 , . . . . PROOF. Choose any c o m p a c t and n-clean submanifold A of M . By L e m m a 4.4.17, it suffices to find a c o m p a c t and n-clean submanifold B of M such t h a t A C_ B 5(B) and the inclusion 8(B) ~-~ ( M - B ) U 5(B) is an nh o m o t o p y equivalence. By L e m m a 4.4.16, the # n + l - m a n i f o l d ( M - A)8(A) is n - h o m o t o p y equivalent to some c o m p a c t # n + l - m a n i f o l d X. Fix the corresponding n - h o m o t o p y equivalence r ( M - a ) U S ( A ) + X and its n - h o m o t o p y inverse ~o1" X ~ (M - A ) U 5(A). Obviously there is a m a p ~92" (M - A ) 5 ( A ) ~ X such t h a t r 5(A) --~ X is a Z - e m b e d d i n g a n d r is as close to r as we wish. Similarly, there is a Z - e m b e d d i n g ~o2" X ~ ( M - A ) U 5(A) which is as close to ~Ol as we wish. In particular, we can assume t h a t r and ~o2 are n - h o m o t o p y equivalences. If r and ~o2 were chosen sufficiently close to r and ~ol respectively, then, by T h e o r e m 4.1.18, t h e r e exists a h o m e o m o r phism h" ( M - A ) U 5(A) ~ ( M - A ) U 5(A) which e x t e n d s the h o m e o m o r phism qo2r

5(A) ~

qo2r

and which is sufficiently close to the

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identity map of (M - A ) U 5(A). In particular, we can assume that h is nhomotopic to id(M_A)oh(A ). T h e n the n - h o m o t o p y equivalence ~p - h -1~2" X --~ (iA ) U 6(A) is a Z - e m b e d d i n g and 6(A) C ~ ( X ) -- Y . Since Y is a compact #n+l-manifold, there exists a U V n - r e t r a c t i o n s" Y ~ K onto a finite (n + 1)-dimensional polyhedron K (see T h e o r e m 4.2.23). Similarly, take a proper V Y n - r e t r a c t i o n r" ( M - A) U 6 ( A ) --, T, where T is a polyhedron. Let i" Y "--. ( M - A ) U 6 ( A ) denote the inclusion map and j" K ~ Y be a section of s (i.e. s j - i d g ) . Note t h a t i is an n - h o m o t o p y equivalence. Let p" K --, T be a P L - m a p homotopic to the composition r i j . Form the mapping cylinder M ( p ) -- P of the map p. For the reader's convenience, we again recall t h a t P is the space obtained from the disjoint union ( g x [0, 1]) @ T, by identifying (k, 1) with p ( k ) , k E g . At the same time we identify g with g x {0}. Since p is a P L - m a p , K x {0} and T are s u b p o l y h e d r a of the polyhedron P. Let c" P --. T be the collapse to the base, i.e. the natural retraction defined by sending (k, t) to p ( k ) for each (k, t) E K • [0, 1]. Obviously, c is a proper cell-like map t h a t is a proper h o m o t o p y equivalence. Now consider a proper U V n - s u r j e c t i o n f" N --. P of some # n + l - m a n i f o l d N onto P, satisfying the conditions of Theorem 4.3.6. T h e compact #n+l-manifolds Y and N1 = f - l ( K • {0}) admit U V n - s u r j e c t i o n s s" Y ~ K x {0} and f / N l " N1 ---* K • {0} onto the same polyhedron. Consequently, by T h e o r e m 4.1.21, there exists a homeomorphism n gl" Y ---* N1 such t h a t f g l ~ - s . Similarly, we have two proper U V n - s u r j e c t i o n s r" ( M - A ) U 5 ( A ) --. T and c f " N --~ T. As above, there is a homeomorphism g2" ( M - A ) U 5 ( A ) --. N such t h a t c f g2 ~ p r. By the construction and the corresponding properties of proper U V n - s u r j e c t i o n s , we have n

n

n

c f gl ~-- cs -- ps ~ r i j s ~_ ri ~ c f g2i.

Since c f is a proper n - h o m o t o p y equivalence, we conclude t h a t gl" Y ~ N and g 2 / Y " Y ~ N are n-homotopic. Consider the h o m e o m o r p h i s m

= ~g2--1 /g2(Y)" g 2 ( Y ) ~ N~. Clearly n

--1

~-- g2g2 / g 2 ( Y ) -- idg2(y ).

By the properties of the map f, N1 is a Z-set in N. Note also that, by our construction, g2(Y) is a Z-set in N as well. By T h e o r e m 4.1.15, we can find a h o m e o m o r p h i s m G: N ~ N extending a. Let H -- G g2. Note that H ( Y ) -G g 2 ( Y ) = a g 2 ( Y ) = g l . Finally, let B -- A U H - I ( f - I ( K

• [0, ~])) and 5(B) -- H - I ( f - I ( K

• { })).

It follows from the properties of the map f and Proposition 4.4.18 t h a t B is a compact and n-clean submanifold of M , A C B - 5(B), and the inclusion 5(B) ~ ( M - B ) U S ( B ) is an n - h o m o t o p y equivalence. To see this, observe t h a t

4.4. FURTHER PROPERTIES OF MENGER MANIFOLDS 1 the m a p p, and consequently the inclusion K x {~} ~ n - h o m o t o p y equivalence. [-1

175

P - ( K x [0, ~]), 1 is an

LEMMA 4.4.20. Let a #n+l-manifold M be a Z - s e t in a compact # n + l - m a n i fold N . I f the inclusion i" M --~ N is an n-homotopy equivalence, then there exists a UVn-retraction of N onto M . PROOF. Let j" N --~ M be an n - h o m o t o p y inverse of i. By T h e o r e m 4.1.21, ?1 there is a h o m e o m o r p h i s m h" N ~ M such t h a t h n J. T h e n hi rn~ j i ~_ idM. Consequently, by P r o p o s i t i o n 4.2.2, there is a UV'~-surjection r" N ~ M such t h a t ri = idM. U] T h e following result gives us a c h a r a c t e r i z a t i o n of t i n + l - m a n i f o l d s with b o u n d aries. THEOREM 4.4.21. A #n+l-manifold admits a boundary if and only if it is n-tame at oo. PROOF. Let M be a # n + l - m a n i f o l d which is n - t a m e at cx~. By L e m m a 4.4.19, we can represent M as a union M -- U ~ I M i such t h a t all t h e M i ' s are c o m p a c t a n d n-clean, Mi C M i + l - 5 ( M i + I ) and the inclusion 5(Mi) ~ ( M i + l - Mi) U 5(Mi) is an n - h o m o t o p y equivalence for each i = 1 , 2 , . . . . By L e m m a 4.4.20, for each i there exists a u y n - r e t r a c t i o n fi" ( M i + l - Mi) U 5(Mi) ~ 3(Mi). Let the u v n - r e t r a c t i o n ri" M i + l --* Mi coincide with fi on M i + l - Mi and with t h e identity on Mi. T h e n we have an inverse sequence S = {Mi, ri} consisting of c o m p a c t # n + l - m a n i f o l d s and U V n -retractions. By T h e o r e m 4.1.20, ri is a nearh o m e o m o r p h i s m for each i. By [58], each limit p r o j e c t i o n of the s p e c t r u m S is a n e a r - h o m e o m o r p h i s m as well. Consequently, N = lim S, being h o m e o m o r p h i c to M1, is a c o m p a c t # n + l - m a n i f o l d . Since 5(Mi) is a Z-set in Mi for each i, we conclude t h a t the subset Z = l i m { 5 ( M i + l ) , r i / 5 ( M i + l ) } is a Z - s e t in N. It only remains to note t h a t N - Z is n a t u r a l l y h o m e o m o r p h i c to M . Conversely, suppose t h a t the # n + l - m a n i f o l d M a d m i t s a b o u n d a r y . This m e a n s t h a t there are a c o m p a c t # n + l - m a n i f o l d N and a Z - s e t Z in N such that M = N-Z. Let us show t h a t M is n - t a m e at cx~. Let A be a c o m p a c t subspace of M . As in the proof of L e m m a 4.4.17, t h e r e exists a c o m p a c t and n-clean submanifold B of M such t h a t A C_ B - 5 ( B ) . It suffices to show t h a t ( M - B ) U 5(B) is n - h o m o t o p y equivalent to an at m o s t (n + 1)-dimensional finite p o l y h e d r o n . Indeed, it is easy to see t h a t ( M - B ) U 5 ( B ) is n - h o m o t o p y equivalent to a c o m p a c t # n + l - m a n i f o l d ( N - B ) U S ( B ) . It only remains to a p p l y P r o p o s i t i o n 4.1.10. El Not all # n + l - m a n i f o l d s a d m i t b o u n d a r i e s in the above sense. To see this, consider the 3-dimensional (topological) manifold W ( c o n s t r u c t e d by W h i t e h e a d ) which is defined as the c o m p l e m e n t in S 3 of a c o n t i n u u m W h which, in t u r n , is the intersection of a nested sequence of tori in S 3. T h e manifold W has an

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4. MENGER MANIFOLDS

infinitely generated f u n d a m e n t a l group at c~. Let n >_ 1, and consider a #n+l_ manifold M and a proper U V n - s u r j e c t i o n f " M ---. W . Since n + 1 >_ 2, we see t h a t f induces an isomorphism of f u n d a m e n t a l groups of ends. T h e n it is easy to see t h a t M is not 1-tame at c~ and, therefore, cannot have a boundary. On the other hand, it can shown t h a t the Freudenthal compactification of any connected #l-manifold contains its end as a Z-set. Consequently, the Freudenthal compactification of any connected #1manifold is homeomorphic to #1. In other words, any connected # l - m a n i f o l d has a boundary. We also mention two related results. PROPOSITION 4.4.22. I f the compactum X is a boundary for a #n+l-manifold M , then the compactum Y is also a boundary for M if and only if dim Y < n + 1 andn-Sh(Y)=n-Sh(X). Two compactifications N and T of the same space M are said to be equivalent if for every c o m p a c t u m A C M there is a h o m e o m o r p h i s m of N onto T fixing A point wise. Of course, if the #n+l-manifolds N and T are compactifications of a #n+l_ manifold M , then the inclusions M ~ N and M --~ T are n-homotopy equivalences (because, N - M and T - M are Z-sets in N and T respectively). Consequently, N and T are h o m e o m o r p h i c as n - h o m o t o p y equivalent compact #n+l-manifolds (Theorem 4.1.21). A stronger result can be obtained. PROPOSITION 4.4.23. Every two # n + l - m a n i f o l d compactifications of a given # n + l - m a n i f o l d are equivalent in the above sense. T h e problem of w h e t h e r a # n + l - m a n i f o l d has a b o u n d a r y which is itself a # n + l - m a n i f o l d was also considered in [104]. DEFINITION 4.4.24. A proper map f " Y ---, X between at most (n + 1)-dimensional locally compact spaces is an n - d o m i n a t i o n near c~ provided that there exists a cofinite subspace X1 of X ( i.e. X1 is closed and X - X1 has compact closure) and a proper map g" X1 ---* Y such that f g is properly n-homotopic to the inclusion map X1 ~-* X . If, in addition, for some cofinite subspace Y1 of Y the composition g f /Y1 is properly n - h o m o t o p i c to the inclusion map ]I1 ~ Y , then we say that f is an equivalence near c 0 is a sufficiently small number. Let U be a n e i g h b o r h o o d of i d x in A u t h ( X ) with d i a m U < e. Let x , y E X with d(x, y) -- e, and let A be a small d i a m e t e r arc from x to y in X. One can define, using local setwise homogeneity, a convergent sequence {h~} in A u t h ( X ) "sliding x towards y along A" and such t h a t h -limh~ E A u t h ( X ) and h E Bd(U). Thus, each e - n e i g h b o r h o o d of i d x has none m p t y b o u n d a r y , and so dim A u t h ( X ) >_ 1. Therefore we have the following statement. THEOREM 4.5.2. Let X

be a locally setwise homogeneous continuum. A u t h ( X ) is at least 1-dimensional.

Then

S t r o n g local h o m o g e n e i t y of # " - m a n i f o l d s (Corollary 4.1.17), implies their local setwise homogeneity. Consequently, we have the following. COROLLARY 4.5.3. If M is a compact #"-manifold, then dim A u t h ( M ) >_ 1. Recall t h a t a space X is almost O-dimensional if it has a basis B of open sets such t h a t for each B E B, X - c l B -- U(Ui 9 i E N } where each Ui is b o t h open and closed. Clearly, each 0-dimensional space is almost 0-dimensional. Also, it is easy to see t h a t every almost 0-dimensional space is totally disconnected. T h e c o m p l e t e Erdhs space ~ = (x E 12" xi is irrational for each i} is a 1-dimensional space which is almost 0-dimensional. PROPOSITION 4.5.4. Each almost O-dimensional space is at most 1-dimensi-

onal. SKETCH OF PROOF. Let X be an almost 0-dimensional space and let B be a c o u n t a b l e basis witnessing this fact. Let ~ = ( f i " i E N } be a collection of continuous functions f i " X ~ (0, 1} such t h a t if B , B ' E 13 with cl(B) Mcl(B') -0, there is a f i E ~ ' w i t h f~(B) = 0 and f i ( B ' ) = 1. Let p be the metric o n X given by p(x, y) = ~-~i 2 - i I f i ( x ) - f i ( y ) ]. Let d be a t o t a l l y bounded metric on X. It suffices to show t h a t the metric dimension # d i m ( X , d ) < 1. Now, d ~ -- d + p is also a t o t a l l y b o u n d e d metric on X. Let Y be the completion of X with respect to d ~. It suffices to show t h a t for each t > 0, there is an open set U of Y containing X such t h a t each c o n t i n u u m in U has d i a m e t e r less t h a n t. Let 34 - {U open in Y 9 diam U < t / 3 and U MX E B}. Let C be any continu u m in W - U/d. I f B , B ' E B with B M C ~ 0 ~ B ' M C , then cl(B) N c l ( B ' ) =fi 0, for otherwise there is fi E 9r with f~(B) = 0 and f i ( B ' ) = 1. But f~(X) = {0, 1} and so, by the definition of d', Y = c l y ( f ~ - l ( O ) ) U c l y ( f ~ - l ( 1 ) ) , where cly(f/--l(0)) and c l y ( f ~ - l ( 1 ) ) are disjoint closed sets. Hence C, being a continuum, c a n n o t meet b o t h c l y ( f / - l ( 0 ) ) and c l y ( f ~ - l ( 1 ) ) which is a contradiction. It follows t h a t d i a m C < t. V1 PROPOSITION 4.5.5. If M is an Mkn-manifold with 0 ~_ n < k < oe, then A u t h ( M ) is almost O-dimensional.

4.5. HOMEOMORPHISM GROUPS

179

SKETCH OF PROOF. Let g E A u t h ( M ) and ~ > 0. Let h E A u t h ( M ) with d(g,h) > e. We shall show that there is an open and closed set U containing h such that d ( g , j ) > e for each j E U. Now, d(g, h) = c + 45 for some 5 > 0, and there is x E M so that d(g, h) = d(g(x), h(x)). Choose an n-sphere S in M such that g(S) c_ N ( g ( x ) , 5 ) and h(S) C_ N ( h ( x ) , 5). Since dim M = n, there is a retraction r : M ~ h(S) such that r ( M - N ( h ( x ) , 2 5 ) ) is constant. Let U -- {f E A u t h ( M ) : r f / S ~ .}. Then U is both open and closed in A u t h ( M ) because close maps into S are homotopic. Also h E U. Let f E N(g, e). T h e n f ( S ) C N ( g ( x ) , c + 5). Hence, r . f ( S ) is a point and f ~ U. D COROLLARY 4.5.6. Let 1 < n < cx~. If M is a compact #n-manifold, then d i m A u t h ( M ) - - 1. PROOF. Apply Corollary 4.5.3 and Propositions 4.5.4 and 4.5.5.

[:]

4.5.2. S i m p l i c i t y . Anderson [16], [17] originated a technique for identifying minimal, non-trivial normal subgroups of A u t h ( X ) for spaces with certain dilation and homogeneity properties. DEFINITION 4.5.7. Let X be a space. A subset A of X is deformable if for every non-empty open set U in X , there is h E A u t h ( X ) with h(A) C_ U. Let V be an open set. A collection ({Bi : i E N } , h) is called a dilation system in U if {B~} is a sequence of disjoint non-empty open sets in U with lim Bi -- {p} for some p E U and h E A u t h ( X ) supported on U such that h(B~+l) ---- B~ for each i. PROPOSITION 4.5.8. Let X be a metrizable space in which each non-empty open set contains a dilation system. Let G be a subgroup of A u t h ( X ) generated by all homeomorphisms which are supported by deformable subsets of X . I f G :fi {e}, then G is the smallest non-trivial normal subgroup of A u t h ( X ) . If X is a finite-dimensional manifold without boundary, then A u t h 0 ( X ) , the subgroup of homeomorphisms isotopic to the identity, is simple (see [151] and [141]). It is also known (see [220] and [323]) that Auth(/2) and A u t h ( I ~ ) are simple. DEFINITION 4.5.9. Let M be a #n+l-manifold. A pair (W, 5 ( W ) ) is an nclean pair if W is n-clean with respect to 5 ( W ) in the sense of Definition 4.4.15 and if, in addition, both W and 5 ( W ) are homeomorphic to #n+l. By Theorem 4.1.15 and the existence of n-clean pairs in #n+l, (compare with [184]), it follows that every open set in #n+l has a dilation system. Also, every proper closed set in #n+l is deformable. Since every element of A u t h ( # n+l) is stable (see Theorem 4.5.13 below) we have the following. THEOREM 4.5.10. A u t h ( # n+l) is simple.

180

4. MENGER MANIFOLDS

4.5.3. Stability of homeomorphisms. An a u t o h o m e o m o r p h i s m of a space X is said to be stable [32] if it can be expressed as the composition of finitely m a n y a u t o h o m e o m o r p h i s m s each of which is the identity on some n o n - e m p t y open subspace of X. It is well-known t h a t all a u t o h o m e o m o r p h i s m s of the Hilbert cube I ~ and the Hilbert space 12 are stable (see, for example, [32]). Every orientation-preserving h o m e o m o r p h i s m of R n is stable [194]. LEMMA 4.5.11. For each #n+l-manifold M , there is a Z-embedding a" M ---+ M which is properly n-homotopic to idM and which satisfies the following condition: (.) If F e A u t h ( M ) and F / a ( M ) = id, then F can be expressed as the composition of two autohomeomorphisms of M each of which is the identity on some open subspace of M . PROOF. Take a proper UVnsurjection g" M ---+ K , where K is an at most (n + 1)-dimensional locally compact p o l y h e d r o n ( T h e o r e m 4.2.23). Consider also a p r o p e r UVn-surjection f" MI" K x [-1, 2], satisfying conditions of T h e o r e m 4.3.6. It can easily be checked t h a t the inverse image M -- f - l ( g x [0,2]) is also a copy of the # n + l - m a n i f o l d M. Moreover, since the composition 7 r l f / M --* g x {0}, where ~rl" g x [0, 2] --. g x {0} denotes the projection, is a proper U V nsurjection, we conclude t h a t there exists a h o m e o m o r p h i s m a" M --, f - l ( K x {0}) such t h a t f a ..~n 7 r l f / M o T h e n a is properly n - h o m o t o p i c to idM. By the mp properties of f , we see t h a t a" M --~ M is a Z-embedding. Consequently, it only remains to show t h a t if F E A u t h ( M ) and F / f - l ( g x {0}) -- id, then F can be expressed as the composition of two a u t o h o m e o m o r p h i s m s of M each of which is the identity on some open subspace of M . Let U - F - l ( f - l ( g x (1,2])). Since F / a ( M ) - id we conclude t h a t U n f - l ( K x {0}) = 0. Consequently, there exists a sufficiently small tl > 0 such t h a t f - l ( K x [O, t l ] ) N ( U U F ( U ) ) = O. Let V : f - l ( K x [0, t0)), where to = l t l l + t1l . Consider the h o m e o m o r p h i s m ~" [0, tl] --* [ - 1 , t l ] defined as v~(x) = t-LL--x--1 1+tl Consider also an open cover ZX=

{ [ K x

--1,--

,K x

--~,tl

]}

of the p r o d u c t K x [-1, tl]. Again, by the properties of f , the inverse images N1 = f - l ( K x [0, tl]) and N2 = f - l ( K x [-1, tl]) are #n+l-manifolds. Clearly these a d m i t proper u v n - s u r j e c t i o n s f l = (id x t p ) f / N l " N1 ---+ K x [-1, tl] and f2 = f / N 2 " K x [-1, tl] onto the polyhedron K x [-1, tl]. T h e n we see t h a t the inverse image f - l ( K x {tl}) is a Z-set b o t h in N1 and N2. Moreover, since, v~(tl) = tl, we conclude t h a t the identity h o m e o m o r p h i s m h o f . f - l ( K x {tl}) satisfies the equality f2h = fl/f-l(K x {tl}). Consequently, there exists a h o m e o m o r p h i s m T" N1 ---+ N2 such t h a t T / f - I ( K x {tl}) = id and the composition f 2 T is U-close to f l . Now

4.5. HOMEOMORPHISM GROUPS

181

consider the homeomorphism H" M ~ M1 such that H / f - I ( K x [0, tl]) = T and H / f - I ( K x [tl,2]) = id. It is easy to verify that H ( V ) N M = 0 and H/(UUF(U))=id. Let f

a(x) = ~F(x),

[ x,

ifx eM ifxEM1-M.

Since F / f - I ( K x {0}) = id, we conclude t h a t G E Auth(M1). Now let F2 -H - 1 G H e A u t h ( M ) . If x e U, then H ( x ) - x. Hence, G H ( x ) - F ( x ) e F ( V ) . Consequently, H - 1 G H ( x ) -- F ( x ) . In other words, F2/U -- F l U . Similarly, if x e V, then H ( x ) e M1 - M . Hence G H ( x ) = H ( x ) and H - 1 G H ( x ) = H - 1 H ( x ) = x. This means that F 2 / V = id. Obviously, F -- F2F1, where F1 = F 2 1 F . It only remains to note t h a t F 1 / U -- id and F 2 / V -- id. 0 LEMMA 4.5.12. Let M be an #n+l-manifold and ~" M ~ M be a Z-embedding properly n-homotopic to idM. I f G E A u t h ( M ) and G / ~ ( M ) -- id, then G can be expressed as the composition of two autohomeomorphisms each of which is the identity on some open subspace of M . PROOF. Take a Z-embedding a satisfying condition (.) of L e m m a 4.5.11. Then the homeomorphism / 3 a - l : a ( M ) ~ ~ ( M ) is properly n-homotopic to the inclusion a ( M ) ~ M. Consequently, by Theorem 4.1.15, there exists a homeomorphism h: M ~ M extending ~ a -1. Consider the homeomorphism h - l G h E A u t h ( M ) . Evidently, h - l G h / a ( M ) -- id. By the choice of a, there exist F1, F2 E A u t h ( M ) and open subspaces 1/1 and 1/2 of M such that F i / V i = id, i -- 1, 2, and h - l G h -- F2F1. Then G -- G2G1, where Gi -- hFih -1, i -- 1, 2. It only remains to note t h a t G i / h ( V i ) -- id, i -- 1, 2. V] THEOREM 4.5.13. Let M be a #n+l-manifold and F E A u t h ( M ) . properly n-homotopic to idM, then F is stable.

I f F is

PROOF. Let ~" M --~ M be a Z-embedding properly n-homotopic to idM. Let a" M ---+ M be another Z-embedding properly n-homotopic to idM and such that a ( M ) N ( ~ ( M ) U F ( ~ ( M ) ) ) = 0. Consider the homeomorphism S" a ( M ) U f l ( M ) ~ a ( M ) U F ( ~ ( M ) )

which coincides with the identity on a ( M ) and with F on /~(M). Clearly, f is properly n-homotopic to the inclusion a ( M ) U ~ ( M ) ~ M . Therefore there exists an extension F2 E A u t h ( M ) of f . Let F1 = F 2 1 F . By L e m m a 4.5.12, F = F2F1 can be expressed as the composition of four a u t o h o m e o m o r p h i s m s each of which is the identity on some open subspace of M. V1 Since each map of #n+l into itself is (properly) n-homotopic to the identity, we obtain the following statement.

COROLLARY 4.5.14. Every autohomeomorphism of#n+1 is stable.

182

4. MENGER MANIFOLDS

4 . 5 . 4 . G r o u p a c t i o n s o n M e n d e r m a n i f o l d s . T h e well-known HilbertSmith conjecture asks w h e t h e r every c o m p a c t group acting effectively on a manifold is a Lie group. This is equivalent to asking w h e t h e r the group A n of p-adic integers acts effectively on a manifold. This long s t a n d i n g problem is still open. T h e situation is r a t h e r different for Mender manifolds. For instance, it is known [13] t h a t any c o m p a c t metrizable zero-dimensional topological group G acts freely on #1 so t h a t the orbit space # I / G is h o m e o m o r p h i c to tt 1. There are several constructions of group actions on #n-manifolds. THEOREM 4.5.15. Let M be a # " - m a n i f o l d . T h e n : (i) E v e r y c o m p a c t z e r o - d i m e n s i o n a l metrizable group G acts on M so that the orbit space M / G

is h o m e o m o r p h i c to M

(ii) Ap acts freely on M so that d i m M / A p -- n + 1 (iii) Ap acts on M so that dim M / A p = n + 2 T h e r e are u n e x p e c t e d ties between group actions on Mender c o m p a c t a and the H i l b e r t - S m i t h problem mentioned above. Namely, a positive solution to the following conjecture would prove t h a t there is no free Ap-action on a connected (topological) manifold M with dim M / A p < 0r

CONJECTURE 4.5.16. Let m and n be positive integers and G be a zero-dim e n s i o n a l c o m p a c t m e t r i c group. I f #mWn and #n are free G-spaces, then there is no equivariant map #mTn ~ #n.

Historical and bibliographical notes 4.5. T h e o r e m 4.5.2 was proved in [54]. Corollary 4.5.6 a p p e a r s in [244]. Corollary 4.5.14 is due to the author [94]. It was shown later t h a t all a u t o h o m e o m o r p h i s m s of all connected Mender manifolds are stable [266]. Item (i) of T h e o r e m 4.5.15 was proved in [131] (see also [265]). Items (ii) and (iii) a p p e a r in [219] (the last s t a t e m e n t is based on the work [257] ). Conjecture 4.5.16 appears in [1].

4.6. w-soft map

of a onto

E

In this Section, using T h e o r e m 4.2.21, we show t h a t there exists an "almost soft" m a p from a onto E. DEFINITION 4.6.1. A m a p f : X c o u n t a b l e - d i m e n s i o n a l I space B , g: A ---, X

~

Y

is called w - s o f t if f o r each strongly

closed subspace A

and h: B ---, Y with f g -- h / A ,

of B

and any two m a p s

there exists a m a p k ~

B ---, X

such that k / A -- g and f k -- h, i.e. if the following diagram

1Recall that a space is strongly countable-dimensional if it can be represented as the countable union of finite-dimensional closed subspaces.

4.6. w-SOFT MAP OF a ONTO E

X

A

183

~Y

~

,.-B

commutes.

Obviously, e v e r y w-soft m a p is n-soft for each n E w. In p a r t i c u l a r , e v e r y wsoft m a p is s u r j e c t i v e a n d o p e n ( c o m p a r e w i t h t h e p r o o f of i m p l i c a t i o n (ii) ~

(i)

in P r o p o s i t i o n 2.1.34).

LEMMA 4.6.2. Let

T

,-X

i

,~ X I

,~Y

be a c o m m u t a t i v e diagram, consisting of compact spaces, where i is an embedding and d i m T < n. T h e n there exists a c o m m u t a t i v e n - s o f t diagram 2

2This means that the diagonal product k~Ah ~, considered as the map of T ~ onto its image, is n-soft (see page 159).

184

4. M E N G E R M A N I F O L D S

TI

\

h!

~

T

,

~

XI

h

f

Z

g

~Y

where j is an embedding and d i m T ~ _ 1, then the class of Polish A N E ( n ) - s p a c e s (respectively, A E ( n ) - s p a c e s ) coincides with the class of Polish L C n - l - s p a c e s (respectively, L C n - I N Cn-l-spaces). PROOF. Let X be a Polish space and Z0 be a closed subset of a zero-dimensional r e a l c o m p a c t space Z. Consider a m a p f : Z0 --~ X satisfying the inclusion C ( f ) ( C ( X ) ) c_ C ( Z ) / Z o . B y T h e o r e m 1.3.10, t h e space Z can be represented as t h e limit space of a factorizing w - s p e c t r u m S = { Z a , p ~ , A } consisting of zerod i m e n s i o n a l Polish spaces. Since Z0 is closed in Z, t h e limit space of the induced

6.1. ANE(n)-SPACES AND n-SOFT MAPPINGS

231

spectrum

A}

So=

coincides with Z0 (Corollary 1.2.6). By Proposition 1.3.13, there exist an index a EA andamap

f~: clz~ p,~(Xo)

---, x

such that f = f a p a / Z 0 . Since d i m Z a = 0, the map f a has an extension ga onto Za (see Proposition 2.1.13). Then the composition g = gap~ is the desired extension of f onto Z. Consequently, by Proposition 6.1.9, X is an AE(0)-space. The remaining part of our s t a t e m e n t can be proved in a completely similar way (compare with Theorem 2.1.12). E] PROPOSITION 6.1.11. Let X be a compact space. Then X is an A ( N ) E ( n ) space if and only if the following condition is satisfied: 9 For each at most n-dimensional compactum Z and for each closed subspace Zo of Z, any map f : Zo ---* X has an extension to (a neighborhood of Zo in Z ) Z. PROOF. Obviously, if X is an A ( N ) E ( n ) - c o m p a c t u m , then it satisfies the above formulated condition. Let us show t h a t the converse is also true. Suppose t h a t a c o m p a c t u m X satisfies the above condition. We need to show t h a t X is an A ( g ) E ( n ) - s p a c e in the sense of Definition 6.1.3. Consider an arbitrary space Z of dimension at most n, its arbitrary subspace Z0, and a map f : Z0 ---* X such t h a t C ( f ) ( C ( X ) ) C C ( Z ) / Z o . Denote by A0 the closure of Z0 in the Stone-Cech compactification flZ of the space Z. Following the proof of Proposition 1.3.12, we can conclude t h a t the map f can be extended to a map g: A0 ---* X. By our assumption and the equality d i m f l Z - d i m Z , the map g can be extended to a map h: V ~ X, where V is a functionally open neighborhood of the set A0 in flZ. Let U -- V NZ. Clearly, U is a stable functionally open neighborhood of Z0 in Z and the map h / U : U --, X extends the originally given map f. V1 As follows from the last two statements, the above defined notion of A ( g ) E ( n ) space is completely compatible with the known definition of this notion in the classes of Polish and compact spaces. Of course, within Polish spaces (and, in general, in the class of metrizable spaces) these concepts have been exploited for decades, and an extensive literature is devoted to their study. On the other hand, until the middle of the nineteen seventies almost nothing, except the definition itself, was known about non-metrizable A ( N ) E ( n ) - c o m p a c t a . The first serious investigation in this direction was the work of Haydon [164] in which the spectral characterization of non-metrizable A E ( 0 ) - c o m p a c t spaces was obtained. Moreover, it was shown in [164] t h a t the class of A E ( 0 ) - c o m p a c t a coincides with the class of Dugundji compacta (the latter class, in different terms, was defined in [249]). In Section 6.3 we consider this problem more formally.

232

6. ABSOLUTE EXTENSORS

Concluding this Subsection, we remark that the compactness of X is absolutely essential in Proposition 6.1.11. More formally: 9 There exists a Polish space, satisfying the condition of Proposition 6.1.11 (for n = c~), which is not an absolute retract. This fact follows from Corollary 3.2.16. As already mentioned in Subsection 2.1.2, this was observed by van Mill (see [22]).

6.1.3. D e f i n i t i o n of n - s o f t m a p p i n g s . The following definition shows that the notion of an n-soft map is a categorical counterpart of the notion of A E ( n ) space. D E F I N I T I O N 6 . 1 . 1 2 . A m a p f : X --+ Y is s a i d to be n - s o f t , n -- 0, 1 , . . . , c~, if for each at most n-dimensional space Z, its two subspaces Zo and Z1 with Zo C Z1 and any two maps g: Zo ~ X and h: Z1 ~ Y such that C ( g ) ( C ( X ) ) C C ( Z ) / Z o , C ( h ) ( C ( Y ) ) C C ( Z ) / Z 1 and f g = h/Zo, there exists a map k: ZI: X such that f k = h, g = k/Zo and C ( k ) ( C ( X ) ) C_ C ( Z ) / Z 1 . Maps that are c~-soft are called soft The following diagram helps to understand the situation described in the above Definition.

X

,~Y

go t

,~Z1 t

~Z

We are now going to establish some elementary properties of n-soft maps. We begin with the following statement, the proof of which is trivial and hence omitted. LEMMA 6.1.13. Each n-soft map is surjective. LEMMA 6.1.14. A finite composition of n-soft maps is n-soft. PROOF. It suffices to consider the case of two maps. Let f l : X1 ---+ X2 and f 2 : X 2 ---* X3 be n-soft maps. Consider an at most n-dimensional space Z, its subspaces Z0 and Z1 with Z0 c Z1, and two maps g: Z0 ---* X1 and h: Z1 --~ X3 satisfying the conditions

C ( g ) ( C ( X l ) ) C C ( Z ) / Z o , C ( h ) ( C ( X 3 ) ) C C ( Z ) / Z 1 and f 2 f l g = h/gO.

6.1. ANE(n)-SPACES AND n-SOFT MAPPINGS

233

Straightforward verification shows that C ( f l g ) ( C ( X 2 ) ) c_ C ( Z ) / Z o . Consequently, by the n-softness of the map f2, there exists a map k~: Z1 ---, X2 such that f 2 k ' = h, f i g -

k ' / Z o and C ( k ' ) ( C ( X 2 ) ) C

C(Z)/Zl.

Further, the n-softness of the map f l guarantees the existence of a map k: Z1 X1 satisfying the following conditions: f l k = k' k / Z o = g and C ( k ) ( C ( X l ) ) C C ( Z ) / Z 1 . It only remains to observe that f 2 f l k -- f2 k t = h.

D

LEMMA 6.1.15. Let f = f 2 f l and f l and .f be n-soft maps. an n-soft map.

Then .f2 is also

PROOF. Consider two maps f l : X1 ---* X 2 and f 2 : X 2 ~ X 3 and assume t h a t f l and f - f 2 f l are n-soft. Consider also an at most n-dimensional space Z, its subspaces Z0 and Z1 with Z0 C Z1, and two maps g: Z0 ~ X2 and h: Z1 ~ X3 satisfying the following conditions: C(g)(C(X2)) C C(Z)/Zo,

C(h)(C(X3)) C C(Z)/Z1

and f2g = h/Zo.

Consider an arbitrary point z0 E Z0. By L e m m a 6.1.13, there is a point xl E X1 such that .fl(Xl) = g(zo). By the n-softness of the map f l , there exists a map g~: Z0 ~ X1 satisfying, in particular, the following conditions: C(gt)(c(x1)) C C(Z)/Zo,

and f l g ' = g.

The n-softness of the map f guarantees the existence of a map k~: Z I : X1 such that C(k')(C(Xl)) C C(Z)/Z1,

k ' / Z o = g' and f k ' =

h.

Finally observe t h a t the map k = f l U : Z1 ~ X2 has all the required properties from Definition 6.1.12 with respect to the map f2. [--1 PROPOSITION 6.1.16. A map f : X ~ Y of a realcompact space X onto an A N E ( n ) - s p a c e Y is n-soft if and only if the following condition is satisfied: 9 For each at most n-dimensional realcompact space Z, for its closed subspace Zo, and for any two maps g: Zo -+ X and h: Z ~ Y such that f g = h/Zo and C ( g ) ( C ( X ) ) C C ( Z ) / Z o , there exists a map k: Z ~ X such that k / Z o = g and f k = h. PROOF. Obviously every n-soft map satisfies the above stated condition. Suppose now that f is the map given in our statement. Let us show its n-softness. Let Z, Z0, Z1, g and h satisfy the conditions of Definition 6.1.12 with respect to the map f . Consider the Hewitt realcompactification u Z of the space Z, and the closures A0 and A1 of Z0 and Z1, respectively, in uZ. Since every space

234

6. ABSOLUTE EXTENSORS

is C-embedded in its Hewitt realcompactification, we have the following two inclusions:

C ( g ) ( C ( X ) ) C C ( v Z ) / Z o and C ( h ) ( C ( Y ) ) C C ( v Z ) / Z 1 . By Propositions 6.1.7 and 1.3.12, the maps g and h have extensions ~" A0 ~ X and h" A1 ---* Y such that

C ( ~ ) ( C ( X ) ) C_ C ( v Z ) / A o and C ( h ) ( C ( Y ) ) C_ C ( v Z ) / A 1 . Since f ~ / Z o = h/Zo, and since Z0 is dense in A0, we conclude that f[7 = h/Ao. Since Y is an ANE(n)-space, and since dim vZ - dim Z < n, there exists a map h I" U ---, Y, defined on a stable functionally open neighborhood U of A1 in vZ, extending h. Corollary 1.1.15, Lemma 1.1.18 and Proposition 1.3.18 imply that U is an at most n-dimensional realcompact space. Observe also that A0 is closed in U. Consequently, according to our assumption, there exists a map k t" U --* X such that f k ~ = h ~ and k~/Ao = ~.. The restriction k~/Z1 will be denoted by k. It follows from the construction that f k -- h and k/Zo -- g. All that remains to be verified is the inclusion C ( k ) ( C ( X ) ) C C(Z)/Z1. Let E C ( X ) . By the construction, U is a stable functionally open neighborhood of the set A1, and hence of the set Z1, in vZ. Take a functionally closed subset F o f v Z such that Z1 C F C U. The f u n c t i o n ~ k E C(Z1) can be extended to a function ~k ~ E C(U). In this situation, Proposition 1.1.23 guarantees that the function ~k~/F E C(F) has an extension onto vZ. Denote this extension by r Obviously, the restriction r is an extension of ~k onto Z. This completes the proof. E1 COROLLARY 6.1.17. Let f" X ---, Y be an n-soft map between realcompact spaces and Iio be an ANE(n)-subspace of Y. Then the restriction

/ / / - l ( y 0 ) ' / - l ( y 0 ) - ~ Yo is also n-soft. PROOF. It is easy to see that the inverse image f - l ( Y 0 ) is realcompact. Therefore we can use Proposition 6.1.16. Let Z0 be a closed subspace of an at most n-dimensional realcompact space Z. Consider two maps g" Zo ~ f - l ( y o ) and h" Z ~ II0, satisfying the conditions

f g = h and C ( g ) ( C ( f - l ( y o ) ) ) C_ C ( Z ) / Z o . One can easily verify that

c(g)(c(x)) c c(z)/zo. Consequently, by the n-softness of the map f, there exists a map k" Z ~ X such that k/Zo = g and f k = h. The last equality guarantees that k(Z) C

/-~(v0). D

6.1. ANE(n)-SPACES AND n-SOFT MAPPINGS

235

Combining the proofs of Proposition 6.1.11 and 6.1.16, we obtain the following statement. PROPOSITION 6.1.18. A map f : X ~ Y between compact spaces is n - s o f t if and only if the following condition is satisfied: 9 For each at m o s t n - d i m e n s i o n a l compactum Z , its closed subspace Zo, and any two maps g: Zo ~ X and h: Z ~ Y with f g = h / Z o , there exists a map k ~ Z ~ X such that k / Z o = g and f k = h.

Combining the proofs of Proposition 6.1.10 and 6.1.16 we get PROPOSITION 6.1.19. A map between Polish spaces following condition is satisfied: 9 For each at m o s t n - d i m e n s i o n a l Polish space and any two maps g: Zo ~ X and h: Z --+ exists a map k ---, Z ~ X such that k / Z o = g

in n - s o f t if and only if the Z , its closed subspace Zo, Y with f g = h / Z o , there and f k = h.

R e m a r k 6.1.20. If the compact spaces X and Y in Proposition 6.1.18 are metrizable, then we can assume that the compacta Z and Zo are also metrizable. Therefore the last two Propositions, coupled with the fact just mentioned, show that in the class of Polish spaces the definition n-soft map (Definition 6.1.12) coincides with the definition of the same notion given in Chapter 2 (see Definition 2.1.33). It is also useful to observe the following simple fact: 9 in Corollary 6.1.17 the assumption that Y0 is an A N E (n )-space can be removed if (a) X, Y and Yo are compact spaces, or (b) X, Y and ]Io are Polish spaces.

PROPOSITION 6.1.21. Let f " X ~ Y be an n - s o f t map. A ( g ) E ( n ) - s p a c e if and only if Y is an A ( g ) E ( n ) - s p a c e .

Then X

is an

PROOF. First assume that X is an A ( N ) E ( n ) - s p a c e . Consider an arbitrary at most n-dimensional space Z, its subspace Z0, and a map g" Zo ~ Y such t h a t C ( g ) ( C ( Y ) ) c C ( Z ) / Z o . Take a point z0 e Z0 and, using L e m m a 6.1.13, consider a point x0 E X such that f ( x o ) -- g(zo). Let go(zo) -- xo. The nsoftness of the map f guarantees the existence of a map gl. Zo ~ X such that fg'g and C ( g ) ( C ( X ) ) C_ C ( Z ) / Z o . By our assumption, there exist a stable functionally open neighborhood U of the set Z0 in Z and a map h/" U ~ X, extending g~. Finally, observe that the composition h -- f h t" U --. Y is an extension of the originally given map g. Therefore Y is an A ( N ) E ( n ) - s p a c e . Conversely, assume that Y is an A ( N ) E ( n ) - s p a c e . Consider an arbitrary at most n-dimensional space Z, its subspace Z0, and a map g" Z0 --* X such t h a t C ( g ) ( C ( X ) ) c_ C ( Z ) / Z o . Then, by our assumption, there exist a stable functionally open neighborhood U of the set Z0 in Z and a map h ~" U --* Y such that h~/Zo = f g . By Lemma 1.1.18 and Proposition 1.3.18, we have dim U _ n.

236

6. ABSOLUTE EXTENSORS

Consequently, the n-softness of the map f guarantees the existence of a map h" U --~ X with h / Z o = g. This shows t h a t X is an A ( N ) E ( n ) - s p a c e . [:] Next we introduce some additional concepts needed in the sequel. DEFINITION 6.1.22. A map f " X ~ Y is called n-invertible if for any map h" Z ---, Y of any at most n - d i m e n s i o n a l space Z into Y , there is a map g" Z ---, X such that f g = h. Obviously, every n-soft map is n-invertible. In turn, every n-invertible map is surjective. It is known [171] t h a t each 0-invertible map between compact spaces has a so-called regular averaging operator (see Definition 6.1.24) and t h a t for maps between metrizable c o m p a c t a the converse is also true. Every metrizable c o m p a c t u m is a 0-invertible image of the Cantor cube D • (see [233], [171]). In C h a p t e r 4 we have already seen t h a t a more general s t a t e m e n t is true: each metrizable c o m p a c t u m is an n-invertible image of an n-dimensional c o m p a c t u m (the universal Menger c o m p a c t u m #n for example; see T h e o r e m 4.2.21). In the meantime, the map constructed in T h e o r e m 4.2.21 has several additional properties (for instance, t h a t m a p is open for each n _> 1). If we do not require the presence of these additional properties, then the above fact can be obtained as an immediate consequence of T h e o r e m 1.3.10. Indeed, take an arbitrary metrizable c o m p a c t u m X and consider all possible maps ga" Za ~ X of at most n-dimensional metrizable c o m p a c t a Za into X. Let Z denote the discrete sum of these c o m p a c t a Za. T h e maps ga naturally induce the map g" Z --~ X. Consider the Stone-(~ech extension g~" f~Z ---, X of g. Since dim f~Z = n we can, applying T h e o r e m 1.3.10, find an n-dimensional metrizable c o m p a c t u m Y and two m a p s p " f~Z --~ Y and f" Y ---, X such t h a t g~ = f p . It only remains to observe t h a t the map f is n-invertible. DEFINITION 6.1.23. A map f : X --~ Y is said to be functionally open (respectively, functionally closed) if the image of any f u n c t i o n a l l y open (respectively, f u n c t i o n a l l y closed) subset of X is functionally open (respectively, functionally closed) in Y DEFINITION 6.1.24. We say that a surjection f " X - , Y admits a regular averaging o p e r a t o r if there exists a linear continuous I map v" C ( X ) --+ C ( Y ) satisfying the following conditions" (a) v ( 1 x ) = 1y. (b) /f ~ >_ 0, then v(~ >_ O. (c) v . C ( f ) = i d c ( y ) . T h e following s t a t e m e n t will be exploited below. 1Here C(X) and C(Y) are endowed with the compact-open topology.

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237

PROPOSITION 6.1.25. For each infinite cardinal n u m b e r T, there exists a functionally closed, proper and O-invertible m a p g" N r ---. R r admitting a regular averaging operator. PROOF. For each a 6 r , t a k e a 0-invertible m a p ha" D ~a ~

[0, 1]a of t h e

C a n t o r cube D ~ o n t o t h e closed interval [0, 1]a (see t h e a b o v e discussion). Let L a = h~-l((0, 1)a) a n d ga = h a / L a , ~ 6 T. D e n o t e by g t h e p r o d u c t of ga's, i.e. g = •

~ e ~}.

Since L a is a z e r o - d i m e n s i o n a l , n o n - c o m p a c t , locally c o m p a c t space w i t h a c o u n t a b l e basis, we c o n c l u d e (recall t h a t T is infinite c a r d i n a l ) t h a t t h e p r o d u c t l-I{n~" ~ 9 r } is h o m e o m o r p h i c to N r (use T h e o r e m 1.1.5). In t u r n , t h e p r o d u c t YI{(o, 1)a. a 6 T} is obviously h o m e o m o r p h i c to R r . T h e 0-invertibility of t h e m a p g follows from t h e 0-invertibility of t h e ga's, a 6 T. It is easy to see t h a t t h e m a p g is p r o p e r a n d f u n c t i o n a l l y closed. T h u s , it only r e m a i n s to show t h a t g admits a regular averaging operator. It easily follows from t h e a b o v e c o n s t r u c t i o n t h a t we are in t h e following s i t u a t i o n . A 0-invertible m a p ( n a m e l y t h e p r o d u c t of h a ' s , a 6 7-) h" D r ~ I r is given, R r C I r , h - I ( R r ) is h o m e o m o r p h i c to N r a n d t h e m a p g" N r ~ R r coincides w i t h t h e r e s t r i c t i o n h / h - I ( R

r ).

Take a r e g u l a r a v e r a g i n g o p e r a t o r

u for t h e m a p h. It follows from e l e m e n t a r y p r o p e r t i e s of r e g u l a r a v e r a g i n g o p e r a t o r s [249] t h a t for each c o m p a c t u m K in R r , t h e m a p h g = h / h - I ( K ) also a d m i t s a r e g u l a r a v e r a g i n g o p e r a t o r UK. T h e s e o p e r a t o r s are c o m p a t i b l e in t h e following sense: 9 If K1 a n d K2 are c o m p a c t a in R r a n d K1 C K2, t h e n for each f u n c t i o n 6 C ( h - l ( g 2 ) we have

UK2(~p)/gl -- UKl ( C f l / h - l ( g l ) ) . U s i n g these p r o p e r t i e s of o p e r a t o r s u g , a n d also t h e fact t h a t b o t h spaces N r a n d R r are f g - s p a c e s 2, it is not h a r d to see t h a t by l e t t i n g

v(~)(y) = u{u}(~/g-l(y)),

y 6 Rr

a n d ~p e C ( N r )

we define a r e g u l a r a v e r a g i n g o p e r a t o r for t h e m a p g.

[-1

PROPOSITION 6.1.26. Every O-soft m a p between A E ( O ) - s p a c e s is f u n c t i o n a l l y open. PROOF. L e t f" X ~ Y be a 0-soft m a p a n d Y be an A E ( 0 ) - s p a c e . W i t h o u t loss of g e n e r a l i t y we can a s s u m e t h a t for s o m e infinite c a r d i n a l n u m b e r T t h e s p a c e Y is C - e m b e d d e d into R r , the space X is C - e m b e d d e d into t h e p r o d u c t R r x R r , a n d t h e m a p f coincides w i t h t h e r e s t r i c t i o n of t h e p r o j e c t i o n ~1" R r • R r ~ R r onto X . Take a m a p gl" N r --~ R r w i t h t h e p r o p e r t i e s of P r o p o s i t i o n 2X is an fK-spa~e if any real-valued function, defined on X and having continuous restrictions on every compact subspace of X, is continuous on X; see [193]

238

6. ABSOLUTE EXTENSORS

6.1.25. It follows from t h e c o n s t r u c t i o n s p r e s e n t e d in the proof of P r o p o s i t i o n 6.1.25 t h a t t h e p r o d u c t g2 -- gl

•

gl : N~

x N T

-~

R T

x R ~

is also p r o p e r a n d functionally closed (moreover, g2 is 0-invertible and a d m i t s a regular averaging o p e r a t o r ) . Observe also t h a t 7rig2 - glA1, where A I : N r x N r --, N r d e n o t e s the p r o j e c t i o n onto t h e first coordinate. Since Y is a Ce m b e d d e d A E ( 0 ) - s u b s p a c e of R r , there exists a m a p h i : N r --~ Y coinciding w i t h gl on t h e set g ~ l ( y ) . Similarly, since X is a C - e m b e d d e d A E ( 0 ) - s u b s p a c e of t h e p r o d u c t R r x R r , and since t h e m a p f is 0-soft, we conclude t h a t there is a m a p h2: N r x N r --, X coinciding with g2 on the set E g 2 1 ( X ) and such t h a t f h 2 -- hlA1.

Now consider c o m p a c t - v a l u e d and u p p e r semi-continuous

retractions

rx--h2g21:

R r x R r ---,X and r y =

h l g l l : R r ---,Y.

It follows from t h e c o n s t r u c t i o n s t h a t f r x - ryTrl. In particular, since r y is t h e identity on Y, we see t h a t for each point b E r ~ - l ( Y ) we have the equality f ( r x ( b ) -- ~rl(b). T h e functional closedness of t h e m a p g2 g u a r a n t e e s t h a t the set ~(v)

= (b e R ~ • R ~ : ~ x ( b ) c U }

is functionally open in the p r o d u c t R r • R r for each functionally open subset U of X . It is also easy to see t h a t the projection r l is functionally open. Consequently, t h e set v = ~(~x~(V)) n Y is functionally o p e n in Y. T h u s , in order to c o m p l e t e the proof, it only remains to show t h a t V = f (U). Since U c_ r x l ( U ) N l r ~ l ( Y ) , we see t h a t

f ( u ) = ~ ( u ) c_ ~ (~x~(U) n ~ ( y ) )

= ~(~x~(U))n

If y E V, t h e n there is a point b E r x l ( U ) N

Y = v.

zr~-l(Y) such t h a t 7rl(b) -- y.

As r e m a r k e d above, for t h e point b we have f r x ( b )

-- 7rl(b). B u t r x ( b ) C_ U.

Consequently,

y = ~ ( b ) = f ~ x ( b ) c_ f ( u ) . This implies the desired equality V = f ( U ) .

D

It is useful to c o m p a r e the following i m m e d i a t e consequence of P r o p o s i t i o n 6.1.26 w i t h P r o p o s i t i o n 2.1.34. COROLLARY 6.1.27. A surjection between Polish spaces is O-soft if and only

if it is open.

6.2. MORPHISMS OF SPECTRA AND SQUARE DIAGRAMS

239

Historical and bibliographical notes 6.1. Definitions 6.1.22 and 6.1.24 appeared in [171] and [39] respectively. The other notions and results of this Section are due to the author (see, for example, [79], [84], [80]).

6.2. M o r p h i s m s of s p e c t r a and square d i a g r a m s Every morphism between spectra induces the limit map between a limit spaces of given spectra (see Subsection 1.2.2). On the other hand, for "good" spectra the converse is also true: every map between the limit spaces of these spectra is induced by some morphism (see Section 1.3). Consequently, every topological property of these limit maps can be described in terms of morphisms of these spectra. From this point of view, the study of the structure of morphisms of spectra themselves becomes very important. There are several possible approaches to this problem. One of them deals with the description of properties of morphisms using the language of square diagrams, consisting of elements of morphisms and short projections of the corresponding spectra. We begin with the following definition. DEFINITION 6.2.1. A characteristic map of a commutative square diagram

/2 X~

Xl

"-Y2

fl

~-Yl

is the diagonal product p A f2 , considered as a map from the space X2 into the fibered product (i.e. pullback) of the spaces X1 and Y2 with respect to the maps

f l : X1 ---* Y1 and q: Y2 ---+ Y1. Square diagrams with surjective characteristic maps are called bicommutative (see, for example, [202], [245], [278]). A much finer way of distinguishing square diagrams is contained in the following definition. DEFINITION 6.2.2. A commutative square diagram is called n-soft, n - 0, 1, .. , co, if its characteristic map is n-soft. The co-soft diagrams are called soft. DEFINITION 6.2.3. A commutative square diagram is said to be a Cartesian square if its characteristic map is a homeomorphism.

240

6. ABSOLUTE EXTENSORS

LEMMA 6.2.4. Suppose that the spaces Y1 and ]I2 in the above square diagram, consisting of realcompact spaces, are A N E ( n ) - s p a c e s . I f the map f l is n-soft and the diagram is a Cartesian square, then f2 is also n-soft. PROOF. By P r o p o s i t i o n 6.1.16, it suffices to consider an a r b i t r a r y at most ndimensional r e a l c o m p a c t space Z, its closed subspace Zo, and two maps g2 : Zo ---+ X2 and h 2 : Y 2 satisfying the following conditions

f2g2-

h2/Zo and C ( g 2 ) ( C ( X 2 ) ) C_ C ( Z ) / Z o ,

and to prove the existence of a m a p k2: Z ---. X2 such t h a t f2k2 -- h2 and

g2 = k2/Zo. Let gl - p g 2 and hi - - q h 2 . Clearly fig1-

h l / Z o and C ( g l ) ( C ( X 1 ) ) C C ( Z ) / Z o .

T h e n-softness of the m a p f l g u a r a n t e e s the existence of a m a p k l : Z --+ X1 satisfying the following two conditions: f l k l -- hi and gl -- k l / Z o . Since our d i a g r a m is a C a r t e s i a n square, the diagonal p r o d u c t k -- k l A h 2 of the maps kl and h2 m a p s the space Z into the space X2. It only remains to observe t h a t

f2k2 -- h2 and g2 - k2/Zo.

E]

LEMMA 6.2.5. Let an n-soft square diagram, consisting of realcompact spaces, be given. I f the spaces Y1 and Y2 are A N E(n)-spaces and the map f l is n-soft, then f2 is also n-soft. Moreover, for any at most n-dimensional realcompact space Z , its closed subspace Zo, and any three maps g2: Zo ---* X2, h2: Z ---, ]I2 and kl : Z ---, X1 satisfying the conditions

f l k l = qh2, f2g2 -- h2/Zo, Pg2 -- k l / Z o and C ( g 2 ) ( C ( X 2 ) ) C C ( Z ) / Z o , there exists a map k2: Z ---, X2 such that g2 - k2/Zo, h2 -- f 2k2 and pk2 - kl. PROOF. Let X denote the fibered p r o d u c t of the spaces X l and ]I2 with respect to the m a p s f l : X 1 ~ Y1 and q: Y2 ~ ]I1. Let ~ : X --. X1 and r X ~ Y2 d e n o t e the n a t u r a l projections of the fibered product. Since the diagram

X1

fl

~Y1

6.2. MORPHISMS OF SPECTRA AND SQUARE DIAGRAMS

241

is a C a r t e s i a n square, we conclude, by L e m m a 6.2.4, t h a t the m a p r is n-soft. T h e characteristic m a p h = p/k f2 of the given d i a g r a m is n-soft by our assumption. Consequently, by L e m m a 6.1.14, the m a p f2 (being the c o m p o s i t i o n of the maps h and r is also n-soft. T h e validity of the second part of our s t a t e m e n t can be shown as follows. First, observe t h a t the diagonal p r o d u c t k l / k h 2 : Z ---+ X satisfies the following equalities: ~o(kl/kh2) = kl,

h2 and hg2 = ( k l / k h 2 ) / Z o .

r

T h e n-softness of the m a p h g u a r a n t e e s the existence of a m a p k2: Z --+ X2 such t h a t g2 = k2/Zo and hk2 = ( k l A h 2 ) . It only r e m a i n s to note t h a t

f2k2 = Chk2 = r

= h2

and

pk2 = Chk2 = ~ ( k l A h 2 ) = kl. This completes the proof.

I-1

LEMMA 6.2.6. Let S x -- { X ~ , p ~ , T} be a well ordered continuous spectrum, consisting of realcompact spaces X a and n-soft short projections paa + l . I f Xo is

an A N E ( n ) - s p a c e , then the limit projection p0: l i m S x ~ Xo is n-soft. PROOF. T h e limit space of the s p e c t r u m X = S x is r e a l c o m p a c t (see Subsection 1.1.2). Therefore we can use P r o p o s i t i o n 6.1.16. Take an a r b i t r a r y at most n - d i m e n s i o n a l r e a l c o m p a c t space Z, a closed s u b s p a c e Z0 of Z, and any two m a p s g: Z0 --~ X and h: Z ~ X0 such t h a t

Pog = h / Z o and C ( g ) ( C ( X ) ) C_ C ( Z ) / Z o . Let k0 = h. Suppose t h a t for each a < /~ we have a l r e a d y c o n s t r u c t e d m a p s k~: Z ~ X a such t h a t

pag = k a / Z o whenever a < and

ka=p~k.y

whenever a < 7

0 and for each T >_ w, t h e r e exists an (n 4- 1)-dimensional c o m p a c t u m Dnr+l of weight T a d m i t t i n g an n-soft m a p onto the T y c h o n o v cube I r . Obviously, this c o m p a c t u m serves as an e x a m p l e of an (n 4- 1)-dimensional AE(n)-compactum of weight T. It can be shown [125] t h a t we cannot lower the dimension of such c o m p a c t a . More precisely: 9 I f n >_ 1, t h e n each n - d i m e n s i o n a l AE(n)-compactum is metrizable (if n -- 0 this s t a t e m e n t is not true take the C a n t o r cube D r of any weight T). T h e reason for this lies in the m e t r i z a b l e case. N a m e l y in T h e o r e m 4.2.25, which states t h a t t h e r e is no n-soft m a p of an n - d i m e n s i o n a l metrizable comp a c t u m onto the Hilbert cube I W . On the o t h e r hand, as shown in T h e o r e m 5.1.10, every Polish space is an n-soft image of an n - d i m e n s i o n a l Polish space. Analysis of this s i t u a t i o n leads us to the hypothesis t h a t perhaps, in contrast to t h e c o m p a c t case, for each n > 0 and for each T _> w there exist n-dimensional AE(n)-spaces of weight T. In order to establish this principal fact, we need two p r e l i m i n a r y results. LEMMA 6.4.13. Let s -- (Xk, "Pk k+l ,w} be an inverse sequence, consisting of n-dimensional AYE(n)-spaces and n-soft projections. Then d i m ( l i m 8 ) < n. PROOF. In our s i t u a t i o n each of the spaces X k, k E w, can be C - e m b e d d e d in X - l i m 8 . Consequently, R - w(Xk) < R - w ( X ) for each k E w. Having in m i n d this observation, we can, based on P r o p o s i t i o n 6.3.5, represent the spaces Xk, k E w, as well as the space X , as the limit spaces of factorizing w-spectra

S I c - (X~,q~a,k,A} and S x - - (X~,p~a,A}, consisting of A N E(n)-spaces and n-soft limit projections, and with the same indexing set A. F u r t h e r , by T h e o r e m s 1.3.6 and 1.3.10, we can w i t h o u t loss of generality assume t h a t all spaces in the s p e c t r a 8k are at most n-dimensional. A p p l y i n g T h e o r e m 6.3.2 to each of the short projections of the sequence 8, one can see t h a t for each k E w there is a b i c o m m u t a t i v e (even n-soft) m o r p h i s m M k+l

k

e

k4-1,a

= lPk

}" S k + ~ - ~ S k

the limit m a p of which coincides with the projection pk+l In this situation, for k 9 each index a E A we get an inverse sequence k+l,a

consisting of n - d i m e n s i o n a l Polish spaces (and n-soft projections). T h e bicomm u t a t i v i t y of all n a t u r a l l y occuring square d i a g r a m s g u a r a n t e e s t h a t X a = lim 8 a for each a E A. Consequently, each of the spaces Xa, a E A, as the limit of the sequence 8a, consisting of n - d i m e n s i o n a l Polish spaces, is n-dimensional. A p p l y i n g L e m m a 1.3.7, we conclude t h a t dim X < n. T h e reverse inequality

6.4. FURTHER PROPERTIES OF AE(0)-SPACES

269

follows from the remark made at the beginning of the proof (that X contains a C-embedded copy of each Xk). [-q LEMMA 6.4.14. Let S x = {Xa,pBa,~"} be a continuous well ordered spectrum, consisting of n-dimensional ANE(n)-spaces and n-soft projections. Then dim(lim S x ) = n. PROOF. We consider two cases. First, if the spectrum S x contains a countable cofinal subspectrum, then our s t a t e m e n t follows from L e m m a 6.4.13. Suppose now than there is no countable cofinal subspectrum in S x . Then this spectrum is w-complete. By L e m m a 6.2.6 and Proposition 6.1.21, the limit space lim S x is an A N E (n )-space. Consequently, by Proposition 6.1.8, its Suslin number is countable. Therefore, based on Proposition 1.3.3, we can conclude t h a t S x is a factorizing spectrum. Now the inequality dim(lira S x ) _w. Each ANE(n)-space of weight T is an n-soft image of n-dimensional space of weight n. PROOF. We proceed by induction on T. For ANE(n)-spaces of countable weight, i.e. for Polish A N E (n )-spaces (see Proposition 6.1.10), our s t a t e m e n t is contained in Theorem 5.1.10. Suppose t h a t the theorem has already been proved for AgE(n)-spaces of weight < ~- and consider an arbitrary AgE(n)-space Z of weight T. By Proposition 6.3.4, X can be represented as the limit space of a continuous well ordered spectrum

Sx = {x~,pg +~, ~} consisting of ANE(n)-spaces of weight < T and n-soft projections. By induction, let us now construct a continuous well ordered spectrum

s v = {y~, q~+~, ~ } "parallel" to the given one and consisting of n-dimensional ANE(n)-spaces. We also construct a morphism {f~}: S y -:-+ SX, consisting of n-soft maps. By the inductive hypothesis, there exist an n-dimensional A N E (n )-space ]Io of weight w(Yo) = w(Xo) and an n-soft map fo: Yo ~ Xo. Suppose t h a t for each a < 3', where 3' < A, we have already constructed: (a) an n-dimensional ANE(n)-space Y~ of weight w ( Y a ) = w(X~). (b) an n-soft map f a : Ya ~ X~. (c) n-soft maps q ~ ' Y 3 ~ Y~ (a < 3 < 7, satisfying the following two conditions: _ q~=q~qt~ for e a c h a 5 , 3 w i t h a < 5 < 3 < - y

270

6. ABSOLUTE EXTENSORS All square diagrams of the type

Y.+~

* Xa+l

p~,+~

qg+l

Y~

~X~

are n-soft (a + 1 < "r). We now construct the space X r and maps f r and qa7 with a < 7. If "r is a limit ordinal, then X r --lim{Y~,q~,'r}

and f r -

lim{f~" a < "r}.

By L e m m a s 6.2.6 and 6.4.14, X r is an n-dimensional ANE(n)-space of weight w(Yr) -- w(Xr). L e m m a 6.2.7 guarantees the n-softness of the map f r : Yr X r. We define the maps qa7 as the limit projections of the spectrum the limit space of which coincides with Yr" These maps are n-soft by L e m m a 6.2.6. It is easy to see t h a t all naturally occuring square diagrams are n-soft. Next we consider the case "r - a + 1. Consider the fibered p r o d u c t X of spaces Y~ and X r with respect to the maps fa and p7a. Denote by ~o" Z ~ Ya and r Z ~ X r the canonical projections of this fibered product. By L e m m a 6.2.4, ~o and r are n-soft maps. Consequently, the space Z is an A N E (n )-space (Proposition 6.1.21). Since the space X is contained in the p r o d u c t Y~ x X r , its weight is less t h a n r. Consequently, by inductive hypothesis, there exist an n-dimensional ANE(n)-space Yr of weight w(Z) and an n-soft map h ' Y r ~ Z. Let f r - Ch and q~ - ~oh. Obviously, these maps are n-soft ( L e m m a 6.1.14) and the newly arising square diagram is also n-soft (because its characteristic m a p coincides, by construction, with nsoft map h). Thus the s p e c t r u m 8 y and the morphism {f~}, satisfying all the required properties, is constructed. Finally, let Y - lim Sy and f -- lim{fa }. Applying L e m m a s 6.2.6, 6.2.7 and 6.4.14, we conclude t h a t Y is an n-dimensional V1

ANE(n)-space of weight w(X) and f : Y --. X is an n-soft map.

Consider now an n-dimensional space R~ of weight T which admits an nsoft map onto R r (we have just proved the existence of such spaces in T h e o r e m 6.4.15). If n -- 0 we can, by T h e o r e m 6.4.2, assume t h a t R~ = N r . These

6.4. FURTHER PROPERTIES OF AE(0)-SPACES

271

spaces have specific universality properties. We will investigate these in the next Section. Thus, in contrast with the c o m p a c t case, for each n _ 0 there are n-dimensional A E (n )-spaces of any weight. Is it possible to lower the dimension of such spaces? In order to answer this question we need the following lemma. LEMMA 6.4.16. Let F" X --~ Y be an n-soft map between Polish spaces and d i m Y -- k < n. I f [ f - l ( y ) [ >_ 2 for each point y 6 Y , then d i m X > k. PROOF. Since dim Y -- k, there is a point y0 6 Y such t h a t every neighborhood of y0 in Y is exactly k-dimensional. By our assumption, the fiber f - l ( y o ) contains at least two points, say x0 and Xl. By the n-softness of the m a p f , we can find two maps i0, i1" Y ~ X satisfying the following conditions: (a) fire = i d y for each m = 0, 1. (b) im(YO) -- Xm for each m -- 0, 1. Let U - {y 6 Y" io(y) 7~ il(y)}. Obviously, U is an open n e i g h b o r h o o d of the point y0 in Y. By the choice of y0, we have d i m U - - k. Let V - - f - l ( u ) and g - f / V . T h e n g" V ~ U is an n-soft m a p with two sections (namely: j m - ira~U, m = O, 1) with disjoint images. Since dim U = k, a t h e o r e m on partitions (see, for example, [146, page 67]) shows t h a t there exists a collection T -- ((Ap, Bp)" 1 < p < k} of closed subsets of V such t h a t Ap N Bp -- q) for each p - 1, 2 , . . . , k and for any choice of partitions Cp (in U) between Ap and Bp their intersection N{Cp" 1 < p _ dim V > k. [:] PROPOSITION 6.4.17. If X is an non-metrizable ANE(n)-space, then d i m X

>n. PROOF. Obviously, our s t a t e m e n t is true for n - 0. It is also clear that it suffices to consider only the case of spaces of weight wl. Thus, let X be an ANE(n)-space of weight wl, n > 1 and suppose the contrary, i.e. assume t h a t dim X - k < n. W i t h o u t loss of generality, according to Corollaries 6.4.4 and 6.4.5, we may assume t h a t there is no point of countable pseudo-character in X. Represent X as the limit space of a factorizing Polish spectrum S x = { X ~ , p ~ , A } , consisting of Polish ANE(n)-spaces and n-soft limit projections (Proposition 6.3.5). By T h e o r e m s 1.3.6 and T:3.2.23, we may additionally assume t h a t dim X a - k for each c~ E A. Take an arbitrary index c~ E A. If x is a point in X a , then the fiber p-gl(x) is functionally closed in X. Consequently, the absence of points of countable pseudo-character in X guarantees t h a t this fiber contains at least two points. But then there is an index/~x E A with/3x > c~ such t h a t the fiber (pflax)-l(x)is also non-trivial. The openness of the projection p~x guarantees the existence of an open neighborhood Vx of the point x in X a such t h a t I ( p ~ ) - l ( y ) ! >_ 2 for each point y E Vx. Take a countable subcover {Vx,~ " m E w} of the open cover {Vx" x E Z a } of the Polish space Xa and let /~ - sup/~x.~. Next observe t h a t for each point x E X a we have ](p~)-l(x)] > 2. By L e m m a 6.4.16, dim X~ > k. This contradiction completes the proof. D

COROLLARY 6.4.18. Every finite-dimensional ANR-space is metrizable.

Historical and bibliographical notes 6.4. T h e results of this Section are due to the a u t h o r [80], [84], [85]. A particular case of Corollary 6.4.3 can be found in [251]. In the compact case Corollaries 6.4.3, 6.4.5, and 6.4.18, as well as L e m m a 6.4.16 were obtained in [275], [276]. For the same class of spaces, Proposition 6.4.9 appears in [171]. T h e metrizability of 1-dimensional A N R-compacta was established earlier by Isbell [179].Characterizations of AE(0)-spaces in terms of many-valued maps are given in [109].

6.5. STRONGLY UNIVERSAL SPACES

273

6.5. S t r o n g l y u n i v e r s a l s p a c e s

6.5.1. S p a c e s o f m a p s ( c o n t i n u a t i o n ) . Let X and Y be arbitrary Tychonov spaces and T be an arbitrary infinite cardinal number. We are going to introduce a topology, depending on T, on the set C(Y, X) of all continuous maps from Y into X. The space thus obtained will be denoted by Cr(Y, X ) . We recall that coy(X) denotes the collection of all countable functionally open covers of the space X. For each map f : Y --~ X the sets of the form

B(f,{blt: t 9 T } ) = {g 9 C r ( Y , X ) : g is L/t - close to f for e a c h t 9 T}, where ITI < T and L/t 9 coy(X) for each t 9 T, are declared to be open basic neighborhoods of the point f in C~(Y, X). The maps, contained in the neighborhood B ( f , {L/t: t 9 T}) are called {L/t: t 9 T}-close to f. If a space X has a countable basis, then, obviously, the space C~o(Y,X) (for any space Y) coincides with the space C ( Y , X ) , endowed with the topology introduced in Subsection 2.1.1. We will make use of the following simple lemma. LEMMA 6.5.1. Let X be an AE(o)-space of infinite weight 7. Then there exists a subcollection {L/t: T} of the collection coy(X) such that for any space Y and for any two maps f, g: Y - . X the following conditions are equivalent: (i)

f

= g.

(ii) The maps f and g are {L/t: t 9 T}-close. PROOF. If X is a Polish space, then take a metric d generating the topology of X, and denote b y / i n an arbitrary countable open cover of X, the diameters of whose elements are less than 1_. Obviously, {b/n} is the desired collection. n Next consider the case when X is a non-metrizable AE(0)-space. Represent X as the limit space of a factorizing Polish spectrum S x = { X t , P tt ~, T}. Here the indexing set T of the spectrum S x satisfies the equality IT] = w ( X ) . In each of the Polish spaces Xt, take a countable collection {L/~" n 9 N } as considered above. Then the collection {ptl(Unt)" n e N , t 9 T} has all the required properties.

El

The above topology on the set of continuous maps allows us to give a precise meaning to the notion of strong universality in the general case (compare with Definition 2.3.12). Consider an arbitrary subclass P of the class of all Tychonov spaces. If r _ w and n = 0, 1 , . . . , c o , we let

"Pr,n= { X C'I9" R - w ( X )

_ w.

Therefore we have

T h e n the T y c h o n o v cube I r is strongly Br,oo-

universal.

In a similar way, one can prove the 0-dimensional counterpart of the above result. LEMMA 6.5.3. Let T >_ w. T h e n the C a n t o r cube D r is strongly Br,o-universal. Next we consider the non-compact case.

6.5. STRONGLY UNIVERSAL SPACES

275

LEMMA 6.5.4. Let T > w. T h e n the space N r is strongly Jtr,o-universal. P R O O F . F i r s t c o n s i d e r t h e c a s e T -- w.

L e t f " Y -+ N w b e a m a p of a n

a r b i t r a r y z e r o - d i m e n s i o n a l P o l i s h s p a c e Y. L e t / 4 -

{Un" n E w} be a c o u n t a b l e

o p e n cover of N ~ . O u r g o a l is to c o n s t r u c t a closed e m b e d d i n g g" Y --. N W , /4-close to f .

Without

loss of g e n e r a l i t y , we m a y a s s u m e t h a t /4 c o n s i s t s of

p a i r w i s e d i s j o i n t sets. L e t Yn = f - l ( U n ) , s i o n a l P o l i s h s p a c e , n E w.

n E w. T h e s p a c e Yn is a z e r o - d i m e n -

T h e s p a c e Un (as a n o p e n s u b s p a c e of N ~ ) d o e s

not contain an open compact subspace and, consequently, by Theorem is h o m e o m o r p h i c

to N ~ , n E w.

By Proposition

1.1.5,

1.1.6, t h e r e e x i s t s a c l o s e d

e m b e d d i n g gn" Yn --'+ Un for e a c h n E w. I t o n l y r e m a i n s to n o t e t h a t t h e m a p g" Y --~ N ~ , c o i n c i d i n g w i t h gn on Yn, n E w, is a c l o s e d e m b e d d i n g / g - c l o s e

to

S. L e t us now c o n s i d e r t h e case T > w. T a k e a n a r b i t r a r y set A of c a r d i n a l i t y r a n d let Y be a z e r o - d i m e n s i o n a l s p a c e o f R - w e i g h t < T. C o n s i d e r also a m a p

f" Y ~

N A a n d its n e i g h b o r h o o d O f in t h e s p a c e C r ( Y , N A ) . W i t h o u t

loss of

g e n e r a l i t y we m a y a s s u m e t h a t

o f = {g ~ C~(Y, N A ) 9 ~Bg = ~ . f } w h e r e B is a s u b s e t of A of c a r d i n a l i t y < T a n d r B " N A ---+ N B s t a n d s for t h e natural projection onto the corresponding subproduct. space Y admits a C-embedding

B y T h e o r e m 6.4.2, t h e

i n t o N ~ 4. Since IA - B I = T, it follows t h a t

t h e r e is a C - e m b e d d i n g j" Y --~ N A - B .

Then the desired C-embedding

of Y

into N A can be defined by letting

i - ~vBf/kj. This completes the proof.

[-1

LEMMA 6.5.5. Let r >_ w. T h e n the space R r is strongly .Ar,oo-universal. P R O O F . If

r

--

w, t h e n t h e c o n c l u s i o n follows f r o m P r o p o s i t i o n 2.3.14.

An

a r g u m e n t s i m i l a r t o t h a t u s e d in L e m m a 6.5.4 verifies t h e r e m a i n i n g c a s e s (i.e.

r>w).

r-1

T h u s , if n = 0 or n = oo, s t r o n g l y A r , n - u n i v e r s a l s p a c e s e x i s t for e a c h r _> to. I n t h e m e a n t i m e , as s h o w n in T h e o r e m

5.1.12, s t r o n g l y A w , n - u n i v e r s a l s p a c e s

e x i s t for e a c h n E to. D o we h a v e a s i m i l a r s i t u a t i o n in t h e n o n - m e t r i z a b l e c a s e ? The following statement answers this question. PROPOSITION 6.5.6. Let T > w a n d n -- 0, 1 , . . . , c~.

Each n-dimensional

space o f weight r , a d m i t t i n g an n - s o f t m a p onto R T , is strongly ~4r,n-universal. 4Indeed, consider a 0-soft map ~a" N r --~ R r . Since R - w(Y) ~_ r, we may assume that Y is C-embedded in R r . Since dim Y -- 0, we conclude, by the 0-softness of ~, that there is a map r Y --~ N r such that ~0r = idy. It only remains to note that r is a C-embedding.

276

6. ABSOLUTE EXTENSORS

PROOF. Let f " X ~ R r be an n-soft map, where X is an n-dimensional space of weight T. Consider an arbitrary at most n-dimensional space Y of R-weight _ r. Consider also a map g" Y ~ X and its neighborhood B ( g , {lgt" t e T}), [T I < r, /At E c o y ( X ) ,

t E T

in the space C r ( Y , X ) . Let A = max{w, [T[}. Since r is uncountable, we have A < T. Represent the space R ~ as the limit space of the s t a n d a r d (factorizing) A-spectrum S = {Za, lrO, A} where each of the spaces Za is homeomorphic to R x and each of the limit projections 7ra" R r --. Z a is the trivial bundle with fiber R r . Since X, as an n-soft preimage of R r , is an A E ( n ) - s p a c e , we conclude, applying Proposition 6.3.3, t h a t there exist a factorizing A-spectrum S x = { X a , p ~ , A } , consisting of A E ( n ) - s p a c e s of weight A, and a morphism {f~}" S x -+ S, consisting of n-soft maps, such t h a t (i) Z - - l i m S x and f = lim{fa}. (ii) All the naturally occuring 5 limit square diagrams are n-soft. Since S x is a factorizing ,k-spectrum and [T I < ,X, there exist an index a E A and functionally open countable covers/A~ E c o v ( X a ) , t E T , such that /At = pffl(L/~) for each t E T. Consider the corresponding limit square diagram:

Y X

Pc~

X0

,R r

Z

for

lr~

* Za

Here Z denotes the fibered p r o d u c t of the spaces X~ and R r with respect to the maps fa and 7ra. As usual r Z -~ R r and ~o: Z ~ X a denote the canonical projections of this product. T h e map h: X ~ Z is the characteristic map of the indicated diagram, which is n-soft by condition (ii). T h e m a p 7ra is the trivial bundle with fiber R r . Consequently, by the inequality R - w ( Y ) < T, there is a C - e m b e d d i n g i: Y ~ R r such t h a t 7 r a i - 7raf g. T h e n f a P a g = 7raf g = 7rai.

5Generated by the limit projections of the spectra Sx, S, elements of the morphism { f a } and by the map f.

6.5. STRONGLY UNIVERSAL SPACES

277

B u t in this s i t u a t i o n the diagonal p r o d u c t j = p a g / k i maps the space Y into Z. Since i is a C - e m b e d d i n g , we see t h a t j is also a C - e m b e d d i n g . O b s e r v e also that Cj--i and ~ j - p a g . Since d i m Y < n, the n-softness of the characteristic m a p h g u a r a n t e e s the existence of a m a p g~" Y -+ X such t h a t hg ~ -- j . Clearly, g~ is a C - e m b e d d i n g . It only remains to show t h a t !

g ~_ B ( g , { b l t ' t e T}).

This fact follows i m m e d i a t e l y from the following equalities I

P a g = ~ h g I = ~ J = Pag

and from the choice of t h e index a E A. T h e proof is completed.

[-]

A similar a r g u m e n t proves t h e following s t a t e m e n t . PROPOSITION 6.5.7. L e t r > w.

Each compactum

of weight r, admitting a

soft m a p o n t o I r , is s t r o n g l y B r , c ~ - u n i v e r s a l .

THEOREM 6.5.8. L e t r >_ w a n d n = O, 1 , . . . , c~. T h e n there e x i s t s a s t r o n g l y .Ar,n-universal AE (n)-space R~.

PROOF. If n = 0 or n = oc, t h e n the result follows from L e m m a s 6.5.4 and 6.5.5. If T = w and n is a positive integer, it suffices to a p p l y T h e o r e m 5.1.12. Finally, if T > w, apply T h e o r e m 6.4.15 and P r o p o s i t i o n 6.5.6. El

H i s t o r i c a l a n d bibliographical n o t e s 6.5. All t h e results of this Section are due

to the a u t h o r [84], [83].

CHAPTER

7

T o p o l o g y of N o n - M e t r i z a b l e M a n i f o l d s

7.1. N o n - m e t r i z a b l e

manifolds

7.1.1. D e f i n i t i o n o f m a n i f o l d s . We are now ready to define the f u n d a m e n tal concepts of R r -manifolds and I r -manifolds for r > w. First let us ask: is it possible to extend the usual definition of R ~ - (or I W-) manifold to the nonmetrizable case? In other words, is the following definition, stating t h a t R r - ( I r - ) manifolds are precisely those spaces locally h o m e o m o r p h i c to R r (respectively, I r ), satisfactory? Let us discuss this question more formally. Consider the simplest case. Let T > w and x be an a r b i t r a r y point in R r . Let U = R r - x . Obviously, U is locally h o m e o m o r p h i c to R r . In the meantime, U is not an A N R - s p a c e and, moreover, is not realcompact 1. But u n d e r a "good" definition, each R r -manifold must be realcompact (since the model space R r is an absolute retract; compare with Corollary 6.1.6 and Proposition 6.1.7). Therefore the s t a n d a r d definition has to be a d a p t e d to the non-metrizable case. In order to u n d e r s t a n d exactly w h a t modifications have to be performed, we continue our analysis and ask the next question: which open subspaces of R r are realcompact? T h e complete answer to this question is provided by Corollary 6.4.11 these are precisely the functionally open subspaces of R r . In turn, each functionally open subspace U of R r can be represented as the p r o d u c t V x R r , where V is an open subspace of R ~ , i.e. an R ~ -manifold. Obviously, V has a countable functionally open cover each element of which is h o m e o m o r p h i c to R ~ . Therefore, the space U also admits a countable functionally open cover each element of which is h o m e o m o r p h i c to R ~ x R r ~ R ~ . W i t h these observations in mind, we arrive at the following definition. DEFINITION 7.1.1. A s p a c e is c a l l e d a n R r - m a n i f o l d countable

functionally

open

cover

each

element

of which

R r .

1The last fact immediately follows from Proposition 1.1.24. 279

(T >_ w ) ,

if it has

is h o m e o m o r p h i c

a to

280

7. NON-METRIZABLE MANIFOLDS

Of course, in t h e case ~- -

w t h e a b o v e definition coincides with t h e u s u a l

definition of R ~ -manifolds. In a d d i t i o n , it is obvious t h a t each functionally o p e n s u b s p a c e of R r is an R r -manifold. A similar discussion leads to t h e following definition. DEFINITION 7.1.2. A space is called an I r -manifold (T > w), if it has a countable functionally open cover each element of which is homeomorphic to a f u n c t i o n a l l y open subspace of I r . Obviously, each I r - m a n i f o l d is locally c o m p a c t a n d LindelSf. O u r n e x t goal is to show t h a t R r - a n d I r -manifolds are A N R-spaces. This allows us to a p p l y the s p e c t r a l t e c h n i q u e d e v e l o p e d in C h a p t e r 6. T h e p r o o f of t h e following s t a t e m e n t is trivial and is left to the reader. LEMMA 7.1.3. A functionally open subspace of an A N R-space is an A N R space. PROPOSITION 7.1.4. I f a space X is represented as a countable union of its f u n c t i o n a l l y open subspaces, each of which is an A N R-space, then X is also an A N R-space. PROOF. If X is m e t r i z a b l e , t h e n t h e s t a t e m e n t follows from the well-known result of [161]. C o n s i d e r t h e case of n o n - m e t r i z a b l e X . It follows from L e m m a 1.1.18 a n d P r o p o s i t i o n 1.1.26 t h a t the space X is r e a l c o m p a c t . Therefore, we can a s s u m e t h a t X is a closed a n d C - e m b e d d e d s u b s p a c e of t h e space R r for s o m e u n c o u n t a b l e T. Let X ---- U{Xi" i E N } , w h e r e each Xi is a functionally o p e n A N R - s u b s p a c e of X . T h e p r o o f of t h e s t a t e m e n t is di vi ded into t h r e e parts. Case

1. First consider t h e case w h e n t h e r e are only finitely m a n y m e m b e r s

in t h e a b o v e r e p r e s e n t a t i o n of X .

In this case, w i t h o u t loss of generality, we

can a s s u m e t h a t X - X1 U X2. Let ~" X ~ [1,2] be a c o n t i n u o u s function such t h a t ~ - 1 ( i ) - X - X i , i - 1,2. Since X is C - e m b e d d e d in R r , t h e r e is a f u n c t i o n ~" R ~ --* [1,2] such t h a t ~ / X - ~. T h e sets ~ - 1 ( i ) , i - 1,2, are disjoint f u n c t i o n a l l y closed s u b s e t s of R r . C o n s e q u e n t l y , t h e r e exist functionally o p e n s u b s e t s O1 a n d 0 2 in R ~ such t h a t ~ - 1 ( i ) _c Oi a n d O1 M 0 2 - ~, i = 1, 2. Let Zi - R r - Oi, i -- 1, 2. E v i d e n t l y t h e sets Z1 a n d Z2 are functionally closed in R r a n d satisfy t h e following conditions:

Z1 U Z2 -

R r a n d Z0 M X C_ X0,

w h e r e Zo - Z1 M Z2 a n d Xo - X1 M X2. Let us show t h a t X0 is C - e m b e d d e d in Xo U Z0. In o r d e r to do this we have to verify t h e following two properties: (i) X0 is z - e m b e d d e d in X0 U Z0. (ii) If Z is f u n c t i o n a l l y closed in X0 U Z0 a n d Z M X0 - 0, t h e n Z and X0 can be f u n c t i o n a l l y s e p a r a t e d in Xo U Zo. Let Z be a f u n c t i o n a l l y closed s u b s e t of Xo. Since X0 is z - e m b e d d e d in X , t h e r e is a f u n c t i o n a l l y closed s u b s e t Z ~ in X such t h a t Z = Z ~ M X0. Since X is even C - e m b e d d e d in R ~ , t h e r e is a f u n c t i o n a l l y closed s u b s e t Z ~ in R r such that Z ~MX

-

Z ~. Let F -- Z ~ M ( X O U Z 0 ) .

It can easily be seen t h a t F is

7.1. NON-METRIZABLE MANIFOLDS

281

functionally closed in Xo U Zo and F M Xo = Z. This finishes the verification of property (i). Let us now verify property (ii). If Z is a functionally closed subset of Xo U Zo and Z M X0 = 13, then Z C Z0 - X. Since, by Corollary 6.4.8, every functionally closed subset of R r is C - e m b e d d e d in R r , we conclude t h a t Z is functionally closed in R r . But X is C - e m b e d d e d in R r . Consequently, X and Z are functionally separated in R r . This shows t h a t Xo and Z are functionally s e p a r a t e d in X0 U Z0 and finishes the verification of p r o p e r t y (ii). Thus, X0 is C - e m b e d d e d in X0 U Zo. By L e m m a 7.1.3, X0 is an A N R - s p a c e . Therefore there exists a retraction r0" V ~ X0, where V - is a functionally open neighborhood of Xo in X0 U Z0. Obviously the set (X0 U Z0) - V is functionally closed i n X o U Z o and ( X o U Z o ) - V c Z0-X. Therefore, ( X o U Z o ) - V is functionally closed in R r . Once again using C - e m b e d d e d n e s s of X in R r, we can conclude t h a t there exists a functionally open subset U of R r such t h a t X C U and U M (X0 U Zo) c_ V. It can now be easily seen t h a t by setting

ri(a) = { a,~'~

ifif aa Ee xi(X~176

CI Zi M U

we obtain a m a p ri" ( X M Zo) U (Zo M U) ~ X i such t h a t C ( r i ) ( C ( X i ) ) C C ( U M Z i ) / ( ( X N Zi) U (Z0 M U)), i = 1,2. Since X i is an A N R - s p a c e , there exists an extension gi" Gi ~ X i of the m a p ri, where Gi is a functionally open neighborhood of the set ( ( X M Z ~ ) U ( Z o M U ) ) in UMZ~, i = 1,2. A straightforward a r g u m e n t shows t h a t the set G = G1 U G2 is a functionally open n e i g h b o r h o o d of X in R r . It only remains to observe t h a t the map r" G --. X , defined by setting r(a) = ri(a) for a e Gi (i = 1,2), is a retraction. C a s e 2. Now we consider the case when X = U { X i " i E N } and Xi M X j = 13 for i ~ j , i , j E N . Consider a function ~o" X --+ R defined by setting ~o(Xi) = i, i E N. Since X is C - e m b e d d e d in R r , there is a function r R r --. R e x t e n d i n g ~o. T h e pairwise disjoint sets Ui = r 89 i + 1)) are functionally open in t:l r , and U i M X = X i for each i E N. Obviously, Xi is C - e m b e d d e d in Ui, i E N. Consequently, since each Xi is an A N R-space, there is a retraction ri" Vi ~ X i , where 1I/ is a functionally open in Ui (and, hence, in R r ) n e i g h b o r h o o d of Xi. Let V = U{Vi" i E N } and r(a) = ri(a) for each a E 1I/ and i E N . Clearly, r ~ V --. X is a retraction and V is a functionally open n e i g h b o r h o o d of X in Rr . C a s e 3. We are now in position to consider the general case. Let Gi = X1 U 9 .. U X i , i E N . T h e sets Gi are functionally open in X and U{Gi" i E N } = X . By (a), Gi is an A N R - s p a c e , i E N . T h e C - e m b e d d e d n e s s of X in R r g u a r a n t e e s the existence of a countable subset T C T and open subsets Oi in ~rT(X) such t h a t ~rTl(Oi)gl X = Gi (~rT" R r --* R T denotes the n a t u r a l projection onto the corresponding s u b p r o d u c t ) , i E N. Since T is countable, l r T ( X ) has a countable base and, consequently, we can fix a metric d which induces the topology of

282

7. NON-METRIZABLE MANIFOLDS

lrT(X).

Let

f~ = (~ 9 -r(X)" d ( ~ , ~ r ( X ) - O~)> 1}. It is easy to see t h a t cl(F~) C F i + l and ~ r T ( X ) -- U{Fi" i E N}. Let { F~, K i --

ifi=l,2 if i = 3, 4 , - . .

Fi - c l ( F i _ 2 ) ,

Let Li -- ~rTI(Ki) MXi, i E N. T h e n X - U{Li" i E N}, and L j M L i + 2 = 0 for each i E N and j _ i. Obviously, L~ is functionally open in G~ and consequently, by L e m m a 7.1.3, is an A N R - s p a c e , i E N . By Case 2, the spaces A ---- U{L2i-I: i E N } and B ---- U{L2~: i E N} are A N R - s p a c e s . Observe now t h a t A and B are functionally open subspaces of X and X - A U B. Therefore, by Case 1, X is an A N R - s p a c e . [-1 COROLLARY 7.1.5. E v e r y R r - m a n i f o l d is an A N R - s p a c e . COROLLARY 7.1.6. E v e r y I r - m a n i f o l d is an A N R - s p a c e . LEMMA 7.1.7. L e t Y E { R r, [-1, 1]r}, T >_ w. T h e n there exists a h o m e o m o r p h i s m f " y N • I ---, Y N

• I s u c h t h a t the f o l l o w i n g c o n d i t i o n s are satisfied:

(i) ~ r l f ( y , t ) -- t f o r each ( y , t ) e y g

• I , w h e r e 7ri" y g

• I --~ I d e n o t e s

the p r o j e c t i o n o n t o the s e c o n d f a c t o r .

(ii) f ( ( 0 , . . - , 0 , . . - ) , t ) = ( ( 0 , . - . , 0 , . . - ) , t ) f o r each t e I . (ii) f (y, O ) = (y, O ) / o r each y e y g . (iv) /((Yl,Y2, Y3,"" , Y n , ' " ) , 1) = ((Y2, Y3,"" , Y n , ' " ) , 1) / o r each { Y n } e yN.

PROOF. Let g" Y x Y x I ~ Y x Y be a reflection isotopy from L e m m a 2.3.4. T h e desired h o m e o m o r p h i s m / " Y N x I --~ Y N X I can be defined as follows: f((Yl,Y2, Y3,''" , Y n , ' ' ' ) , t )

-- ((g(Yl,Y2, t ) , y 3 , ' ' ' , Y n , ' ' ' ) , t ) .

[3 T h e following L e m m a is an i m m e d i a t e consequence of the above statement. LEMMA 7.1.8. L e t Y ~ . y g x y N x I --+ Y N

(i)

7ri~o(x , y,

E { R r, [-1, 1]r}, T > w.

T h e n there exists a m a p

x I s u c h t h a t the f o l l o w i n g c o n d i t i o n s are satisfied:

t ) - - t f o r each ( y , x , t) E y g x y N x I .

(ii) T h e r e s t r i c t i o n ~ / Y g x Y g

X (0,1)'yNxyN

x (0,1) ~ y N x

(0,1)

is a h o m e o m o r p h i s m .

(ii) ~ ( y , x , 0 ) = (y, 0) f o r each (y,x) e y Y x y g . (iv) ~ ( y , x , 1 ) = (x, 1) f o r each ( y , x ) E y N x y g . (v) ~o((0, . . . , 0, . . . ), (O, . . . , O, . . . ), t) -- ((O, . . . , O, . . . ), t) f o r each t e I .

7.1. NON-METRIZABLE MANIFOLDS

283

(vi) For each n >_ 1, there exists e E (0, 1) such that if t E [0, e] and 1 1. The softness of the

290

7. NON-METRIZABLE MANIFOLDS

projection Paoal guarantees t h a t it has two sections i ~ i~- Xao -~ X a l such that ~ ill(Z). Let

i~

=

9

Clearly, Vx is a n o n - e m p t y 2 open subset of Xao. Let 7 < v+ (by v + we denote the minimal cardinal n u m b e r greater than T). Suppose t h a t for each f~, 1 < f~ < 7, we have already constructed an index a 0 e A, an open subset Vf~ in Xao and two sections i~, i~" Xao ~ Xf~ of the projection p~" X~ ~ Xao, satisfying the following conditions: (i) a~ < a~, whenever 1 _ 6 < / 3 < 7. (ii) a ~ - sup{a6" 6 < f~}, whenever f~ is a limit ordinal. (iii) V6 c Vf~, whenever 1 _ ~ < f~ < 7. (iv) V0 -- U{V~" 6 < f~}, whenever/3 is a limit ordinal. (v) = p,~, z~, whenever 1 s 6 < f~ < 7 and k = 0, 1. (vi) i~ = A{i~: ~ < f~}, whenever f~ is a limit ordinal and k = 0, 1. (vii) Va = {x ~ X,~ o " i~ ~ i~(x)}. We shall now construct an index a. r ~ A, an open subset V7 of X~ o and two sections iTk" X~ o ~ XT, k = 0, 1, of the projection Pao ~ , satisfying all the required properties. First assume t h a t 7 is a limit ordinal. It follows from conditions (i) and (ii) t h a t the collection { a a - f~ < 7} forms a chain in the indexing set A. Since 7 < r + , the cardinality of this chain does not exceed r. By the r-completeness of A there exists an index a7 = sup{aft" f~ < 7} E A. By the continuity of the s p e c t r u m 3 x , the space X 7 is naturally homeomorphic to the limit space of the s p e c t r u m {Xaa , pa~,/3, as 6 < 7 }. Therefore, by conditions (v) ~ .k __ i ~ and (vi), the maps z9k" Xao --* Xa~, k -- 0, 1, satisfying the equalities paaz~ for each k -- 0, 1 and each/3, with 1 < f~ < 7, are well defined. In particular, the 9k k 0, 1 are sections of the projection P(~0 a~ m a p s ~7, Let V.~- {x e Xao" i~ ~ i~(x)}. We shall verify conditions (iii) and (iv). Let/3 < 7. Consider an arbitrary point x e Vf~. By condition (vii), i~ ~ i~(x). But, by construction, pan.~Ta':k ~ = i~, k -- 0, 1. Consequently, Pa~Z7

91 (x) . This proves the inclusion V~ C V~ (condition (iii)). and obviously i~ ~ z.y Consequently, 117 D U{V~" fl < 7}. Let us prove the converse inclusion (and 9 # finish the verification of condition (iv)). Let x E V.r. By definition, z~ It is not hard to see t h a t there exists an index f~ < 7 such t h a t i~ ~ i~(x). Consequently, by condition (vii), x E V~ _c U{V~" f~ < 7}. Thus the above constructed objects satisfy all the required properties. 2Since z E 171.

7.2. Ir -MANIFOLDS Consider now the case ~ - f l + l . z E X(, o - V~, and let

291

Suppose t h a t V~ ~ X a o. Take a point

9 ~ = ~(~) ~ ~(X,o) c x,~. Consider the fiber p-l(x/~). Since the weight of the space Xao does not exceed af~ T, and the space X is homogeneous with respect to pseudo-character and has -1(~) weight strictly more than T, we conclude, by L e m m a 7.2.2, t h a t the fiber Pa~ contains at least two different points. Then there is an index a~ E A such t h a t a~ > a~ and

I(p~,)-~(~)l > 1 f~ Softness of the projection Pa~ a~ guarantees the existence of sections

9o ,.;1 . X a ~ ._~ Xa.r of the projection such that j~(xf~) ~ j ~ ( x z ) . The sections ,~, o~,,t Pao can now be defined by letting 9k

.k .k

~,.y = 3 / ~ / ~ ,

k

=

0,

1.

As above, if we let

0 v~ = {~ e X.o" ~(~) # ~~(~)}, then Vf~ _C VT. Observe also that, by construction, z E V ~ - V#. Thus the construction of the above objects, satisfying conditions (i) (vii), can be continued and carried out for each/~ < r +. Then we obtain an increasing (by conditions (iii) and (iv)) collection {V#:/~ < T +} of open subsets of Z o o . Since w ( X a o ) ~ ( g ) >_ ~ , t h ~ f has two sections with disjoint images.

x i~ ho.~og~eo~ ~ith ~ p ~ a to p ~ d o - ~ h a ~ a ~

PROOF. Let T = w ( Y ) and represent X as the limit space of a factorizing T-spectrum S x -- { X a , p ~ , A } , consisting of A N R - s p a c e s and soft limit projections (Proposition 6.3.5). By T h e o r e m 1.3.4, there exist an index a E A and a map f a "* Y such t h a t f -- f ~ p a . It follows from L e m m a 7.2.3 t h a t the projection pc" X ---. Xo, of the spectrum S x has two sections i0 and il with disjoint images. Softness of f implies softness of the map f a (Lemma 6.1.15). Therefore f a has a section i: Y ~ X a . T h e n the desired sections of the m a p f can be defined as the compositions j0 = ioi and j l = ill. r]

292

7. NON-METRIZABLE MANIFOLDS

LEMMA 7.2.5. Let a square diagram

X

~Y

qo

Xo

Po

'~Yo

be a Cartesian square. I f q is a retraction and the map p has two sections with disjoint images, then the map po also has two sections with disjoint images. PROOF. Let r : Yo ---* Y be a section of the r e t r a c t i o n q and il, i2: Y ~ X be sections of the m a p p with disjoint images. Let j k -- qoikr, k -- 1, 2. Obviously t h e m a p s jk, k -- 1, 2 are sections of the m a p p0. Suppose t h a t j l (Y0) CI j2(Yo) 7~ 0, a n d take a point x0 from this intersection. T h e n there is a point y0 E Y0 such t h a t Jl(yo) -- x0 - J2(y0). Since the d i a g r a m is a C a r t e s i a n square, there is a unique point x E X with the p r o p e r t i e s qo(x) -- xo and p(x) - r(yo). At the same time, it is easy to check t h a t the two different points i l ( r ( y o ) ) and i2(r(yo)) of X also have the p r o p e r t i e s qo(ik(r(yo))) -- Jk(YO) -- xo and p ( i k ( r ( y o ) ) ) -- r(yo), k -- 1, 2. This c o n t r a d i c t i o n s shows t h a t t h e sections j l and j2 have disjoint images.

[-1

LEMMA 7.2.6. Let S -- { X n , p ~ + l , w } be an inverse sequence, consisting of locally compact and LindelSf A N R-spaces and soft proper short projections with Polish kernels. I f each of these projections has two sections with disjoint images, then the limit projection P0: l i m b -~ X0 is the trivial bundle with fiber the Hilbert cube I ~ . PROOF. If w ( X o ) ----w, the s t a t e m e n t has already been proved in Subsection 2.4.4 ( P r o p o s i t i o n 2.4.38). Therefore, we may assume t h a t w ( X o ) - T > w. Each of the spaces X n , n E w can be r e p r e s e n t e d as the limit space of a factorizing w - s p e c t r u m Sn - { X ~ , q ~ ' n , A } consisting of locally c o m p a c t A N R-spaces and soft limit projections ( P r o p o s i t i o n 6.3.5 and R e m a r k 6.3.6). In the m e a n t i m e , observe t h a t these s p e c t r a have the same indexing set A. This follows from the

7.2. /r-MANIFOLDS

293

equalities w ( X n ) = r , n E w. Since all short projections pnn+l of t h e s p e c t r u m ,9 are proper, soft and have Polish kernels, we m a y assume w i t h o u t loss of generality (by T h e o r e m 6.3.2) t h a t for each n E w the p r o j e c t i o n pr~+1 is t h e limit m a p of a Cartesian morphism

Mnn+l= {pn-bl,o. xn+l _.+ X n a , A } .

,gn+l --+ ,gn

consisting of p r o p e r and soft m a p s between locally c o m p a c t A N R - s p a c e s with c o u n t a b l e bases. Take an a r b i t r a r y index a E A and consider the c o u n t a b l e inverse s p e c t r u m , n + l , o , w } . Let X o = l i m S o and X = lim,9. T h e limit m a p of the s.= m o r p h i s m {qan} " ,9 --* ,5,, is d e n o t e d by qo. T h u s we have the following infinite commutative diagram X

~

~ Xn+l 9

"

*- X n

;-

9

"

"

~-XO

qn+l

qo

Xo

pn+l

"

qn

O

~

~ X~+l

9 ..

O

~n+l,o

~n

~._ X

?1' O

~

" " "

I'-Xo

0

S t r a i g h t f o r w a r d verification shows t h a t all t h e square s d b d i a g r a m s of t h e above d i a g r a m are C a r t e s i a n squares. In particular, the d i a g r a m P0 X -"

*-X0

qO

qo

x. is also a C a r t e s i a n square.

.x ~

By L e m m a 7.2.5, each of t h e m a p s Pn ..n+l 'o , n E W , has two sections with disjoint images. In this situation, the limit p r o j e c t i o n p~" X o -+ X ~ of the s p e c t r u m So is the trivial b u n d l e with fiber t h e Hilbert cube I "~ ( P r o p o s i t i o n 2.4.38). Since t h e last d i a g r a m is a C a r t e s i a n square, we easily conclude t h a t t h e limit p r o j e c t i o n p0: X -+ X0 of t h e s p e c t r u m ,9 is also t h e trivial bundle with fiber I w . [--I

294

7. NON-METI:tIZABLE MANIFOLDS

We need one more auxiliary lemma. LEMMA 7.2.7. Let X be a locally compact and L i n d e l 6 f space of weight r > w. T h e n X a d m i t s a closed embedding into a p r o d u c t P x I r , where P is an I ~ manifold. PROOF. Consider the o n e - p o i n t compactification a X of t h e space X. Identify a X with a (closed) subspace of the cube I r . Obviously X is functionally open in X . Therefore t h e r e is a functionally open subset V of I r such t h a t a X A V - X. T h e n X is closed in V. It only remains to note t h a t every functionally open s u b s p a c e of I r a d m i t s a r e p r e s e n t a t i o n as a p r o d u c t P x I r , where P is a I wmanifold. El We are now r e a d y to prove the following c h a r a c t e r i z a t i o n theorem. THEOREM 7.2.8. Let r >_ w. T h e n the following conditions are equivalent f o r every locally c o m p a c t and L i n d e l h f A N R - s p a c e X o f weight r" (i) X is an I r - m a n i f o l d .

(ii) X is strongly Br,oo-universal. (iii) F o r each c o m p a c t u m Y o f weight < r, the set of embeddings is dense in the space C r ( Y , X ). (iv) T h e set of embeddings is dense in the space C r ( D , X ).3 (v) X is h o m o g e n e o u s with respect to the p s e u d o - c h a r a c t e r . PROOF. T h e validity of t h e implications (i) ~ (ii) has been established in Corollary 7.1.26. I m p l i c a t i o n s (ii) ==~ (iii), (iii) ==~ (iv) and (iv) = ~ (v) are obvious. Let us prove t h a t (v) ~ (i). Recalling L e m m a 7.2.7, we may assume t h a t X is a closed subspace of t h e p r o d u c t P x I A, where P is an I ~ -manifold and A is an a r b i t r a r y set of c a r d i n a l i t y r. Since every closed subset of any n o r m a l space in C - e m b e d d e d , we see t h a t there is a r e t r a c t i o n r" V ---, X, where V is a functionally open n e i g h b o r h o o d of X in the p r o d u c t P x I A. T h e space V can be r e p r e s e n t e d in the same m a n n e r as the p r o d u c t L x I A, where L is a I ~ -manifolds. T h u s we have the following situation: T h e space X is closed in the p r o d u c t L x I A (where L is a I w -manifold) and t h e r e is a r e t r a c t i o n r" L x I A ~ X . We need some a d d i t i o n a l notations. If C c B C A, then A~" L x I B ~ L x I C and AB" L x I A ---. L x I B

d e n o t e the m a p s idL • r ~ and idL • 7rB respectively. Here r B" I B ---, I v and r B " I A ---+ I B d e n o t e t h e n a t u r a l projections onto the c o r r e s p o n d i n g subprod-

ucts. T h e subspace AB(X) of the p r o d u c t L x I B shall be denoted by X B . T h e properness of the m a p AB g u a r a n t e e s t h a t X B is closed in L x I B. Let 3Recall that D stands for the two-point discrete space.

7.2. I r - M A N I F O L D S

295

X ( B ) -- ABI(XB). The restriction of the map A~ onto X B is denoted by p~. Similarly, pB -- AB / X. A subset B C A is said to be admissible if for each point x E X (B) we have

pB~(~) = A~(x). As in the proof of Theorem 6.3.1 one can see t h a t admissible subsets of A have the following two properties: (a) the union of an arbitrary collection of admissible subsets is admissible. (b) each countable subset of A is contained in a countable admissible subset of A. (c) if C C_ B C A, and C and B are admissible subsets of A, then the maps PB" X --+ X B and p~" X B ---+X c are proper and soft. Next consider an arbitrary admissible subset B of A of cardinality [B[ < -i-. We are now going to show t h a t there is a countable admissible subset B' of A such t h a t the map p B u B ' . X B U B , '+ X B is the trivial bundle with fiber the Hilbert cube I • . Since the map PB" X ---+X B is soft, and since the space X is homogeneous with respect to pseudo-character, we conclude by L e m m a 7.2.4, t h a t there are two sections i0, i l" XB ' + X o f the map P B such t h a t

io(XB)ni~(x~)=O. Obviously, the sets io(XB) and i l ( X B ) are closed in the product L x I A. Consequently, there is a countable subset B1 of A such t h a t

~,B~ (~o(XB)) n ~B~ ( ~ ( x ~ ) ) = O. By property (b) of admissible sets, we may assume t h a t B1 is an admissible subset of A. In this situation, the map PBuBoBUBz" X BUB1 -'+ X BUBo

has two sections (namely, PBuBlio and PBuBIil) with disjoint images (here we assume that B0 -- 0). Continuing this process by induction, we construct countable admissible subsets Bn of A such t h a t the map B U BoU...U B,,U B,,+ z PBUBoU...UB, " X BUBoU...UB,+~ --* X BUBoU...UB,,

has two sections with disjoint images, n E w. The countability of each of the sets Bn ensures t h a t each of the indicated maps P BUBoU...UB,~+I BUBoU...UB,~ has Polish kernel and, by properties (a) and (c), are proper and soft. Therefore we obtain the inverse sequence BUBoU-..UBB+I {XBuBoU'"UB,,PBUBoU...UB,

,n E u)}

consisting of locally compact and Lindelhf A N R-spaces and proper and soft short projections with Polish kernels admitting t w o s e c t i o n s with disjoint images.

296

7. NON-METRIZABLE MANIFOLDS

Let B ~ -- U { B n : n E w}. T h e limit space of t h e above inverse sequence is obviously h o m e o m o r p h i c to the space X B u B ' , and the limit projection 4 of t h a t sequence coincides w i t h the m a p pgOB'.

X BUB ' ~

X B.

L e m m a 7.2.6 allows us to c o n c l u d e t h a t the m a p pBBUB' is the trivial bundle with fiber I W. Observe also t h a t the set B U B ~ is, by p r o p e r t y (a), also an admissible subset of A. F u r t h e r , by transfinite induction ( r e p e a t i n g the above described process of passing from B to B U B ~) we c o n s t r u c t a continuous well ordered s p e c t r u m ,.q = { X a , p aa + l , T} of length r, satisfying the following conditions: (i) X = l i m S . (ii) all spaces X a are locally c o m p a c t and Lindel6f ANR-spaces. (iii) all short projections p~+l are trivial bundles with fiber I W . (iv) the space X0 is a locally c o m p a c t A N R-space of c o u n t a b l e weight. In this situation, the space X is h o m e o m o r p h i c to the p r o d u c t X0 x I T . By condition (iv), the p r o d u c t X0 x I W is a n / W - m a n i f o l d ( T h e o r e m 2.3.21). It only remains to note t h a t , by T h e o r e m 7.1.19, the space X is an I T-manifold. V1 T h e following s t a t e m e n t , providing a topological c h a r a c t e r i z a t i o n of the Tychonov cube I T , follows directly from T h e o r e m 7.2.8. THEOREM 7.2.9. Let T > w. The following conditions are equivalent for each compact AR-space of weight 7": (i) X is homeomorphic to I T . (ii) X is strongly Bv,~-universal. (iii) for each compactum Y of weight < T, the set of embeddings is dense in the space Cv(Y, X ) . (iv) the set of embeddings is dense in the space C v ( D , X ) . (v) X is homogeneous with respect to pseudo-character. It follows from T h e o r e m 2.4.18 t h a t the equivalences (i) r (ii) in T h e o r e m s 7.2.8 and 7.2.9 are valid in the case ~- = w as well. T h e o t h e r conditions from T h e o r e m s 7.2.8 a n d 7.2.9 are not equivalent to condition (i). For instance, the closed unit interval is homogeneous with respect to p s e u d o - c h a r a c t e r a n d is an AR-compactum. T h e t h e o r e m s proved above have several corollaries. COROLLARY 7.2.10. Let T >_ w. The product X x I T is an I T-manifold if and only if X is a locally compact and LindelSf A N R - s p a c e of weight < T. PROOF. If T -- W, then the s t a t e m e n t is true ( T h e o r e m 2.3.21). Assume t h a t T > w. If the p r o d u c t X x I r is an I r - m a n i f o l d , t h e n the space X, as a r e t r a c t of an I r -manifold, is a locally c o m p a c t and Lindel/if A N R-space of weight __ T. 4which maps X B U B, onto the space Z B

7.2. I ~ -MANIFOLDS

297

Conversely, since the p r o d u c t X x I r is a locally c o m p a c t and Lindelbf A N R space of weight T, it suffices, by T h e o r e m 7.2.8, to verify the h o m o g e n e i t y with respect to pseudo-character of the p r o d u c t X x I ~ . The last fact is an i m m e d i a t e consequence of an easy observation t h a t the Tychonov cube I r is h o m o g e n e o u s with respect to pseudo-character. F-I Similarly we have the following.

COROLLARY 7.2.11. L e t T >_ w. T h e p r o d u c t X x I v is h o m e o m o r p h i c to the cube I v i f and only i f X

is a c o m p a c t A R - s p a c e o f w e i g h t w. I f the p r o d u c t o f two spaces is an I V - m a n i f o l d (respectively, is h o m e o m o r p h i c to the cube I r ), t h e n at least one o f these spaces is an I r - m a n i f o l d (respectively, is h o m e o m o r p h i c to I ~" ).

PROOF. Suppose t h a t X1 x X2 is a n / r - m a n i f o l d . T h e n each of the spaces X I and X2 is a locally c o m p a c t and Lindelbf A N R - s p a c e of weight < T. Assume, contradicting our hypothesis, t h a t neither X1 nor X2 is an I r -manifold. Then, by T h e o r e m 7.2.8, there are points xz E X1 and x2 E X2 such t h a t r X1) < T and r X2) < T. It only remains to note t h a t in this situation the pseudo-character of the point (Xl, x2) in X1 x X2 is strictly less t h a n T. This contradiction completes the proof. [-! COROLLARY 7.2.13. I f the square o f a space X is h o m e o m o r p h i c to the Tyc h o n o v cube I r , T > w, t h e n X

is h o m e o m o r p h i c to I r .

We are now in position to prove the converse to Corollary 7.1.22. COROLLARY 7.2.14. I f the p r o d u c t X x [0, 1) can be e m b e d d e d in the T y c h o n o v cube I v , T > w, as a f u n c t i o n a l l y open subspace, t h e n X

is an I v - m a n i f o l d .

PROOF. Obviously the p r o d u c t X x [0, 1), as a functionally open subspace of I r , is an I r -manifold. By Corollary 7.2.12, X must be an I v -manifold. E] We conclude this Section by presenting the following s t a t e m e n t .

COROLLARY 7.2.15. L e t T > w.

E a c h c o m p a c t u m o f w e i g h t T, a d m i t t i n g a

soft m a p onto the T y c h o n o v cube I r , is h o m e o m o r p h i c to I r .

PROOF. It suffices to apply P r o p o s i t i o n 6.5.7 and T h e o r e m 7.2.9.

E]

H i s t o r i c a l a n d bibliographical n o t e s 7.2. The results of this Section are basically obtained in [277]. T h e equivalence of conditions (ii) and (iii) with conditions (i) and (iv) in T h e o r e m s 7.2.8 and 7.2.9 was observed in [83]. Corollaries 7.2.14 and 7.2.15 also are due to the author.

298

7. NON-METRIZABLE MANIFOLDS 7.3. T o p o l o g i c a l c h a r a c t e r i z a t i o n

of R r -manifolds

We begin with the following statement. LEMMA 7.3 91 9 L e t S = { X n , e"n+l n , w } be an inverse sequence, consisting of ANR-spaces

and soft short projections with Polish kernels.

Suppose that f o r

each n E w, the space X n + l contains a C - e m b e d d e d copy of the product X n x R ~ such that p n + l / ( x n

x R ~ ) = ~rx~, where r x , "

onto the first coordinate.

X n • R ~ -+ X n is the projection

T h e n the limit projection po: X = l i m S --~ X o is the

trivial bundle with fiber R ~ .

PROOF9 The case w ( X o ) = w was considered in Chapter 2 (see Proposition 29 Consider the case w ( X o ) = T > w. Since each sort projection of the spectrum S has Polish kernel, we conclude t h a t w ( X n ) = r for each n E w. Represent the space Xn, n E w, as the limit space of a factorizing w-spectrum S n - { X ~ , q~,n, A } , consisting of Polish A N R - s p a c e s and soft limit projections. Observe, in the meantime, that the indexing sets of all these spectra coincide with A (which has cardinality r). Since all short projections pn+l of the spectrum S are soft and have Polish kernels, we see, by T h e o r e m 6.3.2, that pr~+1, n E w, is the limit of some Cartesian morphism

Mnn + l = {pnn+l'~149 Xan+l --+ X n , A } " Sn+l --+ Sn, consisting of soft maps between Polish A N R - s p a c e s . For each a E A, consider the inverse sequence S,, - {X~,,pnn+l'a,w}, and let Xo = lim So. If/~ > a, then there is a Cartesian morphism M ~ = {q~,n. X ~ --+ x n , w } " S 8 --+ S o ,

consisting of soft maps q~,n, n E w. Denote by q~ the limit map of the morphism M~. Thus, the following infinite commutative diagram 5 arises" X

Dr "'"

pnn+l D- X n

~ Xn+l

,~ "'"

*-Xo

qn+l

qo

O

,...n+l,a

Xa

~

9 ..

,. x

n+l

Pn

5Compare with the proof of Lemma 7.2.6.

~

X

n o

o ~

...

7.3. Rr -MANIFOLDS

299

S t r a i g h t f o r w a r d verification shows t h a t the limit space of the s p e c t r u m S ' = { X a , q ~ , A } coincides with the space X , and t h a t all newly arising square diag r a m s are also C a r t e s i a n squares. B y a s s u m p t i o n , the space X n + I contains a C - e m b e d d e d copy of the p r o d u c t X n x R ~ so t h a t p n + l / ( x n x R ~ ) = 7rx,. Consider the factorizing w - s p e c t r u m S " = { X ~ x R ~ ,q~,n x idRw , A } ,

t h e limit space of which is t h e p r o d u c t X n x R "~ . Since X n x R "~ is C - e m b e d d e d in

Xn+l,

we

see t h a t the s p e c t r u m Sn+l/(Xn

x R w ) -- {clx2+1 q n + l ( x n x R w ), ua"B'n+l, A } ,

also having the p r o d u c t X n x R ~ as its limit, is factorizing. Consequently, by T h e o r e m 1.3.6, we m a y a s s u m e t h a t t h e s p e c t r a S " and 3 n + l / ( X n • R ~ ) are isomorphic. A n a l y z i n g the m a p s pnn+l and ~rx~ (using results of Section 6.3), we conclude t h a t the indexing set A contains a cofinal and w-closed subset A n , n E w, such t h a t for each c~ E A n t h e space X n + l contains a closed copy of the p r o d u c t X an x R ~ so t h a t p n T l , ~ jI ( x ~n

x

R w ) = 7rx:,,

w h e r e ~rx2" X ~ • R ~ -+ X an d e n o t e s the projection onto t h e first coordinate, n E w. Let A' -- n { A n : n E w}. B y P r o p o s i t i o n 1.1.27, t h e set A' is cofinal a n d w-closed in A. In particular, A' ~ 0. Take a E A'. T h e n t h e limit m a p p~" X a -+ X ~ of the s p e c t r u m S a is, by the case considered above (recall t h a t w ( X ~ - w ) , the trivial bundle w i t h fiber R "J . Using t h e fact t h a t the d i a g r a m po X

*X0

qO

q~

Xa

P~

~ X ao

is a C a r t e s i a n square, we see t h a t the limit p r o j e c t i o n p0 of t h e s p e c t r u m S is also the trivial b u n d l e with fiber R ~ .

KI

LEMMA 7.3.2. Let X be an A N R - s p a c e of weight T > w, and suppose that f o r each space Y of R - w e i g h t < T the set of C - e m b e d d i n g s is dense in the space C r ( Y , X ) . Suppose, in addition, that a soft m a p p" X -+ Z , where w ( Z ) < T, is also given. T h e n X contains a C - e m b e d d e d copy of the product Z • R ~ so that p / ( Z x R ~ ) -- ~rg, where ~rz" Z x R ~ -+ Z denotes the projection onto the first coordinate.

300

7. NON-METRIZABLE MANIFOLDS

PROOF. Since the map p is soft, Z is an A N R - s p a c e (Proposition 6.1.21). T h e n the p r o d u c t Z • R ~ is also an A N R - s p a c e and, consequently, by Corollary 6.3.8, the R-weight of the p r o d u c t Z • W is strictly less t h a n T. By L e m m a 6.5.1, there is a collection {/g~: t E T}, IT] -- w ( Z ) , of countable functionally open covers of the space Z, satisfying the conditions of t h a t Lemma. Let Ht - p-1 (L/~), t E T. By the softness of the m a p p, there is a section i: Z --~ X of p. T h e n the neighborhood B(i~rz, {/act: t E T})

of the point iTrz: Z x R '~ --. X , in the space C r ( Z x R '~ , X ) , contains at least one C-embedding. Denote it by g. By the choice of the above neighborhood, we have pg = lrz. Therefore, g is the desired C - e m b e d d i n g of the product Z x R ~~ into the space X. V! Now we are ready to prove the characterization theorem for R r -manifolds (and, in particular, of the space R r itself). THEOREM 7.3.3. Let r > w. The following conditions are equivalent for any A ( N ) R - s p a c e X of weight r:

(i) x i~ ho~omo~phi~ to n " ( ~ p ~ c t i ~ d y , x ~ ~n n " - m ~ i f o l d ) . (ii) X is strongly ~4r,~-universal. (iii) For each space Y of R - w e i g h t < r, the set of C-embeddings is dense in the space C r ( Y , X ) . PROOF. Implication (i) - - ~ (ii) was proved in Corollary 7.1.18. Implication (ii) ~ (iii)is trivial. Let us prove implication (iii) ==~ (i). We consider only the absolute case (the remaining case can be proved in a similar way). Since X is an AR-space of weight T, we may assume t h a t X is a retract of the space R A, where A is an a r b i t r a r y set of cardinality r. Let B be an arbitrary admissible subset of A (see the proof of T h e o r e m 6.3.1) of cardinality < T. T h e n the map 7 r s / X : X ~ i t s ( X ) , where ~rs: R A ~ R s is the projection onto the corresponding subproduct, is soft. Consequently, by L e m m a 7.3.2, the space X contains a C - e m b e d d e d copy of the product 7 r s ( X ) x R ~~ so t h a t 7rB/(TrB(X) • R w ) = 7r,rB(x), where 7r,rB(x) : 7rB(X) x R ~~ --. 7rB(X) is the projection onto the first coordinate. Since the R-weight of the product 7rB(X) x R ~ is less t h a n T, we see (using results of Section 6.3) t h a t there is a countable admissible subset B ~ of A, satisfying the following conditions: (a) T h e restriction of the projection 7rBUB, OIltO 7rB(X ) • R w (considered as a C - e m b e d d e d subspace of the space R A) is a homeomorphism. (b) T h e subspace 7rBuB,(TrB(X) X R ~~) is C - e m b e d d e d in 7rBuB,(X). In this situation the space 7 r B u s , ( X ) contains a C - e m b e d d e d copy of the p r o d u c t 7rS(X) X R ~~ so t h a t

~ "

/(~.(x)

• n~

)= ~.(~),

7.3. Rr -MANIFOLDS

301

where lrguB'. RBUB ' _._+R B is t h e p r o j e c t i o n onto the c o r r e s p o n d i n g s u b p r o d u c t . K e e p i n g in m i n d t h a t t h e union of an a r b i t r a r y collection of admissible subsets is admissible, by transfinite i n d u c t i o n ( r e p e a t i n g the c o r r e s p o n d i n g p a r t of the proof T h e o r e m 7.2.8) we c o n s t r u c t a continuous well ordered s p e c t r u m S x = { X ~ , p ~ , T} of length T SO t h a t the following conditions are satisfied: (1) X - lira S x . (2) All spaces X a are AR-spaces. (3) All short projections p ~ + l are soft and have Polish kernels (since the above c o n s t r u c t e d sets B ' are countable). (4) For each a < T the space X a + l contains a C - e m b e d d e d copy of the p r o d u c t X a x R ~ so t h a t p,~+l/(x,~ x R ~ ) = 7rx~, where ~rx~" X a x R w ~ X a is the projection onto the first coordinate. (5) X0 is a Polish AR-space. A p p l y i n g transfinite i n d u c t i o n one more time and using L e m m a 7.3.1, We see t h a t X is h o m e o m o r p h i c to the p r o d u c t X0 x R r ~ R r . T h e p r o o f is complete. I--1 This result has several consequences. COROLLARY 7.3.4. The product X x R r , T > w, is an R r -manifold if and only if X is an A N R-space of weight w. If the p r o d u c t X x R r is an R r -manifold, t h e n the space X , as a r e t r a c t of R r -manifold, is an A N R-space of weight _ r. Let us prove the sufficiency of our condition. Thus, let X be an A N R - s p a c e of weight < ~-. Obviously, the p r o d u c t X x R r is an A N R - s p a c e of weight < T. Consequently, by T h e o r e m 7.3.3, it suffices to verify t h a t for each space Y of R-weight < r, the set of C - e m b e d d i n g s is dense in the space C r ( Y , X x R r ). Take an a r b i t r a r y set A of cardinality T, and consider a m a p f " Y ---+ X x R A and an a r b i t r a r y basic n e i g h b o r h o o d 0 - B ( f , {Lit" t E T}), ITI < T, of the point f in the space C r ( Y , X x R A ) . Let A = max{w, ITI}. T h e uncountability of T implies t h a t A < T. R e p r e s e n t X as the limit space of a factorizing A-spectrum S z = { X B , p ~ , e x p ~ A } , consisting of A N R - s p a c e s of weight A and soft limit projections. T h e space R A can also be r e p r e s e n t e d as the limit space of the s t a n d a r d factorizing A-spectrum S = { R B, 7rB, e x p ~ A }. T h e n the p r o d u c t X x R A is the limit space of the factorizing A-spectrum

• S = {X.

• R',p

•

In this situation, by L e m m a 6.5.1, there is an element B E e x p ~ A such t h a t

o'=

{g e c

(Y,X x RA) 9 (pB x

B)g = (;B x

B)f}

302

7. NON-METRIZABLE MANIFOLDS

is a neighborhood of f in the space C r ( Y , X • R A ) , contained in O. R - w ( Y ) w. E v e r y space o f w e i g h t r , a d m i t t i n g a soft m a p o n t o R r , is h o m e o m o r p h i c to R r .

PROOF. Apply Theorem 7.3.3 and Proposition 6.5.6.

E!

H i s t o r i c a l a n d bibliographical n o t e s 7.3. All results of this Section are due to the author [84]. As already shown (Corollary 7.1.14; compare with Corollary 7.1.22), the class of functionally open subspaces of R r coincides with the class of R r -manifolds. The problem of the satisfactory topological characterization of open (not necessarily functionally open) subspaces of R r is still open. Corollary 6.4.10 and Proposition 6.1.7 show that if an open subspace of R r is not functionally open, then it is not even an AE(0)-space and, consequently, we cannot apply the spectral theory developed here. A similar problem with respect to open subspaces of Tychonov cubes also remains open.

7.4. T r i v i a l b u n d l e s In this Section we extend Theorems 2.4.32 and 2.4.33 to the non-metrizable case.

7.4. TRIVIAL BUNDLES

303

7 . 4 . 1 . T r i v i a l b u n d l e s w i t h f i b e r R r . Let a cardinal n u m b e r T > w and a m a p f" X ~ Y be given. T h e n for each m a p g" Z ---, X , we denote by C ~ ( Z , X ) the subspace of the space C r ( Z , X ) , consisting of maps h" Z ~ X such t h a t fh=

fg.

THEOREM 7.4.1. Let T > w and f " X ~ Y be a soft map between A N R spaces with w ( X ) = 7-. Then the following conditions are equivalent: (i) f is the trivial bundle with fiber R r . (ii) For each space Z of R - w e i g h t a such t h a t the projection t~ (a)" Zf~(a) --* Za of the s p e c t r u m S has a Polish kernel and the space Z~(a) contains a C - e m b e d d e d copy of the p r o d u c t Z a x R • so t h a t tfla(a)/(Za x R " ) = ~rz~, where, as usual, 7rz." Za x R ~ ---, Z a is the projection onto the first coordinate. As follows from the proof of T h e o r e m 6.3.1 in order to show the last fact it suffices to prove the existence in X of a C - e m b e d d e d copy of the p r o d u c t Za x R w so t h a t h a / ( Z a x R ~ ) =Trz~. Since ha i s a s o f t map, there is a m a p i" Za x R ~ --, X such t h a t hai = 7rz.. By L e m m a 6.5.1, t h e r e is a subcollection {U~ e T e T} of cardinality [T[ = w ( X a ) < T of the collection c o v ( X a ) such t h a t the conditions of L e m m a 6.5.1 are satisfied. Let /.It = p~l(b/;), for each t c T. T h e n , by our a s s u m p t i o n , there is a C - e m b e d d i n g g" Z a x R ~ ---, X such t h a t the following conditions hold: (h) T h e maps g and i are { U t ' t e T}-close.

(i) /g = / i . We claim t h a t g is the desired C - e m b e d d i n g . that

Indeed, it suffices to observe

hag = ( p a A f )g = p a g A f g = p a i A f i = ( p a A f )i = hai = ~rz,:,.

7.4. TRIVIAL BUNDLES

305

Thus indexes f~(a), satisfying the above conditions, exist for each a. In this situation, applying L e m m a 7.3.1, by straightforward transfinite induction (compare with the proof of Theorem 7.3.3) we reach the final c o n c l u s i o n - f is the trivial bundle with fiber R r . D The following s t a t e m e n t is an immediate consequence of the above Theorem. COROLLARY 7.4.2. L e t T > w, .f" X spaces, a n d w ( X )

~_ r .

~

Y

be a s o f t m a p

T h e n the c o m p o s i t i o n f ~ z ,

between AN R-

w h e r e l r x " X x R r ---. X

is

the p r o j e c t i o n o n t o the f i r s t c o o r d i n a t e , is the trivial b u n d l e w i t h fiber R r .

COROLLARY 7.4.3. L e t T > w a n d f : X ---. Y be a s o f t m a p o f an R r - m a n i f o l d X

o n t o a space Y o f w e i g h t < r .

T h e n f is the trivial b u n d l e w i t h f i b e r R r .

PROOF. By Theorem 7.1.11, the R r -manifold X is homeomorphic to the product P • R r , where P is an R ~ -manifold. Represent the product P • R r as the limit space of the s t a n d a r d factorizing A-spectrum S-

{ P • (R;~) a, i d p • ~ ,

exp;~r}

where r~" (R)') f~ ~ (R)') a denotes the projection onto the corresponding subproduct and A = m a x { w , w ( Y ) } . Then there are an index a e e x p ~ r and a map f a : P x (R'X) a ~

Y

such t h a t f = f a ( i d p x 7ra), where ~a: R r ~ ( R X ) a is the projection. The softness of the maps f and i d p x ~a guarantees the softness of the map f~ (Lemma 6.1.15). Since the map i d p x ~a is the trivial bundle with fiber R r , Corollary 7.4.2 completes the proof. W1

7.4.2. T r i v i a l b u n d l e s w i t h f i b e r I r . Replacing Lamina 7.3.1 by Lamina 7.2.6 in the proof of T h e o r e m 7.4.1, we obtain the following result. THEOREM 7.4.4. L e t T > W a n d f " X spaces w i t h w ( X )

---+ Y

be a s o f t m a p

between ANR-

-- T. T h e n the f o l l o w i n g c o n d i t i o n s are e q u i v a l e n t :

(i) f is the trivial b u n d l e w i t h fiber I r . (ii) F o r each c o m p a c t space Z o f w e i g h t ~_ r a n d f o r a n y m a p g" Z ---. X , the set o f e m b e d d i n g s is d e n s e in the space C g ( Z , X ) .

(iii) T h e set o f e m b e d d i n g s is d e n s e in the space C ~ ( X x D , X ), w h e r e ~r" X x D ~ X

is the p r o j e c t i o n .

COROLLARY 7.4.5. L e t T > w, f : X ---* Y be a p r o p e r a n d s o f t m a p b e t w e e n locally c o m p a c t a n d L i n d e l h f A N R - s p a c e s , f rx,

where r z: X x I r ~

trivial b u n d l e w i t h fiber I r .

X

and w(X)

~_ T. T h e n the c o m p o s i t i o n

is the p r o j e c t i o n o n t o the f i r s t c o o r d i n a t e , is the

306

7. NON-METI:tIZABLE MANIFOLDS

COROLLARY 7.4.6. L e t T > w a n d f " X --~ Y be a p r o p e r a n d soft m a p o f an I r -manifold X

o n t o a space Y o f w e i g h t < T. T h e n f is the trivial b u n d l e w i t h

fiber I r .

In connection with T h e o r e m 7.4.4, the following question arises naturally [277]: 9 Is the soft map between c o m p a c t A N R-spaces, whose fibers are all homeomorphic to I r and which satisfies condition (iii) for T - - w , the trivial bundle with fiber I T . T h e o r e m 7.4.4 allows us to answer the above question negatively. Indeed, consider an arbitrary soft map f : I ~1 ~ I wl , all fibers of which are homeomorphic to I ~1 and which does not a d m i t two sections with disjoint images (such maps exist: take an wl power of the soft map ~: I W ~ I ~ of the Hilbert cube onto itself, all fibers of which are the copies of the Hilbert cube and which, nevertheless, is not the trivial bundle with fiber I "~ [300]). Let r : I wl x I ~ ~ I wl be the projection onto the first coordinate. We claim the desired map is the composition f~r. Obviously, all fibers of the map fir are copies of the Tychonov cube I ~1. Since the projection lr satisfies condition (iii) of T h e o r e m 7.4.4, the composition f r also does. We are now going to outline why f ~ is not the trivial bundle. Assume the contrary, i.e. suppose t h a t f r is the trivial bundle with fiber I v . Take a section i: I ~1 ---. I ~1 of the map f and a section j : I "~1 ---, I "~ • I "~ of the projection ~r. Also take a countable collection {Lli: i E w} of open covers of I "~ satisfying the conditions of L e m m a 6.5.1. Let ))i - {I ~ x 3/1, i E w}. Then, by condition (iii) of T h e o r e m 7.4.4, there exist maps hk : I ~ ~ I ~1 • I ~ , k ---- 1,2, {~)i: i E w}close to the composition j i such t h a t h l ( I ~ ) M I ~1 = 0 and f~Thk -- f r j i = i d x ~ , k : 1, 2. Let gk : 7rhk, k -- 1, 2. Obviously, the maps gl and g2 are sections of the m a p f. It is also not hard to see t h a t g l ( I w ~ ) n g 2 ( I ~1) - 0 . This contradicts the choice of f. Therefore, the composition f r is not the trivial bundle. 7.4.3. A p p r o x i m a t i o n by h o m e o m o r p h i s m s . A map f : X ~ Y is said to be a T - n e a r - h o m e o m o r p h i s m , T > w, if f is in the closure of the set of homeom o r p h i s m s in the space C T ( X , Y ) . In Other words, f is a T-near-homeomorphism if for each collection {L/t: t E T} _C c o y ( Y ) with IT[ < r there is a homeomorphism h: X ~ Y {/it: t E T}-close to f . In Subsection 2.1.1 (see T h e o r e m 2.1.8) we have already discussed w-near-hom e o m o r p h i s m s (i.e. n e a r - h o m e o m o r p h i s m s in the usual sense). Near-homeomorphisms between R ~- and/W-manifolds also have been characterized (Propositions 2.4.20 and 2.4.21 respectively). In order to describe the general situation we need the following definition. DEFINITION 7.4.7. L e t T >_ w. A m a p f " X ~ Y b e t w e e n r e a l c o m p a c t spaces is called T-approximatively n-soft

(n

dimensional

r e a l c o m p a c t space Z ,

a r b i t r a r y closed s u b s e t Zo o f Z ,

{blt" t E T }

C coy(Y),

=

O, 1 , . . . , c r

i f f o r each at m o s t n -

w h e r e IT ] < T, a n d m a p s g" Zo ~

X

collection

and h" Z ~

Y

7.4. TRIVIAL BUNDLES

307

such that f g = h l Z o and C ( g ) ( C ( X ) ) C_ C ( Z ) / Z o , there is a map k" Z --+ X such that g = k / Z o and the composition f k is { b t t ' t E T } - c l o s e to h. The T-approxitively oo-soft m a p s are called T-approxitively soft. Obviously, every n-soft m a p is T - a p p r o x i m a t i v e l y n-soft for each 7 _> w. Observe also t h a t if, as in t h e case of n-soft m a p s (see P r o p o s i t i o n 6.1.18), the spaces X and Y in Definition 7.4.7 are also c o m p a c t , then the spaces Z and Z0 also can be a s s u m e d to be compact. In this ease the inclusion C ( g ) ( C ( X ) ) C_ C ( Z ) / Z o is a u t o m a t i c a l l y satisfied. LEMMA 7.4.8. A map f " X ~ Y between Polish spaces is w-approximatively n-soft (n = O, 1 , . . . , oo) if and only if the foUowing condition is satisfied: 9 For each at m o s t n - d i m e n s i o n a l Polish space Z , for each closed subset Zo of Z , for each open cover lg E c o y ( Y ) and for any two maps g" Zo --+ X and h" Z --, Y with f g -- h / Z o , there is a map k" Z -+ X such that g -- k / Z o and the composition f k is U-close to h. PROOF. Obviously every w - a p p r o x i m a t i v e l y n-soft m a p b e t w e e n Polish spaces satisfies the above condition. Let us prove the converse. Thus, suppose t h a t a m a p f - X --+ Y has the above property. Take an a r b i t r a r y at m o s t n - d i m e n s i o n a l r e a l c o m p a c t space Z, closed subset Z0 of Z, and a collection {b/t" t c T} C c o y ( Y ) with IT] < w. Also take two maps g" Z0 -+ X and h" Z --+ Y such t h a t f g - h / Z o and C ( g ) ( C ( X ) ) C C ( Z ) / Z o . R e p r e s e n t the space Z as the limit space of factorizing w - s p e c t r u m

8-

{Za,p~,A}

consisting of at most n - d i m e n s i o n a l Polish spaces ( T h e o r e m 1.3.10). Since Z0 is closed in Z, the limit space of the induced s p e c t r u m 80 = {clz~, p~, A} coincides with Z0. By T h e o r e m 1.3.6, t h e r e exist an index a l E A and a m a p h a l " Z a l ~ Y such t h a t h -- halpa~. F u r t h e r , by P r o p o s i t i o n 1.3.13, t h e r e exist an index a2 E A and a m a p ga2" clz= 2 --+ X such t h a t g : ga2pa2/Zo. W i t h o u t loss of generality we m a y a s s u m e t h a t a l = c ~ 2 : C~. It is not h a r d to see t h a t the m a p s ha = hal and ga -- ga2 have the following properties"

h -- hapa,

g -- g a P a / Z o and f g a : h a l clz,~ p a ( Z o ) .

By our a s s u m p t i o n , there is a m a p ka" Z a ~ X such t h a t ga = k a / c l z , p a ( Z o ) and t h e c o m p o s i t i o n f k a is b/-close to ha, where b/ refines each of the covers b/t, t E T (recall t h a t T is finite). T h e n the desired m a p k can be defined as k -- kapa. [7 THEOREM 7.4.9. Let T >_ w. A map between R r -manifolds is a T-near-homeo m o r p h i s m if and only if it is ~--approximatively soft.

308

7. NON-METRIZABLE MANIFOLDS

PROOF. If ~- = w, t h e n the s t a t e m e n t follows from L e m m a 7.4.8 and Proposition 2.4.20. Consider the case T > w. Let f : X --* Y be a T-approximatively soft m a p between R r - m a n i f o l d s . Consider an a r b i t r a r y collection {b/t: t E T} C_ coy(Y) with IT] < T. We shall c o n s t r u c t a h o m e o m o r p h i s m h: X -~ Y which is {L/t E T}-close to f . Let ~ -- max{w, ITI}. Clearly, ~ < T. By T h e o r e m 7.1.11, the R r -manifolds X and Y are h o m e o m o r p h i c to the limit spaces of factorizing ,~-spectra 5'x = {Xa, p~, A } and 8 y = {Ya, q~, A } respectively. Moreover, these s p e c t r a consist of R~-manifolds and soft limit projections (even trivial bundles with fiber R r ). By T h e o r e m 1.3.6, the m a p f is the limit m a p of the m o r p h i s m { f a : X a ~ Ya, A } : ,Sx ~ S y . A s t a n d a r d a r g u m e n t shows t h a t there exist an index a E A and open covers L/~ E cov(Ya) such t h a t L/t = q~-l(L/~) for each t E T. Take a subcollection {V~: t E T t} of the collection cov(Ya) of cardinality IT'I = w(Ya) = ~ < T, satisfying the conditions of L e m m a 6.5.1. Let ~)t = q~-1(1)~), t E T ~. Let us show t h a t the m a p f a : X a -* Ya is soft. Take an a r b i t r a r y realcompact space Z, a closed subspace Z0 of Z, and two m a p s ga: Z0 --* X a and ha : Z --. Ya satisfying the conditions

faga = h a / Z o and C ( g a ) ( C ( X a ) ) C_ C ( Z ) / Z o . Take a section i: X a ---* X of the limit projection pa of the s p e c t r u m S x and let g -- iga. Obviously,

q a f g -- h a / Z o and C ( f g ) ( C ( Y ) ) C_ C ( Z ) / Z o . Consequently, by softness of the limit projection qa of the s p e c t r u m S y , there is a m a p h: Z --~ Y such t h a t f g = h/Zo and gah = ha. By our assumption, f is T - a p p r o x i m a t i v e l y soft. Therefore there is a m a p k: Z ~ X such t h a t g = k/Zo and the composition f k is {Yt: t E Tt}-close to h. Let ka = pak. Keeping in mind the choice of the collection { P t : t E Tt}, one can easily verify t h a t ga = k a / Z o and f a k a = ha. This shows the softness of the m a p fa. But then, by L e m m a 6.1.14, the composition faPa : X ---* Ya is also soft. By Corollary 7.4.3, this composition is the trivial b u n d l e with fiber R ~ . Similarly, the limit p r o j e c t i o n qa of the s p e c t r u m 8y is also the trivial bundle with fiber R ~ . T h e bases of these trivial bundles coincide with the space Ya. Therefore there is a h o m e o m o r p h i s m s: X ~ Y such t h a t qas = fapa. T h e last equality shows, by the choice of the index a, t h a t the h o m e o m o r p h i s m s is {L/t: t E T}-close to the m a p f . Thus, f is a T - n e a r - h o m e o m o r p h i s m . Let us prove the converse. Let f : X ~ Y be a T - n e a r - h o m e o m o r p h i s m between R r -manifolds. We are going to show t h a t f is 7 - a p p r o x i m a t i v e l y soft. Consider an a r b i t r a r y r e a l c o m p a c t space Z, a closed subspace Z0 of Z, a collection {L/t: t E T } C_ cov(Y) with IT[ < T, and two m a p s g: Z0 --~ X and h: Z --. Y such t h a t f g = h/Zo and C ( g ) ( C ( X ) ) C_ C ( Z ) / Z o . As above, let A = max{w, IT[}. Obviously A < T. R e p r e s e n t the space Y as the limit space of

7.4. TRIVIAL BUNDLES

309

a factorizing A-spectrum ,gy -- {Ya, q~, A} consisting of RA-manifolds and soft limit projections. As above, we can find an index c~ E A and an open covers lg~ E c o v ( Y ~ ) such t h a t b/t - q~-l(b/~) for each t E T. Applying L e m m a 6.5.1, we get a collection { ~ : t E T ' } C c o v ( Y a ) , I T ' I - A < T with the properties of t h a t Lemma. Let ))t -- q~-l(V~), t E T ~. By our assumption, there is a homeomorphism s: X ~ Y {])t: t E T~}-close to f. By the choice of the collection {~)t: t E T~}, we have q a f -- qas. Therefore, the map q a f , as the composition of soft maps s and qa, is soft ( L e m m a 6.1.14). T h e n there is a m a p k: Z ~ X such t h a t g - k l Z o and q a f k -- qah. The last equality, and the choice of the index c~, guarantee t h a t the composition f k is (b/t: t E T}-close to h. This completes the proof. V1 A similar a r g u m e n t proves the following statement. THEOREM 7.4.10. Let T >_ w.

A proper m a p between I v - m a n i f o l d s is a T-

n e a r - h o m e o m o r p h i s m if and only if it is T - a p p r o x i m a t i v e l y soft.

COROLLARY 7.4.11. Let T >_ w and X be an R V - m a n i f o l d .

T h e n the projection

r x " X • R r --~ X is a T - n e a r - h o m e o m o r p h i s m .

COROLLARY 7.4.12. Let T >_ w and X be an I V - m a n i f o l d .

T h e n the projection

7rx" X • I v --~ X is a r - n e a r - h o m e m o r p h i s m .

Historical a n d bibliographical notes 7.4. T h e results of this Section were ob-

tained by the a u t h o r [83], [91].

Corollaries 7.4.5 and 7.4.6 can be found in

CHAPTER

8

Applications

8.1. U n c o u n t a b l e

In this D r for T > Theorems the proofs

powers of countable discrete spaces

Section we give topological characterizations of the spaces N r and w. Recall t h a t characterizations of the spaces N ~ and D ~ are given in 1.1.5 and 1.1.10, respectively. We also suggest t h a t the reader compare presented below to those of T h e o r e m s 7.3.3 and 7.2.8.

8.1.1. C h a r a c t e r i z a t i o n

o f N r . We begin with some auxiliary statements.

LEMMA 8.1.1. Let X be a z e r o - d i m e n s i o n a l Polish space and let G be open and closed in the product X x N ~ . T h e n the m a p ~r/G" G --. It(G) is the trivial bundle with fiber N ~ , where lr" X • N ~ ~ X denotes the projection onto the first factor.

PROOF. W i t h o u t loss of generality it can be assumed t h a t ~r(G) -- X. Fix complete b o u n d e d metrics pl and p2 on X and N ~ , respectively. On X • N ~ we consider the metric p defined by the formula p ( ( x l , a l ) , (x2, a2)) - 2 - 1 p l ( x l , x2) + 2 - 2 p 2 ( a l , a 2 )

where (xk, ak) E X • N w , k --- 1,2. Using a s t a n d a r d a r g u m e n t we construct a collection ( G n ' n E N ) of open and closed subsets in G such t h a t (a) G - - U { G n ' n e N } , (b) Gn N G m : 0 whenever n ~ m, (c) ~ r ( G n ) = X for each n e N, and (d) d i a m p ( l r - l ( x ) N Gn) < 2 -1 for any n e N and x e X . Let/~1 = {Gn" n e N}. R e p e a t i n g the procedure described above for each of the sets G n separately, and using induction, we construct a sequence {/gk" k E N } of c o u n t a b l y infinite open and closed disjoint covers of G with the following properties (compare with the proof of L e m m a 6.4.1)" (i) b / k - { a n , , . . . , n k " ni e g , i

e g},

k e g.

(ii) Gnl,...,nk --U{Gnl,...,nk,nk+~ " n k + l e g } . (iii) r ( G n ~ , . . . , n k ) - X for any k-tuple ( n l , - ' - , n k ) . 311

312

8. APPLICATIONS

(iv) d i a m p ( T r - l ( x ) N Gn~,...,nk) < 2 - k for any x 9 X and any k-tuple (nl,'"" ,nk). We define a m a p h: X x N W --+ G as follows: h(x,a)-- r-l(x)n

n(a.

k e

N}

for any point ( x , a ) - (x, ( n l , ' " , n k , ' " )) 9 X x N W . It follows from the completeness of p and properties (ii) and (iv) t h a t h(x, a) is n o n - e m p t y and consists of precisely one point. Direct verification shows t h a t h is a homeomorphism. It remains to observe that, by the construction, r h -- r . [-1 LEMMA 8.1.2. spaces is a trivial G of X admits a p ( G n ) - p(G) f o r

A n open surjection p: X ~ Y between zero-dimensional Polish bundle with fiber N W if and only if any open and closed subset countably infinite disjoint open cover {Gn : n 9 N } such that each n 9 g .

PROOF. T h e necessity follows from L e m m a 8.1.1. In order to prove the sufficiency we follow the a r g u m e n t used in the proof of L e m m a 6.4.1. As in t h a t proof (using, in addition, the zero-dimensionality of our spaces and the above condition) we can take a complete metric d on X and construct a sequence {b/k} of countably infinite disjoint open covers of X with the following properties: (i) U k - {Vii ..... ik" ij e g } , j E N , k e N . (ii) Uil ..... i~,ik+l C_ Uil ..... ik, ik+l E N . (iii) Uil ..... ik --U{Ui~ ..... ik,ik+~" ik+l E N } . (iv) diamd(Ui, ..... ik N p - l ( y ) ) < ~ , y 9 y . (v) p ( V i , ..... i k ) - Y , ij 9 N , k 9 g . T h e desired fiber preserving h o m e o m o r p h i s m h" Y x N • --+ X can be defined by letting for each point ( y , a ) -

(y, ( i l , . . . , i k , . . . ) )

9 Y x g ~ .

N

LEMMA 8.1.3. Let S -- { X n , p ~ + l , w } be an inverse sequence consisting of zero-dimensional AE(O)-spaces and O-soft short projections with Polish kernels. Suppose that f o r each n 9 w, the space X n + l contains a C-embedded copy of the product X n • ~ such t h a t p n n + l / ( X n x N ~ ) = 7rx,, w h e r e r x ~ " X n x N ~ --~ X n is the projection onto the first coordinate. Then the limit projection po" X -lim,.q --~ X o is the trivial bundle with fiber N W . PROOF. Apply the a r g u m e n t presented in the proof of L e m m a 7.3.1.

[:]

Combining L e m m a s 6.5.4, 8.1.3 and the proof of T h e o r e m 7.3.3 we obtain the following characterization of the space N r ,T > w. THEOREM 8.1.4. Let T > w. The following conditions are equivalent for any zero-dimensional AE(O)-space X of weight T: (i) X is homeomorphic to N r . (ii) X is strongly A t , o-universal.

8.1. UNCOUNTABLE POWERS OF COUNTABLE DISCRETE SPACES

313

(iii) For each z e r o - d i m e n s i o n a l space Y of R - w e i g h t < T the set of C-embeddings is dense in the space C r ( Y , X ) .

Applying Theorem 1.1.5 and the above characterization we get (compare with Corollary 7.3.4) the following statement. COROLLARY 8.1.5. The product X x N r , T >_ w, is h o m e o m o r p h i c to N r if and only if X is a z e r o - d i m e n s i o n a l A E ( O ) - s p a c e of weight ~_ r.

8.1.2. C h a r a c t e r i z a t i o n of D r . Recall that the Cantor cube D r is strongly Br,0-universal (Lemma 6.5.3). In this Subsection we are going to show that this property in fact characterizes the space D r . THEOREM 8.1.6. Let r > w. Then the following conditions are equivalent f o r every zero-dimensional compact (locally compact and i i n d e l S f ) A E ( O ) - s p a c e X of weight T: (i) X is h o m e o m o r p h i c to D r (to D r x N ) .

(ii) X is strongly Br,o-universal. (iii) For each z e r o - d i m e n s i o n a l compactum Y of weight < T the set of erabeddings is dense in the space C r ( Y , X ) .

(iv) The set of embeddings is dense in the space C r ( n , X ) . (v) X is homogeneous with respect to pseudo-character. The proof of this Theorem involves Lemma 6.5.3, and follows the proof of Theorem 7.2.8 with obvious changes of a technical nature (identical to those between the proofs of Theorems 7.3.3 and 8.1.4). In particular, the proof is based on the following statement (compare with Lemma 8.1.2). LEMMA 8.1.7. Let f : X - . Y be an open surjection between zero-dimensional metrizable compacta. T h e n the following conditions are equivalent: (i) f is a trivial bundle with fiber n "~ . (ii) A l l fibers of f are h o m e o m o r p h i c to n ~ . Theorem 8.1.6 has several consequences. COROLLARY 8.1.8. The product X x D r , r >_ w, is h o m e o m o r p h i c to D r if and only if X is a z e r o - d i m e n s i o n a l A E ( O ) - c o m p a c t u m of weight ~_ r.

COROLLARY 8.1.9. The C a n t o r cube D ~1 is h o m e o m o r p h i c to its hyperspace e x p D W~.

PROOF. Observe that if f" X -~ Y is an open surjection between zerodimensional metrizable compacta, then the corresponding map e x p f " e x p X -+ e x p Y

314

8. APPLICATIONS

is also an open surjection. Next verify that e x p ~ is equivalent to ~, where r : D ~ x D ~ --+ D ~ denotes the projection onto the first coordinate. Finally apply Theorem 8.1.6. rl COROLLARY 8.1.10. T h e C a n t o r cube D ~2 is n o t h o m e o m o r p h i c to its hyperspace e x p D ~2 .

PROOF. Assuming the contrary and applying Theorems 8.1.6 and 1.3.4 we can easily conclude that the map e x p r , where ~r: D ~1 x D ~1 ~ D ~1 must be equivalent to ~r. Further observe t h a t not all fibers of the map e x p r are homeomorphic to the Cantor cube D ~ . This contradiction proves our statement. [::]

H i s t o r i c a l a n d bibliographical n o t e s 8.1. The results of Subsection 8.1.1 are due to the author. The equivalence of conditions (i) and (v) of Theorem 8.1.6 has been proved in [272]. The related result, Lemma 8.1.7, providing a characterization of trivial bundles with fiber D "~ was obtained in [286]. One can prove this statement by simply "parameterizing" the well known proof of Theorem 1.1.10. Another possible way involves Michael's Selection Theorem 2.1.14. Corollary 8.1.9 also appears in [286]. Corollary 8.1.10 was proved in [271].

8.2. Spectral representations of topological groups A classical result of Pontrjagin [253] asserts that each compact topological group can be expressed as the inverse limit of some transfinite inverse spectrum in which all short projections are continuous homomorphisms whose kernels are compact Lie groups (so-called Lie series). Here kernel (i.e. the inverse image of the unit element) has algebraic meaning. Recall also Proposition 6.3.5 (and Remark 6.3.7) which states, in particular, that the AE(0)-compacta are precisely those t h a t admit special spectral representations (the so-called Haydon series). A simple comparison of these two results immediately shows that Haydon's theorem is, in some sense, the analog of Pontrjagin's theorem in the category of compacta and continuous maps. This comparison can be extended to yield a proof of the fact t h a t each compact topological group is an AE(0)-compactum. Moreover, noticing that each locally compact topological group is LindelSf and applying R e m a r k 6.3.6, one can easily see that every locally compact topological group is an AE(0)-space. This allows us to study locally compact topological groups from the point of view of the general theory of AE(0)-spaces and 0-soft maps. For instance, it immediately follows from Theorem 8.1.6 that each uncountable, zero-dimensional compact (locally compact and LindelSf) group is topologically equivalent to D r (to the product N • D r ) [181], [204]. In this Section we extend these characterizations to the non-locally compact case. We begin with the following useful statement.

8.2. TOPOLOGICAL GROUPS

315

LEMMA 8.2.1. Let A >_ w. Each AE(O) topological group X of weight T > A is isomorphic to the limit space of a factorizing A-spectrum S x -- { X a , p ~ , A } all spaces of which are AE(O)-groups of weight A and all projections of which are O-soft homomorphisms. PROOF. By Proposition 6.3.5, the space X can be represented as the limit space of a factorizing A-spectrum 8 x - { X a , p ~ , A } consisting of AE(0)-spaces of weight A and 0-soft limit projections. Let us show that this spectrum contains a A-closed and cofinal subspectrum consisting of topological groups and limit projections that are (continuous) homomorphisms. Since X is a topological group, there is a continuous multiplication #: X x X ~ X. Clearly, X • X is the limit space of the spectrum

All projections of the spectrum S x x S x are 0-soft, and hence open (Proposition 6.1.26). The Suslin number of the product X x X is obviously countable. Consequently, by Proposition 1.3.3, the spectrum S x x S z is factorizing. Next we apply Theorem 1.3.4 to the spectra S x x S x , S x and to the map # between their limit spaces. Then we get a A-closed and cofinal subset B of A such that for each a E B we have a continuous map # a : X a x X a --* X a such that =

• p.).

Now we can define a continuous multiplication operation in Xa by letting xa . y a = # a ( x a , Ya) whenever (xa, y~) e Xa x Xa. It is easy to see that Xa, c~ E B, becomes a topological group with respect to this operation. Moreover, for all c~,/~ E B with a < f~, the projection p~" Xt~ --~ Xa becomes a homomorphism with respect to the above defined operations. [:] COROLLARY 8.2.2. I f (in the notations of L e m m a 8.2.1) X is n-dimensional, then all elements of the spectrum S x = { X a , p ~ , A } can also be assumed to be n-dimensional. PROOF. Apply Lemma 8.2.1 and Theorem 1.3.10.

[-1

We also need the following two simple statements. LEMMA 8.2.3. Let p: X ~ Y be a O-soft h o m o m o r p h i s m between zero-dimensional AE(O) topological groups. Then there exists a homeomorphism h: Y x kerp ~ X such that ph -- ~1, where r l : Y x kerp --~ Y denotes the natural projection onto the first factor. PROOF. Since p is 0-soft and dim Y - 0, there exists a continuous map i" Y X such that pi = i d y and i ( e y ) - e x ( e x and e y denote neutral elements of

316

8. APPLICATIONS

the groups X and Y respectively). Define a h o m e o m o r p h i s m h: Y • kerp ~ X by letting

h(y,a) = i(y).a,

(the dot r l . [::]

9

whenever (y,a) e Y • k e r p

denotes the m u l t i p l i c a t i o n o p e r a t i o n of the group X ) . Clearly ph

=

LEMMA 8.2.4. A n y zero-dimensional space of R-weight ~" admits a C-embedding into N r .

PROOF. 1 Since any zero-dimensional completely m e t r i z a b l e space can be considered as a closed subspace of the space N "~ ( P r o p o s i t i o n 1.1.6), the L e m m a is true for T -- w. Consider now a zero-dimensional space X of R-weight T > w. Obviously the Hewitt realcompactification v X of X is zero-dimensional and has the same R-weight as X . By T h e o r e m 1.3.10, v X can be represented as the limit space of some factorizing w - s p e c t r u m S = {Ya, q~, A} consisting of zerodimensional Polish spaces. Since S is factorizing, it is easy to see t h a t v X is C - e m b e d d e d (and closed) in the p r o d u c t 1-'[{Y~ : c~ e A}. This product, in turn, can be C - e m b e d d e d into N r (since each Ya can be e m b e d d e d into N • as a closed subspace and IAI = T). Consequently, X can be C - e m b e d d e d into N r . D

In the following T h e o r e m , k ( X ) denotes the minimal cardinal ~ such t h a t X contains a c o m p a c t s u b s p a c e K whose p s e u d o - c h a r a c t e r in X , r is T. Note t h a t if, for a topological group X , we have k ( X ) < w, t h e n X is locally

1One might formally apply Lemma 6.5.3.

8.2. TOPOLOGICAL GROUPS

317

compact. THEOREM 8.2.5. Let X be a non-locaUy compact zero-dimensional AE(O) topological group. Then X is homeomorphic to the product N a(X) x D w(X). PROOF. We consider two cases separately. Case 1" k ( X ) -- w ( X ) . We use transfinite induction. T h e s t a r t i n g point of the i n d u c t i o n (i.e. the case k ( X ) -- w ( X ) = w) is trivial, because topologically X is a zero-dimensional Polish space w i t h o u t open c o m p a c t subspaces. Consequently, by T h e o r e m 1.1.5, X is h o m e o m o r p h i c to the space of irrationals N ~ ..~ N W • D ~ . T h u s we can suppose t h a t in Case 1 our s t a t e m e n t is t r u e for all spaces of weight A, where w _< A < T, and consider a zero-dimensional AE(O) topological g r o u p X with k ( X ) = w ( X ) = T. To prove t h a t X is h o m e o m o r p h i c to the p r o d u c t N v x D r ~ N r ' we can use a topological c h a r a c t e r i z a t i o n of the l a t t e r space ( T h e o r e m 8.1.4). T r a n s l a t i n g condition (iii) of T h e o r e m 8.1.4 into s p e c t r a l language, we see t h a t it suffices to show the following: Consider a zero-dimensional space Y of R-weight A < ~- ( w i t h o u t loss of generality, we m a y a s s u m e t h a t A _ w), fix an a r b i t r a r y m a p f " Y --. X , fix a factorizing A-spectrum S x -- { X a , p ~ , A} , satisfying the conditions of L e m m a 8.2.1, and any index a E A. Our goal is to prove the existence of a C - e m b e d d i n g g" Y ---+ X such t h a t p a f = pad. First, let us t r y to find an index fl > a such t h a t the p r o j e c t i o n p~" X~ --~ X~ of the s p e c t r u m S x is (topologically) a trivial bundle with fiber N>'. Let a0 = a. Since the weight of the space X a o is strictly less t h a n the weight of the space X , we can conclude t h a t the fiber kerpao (i.e. the inverse image P~o -1 (e~o), where eao denotes the n e u t r a l element of the group Xao) is n o n - c o m p a c t . Indeed, otherwise we would have

k ( Z ) a0, such t h a t kerpg~ is n o n - c o m p a c t . Suppose now t h a t for every ordinal 7 with 7 < 5, where 5 < A, we have a l r e a d y chosen indexes a 7 E A in such a way t h a t the following conditions are satisfied: (a) I f # < v < 6 , thena, w. W i t h o u t loss of generality we may suppose t h a t each space Ea, a 9 A, contains at least two points, and consequently the space E is h o m e o m o r p h i c to R r (apply Proposition 2.4.23). Consider a functionally closed, proper and 0-invertible m a p f" N A ----+E having a regular averaging o p e r a t o r (see Proposition 6.1.25). Since dim N A = 0 and K is a C - e m b e d d e d A E ( 0 ) - s u b s p a c e of E, there exists a m a p h" N A ~ K such t h a t h/f-l(g) = f /f-l(g). We now introduce some notation. If C C_ B C_ A, then

~r~" H { E a " a 9 B}---+ H { E a " a 9 C} and

7rB'E-+H{Ea'aEB

}

denote the n a t u r a l projections onto the corresponding subproducts. Also

K B "- 1rB(K), K ( B ) - 7rBI(KB), PB -- 7rB/K and p~ -- 7r~/KB. We say t h a t a subset B of A is admissible if

lrB(x) - P B h f - l ( x )

for each point x 9 K ( B ) .

As in the proof of T h e o r e m 6.3.1, we can verify the following properties of admissible subsets: (a) T h e union of an a r b i t r a r y collection of admissible subsets is again an admissible subset. (b) For any admissible subset B, the set KB is closed and C - e m b e d d e d in the p r o d u c t YI{Ea" a E B}. (c) For any admissible subset B the space KB is an AE(0)-space. (d) For any two admissible subsets C and B with C C_ B, the maps PB and p ~ are 0-soft. (e) For any admissible subset B, the set K B is convex in the space

rI{Ea" a e B}. (f) T h e family of all countable admissible subsets of A is w-closed and cofinal in the w-complete set exp,,A. Since IAI = T, we may write A = {as" c~ < T}. By (f), each point a s of A is contained in some countable admissible subset B s . P u t A s = LJ{Bt~"/~ _< c~}, K s = KA,~ and p g + l = PA,~A'~+I (a < T).

8.3. LOCALLY CONVEX SPACES

323

It follows, from the properties of admissible subsets listed above, t h a t the limit space of the n a t u r a l l y induced well ordered continuous s p e c t r u m S - - { g a , p g +I,T} coincides with the given space K. T h e countability of the sets B a guarantees, by L e m m a 8.3.4, the softness of the short projections of this spectrum. Obviously, the space B0 is an absolute retract ( T h e o r e m 2.1.17). In this situation, Proposition 6.1.21 and L e m m a 6.2.6 conclude the proof. Vl We are now ready to prove the following s t a t e m e n t , which extends Proposition 2.4.30 to the non-metrizable case. THEOREM 8.3.6. Let T be an uncountable cardinal.

Then the following con-

ditions are equivalent f o r each locally convex linear topological space E of weight T"

(i) E is h o m e o m o r p h i c to R r (ii) E is an absolute retract. (iii) E is an AE(O)-space. PROOF. T h e implications (i) ==~ (ii) and (ii) ==~ (iii) are obvious. Let us prove the implication (iii) ~ (i). We use transfinite induction. If ~- - - w , then E is an infinite-dimensional separable Frgchet space and the conclusion follows from Proposition 2.4.30. Therefore we may suppose t h a t our s t a t e m e n t is true for all infinite-dimensional locally convex A E ( 0 ) - s p a c e s of weight A, w _ A < T, and consider a locally convex A E ( 0 ) - s p a c e E of weight r. By Proposition 8.3.5, E is an absolute retract. Consequently, by T h e o r e m 7.3.3 it suffices to show t h a t for any space Z of R-weight < T, the set of C - e m b e d d i n g s is dense in the space Cr(Z,E). Consider an a r b i t r a r y space Z of R-weight A (where w __ A < r), a map g" Z --. E, a factorizing A-spectrum A s -- { E a , p ~ , A } the limit space of which coincides with E, and an index a E A. We wish to construct a C - e m b e d d i n g h" Z --, E such t h a t p a h -- pag. By Proposition 8.3.2, we may suppose w i t h o u t loss of generality t h a t all spaces E a are locally convex AE(0)-spaces of weight A and all limit projections pa are linear 0-soft maps. Moreover, by Proposition 8.3.5, we see t h a t all spaces E a are absolute retracts. Finally, by Proposition 6.3.5, we may assume w i t h o u t loss of generality t h a t all limit projections are (linear and) soft. Let us find an index/~ E A such t h a t f~ > c~ and the projection p~" Et~ ~ E a is a trivial bundle with fiber R )'. P u t a0 -- a. Since the weight of E a o is strictly less t h a n the weight of E and the projection Pao" E --~ E a o is linear and open (Proposition 6.1.26), we conclude t h a t the fiber k e r p a o contains at least two points. T h e n we can choose an index a l > s0 such t h a t ker Paoal also contains at least two points. Suppose now t h a t for every ordinal "r with ~/ < 5, where 5 < A, we have already chosen indexes a~ E A in such a way t h a t the following conditions are satisfied:

324

8. APPLICATIONS

(a) If ~ < u < 5, then c~ < c~v. (b) If v is a limit ordinal, then a~ - s u p { ~ : # < v}. (c) T h e fiber k e r p ~ , contains at least two points (# < v < 5). Now we c o n s t r u c t an index c~ with the desired properties. If 5 is a limit ordinal, then we p u t a~ - sup{c~.y: ~/ < 5}. By the inequality 5 < A and the A-completeness of the s p e c t r u m S, we have t h a t c~ E A. If 5 - ~ + 1, then the index a~ can be c o n s t r u c t e d in precisely the same way as the index c~ - 1. Now consider fl - sup{c~.y: ~, < A} and note t h a t fl E A. Consider the well ordered spectrum , ~ ' = {E.y,- a~+l A}. T h e A-continuity of the s p e c t r u m S g u a r a n t e e s t h a t the limit space of the s p e c t r u m 8 ' is naturally isomorphic to the space Eft, and the limit projection l i m S ' ---+ Eao -- E a is isomorphic to the short projection p~" Ef~ ---+ E a of the s p e c t r u m S. As in L e m m a 8.2.3, we can easily see t h a t the m a p p~ is topologically a trivial bundle with fiber kerp~. It is also easy to verify t h a t the fiber kerp~ is h o m e o m o r p h i c to the p r o d u c t H { k e r , ~r ~

9 ~ < A}.

Consequently, by conditions (a) - (c), the weight of kerp~ is equal to A. By the same conditions we can conclude t h a t even if A -- w, the space is infinite-dimensional (otherwise we get a contradiction to condition (c)). Consequently, by our hypothesis, the space kerp~ is h o m e o m o r p h i c to R x. Moreover, in this case the projection p~ can be topologically identified with the natural projection 71"1: Ec~

X

R A ----+Ec~.

Since the R-weight of the space X is equal to A, there exists a C - e m b e d d i n g i: Z ~ R x. T h e n the diagonal p r o d u c t j -pof/ki:

Z ---+ E ~ • R ~ -

El3

is a C - e m b e d d i n g satisfying the equality p ~ j - p a l . Take an a r b i t r a r y section k: Ef~ ~ E of the limit projection p• of the s p e c t r u m S (recall t h a t pf~ is a soft map). T h e n the composition h -- k j is the desired C - e m b e d d i n g of Z into E. [-I COROLLARY 8.3.7. E v e r y closed c o n v e x body in a locally c o n v e x A E ( O ) - s p a c e o f w e i g h t v > w is h o m e o m o r p h i c

to R r .

PROOF. Let X be a closed and convex b o d y in a locally convex A E ( 0 ) - s p a c e E of weight T > w. Since int X =-~ 0 we may conclude t h a t X - cl(int X ) . In particular, w ( X ) - T. By T h e o r e m 8.3.6, E is h o m e o m o r p h i c to R r and consequently, by Proposition 6.1.8, X is functionally closed in E. By Corollary 6.4.8, X is C - e m b e d d e d in E. By Propositions 6.4.9 and 8.3.5, X is an absolute retract. Corollary 7.1.17 finishes the proof. !-7

8.4. SHAPE PROPERTIES

325

COROLLARY 8.3.8. Every linear G6-subspace of a locally convex AE(O)-space of weight T > w is homeomorphic to R r . PROOF. Let X be a linear G6-subspace of a locally convex AE(0)-space E of weight T > w. P u t Y -- c l X . As in the proof of Corollary 8.3.7, we can conclude t h a t Y is h o m e o m o r p h i c to R r . Consequently it is sufficient to show t h a t X - Y. Indeed, if there exists a point y E Y - X, then the translate X + y would be a dense G6-subspace of Y and (X + y) N Y = 0, which contradicts the Baire property in Y ~ R r . D COROLLARY 8.3.9. The closed convex hall of a compact subset of a locally convex AE(O)-space is compact. PROOF. We only have to observe t h a t each locally convex AE(0)-space, by Proposition 8.3.2, is complete. T h e rest is well known (see [258]). [-1 T h e concept Recall t h a t a Obviously, the a r g u m e n t used

of projector is i m p o r t a n t (continuous) linear map image f ( E ) is a linear in the proof of T h e o r e m

in the theory of linear topological spaces. f:E --~ E is a projector if f2 = f . continuous retract of E. Applying the 8.3.6 we obtain the following s t a t e m e n t .

PROPOSITION 8.3.10. The image of a continuous projector, defined on a product of separable Frgchet spaces, is isomorphic to a product of separable Frgchet spaces.

Historical and bibliographical notes 8.3. All results of this Section are due to the a u t h o r [93]. Corollaries 8.3.7 and 8.3.8 e x t e n d to the non-metrizable case the corresponding results of Slee-Corson and M a z u r - S t r e n b a c h (see [32]).

8.4. Shape properties of non-metrizable compacta 8.4.1. Spectral theorem in Shape category. We start with the following definition. DEFINITION 8.4.1. A n inverse spectrum s -- ( X a , p ~ , A } is said to be homotopically stable with respect to the space Y if for each index a E A, each closed subspace F of X a , and maps f , g : F --~ Y , the relation f p a / p ~ l ( F ) ~_ gpa/p-~l(F) implies the relation f ~_ g. We say that S x is (absolutely) homotopically stable if 3 x is homotopically stable with respect to any metrizable A N R-compactum. LEMMA 8.4.2. There exists a countable collection (Gn: n E w} of metrizable locally compact A N R-spaces such that any inverse spectrum which is homotopically stable with respect to each space Gn, n E w, is absolutely homotopically stable.

326

8. APPLICATIONS

PROOF. As the desired collection, we may take an arbitrary countable open basis of the topology of the Hilbert cube containing unions of its finite subcollections. The verification of this fact is trivial and is left to the reader. W1 LEMMA 8.4.3. Let p: X ---. Y be a surjection and .~ = {Fa} be a closed basis

of the topology of Y containing intersections of its finite subfamilies. an A N R-space and suppose that .for each Fa E jz and any maps f, g: the condition f p / p - 1 (Fa) "" g p / p - l ( F a ) implies the condition f ~_ for each closed subspace F of Y and any maps f , g : F ---. Z, the

Let Z be Fa ~ Z, g. Then condition

PROOF. Since Z is an A N R-space there exists a neighborhood V of F in Y and extensions f , ~ " V ~ Z of f and g respectively. By our assumption, the restrictions f p / p - l ( Y ) and ~ p / p - l ( Y ) to p - l ( F ) are homotopic. Since Z is an ANR-space, we can find a neighborhood U of p - l ( F ) in X contained in p - l ( Y ) such that the restrictions ] p / U and ~p/U are homotopic. Clearly, we may suppose without loss of generality that the set U has the form U - p - l ( w ) , where W is a neighborhood of F in Y. Then the compactness of Y and the corresponding property of ~" imply that there exists an element F~ E $" such that F C Fa C W. By our assumption ] / p - l ( F a ) ~_ ~ / p - l ( F a ) . Consequently,

f~--g.

D

THEOREM 8.4.4. Each w-spectrum, consisting of metrizable compacta, contains a homotopically stable w-closed and co final subspectrum. PROOF. Let S x = {Xa,P~a,A} be an arbitrary w-spectrum consisting of metrizable compacta. Fix an integer n E w. We shall show that the indexing set A of our spectrum contains an w-closed and cofinal subset An such t h a t the corresponding subspectrum s~/A.

=__s .

=

{x~,p~.,A.}

is homotopically stable with respect to the space Gn from Lemma 8.4.2. Let •n

-

{ ( a , ~ ) E A 2" a _~ f~, and for each closed subspace F of X~ and any two maps f , g " F ~ Gn the conditionfp~/p-~l(F) "~ g p a / p ~ l ( F ) implies the c o n d i t i o n f pf~a / ( p ~ ) - l ( f ) ' ~ gpa/(p~)

l(f)}

Let us verify that the existence condition of Proposition 1.1.29 is satisfied with respect to the relation s Fix a E A. Our goal is to show that there exists an index f~ E A such that (a, f~) c s First of all, we shall fix a closed subspace F of the compactum Xa and construct an index a F _> a such that the following condition is satisfied: (*)F for maps f, g" F ~ Gn the condition fpa/p-~l(F) ~_ g p a / p ; l ( F ) implies the condition f p ~ F / ( p ~ F ) - l ( f ) ~-- g p ~ F / ( p ~ F ) - - l ( f ).

8.4. SHAPE PROPERTIES

327

Fix a countable dense subset {hi" i E N } of the space C(F, Gn) (the compactopen topology is considered). Let M denote the set of those pairs (i,j) E N 2 for which the conditions hipa/p'~l(F) ~ hjp,~/p~l(F) are satisfied. For each (i,j) E U we can choose an index a(i,j) >_ ~ such t h a t (see, for example,

[214]): hipg(i,J) / (pg(i,J))-l (F) ~_ hjpg(i'J) / (pg(i,J))-l (F). It follows easily from the construction t h a t for each (i,j) E M and each 9' E A we may suppose t h a t a(i,j) >_ .~. Consequently, well-ordering M in an a r b i t r a r y way, we may assume t h a t {c~(i,j)" (i,j) E M } forms a chain in A. Let us verify t h a t the index a F = m a x { a ( i , j ) " (i,j) E M } is the desired one. Consider two maps f,g" F --, Gn such t h a t f p a / p ~ l ( F ) ~ gpa/p-~l(F). By our assumption, Gn is an ANR-space and, by the construction, {hi" i E N } is a dense subset of the space C(F, Gn). Consequently, there exist integers i,j E N such t h a t f ~_ hi and g "-~ hi. T h e n

hip,/p-~l(F) ~_ fp,/p-~l(F) ~ gp,/p-~l(F) ,.~ h j p , / p ~ l ( F ) . In the other words, (i,j) E i . have

Hence, by the definition of the index a(i,j), we

h i p g F / ( p ~ F ) - l ( F ) _ h.~a(i,j).~aF ~t'a ~'a(i,j)/(P~ F)

--1

(F) ~ hjpg(~'3)paa~i,j)/(pgF)-l(F ) ""

= hjpg~/(p~)-~(F). Then

fp~/(p~F)-l(F)

~_ hip~F/(p~F)-l(F) ~ h j p ~ / ( p ~ F ) - l ( F )

Thus, for each closed subspace F of the c o m p a c t u m Xa we have an index a F satisfying the above condition (*)F. Consider now an a r b i t r a r y countable basis {Fk" k E w} of the topology of X a containing the intersections of its finite subfamilies. For each k E w, fix an index ak -- a F k. Obviously, by Corollary 1.1.28, there exists an index f~ E A such t h a t f~ > c~k for each k E w. It follows from the above construction, and from L e m m a 8.4.3, t h a t (c~,f~) E L:n. T h e verification of the existence condition is completed. T h e verification of the m a j o r a n t n e s s condition from Proposition 1.1.29 is trivial. Finally we must verify the w-closeness condition from Proposition 1.1.29. This means t h a t if we have a countable chain {OLm'm E w} in A and condition (am, f~) E s is satisfied for each m E w and some f~ E A, t h e n the condition (c~, f~) E s where a = sup{c~m" m e w}, is also satisfied. Indeed, since our s p e c t r u m is w-continuous, the c o m p a c t u m X a is naturally O~m-{-1 h o m e o m o r p h i c to the limit space of the inverse sequence {Xa.~,~'am ,W}. Conpa -1 sequently, the sets of the form (a,,.) (Fro), where Fm is closed in Xa~ and

328

8. APPLICATIONS

m E w, form a closed basis of the topology of X a which contains the intersections of its finite subfamilies. Consequently, if we wish to prove the inclusion (c~,~) E s it suffices, by L e m m a 8.4.3, to show that if Fm is closed in Xa.~, m e w, and the maps f,g: F -- (p~,~)-l(Fm) -+ Gn satisfy the condition f p a / p ~ l ( F ) ~ _ gpa/p~l(F), then the condition fp~/(p~a)-l(F)~_ g p ~ / ( p ~ ) - l ( f ) will also be satisfied. Since Gn is an ANR-space, there exist an integer k _> m and maps fl,gl" paok(F) - , Gn such that f ~ flp~)k and g ~- glpg k. Then we have

flpak/p~:(Fk) ~-- glpak/p~:(Fk) where F k - P~k (F). By the assumption, (ak,/~) E s

/

(Fk) "~ glp~k

Consequently, we have

I

(f k).

It only remains to note that ( p ~ ) - l ( F ) = (p~)-l((p~m)-l(Fm)) = (p~k)-l(Fk) and that a

f~

--1

~_ glpgkp~/(p~a)-l(F)

--1

"~ gp~a/(p~)-l(F).

Thus the w-closeness condition is also satisfied. Now, by Proposition 1.1.29, we can conclude that the set An of all s indexes from A is w-closed and cofinal in A. At the same time we note that, by the corresponding definitions, this means that the subspectrum Sn of S x is homotopically stable with respect to the space Gn. Finally, let

A ' - - N{An: n E w}. By Proposition 1.1.27, this set is w-closed and cofinal in A as well. Consequently, the corresponding subspectrum S' -- {Xa,p~, A'} is homotopically stable with respect to each space Gn, n E w. L e m m a 8.4.2 finishes the proof. [-1

Remark 8.4.5. It is easy to see that each homotopically stable spectrum is homotopically stable with respect to every (:W-complex. Let S x - {Xa,p~a, A} be any homotopically stable spectrum and K be a (:W-complex. Suppose that for some index a E A and for some maps f , g : F --, K, where F is a closed subspace of Xa, the condition f p a / p ~ l ( F ) ~_ g p a / p ~ l ( F ) i s satisfied. Then, by compactness of lim S x and by the properties of the topology of C1/Y-complexes, there exists a metrizable A N R-compactum K' lying in K, such that the maps f p a / p ~ l ( F ) and gpa/p-~l(F) are homotopic in g ' . By our assumption, we can conclude that f and g are homotopic in K ' and, consequently, in K as well.

8.4. SHAPE PROPERTIES

329

Let us now recall the definition of the S H A P E category over compacta. T h e class of objects of the S H A P E category is the class of all compacta. T h e morphisms, or more precisely, the elements of the set M O r S H A P E ( X , Y ) are defined to be the maps a" U { [ Y , P ] " P is an A N R }

-+ U { [ X , P]" P is an A N R }

satisfying the following two conditions: . If P is an A N R - c o m p a c t u m , then a([Y, P]) _C IX, P]. . If P and P ' are A N R - c o m p a c t a , r E [Y, P], ~ E [Y, P'] and -y E [P, P'] are such t h a t 7 r = ~, then -ya(r = a ( ~ ) . T h e n a t u r a l (fundamental) functor Jz: H O M O T

~ SHAPE

acts in the following fashion: . ~ ' ( X ) = X for each c o m p a c t u m X. - If ~o E IX, Y], then 9r(~): X ~ Y is given by 9r(~o)(r r E [Y, P] where P is an A N R - c o m p a c t u m . T h e composition of the functors 7-/and ~" is d e n o t e d by Sh: COMP

= r

for any

~ SHAPE

and called the shape functor. It is known [175], [214] t h a t (unlike the h o m o t o p y functor) the shape functor is continuous. LEMMA 8.4.6. Let S x = { X a , p ~ a , A } be an w-spectrum consisting of metrizable compacta and ~: l i m S x --~ Y be a shape m o r p h i s m , where Y is a metrizable compactum. Then there exist an index a E A and a shape m o r p h i s m ~a: X a --* Y such that ~o = ~ a Sh(pa). PROOF. W i t h o u t loss of generality, by T h e o r e m 8.4.4, we may assume t h a t the s p e c t r u m 8 x is homotopically stable. Represent Y as the limit space of some inverse sequence S y = {Yn, qnn+l,w} consisting of compact polyhedra. T h e shape m o r p h i s m ~: X ~ Y, where X = lim 8 x , naturally induces a homotopy class ~([ql]) : X --. ]"1. Denote by gl : X --~ Y1 any representative of this class. Fix an index a l E A and a map gal: X a l ~ Y1 such t h a t gl --- galPal. Now consider the map q2: Y --* ]I2. As above, let g2: X --~ Y2 be any representative of the h o m o t o p y class T([q2])" X --~ ]I2. Since ql - q2q2 and Y1 and Y2 are compact polyhedra, we can conclude, by the definition of shape m o r p h i s m [214], t h a t the condition ~([ql]) - [q12]~o([q2]) is satisfied. Consequently, gl - q2g2. As above, we choose an index a2 >__ a l and a m a p g a 2 : X a 2 -* Y2 such t h a t g2 -- ga2Pa2. T h u s we have c~2

2

2

galPalPa2 -" galPal - gl ~ q g2 = qlga2Pa2" Since Y1 is an A N R - c o m p a c t u m and our s p e c t r u m is homotopically stable, we conclude t h a t g ~ p ~ "~ q21g~2.

330

8. APPLICATIONS

Continuing this process, we get, for each n E w, an index C~n E A and a map dan" X~,~ ---* Yn satisfying the following conditions: (i) an a0, define a shape morphism ~a" X a --+ Y by letting ~a ~ao Sh( so)" Clearly, ~ ~a Sh(pa) for each a > a0 Since the c o m p a c t u m X is the limit space of the w-spectrum S' = { X a , p ~ , A o } , where A0 = {a E A" a _> a0}, it suffices to show that each limit projection pc" X --~ X a , a >_ co, is a shape equivalence. This means that for each metrizable A N R c o m p a c t u m Z, the natural correspondence [Xa, Z] --~ IX, Z], induced by the limit projection pa by the rule If] --~ [fpa], is a bijection. Since the spectrum S is homotopically stable, we immediately conclude that this correspondence is injective. Let us show that it is surjective as well. Consider an arbitrary homotopy class [g] E [X, Z]. Then ~-l([g]) e [Y,Z] and ~ a ( ~ - l ( [ g ] ) ) E [Xa, Z]. It only remains to note that

v.(v-~([g]))[p.] v(v-l([g]))= [g].

8.4. SHAPE PROPERTIES

333

(iii) =:~ (iv). As the desired surjection, we m a y take any limit projection of the s p e c t r u m satisfying the properties of condition (iii). T h e implication (iv) ~ (i) is trivial. E] PROPOSITION 8.4.10. Let X be a compactum. Then the following conditions are equivalent: (i) For each closed G~-subset Z of X and each metrizable A N R - c o m p a c t u m P, the set [Z, P] is countable. (ii) X can be represented as the limit space of some w-spectrum, all limit projections of which are hereditary shape equivalences. PROOF. (i) ~ (ii). By T h e o r e m 8.4.4, we can represent X as the limit space of a homotopically stable w-spectrum S x = {X~, p~, A} . Let _

_

{(c~, f~) E A 2" a _< f~, and for each closed subspace F of X a the m a p

~/p~l(F)"

p-~l(F) ~ ( p ~ ) - l ( F ) is a shape equivalence}.

First, let us verify the existence condition of P r o p o s i t i o n 1.1.29. Fix an index a E A and a countable closed basis {Fn" n E w} of the topology of metrizable c o m p a c t u m X a containing the intersections of its finite subcollections. By (i), for each n E w and for each metrizable A N R - c o m p a c t u m P, the set [p~l(Fn)], P] is countable. Consequently, by Proposition 8.4.9, for each n E w the c o m p a c t u m p~l(Fn) can be represented as the limit space of some w - s p e c t r u m all limit projections of which are shape equivalences. By T h e o r e m 1.3.4, there exists an w-closed and cofinal s u b s p e c t r u m with the above p r o p e r t y in every w - s p e c t r u m representing the c o m p a c t u m p-~l(Fn). In particular, this is true for the n a t u r a l l y induced (by S x ) s p e c t r u m {(p~a) - l ( F n ) ,

5 5 -1 (F~),5 >_ ~ _> .}. p~/(v~)

Hence there exists at least one index an _> a such t h a t the m a p v../v-~(Fn)

9V;~(F.)

~

(p."~)-l(F~)

is a shape equivalence. W i t h o u t loss of generality, we m a y assume t h a t {an" n E w} is a chain in A. Let f~ = sup{an" n E w}. Note t h a t for each n C w, the m a p

p/3/p-~l(Fn) " p-~l(Fn) --+ (p~a)-l(Fn) is a shape equivalence. Let us show t h a t ( a , ~ ) E s Let F be an a r b i t r a r y closed subset of X a and f" p ~ l ( F ) --~ P be a m a p into a metrizable A N R c o m p a c t u m . As in the proof of L e m m a 8.4.3, we can find an integer n E w and a m a p f " p~l(Fn) ~ P where F C_ Fn and .f = f ' / p - ~ l ( F ) . By the definition of ~, there exists a m a p g " ( p ~ ) - l ( F n ) --~ P such t h a t f ' ~ g'p~/p~l(Fn). P u t g = g ' / ( p ~ ) - l ( F ) . Obviously, f ~ gp/3/p-~l(F). T h u s we have shown t h a t the n a t u r a l correspondence [(p~)-I(F),P]--~ [p~I(F),P], induced by p z / p ~ l ( F ) , is surjective. Since the s p e c t r u m S x is homotopically stable we conclude t h a t t h e above correspondence is bijective.

334

8. A P P L I C A T I O N S

The verification of the majorantness condition from Proposition 1.1.29 is trivial. Now let {~n: n E w} be a countable chain in A, and let (an, ~) E E for each n E w and for some ~ E A. Let us show t h a t (a, ~) E L, where a - sup{an : n E w}. By the w-continuity of the spectrum S x , the c o m p a c t u m X a is naturally homeomorphic to the limit space of the inverse sequence {Xa~ ,pa~ _a~+l ,w}. Let F be a closed subset of X a and f : p ~ l ( F ) ---. P be an arbitrary map into a metrizable A N R - c o m p a c t u m . Fix a neighborhood V of p-~l(F) in X and an extension f ' : V ~ P of f . W i t h o u t loss of generality, we may assume that V - p ~ l ( U ) , where U is a neighborhood of F in Xa. There exists an integer n E w such t h a t

F c where F n -

c U,

(p~,)(F). Hence c

C Y.

Since ( a n , ~ ) e s there exists a map g" (p~.)(Fn) --* P such that P

--1

P

--1

f / P a ~ (Fn) ~-- g PZ/Pa~ (Fn). Clearly the map g - g ' / ( p ~ ) - l ( f ) satisfies the condition f ~_ gp~/p~l(F). This shows that the natural correspondence [(p~)-l(F),P] --+ [ p ~ l ( F ) , P ] is surjective. The homotopy stability of S x implies that this correspondence is bijective. Thus (a, B) e L. Now, by Proposition 1.1.29, the set A' of L-reflexive indexes of A is w-closed and cofinal in A. It only remains to note t h a t the L-reflexibility of an index a E A means precisely t h a t the limit projection pc: X ---, X a is a hereditary shape equivalence. The proof of the implication (i) ~ (ii) is finished. The verification of the converse implication is trivial. V1 PROPOSITION 8.4.11. Let X be a finite-dimensional 1-UV-compactum. the following conditions are equivalent: (i) X is shape equivalent to a finite polyhedron. (ii) The C ech cohomology groups of X are finitely generated.

Then

PROOF. For metrizable X this s t a t e m e n t was proved in [157]. Therefore, we may assume below that X is a non-metrizable compactum. (ii) ===>(i). Let 8 x - {X~, p~, A} be an w-spectrum the limit space of which coincides with X. By (ii), Proposition 8.4.9 and Theorem 1.3.4, there is an wclosed and cofinal subset A1 _C A such that the limit projection pc: X --+ X a is a shape equivalence whenever a E A1. In particular, the (~ech cohomology groups of X~, a E A 1, are finitely generated [4"7]. For the same reason, X a is 1 - UV for each a E A1. By Theorem 1.3.10, there is an w-closed and cofinal subset A2 C_ A such that d i m X a - d i m X < c~ for each a E A2. Now, by Proposition 1.1.27, the set B -- A1 M A2 is also w-closed and cofinal in A. In particular, B :fi 0. Let E B. Since X a is metrizable, we conclude (see the beginning of the proof)

8.5. FIXED POINT SETS

335

t h a t X a is shape equivalent to a finite p o l y h e d r o n . It only remains to note t h a t X is s h a p e equivalent to X a. T h e implication (i) = ~ (ii) is trivial. [::]

Historical and bibliographical notes 8.4. T h e results of this Section are t a k e n from [99]. T h e equivalence of conditions (i) and (ii) in P r o p o s i t i o n 8.4.9 was proved in [310]. P r o p o s i t i o n 8.4.11 for m e t r i z a b l e c o m p a c t a was o b t a i n e d in [157]. T h e axiomatic description of the s h a p e functor a p p e a r s in [174].

8.5. F i x e d p o i n t s e t s o f T y c h o n o v

cubes

T h e p r o b l e m we are interested in is the following: W h i c h closed subsets of t h e T y c h o n o v cube I r coincide with the set of fixed points of continuous self-maps of I r ? T h e p r o b l e m is m o t i v a t e d by the following two facts. First of all, each n o n e m p t y closed subset of the Hilbert cube I • coincides with the set of fixed points of some self-mapping (actually, a u t o h o m e o m o r p h i s m ) of I ~ [216]. On t h e o t h e r hand, for each u n c o u n t a b l e r there is a closed zero-dimensional subspace K~ of the T y c h o n o v cube I ~ which does not coincide with the set of fixed points of any self-map of I r [197]. T h e first fact says t h a t the Hilbert cube I ~ has the CIP (complete invariance property [308]), whereas t h e second tells us t h a t the T y c h o n o v cube I r (T > w) does not. B o t h results express a p r o p e r t y of t h e a m b i e n t spaces I ~ and I ~ , T > w, respectively. B u t obviously, t h e r e is a n o t h e r dual point of view of this situation. Before we s t a r t a more formal discussion, let us (for simplicity) i n t r o d u c e a p p r o p r i a t e terminology. Let us say t h a t a c o m p a c t u m X can be fixed in a c o m p a c t u m Y if t h e r e is an e m b e d d i n g i" X ---. Y such t h a t i ( X ) = f i x ( f ) for some m a p f " Y ~ Y ( f i x ( f ) denotes the set of all fixed points of f ) . Respectively, we say t h a t X cannot be fixed in Y if t h e r e is no such e m b e d d i n g of X into Y. Of course, if we fix an a m b i e n t c o m p a c t u m Y, t h e n the above properties become topological p r o p e r t i e s of X . Therefore, it does not follow from the above cited result of [197] t h a t the c o m p a c t u m K r c a n n o t be fixed in I r . So t h e r e still is a possibility t h a t any c o m p a c t u m of weight w, we strengthen the above mentioned result of [197] by showing t h a t the compactum K,, 1 cannot be fixed in any non-metrizable ANR-compactum. Representing K,~I as the limit space of a transfinite inverse spectrum consisting of zero-dimensional metrizable compacta, and applying our characterization theorem together with a spectral theorem, we conclude t h a t among the projections of this spectrum, only countably many can be stable. All others serve as examples of surjections between zero-dimensional metrizable compacta which are not stable. Obviously, these maps are not open (open surjections between zero-dimensional metrizable compacta are retractions, and therefore are stable). We also present an example of a stable retraction between metrizable AR-compacta (which is not open).

8.5.1. E x t e n s i o n of h o m e o m o r p h i s m s . The following definition is motivated by the definition of (usual) Z-sets, and is adapted to the case of uncountable products. DEFINITION 8.5.1. Let T > W. We say that a closed subset Z of a space X is a Zr-set in X if for each collection {lilt" t E T } , where Ltt E c o y ( X ) and IT I < T, there is a map f " X ---, X such that f ( X ) M Z = 0 and f is {Lit" t E T}-close to idx. In other words, Z is a Zr-set in X if idx E clc~.(x,x){f E Cr(X,X)"

f ( X ) M Z = 0).

Note that the notion of Z~-set coincides with the usual notion of Z-set. LEMMA 8.5.2. Let X -- Y I { x a : a E A } be a product of compact I ~ - m a n i f o l d s , ]A] -- r > w, and suppose there is an index b E A such that X a is a copy of the Hilbert cube I ~ f o r each a ~ b. Suppose that Z and F are closed subsets of Z such that 7ra(Z) and 7~a(F) are Z - s e t s in X a for each a E A. Then f o r each h o m e o m o r p h i s m g: Z --~ F which is homotopic (in X ) to the inclusion map of Z , there exists an a u t o h o m e o m o r p h i s m G: X ---, X such that G / Z -- g and G ~_ i d x . PROOF. Represent X as the limit space of the standard w-spectrum S x = { X B , lr~,exp,,,A} consisting of countable subproducts X B = Y I { X a ' a E B} of X and natural projections 7r~" X B ~ X c . Consider also the induced w-spectra Sz--

{ZB,p~,exp~A}

and SF = {FB, q ~ , e x p ~ A }

8.5. FIXED POINT SETS

337

where Z B = lrB(Z) , p ~ -- 1r~/ZB, FB -- 7rB(F) and q~ = l r ~ / F B ( C , B E exp,.,A, C C_ B ) . Obviously, Z = lim S z and F = lim SF. By T h e o r e m 1.3.4, the h o m e o m o r p h i s m g is the limit map of some m o r p h i s m g :

{gB: F B -'+ Z B , B E )i~l}: ,SZ/]~I ---+5F/)i~I

consisting of h o m e o m o r p h i s m s gB's, where 1~1 denotes a cofinal and w-closed subset of expwA. Consider an w-spectrum B 5 Z X 51 = { Z B X I , p C X i d i , e x p w A }

where I = [0, 1]. Obviously l i m ( S z x 5x) = Z x I. Fix a h o m o t o p y H : Z x I --, X connecting g and the inclusion map i: Z --, X. Applying T h e o r e m 1.3.4 to the spectra S z x 5 i , 5 x and the h o m o t o p y H, we can represent H as the limit map of some morphism ~'L: { H B : Z B • I ---+ X B , B E ~ 2 } : (SZ • 5 I ) / ~ 2 ~ 5X/]C2 where )E2 denotes a cofinal and w-closed subset of e x p ~ A . By Proposition 1.1.27, -- ~1N)i~2 is again a cofinal and w-closed subset of expwA. It is easy to see t h a t for each B E )E, the h o m e o m o r p h i s m gB and the inclusion map iB: Z B ~ X B are connected by the h o m o t o p y H B. Fix a well-ordering A = { a s : a < T} of A such t h a t a0 = b. Let B0 E /C and a0 E B0 9 By the properties of Z and F , their projections Z Bo and FBo are Z-sets in the Hilbert cube X Bo. T h e h o m e o m o r p h i s m go -- gBo: Z Bo ~ FBo is homotopic to the inclusion map io - i Bo (via the h o m o t o p y Ho -- H Bo). Consequently, by T h e o r e m 2.3.19, go can be e x t e n d e d to an a u t o h o m e o m o r p h i s m Go: XBo --~ XBo which is homotopic (via some h o m o t o p y R0: X S o x I ~ X S o ) to the identity map of XBo. Suppose now t h a t for each/3 < a, where a < T and a > 0, we have already constructed subsets BO of the index set A, h o m e o m o r p h i s m s gf~: Z B , ---* FB~, a u t o h o m e o m o r p h i s m s G~ of X B, and homotopies RZ: X B~ x I ---. XB~ in such a way t h a t the following conditions are satisfied: (a) { a 5 : 6 < ~ } C_ Bfl. (b) B5 C B~ whenever 6 w. PROPOSITION 8.5.6. Let X = 1-I{xa: a E A ) , ]A I = 7 > w, be a product of metrizable compacta such that all X a ' s with a ?~ b (b E A ) are homogeneous. Then the following conditions are equivalent for each closed subset Z of X : (i) r X ) = T for each closed subset F of Z . (ii) Z is a Z r - s e t in X . PROOF. (i) ~ (ii). Suppose t h a t the collection {/It" t E T}, IT I < T, of open covers of X is given. Our goal is the construction of a map f" X --~ X which is {b/t" t E T}-close to i d x and which satisfies the equality f ( X ) M Z - 0.

342

8. APPLICATIONS

Since ITI < T, t h e r e exist a subset B C A of cardinality ~ -

max{w, ]TI} and

a collection {1)t" t e T} of open covers of X B such t h a t PiBl()2t) refines hot for each t E T . W i t h o u t loss of generality we may assume t h a t b E B. Note also t h a t ~, < T. In order to proceed, we need the following s t a t e m e n t .

Claim. There exist a subset D C A such that B C_ D, IDI - ~, and a section i" XB---* XD such that i(ZB) N 7 t o ( Z ) - - 0 . Proof of Claim. Let B0 -- B and let i0 d e n o t e the identity map of X Bo. Suppose t h a t for each fl < a, where a < ~+2, we have already c o n s t r u c t e d subsets Bf~ of A and sections i~ 9 XBo ~ XB~ of the projection rBo satisfying the following conditions:

(a) IBI-

XB~ --+ XBo

,~.

(b) B6 C_ B~ whenever 5 _< ft. (c) B~ = U{B6" 5 < ~} w h e n e v e r fl is a limit ordinal. (d) 7rB6B~if~ = i6 whenever 5 _< ft. (e) i~ = lim{i6" 5 < fl} w h e n e v e r fl is a limit ordinal. For each fl < a, let

v~ = {= e XBo" i~(=) r ~B~(z)). Clearly, V~ is an open subset of Xuo. It follows from the above conditions that" (f) V6 c Vf~ whenever 5 X be any section of the projection 7tO" X ~ XD. T h e n f -- jiTrB" X --+ X is {Llt" t E T}-close to i d x and f (X) N Z -- 0. (ii) ===~ (i). This is straightforward and so is omitted. [El COROLLARY 8.5.7. Let T > w, and let X be a closed subset of a compact I ~ -manifold Y . If d i m X < c~ or w ( X ) < T, then X is a Z r - s e t in Y . PROOF. Suppose t h a t X is not a Z~-set in Y. By Proposition 8.5.6, there must exist a closed subset F of X such t h a t r Y) < T. In this case, F d e p e n d s on fewer t h a n T coordinates and, consequently, contains a copy of I r . In either case this is impossible. [El

8.5.2. F i x e d p o i n t s e t s o f T y c h o n o v c u b e s . T h e main result of this Subsection ( T h e o r e m 8.5.9) gives a complete characterization of c o m p a c t a which can be fixed in Tychonov cubes as limit spaces of transfinite inverse spectra whose first elements are metrizable c o m p a c t a and all short projections of which are stable in the sense of Definition 8.5.8. Here is a simple scheme illustrating how this result might be used. Suppose t h a t X is a c o m p a c t u m of weight wl. Consider any w-spectrum S x whose limit space coincides with X . Converting S z into a transfinite spectrum, we may assume from the outset t h a t S x -- { X a , p ~ +1, wl}

344

8. APPLICATIONS

is a transfinite spectrum of length wl consisting of metrizable compacta. We now investigate the short projections of S x . Theorem 8.5.9 says that if for "many" a's, the projection p~+l is stable, then X can be fixed in the Tychonov cube of weight wl. This is in some sense the obvious part. The more important fact is t h a t if for "many" a's, the projection aa+l is not stable, then X cannot be fixed in t h a t cube. Let us emphasize this point again. The spectrum 8 x is chosen arbitrarily (there are many others representing X). Nevertheless, the behavior of the short projections (from the point of view of stability) of a given spectrum allows us to conclude the existence or non-existence of an embedding of X into I W1 whose image is the fixed point set of some self-mapping of I ~1 . For example, any (uncountable) product of metrizable c o m p a c t a can be represented as the limit space of the standard spectrum consisting of the corresponding subproducts and natural projections between them. Any closed subspace X of such a product is then the limit space of an induced spectrum ,~x 9 If, additionally, X is a retract of that product, then we can conclude that "many" projections of 8 x are open retractions (because the projections of the spectrum representing the whole product are obviously open retractions). Therefore they are stable (Proposition 8.5.13) and X can be fixed in the Tychonov cube of the corresponding weight. This proves Proposition 8.5.14. DEFINITION 8.5.8. W e s a y t h a t a m a p p: X embedding i : X ~ ryf

-- r y

Y • I~

and fix(f)

~

Y

is stable if there is an

a n d a m a p f : Y • I ~ ---+ Y • I ~ such t h a t r y i -- p,

= i(X),

where ~ry: Y • I W ---+ Y is the projection.

THEOREM 8.5.9. L e t r > w. T h e f o l l o w i n g c o n d i t i o n s are equivalent f o r a n y compactum X :

(a) (b) (c) (d)

X

can be fixed in the T y c h o n o v cube I v .

X

can be fixed in a n y c o m p a c t I v - m a n i f o l d .

X

can be fixed in s o m e c o m p a c t I v - m a n i f o l d .

X

can be r e p r e s e n t e d as the l i m i t space o f s o m e t r a n s f i n i t e i n v e r s e spec-

trum Sx

-

( X a , p a a+l, v } w h o s e s h o r t p r o j e c t i o n s p~+l.a X~+I --+ X~

are stable and w h o s e f i r s t e l e m e n t X o is m e t r i z a b l e .

PROOF. (a) ==~ (b). Any c o m p a c t / r - m a n i f o l d Y contains a copy of I v . Fix a r e t r a c t i o n r : Y ~ I v and a m a p f : I r ---, I v such that f i x ( f ) - X . Then fix(g)X , where g - f r. (b) ==~ (c). Trivial. (c) ===> (d). Suppose that Y is a compact I v -manifold and f i x ( f ) - X for some map f : Y --+ Y. Represent Y as a product Y = 1-I(Ya: a E A } of compact IW-manifolds, where all Y a ' s with a ~ b are copies of the Hilbert cube I ~ (b E A) and A is a set of cardinality r. Fix a well ordering ( a s : (~ < T} of A such that a0 = b. By Theorem 1.3.16, there is a countable subset A0 of A, containing a0, and a map f0: YAo ~ YAo such that 7rAo f -- foTrAo. Let us show that 7rAo(X ) -- f i x ( f o ) . Indeed, if x E 7rAo(Z),

8.5. FIXED POINT SETS

345

then there is a point y E X such t h a t 7rAo(Y ) -- x. Therefore

So(~) = s0(~Ao(y))= ~Ao(S(Y)) = ~Ao(Y)= and XAo C f i x ( f o ) . Conversely, if x E f i x ( f o ) , then the same equality 7rAof -fOzrAo shows t h a t f(TrAlo(X)) C 7r-l(x) Being a topological copy of I • the fiber 71-Ao - 1 (x) has the fixed point p r o p e r t y (note t h a t b r A - A0). Therefore there is a point y E 7r-l(X)Ao such t h a t f ( y ) = y. T h e n y E f i x ( f ) = X . It only remains to note t h a t x = 7rAo(Y) e 7rAo(X). Therefore 7rAo(X) = f i x ( f o ) . Suppose now t h a t for each/~ < a, where a < T, we have already c o n s t r u c t e d a subset At~ of A and a m a p f~: YAm ---* YAm in such a way t h a t the following conditions are satisfied: (i) At~ C_ A~+x and [A~+I - A~[ _< w. (ii) At~ = U { A 6 : 5 < fl} whenever ~ is a limit ordinal. (iii) {a~: 5 < fl} C_ A~ whenever ~ > 0. (iv) 7rAm.f = :~TrAa. --

(v)

Am 7r A , 5 f fl =

An

Ao

f 57rA,5 whenever 5

"

_ r > 0 so t h a t Sr = L ( p ) - l p - l ( [ - r , r] J) satisfies S 2 r N 7/~J -- {0} and p(exPG St) = exp T L ( p ) ( S r ) C_ p ( Y ) . T h e n p exPG maps Sr homeomorphically into T and t h e n a f o r t i o r i e x p a maps Sr homeomorphically into G. But V -- V A = p - l p ( V ) C U. Hence expG Sr C U, and e = exPG ISr" Sr --* U is a h o m e o m o r p h i s m onto the image. T h e n h - l e 9 Sr --~ D x C n is a h o m e o m o r p h i s m onto the image. If (d, c) = h - l ( 1 ) , then h({d} • C n) is the connected c o m p o n e n t of 1 in U, and thus h - l e ( S r ) is a h o m e o m o r p h i c copy of Sr contained in the n-cell {d} • C n. B u t Sr is h o m e o m o r p h i c to [ - r , r] J hence to [-1, 1] J. Since [-1, 1] J contains [-1, 1] m for m = 0, 1, 2 , . . . , lYl, this entails IJI _< n by the invariance of domain. def

T h u s m -- dim L(G) = IJI _< n. T h e n by " ( 6 ) = ~ (8)", there are arbitrarily small identity neighborhoods h o m e o m o r p h i c to D ~ • C m with a totally disconnected compact space D ~ and an m-cell C m. Thus, by hypothesis (8), an n-cell must be contained in an m-cell, and this implies n _< m by invariance of domain. El DEFINITION 8.6.6. Let G be a compact abelian group. Then we set d i m G = d i m Q ( Q | G) and call this cardinal the dimension of G. I f dim G is finite, then G is called finite-dimensional and otherwise infinite-dimensional. C O R O L L A R Y 8.6.7. I f for a compact abelian group G there is a natural number n such that the equivalent conditions of Theorem 8.6.5 are satisfied, then n -dim G.

Suppose t h a t DIM is a dimension function defined for compact topological spaces such t h a t 9 it assigns to a product D x C n of a c o m p a c t totally disconnected space and a compact n-cell C n the dimension n, 9 it assigns to a homogeneous space containing such a subspace with n o n e m p t y interior the dimension n, and 9 it assigns to a compact space the dimension c~ if it contains a homeomorphic copy of [-1, 1] ~. T h e n we will have DIM(G) = {dimac~

if G is finite-dimensional, if G is infinite-dimensional.

PROPOSITION 8.6.8. (Pontryagin) For a compact abelian group G one has w(G) --IGI, where w(G) denotes the weight of the space G. A

356

8. APPLICATIONS

From Theorem 8.6.5 for an infinite compact group we note that

IG/tor(G)[ = IQ|

GI = max{w, d i m G } .

We derive PROPOSITION 8.6.9. For an infinite compact abelian group G,

w ( a ) = w ( a / a o ) max{w, dim G}. In particular, a finite-dimensional connected compact abelian group is metric. More generally, a finite-dimensional compact abelian group is metrizable iff w(G/Go) w). S u p p o s e t h a t T h e o r e m has b e e n proved in the cases when w ( Z ) < ~ w h e r e w n. Let F be a closed s u b s e t of t h e p r o d u c t ]I}r x G ~ such t h a t r]i)~(F) = ]I}r a n d F is n o t a G ~ - s u b s e t of t h a t p r o d u c t . S u p p o s e t h a t F -- f i x ( h ) for s o m e m a p h" IDr x G ~ --~ ]]])r x G ~. By L e m m a 8.6.41, t h e r e is a p o i n t (x, s) e]I} ~ x G ~ - F such t h a t x ~ x', w h e r e (x', s') = h ( x , s).

374

8. APPLICATIONS

Consider t h e c o n n e c t e d set S ' = zrI[])~(h({x } x G~)). Clearly, x' e S t. On t h e o t h e r h a n d , since 7 r i ~ ( F ) -- D r we see t h a t x E S ~. Therefore, S I is a c o n n e c t e d (as a c o n t i n u o u s image of G ~) and zero-dimensional (as a subspace of IDr) set c o n t a i n i n g two distinct points. This c o n t r a d i c t i o n shows t h a t there is no m a p h w i t h F = f i x ( h ) . T h e r e f o r e the p r o d u c t IDr x G ~ does not have CIP.

[3

Historical and bibliographical notes 8.6. T h e results of Subsections 8.6.1 a n d 8.6.2 are t a k e n from [103].

8.7. G r o u p actions In this Section we investigate actions of n o n m e t r i z a b l e groups. Of course, each principal G - b u n d l e g e n e r a t e s a canonical free G - a c t i o n (and vice versa for G a Lie group). Consequently, one m i g h t ask: 9 W h a t sort of p r o p e r t i e s of a space X , in t h e presence of a free action of a "nice" group G on X , g u a r a n t e e t h a t X has t h e C I P H ? This q u e s t i o n r e m a i n s o p e n even for p a r t i c u l a r l y chosen groups. On the o t h e r hand, if G -= 71" (or G - R) the question is answered for any metrizable space [216]. T h e l a t t e r fact suggests t h a t the difficulties involved in e x t e n d i n g our result, c o n c e r n i n g principal G - b u n d l e s (see T h e o r e m 8.6.40), to free G-actions are p e r h a p s of a s e t - t h e o r e t i c a l n a t u r e . For instance, we show (Corollary 8.7.3) t h a t there is no n o n - m e t r i z a b l e (locally) c o m p a c t group G, acting freely on any locally c o m p a c t ANR-space (note t h a t the t o t a l space of any principal T r - b u n d l e is n o t an A N R ) . This answers a n o n - m e t r i z a b l e version of R . D . E d w a r d s ' s question 3 [140] asking w h e t h e r t h e r e is a c o m p a c t Hilbert cube manifold a d m i t t i n g a free action of t h e g r o u p Ap 7.

8.7.1. Spectral representations o f g r o u p a c t i o n s . By an action of a topological group G on a space X we m e a n a c o n t i n u o u s m a p A: G x X --~ X satisfying the following two conditions: 9 A ( g , A ( h , x ) ) = A(gh, x) w h e n e v e r g,h e G and x e Z . 9 A(e,x) = x for each x E X , where e denotes t h e unit element of t h e g r o u p G. We prefer to use t h e l a n g u a g e of diagrams. If #: G x G --* G is a continuous m u l t i p l i c a t i o n o p e r a t i o n on G, t h e n the validity of the first condition is equivalent to the c o m m u t a t i v i t y of t h e following square diagram:

3To the best of our knowledge, Edwards's question is still open. Obviously it is a weaker version of the well-known Hilbert-Smith conjecture which asks whether a locally compact group acting effectively on an n-manifold must be a Lie group.

8.7. GROUP ACTIONS

375

ida x A GxGxX

,.- G x X

# x idx

A

GxX

,~X

T h e second condition is j u s t the equality

~/({~} where 7rx" G

x

x x)=

~x/({~} x x)

X ~ X denotes the p r o j e c t i o n onto the second coordinate.

THEOREM 8.7.1. Let r >_ w and A" G x X ~ X

be an action. Suppose that 8 x = { X s , p ~ , A } is a r-spectrum, the limit space of which is h o m e o m o r p h i c to X , and SG = { G s , s ~ , A } is a T-spectrum consisting of topological groups and continuous h o m o m o r p h i s m s , the limit of which is isomorphic to G. I f X and G are both locally compact and Lindelbf, or if both are AE(O)-spaces (topologically), then A is induced by a cofinal and T-closed m o r p h i s m of "level actions"

{~.

a ~ x x ~ ~ X~; B}" 8 c / B

x Sx/B ~ 8x/B.

Moreover, if X and G are both locally compact and Lindel6f and A is an effective 4 action, we m a y assume that all actions As, a E B , also are effective. PROOF. T h e conditions imposed on G and X are needed to ensure t h a t not only the given s p e c t r a 8 a and S x , b u t also the s p e c t r u m

s~ x &

= { a . x x . . ~ x p~. A}

is factorizing. A p p l y i n g T h e o r e m 1.3.6 to the m a p A (which m a p s the limit space of the s p e c t r u m S a x ,~x into the limit space of the s p e c t r u m 8 x ) , we conclude t h a t t h e r e exists a cofinal and T-closed subset B of the indexing set A a n d m a p s

As" G s X X s --* X s ,

a E B

such t h a t A = lira{As}. Since A is an action, we easily see t h a t each m a p As, a E B, is an action of t h e group G s to the space X a . Indeed, let e s d e n o t e the unit e l e m e n t of G s a n d

x s E X s . Take a point x E X such t h a t p s ( x ) = x s . T h e n

a~(~., ~.)

= a~(~(~),v~(~))

= a~((~

• p ~ ) ( ~ , ~ ) ) = v ~ a ( ~ , ~) = , ~ ( ~ )

= x~.

4Recall that an action A" G x X --~ X is effective if for each 9 G G, g r e, there exists a point x E X such that g(x) :/: x.

376

8. APPLICATIONS

This proves the commutativity of the first diagram from the definition of an action. The second diagram from that definition appears as the bottom of the following cubic diagram: idc

GxGxX It X i

x h

~GxX

sa X sa X pa

GxX

,~X

sa x Pa

Pa

sa

x pa

Ga

x

/

Ga

idc~ x ha

x Xa

*-Ga x

Xa

/ , " ~ a x idxo

ha

Ga x X a

,-Xa

The commutativity of the top (h is an action) and of all the sides ({ha} is a morphism) of the above cube allow us to conclude the commutativity of the ! bottom. Indeed, let (ga, g a , x a ) e. Ga x Ga • X a . Choose a point (g,g',x) E ! G • G x X such t h a t s a ( g ) - ga, s a(g ~) = g,~ and p c ( x ) = x a. Then

Aa((idco xAa)(ga, ga, ' xa)) = A a ( ( i d c , xAa)((sa x sa • p a ) ( g , g ' , x ) ) ) =

x p.)((idc

p.(

((idc

p c ( h ( ( # x i d x ) ( g , g ' , x ) ) ) - Aa((sa x Pc)((# x i d x ) ( g , g ' , x ) ) ) h a ( ( , a X i d x . ) ( ( s a x sa x p a ) ( g , g ' , x ) ) ) -

h a ( ( , a x idx~)(ga, g'~,xa)).

Therefore, ha is an action for each a E B. This proves the first part of theorem. In order to prove the second part of theorem consider the following relation LCB2: L--

{ ( a , f ~ ) e B 2" a__f~ and for each g e G ~ - k e r ( s ~ ) there exists a point x e X~ such that hE(g, x) ~ x}.

Let us verify conditions of Proposition 1.1.29 for the above relation L. Existance. Let a E B and g E G - ker sa. Since A is an effective action, there exists a point xg e X such that h(g, xg) ~ xg. Let O[ and O~ be disjoint open neighborhoods in X of the points h(g, xg) and xg respectively. The continuity of h implies that the set h -1(O~) is open in the product G • X. Consequently there exist open subsets Ug _C G and Vg c_ 02 such that (g, xg) E U~ x Vg c_ h - l ( O ~ ) .

8.7. GROUP ACTIONS

377

Clearly A(Ug x Vg)M Vg = O. This shows t h a t A(g',x') -~ x' for each (g',x') E Ug x Vg. Observe t h a t the open cover {Ug: g E G - k e r s o } contains an open refinement {Ut: t E T} of cardinality at most T, i.e. ITI _< T (to see this note t h a t the restriction s o / ( G - k e r so): ( G - k e r so) ~ ( G o - { c o } ) i s a proper m a p and t h a t the space Go - {co} can be w r i t t e n as an union of at most T c o m p a c t subsets of the group Go). If t E T, then Ut C_ Ug for some g E G - k e r s o . By Vt we denote the set Vg constructed above. W i t h o u t loss of generality we m a y assume t h a t for each t E T there exist an index ~t E B and open sets UEt C GEt and VEt C_ XE~ such t h a t Ut = sE-tl(UEt) and Vt = p-~(VEt). T h e r e exists an index f~ E B such t h a t / 3 _ c~ and f~ >_ ~t for each t E T. Let U~ - (s~t)-l(UE~) and V~ = (p~t)-l(VEt) for each t E T. PE- l ( v t ~ ) = Vt for each t E T.

T h e n we see t h a t s~l(Ut ~) = Ut a n d

Since AE(sf~ x PE) = PE A we conclude t h a t

)~E(Ut~ • V~)M Vt~ -- 0. Finally observe t h a t G E - ker(s~) = U { u t ~ . t E T}. It only remains to note t h a t in this situation for each g E G E - ker(s~) there is a point x E Xt~ such t h a t AE(g,x) ~ x. This proves t h a t ( a , ~ ) E L. Majorantness. Let (a, fl) E L and ~, _> f~ for some 7 E B. consider an element g E G ~ - ker(s~). Clearly s~(g) E G E - ker(s~). Since (c~,f~) E L, there exists a point x E X E such t h a t )~E(s~(g),x) ~ x. Consider a point y E X.y such t h a t p ~ ( y ) - x. T h e n )~(g, y) -~ y. T h u s (c~, ~,) E L. T-closedness. Let P = {a~: -~ < T} be a chain of element of B of cardinality at most T. Suppose also t h a t (c~.y,f~) E L for each ~, < T and for some element E B. We need to show t h a t (c~,/?) E L, where a = sup P. Let g E G E - k e r ( s ~ ) . Since the s p e c t r u m SG is a Y-spectrum, the space Go is canonically h o m e o m o r phic (even isomorphic in our situation) to the limit space of the s p e c t r u m S c / P . Therefore {co} -- N{ker(saa~): "), < T}. This allows us to find an ordinal ~ < T such t h a t g E G E - k e r ( s ~ ) . Since (c~.y,f~) E L, there exists a point x E X E such t h a t ~E(g,x) ~ x Thus the relation L satisfies all three conditions of Proposition 1.1.29. Consequently, by t h a t Proposition, the set of all L-reflexive indexes, denote it by C, is cofinal and T-closed in B. It only remains to note t h a t L-reflexibility of an index a means t h a t the action Ao is effective. This completes the proof of theorem. [::] Recall t h a t for a given action A: G • X --~ X and a point x E X the closed subgroup Gx -- {g E G: g(x) - x} is called a stabilizer of x. An action ~ is said to be semi-free if the stabilizer of each point of X coincides with G or is trivial. If only the second possibility occurs for every point x E X , then A is said to be

free. COROLLARY 8.7.2. If an AE(O)-group G acts semi-freely on a non-metrizable (locally) compact A N R-space X , then (a) ~ is the trivial action or

378

8. APPLICATIONS

(b) ~(a)< ,~(x). PROOF. Let w ( X ) = T > w and suppose that w(G) > T. First consider the case w(G) > T. By Lemma 8.2.1, G is isomorphic to the limit space of a factorizing r - s p e c t r u m SG = { G , , s ~ , A } consisting of AS(O)groups of weight T and 0-soft limit homomorphisms. Applying Theorem 8.7.1 to this situation, we find an index a E A and an action

Ao~" Go~ • X --~ X such that

A-

(8.7.1)

Aa(sa • i d x ) .

Since w(G) > T -- w(Ga), we see that kers~ contains at least two elements (otherwise sa" G ---, Ga is an isomorphism). Let g 6 kersa be a non-trivial element of G. Then, by (1),

for each point x 6 X (here ea denotes the trivial element of the group G~). This shows that A is the trivial action. Next consider the case w(G) - T. The action A is trivial if (and only if) the set X c -- {x 6 X" Gx - G} coincides with X. Obviously, X c is closed in X. Therefore, we only need to show that X G is dense in X. By Lemma 8.2.1, G is isomorphic to the limit space of a factorizing w-spectrum 8G = {Ga, s ~ , A } consisting of Polish groups Ga and 0-soft limit homomorphisms sa. Similarly, by Proposition 6.3.5 and Remark 6.3.6, X is the limit space of a factorizing w-spectrum S x - {Xa, p~, A} consisting of locally compact A N R - s p a c e s (with countable bases) X a and proper and soft limit projections pc. By Theorem 8.7.1, we may assume, without loss of generality, that A is the limit of "level actions" As" Ga • Xa --, Xc~, a 6 A. Let U be a non-empty open Fa-subset of X. Since the spectrum S x is an w-spectrum, there is an index a 6 A such that U - p ~ l ( p a ( U ) ) . As above, there is a non-trivial element g 6 ker sa. Let xa 6 pc(U). We are going to show that the map Ag' X --~ X (defined by letting Ag(x) -- A ( g , x ) ) preserves the fiber p ~ l ( x a ) of the limit projection pc. Indeed, take a point x 6 p ~ i ( x a ) . Then

Therefore, Aa(p-~l(x,)) C_ p ~ l ( x , ) . The softness and properness of the projection pa guarantees that the fiber p ~ l ( x a ) is an AR-compactum and, hence, has the fixed point property. Therefore there is a point z 6 p-~l(xa) such that g(z) = z. Since A is a semi-free action, the latter means that z E X G. It only remains to note that, by the choice of the index a, z 6 U. Therefore,

unxa#O.

[3

8.7. GROUP ACTIONS

379

COROLLARY 8.7.3. There is no non-metrizable A N R - c o m p a c t u m admitting a non-trivial semi-free action of any non-metrizable compact group. PROOF. Let A: G • X --~ X be a non-trivial semi-free action of a c o m p a c t group G of weight T > w onto a n o n - m e t r i z a b l e c o m p a c t u m X of weight a > w. Since A is non-trivial, we have, by C o r o l l a r y 8.7.2, T < a. R e p r e s e n t X as t h e limit space of a factorizing T - s p e c t r u m S x -- { X s , p ~ , A } consisting of A N Rc o m p a c t spaces of weight T and soft limit projections. By T h e o r e m 8.7.1, we m a y assume t h a t A is the limit of a m o r p h i s m {As: G • X s --~ X s ; A } of "level actions". Since A is non-trivial action, t h e r e is a pair ( g , x ) E G • X such t h a t A(g, x) -~ x. Take an index a E A such t h a t

(8.7.2)

p.(~(g, ~)) # p,(~).

We claim t h a t the action Aa: G x X s ~ X s is also non-trivial and semi-free. Indeed, since p s A - A s ( i d a • p s ) , we see t h a t , by (2),

This shows t h a t the element g E G moves the point p s ( x ) and, hence, As is a non-trivial action. Next we show t h a t As is semi-free. Take any point x s E X s , and s u p p o s e t h a t for some non-trivial element g E G we have As(g, x s ) = x s . Take any point x E p-~l(Xs). T h e n we have

This shows t h a t the m a p Aa: X --~ X (defined, as above, by l e t t i n g Ag(x) -A ( g , x ) ) m a p s t h e fiber p ~ l ( x s ) over the point x s into itself. T h e softness of the limit p r o j e c t i o n p s of the s p e c t r u m S x g u a r a n t e e s t h a t this fiber is an A R c o m p a c t u m . Therefore, t h e r e is a point y E p ~ l ( x s ) such t h a t y = Ag(y) A(g,y). Since g is a non-trivial element of G, the stabilizer Gy of the point y E X is non-trivial. Since A is a semi-free action, this implies t h a t G y -- G, i.e. A(g ~, y) = y for each element g~ E G. B u t t h e n

~.(g'. ~ . ) = ~.(idG • p.)(g', y) = p.~(g', y) = p.(y) =

~..

This shows t h a t the stabilizer G x . of any point x s E X a coincides w i t h G whenever this stabilizer contains at least one non-trivial element. This m e a n s t h a t As is a semi-free action. It only remains to observe t h a t the e q u a l i t y w ( G ) = T = w ( X s ) , coupled w i t h Corollary 8.7.2 (applied to t h e non-trivial semi-free action As: G x X s -~ X s ) , leads us to a contradiction. [:]

380

8. APPLICATIONS

Remark 8.7.4. Note that the assumption of non-metrizability in Corollary 8.7.2 is essential. Indeed, the group A n acts on the Hilbert cube I ~ fixing a single point and acting freely off that point [140]. Hence Ap acts freely on the non-compact Hilbert cube manifold I ~ - {pt} ~ I ~ x [0, 1). We also remark that the product M x [0, 1) has the CIPH for each/W-manifold M [216]. 8.7.2. E x a m p l e s . PROPOSITION 8.7.5. Let f : X --, Y be a principal R-bundle between metrizable spaces. If F is a nonempty closed subset of X , then there is a fiber preserving autohomeomorphism of X whose fixed point set is F (in particular, X has the CIPH). PROOF. Let [-[ denote the usual norm metric on the real line R and let d be a metric for X such that d ( x , y ) - - I x - y [ whenever x , y 9 Z and f ( x ) = f ( y ) (see [295] or [300]). Given a nonempty closed subset F of X, set r ( x ) - 89 Consider (we maintain the notation in the proof of Theorem 8.6.40) a collection {Ui: i 9 I} of open subsets of Y, a collection

hi: U i x R--+ f - l ( U i ) ,

i 9 A,

of fiber-preserving homeomorphisms and a collection

g i,j: U i M U j___,R, i , j 9 of continuous maps such that

h i ( x , t ) = h J ( x , t +gi'J(x)) for each (x,t) e (V i M U j) x R and i , j 9 A. Here the plus sign represents the usual addition operation in the additive group R and A stands for an indexing set. Let

Hi(x)-

hi(Tr~((hi)-l(x)),Tr~((hi)-l(x)) + r ( x ) ) whenever x e f - l ( U i ) , i E A.

Here 7rl and 7r~ denote the natural projections of the product U i x R onto the first and the second coordinates respectively. Note that if x E f - l ( U i M UJ), i , j E A, then

Hi/f-l(ui

M U j) = H J / f - I ( u

i M UJ).

Therefore, the map H : X --, X defined by letting H ( x ) -- H i ( x ) for each x E f - l ( U i ) , i E A, is well defined. Obviously, f H -- f and the set of all fixed points of H coincides with F. A straightforward verification (using a slight modification of the argument presented in the proof of [216, Theorem 2.2]) shows that H is a homeomorphism. V] Another important class of spaces with the C I P H spaces.

is the class of normed linear

8.7. GROUP ACTIONS

381

PROPOSITION 8.7.6. Any normed linear space has the CIPH. PROOF. Let (X, I1" II) be a n o r m e d linear space. By d we denote the metric generated by the norm, i.e. d(x, y ) = ]ix - y i [ for each x, y E X. Let A be a closed subset of X. W i t h o u t loss of generality we may assume t h a t 0 E A. Define a m a p h: X ~ X as follows:

h(x) = ~ x + {d(x, A ) . I-~x

(

for x :~ 0, for x = 0.

0

Obviously, f i x ( h ) = A. Let us show t h a t h is a h o m e o m o r p h i s m . Let x, y E X and h(x) = h(y). T h e n 1

x + -~d(x,A).~

x

1

= y + -~d(y,A).

Ilvll

Consequently,

1

1

(1 + 21lxlld(X,A))llxll = (1 + Then

1 I1~11+ -~d(x,A)

2IlyiI)

d(y,A))llYll.

1

=

Ilyll + -~d(y,A)

and

IlY-~11- II1~11- IlYlII = I~1 (d(x, A) - d(y , A)I 9 On the other hand, the triangle inequality applied to x, y and A implies

Ily - ~11 ~ Id(x,A) d(y,A)l. Id(x,A)- d(y,A)l = 0 and x -- y. This shows -

Therefore, t h a t h is an injection. We now show t h a t for each point x E X , there is a point y E X such t h a t h(y) = x (i.e. h is a surjection). Since, by definition, h(0) = 0 we may assume t h a t x -~ 0. Since 0 E A by our assumption, we conclude t h a t Ilxi[ >_ d(x, A). Choose a real n u m b e r t such t h a t t _ I i x i I - d ( x , A ) . T h e n d(x,A) w. Let f" X Y be a first level Baire isomorphism and dim X _< n. Represent X and Y as the limit spaces of (factorizing) w-spectra 8 x = { X a , p ~ , A } and 8 y - {Ya, q~,A} respectively. By Theorem 1.3.10, we may additionally assume that dim Xa _< n for each a E A. By Theorem 8.8.1, f is the limit of the morphism

{f~" X~ ---+Ya;n}" 8 x ---+8 y consisting of first level Baire isomorphisms. By the above cited result from [259], dim Ya 2 •. D r+ and expDr+.

Then there is no Baire isomorphism between

PROOF. Consider the Cantor cube D r+ of weight T +. limit space of the T-spectrum

Clearly D r+ is the

S -- {D A, p~, exprT + } where exprT + denotes the collection of all subsets of T + of cardinality _ T and p~ is the natural projection of D B onto D A. Observe that all limit projections PA of S are homeomorphic to the natural projection of D r+ onto D r. Consider also the T-spectrum

e x p 3 -- {expD A, e x p ( p ~ ) , exprT + } whose limit space is homeomorphic to e x p D r+. Suppose now that there exists a Baire isomorphism f 9 D r+ ---+ expD r+ . By Theorem 8.8.1, f is the limit map of a Baire morphism

{ f A" DA ---* expDA; exprT+}" S ---+expS. This simply means that f APA ~- e x p ( p A ) f for each A C exprT +. Consequently, for each point F of expD A, the fiber e x p ( p A ) - l ( F ) is Baire isomorphic to the corresponding fiber p A l ( f A l ( F ) ) , which in turn is homeomorphic to D r+. Let T be a subspace of D A such that T is discrete in the relative topology and ITI -- T. Let F denote the closure of T in D A. Obviously, F is a point of expD A and there exists a pairwise disjoint collection of cardinality T of open subsets of F. One can easily check t h a t the fiber e x p ( p A ) - l ( F ) also contains a pairwise disjoint collection of cardinality T of open subsets. T h e n we can conclude that there exists a pairwise disjoint collection of cardinality T of Baire sets in p A l ( f n l ( F ) ) -- D r+. Since each Baire set is the union of functionally closed sets, we see that there exists a pairwise disjoint collection of cardinality

386

8. APPLICATIONS

r of functionally closed subsets of D r+. r > 2 ~. V1

This is impossible by the inequality

COROLLARY 8.8.5. A s s u m e wl = 2 ~. between D ~3 and e x p D ~3.

Then there is no Baire isomorphism

We conclude this Section by proving the following two statements. PROPOSITION 8.8.6. Let X be a compact space and Y a Baire subset of the Stone-C ech compactification ~ Y . I f X and Y are first level Baire isomorphic, then Y is a-compact. PROPOSITION 8.8.7. Let X be a Baire subset of its Stone-Cech compactification ~ X and Y a Suslin subset of f l y (generated by the collection of Z o ( ~ Y ) ) . I f X and Y are Baire isomorphic, then Y is a Baire subset of ~ Y . Both statements are known to be true for metrizable X and Y (see, for example, [260]). Our goal is to illustrate how Theorem 8.8.1 can be applied in order to get the general statements. The proof of Propositions 8.8.6 and 8.8.7 are similar. For this reason, below we sketch them simultaneously. Sketch of proof of Propositions 8.8.6. It can be shown, using the standard argument (see [260, Theorem 5.8.7]), that Y embeds as a closed subspace of the product I T • B of the Tychonov cube I T and a Baire subset B of some Polish space. Consider the images of Y under the natural projections of this product onto the subproducts of the form I g • B, where K is a countable subset of T. Obviously, each of these images is closed in the corresponding subproduct, and the restrictions of the indicated projections onto Y are proper. Therefore, we can conclude that Y is homeomorphic to the limit space of the inverse spectrum ,.qy - {Y~, q~, A) consisting of the indicated images and restrictions of natural projections. It is also easy to see that this spectrum is an w-spectrum. Furthermore, the closedness of Y in the normal space ! T • B guarantees that the spectrum 8 y is factorizing. W i t h o u t loss of generality we may also assume t h a t the compactum X is the limit space of a factorizing w-spectrum $ x = { X a , p ~ , A } with the same indexing set. Let f" X ---, Y denote a Baire isomorphism of the first level. Applying Theorem 8.8.1 to our situation, we see that the set A contains a cofinal and w-closed subset consisting of indexes c~ such that faPa - q ~ f , where f a : Xa --- Y~ is a Baire isomorphism of the first level. Since the spaces X a and Ya are separable and metrizable, we conclude that Ya is a-compact. Since the limit projection qa: Y ~ Ya is proper we see that Y is also a-compact. This completes the proof of Proposition 8.8.6. Sketch of proof of Proposition 8.8.7. According to [260, Theorem 5.8.9], Y can be identified with a closed subspace of the product I T • B of a Tychonov cube I T and an analytic subspace B of a Polish space. Repeating the above argument, we easily conclude t h a t Y is the limit space of a factorizing w-spectrum $Y = {Ya, q~, A} consisting of analytic subspaces of Polish spaces and proper

8.9. DOUBLE SPECTRA

387

limit projections.

Similarly, X is the limit space of a factorizing w-spectrum consisting of Baire subspaces of Polish spaces and proper limit projections. By Theorem 8.8.1, there exists at least one index a E A such that the spaces Xa and Ya are Baire isomorphic. This obviously implies that Y~ is also a Baire subset of a Polish space. Then Y is homeomorphic to a closed subspace of the product of a c o m p a c t u m and a Baire subspace of a Polish space (here we have used properness of the limit projection q~). This is enough (by Theorem 5.8.7 of [260]) to conclude that Y is a Baire subset of its own Stone-Cech compactification. This finishes the proof of Proposition 8.8.7.

S x -- { X a , p ~ , A }

Historical and bibliographical notes 8.8. Theorem 8.8.1 and all other results of this Section were proved by the author [87], [82], [95]. Corollary 8.8.5, without assuming the Continuum Hypothesis, and also a related result (stating that there is no Baire isomorphism between D ~2 and expD W2) was independently obtained by L. B. Shapiro. Propositions 8.8.6 and 8.8.7 solve the corresponding questions from [260].

8.9. D o u b l e s p e c t r a 8.9.1. D i r e c t s p e c t r a . We consider direct spectra S x = {Xa, i~, A} all projections i~" X a ~ XZ, c~ at, t E T. We may assume without loss of generality that ~ >_ a. Obviously,

f ( X a ) = f ( c l l i m S x Ba) C cllimS Y ( f ( B a ) ) C_ Cllim8 Y Y~ = Y~. This shows that (a, ~) E L. The majorantness and T-closedness conditions are trivial. Therefore, by Proposition 1.1.29, the set of all L-reflexive elements is cofinal and T-closed in A. It only remains to note that the index a E A is L-reflexive (i.e. (a, a) E L) if and only if fo~(Xa) C Ya. VI

8.9. DOUBLE SPECTRA

389

8 . 9 . 2 . E - p r o d u c t s . Let P be the Hilbert space (or any other Polish A R space) and q* E P. Let T be an uncountable indexing set with ITI = T > w. For e a c h t E T, let Pt = P a n d q ~ = q*. We denote the product y I { P t . t E T} by p T . For a point x ---- {xt" t E T} E p T let us set

supp(x) = {t E T" xt ~ q*}. The subspace E(T, q*) = {x E P T. Isupp(x)l w. The following conditions are equivalent for any space X : (i) X is homeomorphic to a closed subspace of R r embeddable into the Eproduct E('r, q*). (ii) X can be represented as the limit space of a well-ordered E - s y s t e m of length T. PROOF. D u r i n g the proof we will use the following notations: E - E ( r , q*) and E a = E a ( r , q*) for each a < T. (ii) ~ (i). Suppose t h a t X is the limit space of a well-ordered E-system Sx

=

For each ordinal a < T we are going to construct a closed e m b e d d i n g f a : X a ---* E a so t h a t the following conditions are satisfied: (i) f a p ~ - r~f/3 for each a < T, whenever a < fl < r. (ii) f t ~ - l i m { f a : a < fl} whenever fl < r is a limit ordinal. (iii) f a / X a - - f / 3 / X a whenever a < fl < T.

8.9. DOUBLE SPECTRA

395

Since E0 is the Hilbert space R ~ , there exists a closed embedding f 0 : X 0 ---* Eo. Suppose now t h a t the closed embeddings f a : X a ~ Ea, satisfying the conditions (i)-(iii) for appropriate ordinals, have already been constructed for each c~ < /~, where ~ < T. We now construct a closed embedding f~: X~ --. E~. C a s e 1 (~ -- a + 1). By (g)x, there is a Polish space g such t h a t X a + l C X a x K and raa+l = ~rl/Xa, where lrl" X a x K ~ X a denotes the first projection. Since K is a Polish space, there exists a closed embedding g: K ---. Pt,~+l. Clearly the map f a x g: X a x K ~ Ea x Pt~+~ is a closed embedding. Therefore its restriction f a + l = (fa x g ) / X a + l : X a + l ~ E a + l also is a closed embedding. By the construction, r~+lfa+l = fap~ +1. Changing f a + l if necessary (and preserving the first coordinate) we may additionaly assume, by (h)r~, t h a t

f ~ + ~ / x ~ = f~. C a s e 2 (~ is a l i m i t o r d i n a l ) . In this case, simply let f - l i m ( f a : c~ < g ) . Condition (i) implies t h a t f maps the limit space of the spectrum {:Xa, p~+l, ~ ) into the limit space of the spectrum ( E a , r~ +1, ~). Since, by our constructions, all fa's, c~ < g, are closed embeddings, we conclude that f is also a closed embedding. By (g)x, f ( X z ) c_ l i m S ~ ( E ) . Recall also t h a t E~ c_ l i m S ~ ( E ) . Further, suppose t h a t x E tJ{Xa : a < ~ ) and choose an index c~ < ~ such t h a t x E Xa. By condition (iii), one can easily conclude t h a t f ( x ) E Ea. Therefore, t2(Xa: a < ~ ) C_ U{:Ea: c~ < ~). By (c)x and (c)~, we have / ( x ~ ) = f ( d x ~ ( U { X ~ : ~ < ~))) c_ d ~ , ~ ( f ( u { x ~ : ~ < ~ ) ) =

~l~,~(u{f(x~). ~ < Z}) c_ ~ , ~ ( u ( r ~ .

~ < Z } ) = r,~.

Therefore the map fF -- f/X/3 satisfies all the required conditions. This finishes the construction of the maps fa for each c~ < T. Now consider the inverse spectra $ x -- { Z a , p~, T} and $r~ = {Ea, r~, T}. By condition (i), the limit map g - l i m { f a : t~ < T} maps l i m S x into limSr.. Also note t h a t rag = faPa for each a < r (here pa : lim S x ~ X a and ra : lim St. Ea denote the a - t h limit projections of these spectra). It can easily be seen t h a t X C_limSx and E C _ l i m S r . . It also follows from the construction t h a t g(X) c_ E. It only remains to note t h a t the map f = g / X : X ~ E is the desired closed embedding. (i) ~ (ii). Suppose now t h a t X is a closed subspace R r contained in the Eproduct E. We are going to define closed subspaces X a , a < T, and retractions r~" X/3 ---* Xa satisfying conditions (a)x - ( g ) x . Let us call a subset A of T admissible if

~A ( x ) = ~A ( x n r~A).

396

8. APPLICATIONS

Applying L e m m a 8.9.3 to the double spectrum S~, and its closed subspace X, we see that the collection of countable admissible subsets of T is w-closed and cofinal in expwT. Now we proceed as in the final part of the proof of Theorem 6.3.1. Since ITI-- T, we can write T -- {ta" c~ < T}. Since the collection of countable admissible subsets of T is cofinal in expwT, each point ta E T is contained in a countable admissible subset Ba C T. Let As -- U{Bf~"/3 _< c~}. Finally, let Xa -- X n EA,~ and pg+l _. rA:+l. It follows from the properties ( a ) ~ - ( h ) ~ , and the definition of admissible subsets that the spectrum

sx = has all the required properties. This completes the proof of the Theorem.

El

In particular we have the following. PROPOSITION 8.9.11. Let T >_ w. Then the following conditions are equivalent: (i) X is a Corson compactum of weight a we see t h a t f / ( A M Ea) = id. Since f~ _ q, one can easily observe that f E U. Thus we have the following statement. PROPOSITION 8.10.1. The collection {Ea" a < wl} has the following property: (*)(w,w~) For each compactum A C_ I W~ of weight a and an embedding f " A --~ I W~ such that f/(AMEa)--id

and f E U.

It is easy to see t h a t E is a countably compact space. Therefore E is not realcompact and, consequently, by Proposition 6.1.7, cannot be an A N R - s p a c e . Nevertheless the following statement shows t h a t E is an absolute extensor with respect to the class of metrizable compacta. PROPOSITION 8.10.2. Let B be a metrizable compactum and let a map f " A be defined on a closed subspace A of B .

Then f

can be extended to a map

f'B----~E.

PROOF. The image f ( A ) , being a metrizable compactum, is contained in Ea for some countable ordinal a. But Ea is the copy of the Hilbert cube I W . Therefore there exists an extension

].B--+E.c_E off.

!"7

The next statement shows t h a t E has some sort of strong universality property with respect to the class of metrizable compacta. PROPOSITION 8.10.3. Let A be a closed subspace of a metrizable c o m p a c t u m B and f : B --+ E be a map such that the restriction f / A : A ~ E is an embedding. Let U be a neighborhood of f in the space C w ( B , E). T h e n there exists an embedding g : B ---+ E such that g / A -- f / A

and g E U.

PROOF. Since E is pseudocompact and a dense subspace of the Tychonov cube I W1, it follows easily that I W~ serves as the Stone-(Tech compactification of E, i.e. I W~ - j3E. This in turn guarantees t h a t any neighborhood U of f in the space C w ( B , E) uniquely determines a neighborhood 0 of the same map f in the

400

8. APPLICATIONS

space C ~ ( B , I ~ I ) . Further, we may assume without loss of generality that the n e i g h b o r h o o d D has the form D = {~ ~ C~(B,•

9~

= f.~}.

As above we can find a countable ordinal f~ such t h a t f ( B ) max{f~, 7}. Since the projection

C Eft. Let a =

r g + l . E~+I --+ E~ is the trivial bundle with fiber I ~ (see the property (i)~ and Ea is a fibered Z-set in E a + l with respect to r aa+l (see the property (j)~) we immediately conclude t h a t there is a map g" B ~ Ea-F1 such t h a t g/A =//A

and r ~ + l g = r~+l f .

It only remains to observe t h a t g E U (which is g u a r a n t e e d by the inequality a >__~ and the construction of g). I-l It is not hard to see t h a t above s t a t e m e n t s 5 remain valid for E-products associated with the representation I ~ = (I ~ )~. Do these properties characterize E-products? At least as subspaces of the corresponding Tychonov cubes? There is a stronger version of this question" Is there a corresponding theory of skeletoids in Tychonov cubes? More precisely, we ask" PROBLEM 8.10.4. A r e any two skeletoids of I r equivalent as subspaces of I r , i.e. if A and B are skeletoids in I r , is there a h o m e o m o r p h i s m f " I r --~ I v (as close 6 to the identity as we wish) such that f ( A ) =

B ?

PROBLEM 8.10.5. Let K be a m e t r i z a b l e c o m p a c t u m in I v and U be a neighborhood of i d l r

in C r ( I ~" , I v ). Does there exist a h o m e o m o r p h i s m h" I v ---+I r

such that h is U - c l o s e

to i d I r

and h ( E U K ) - - E .

In particular, we ask the following.

PROBLEM 8.10.6. I n the n o t a t i o n of P r o b l e m 8.10.5, are E U K and F. h o m e omorphic ?

5Recall that everywhere in this Section r denotes an uncountable cardinal. 6Of course, closeness of f and id must be measured in the space C~.(I r , I ~ ).

8.10. SKELETOIDS IN TYCHONOV CUBES PROBLEM 8.10.7. L e t U be a n e i g h b o r h o o d o f i d l r Does there exist an embedding that f (I r)

f" I r

---, I r

401

i n t h e s p a c e C r ( I r , I r ).

w h i c h is U - c l o s e

to i d i r

and such

C_ I r - ~ ?

All of these questions remain open at this point. The next possible step in this direction is the investigation of various types of skeletoids of Tychonov cubes. For example, are there n-dimensional skeletoids in I r ? In particular, what about the easiest (perhaps) case n = 07 Of course, it is interesting to investigate the complements of the above skeletoids, i.e. versions of pseudo-interiors of Tychonov cubes. The situation is not trivial at all. For instance, let us ask the following question: is the complement I r - E homeomorphic to R r ? The answer is negative. Moreover, the complement I r - ~ is not even realcompact. Indeed, assuming that I r - ~ is realcompact, we conclude, by Proposition 1.1.24, that E contains a functionally closed subset of I r . Since every functionally closed subset of I r contains, in turn, a copy of the Tychonov cube I r , we see that a copy of the Tychonov cube must be contained in E. This implies that I r is a Corson compactum, which is an obvious contradiction (see Corollary 8.9.14). Thus I r - E ~ R r . This suggests the following problem. PROBLEM 8.10.8. F i n d a t o p o l o g i c a l c h a r a c t e r i z a t i o n

of I r -~.

Of course, the positive solution of Problem 8.10.4 would imply the topological homogeneity of the space I r - ~ . But at this point the following question remains unanswered. PROBLEM 8.10.9. I s t h e s p a c e I r - E t o p o l o g i c a l l y h o m o g e n e o u s ? A positive solution of Problem 8.10.7 would guarantee that the space I r - E contains a copy of the Tychonov cube I r and, consequently, that I r - E is universal with respect to the class of spaces of weight _< T. A more delicate problem is: PROBLEM 8.10.10. C h a r a c t e r i z e t h e c l o s e d s u b s p a c e s o f I r - ~ .

H i s t o r i c a l a n d b i b l i o g r a p h i c a l n o t e s 8.10. This concluding Section consists of questions suggested for the reader. Of course, there are several others closely related to those presented in the text. We hope that the interested reader will formulate (and solve!) them independently.

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Subject Index

absolute (neighborhood) extensor 38,228 in dimension n 38, 228 in integral cohomological dimension n 118 retract 42, 228 action free 377 semi-free 377

compactum cell-like 45 Corson 394 Dugundji 231 locally setwise homogeneous 177 Menger 127 of trivial shape 45 U V '~- 45 complete invariance property 335 with respect to homeomorphism 350

Baire function 383 isomorphism 383 a - t h level 383 of class (a, ~) 383 map 383 of class a 383 property 2 set 383

diagram bicommutative square 239 n-soft commutative 239 soft commutative 239 diameter of a map binary (n, k)-dimensional 100 n-dimensional 100

Borel function 383 map 383 of class a 383 set 383

Estimated Extension Property 55 exactness axiom 110 filtration 112

Bott periodicity 111 fine homotopy equivalence 43 boundary 171 functor fundamental 329 shape 329

chain 9 clean I ~ -manifold 171

H-cogroup 110 cofinal subspectrum 13 Hewitt realcompactification 8 cohomology theory generalized 110 reduced 110 unreduced 111

homotopy groups of ends epimorphism 132 monomorphism 132

419

420

SUBJECT INDEX isotopy displacement 63 reflection 62 limit inverse 12 projection 12 of morphism 16

near-homeomorphism 36 n-homotopy domination 133 near c~ 176 n-homotopy kernel 163 non degenerate value point 72

manifold strictly Y-stable 61

n-tameness at c~ 171

map approximately invertible 115 n-soft 45 polyhedrally n-soft 45 soft 43 bonding 12 cell-like 45 characteristic 159, 239 convex 41 functionally open 236 closed 236 many-valued 40 n-full 193 n-invertible 236 n-soft 46, 232 non-stretching 103 polyhedrally n-soft 140 semi-continuous lower 40 upper 40 soft 46, 232 stable 342 strongly .A~,n-universal 189 T-approximatively n-soft 305 soft 305 U V ~ - 45 U V n - 45 with metrizable kernel 257 with Polish kernel 249 (Z, n)-invertible 117 (Z, n)-soft 117

perfect collection 55

metric convex 86 invariant 86

product reduced 61 ]E- 389 smash 112 projector 325 projection 12 pseudo-boundary 198 of I ~ 56 pseudo-interior 198 of I w 56 polyhedral n-dimensional 190 universal n-dimensional 190 reduced cone 110 suspension 110 reflective isotopy property 62 reflexive element 10 regular averaging operator 234 R-weight 28 sequence Cauchy 1 inverse 12 Mayer-Vietoris exact 111 selection 40 set

morphism bicommutative 241 of inverse spectra 15 n-clean Menger manifold 171

absorbing 222 admissible 252 cofinal 9 deformable 179 directed 9

SUBJECT INDEX (F,/C)-absorbing 52 strong Z- 49 r-closed 9 r-complete 9 thin 55 Zn-47 Z- 48 Zr- 334 shape category 329 skeleton 54 space adjunction 61 almost 0-dimensional 178 classifying 111 complete 1 countable dimensional 384 strongly 182 finitely n-dominated near c~ 176 fK- 237 Frgchet 41 homogeneous with respect to pseudocharacter 289 Moore 112 perfectly x-normal 6 Polish 1 realcompact 4 spectrally complete 245 strongly 7~-universal 66 7~r,n-universal 274 universal NSbeling 189 spectrum direct T-cintinuous 387 double 390 factorizing 17 Haydon 259 homotopically stable 323 homotopically stable with respect to space 325 induced 13 inverse 12 Polish 247 T-continuous 19 T- 19 transfinite 19 stabilizer 377 stable homeomorphism 180 standard embedding 247

stearing function 64 Sullivan Conjecture 115 thread 12 r-near-homeomorphism 306 topology limitation 33 of d-uniform convergence 35 of uniform convergence 35 transfinite dimension 384 U-close maps 33 //-homotopic maps 39 U-map 33 uniformly locally connected collection 40 unitary group 111 z-embedded subspace 5

421