Vector Bundles VOLUME1 Foundarions and Stiejel Whitney Classes
This is a volume in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: SAMUEL EILENBERG AND HYMAN BASS A list of recent titles in this series appears at the end of this volume.
Vector Bundles VOLUME1 Foundations and Stiejel Whitney Classes
Howard Osborn Depurtment of Muthemutics tin iv ers ity of IIlino is u t tirhunu  Chumpuign tirbunu, lllino is
@
1982
ACADEMIC PRESS A
Sub\ic/iriii
u/ Huitouir Biute J o ~ u n o ~ r t Publishus h.
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Linrary o f Corgress Cataloqiy in Fualication Data Osborn, Howard. Vector bundles. (Pure and applied mathematics;
)
Incluaes oiDliograpnica1 references and index. Contents: v. 1. Foundations and Stiefelwnitney classes. I. Vector ounales. 2. Cnaracteristic classes. I. Title. 11. Series: Pure and applied mathematics (kaoemic Press) ; PA3.P8 LQA612.631 ISBN 0125293011
510s [514'.224] ( v . I)
82163Y
PRINTED IN THE UNITED STATES OF AMERICA 82 83 84 85
9 8 7 6 5 4 3 2 1
To Jean, and lo our children, Mark, Stephen, Adrienne, and Emily
This Page Intentionally Left Blank
Contents
xi
Prejuce
i
Introduction Chapter 1 Base Spaces 0. I. 2. 3. 4. 5. 6. 7. 8. 9.
Introduction The Category of Base Spaces Some Simplicial Spaces More Simplicial Spaces Weak Simplicial Spaces CW Spaces Smooth Manifolds Grassmann Manifolds Some More Coverings The MayerVietoris Technique 10. Remarks and Exercises
Chapter 11 0. 1, 2. 3.
5
5 8 II 15 19 23 34 39 41 41
Fiber Bundles
Introduction Fibre Bundles and Fiber Bundles Coordinate Bundles Bundles over Contractible Spaces 4. Pullbacks along Homotopic Maps 5. Reduction of Structure Groups
57 60 66 13 15 80
vii
...
Contents
Vlll
86 93 91
6 . Polar Decompositions 7. The LerayHirsch Theorem 8. Remarks and Exercises
Chapter III Vector Bundles Introduction Real Vector Bundles Whitney Sums and Products Riemannian Metrics Sections of Vector Bundles Smooth Vector Bundles Vector Fields and Tangent Bundles Canonical Vector Bundles The Homotopy Classification Theorem More Smooth Vector Bundles 10. Orientable Vector Bundles I I . Complex Vector Bundles 12. Realifications and Complexifications 13. Remarks and Exercises 0. I. 2. 3. 4. 5. 6. 7. 8. 9.
Chapter IV 0. I. 2. 3. 4. 5. 6. 7. 8. 9.
105 105 108 112
I I6 1 I9 126 140 147 158 160 169 172 178
2/2 Euler Classes
Introduction The 2/2 Thom Class Ute H"(E, Ec) Properties of 212 Thom Classes E / 2 Euler Classes Gysin Sequences and H*(RPa ; 212) The Splitting Principle Axioms for hi2 Euler Classes Z/2 Euler Classes of Product Line Bundles The 212 Thom Class VcE H"(P,, , Ps) Remarks and Exercises
,
191
192 194 196 199 201 205 206 209 21 1
Chapter V StiefelWhitney Classes 0. I. 2. 3. 4. 5. 6. 7.
Introduction Multiplicative 212 Classes Whitney Product Formulas Orientability The Rings H*(Cm(Wm); Z/2) Axioms for StiefelWhitney Classes Dual Classes Remarks and Exercises
215 216 219 226 227 233 237 238
ix
Contents
Chapter VI
Unoriented Manifolds
0. Introduction I . H / 2 Fundamental Classes 2. h / 2 PoincareLefschetz Duality 3. Multiplicative 212 Classes of RP" 4. Nonimmersions and Nonembeddings 5. StiefelWhitney Numbers 6. StiefelWhitney Genera 1. HI2 Thorn Forms 8. The ThornWu Theorem 9. Remarks and Exercises
249 25 1 256 258 260 265 274 216 285 281
References
303
Glossary of Notation Index
351 359
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Preface
Vector bundles provide the background for the proper formulation of many classical and modem problems of differential topology, whose solutions via characteristic classes and Ktheory are among the mathematical triumphs of the past few decades. These volumes are an introduction to vector bundles, characteristic classes, and Ktheory and to some of their applications : Volume 1 : Foundations and Stiejel Whitney Classes Volume 2 : Euler, Chern, and Pontrjagin Classes Volume.3 : K  Theory and Integrality Theorems The exposition is based on various courses the author has presented at the University of Illinois, with a onesemester course in singular homology and cohomology as the only prerequisite; no further background is necessary. Appropriate portions of differential topology, Lie groups, and homotopy theory, for example, are explicitly introduced as needed, with complete proofs in most cases; in a few exceptional cases, specific references are substituted for proofs. Each chapter ends with illustrative remarks and exercises. The remarks constitute a guide to much of the literature on vector bundles, characteristic classes, and Ktheory from 1935 to 1981. This first volume is designed both for selfstudy and as a classroom text about real vector bundles and their 2/2 characteristic classes. For classroom use, one should perhaps not dwell on the background details of Chapter I, whose results are often taken for granted in any event; only some scattered definitions and the MayerVietoris technique itself need any special emphasis. The remaining chapters can then easily be adapted for presentation xi
xii
Preface
either in a onequarter course or in a onesemester course, with very few omissions in the latter case. The author is indebted to many mathematical friends for their direct and indirect contributions to this work: to Professor S . S. Chern for an exciting and informative 1953 graduate course based on the newly published Steenrod [4]; to Professor John Milnor, who has been an inspiration since shared undergraduate days at Princeton University, and who provided Milnor [3] ; to Professor Rene Thorn, whose pithy and patient explanations turned mathematical abstractions into virtually tactile objects ; to Professor Emery Thomas, whose clearly delivered lectures (Thomas [2]) rekindled the author’s interest in vector bundles; to Professor Peter Hilton and Professor Raoul Bott, who provided much encouragement and moral support during early stages of the project; to Professor Felix Albrecht and Professor Philippe Tondeur, who read large portions of the completed typescript and provided many constructive suggestions; to Professor Wu WenTsun, who addressed himself to the details of Chapter VI with the astonishing enthusiasm one expects only of someone discovering a new world (which he had in fact helped to create); to Professor Hassler Whitney, who created vector bundles in the first place, and who warmly shared several days of companionship and chamber music; and to many other mathematical friends and colleagues who helped clarify questions as they arose. The author is equally grateful to Mrs. Doris Bartle for her assistance in proofreading the bibliography and to the staff of Academic Press for their cooperation during the production of the book. I thank my wife, Jean, and our children, Mark, Stephen, Adrienne, and Emily, for their patience, understanding, and love throughout the entire project. University ?f Illinois at UrbanaChampaign June, I982
HOWARDOSRORN
Introduction
In the first half of this volume both real and complex vector bundles are introduced and classified, and many examples are given which will play a role in later applications. The second half of the volume concentrates exclusively on certain 2/2 cohomology classes assigned to real vector bundles; there are several applications to classical problems in differential topology. (Complex vector bundles reappear in the next volume, where further cohomology classes are assigned to both real and complex vector bundles.) Let E X be a continuous map from a topological space E (the total space) onto a topological space X (the base space). Suppose that there is an open covering .( Uil i E I } of X such that each inverse image 7c'(Ui) c E is homeomorphic to the product U ix R". The homeomorphisms n'( U , )5 U ix R" need not be unique; however, they will belong to a certain family of homeomorphisms whose composition with the first projection U i x R" + U iis the restriction of 7c to 7c'(Ui).If an intersection U i n U j of two sets in the covering U iI i E I } is nonempty, then for each x E U i n U j one has a composition ( x } x R" n ' ( { x } ) (x) x R", hence a homeomorphism R" + R". If all the latter homeomorphisms result from the usual action GL(m,R) x R" R"' of the general linear group GL(m,R), then the projection E 3 X is a coordinate bundle representing a real vector bundle 5 of' rank m over the base space X . The bundle 5 is also called a real mplane bundle over X , itsjber being R". For example, any smooth mdimensional manifold X is covered by open coordinate patches U i , each of which has coordinate functions x', . . . , x" +
1
2
Introduction
used to describe a homeomorphism from U ito an open set in R"; the partial differentiations a / ? x ' , . . . , d/ax" at each point of U i form a basis of a vector space R". Let U j be another open coordinate patch on X such that the intersection U i n U j is nonempty, and let y', . . . , y"' be coordinate functions on U j . Then over each point of U in U j one has a/dyj = ( i J x ' / d y j )a/dx' . . . + (dx"/dyJ)a/dx" for j = I , , . . , m, so that the two copies of R" over each point of U i n U j are related by the usual linear action of the jacobian matrix (dxi/ayJ)E GL(m,R). This suggests that X has a welldefined real mplane bundle associated to its differentiable structure: the tangent bundle
+
T ( X )Of
x.
Now let 2/2 be the field of integers modulo 2, and let 2/2 [[t]] be the ring of formal power series over 2/2. To each element f ( t ) E 2/2 [ [ t ] ] with leading term 1 E 2/2, and to each real vector bundle over a base space X , there is a naturally defined cohomology class us( 0. One starts with a singleton space X , = (*) and attaches no cells of dimension less than n, giving X , = X I = . . . = X, ; finally one attaches a single ncell D"  S" ' via the only possible attaching map S" I + X, = ( * ) .(Even the 0sphere So can be constructed in this fashion as a CW space, with the obvious meaning of attaching a 0cell.) For any CW complex X , + X , + X , + . . . and any n 2 0 there is an obvious inclusion of each X , into the CW space limnX,; X , is the nskeleton of the CW complex, and its image is the nskeleton of the CW space. A CW complex and the corresponding CW space arejnitedimensional if the inclusion of the nskeleton is a homeomorphism for n sufficiently large; in this case the dimension is the least m 1 0 such that the inclusion of X , is a homeomorphism whenever n 2 m. 5.1 Proposition: Any weuk simplicial spuce is a CW space.
PROOF: For any abstract simplicia1 complex K and any n 2 0 let K , be the subcomplex obtained by deleting all psimplexes with p > n. The sequence K O + K + K 2 + . . . of inclusions then induces a corresponding sequence lKol lKll lK21 +. . . of inclusions of weak simplicial spaces with IKI = lim,,[K,,as a set. As a topological space lKol is discrete, and since any geometric nsimplex 111 c R"' ' is trivially homeomorphic to the ndisk D" c R", it follows that lKol + lKll + lK21 +. . . is a cell complex; closurefiniteness is obvious. The weak topology of IKI was described in terms of closed sets, a set W c (KI being closed if and only if W n 111 is closed in 111 for each 111 c 1KI. However, one can equally well replace each geometric nsimplex [iO, . . . , in[= 111 by its interior IZI', consisting of those points xodiO + . . . + x,Sin with all coefficients xo, . . . , xf, positive, which is canonically homeomorphic to the (open) ncell D"  S"'; then U c IKI is open in the weak topology of IKI if and only if U n 111 is open in 111' for each I E K . +
+
21
5. CW Spaces
There is a nontrivial partial converse to Proposition 5.1, which we shall not prove. We formulate the converse primarily to indicate the size of the category of base spaces. However, in order to keep the exposition as nearly selfcontained as possible, we shall not apply the converse as such. 5.2 Theorem (J. H. C. Whitehead [4]): alent to u weak simplicial space.
Any C W space is homotopy equiu
The proof of Whitehead’s theorem involves two steps. One first shows that a map Y ,X of CW spaces is a homotopy equivalence whenever it satisfies an apparently weaker condition; this step can be found in Gray [l, p. 1391, Lundell and Weingram [l, p. 1251, Maunder [l, pp. 2983001, or Switzer [l, pp. 8790], for example. The second step consists in assigning to any CW space X a weak simplicial space IKI and a map IKI , X satisfying the preceding condition; this step can be found in Gray [l, pp. 1451521 and Lundell and Weingram [l, pp. 1021031, for example. The conclusion that the two preceding steps prove Theorem 5.2 is expressed in Gray [l, p. 1491 and Lundell and Weingram [l, pp. 1261271.
5.3 Corollary: The category W of spaces of the homotopy types of metric simpliciul spuces lKlm is identical to the category of spaces of the homotopy types of C W spaces. PROOF: Corollary 4.7 asserts that W is identical to the category of spaces of the homotopy types of weak simplicial spaces, and Proposition 5.1 and Theorem 5.2 imply that the latter category is identical to the category of spaces of the homotopy types of CW spaces. (The category I was + initially . introduced in Milnor [8] as the category of spaces of the homotopy types of CW spaces. However, the equivalent characterization of Definition 3.8 is implicit in Milnor’s paper.)
5.4 Corollary : Every C W space is a base space. PROOF: We showed in Theorem 3.7 that every metric simplicial space is a base space, base spaces being defined in terms of homotopy type. One can obtain a shorter proof of Corollary 5.4 by using just the material needed for the first step of the proof of Theorem 5.2; a sketch is given in Remark 10.6. In the remainder of this section we describe CW structures of some other useful CW spaces. We shall later give independent proofs that these spaces are base spaces, without invoking Theorem 5.2 or its corollaries.
22
1. Base Spaces
We shall describe the CW structures of real and complex projective spaces, which we now define. For any n 2 0 let R(n+' ) * be the space R"+ { 0 ) consisting of Iw"+I without its origin, in the relative topology. There is an in R'"+')* with x y whenever x = uy for some equivalence relation u E R*. The real projective space RP" is the quotient R("+'I*/ in the quotient topology. The space RP" can equally well be regarded as a quotient of S"c R"+ ' by the same equivalence relation.
'



5.5 Proposition: RP" is un ndimensional CW space, with u cell structure consisting qj'one cell in each dimension q = 0, 1, . . . , n.
PROOF: Trivially RPo is a singleton space {*}. Let RPo
+
RP'
+
RP2
+
. . . be the sequence ofinclusions induced by the natural inclusions R' + Rz R31... ; that is, for any q = 1 , . . . , n and any x = (xo,. . . , x Y  ' ) E S"' themapRP' RP"carriesthepoint[x] =[(.yo,. . . , . x ~  ~ E) ]R P Y  l into +
+
] Let S 4  ' carry the point [ x @ 01 = [(xo,. . . , X ~  ~ , OE) RP". S4' ( cDq c Ry) into [x] E R P  ' , and let D43 R P carry .Y E Dq into [x 0d 1E RP"; t h e n 1 and g are continuous, and the inclusions S"' + D4 and RP4' + RP4 provide a commutative diagram
.Y E
sq1
I
.
RF1
I
Dq A R F .
Jm
Since is positive whenever llxll < 1, the restriction g1D4  S 4  ' is a homeomorphism onto R P q  RP4', and it follows that j' is an attaching map with adjunction space R F , as required. Projective spaces need not be finitedimensional. Let iw' + R2 + R3 + . . . be the sequence of canonical inclusions and set R" = lim,R" in the weak fopology; that is, a set W c R' is closed whenever each W n R" is closed. Observe that R" is also a direct sum of real linear spaces R', hence itself a real linear space. Specifically, points of R " are sequences ( x o , x 1 , x 2 . .. .) of real numbers, only finitely many of which are nonzero, and addition and multiplication by scalars are defined as in the finitedimensional spaces R". Let R x * = R'  (0) and observe that there is an equivalence relation in R'* with .Y y whenever x = aj' for some u E R*.The real projective space RP' is the quotient R2*/, in the quotient topology.

5.6 Corollary: RP'
cell in euch dimension.

is a CW space, with u cell structure consisting of'onr
23
6 . Smooth Manifolds
PROOF: The inclusions RPo + RP' + RP2 +. . . induced by the inclusions R ' + R2 + R3 + * * . provide a limit limnRP", in the weak topology, which is canonically homeomorphic to the definition R&*/ of RP". One can replace the real field R by the complex field C throughout the definition of RP" to obtain the complex projectioe space CP" = C=("'l)*/, which can equally well be regarded as a quotient S2"+l/ of the (2n 1)sphere SZn+I c C"+ [S"" consists of those points z = ( z o , . . . , z,) E C"' such that 11z)12= 1z01*+ . . . + Iz,,IZ = 1.1
+
'.
'
5.7 Proposition: CP" is a 2ndimensional CW space, with a cell structure consisting yfone cell in each even dimension 29 = 0, 2, , . . , 2n. PROOF: By analogy with Proposition 5.5 one uses the map 0 ' 4 3 CP4 carrying z E D Z 4(cCq)into [: 0 E C P , where 11 11 is the usual norm on C4. Since 41  llzll' is real and positive for llzll < 1, the restriction yID2Y  S 2 4  l IS a homeomorphism onto CP4  C P  ' , so that the restriction g l S Z 4  ' is an attaching map S Z 4  ' CP4' with adjunction space C P .
41
'
The definitions of the projective spaces RP" and CP" suggest the obvious definition of the complex projective space CP", and the proofs of the preceding two results suggest the obvious proof of the following result: 5.8 Corollary: CP' is a CW space, with a cell structure consisting of one cell in each even dimension. Many other expositions of the study ofCW spaces in general are indicated in Remark 10.8.
6. Smooth Manifolds In this section we briefly describe the category L4tof (smooth) manifolds and we formulate some standard basic results of differential topology. These results quickly imply that any (smooth) manifold is a base space. A closed ndimensional manijbld is a compact hausdorff space X with an I i E I } of homeomorphisms open covering f U iI i E I } and a family {Oi U i2Oi( Ui) onto open sets Oi( Ui) c R". The open covering is a coordinate covering of X by coordinate neighborhoods, and the family {Oi I i E I } of maps is an atlus for X . Suppose that the intersection U in U j c X of two coordinate neighborhoods is nonempty. Then Oiand O j restrict to homeomorphisms of U in U j onto nonempty open subsets Oi( U in U j ) c R" and Oj(U in U j )c [w". A
24
I. Base Spaces
closed ndimensional manifold X is smooth whenever all the induced ho8 meomorphisms Qi(Uin U j ) Oj(Uin U j ) are diffeomorphisms of open subsets of R"; that is, the realvalued functions on Qi(Ui n U j )c R" and Oj(U i n U j )t R" which define O j Ql: and Qi 0 QJ:I , respectively, all have partial derivatives of all orders. In this case the coordinate covering and atlas form a smooth structure on X. Let X be a smooth closed mdimensional manifold with coordinate covering { Uil i E I } and atlas {OiliE I},let Y be a smooth closed ndimensional manifold with coordinate covering { blj E J } and atlas { Y j \j E J}, and let X Y be a map. Then f induces maps Y j f O; of open subsets Qi(U i nf  '( 5))c R" into R", and f is itself smooth whenever all the induced maps Y j o f 01: are smooth; that is, all partial derivatives exist as continuous functions on Qi(U i nf '( 6)). The category of smooth closed manifolds consists of smooth closed manifolds and smooth maps. If a smooth map X Y has a smooth inverse Y % X, then f is a dgeomorphism; in this case we usually do not distinguish between X and Y . One example of a smooth closed ndimensional manifold is the real projective space RP", whose cell structure was given in Proposition 5.5. To define the smooth structure of RP", for each i = 0, . . . ,n let U i c RP" be the open subset of all points [(x,, . . . , x,)] E RP" with xi # 0. There is a homeoR" with morphism Ui 0 Q  I
0
0
0
0
a
@i[(X,,
. . . , x,)]
=x
i YXO, . . . , xi I ,xi+ 1, . . ,X"), '
is the diffeomorphism carrying points and each composition O j @:, ( y o , . . . , yi l , y i + . . . , y,) E R" with yj # 0 into 0
Y ~ ' ( Y O. ., * Yil?l,Yi+l, *
1 . 9
YjliYj+l, * .
. ¶
Y n ) E [W".
Suppose that one replaces R" in the definition of smooth closed ndimensional manifolds by the closed subset (R")+ c R" of points (xI, . . . , x,) with x, >= 0; if x, > 0, then ( x l , . . . , x,) is an interior point of (R")', and if x, = 0, then ( x l , . . . , x,) is a boundary point of (IF!")+. The result of this substitution is the category of smooth compact manifolds. If { Ui I i E I} and {Oi1 i E I } provide the smooth structure of such a manifold, and if Oi(x)is a boundary point of (R")' for some i E I and some x E Ui, then necessarily Qj(x) is a boundary point of (R")' for all j E I such that x E U j . The set of such points is the boundary of X. One easily verifies that the smooth structure of X induces a smooth structure for which X is a smooth closed ( n  1)dimensional manifold. The boundary X of a compact manifold X may be void, in which case X is a closed manifold. We occasionally use the language compact manifold with boundary to identify manifolds which are compact but not closed.
25
6 . Smooth Manifolds
An arbitrary hausdorff space X , not necessarily compact, is an ndimensional manifold if it can be covered by denumerably many compact ndimensional manifolds. Suppose that the manifolds in such a covering are smooth, and let lo,I i E I ) and {'Pj Ij E J ) be the atlases of two such smooth manifolds. Then X is a smooth ndimensional manifold whenever every nontrivial If every point restriction Y j ' . 0; is a diffeomorphism of open sets in (IT)+. of a manifold X is an interior point of one of the manifolds in its denumerable covering by compact manifolds, then X is an open manifold; for example, [w" is itself a smooth open ndimensional manifold. Smooth maps in the of' smooth manifolds are defined as before in terms of restrictions category to coordinate neighborhoods. Since a smooth manifold is covered by denumerably many compact manifolds of the same dimension, its underlying topological space X is necessarily second countable in the usual sense that there is a denumerable basis of the open sets of its topology. Conversely, any second countable hausdorff space X with a smooth structure, { U iI i E I } and {Qi 1 i E I } , is a smooth manifold. Equivalently, one can characterize smooth manifolds as hausdorff spaces with denumerable smooth structures, { U i1 i E N} and (Oi I i E N}, where N = {O, 1,2, . . . ). This implies that any smooth ndimensional manifold is of the form lim, X i for a sequence X , + X , + X , + . . . of inclusions of compact ndimensional manifolds X , , X I , X , , . . . . Given an open covering ( U , [ iE I } of a topological space X , another open covering [ V,Ij E J } of X refines { Ui I i E I ) whenever there is a function J 1: I such that 5 c U p ( j for ) every j E J . A hausdorff space X is puracompact if any open covering of X has a locally finite refinement. The conclusion of the following lemma is somewhat stronger than paracompactness. 6.1 Lemma: Any open covering { Uil i E I } of a manifold X has a countable locally jinitr rejinement [ 51j E N} such that each closure is compact and satisfies c U p ( j , .
6
PROOF: Let X = lim, X , for a sequence X , + X , + K 2 + . . . of inclusions of compact manifolds of the same dimension, and let X, denote the interior X ,  Xk0ofX k . For any open covering { Uili E I ) of X there is a refinement ( U in ( X , , ,  Xk,)l(!, k ) I ~x N ) , where X  , = X  , = 0. For each k E N the space X ,  X k  is compact, and it can therefore be covered by finitely many of the open sets U , n (kk+  x k  ,). The latter sets, for each k E N, form the desired refinement { 51j E N}.
,
,
Suppose that { h j l j E J } is a partition of unity subordinate to a locally finite cover [ j E J J of a smooth manifold X ; then {hjl j E J } is a smooth partition of' unity whenever each X 3 [0,1] is smooth. If { V j l j E J } happens
26
I . Base Spaces
to refine another open covering { Uili E 1). of X , then [hjlj E J ) is also regarded as subordinate to { U iI i E I ).
For unj1 open covering [ U ,I i E I ) of' u smooth mangold X tkrrr countuhle smooth purtition ojunity (hjlj E N} subordinate to fUili E I ) .
6.2 Lemma: is u
PROOF: Let R" s R be given by
where R" % R is the usual euclidean norm. Then j is a smooth function which is positive for llxll < 1 and zero for llxll 2 1. For any r > 0 and any point y E R" one easily adjusts f' to yield a smooth function R" k [0,1] which is nonzero precisely in the open disk DY,,of radius r about y. Let U be any open subset of R", and let I/ be any open subset with compact closure V c U . Then B can be covered by finitely many such open disks Dy., c U , and the sum of the corresponding finitely many functions&, is a nonnegative smooth function R" + R which is positive on V and vanishes outside of U . One can refine the given open covering { Ui I i E I } of X by taking intersections with coordinate neighborhoods, so that without loss of generality one can assume that f U iI i E I } is a coordinate covering; furthermore, since Lemma 6.1 provides a locally finite refinement of the latter covering, one may as well assume in addition that { Uil i E I } is itself a locally finite coordinate covering of X , with atlas ( Q i l i E I ) . A second application of Lemma 6.1 gives a countable locally finite refinement {vjlj E N) of {UiI i E I } such . there are open that each closure is compact and satisfies c U p ( j ,Thus sets @ p ( j ) ( Fand ) @ p ( j ) ( u p ( j ) ) in R" with @ p ( j ) ( v j ) = @ p ( j ) ( V j ) c @ p ( j ) ( U p ( j ) ) , SO that by the result of the preceding paragraph there are smooth nonnegative functions R" 2 R such that each gj is positive on (Dp(j,( and vanishes outside of U p ( j ) ) .Each composition U p ( j ) @'"' @ p ( j ) ( U p ( j ) ( U p ( j J R R that is positive on extends to a smooth nonnegative function X and vanishes outside of U p ( j ) Since . .(UiliE 1). and { vjlj E N} are locally finite one can compute the sum f' = CjE fj, and since .( 6 1 j E N ), covers X , it follows that f' is everywhere positive. The desired smooth partition of unity {hjlj E N) is then obtained by setting hj = j J j for eachj E N.
4
5
5)
'5
Suppose that a smooth map X & Y is a homeomorphism; then f' need not be a diffeomorphism. For example, if R R is given by f'(x) = x3, then f' is clearly a smooth homeomorphism: but since the derivative off vanishes at x = 0, the inverse is not smooth. The difficulty can be overcome by imposing the obvious condition on .f, which we do in a more general situation.
27
6. Smooth Manifolds
Let X and Y be smooth manifolds of dimensions m and n with smooth structures ( U i l i € I ) , ( @ i l i ~ I and ) ( v j l j ~ J ){Yjlj~J),respectively,and , let X 3 Y be a smooth map. By definition, the maps ' Y j a f o @ i ' carrying open subsets U in f  ' ( v j )c R" into R" are smooth in the sense that all partial derivatives exist as continuous functions on Ui n1''( Vj); in particular, one can compute the m x n jacobian matrix of Yj ;c ,f @il at each The map f is an immersion whenever each of these point of U i nf  '( jacobian matrices has rank m. Next suppose that X L Y is any injective map, not necessarily smooth. Then the image f ( X ) c Y , in the relative topology, need not be homeomorphic to X . For example, one easily constructs a (smooth) injective map R R2withj'(0) = (0,O)andf(x) = (l/x, sin(l/x))for x 2 1. In this example the inverse image of every neighborhood of (0, 0) contains arbitrarily large real numbers so that the relative topology of the image f(R) c R2 is not the usual topology of R; thus 1'is not a homeomorphism onto its image. An embedding of a smooth manifold X into a smooth manifold Y is a smooth injective immersion X I, Y that induces a homeomorphism of X onto f ( X ) c Y , in the relative topology. (A map X Y of topological spaces is proper if and only if the inverse image of every compact set in Y is compact in X . One easily verifies that a smooth injective immersion X L Y is an embedding if and only if it is proper.) We shall extend the definition of embeddings somewhat. Recall from the preceding section that R" denotes the limit lim,R" of the sequence R' + R2 + R3 +.. . of canonical inclusions, in the weak topology. For any smooth manifold X a map X R" is smooth whenever the composition X R" + R with each projection R" + R is smooth. If one regards points of R as row vectors with countably many entries, then for each coordinate neighborhood of X the jacobian matrix o f f consists of countably many columns, the number of rows being the dimension of X . A smooth map X R" is an immersion whenever the rank of each such jacobian matrix is the dimension of X . A smooth injective immersion X 3 RE' is an embedding whenever it induces a homeomorphism of X onto f ( X ) c R"', in the relative topology.
v).
6.3 Proposition: For. uny smooth munifold X there is at least one embedding
X+R'. PROOF: Lemma 6.1 provides a locally finite coordinate covering { Uil i E I ) of X such that each closure Biis compact. Let (Oi I i E I } be the corresponding atlas; if X is ndimensional, then each homeomorphism Qifrom U i to the open set Q i ( U i )c R" is given by coordinate functions U j A R, . . . ,
28
I . Base Spaces
Ui R. According to Lemma 6.2 there is a countable smooth partition of unity (hjl j E N} subordinate to { U i l i E I } ; specifically, there is a funch tion N 5 J such that X 3 R vanishes outside of some open set vj c X , and such that c where ( V , l j E N} is also a locally finite covering of X. For each j E N and each k = 1 , . . . , n the product vj R therefore extends to a map X 3 R that vanishes outside of vj, and since { vjl j E N] is locally finite, it follows that the denumerably many maps yr induce a map X + R”. One easily verifies that the latter map is an embedding.
5
hJXk,cJ’~
6.4 Corollary:

Any smooth manifold X i s metrizable.
PROOF: As in (32there is a norm R“ II II R for which each \l(x,,, x x 2 , . . . ) 1) is the (finite) sum cjsNIxjl. For any embedding X R” there is then a metric d on X with d(x,y) = Ilf(x)  f(y)II. 6.5 Proposition: For any smooth compact manifold X there is at least one natural number N 2 0 for which there exists an embedding X + RN.
PROOF: If X is compact, then the modifiers “denumerable” or “countable” become “finite” throughout the proofs of Lemmas 6.1, 6.2, and Proposition 6.3. Propositions 6.3 and 6.5 are definitely not bestpossible results. For certain values of n the following result is best possible. 6.6 Theorem (Whitney Embedding Theorem): Any smooth ndimensionul manijdd X has at least one embedding X + R2”.
This theorem appears in Whitney [7, p. 2361; its proof is long. A detailed discussion of Theorem 6.6 and related embedding theorems is given in Remark 10.11. Any ndimensional sphere S” has at least one smooth structure, constructed in the obvious way from two coordinate neighborhoods homeomorphic to R” itself. Since one can compose an embedding X + R2” with the inclusion Rz” Sz” of either coordinate neighborhood of S2”the following assertion is clearly equivalent to the Whitney embedding theorem: any smooth ndimensional manifold X has at least one embedding X + S2”. +
6.7 Theorem (CairnsWhitehead Triangulation Theorem): Any smooth ndimensional manijdd X is homeomorphic to un ndimensional metric simpliciul space K
I I.
In fact X has a “smooth triangulation” in the obvious sense. References for proofs of Theorem 6.7 are given in Remark 10.14.
29
6 . Smooth Manifolds
6.8 Corollary: Any smooth munijold is u base space; hence the category of'smooth munijblds is a f u l l subcategory ojthe category B of base spaces. c&'
PROOF: According to Corollary 2.2 any finitedimensional metric simplicia1 space IKI is a base space. A strengthened version of Corollary 6.8 is described in Remarks 10.17 and 10.18; any (second countable) topological manifold whatsoever is a base space. The proof avoids the CairnsWhitehead triangulation theorem, which is not available for topological manifolds. Recall that an embedding is a specialized immersion. Immersions are themselves of interest, however. For certain values of n the following result is bestpossible, where SZn' is the usual smooth sphere ofdimension 2n  1. 6.9 Theorem (Whitney Immersion Theorem) : Any smooth ndimensional munijbld X hus ut least one immersion X + Sz"
'.
References to two proofs of Theorem 6.9 can be found in Remark 10.13, along with a detailed discussion of related results. As in the case of the Whitney embedding theorem there is a trivially equivalent alternative version of Theorem 6.9, but with a dimensional restriction: if n > 1, then any smooth ndimensional manifold X has at least one immersion X , R2"' . The reason for the restriction is that S' itself clearly does not immerse in R'. Suppose that X is a smooth manifold of even dimension 2n, with smooth structure ( U i ( iE I ) and (OiliE I ) . The space Rz" is homeomorphic (in many ways) to C", so that the maps
a i ( U in U j )
8, 0 ; ' +
Q j ( U in U j )
can be regarded as maps from sets in C"to sets in @".The manifold X is a complex manifold of complex dimension n whenever there is a such a homeomorphism for each i E I for which all the maps aj 01: are holomorphic; that is, the n complexvalued functions on Qi(U i n U j )c @" which describe Oj QIT have local expansions about each point as Taylor series in n complex variables. We shall later show that are are many real evendimensional smooth manifolds which have no complex structure in the preceding sense; for example, the 4dimensional sphere S4 has no such structure. However, we have already encountered some topological spaces which do have such structures: the complex projective spaces CP",whose cell structures were described in Proposition 5.7. To construct a complex structure for CP" one merely substitutes C for R in the construction of a smooth structure for the real projective space RP", given at the beginning of this section. Incidentally, 0
'
'
30
I. Base Spaces
as topological spaces the projective spaces RP' and CP' are the circle S' and the 2sphere S2, respectively; in particular, S2 does have the complex structure it receives from its identification with CP'. For later convenience we record an obvious consequence of Lemma 6.1. It is in part a specialized version of the shrinking lemma of Dieudonne [11, which also appears in Dugundji [2, pp. 1521531. 6.10 Proposition: Any manifold X has a countable locally jinite covering [ U,In E N} by open sets U n c X with compact closures On, refining a given open covering of X . Furthermore, for such a covering {U,In E N], there is another countable locally finite covering { V, I n E N} by open sets V, with compact closures V, c U , for each n E N. PROOF: Lemma 6.1 itself guarantees the existence of (U,ln E N) as well as a countable open covering { W,In E N} and a map N 4 N such that W,c Up(,)for each n E N. Let p  ' ( n ) denote the set of those m E N with p(m) = n, and set V, = UrnEp+,) W,, for each n E N. Then V, c U , and {V,ln E N] is an open covering of X . Since each V, is thus a closed subset of the compact set Dn,it is itself compact, and since { U,I n E N} is locally finite, so is { V,ln E
N 1. 6.11 Corollary: Let X be a smooth manifold with countable locally jinite open coverings { U , I n E N} and { V, I n E N} as in Proposition 6.10; then there is a smooth partition of unity { h, I n E N } such that each h, is positive on V, and vanishes on X  U,.
PROOF: One simply repeats the proof of Lemma 6.2, observing that the identity map N * N now replaces the map N f+ 1 used in Lemma 6.2 to describe { vjlj E N } as a refinement of { U i l i E I ) . Proposition 6.10 will be used to prove several approximation theorems, whose proofs also depend on the following bestknown elementary approximation theorem. 6.12 Theorem (The StoneWeierstrass Theorem): Let A ( 0 ) be any real algebra of continuous realvalued functions 0 3 R on a compact hausdorff space 0, and suppose that (i) A( 0) contains all realvalued constant junctions and that (ii) if x # y in 0 then there is at least one function g E A( 0) such that g(x) # g(y) in R. Then for any continuous function 0 R whatsoever, and for any constant E > 0, there is a g E A( 0 )such that Ig(x) f ( x ) l < E for every XE
0.
PROOF: See Bartle [l, pp. 1851861 or Royden [l, p. 1741, for example.
31
6 . Smooth Manifolds
6.13 Corollary: L e t X he any smooth manijbld, and let X (R")' he any (continuous)rnup.from X to the euclidean halflspace (R"')' ; then,for the usual R + and any c > 0, there is a smooth map X euclidean norm (R"') (R")' .suc/i [hut Ilg(s)  j'(x)ll < I: ,fiw t r n j ~.Y E X .
 
J PROOF: The map ,f consists of m realvalued functions X A R, . . . , I Jrn R, X R'. In case X is a compact smooth manifold 0,the smooth realvalued functions D 4 R satisfy the conditions required of the algebra A ( 0 ) in the StoneWeierstrass theorem, so that there are smooth functions 03 R, . . . , 1!7%R such that lgj(x)  fj(x)(< E/m for every x E 0,where j = I , . . . , m  1; a minor modification of the StoneWeierstrass theorem also provides a smooth function V L R ' such that one has lg,,(s)  j",(.~)l < E/m for every x E 0,and it follows that (g,, . . . , g), is a smooth map il % (R"')' such that Ilg(x)  ,f'(x)Il< E for every x E 0. Now let X be any smooth manifold whatsoever. By Proposition 6.10 there is a locally finite covering [ Ur'InE N; of X by open sets U,, c X with compact closures Or,,so that the preceding argument provides a smooth map 0, (R"')+for each IZ E N such that Ilg"(x)  j'(x)ll == c for each x E Utn.However, according to Corollary 6.1 1 there is a smooth partition of unity {/i,,ln E N j subordinate to [U,llnE N;,and CneN /i,g" is then a welldefined map X 3 (R")' such that Ilg(x)  f(x)ll < E for every .Y E X , as desired. ./'8>1
Corollary 6.13 is a special case of a more general approximation theorem that implies that any (continuous) map X 5 Y of smooth manifolds is homotopic to a smooth map; we need only the latter implication. To start the proof one first uses Proposition 6.10 to find a countable locally finite covering [ U,"In E Nj of Y by open sets U,",each of which has a compact closure in some coordinate neighborhood of Y. Thus, if Y is rndimensional, there is an atlas {@,,InE N; of diffeomorphisms of open sets in Y onto open sets in (R"')' such that = @,,(U:) c (Rm)' for each n E N. One then applies Proposition 6.10 repeatedly to find a sequence { U: I n E Nj, UA In E Nj, [ U,Z I n E N), . . . of countable locally finite open coverings of Y such that each U;" has compact closure satisfying c U4,. The sets U ; and their closures appear throughout the following lemmas. Two maps X Y and X 5Y are homotopic relative to a subset X' c X if they are the restrictions to X x (0) and X x {l},respectively, of a map X x [0,1] 5 Y such that F(x,t ) is independent o f t E [0,1] whenever x E X'.
@,,(q)
6.14 Lemma: Let X 1; Y he a (continuous) map of smooth manifolds, and jbr each y E N let [ U ; I n E N) be the covering of' Y described above. Then there
32
I . Base Spaces
is a sequence (fo,fl,f2,. . .) of maps X kY , beginning with fo = f,such that the following conditions are satisjied for each p E N: (i) fp is homotopic to fp+ relative to X  f; '( U ; + I), (ii) the restriction of fp+ to f; ' ( U i ) u . . * u f; 1(U;+2)is smooth, (iii) i f 0 sq Sn+2p,thenf,:l(UY,f2)cf;1(UY,+1) f o r a l l n E N,and (iv) if 0 5 q 5 n + 2p, then f;'(UY,+*) c .f;:l(UY,+l) for all n E N.
PROOF: We shall show that if a finite sequence (fo, . . . ,fp) satisfies all four conditions, then there is anfp+I such that (fo, . . . ,fp,fp+ ') satisfies the same conditions. Since is compact, it meets only finitely many members of the finite covering { U,"ln E N}. Hence there is an n p E N such that locally U," n UY,is void whenever n > n p ,for any q E N. Let {Onln E N} be the atlas described earlier, and for each U4, c Y set VY, = Op(U,"n UY,)c (!Rim)+. For each n E N and q E N the sets V:+ and V:  VY, are disjoint closed subsets of the compact closure V," = = Op(O,")c ([W"')', and they are therefore separated by a positive euclidean distance E: > 0, where EY, = 1 if one or both of the subsets is void. One then defines E > 0 by setting
' w)
E =
s
min { E Y , ~ O n Ow)
np, O 5 q 5 np
+ 2p + 2).
V," c (Rm)+ be the composition f; '(0;) Up0 Let J"i'(0;) !%V:. One applies Corollary 6.13 to find a smooth map f; '(0,") (!Rim)+ such that IIgp(x) gp(x)ll < E for any x E f; '(0:).Since and V,"  V ; + are disjoint closed sets in V," the method of Lemma 6.2 yields a smoothmap Po * [0,1] with restrictions h I = 1 and h 1 V :  V ; + ' = 0. Since V;+' and V :  Vg+' are separated by euclidean distance at least E > 0, the points ( 1  th(g,(x)))g,(x)+ th(g,(x))g',(x) of (Rm)+ all lie in whenever ( x ,t ) E f; ' ( 8 : )x [0,1], and one thereby obtains a map J"; '(0;) x [0,1] 9 V,".By construction, G ( x ,t ) is independent of t E [0,1] whenever x E .f; (0:) f ; '( UF+'), so that G is a homotopy from g p to a V:, relative to f;'(0:)  J;'(u;+'1. Consequently map .f; '( 0:) there is a unique homotopy X x [0,1] 5 Y from , f p to a map X 3Y, relative to X  .f; '(U;"), such that Flf;'(U:) x [0,1] = O;' G . The map f p + satisfies condition (i) by its very construction. Since g p + is clearly smooth on the union off; '(UP,") with any open set in V : on which g p is smooth, fp+ satisfies condition (ii). Finally, the definition of E and the property llgp  gpll < E imply both
'
'
,f;:l(O,"
n V Y , +cf;'(0: ~) n UY,+')
and
j';'(0: n U Y , +cf;:,(0,0 ~) n UY,+')
33
6 . Smooth Manifolds
whenever 0 5 VI 5 lip and 0 S q n p + 2p + 2 ; since 0;n Uj is void for n > n p , the same conclusions apply whenever n > n p , for any q E N. Since F is a homotopy relative to X  f ; ' ( U ; + ' ) , a fortiori relative to X 1'; '( the inclusions U j + 2 c UY," themselves imply both
o;),
0;) n ~j+l)
j';+!'((Y 
0;) n ~ j +c ~~' ; )' ( ( Y
f';'((Y
0;) n 1/j+2)c f ; + ! ' ( (~0:)n ~ j + '

and 
for any )I and q whatsoever. Consequently both f;+'I ( U4,+2 , c f , '( U;+ I ) and f ; '( Uj'2) c f ' i 2 1 ( U j + ' ) whenever 0 5 q 5 n 2p + 2, as required by conditions (iii) and (IV).
+
6.15 Lemma: f;'(LT;+') c j ' ; ' ( U i ) f o rany p E N.
PROOF: This is a string
f;'(U;+')
cf;Qu;) c . .. c j ; ' ( u ; )c f ; ' ( U i )
of applications of condition (iii) of Lemma 6.14. 6.16 Lemma: f ; ' ( U ; p f 2 )c j ' p 1 ( U ; f 2 ) f o r a n y p EN.
PROOF: This is a string f;'(U;p+2)
c f ; ' ( U ; P + ' ) c . * . c f,"(u;+3) cf,'(u;+2)
of applications of condition (iv) of Lemma 6.14. 6.17 Lemma:
Thejbmily { U;P+21pE N) of open sets U ; p + 2couers Y.
PROOF: For any y E Y let ny E N be the largest number such that y E Ufs;ny exists because [ Uf I n E N j is locally finite. Since { U ; ' y + 1 p E N) also covers c Uz whenever q 5 2ny + 2, it follows that Y, and since )!E U i " Y + 2
6.18 Lemma: X.
u
u;n,,+2
u... u
u; u u; u . .. u u;s:+2.
u;syn,.+2c
Thefbmily [ f ; ' ( U ; + 2 ) l p E N} of o p e n s e t s f ; 1(U;+2)covers
PROOF: Sincef'; ' ( Y )= X , this is a consequence of Lemmas 6.16 and 6.17: f ; ' ( u ; u... u u ; P + 2 ) q ; ' ( U ; ) u . .. u f ; ' ( U ; P + 2 ) c .f';
6.19 Theorem: Any (continuous)map X homotopic to a smooth map X + Y .
'(u;)u .
* *
u f, '(Ui+2).
5 Y of' smooth manifolds X
and Y is
34
I . Base Spaces
PROOF: Let ( f o , f , , f 2 , . . .) be the sequence of maps X 2 Y constructed in Lemma 6.14, and regard the homotopy of condition (i) as a map X x F [ p / ( p l), ( p l ) / ( p 2)] 2 Y; specifically, F , is a homotopy relative to X  f ; ' (U ; + ' ) . Sincef; '(U;") c J'; ' ( U ; ) for any p E N, by Lemma 6.15, and since i f ; '( U j ) I p E N} is a locally finite cover of X , it follows for any x E X that there is a px E N such that F,(x, t ) is independent of the choice of t E [ p / ( p l), ( p l)/(p 2)] for p > p x ; hence F , is the restriction to X x [ p / ( p l),(p l)/(p + 2)] of a welldefined homotopy X x [0,1] 5 Y, the restriction X x {O} 5 Y being the initial J' = fb. Condition (ii) of Lemma 6.14 implies that X x { 1 ) 5 Y is smooth on any open set f ; ' ( U ; + ' ) c X , and since { f ; ' ( U P , + ' ) l p ~N} covers X , by Lemma 6.18, X x { l ! . L Y is smooth on X , as required.
+
+
+
+
+
+
+
+
Many other expositions of the study of smooth manifolds in general are indicated in Remark 10.10.
7. Grassmann Manifolds The projective spaces RP", RP", CP", and CP" of $5 can be regarded as spaces of lines through the origins of the vector spaces R"+ ', R", C"+', and C",respectively; that is, their points are the 1dimensional subspaces of the corresponding vector spaces. More generally, for any natural number m > 0 there are spaces whose points are the rndimensional subspaces of R""", R", Cm+",and C".These spaces play a crucial role in the theory of vector bundles, and the primary purpose of this section is to develop some of their properties, especially the fact that they are base spaces. 7.1 Definition: For any natural number m > 0 let V denote any of the real or complex vector spaces Rm+",R", C"'+", C";the usual topology is imposed on Rm+"and ern+", and the weak topology is imposed on R" and C".Let ( V x . . . x V ) * denote the set of mtuples of linearly independent vectors in V , in the relative topology of the mfold product V x * * x V , and let be ( y , , . . . ,y,) the equivalence relation in ( V x . . . x V ) * with (x,,. . . , x),
span the same mdimensional subspace of 3
whenever the vectors x,, . . . , x, V as the vectors y,, . . . , y,. The Grassmann manifold G"(V) is the quotient ( V x . . . x V ) * /  , in the quotient topology.
For example, the Grassmann manifolds G'(R"+ '), G'(R"), G1(C"+') and G ' ( C " ) are precisely the projective spaces RP", RP", CP", and CP". Observe that the sequence R m f l+ R m + 2 R m + 3$.. . of canonical ') Gm(Rm+')+ inclusions induces a corresponding sequence Gm(Rm+ +
+
35
7. Grassmann Manifolds
G'"(R"+3) + . . . of inclusions for which Gm(R")= limn Gm(Rm+") in the weak topology; similarly G"(C") = limn G"'(@"'+").We shall show that each of Gm(C"'+n) and Gm(@m+") is a smooth manifold, a fortiori a base space. More generally, if V is any of the vector spaces Rm+n,R", @"'+", @", then Gm(V) is a CW space, hence a base space; however, it will be more convenient to prove directly that G"(V) is a base space. We show that if V = Rm+n,then G"(V) has the structure of a smooth closed manifold of dimension rnn; the case V = @,+" is similar. Let @V be the tensor algebra generated over R by V , let I c @V be the twosided ideal generated by squares x @ x E V @ V c @V, and let A V be the exterior algebra @V/l. The tensor algebra is graded by assigning degree rn to real linear combinations of elements x , 0 .. . @ x, E @V for any rn vectors x , , . . . ,x, E V , and since I is homogeneous, there is a corresponding grading of The real vector space of elements of degree rn in is the rnfold exterior product f l V ; the image in of x, 0 . . . @ x, E @V is denoted
AV.
AmV
AV
X1 A " ' A X , .
One easily shows that m vectors x , , . . . , x , in V are linearly dependent if and only if x, A . . . A x, = 0 E A" V . A nonzero element of A" V is simple if and only if it is of the form x, A . . . A x, for m linearly independent vectors x,, . . . , x , in V . If (x,, . . . , x,+,) is a basis of V , then the (,mf") simple elements xi,A . . . A xi,, E A" V with i , < . . . < i, form a basis of A" V . If y , , . . . , y , and z , , . . . , z, span the same rndimensional subspace of V , then for some a # 0 one has y , A . . . A y, = az, A . . . A z, so that each point of G"( V ) is identified up to a nonzero factor with a simple element in A" V ; more specifically, there is an injective map G"( V )5 G 1 ( PV ) that is a homeomorphism onto the set Im F c G 1 ( A mV )of 1dimensional subspaces spanned by simple elements of f l V , in the relative topology. Hence it is of interest to develop an algebraic criterion for simplicity of elements of A" V . For m = 1 there is a canonical isomorphism V + A' V , so that every nonzero element of A' V is simple. In case rn > 1 let f l  'V 2 R be any real linear functional on the (m 1)fold exterior product V . One 0J easily verifies that there is then a unique linear map A"' V V with value 0 J (x, A . . . A x,) = ,(  lY'@(x, A . . . A .fi A . . . A x,)xi E V on every simple element x, A . . . A x, E A" V , where Zi indicates that xi is deleted.
17'
7.2 Lemma : 11'm > 1 and X E A" V is nonzero, then X is simple if and only v 5 R. (f x A (0J X ) = o E 1 v jiw eoery linear jimctional +
PROOF:If X = x , A . . ' A X , , then any 0 I X is a linear combination of . . . , x,, and since (x, A . . . AX,) A xi= 0 for i = 1, . . . , rn, it follows that
x,,
36
I. Base Spaces
X A (0_I X) = 0. Conversely, let V, c V be the subspace of all vectors of the form 0 J X.Since X is nonzero, one has dim, V, 2 m, and if X A x = 0 for every x E V,, then dim, Vx S m. Hence dim, V, = m, so that X = X , A . . ' A X , for some basis x,, . . . , x , of V,. For any simple element X E K V the identities X A (0J X) = 0 are the Plucker relations. If {xl, . . . , x,+,} is a basis of V , then any X E A" V is uniquely of the form a(i,, . . . , im)xi,A . * . A xi,
1
i l < . , . 0 let d5(V ) c O( V ) consist of those open sets W c V which are unions of at most s members of the finite family [ f i r ] , . . Then V = U , u . . . u U , E 9,(V ) . We shall show by induction on s that if W EA!s(V), then each (1 IV h"( W ) k"( W ) is an isomorphism.
u,,
,,

Since d , ( V ) is precisely the family (fir),.itself, it follows that any W E d , ( V ) is a disjoint union U 6 of open sets U 6 c V that are spaces of ( n  1)th type. The additivity properties of the given functors and of 0 itself then provide commutative diagrams
u6
low
h"(W)
k"(W)
u6h"(Ud)
]WVd
2Usk"(Ua),
44
1. Base Spaces
and since each 8", is an isomorphism by the hypothesis of the lemma, each 8, is also an isomorphism, for any W E 5,(V ) . Now suppose that each Ow. is an isomorphism for any W' E 2sl ( V ) , and suppose that W EAIs(V). Then W is a union W' LJ W" of some W' E L ~  ~ (and V ) some W " E2 1 ( V ) ( c2 s  l ( V ) ) , and by the de Morgan laws W' n W" is also a member W"' E 2, l(V). Consequently the four homomorphisms labeled either 8,. @ Ow., or Ow... in the commutative diagram 6"' I W',W"+ hm I ( w!) hm I ( w!!)jtY'.W", hm I(w!!f) h"( W 1 @
are isomorphisms by the inductive hypothesis. It follows from the 5lemma that Ow is also an isomorphism, which completes the inductive step. Since V = U , L I ~ ~ * LUI q ~ 5 & V )each , h"(V)%k"(V) is therefore an isomorphism as claimed. 9.3 Theorem (The MayerVietoris Technique): Let O ( X ) be the category of open sets on u space X of finite type, as in Definition 1.1, and let 0 be a natural transjormation of a MayerVietoris ,functor {h"lm E Z } on O ( X ) to a k"( V ) is MayerVietoris jhnctor {k" Im E Z } on O ( X ) . If each h"( U ) an isomorphism whenever U E O ( X )is contractible, then each h"(X) % k"(X) is also an isomorphism. PROOF: We shall show by induction on n that if V E O ( X ) is of nth type, then Ov is an isomorphism. If I/ E O ( X ) is of 0th type, then there is a mutually disjoint family { U6}d of contractible open sets UdE O ( X ) such that V = Ud, so that the additivity properties of the given MayerVietoris functors and 8 itself provide commutative diagrams
ud
h"(V)

h"(U,)
45
9. The MayerVietoris Technique

Since each O,, is an isomorphism by hypothesis, it follows that 8, is also an 0 L' k"( U ) is an isomorphism for isomorphism. Now suppose that h"( U ) each CJ E O ( X ) of ( n  1)th type, and let V E O ( X ) be of nth type. Since V is open in X , it follows that each h"( U ) 3 k"( U ) is an isomorphism for each U E O( V )of ( n  1)th type, so that each h"( V ) % k"( V ) is an isomorphism by Lemma 9.2. This completes the inductive step, and since X is itself of nth type for some n 2 0, this also completes the proof of the theorem. As announced at the beginning of this section there will be several later applications of the MayerVietoris technique. The applications will use slightly embellished versions of Theorem 9.3, and for convenience we present those versions as corollaries. For any space X of finite type one can regard a MayerVietoris functor (h''(y E Zj as a single functor ( ( X ) ,
[email protected] carrying each U E O ( X ) into a direct sum jig( U ) of abelian groups, indexed by the integers 7 . Let R be a Zgraded ring that is commutative in the usual sense that if u E R and L' E R are of degrees p and q, then UL' = (  1 )p40uis degree p q, and suppose that 91lf is the category of 7graded Rmodules. Then 'Jn; c d d @ and we shall consider functors h4 from C ( X ) to 9 J f .
uqEI
+
u,
9.4 Corollary: Let lO(X) be the category of'open sets on a space X ojjinite type, let R he u graded commutative ring as above, and let h4 5 k4 be a degreepreserving natural transformation of' Mayer Vietoris functors from P ( X ) to %IF. I f 9" is an Rmodule isomorphism whenever U E O ( X ) is u con9 tructible open set, then 0 is a naturul equivalence; in particular, h q ( X )A k 4 ( X )is an Rmodule isomorphism.
u,, u, u,
u4
PROOF: Immediate consequence of Theorem 9.3. The next variant of the MayerVietoris technique is closer to Theorem 9.3 itself. 9.5 Corollary: Let H be a natural transformation of MuyerVietoris functors 1 h4 q E Z j und [ kq q E Z)as in Theorem 9.3, und let n E Z be a $xed integer. I / h4(U ) L I P (U ) is un isotnorphism ivhrtirler.q t I and the open set I / E 8(X ) 0 i s c~1ntr*trctihle,then hy(X)* li4(X ) is un isomorphism whenever q 5 ti.
I
I
PROOF: One proceeds exactly as in the proof of Theorem 9.3, observing that if p 5 q S n then the 5lemma produces isomorphisms only when p < q 5 n.
In Chapters VI and VII we shall use a refinement of the MayerVietoris technique, for which we single outoparticular base spaces. Let X be a smooth compact manifold with interior X . According to Proposition 8.4 there is a
46
I . Base Spaces
finite covering [ U l , .. . , U , } of 2 by open cells such that each nonvoid intersection U i n U j is a cell U k in the $overing, and such that the closures in X satisfy Ui n U j = O in U j . Let % ( X ) be the category whose objects are unions of the sets U , , . . . , U , . (One does not need “unions of intersections of the sets U , , . . . , U,” since each nonvoid intersection is automatically a union of the sets U , , . . , , U , . ) Morphisms in %(i) are inclusions. A MayerVietoris functor on %(k) is a family { h41 q E Z}of contravariant functors from 4 ( i ) to a category !JJl of modules for which there is a long exact MayerVietoris sequence for each pair ( U , V )E %(i) x .9(i),as before. The only difference is that we now restrict U and V to 2?(k); the category O ( 2 ) of all open sets in k is not needed as such. Here are two examples. Let H*() denote singular cohomology with coefficients H o ( {*}), and let H * ( X , ) denote relative singular homology with the same coefficients, where X is the given smooth compact manifold. 9.6 Proposition: Suppose that X is a smooth compact manifold; then there is u Mayer Vietoris functor [ h4I q E Z)on %(i) with hq( U ) = H4( f o r every u E d(R).
e)
PROOF: It follows from Proposition 8.4that the closure in X of any element of %(i) is itself finitely triangulable. Hence one can construct the connecting homomorphisms d$,“ of the classical MayerVietoris cohomology sequence
. .. +Hq1(0 n V ) Hq(O H,(U” V)+ . ..
v)& H,( D)0
H4(
V)
for any ( U , V ) E %(g).Proposition 8.4also guarantees that K V = 0 n V , and since one always has U u V = 0 u V , this completes the proof. 9.1 Proposition: Suppose that X is an ndimensional smooth compact mani
I
j d d ; then thew is a MayerVietoris functor { h4 q E Z}on % with hq( U )( = i) H,,(x,x  U ) for every u E % ( X ) ,
PROOF: This time each X  U is itself finitely triangulable, so that one can construct the connecting homomorphisms of the classical relative MayerVietoris homology sequence to complete the proof. We shall construct a third MayerVietoris functor on 2(k)in Chapter VI. Here is the specialized MayerVietoris technique promised earlier. 9.8 Theorem (MayerVietoris Comparison Theorem): Let X be a smooth compact manifold with interior 2, and let 0 be u natural transfurmation of Mayer Vietoris functors { h4I q E H } and { k 4I q E Z}on 2(k).If h4(U i ) % k4(Ui) is an isomorphism for each of the open cells U . . . , U , in 2(2)when
10. Remarks and Exercises
47
ever q S 0, it ,follows that h 4 ( i )3k q ( i ) is an isomorphism whenever 4 5 0. Simi/ur/y,i j ~P ( U J !k kY(Ui) is an isomorphism for each of the open cells U , , . . . , U, it7 d ( i )jbr any q E Z,it follows that h 4 ( i )5k 4 ( X ) is an isotnorphism for un!’ q E Z.
PROOF: The 5lemma applies exactly as in the proof of Lemma 9.2.
10. Remarks and Exercises 10.1 Remark: The category %? was identified in Osborn [6, pp. 745, 7541 as the natural category of topological spaces that serve as base spaces, in the sense described in the next chapter. Indeed, the defining properties of Definition 1.2, the closure properties of Proposition 1.4, the inclusion % c 28 of Corollaries 3.9 and 5.4, and the MayerVietoris technique of Theorems 9.3 and 9.8 will be used not only in the next chapter, but throughout the entire book. The definition of %?, the inclusion W c B, and the MayerVietoris technique were suggested by a construction in Connell [ l , pp. 4995011; Connell attributes the construction to E. H. Brown. It is possible to replace the category by the category W (Definition 3.8 and Corollary 5.3) or by the category Wn (Definition 3.11 and the obvious analog of Corollary 5.3). However, many constructions appearing later in the book are most easily carried out in %? itself, rather than in w‘ or in Wn, and there seems to be little point to frequent appeals to the inclusions %oc 7Y
cJ.
On the other hand, the relative size of %? is not itself a virtue. Although several major existence and uniqueness theorems (Theorems V.5.1, X.4.1, XI.6.1, and XI.7.3) are more easily established for the category 93,one is frequently interested in corresponding results (Theorems V.5.2, X.4.2, XI.6.2, and XI.7.7) for the category 4‘ of smooth manifolds. The inclusion . @ c J of Corollary 6.8 permits one to use the existence assertions of the former results to obtain corresponding existence assertions in the latter results; however, the uniqueness assertions in the latter results are somewhat more delicate. 10.2 Remark:
In Definition 1.1 the terminology “space of finite type” was chosen for convenience; the same phrase undoubtedly appears in many other contexts with different meanings, and no confusion is intended. However, Definition 1 . 1 itself may recall another definition, and there may indeed be some faint relation between the two concepts: a topological space X is of LjusrernikSch,iirelmann category n 2 0 (according to one of the two
48
I . Base Spaces
most common conventions) if X can be covered by n + 1, but not by n, open contractible spaces. The LjusternikSchnirelmann category was introduced in Ljusternik and Schnirelmann [l] as a topological setting for variational problems; it is still used for that purpose, as in Ljusternik [l], Maurin [l], and Palais [l], for example. The homotopy invariance of the LjusternikSchnirelmann category suggests its importance in algebraic topology per se, and it has been investigated in that spirit by Fox [l, 21 and many later authors. Ganea [l] provides a 1962 survey of the subject, and there are more recent developments in Berstein [l], Borsuk [2], Coelho [l], Draper [l], Ganea [2, 31, Hardie [ 1,2,3], Hoo [11, Luft [l, 21, Moran [1,2,3,4], Ono [1,2], Osborne and Stern [l], Singhof [l, 21, and Takens [l, 21 for example; James [3] provides a 1978 survey of the subject.
10.3 Remark: The superficial similarity ofspaces of nth type (Definition 1.1) to spaces of LjusternikSchnirelmann category n has just been considered. However, the LjusternikSchnirelmann category ignores all properties of intersections of the sets in given coverings by open contractible sets, whereas such properties are of paramount interest in Definition 1.1. Intersections in open coverings are considered in Nagata [l, pp. 1331371 in the form of “multiplicative refinements.” For example, a separable metric space is of dimension S n if and only if for every open covering there is a multiplicative refinement of length 5 n + 1 ; the appropriate definitions can be found in Nagata [ 11. (There is a corresponding result in Hurewicz and Wallman [l, pp. 54,66,67], which ignores intersections.) A somewhat more specialized result concerning minimal open coverings of manifolds, with wellbehaved intersections, can be found in Osborne and Stern [l]. 10.4 Remark: There is an omission in the list B, W , Wo of categories considered in Remark 10.1. A hausdorff space X is compactly generated if a subset A c X is closed whenever all intersections A n B c X with compact subsets B c X are closed. The category X of compactly generated spaces ( = Steenrod’s convenient category) was introduced and investigated in Steenrod [6]; the definition and elementary properties can also be found in Cooke and Finney [l, pp. 861051, Gray [l, pp. 50611, and G. W. Whitehead [l, pp. 1820]. For example, any metric space is compactly generated. Some alternate “convenient categories” are introduced in Vogt [1,2] and compared to Steenrod’s convenient category. The categories of Steenrod and Vogt are indeed convenient for much of homotopy theory: Steenrod’s category suffices for more than 700 pages in G. W. Whitehead [l], for example. However, Steenrod’s definition is not itself homotopy invariant, and it would not serve especially well in the very
10. Remarks and Exercises
49
next chapter of this book, which uses frequent homotopy equivalences. One can perhaps replace 93 by the category % of spaces that are homotopy equivalent to compactly generated spaces: 93 c 92 since any metric space is compactly generated, and according to a result in May [l, p. 281, for any X E % there is a homotopy class of weak hornotopy equivalences X ’ + X relating some X’ E w‘ (cg) to X . However, 93 will suffice for the purposes of the present work. 10.5 Remark : Simplicia1 complexes and metric simplicia1 spaces, considered in $2, are also discussed in Cairns [6, pp. 6589], Hilton and Wylie [l, pp. 1452], Maunder [l, pp. 3162], Spanier [4, pp. 1071291, for example. 10.6 Remark: The telescope of Definition 3.3 was suggested to the author by a corresponding construction in Connell [l]. A similar construction is used for a related purpose in Bendersky [2, pp. 1617]. Incidentally, the inclusion w‘ c 9 of Theorem 3.7 (and Corollaries 3.9 and 5.4) was proved more rapidly in Osborn [6], but by a more sophisticated method. Specifically, although Osborn [6] uses a simpler telescope IKI* than the one constructed in Definition 3.3, there is no direct construction of a homotopy inverse to the obvious projection IKI* .+ IKI. Instead, one observes that IKI* +IKI is a weak homotopy equivalence, and according to a classical result of J. H. C. Whitehead [4] it follows that IKI* .+ IKI is a homotopy equivalence in the usual sense, as in this book. Whitehead’s theorem was cited following Theorem 5.2; it can also be found in Gray [l, p. 1391, Lundell and Weingram [I, p. 1251, Maunder [l, pp. 2983001, and Switzer [l, pp. 8790], for example.
10.7 Remark: The proof of Proposition 4.6 (Dowker [l]) can also be found in Lundell and Weingram [l, p. 1311, Milnor [8, p. 2761, and Dold [8, pp. 3543551, for example. 10.8 Remark: CW spaces were first considered in J. H. C. Whitehead [3]. Other expositions can be found in Cooke and Finney [l], Gray [l, pp. 1131211, Hilton [l, pp. 951131, Hu [3, pp. 1111491, throughout Lundell and Weingram [l], Massey [6, pp. 761041, Maunder [l, pp. 2733101, throughout Piccinini [ 11, Rohlin and Fuks [l, Chapter 111, Spanier [4, pp. 4004181, Switzer [ 1, Chapter V], and G. W. Whitehead [ 1, pp. 4695]. There are CW spaces which are not simplicia1 in any sense (other than homotopy equivalence). Specific examples of such spaces can be found in Metzler [ 11 and in Bognar [ 13, for example.
50
I. Base Spaces
10.9 Remark: According to Corollary 5.4 every CW space is a base space, a fortiori homotopy equivalent to a metric space, so that Lemma 8.2 (Stone [I]) guarantees that every CW space is homotopy equivalent to a paracompact space. A stronger statement is available: every CW space is itself paracompact. This was established by Miyazaki [11, following partial results of Bourgin [l] and Dugundji [l]. Proofs of Miyazaki’s theorem can be found in Postnikov [l] and Lundell and Weingram [l,pp. 5455]. 10.10 Remark: There are many elementary introductions to the category M of smooth manifolds, which was described very briefly in $6. See Auslander and MacKenzie [2], Boothby [l], Guillemin and Pollack [l], M. W. Hirsch [4], Hu [6], Lang [l], Milnor [15], Rohlin and Fuks [I, Chapter 1111, or Wallace [5], for example. 10.11 Remark: The Whitney embedding result of Theorem 6.6 is neither the easiest nor the bestpossible embedding theorem for smooth manifolds. The “easy” Whitney embedding theorem asserts that any smooth ndimensional manifold X admits at least one smooth embedding X + R2“+’. This result was announced in Whitney [l] and proved in Whitney [3]. Other proofs of the “easy” Whitney embedding theorem can be found in T. Y. Thomas [l, 21, in Whitney [9], Auslander and MacKenzie [2, pp. 1061 161, de Rham [l, pp. 916], Greene and Wu [l], Guillemin and Pollack [l, pp. 3956], and in the 1966 revised edition of Munkres [l], for example. The “hard” Whitney embedding result X + R2“of Theorem 6.6 appears in Whitney [7, p. 2361; its proof occupies all 27 pages of that paper. A longstanding “bestpossible” embedding conjecture is that any smooth ndimensional manifold X admits a smooth embedding X + [ W Z n  a ( f l ) + 1, where a(n) is the number of 1’s in the dyadic expansion of the dimension n. (See Gitler [l], for example.) One can definitely d o no better: in Chapter VI we shall show for each n > 0 that there is a smooth closed ; no stronger ndimensional manifold which cannot be embedded in R2na(n’ counterexamples are known. Although the “bestpossible’’ embedding conjecture remains unproved, there is at least one faint suggestion of its truth. According to a result of R. L. W. Brown [2, 31, every smooth closed ndimensional manifold is equivalent to one which smoothly embeds in [ W 2 n  a ( n ) + .* th e equivalence is “cobordism,” which will be discussed in Chapter VI. In lieu of a proof of the “bestpossible” embedding conjecture, there have been many improvements upon the “ h a r d Whitney embedding theorem. It is known for every n > 1, with the possible exception of the case n = 4, that every smooth orientable ndimensional manifold embeds in R2” 1 ., the case n = 4 is probably not an exception. It is also known for
51
10. Remarks and Exercises
every natural number n > 1 not of the form 2' that every smooth nonorientable ndimensional manifold embeds in R2" ; the cases n = 2' definitely me exceptions since the real projective spaces RP" do not embed in [W2"' for these values of n. The fact that RP" is an exception for n = 2' will be established in Proposition VI.4.9, and the remainder of the preceding results will be considered in more detail in Remark VI.9.16, with appropriate references. 10.12 Remark: There are several analogs of the Whitney embedding theorems. For example, any ndimensional separable metric space is homeomorphic to a subset of R2"+ ; proofs are given in Hurewicz and Wallman [I, pp. 60631 and Nagata [l, pp. 1011081. For any ndimensional simplicial space IKI there is a simplicial embedding ( K (+ R2"+ ; this analog of the "easy" Whitney embedding theorem is proved in Seifert and Threlfall [ l , German edition, pp. 4446; English edition, pp. 4547], and Cairns [6, pp. 7880]. In case IKI is an ndimensional triangulated manifold there is a simplicial embedding IKI + R2", due to van Kampen [I]. A necessary and sufficient condition for the existence of simplicial embeddings IKI + [W2" of ndimensional simplicial spaces IKI in general is given in Wu [lo, p. 2371, and it is shown to be satisfied by ndimensional triangulated manifolds in Wu [10, p. 2571; this provides another proof of van Kampen's analog of the " h a r d Whitney embedding theorem. Any two simplicial embeddings IKI 5 [ W Z n f 2 and IKI 3 R2n+2of an ndimensional simplicial space IK are always isotopic in the sense that there is a simplicial map IKI x [0,1] + R 2 n + 2such that each IKI % [ W 2 n + 2 is itself a simplicial embedding; if ( K (is a triangulated manifold of dimension n > 1, then any two simplicial embeddings IKI 2 R2"+ and IKI 3 R2"+' are isotopic. These results, also due to van Kampen [l], are proved in Wu [lo, pp. 207, 2581.
'
1,
10.13 Remark: The Whitney immersion result of Theorem 6.9 is neither the easiest nor the bestpossible immersion theorem for smooth manifolds. The "easy" Whitney immersion theorem asserts that any smooth ndimensional manifold X admits at least one smooth immersion X + S2". This result was announced (along with the "easy" embedding result) in Whitney [l] and proved in Whitney [3]. Another proof of the "easy" Whitney immersion theorem can be found in Auslander and MacKenzie [2, pp. 1061331. The "hard" Whitney immersion result X + SZnlof Theorem 6.9 appears in Whitney [8, p. 2701; its proof occupies all 47 pages of that paper. The
52
I . Base Spaces
same result appears in M. W. Hirsch [l, p. 2701, with an alternative proof that has had farreaching consequences. A longstanding “bestpossible” immersion conjecture has recently been proved by Cohen [13, the proof depending upon results and techniques of M. W. Hirsch [I] and Brown and Peterson [4,5]: for any n > 1 any smooth compact ndimensional manifold X admits an immersion X + p” a(“), where a(n) is the number of 1’s in the dyadic expansion of the dimension n. This result will be described in more detail in Remark VI.9.14. One can definitely do no better: in Chapter VI, for each n > 0, we shall show that there is an easily constructed smooth closed ndimensional manifold which cannot be immersed in R 2 n  a ( n ) As for embeddings, a weaker result of R. L. W. Brown [2,3] asserts that every smooth closed ndimensional manifold is equivalent to one which smoothly immerses in R2”‘(”): the equivalence is “cobordism,” which will be discussed in Chapter VI.
’.
10.14 Remark : The CairnsWhitehead triangulation result of Theorem 6.7 was first proved for closed smooth manifolds in Cairns [l, 21. The result was extended to arbitrary smooth manifolds in J. H. C. Whitehead [ 11. Discussions and simplifications of the combined result can be found in Cairns [3,4,5], and other versions of the proof can be found in Whitney [lo, pp. 1241351, in J. H. C. Whitehead [5], and in the 1966 edition of Munkres [ 11, for example.
10.15 Remark: One of the primary goals of differential topology is a reasonable classification (in some sense) of all smooth manifolds. One cannot expect a complete classification, however, even up to homotopy type, for the following reason. At the end of Chapter 7 of Seifert and Threlfall [ 11 one learns that for any prescribed finitely presented group G whatsoever there is a closed oriented 4dimensional manifold X whose fundamental group is G ; the details of the construction can be found in of Massey [4, pp. 1431441, for example. Quite independently of differential topology, Boone [1, 2, 31, Britton [I], and P. S. Novikov [l, 21 showed that the word problem for finitely presented groups is recursively unsolvable; this led Adyan [l] and Rabin [l] to conclude that the isomorphism problem for finitely presented groups is recursively unsolvable. The topological and grouptheoretic pieces of the puzzle were juxtaposed by Markov [1,2,3] to conclude that no algorithm exists for classifying oriented 4dimensional manifolds, a fortiori that no algorithm exists for classifying all smooth manifolds. There are some related results in Boone, Haken, and Poenaru [11. The lack of a universal recipe is no obstacle to useful partial results, however. The classification of compact 2dimensional manifolds is a classical
10. Remarks and Exercises
53
result, for example, which can be found in Massey [4, pp. 10181. Certain 6dimensional manifolds are classified in Jupp [l]. There is a complete classification (up to homeomorphism) of closed ( n  1)connected (2n + 1)dimensional smooth ndimensional manifolds for n > 7 due to Wall [2]; the same result is known for certain smaller values of n as a result of Wilkens [ 11 (and Wall [2]), and related classifications appear in Tamura [ 1, 2, 31. Finally, according to Cheeger and Kister [ 11 there are only countably many closed topologicul rnanifbfd.7; a fortiori, up to homeomorphism there are only countably many closed smooth manifolds. A 1975 survey of the classification problem appears in Sullivan [2], and a 1978 survey appears in T. M. Price [I]. 10.16 Remark : The topological manifolds mentioned in the previous Remark are merely locally euclidean hausdorff spaces whose topologies have a countable basis of open sets. For example, the underlying topological space of a smooth manifold is a topological manifold. Before one deals with the inverse question of finding smooth structures on a given topological manifold it is of interest to consider a broader question: can a topological manifold be triangulated? Onedimensional manifolds pose no problem, and the triangulation of surfaces was established by Rado [ 13, whose proof can be found in Ahlfors and Sario [ 1, pp. 4446]. The triangulation of 3dimensional manifolds was first established by Moise [ 11, and an alternative method was later supplied by Bing [ 11, with the same result. The 4dimensional case remains a mystery. However, for n > 4 the following results were established independently by Kirby and Siebenmann [ l ] (reproduced in Kirby and Siebenmann [2, pp. 2993061), and by Lashof and Rothenberg [2]. If X is any closed ndimensional topological manifold ( n > 4), or any open ndimensional topological manifold ( n > 5), and if H4(X;Z/2) = 0, then X can be triangulated; furthermore, ifH3(X; 2/2) = 0, then any two triangulations of X are equivalent in an obvious sense. The Z/2 cohomology conditions are known to be necessary. In fact, in every dimension n > 4 there is a closed topological manifold with no triangulation (as a manifold). An elementary discussion of the results described in this remark can be found in Schultz [3]. 10.17 Remark: The negative results of the preceding remark are offset by the following homotopy property of arbitrary topological manifolds, which remain locally euclidean hausdorff spaces whose topologies have a countable basis of open sets. Every topological manijold is homotopy equivulent to a s i m p k i d space IKI. More specifically, a simplicia1space IKI is locully$nite if and only if each point of IKI has a neighborhood which meets only finitely many geometric simplexes of IKI; equivalently, IKI is locally Jinite whenever the Dowker
54
I . Base Spaces
homotopy equivalence lKlw + lKlmof Proposition 4.6 is a homeomorphism. Y is proper if and only if the inverse image Furthermore, as in tj6 a map X of any compact set in Y is compact in X ; a proper homotopy equivalence is a homotopy equivalence in a given category of topological spaces and proper maps. According to a result in Kirby and Siebenmann [2, p. 1231 every topological manifold is proper homotopy equivalent to a locallyfinite simplicia1 space IKI.
10.18 Remark: By combining the weaker of the two preceding assertions with Theorem 3.7 one obtains the following generalization of Corollary 6.8: any topological manifold is a base space; hence the category of topological manifolds is a full subcategory of the category of base spaces. 10.19 Remark: The CairnsWhitehead triangulation theorem (Theorem 6.7) precludes smooth structures for nontriangulable manifolds; in fact, there are even triangulable manifolds with no smooth structure, such as the 10dimensional manifold of Kervaire [3]. However, a topological manifold can also have more than one smooth structure, a result first established for the 7sphere S7 in Milnor [l]. Since then, smooth structures on spheres have been thoroughly treated in Milnor [6, 71, Kervaire and Milnor [l], and Eells and Kuiper [3], for example; we shall consider some of this material in Volume 3 of the present work. Most products S P x S4 of spheres have more differentiable structures than the sphere S P + ¶ ,and more differentiable structures than the product of the corresponding numbers for the factors S p and S4. Some recent results concerning such products can be found in de Sapio [ 1,2], Kawakubo [1,2], and Schultz [ 1, 21, for example.
10.20 Remark: Here is a procedure that produces new smooth manifolds from old ones. Let X be a given smooth manifold, and let X + X be a smooth involution that is free in the sense that there are no fixed points; this provides an obvious equivalence relation in X for which the quotient X /  is a new smooth manifold with X as a double covering. For example, for any n > 0 the antipodal map S" S" of the standard nsphere S" is a free involution for which the resulting quotient S"/ is the real projective space RP" of the same dimension. In the special case n = 3 Livesay [ 11 shows that the antipodal map is essentially the only free involution of S3: any free involution S3 + S 3 is smoothly equivalent to the antipodal map. However, according to Milnor [161, Hirsch and Milnor [I], and Fintushel and Stern [ 11, there are "exotic" free involutions of the standard spheres S7,S6, S5, S4 whose quotients S'/are not diffeomorphic to the corresponding projective spaces RP7, RP', R P 5 ,

+
55
10. Remarks and Exercises
RP4. (The "exotic" involution S4 ,S4 is recent, although the quotient S4/had been constructed in Cappell and Shaneson [l] earlier by other means. Although S4/ is not diffeomorphic to RP4, it is homotopy equivalent to RP4.) More generally, Browder and Livesay [13 provide an invariant, further described in Livesay and Thomas [ 11, which leads to the following result of Orlick and Rourke [ I ] : for each k > 0 there are infinitely many distinct free (and of certain exotic spheres C4k+3).The quotients of involutions of S4k+3 S4k+3(and X 4 k + 3 ) provide a large supply of smooth closed manifolds of dimensions n = 4k + 3. There is a criterion for the existence of exotic free involutions of S" (and exotic spheres Y ) ,valid for any n >= 5, in Lopez de Medrano [I, p. 671. One can also use certain smooth involutions X + X that are not free to create smooth quotient manifolds X i  ; however, the results are not always new. For example, complex conjugation induces an involution C P 2 + C P z of the complex projective plane CP', for which one easily establishes that C P z /  is a smooth manifold; however, Kuiper [l] and Massey [5] independently established the disappointing result that C P z /  is merely the standard 4sphere S4. Here are some easily constructed smooth manifolds which will play a role in later remarks; their construction is similar to the construction of RP" from the standard nsphere S". Any odddimensional sphere S2"+ can be regarded as the subspace of those points (zo,zl,. . . , z,) E @ " + I such that Iq,(' + 1z1I2+ . . . + l : n l z = 1. Let q , , . . . ,4. be any integers which are relatively prime to a fixed integer p > 0, and let S Z n + l1:S2"+'be the diffeomorphism given by h ( z o , z , , . . . , 2") = (ezniipzo,e2niq1'p. . , e 2 n r q n / P A, n ) . Then h is periodic with period p , with no fixed points, and the quotient of S Z n + 1 by the resulting equivalence relation is a smooth closed (2n 1)dimensional manifold, the lens space L ( p ; q l , . . . , qn). The 3dimensional lens spaces L ( p ;y) were first constructed in Tietze [ 11. Later work of Reidemeister [ I ] and J . H. C. Whitehead [2] established that L(7; 1 ) and L(7; 2), for example, are homotopy equivalent but not homeomorphic. R. Myers [ I ] generalizes the result of Livesay [ 11 concerning S3 by showing that all involutions of L ( p ;y) are equivalent to those induced by the action of the orthogonal group; similar results are valid for certain involutions of period greater than 2.
+
10.21 Remark: In 1844 Grassmann [l] described the Grassmann maniR""", the embeddings Gm(R"+")4; folds G"(R"'+''), the exterior products G 1 ( KR"'"), and the Plucker relations, all of which are used in Proposition
A"
7.3. Despite the 1862 amplifications in Grassmann [2], Grassmann's work was not fully appreciated until it was reexamined and further developed by
56
I . Base Spaces
Hermann Casar Hannibal Schubert, beginning in 1886; Schubert’s work is briefly cited in Remark V.7.3. By 1897 elementary differential geometry was easily presented in Grassmann’s setting, as in BuraliForti [l]. In 1921 Corrado Segre, who had understood and applied Grassmann’s work as a student during the 1880s, wrote a muchrespected encyclopedia article identifying Grassmann manifolds as the very source of higherdimensional algebraic geometry; Segre’s enthusiasm appears in C. Segre [l, p. 7721. The Plucker relations and Proposition 7.3 can be found in Kleiman and Laksov [I], for example. Alternative proofs of Proposition 7.3 appear in Milnor and Stasheff [l, pp. 5759] and in Lanteri [l]. 10.22 Exercise: Show that any connected base space X E B is pathwise connected. Hint: First prove the property for metric spaces of finite type, then observe that pathwise connectedness is a homotopy property. Some of the materials for this exercise can be found in Hu [3, pp. 8490], for example. 10.23 Remark : The fundamental properties of MayerVietoris sequences
were discovered long before the general notion of an exact sequence existed, in Mayer [l] and Vietoris [l]. Exact sequences as such were introduced by Eilenberg and Steenrod [l], and later by Kelley and Pitcher [l]. 10.24 Remark: Bott and Tu [l, pp. 4253] prove a generalized version of
Proposition 8.4, using Riemannian metrics rather than triangulations, which they apply in their elegant presentation of the MayerVietoris technique for smooth manifolds.
CHAPTER I1
Fiber Bundles
0. Introduction Let E X be a map onto any space X, and suppose for some space F that there is a homeomorphism E 5 X x F for which
commutes, where n 1 is the first projection. Then E 5 X represents a triuiul fiber hundlr. More generally, let E 5 X be locally trivial in the following sense, for some space F : there is an open covering [ U iI i E 1 ) of X and a corresponding family ('4'; I i E I } of homeomorphisms n '( U i )% U i x F for which each f l ( U i ) A U ix F
commutes. Then the projecrion E 5 X represents a j b e r bundle with total spuce E over the base spuce X,the j b e r being F.
57
58
11. Fiber Bundles
The preceding description is incomplete. If the intersection U in U i of two sets in the open covering { UiI i E I } of the base space X is nonvoid, then the restrictions of the corresponding homeomorphisms Y i and Y j to n  ' ( U i n U j )c E induce a homeomorphism Y j c 'I',such :' that ( U i n U j )x F
Y , P; 0
' ( U i n U j )x F
U in U j commutes. Such a homeomorphism is necessarily of the form Y j Y; '(x, j ' ) = (x,${(x)(.f)),where ${ carries each x E U in U j into a homeomorphism *JW F of the fiber; in particular, $)(x) $i(x) is the identity for any x E U i n U j , so that $: and $/ carry any x E U i n U j into inverse homeomorphisms of the fiber. Since any transformations F + F whatsoever obey the associative law, it follows that each ${ can be regarded as a map of U in U j into a group G of homeomorphisms of the fiber, called the structure group of the given fiber bundle; the continuity of the compositions Y j \Y; ' imposes a specific topology on G for which the group operation G x G + G, the group inverse G 0  1 G, and the action G x F + F of G on F are are the transition functions of the continuous. The maps U in U j % G representation E X of the bundle, with respect to the covering { U i I i E I } and corresponding family { Y I i E I } . The structure group G is part of the definition of a fiber bundle, and its choice is critical. For example, if one is interested in preserving given properties of the fiber F , one does not choose G to consist of all homeomorphism F + F ; thus if F = R" and one wants to preserve vector addition in R" and multiplication by scalars, one chooses G to be the general linear group GL(rn,R), or one of its subgroups. On the other hand, if G consists only of the identity map F * F, then any projection E X representing the given bundle is necessarily of the form X x F 3 X , so that the bundle is trivial. The classical Mobius band provides the simplest nontrivial fiber bundle. Let E S' be the projection of the Mobius band E onto the unit circle S',

0

E
0
S'
59
0. Introduction
as indicated in the accompanying figure, the fiber F being the closed unit interval [  1,1]. One can cover the base space S’ by two open sets U o , U 1 homeomorphic to open real intervals, for which there are obvious homeomorphisms f l ( U i ) 5 U ix [  1,1]. The intersection U o n U , consists of two disjoint open sets Vo and V , , which one can regard as subsets of U o , for example. The Mobius band E is completely described by requiring the homeomorphism ( U , n U l ) x [1,1]
‘i’i
Yc i ’
( U , n U l ) x [1,1]
to carry ( x , f ’ )E ( U o n U , ) x [  1,1] into ( x , j ’ )or (x,  f ) according as E V, or x E V , . In this case one takes the structure group G to be 2/2 in the discrete topology, which acts on [  1,1] via multiplication by + 1 or  1. x
One traditional definition of fiber bundles is given in $1, followed in 92 by an equivalent description which is closer to the preceding sketch. Nothing in these two sections requires any properties of the base spaces X ; however, many later results do require some sort of restriction. Accordingly, fibre bundles have arbitrary base spaces X , and fiber bundles have base spaces X in the category $9 of Definition 1.1.2. Portions of the rationale for the eventual restriction to fiber bundles appear in %3,4, 5, and 7; beginning in $7 there are only fiber bundles. Contractible spaces were used in Definition 1.1.1 as building blocks for spaces of finite type, leading to the category B. In $3 it will be shown that fibre bundles over contractible spaces are trivial bundles, the building blocks for fibre bundles in general. In 94 it will be shown for any map X ’ X in the category $9 and for any fiber bundle over X that the corresponding “pullback” fiber bundle over X ’ depends only on the homotopy type off. The proof uses certain properties of the category d,which was defined in terms of homotopy types. We have already observed that the structure group G is an integral part of the description of a fiber bundle, and that it is desirable to “reduce” G to as small a subgroup K c G as possible. In $5 it is shown that if the structure group of a fiber bundle (base space in $9) is a Lie group with finitely many connected components, then one can “reduce” G to a compact subgroup K c G . In particular, in $6 it is shown that one can always “reduce” the linear groups GL(m,R), GL+(rn,R),and GL(n,C) to specific compact subgroups O ( m )c GL(rn,R), O+(rn)c GL+(m,R), and U ( n )c GL(n,C), as structure groups of fiber bundles. In 97 a basic step is provided for assigning cohomology classes in H * ( X ; A) to fiber bundles over base spaces X E 93,where A is an appropriate coefficient ring. The provisions describing the category B are essential for the result, which will be used several times in later chapters.
60
11. Fiber Bundles
1. Fibre Bundles and Fiber Bundles Some of the ingredients of a fiber bundle over a space X are a j b e r F and a structure group G of homeomorphisms F 3 F , as we have just learned. We shall assume that G acts on the lefl of F ; that is, ( g 1 g 2 ) f = g l ( g 2 f ) E F for any g 1 E G, g 2 E G , and f E F. Although the fiber F may be any topological space, the topology of G must be admissible: the group operation G x G + G, the group inverse G G, and the action G x F ,F of G on F must be continuous. The action G x F t F must also be eflectiue: the identity is the only element g E G such that gf' = f for every f E F. The space X is the base space of the bundle; we shall later require X to belong to the category B of base spaces, as in Definition 1.1.2. For any map E $ X of a space E onto a space X , and for any x E X , let E x denote the inverse image x  l ( { x } ) of {x} c X under 7c. A nonvoid set h S, of homeomorphisms E x + F is Grelated whenever for any h E S,, h' E S,, hl and g E G the compositions Ex F 3 F and F Ex 1;F belong to S, and G, respectively. Equivalently, for any fixed h E S,, the set S , consists of all compositions E x 5 F 5 F , where g E G. Given two maps E 5 X and E' 5 X ' onto spaces X and X ' , and given points x E X and x' E X ' and Grelated sets S, and S,. of homeomorphisms h' E x $ F and E',. , F , respectively, a Grelated isomorphism is any homeohW' h' morphism E x + such that every composition F E x + E',. + F belongs to G. If


El,,
E
X
f
+ E'
+ X'
commutes and x E X , then for Ex = x  ' ( { x } ) c E and E;,,, = x''( {f'(x))) c E' it is clear that f induces a map E x + E;,,,. 1.1 Definition: Given a fiber Fand a structure group G of homeomorphisms
F + F , a fumily of' jibers over a space X is a surjective map E 5 X and an assignment to each x E X of a Grelated set S, of homeomorphisms E x + F. If E 3 X and E' 5 X ' are families of fibers with the same fiber F and same structure group G , then a morphism from E 5 X to E' 5 X ' is a pair of maps
61
I . Fibre Bundles and Fiber Bundles
X
X ' and E 1,E such that E
f
X
r
t
E'
t
X'
commutes, and such that for each x E X the induced map E , + E;,,, is a Grelated isomorphism. For any family E 5 X of fibers one calls E the total space, 71 the projection, and X the base space. For the moment the base space X may be any topological space; the restriction X E
[email protected] will be imposed later. For any given fiber F and structure group G it is clear that families of fibers are the objects ofa category V(F,G) whose morphisms are commutative diagrams. Here is a way to construct new such objects and morphisms. 1.2 Definition: Let E' 5 X ' be a family of fibers in the category V(F,G), let X 1; X ' be any map, and let E c X x E' consist of those (x, e') E X x E' with j ( x ) = ~ ' ( e E' )X ' , in the relative topology. The pullback of' E' 5 X ' ulong ,f is the restriction E X to E c X x E' of the first projection X x E' 3 X .
1.3 Lemma: Let E' 5 X ' be a family of'j b e r s in the category W ( F ,G), and X 1; X ' he a map. Then the pullbuck E 5 X also belongs to W ( F ,G), and there is u canonical map f such that let
E
f
X
s
+ E
t
X'
is u morphism in (6( F ,G).
PROOF: This is a direct verification, in which E E' is defined as the restriction to E c X x E' of the second projection X x E' 3 E'.
Here are the most obvious families of fibers. 1.4 Definition: Given a fiber F and a structure group G , as before, the product fumily q/',fiber.sover a space X is the first projection X x F A X ;
62
11. Fiber Bundles
for each x E X the Grelated set S , of homeomorphisms E x , F consists of themapsE,={x} x F = F z F , f o r a l l g E G . 1.5 Lemma: Pullbacks of product families of fibers are product families oJ’
Jibers. PROOF: Let X’ x F % X‘ be a product family, and let X 3 X ’ be any map. The total space E of the pullback along g consists of those (x, (x’, f ) ) E X x ( X ’ x F ) with g(x) = n,(x’,f ) = x’, which is canonically homeomorphic to X x F, and the map E 1: X is the first projection X x F %X . The isomorphisms in the category %?(F,C) are clearly those morphisms E
f
X
s
E’
+
X‘
for which bothf and f are homeomorphisms. However, we shall be interested primarily in those isomorphisms in which X = X’ and f is the identity; such isomorphisms are of the form
for a homeomorphism g. 1.6 Lemma: Let X X‘ be any map. Then pullbacks along f of isomorphic jamilies of ,fibers over X ‘ are isomorphic families of fibers over X .
PROOF: Given a morphism
in which g’ is a homeomorphism, the corresponding diagram
63
I . Fibre Bundles and Fiber Bundles
for the pullbacks is just the restriction of
to E c X x E’, which consists of those (x,e’)E X x E’ with f ( x ) = 7c’(e’). The map E 5 is then the restriction to E of a homeomorphism id x g’, and hence itself a homeomorphism. We now pass from the category V(F,G) of families of fibers, for a given fiber F and structure group G, to the category of isomorphism classes of families of fibers; as before we consider only isomorphisms of the form E
g
+ E
Lemma 1.6 guarantees that if 4 denotes an isomorphism class of families of fibers over a space X ’ , then the pullbacks along any map X 1s X ’ also form ’ such an isomorphism class, denoted f’t.One simply calls f’!( the pullback X’ .% X” is a sequence of maps then for of’ 5 along X X’. Clearly if X any isomorphism class 5” over X ” , then one has f ! g ! ( ” = (g f)’5” as isomorphism classes over X. 0
1.7 Definition: Let X be an arbitrary topological space. The trivial bundle with fiber F and structure group G is the isomorphism class of the product family X x F 3 X of fibers.
5 over X
X’ be 1.8 Lemma: Let 5‘ be u trivial bundle over u space X ‘ , and let X un urbitury nzap; then the pullback f’!
X‘
,fibers 5 is the pullback f ‘5’ of’ 5‘ along f’.
PROOF: The total space E” of the pullback E” X of E’ 5 X‘ consists of those (x,e’) E X x E‘ with f ( x ) = lc‘(e’)E X . Since f n = n’ c f, there is consequently a map E 5 E” with g(e) = (lc(e),f(e))for each e E E . Trivially 0
commutes, where lc” is induced by the first projection X x E‘ 5X . Furthermore, for each x E X the induced map E , + E:’ carries e E E x into (x,f(e))E E:’; therefore, since f induces Grelated isomorphisms E, + E:, the maps E, + E:’ are also Grelated isomorphisms. Hence g is a homeomorphism by Proposition 1.12.
2. Coordinate Bundles The informal description of fiber bundles given in the Introduction to this chapter began with something more concrete than Definition 1.9. Although the informal description was used primarily as motivation for the structure group G, it too can be molded into a formal definition of fiber bundles, equivalent to Definition 1.9.
67
2. Coordinate Bundles
Assume that a fiber F and structure group G are given, where G has an admissible topology and acts effectively on F . Let E 5 X be a map onto a space X . Suppose that there is an open covering { Uil i E I } of X , let El U i= n  ' ( U i ) for each U , , and suppose that there is a corresponding family { Y 1 i E I } of homeomorphisms Y i such that each ElU.
2Ui x
\I
/
F
commutes. If the intersection U i n U j of two sets in the covering is nonvoid, then Y i and Y j induce a homeomorphism Y j 'J 'PI: such that
'
( U i n U j )x F
P
y1
(Uin Uj)x F
commutes, and one necessarily has Y j Y1:'(x,f) = ( x , $ { ( x ) ( f ) )for each ( x , f )E ( U i n Uj) x F , where t,bi carries each x E Ui n U j into a homeomorIL'W F. phism F 4
2.1 Definition: If each of the preceding $ps is a (continuous) map from Ui n U j to the structure group G of homeomorphisms F ,F , then E 5 X is a coordinate bundle with respect to the covering { Ui I i E I } of X . The maps E I Ui 3Ui x F required for Definition 2.1 are the local :X , and the induced maps triuia1i~ation.s of the coordinate bundle E 1 Uin Uj G upon which the definition is based are the transitionfunctions. As before, TL is itself the projection of the total space E onto the space X , and for each .Y E X one lets E , denote the ,fiber n '( {x)) ouer x . In $1 we considered arbitrary families E 5 X of fibers, fibre bundles being locally trivial equivalence classes of such families. A coordinate bundle is clearly just a locally trivial family of fibers.
2.2 Proposition: Let ( be a jibre bundle, consisting of equivalence classes of families E 5 X oj'jibers as in Definition 1.9; then every family E 1:X representing ( is a Coordinate bundle.
68
11. Fiber Bundles
PROOF: Let E 5 X represent 5. By definition there is an open covering { U iI i E I } of X such that each g I Ui is a trivial bundle, so that for E I Ui = 7c '(Ui) one has a homeomorphism Yi such that
EIUi
2U ix F /
\
Ui Y commutes and induces Grelated homeomorphisms E x b ( x ) x F . If Ui n U j is nonvoid, it follows that the homeomorphisms Yi and Y j induce Y,"Yy;' a composition ( U in U j ) x F + (Ui n U j ) x F carrying ( x , f ) into (x, $ { ( x ) ( f ' ) ) ,where t+b{(x)(f)E F depends continuously on ( x , f ) ,and where each ${(x) is a homeomorphism. Since each E x 5 {x} x F is Grelated, Y;' E x 5 {x) x F belongs to G , by definition each composition {x} x F of Grelatedness. Hence ${ is a (continuous) map Ui n U j + G, as required.

Thus if one identifies a fibre bundle over X by choosing a single representative E 5 X of an isomorphism class 5 of families of fibers, that representative is always a coordinate bundle. Proposition 2.2 also provides the desired relation between Definition 1.9 and the informal description of fibre bundles given earlier.
2.3 Corollary : Any j b r e bundle is an equivalence class of' coordinate bundles. PROOF: The equivalence relation is the isomorphism E
r
of families of fibers over the same space X , now restricted only to those families which happen to be coordinate bundles. We already know from Proposition 1.10 that pullbacks of fibre bundles are fibre bundles, so that Proposition 2.2 implies that pullbacks of coordinate bundles are coordinate bundles. Here is a more explicit version of the same result. 2.4 Proposition: Let E' 5 X ' be a coordinate bundle with respect to an open covering {Uili E I ) of' a space X ' , let E 5 X be its pullback along a map X 3 X ' , and let Ui = g  ' ( U ; )for each i E I ; then E 5 X is a coordinate
69
2. Coordinate Bundles

bundle M ' i t h respect t o the open covering [ Ui I i E I } of' X . SpeciJcally, if' Y;' Uj n U > G are the transition functions of E' 5 X', then the compositions $jj g are the transition functions Uj n U j 2G of' the pullback E 5 X . PROOF: Let
E
B
E'
X
Y
X'
be the pullback diagram. Since pullbacks of products are products, the local trivializations E'I Uj 5 Uj x F of E' 5 X' pull back to corresponding local trivializations E I U i% Ui x F of E X . For any nonvoid intersection Uj n U j c X this provides a commutative diagram ( U jn U j )x F
YJ,
YJ
E I U i n U j A ( U in U j ) x F
gxid
gxid
J.
J.
( U j n U;) x F
J
L E ' I U ;n u; A (uj n u;)x F.

Each of the compositions Y j Y ; and Y ; Y j  ' is described via a transition function, t,b! and $fj, as in Definition 2.1, and the outer rectangle of the diagram maps any (x,f)E ( Ui n U j ) x F as indicated: 0
(x,f') T
(x,$!(x)f') T
g x id
g x id
J
J
(dx),.f)t* (g(x),t,bIj(s(x))f).
The result $i(.x)f = $fj(g(x))f' for any (x,f) $:j g, as claimed.
E
(Ui n U j ) x F implies $;

=
Suppose that E 5 X is a coordinate bundle with an open covering Y U iI i E I j of X and a family {Y I i E I ) of local trivializations E I Ui Ui x F . Then if U in U j n uk is nonvoid, the restrictions of Y i , Y j , and Y k triviallysatisfy(Yk Y,: I ) ( Y j Y ; ') = Y k Y ; asselfhomeomorphisms of ( U j n U,i n U , ) x F to itself; consequently the corresponding transition functions satisfy $!(x)$;(x) = $!(.x) in G for every x E uin uj n uk.Conversely, this condition suffices for the construction of a coordinate bundle. I
70
I I . Fiber Bundles
In the following construction we assume as before that a fiber F and structure group G are given, that G acts effectively on F , and that G has an admissible topology.
2.5 Proposition: Let { U i I i E I } be an open covering of a space X , and suppose f o r every nonvoid intersection U i n U j that there i s a map U i n U j +’ L G . The maps ${ are the transition ,functions of a unique coordinute bundle with jiber F and the given action G x F + F i f and only i f for every nonvoid intersection U in U j n Uk and every x E U i n U j n Uk the identity $j”CX,$i(X)
=
4 m
is satisfied in G .
PROOF: We have just learned that the condition is necessary. To prove the converse, observe that if i = j = k, then the condition becomes $i(x)$:(x) = $i(x), so that $i maps any x E U iinto the identity 1 E G ; similarly if one merely assumes i = k, one has $i(x)$:(x) = $$x) = 1, so that $f(x) = (${(x))’ for every x E U i n U j . Let be the relation in the disjoint union u i ( U i x F ) for which (xi,h) ( x j , f j ) whenever x i = x j E U in U j and fj = ${(xi)J E F . The hypothesis of the proposition guarantees that is transitive, and the preceding consequences of the hypothesis guarantee that is reflexive and symmetric. Hence is an equivalence relation, and one sets E = u i ( U i x F ) /  in the quotient topology. The first projections Ui x F A U i induce a map E 5 X , and one easily verifies that E X is the desired coordinate bundle, uniqueness being trivial.

 

In the next chapter we shall use Proposition 2.5 to construct new fibre bundles out of old ones, sometimes changing both the fiber and the structure group.
2.6 Definition: Let G x F +F and G’ x F‘ + F‘ be effective actions of transformation groups G and G’ on topological spaces F and F’, respectively, the topologies of G and G’ being admissible. A morphism of’ transformution groups is a pair (r,@) of maps G G’ and F % F’ such that is a group homomorphism and the diagram G x F  F
G’x F‘ commutes.

F
71
2. Coordinate Bundles
In the following result a morphism of transformation groups is applied to a fibre bundle 5 over a space X to construct a new fibre bundle t' over the same space X . One chooses any coordinate bundle E 5 X representing t, for a family [ Y I i E I j of local trivializations with respect to an open covering [ U ,I i E I ) of X , and one constructs a new coordinate bundle E' 5 X , for a family [ Y i l i I~) of local trivializations with respect to the same open covering. One then verifies that the fibre bundle 5' represented by E' 5 X depends only on 5 itself. r
cg
G', F + F ' ) he u morphism of transjormation groups, und let X he a topological space. Then t o any jibre bundle t over X with structure group G and,fiber F the morphism (r,@) assigns a uniquefibre bundle 5' over X with structure group G' and fiber F , satisfying the following condition: jbr unj' coordinate descriptions qj' 5 and 5' (as above), there is a f projectionpreserving map E +E' whose restriction fl U i to each E I U ic E provides u coniniutative diugram
2.7 Proposition: Let (G
+
Elui
E'IUi
:
+
Ui x F
I Y:
id x 8
U i x F'.
PROOF: Let U i n U.i G be the transition functions corresponding to j Y i ( iI),andlet ~ Ui n Uj+G'bethecompositionsUi *;' n Uj%G5G', for nonvoid intersections U i n U j . Since r is a group homomorphism, the conditions $;(x)$!(x) = $f(x) imply that $F(x)$ij(x)= $ik(x) whenever U i n U j n U k is nonvoid, for any x E U i n U j n uk. By Proposition 2.5, the maps $; are the transition functions of a unique coordinate bundle E' 5 X with fiber F' and group action G' x F' + F'. The total spaces E and E' are quotients of disjoint unions u i ( U i x F ) and u i ( U i x F),and the Y U i x F and E'I U i y; U i x F' arise from the local trivializations E I U i 4 projections of u i ( U i x F ) and u i ( U i x F ' ) onto each U i x F and U i x F', respectively. The relation ( x i , j J ( x j , j j ) in u i ( U i x F ) means that xi = s j E U i n U j and j , = ${(.q),A. E F , and since


GxFF
G' x F'

F'
72
11. Fiber Bundles
commutes, it follows that both x i = xj E U i n U j and qfi = @(t,b{(xi)j;) = (rt,b!(x))(qjJ= t,blj(x)(@h);that is, the relation ( x i ,@jJ (xj,@jJ is satisfied in u i ( U i x F'). Consequently F f F' induces a map u i ( U i x F ) + Ui x F'),which in turn induces a projectionpreserving map

ui(
The remainder of the proof consists of direct verifications.
2.8 Definition: Let (G f, G', F f F') be a morphism of transformation groups. Then for any fibre bundle 4 over a space X , with structure group G and fiber F , the induced bundle with respect to (r,@) and 5 is the bundle 5' of Proposition 2.7. It is clear that the construction of induced bundles is functorial in the following sense, for the morphism (r,0) and bundle 4 of Definition 2.8: for any map 3 X one has ( ~ ' 4 )= ' g'5' over r?. The proof consists of direct verifications, similar to the verifications omitted from the proof of Proposition 2.7. This property will henceforth be used with no further comment. The following result will be used in the next chapter to verify that different morphisms of transformation groups sometimes lead to the same induced bundle.
x
2.9 Proposition: Let (I, 0)be a morphism oj' transjimnution groups consisting of' (I group uutomorphism G 5 G and an uction F % F o j G, und let 5 be any jibre bundle with j b e r F and structure group G over u spuce X ; then the induced bundle 5' over X satisfies 5' = 5.
PROOF: Proposition 2.7 provides a commutative diagram I
E
\
* E'
/
\
/
X of coordinate bundles representing 5 and ial.
w
PROOF: Renumbering W , , . . . , W, ifnecessary, thereareq + 1 real numbers t o , r , , . . . , t, with 0 = t o < r , < . . . < t, = 1 such that [ t i _ , , t i ] c for i = 1,. . . , y. The inclusions X x [ti t i ] c X x i4( imply that each restriction v] I X x [ti ,,ti] is trivial, and the result is then an obvious iteration of Lemma 3.1.
,,
w
74
11. Fiber Bundles
3.3 Lemma: Let X be any space, and let r] be any jibre bundle over the product X x [0,1]. Then there is at least one open covering { U , )i E l } of X such that each restriction r] U i x [0,1] is trivial.
I
PROOF: By Definition 1.9 there is an open covering { $1 j E J } of X x [0,1] such that each restriction r] $ is trivial. Any point (x, t ) E X x [0,1] has a neighborhood basis of sets of the form V x W for an open neighborhood I/ of x and an interval W c [0,1] that is open in [0, I] and contains t in its interior. Hence each (x, t ) E X x [0,1] lies in at least one neighborhood of the form V x W , where V x W c rj for at least one j E J . Since restrictions of trivial bundles are trivial (as in Lemma 1.8) it follows that there is a covering of X x [0,1] by sets of the form V x W such that each restriction r] I V x W is trivial, where W is an interval that is open in [0,1]. For each x E X let ?iY, be the family of all such products V x W for which x E V and I/ x W c rj for some j E J . Since [0,1] is compact, there is a finite subfamily { V, x W,, . . . , V, x W,> of ?iY, such that { W,, . . . , W,} covers [0,1], and for U , = V, n * * * n V, each of the restrictions r] I U , x W,, . . . , r] I U , x W, is trivial. Lemma 3.2 then implies that q ( U , x [0,1] is itself trivial, so that { U , I x E X } is a covering of the desired form.
I
3.4 Proposition: Let X be any space, and let r] be a jibre bundle over the product X x [0,1] such that the restriction r] X x (0) is triuial. Then 11 is itself' trivial.
I
PROOF: Let E + X x [0,1] represent q, and let El X x (0) 5 X x (0) x F be a fixed trivialization of q I X x (0).By Lemma 3.3 there is an open covering { Uil i E I} of X such that each r] U ix [0,1] is trivial, and one can use the procedure of Lemma 3.1 to guarantee that there are trivializations El U ix Y [0,1] U i x [0,1] x F such that each restriction El U i x { O } Ui x {O} x F coincides with the corresponding restriction of Y. Hence the transition functions U in U j x [0,1] A G have the constant value $!(x,O) = 1 E G on each nonvoid intersection U in U j x {O}. For each i E 1 let El U ix [0,1] 3 U i x {O} x F be the composition of Y j with the map U ix [0, I] x F + U i x { 0} x F that carries (x, t, f)into (x, 0, f).Since &(x, 0) = 1 for each nonvoid intersection U i n U j , it follows that the restrictions El U in U j x f [0, I] iU in U j x (0) x F and El U in U j x [0,1] 2 U i n U j x (0) x F agree, so that there is a welldefined map f for which the diagram
I
E

x
x x [O, 11
r
x {O} x F
J
xx
(0)
4. Pullbacks along Homotopic Maps
75
commutes, wheref‘(x, t ) = (x, 0 )for each (x, t ) E X x [0,1]. By Corollary 1.13, q is then the pullback along X x [0, I] X x {O) of the trivial bundle r] I X x { 0}, so that by Lemma 1.8 r] is itself trivial, as asserted. Here is the main result of the section.
3.5 Proposition: Let 5 be any ,fibre bundle over any contractible space X ; then 5 is trivial. PROOF: By hypothesis X is homotopy equivalent to the singleton space (*}, so that there is a composition X 3 { * ) 5 X that is homotopy equivalent to the identity map X + X ; that is, there is a map X x [0,1] 5 X whose restrictions X x {O} X and X x (1) 2 X are the composition h g and the identity map, respectively. Let r] be the pullback f’ 0 the inclusion O(m)+ GL(m,R) is N homotopy equivalence.
PROOF: This is an obvious real analog of Corollary 6.5. Recall that GL+(m,R)is the subgroup of those elements in G L ( m , R ) with positive determinants; the rotation group is the subgroup O+(m)= O(m)n GL+(m,R)c GL+(m,R). 6.9 Proposition (Polar Decomposition of CL+(m,R ) ) : Any element A E GLt(m, R) is uniquely of the jbrm eBC,where eB is positive and C E O+(m).
PROOF: This is an immediate corollary of Proposition 6.6. 6.10 Corollary: For any m > 0 there is a subspace H c GL+(m,R) difleomorphic to Rrn("+' ) I 2 such that every element of GL+(m,R) is uniquely of the form hk for h E H and an element k in the rotation subgroup O t ( m ) c GL+(ni,R).
PROOF: See Corollary 6.7.
91
6. Polar Decompositions
6.11 Corollary: For unjq m > 0 the inclusion O + ( m )+ GL+(m,R) is u homoropy equivulrncr.
PROOF: See Corollaries 6.5 and 6.8. In order to present the main result of this section, it is necessary to know that the subgroups U(n)c GL(n,C), O ( m )c GL(m,R), and O f ( m )c GL+(m,R) are closed; in fact, they are even compact. 6.12 Proposition: For any n > 0 and m > 0 the unitary group U(n), the orthogonal group O(m),and the rotation group O + ( m )are compact.
PROOF: For any matrix (a;) E U ( n )one has a$$ + . . . + a;Zii = 6 for the 5 1, Kronecker delta 6,,, by definition of U(n).For p = q it follows that so that (u;) E C"'lines in the compact subset (0')"' c Cn2,where D2 c C is the closed unit disk. Since limits of points (a;) E (D2)"' satisfying the algebraic relations ajc(; . . . a:c(i = 6,, themselves satisfy the same relations, U ( n ) is a closed subset of the compact set (D2)"',hence compact. Analogous proofs apply to O(m) and O+(rn).
2I ; : [
+
+
6.13 Theorem (Linear Reduction Theorem): Let 4 be any jiber bundle over uny buse spuce X E d whose structure group G is one of' the linear groups GL(n,C ) , GL(m,R), or GL+(m,R); then G can he reduced to one of' the compact subgroups U ( n )c GL(n,e),O(m) c GL(m,R), or O'(m) c GL+(m,R), respectivelj*.
PROOF: By Proposition 6.12 the given subgroups are compact, a fortiori closed, and it remains to apply the general reduction theorem (Theorem 5.10) to Corollaries 6.4, 6.7, or 6.10, respectively. 6.14 Corollary: Let 5 be any fiber bundle over any base space X E 98 whose structure group G is a linear Lie group with only jinitely many connected components; specifically, G is a subgroup OJ' GL(m,R) jbr some m > 0. Then ( C U H be regarded as a ,fiber bundle with a compact structure group.
PROOF: One simply regards 5 as a fiber bundle with structure group GL(m,R), to which Theorem 6.13 applies. Corollary 6.14 is essentially a weak version of Proposition 5.12. However, its proof does not require the IwasawaMal'cev decomposition theorem, and in any event most interesting Lie groups with at most finitely many connected components are linear. (The first example of a connected nonlinear Lie group appeared in Birkhoff [l]; the example appears as an exercise in Hochschild [ I , p. 2251. A method for constructing other nonlinear Lie groups is given in J. F. Price [I, pp. 119121, 1561571.)
92
11. Fiber Bundles
Fiber bundles with structure groups GL(n,C),GL(m,R), GL+(m,R), or equivalently U(n),O(m),O+(m),will occupy the rest ofthe book. It is therefore reasonable to begin looking at the topologies of these groups. For the moment we merely count their connected components.
6.15 Lemma: Let H be a connected closed subgroup of a topological group G such that the homogeneous space G / H is connected; then G is connected. PROOF: If G is covered by two nonvoid open sets U and V , then their images U' and V' in G / H are two nonvoid open sets covering G / H . Since G/H is connected, there is an x E G whose image x' E G / H lies in the intersection U' n V ' . The inverse image W of the set {x'} c G / H is trivially homeomorphic to H , hence connected, so that the two nonvoid sets U n W and V n W , which cover W , have a nonvoid intersection; a fortiori U n V is nonvoid.
6.16 Lemma: The roration group O+(rn)c O(m) is connected for every m > 0. PROOF: We proceed by induction of m, observing that O + ( l )consists of a single point. Regard O+(m)as a transformation group, acting via rotations of the ( m  1)sphere S"' c R", and embed O + ( m 1) in O + ( m )as the subgroup leaving some fixed unit vector e E S" invariant. For any other f E S" there is at least one g E O+(m)with ge = f E Sm',and an easy verification shows for any h E O+(m)that he = f E S" if and only if h and g have the same image in O+(m)/O+(m  1). Since the image S" ' of the induced onetoone map O + ( m ) / O + ( m 1) + S"' is compact, the map is a homeomorphism. Since S"' is connected, so is O+(rn)/O+(m I), and since O+(m 1) is connected by the inductive hypothesis, O+(m)is connected by Lemma 6.15, as required.
'
'
6.17 Lemma: every m > 0.
The orthogonal group O(m)has two connected components .for
PROOF: Multiplication of the connected component O + ( m )c O(m) by an element in O(m) with determinant  1 provides a diffeomorphism from O+(m)to the complement O(m) O + ( m ) .
6.18 Lemma:
The unitury group U ( n )is connected for every n > 0.
PROOF: The construction of Lemma 6.16 is easily modified, beginning with the circle U(1) = S', to yield a homeomorphism of the homogeneous space U(n)/U(n 1) with the (2n  1)sphere St"' c C";since V ( n 1) is connected by the inductive hypothesis, V ( n )is connected by Lemma 6.15.
93
7. The LerayHirsch Theorem
6.19 Proposition: For uny n > 0 und unjl m > 0 the group GL(n, C) has one connected component, and GL+(m,R) is one ~f the two connected components of GL(m,R).
PROOF: By Lemmas, 6.166.18, the statements are valid for the subgroups U(n) c GL(n, C), O(m)c GL(rn,R), and O + ( m )c GL+(m,R), and by Corollaries 6.5, 6.8, and 6.1 1 the inclusions are homotopy equivalences.
7 . The LerayHirsch Theorem We henceforth consider only fiber bundles rather than fibre bundles, both of which appeared in Definition 1.9; that is, all base spaces, now and forever more, belong to the category 98 of Chapter I. Suppose that E 5 X is a coordinate bundle for some X E 98, representing a fiber bundle 5 over X. For any commutative ring A with unit let H * ( X ) and H * ( E ) be the singular cohomology rings H*(X; A) and H * ( E ; A), respectively, with coefficients in A. Since H*() is a contravariant functor, there is an induced ring homomorphism H*(X) 5 H*(E), and one can use n* to regard H * ( E ) as a left H*(X)module, the product p . u E H * ( E ) of a E H * ( E ) by a scalar E H*(X) being the cup product n*P u c( E H*(E). Up to canonical isomorphisms the H*(X)module H * ( E ) depends only on the fiber bundle (, and not upon the particular coordinate bundle E 3 X chosen to represent 5. For if E' 5 X is any other coordinate bundle representing 5 there is a homeomorphism j ' such that
E'
I
+ E
commutes, so that H * ( E ) 1:H*( E') is an isomorphism of H*(X)modules, as claimed. There is even more latitude in the construction of the H*(X)module H * ( E ) , up to canonical isomorphisms. For any map X' 5 X in the category .# of base spaces, and for any coordinate bundle E 3 X, there is a pullback diagram E'
n'
B
I
1
X'
rE
1 4
' x,
94
11. Fiber Bundles
hence a commutative diagram
H*(X) 9 . H*(X’)
rl
H * ( E ) A H*(E’) of ring homomorphisms. One can regard g* as a module homomorphism over the ring homomorphism g*; that is, g * ( p . a) = g*(x*p u a) = g*n*P u g*a = x’*g*p u g*a = ( g * p ) . (g*a) for any a E H * ( E ) and p E H * ( X ) . I f y is a homotopy equivalence, then H * ( X )% H * ( X ’ ) is a ring isomorphism: this is an immediate consequence of the homotopy axiom for singular cohomology.
7.1 Lemma: If X ‘ 5 X is a homotopy equivalence in g, then jor uny courdinate bundle E 5 X the homomorphism H * ( E ) H * ( E ) is an isomorphism over the ring isomorphism H * ( X ) % H * ( X ‘ ) . PROOF: In the pullback diagram E’
B
+ E
X‘
9
+x
the map g is also a homotopy equivalence, by Proposition 4.10. In the following result H*( ) continues to denote singular cohomology with coefficients in a fixed commutative ring A with unit. For any coordinate bundle E 5 X over any X E a, and for any x E X , let E x 2 E denote the inclusion of the fiber E x over x ; then H * ( E ) 4 H*(E,) is a homomorphism of modules over the ground ring A. An element a E H*(E) is homogeneous whenever a E Hq(E) for some fixed index q E Z,in which case j z a E H Y ( E x ) for each x E X .
7.2 Theorem (Absolute LerayHirsch Theorem): Let E 5 X be a coordinate bundle over anj’ buse space X E A!,?, let H*() be singular cohomology H*(; A) with coeflcients in u commutative ground ring A with unit, and suppose that there are finitely many homogeneous elements al, . . . ,a, E H*( E ) such that for each x E X the Amodule H*(E,) is free on the basis ( jza,, . . . ,j;a,} ; then the H*(X)moduleH * ( E ) is free on the basis ( a , , . . . , a,,I .
95
7. The Leray Hirsch Theorem
PROOF: First suppose that X ' % X is any homotopy equivalence, with pullback diagram E'
R
X'
Y
,x
as usual. For any s'E X ' the map g induces a Grelated isomorphism E:. JS Ey,s.j of fibers, hence a Amodule isomorphism H*(E,,,.,) H*(EI,). Since j.T,(g*a)= gfj&.,a for any c1 E H * ( E ) , it follows that each Amodule H*(EI.) is free on the basis (j,*.(g*Cc,),. . . ,j,~(g*Cc,)J.Hence by Lemma 7.1 one can substitute E' 5 X ' for the given coordinate bundle E X whenever X ' .% X is a homotopy equivalence. In particular, since X E %, there is a homotopy equivalence X ' 5 X such that X ' is metrizable and of finite type, by Definition 1.1.2, so that one may as well assume throughout the remainder of.the proof that X is itself of finite type, as in Definition 1.1.1. Suppose that ct1 E H " ( E ) , . . . , c1, E H + ( E ) for integers q l , . . . , q,, let q be any integer, and for each open set U c X let h 4 ( U )be the direct sum H 4  q 1U ( )0. . . @ Hq4v(U ) of Amodules, where H 4  Q i ( U )= 0 for q yi < 0. If R is the cohomology ring H * ( X ) of the base space X itself, then one can use the ring homomorphism R = H * ( X ) + H * ( U ) induced by the e ( )as a graded Rmodule, inclusion U c X to regard the direct sum U q E Z h U via cup product in H*( U ) . It follows from the classical MayerVietoris cohomology exact sequence that one thereby obtains a MayerVietoris on the category O ( X ) of open sets U c X to the category functor UqEZhY 9liF of Lgraded Rmodules, as in Corollary 1.9.4. We now construct another MayerVietoris functor UqEz kY from O ( X ) to 91iz,and we later construct a natural transformation uqezkq to which to apply Corollary 1.9.4. For any open set U c X let El U = n '( U ) as usual. and for any integer y let P ( U ) be the Amodule H4(EI U ) . If El U 3 U is the restriction to El U of the original coordinate bundle E 1:X (over the space X of finite type), one can then combine the homomorphism R = H * ( X ) + H * ( U ) % H*(EI U )with cup product in H*(EI U ) to regard the direct sum UqEZ ky(Li) (=H*(EI U ) )as a graded Rmodule. It follows from the classical MayerVietoris cohomology sequence that one thereby obtains another MayerVietoris functor, k4, from 8 ( X ) to !Ill:, as promised.
UyeZh4
uqEz
96
11. Fiber Bundles
Now recall that hq( U ) = H 4  4 1 ( U0 ) . . . @ H 4  4 r ( U )for each q E Z and U E O(X), and that the hypothesis of the theorem provides elements a , E H 4 ' ( E ) ,. . . , a, E Hqr(E). Let El U 5 E be the inclusion map, and recall that El U 3 U denotes the restriction of E X. Then for each pi E H4"(U) one has n$Pi E H44i(EIV ) and j:ai E H4'(EI U ) , for i = 1, . . . ,r, so that there is a welldefined map h4(U ) + k4( U ) carrying each p10 . . . O p, E h4(U ) into the sum .;Pi u j$ct, E kq(U). One easily verifies that the direct sum of such maps, over all q E Z,is an Rmodule homomorphism U y E z h 4 ( U2 ) U q e m k 4 ( Uand ) that the family {&)" of such homomorh4 5 UqEz k4 of MayerVietoris phisms is a natural transformation functors on O(X) to ! I l l : . The final step in the proof of Theorem 7.2 is to show that 9 is a natural equivalence, a fortiori that Ox is an H*(X)module isomorphism. If U E U ( X) is contractible, then the restriction E I U U represents a trivial bundle over U , by Proposition 3.5, so that there is a homeomorphism f such that
Uqek
commutes, where F is the fiber and n 1 is the first projection. If U contracts to x E U , then UqEH h4(V )2 k4(U) is canonically equivalent to the ) . . O H * ( { x } )+ H*(E,), which carries p1 homomorphism H * ( { x } 0 . . .O p, into RT pi u j,*aiE H*(E,), where H * ( { x ) ) = H o ( { x } )= A. Since H*(E,) is free on the basis { j:alr . . . ,j:a,), by hypothesis, it follows that 6" is an isomorphism for contractible U E O(X). Since X is of finite type, the MayerVietoris technique applies in the form of Corollary 1.9.4, with the consequence that the H*(X)module homomorphism H*(X)0 . . . H*(x) H * ( E ) carrying any p1O . . . 8, E ~ 4  4 " ~ 0) .. . O l F q r ( X ) into n*pl u a1 . . . IT*& u a, E H 4 ( E ) is an isomorphism; that is, H * ( E ) is a free H*(X)module with basis [ a l , .. . , a,), as asserted.
UyEZ
c:=,
o
+
+
o
In a certain sense Theorem 7.2 is a generalization of a Kunneth theorem, as follows.
7.3 Corollary: Let X E 93 be m y base space, and let F be any space whose cohomology H * ( F ) ( = H * ( F ; A)) is a free Amodule on Jinitely many homogeneous generators; then for the projections R , and n2 of X x F onto X and F , respectively, the map H*(X) @, H * ( F ) + H * ( X x F ) currying 0 a into RTP v nrci is rn H*(X)module isomorphism.
8. Remarks and Exercises
97
PROOF: Apply Theorem 7.2 to the trivial coordinate bundle X x F 3 X . The label attached to Theorem 7.2 suggests that there is a corresponding relative LerayHirsch theorem. There is such a theorem, its proof being virtually identical to that of Theorem 7.2 itself. We require only a very specialized relative LerayHirsch theorem, however, which will be formulated in a later chapter.
8. Remarks and Exercises 8.1 Remark: The first organized account of the material of this chapter appeared in Steenrod [4, Part I], in 1951. Steenrod profoundly influenced the development of fiber bundles, and Part I of his book, at least, remains surprisingly modern. More recent introductory accounts of fiber bundles (in general) can be found in Bore1 and Hirzebruch [l, Chapter 111, Auslander and MacKenzie [2, Chapter 91, Holmann [l], Husemoller [l, Part I], Liulevicius [2, Chapter I] and Liulevicius [3, Chapter I], in the beginning pages of Lees [l], Eells [ 13, Porter [2, Chapter 21, and Rohlin and Fuks [1,Chapter IV], for example. Expository accounts of more general fiber spaces (or fibrutions), considered later in these Remarks, can be found in Cartan [3, Expos& 6, 7, 81, Schwartz [l, Part I], Hu [2, Chapter 1111, May [l, pp. 130], Switzer [l, Chapter 41, and G. W. Whitehead [l, pp. 2975], for example. 8.2 Remark: The first explicit definition of a coordinate bundle is due to Whitney [2], and it was further amplified in Whitney [4, 5,6]. The importance of Whitney’s brief original paper and its successors was quickly recognized. Related notions of fiber spaces (without structure groups) were soon independently introduced by Hurewicz and Steenrod [11 and by Eckmann [l], followed by Fox [3]; fiber spaces will be discussed further in Remarks 8.58.8. Simultaneously Ehresmann and Feldbau [13, and later Ehresmann [3,4] considered alternatives to Whitney’s construction, with structure groups. The general definition of coordinate bundles with arbitrary fibers F and appropriate structure groups G had become mathematical folklore before its first appearance as an incidental feature of Steenrod [2], and the equivalence relation providing fibre bundles (and fiber) bundles in the sense of the present chapter was equally well understood before its first publication in Steenrod [4].
8.3 Remark: The distinction between fibre bundles and fiber bundles is a convenient technical device whose introduction is justified by the appearance
98
I I . Fiber Bundles
of the category 9 in several major results of this chapter: Proposition 4.7, the general reduction theorem (Theorem 5.10) and its application (Proposition 5.12), the linear reduction theorem (Theorem 6.13), and the LerayHirsch theorem (Theorem 7.2).These are some of the reasons for introducing the category 9? in the first place. The author apologizes to those readers who might prefer an interchange of the spellings “fibre” and “fiber.”
8.4 Remark: One can achieve some of the results of the preceding remark by restricting the bundles themselves, rather than their base spaces. According to Dold [ 5 ] a fibre bundle over a base space X (not necessarily in a) is numerable if there is a covering { U ii(E I } of X and a corresponding locally finite partition of unity Chili E I } such that (i) the covering i ‘h; ‘(0,1] I i E I } refines { U iI i E I } and (ii) each restriction I U iis trivial. Clearly if X is metric (a fortiori paracompact) and of finite type (as in Definition I.1.1), then Proposition 3.5 implies that any fibre bundle over X is numerable. By Definition 1.1.2 any space in 9? is homotopy equivalent to such a space X , so that one can use Lemma 4.5 to conclude that any fiber bundle whatsoever is automatically numerable. An equivalent definition of numerability occurs in Derwent [3].
0 for all nonzero vectors e E R". We shall establish the existence and demonstrate the usefulness of corresponding "inner products" for any vector bundle over any base space
x E a.
For any real mplane bundle 5 over X E 2 let 5 be the bundle A'q over X with fiber R" x R" and structure group GL(m,R) x GL(m,R), as in Proposition 2.5, for 5' = 5 . If { U iI i E I is an open covering of X,with local trivializations E I U i .% U i x R" of a coordinate bundle E 1:X representing 5, then { U ix U j ) ( ij, ) E I x I } is a corresponding covering of X x X for the bundle q. Since the subfamily { U ix U i ( iE I } covers the image of the diagonal X 5 X x X , it follows that there are corresponding local P;' U ix (R" x R")of a corresponding representation trivializations E" 1 U i JI' E" X of the bundle 5. Specifically, transition functions U i n U j A GL(m,R) describing the coordinate bundle E 5 X provide transition functions U in U j (JI1*JI') * GL(rn,R) x GL(rn,R) describing the coordinate bundle E" 5 X . If Ex is the fiber over x E X of the coordinate bundle E A X , then Ex x E x is the corresponding fiber E:' of the coordinate bundle E" 5 X . Suppose that E" 2R is any map which restricts to an inner product Ex x E x = E:' R over each x E X . If E'"'. X is any other representation of 5, and if E"' 3 X is the corresponding representation of (, then the fiber E:" = EL x Ex over each x E X is related to the fiber E:' = E x x E x by an element in the image of the diagonal GL(m,R) + GL(m,R) x GL(rn,R). Hence if E"' 1,E" is an isomorphism of coordinate bundles representing [,


1 I3
3. Riemannian Metrics

and if E"' R is the composition E"'5E" ' . 1R, it follows that ( , )' also restricts to an inner product EL x EL = E:' ( )k R over each x E X . Consequently the following definition is independent of the coordinate bundle E 5 X chosen to represent 5. 1
3.1 Definition: Let 5 be a real mplane bundle over X E 33,represented by X represent the preceding a coordinate bundle E A X , and let E" bundle [, with fiber R" x R" and structure group GL(m,R) x GL(m, R). A riemannian metric on 5 is any map E" < . ) R that restricts to an inner product E x x E x + R for each x E X .

3.2 Lemma: If X E 9? is paracompact, then there is a riemannian metric on any real vector bundle 5 over X .
z

PROOF: Let E A X and E" X represent 5 and [ as before, and suppose that { Ui I i E I ) is an open covering of X with local trivializations E I U i y , U ix R". Since X is paracompact, one may as well suppose that { Ui I i E I } is locally finite, and that there is a partition of unity { h i ( iE 1 ) subordinate Y;' to { Uil i E I ) . For the corresponding local trivializations E'I U i Ui x (R" x R"), and for any fixed inner product R" x R" R, there is a map h,( U ix (R" x R") R carrying each ( x , ( e l , e 2 )E) U i x (R" x R") into the real number h i ( x ) ( e , e, z ) i .Since hivanishes outside U i ,each composition hi( , )i 3 4' ;' extends to a map E' ha( s) 'Pi' + R that vanishes outside U i , and one trivially verifies that the welldefined sum hi( , )i0 Yi' is a riemannian metric.


#
)I
I
1
xiel
3.3 Lemma: Let ( , ) be a riemannian metric on an mplane bundle 5 over any X E 93,and let X ' 3 X be any map in 99;then there is a riemannian metric ( , )' on the pullback f '5 over X ' E W .
z
PROOF: If E 5 X and E" X represent the bundles 5 and [, as before, n"' X ' represent the pullbacks f '5 and f '[, then and if E' 5 X ' and E"' there is a commutative diagram E"
f
X'
I
E"
+
x,
as in Lemma 11.1.3, for which the composition desired riemannian metric on f '5.
E"'5E"
]IT
PROOF: For any covering of X by open sets U such that each restriction E I U is trivial, Lemma 1.6.1 provides a countable locally finite refinement [ U;,I n E N) by open sets U ; with compact closures 0;, and it follows that there is a smooth local trivialization E I U;,3U ; x R" for each n E N. By Proposition 1.6.10 one can shrink {U:,InE N} to a new open covering (U,ln E N} with compact closures 0,c U ; , and by Corollary 1.6.1 1 there is a smooth partition of unity {h,ln E N; subordinate to {U,ln E N}. For each n E N the local trivialization Y , and the given euclidean norm 11 11 induce on the Co(0,)module 5 1 0,of cona euclidean norm 9I 0,5 Co(0,) y alO,, tinuous sections 0,, On x R", with IIY, 0 ~,,I[,(x) = 1101 Dnll(x) for each restriction 0, 3E I D,and each x E D,.Since 0,is compact, Lemma 5.10 provides a smooth section 0,2 E I D,, such that  (i I ~T,,ll(x)< E ( x ) , for each n E N and each x E On. The partition of unity {h,ln E N} consists of smooth functions X 3 [0,1] such that h, vanishes outside of U n for each n E N, so that each smooth section z, can be extended to a smooth section X 3E that vanishes outside U , . Since { U,ln E N} is locally finite, the sum h,z, is a welldefined smooth section X 5 E of E X . Finally, for any x E X let N , be the finite subset of those indices n E N with x E U , , ,and observe that since h,(x) = 1, one has

GI
\IT,
126
111. Vector Bundles
As in Proposition 4.8, a section X 3 E of a coordinate bundle E X representing a vector bundle over a base space X E is nowherevanishing whenever a(x) # 0 E E x for each x E X.
5.12 Proposition: Let E X be any smooth coordinate bundle representing u vector bundle over a smooth manifold X. Then if there is a continuous nowherevanishing section X 5 E there is also a smooth nowherevunishing section X E . PROOF: Let 9 Co(X)be a euclidean norm as before. Since X E is nowherevanishing, one has Ilall(x) > 0 for every x E X, so that there is a strictly positive function E = 411~~11 E Co(X). By Proposition 5.1 1 there is then a smooth section X A E with  011 < E = fllall, for which the triangle inequality gives llzll 2 llall  IIz  all > tllall > 0, as required.
[IT
6. Vector Fields and Tangent Bundles There is an obvious contravariant functor from the category of smooth manifolds to the category of realvalued function algebras, which we have already used: to any smooth manifold X one assigns the algebra C"(X) of smooth realvalued functions X 4 R, and to any smooth map Y + X one assigns the algebra homomorphism Ca(X),Cm(Y)carrying X iR into the composition Y ,X R. There is also a contravariant functor that assigns a specific C"(X)module b(X) to each smooth manifold X, and that assigns a module homomorphism b ( X ) ,b(Y ) over the ring homomorphism Ca(X) , Cm(Y ) to each smooth map Y ,X. We shall show that if X is mdimensional, then b ( X ) is a locally free C"(X)module of rank m, so that Proposition 5.6 provides a canonical smooth mplane bundle z(X) over X , assigned to X itself. The properties of z(X) characterize much of the differential topology of X itself. The next few paragraphs are devoted to the construction of the dual 8 * ( X ) of &'(X),which will lead to &(X)itself, and to some elementary properties of the resulting bundle z(X) over X. 6.1 Definition: For any smooth manifold X , a smooth vector ,field on X is any real linear map CyJ(X)5 P ( X ) such that L(fg) = ( L f ) g + f ( L g )for any f E C n ( X )and g E Ca(X).
For example, if X is the smooth manifold R", and if xl, . . . , X" E Cm(Rm) are the projections Rm + R that provide the usual coordinate functions on
127
6 . Vector Fields and Tangent Bundles
R", then the partial derivations ?/ax', . . . , d/dx" are vector fields on R". Similarly, if X is the smooth submanifold (Rm)+c R" with nonnegative mth coordinate, then (7/?x', . . . , (7/i)xmare vector fields on (R"')'. If L and M are vector fields on a given smooth manifold X, and if g E Cx(X),then there are vector fields L + M and gL on X that carry any f ' C~x ( X )into ( L f ) ( M f )E C " ( X ) and g ( L f )E C5(X), respectively. It is clear that the set of all vector fields on X forms a C" (X)module with respect respect to these definitions. For example, if y', . . . , g" E Ca(R"'), then y1 ?/?x' + . . . + y" (7/dx"' is a vector field on R".
+
6.2 Definition: For any smooth manifold X, the C"(X)module 8*(X) of' smooth ~iectorjields on X consists of smooth vector fields L , with respect to the preceding addition and scalar multiplication. The goal of the next few lemmas is to show that if X is an mdimensional smooth manifold, then &*(X)is a locally free C"(X)module of rank m.
6.3 Lemma : Vector jields annihilute constant junctions. PROOF: Let 1 E C'(X) be the constant function on X with value 1 E R. ForanyvectorfieldC'(X)fi C"(X)onXonethenhasL(l)= L(12) = 2L(1); hence L( I ) = 0, so that L(c1) = cL(1) = 0 for any c E R. 6.4 Lemma: For uny smooth munijbld X und uny open subset U c X, let E C" (X) have common restrictions f 1 U = y I U E C" ( U ) . Then jix uny vector ,field C" (X) fi C" (X) on X the images L j E C r ( X ) and ~g E C ' ( X ) huve common restrictions L f I u = Lgl U E C = ( U ) .
,f' E C ' ( X ) und g
PROOF: For any x E U Lemma 1.6.2 provides an h E C x ( X ) that vanishes outside U and satisfies h ( x )= 1. Then (.f  g)h = 0, so that (Lf  Lg)h (1'g)Lh = 0. Evaluation at x yields ( L f ) ( x ) (Ly)(x)= 0, and since x is an arbitrary point of CJ, one has LfI U = Lgl U .
+
Lemma 6.4 asserts that any vector field L is local in the sense that & f I U depends only on f I U ; we shall rephrase the result with this in mind. If U is any open set of a smooth manifold X, and if I ( U ) c Ca(X) is the ideal of smooth functions X + R that vanish on U , then the inclusion U c X induces a homomorphism CJ'(X) Cz(CJ) whose image is the quotient algebra C'(X)/I(U ) . One defines smooth vector jie1d.s C" (X)/I(U ) 3 Cx(X)/I(U) by the obvious requirement that M ( ( f 1U ) ( g lU ) )= W f I U ) ( g (U ) + ( j ' l U)M(yI U ) for anyf U and 81 U in C"'(XVI(U). .+
6.5 Lemma : Let U br uny open set of (1 smooth munifold X, and let C" (X) C"(X)/I(U) he the restriction mup currying any f EC"(X) into f I U
.+
E
128
Ill. Vector Bundles
C m ( X ) / I ( U ) Then . any vector jield C z ( X )1; C a ( X )on X induces a unique vector jield Ll U such that L
Ca(X)
'CX(X)
commutes.
PROOF: This is just a rephrasing of Lemma 6.4, as promised. 6.6 Lemma: Let xl, . . . , xmE C" ((R"')') he the usual coordinate functions on the submanifold (Rm)+c R" with nonnegative mth coordinate. Then for any vector j e l d C30((Rm)')1; CZ((Rm)+) one has L = (Lx')d/dx' + . . . (Lx")d/dx"for the functions Lx', . . . , LxmE C"((Rm)+).
+
PROOF:The line segment joining any two points xo E (Rm)+and x1 E (Rm)+ lies entirely in (Rm)+,so that for any f E Cm((Rm)')and each index i = 1, . . . , m one can define a function f; E C"((Rm)+ x (Rm)+)by setting
for xo = (x;, . . . , xt ) and x1 = (x!, . . . , xy). For fixed xo E (Rm)+the latter identity becomes I
c (xi m
.f

f(x0) =
 x~,f)(xo;)
i= 1
in Ca(Rm)+), to which one applies L and Lemma 6.3 to conclude that m
m
129
6. Vector Fields and Tangent Bundles
for any x 1 E ( R m ) + In . particular, ifx, is any point of(Rm)+and x 1 = x o , then
so that Lf = xy=l(Lx')(dJ/dx')for any f I ( L . ~(J/dx') ') as claimed.
E
C o c ( ( R m ) + that ); is, L =
I f U and U' are open sets of a smooth manifold X such that the closure
0 of U satisfies D c U', then every restriction f I U E C a ( U )of an element
C r ( X ) is trivially the restriction gI U E C a ( U ) of the element g = f I U' E C m( U ' ) . The following lemma provides a useful converse assertion. J
E
6.7 Lemma: Let U and U' be open sets of a smooth manijold X such that D c U ' ; then the inclusion U' t X induces an isomorphism C m ( X ) / l ( U+ ) C' (U')/I(U ) of the restrictions to U of the ulgebras of smooth functions on X and U ' .
PROOF: The induced homomorphism C"(X)/I(U) + C%(U')/I( U ) is trivially injective, and it remains to show that the restriction g I U E Coo( U ) of any y E C" ( U ' )is also the restriction ,fl U E Ca(U ) of some f E Ca(X).Since X is paracompact, by Lemma 1.6.1, it is also normal. (See page 163 of Dugundji [2], for example.) Hence there is an open set U" c X such that 0 c U" and 0"c U'. By Lemma 1.6.2 there is a smooth partition of unity subordinate to the covering of X by the two open sets U" and X \ U ; in particular, there is an h E P ( X ) such that hl U = 1 and h(X\D") = 0. For any g E C r (U ' )the product ( h I U ' ) g E C"(U ' )vanishes on the open set U'\D", and since U'' c U', the function ( hI U')g is the restriction f(U' of a function j" E C" (X) that vanishes on X\D". Since hl U = 1, it follows that U = g1 U as required.
fl
6.8 Proposition: For any smooth mdimensional manifold X the Cm(X)module 8*(X) of' smooth vector Jields on X is locally ,free of rank m.
PROOF: By Proposition 1.6.10there is a countable open covering { Ubl n E N} of X and an atlas {@,,InE N}, which consists of homeomorphisms Ui, % VL onto open sets @,(UiI)= V:, c ( R m ) + ;the homeomorphisms @, are in fact diffeomorphisms, by definition of the smooth structure of X. By Proposition 1.6.10 one can also shrink [UillnE N} to a new open covering (U,llnE N) with 0,c U ; for each n E N, and the diffeomorphisms 0, provide new open subsets On(U,J= V, c ( R m ) +with c Vk for each n E N. We shall show for each n E N that the C"(X)/I(U,)module d*(X)/ I(U,)€*(X) is free of rank m. Since C a ( X ) / I ( U , ) + Cw(Ub)/I(U,) is an isomorphism, by Lemma 6.7, and since vector fields are local, by Lemma 6.5,
r,
130
Ill. Vector Bundles
it suffices to show for each n E N that the C7(Ub)/I(U,,)module (s.*(U',)/ I ( U,)c"*( Uh) is free of rank m. Since the diffeomorphisms U ; 3 V:,c (R"')' restrict to diffeomorphisms U , % V, c (R")', it therefore suffices to show for any open sets V c ( R m ) + and V ' c ( R m ) + with V c V' that the C x (V ' ) / I (V)module &*( V')/I(V ) 6 * (V ' ) is free of rank m. Since Cm((Rm)+)/Z(V ) + C"( V ' ) / I (V ) is an isomorphism, by Lemma 6.7, and since vector fields are local, by Lemma 6.5, we are left with the task of showing for any open set V c (Rm)+ that the Cm((R")+)/I( V)module 6*(( R m ) + ) / IV)&( ( (R")') is free of rank m. However, this is an immediate consequence of Lemma 6.6, which asserts that the Ca((R")+)module &*((R")+) is free of rank m, one basis being J/ax',. . . , c?/dx". According to Proposition 5.6, if X is a smooth manifold, then any locally free C"(X)module 9 of rank rn determines a unique real mplane bundle 5 over X ; specifically, the elements of 9 are the smooth sections of some coordinate bundle E: X representing 5. According to Proposition 6.8, if X is of dimension m,then a particular locally free C"(X)module 6'*(X) of rank m is available for such an application. For any smooth manifold X the tangent bundle 7(X) is the smooth real vector bundle over X represented by the smooth coordinate bundle whose sections are vector fields L E a*(X).
6.9 Definition:
We shall give a more direct construction of a particular smooth coordinate bundle E X whose sections are vector fields, which will give a better understanding of the tangent bundle 7 ( X )it represents. Let { U i I i E I } be an open covering of the smooth mdimensional manifold X by coordinate neighborhoods U i , chosen as in Proposition 6.8, and let { Oi I i E I } be a corresponding atlas of diffeomorphisms U i5 ai(Ui),where each Oi(Ui)is open in (Rm)+. For any nonvoid intersection U i n U j the @, @; Ui n Uj) * Oj(U i n U j ) is a diffeomorphism of open composition Oi( sets in (R")' that can be described directly in terms of coordinate functions: for any (x', . . . ,x") E Oi(Ilin U j )one has ,
. . . , x"), . . . , ~ " ( x ' ,. . . , x")) E Oj(ui n U j ) for uniquely defined smooth realvalued functions y', . . . ,7" on Qi( U i n U j )c Oj c O,: '(x',
. . . , x")
= (~'(x',
@. .@:
I
(R"')'. The inverse diffeomorphism Oj(U i n U j ) ' ' + Oi(Uin U j )has a similar description: for any ( y ' , . . . ,y") E O j ( U in U j )one has Oi ') @ :, ' ( y ' , . . . ,y") = ( ~ ' ( y ' ., . . , y"),
. . . , ~ " ( y '.,. . ,y"))E 0 and q > 0, set n = mq. Then any real mplane bundle 5 over any qdimensional metric simplicial space IKI is a pullback f l y : of' the canonical mplane bundle :y over the Grassmann along some map IKI Grn(Rrn+'). manijold Grn(Rrn+'),
PROOF: Since there is a canonical homeomorphism IK'( 2 IKI, for the first barycentric subdivision K' of the simplicial complex K , one may as well regard 5 as a bundle over IK'I. According to Proposition 1.2.1, IK'I is of first type. Specifically, since K is qdimensional, there is a covering (UO,ar}a,. . . , (U,,y}y of IK'I by q 1 families of contractible open sets U p , Dc IK'I, the sets in each family being mutually disjoint. Let U p= c IK'I for each p = 0, . . , ,q, so that { U , , . . . , U , ) is a finite open covering of (K'I. Since any fibre bundle over a contractible space is trivial, by Proposition 11.3.5, it follows that each restriction 5 1 U p , Dis trivial, and since each family { U p , , } s is mutually disjoint, it follows that each restriction 5 1 U p is trivial. Thus if E J IK'I represents (, then one can choose q + 1 specific trivializations E I u03 u0 x I W.~. .,, E 1 U , LU , x and a partition of unity ( h o , .. . , h4) subordinate to { U o ,. . . , U,j to obtain ho'€'o +. . . + hqYq a Gauss map E + R r n ( , + l )as in Lemma 8.8. The analog of Lemma 8.4 then provides a corresponding morphism
+
up
E
f
+ E'
(K'(J ' Grn(R"'+''),
where E' 5 Grn(Rrn+") represents 7:. Since Proposition 1.4.6 (Dowker [I]) provides a homotopy equivalence lKlm , lKlw from any metric simplicial space IK(, to its weak counterpart IKlw, Proposition 9.1 applies equally well to qdimensional weak simplicial spaces. Hence, one may as well omit the metric condition from the statement of Proposition 9.1.
9. More Smooth Vector Bundles
159
9.2 Corollary: Let 5 be anj1 real mplane bundle over any finitedimensional simplicia1 space IKI; then for some n > 0 there is an nplane bundle q over IKI such that the Whitney sum [ @ q is the trivial bundle E ~ + "over IKI. PROOF: Substitute Proposition 9.1 for Proposition 8.12 in the proof of Corollary 8.13. If the simplicial space IK( in Corollary 9.2 is qdimensional, then Proposition 9.1 permits one to set n = mq. However, there is also a direct proof, using a general position argument, in which one can set n = q. The second step toward the main theorem of this section combines Proposition 9.1 with Theorem 1.6.7 (the CairnsWhitehead triangulation theorem) and Theorem 1.6.19 (that any map of smooth manifolds is homotopic to a smooth map); one also uses Proposition 1.7.3 (that the Grassmann manifold Gm(Rm+") is a smooth closed mndimensional manifold).
9.3 Proposition: Given natural numbers m > 0 and q > 0, set n = mq. Then any real mplane bundle 5 over any smooth qdimensional manifold X is a pullback f;;~:of the canonical mplune bundle :y over Gm(Rm+")along a smooth map x ~ m ( ~ m ). + n
a
PROOF: By the CairnsWhitehead theorem X is homeomorphic to a qdimensional metric simplicia1 space IKI, so that by Proposition 9.1 4 is a By Proposition 11.4.7 one pullback f a y : along some map X 5Gm(Rm+"). can replacef, by any map X 2Gm(Rm+" ) homotopic to f,, and by Theorem 1.6.19 one can choose a smooth such map j ; . 9.4 Corollary: Let 5 be any real mplune bundle over any smooth manifold X ; then j b r some n > 0 there is an nplane bundle q over X such that the Whitney sum 5 @ i s the trivial bundle em over X .
PROOF: Substitute Proposition 9.3 for Proposition 8.12 in the proof of Corollary 8.13. If the smooth manifold X in Corollary 9.4 is qdimensional, then Proposition 9.3 permits one to set n = my. However, there is also a direct proof, using the transversality theorem, in which one can set n = q. (See pages 99101 of M. W. Hirsch [4], for example.) The main result of this section is another corollary of Proposition 9.3.
9.5 Theorem: Given a smooth manifold X , any real vector bundle 5 over X can be represented by a smooth coordinate bundle E X ; that is, any real vector bundle over X is itselfsmooth, as in DeJinition 5.2.
160
111. Vector Bundles
PROOF: If 5 is an rnplane bundle, then 5 = f ! ? ; for some n > 0 and some smooth map X i G"(R"+'), by Proposition 9.3. Since 7:: is smooth by Lemma 7.1, it follows from Proposition 5.3 that its pullback f ' y ; is smooth.
10. Orientable Vector Bundles The familiar parlortrick "onesidedness" of the Mobius band E reflects the failure of the tangent bundle r ( E )to be orientable, in the sense described in this section. To define orientability of any real rnplane bundle 5 over any X E &9 one first constructs a real line bundle A" 5 over X , from which one obtains a fiber bundle o(5) over X with structure group 0(1)( = Z j 2 ) and fiber So ( = Z/2); the bundle 5 is orientable if and only if o(5) is trivial. Following the construction of o(5) we shall show that 5 is orientable if and only if it is of the form f ' y " for a map X 3 Gm(Ral)that factors through the total space of the bundle o(y") over Gm(Ra').Finally we show that the canonical line bundle y : over RP' (= S ' ) is not orientable, and that the tangent bundle T ( E )of the Mobius band E is not orientable. Let E 5 X represent a real mplane bundle 5 over X E 98, and for any natural number p > 0 let E x . . . x E X x . . . x X be the product of p copies of E .!!,X . The pullback of the product along the diagonal map X 5 X x . . . x X is a coordinate bundle E' 1;X with fiber R" x . x R"' and structure group GL(m,R) x . . . x GL(m,R), and we recall from Proposition 2.5 that one can construct the Whitney sum 0 .* * 0 5 and product 5 0 .. . 0 5 by applying the morphisms (re,@@)and (T@,@@) to E' 5 X ; these bundles have structure groups GL(mp,R), GL(mP,R) and fibers R" 0 . . .0 R", R" 0 * . 0 R", respectively. We now modify E' 5 X for another purpose. Let GL(m,R) itself act on the left of R" x * . . x R", with g(.x,. . . . , x,) = (yx,, . . . , gx,) for every g E GL(rn,R) and (x,, . . . , x,) E R" x . . . x R". If the original coordinate bundle E X is defined with respect to a covering *J (U,li E 1 ; and transition functions U in U j r GL(rn,R), one can then 1"' construct a new coordinate bundle E" X with respect to [ Uil i E 11, with fiber R" x . . . x R" and structure group GL(rn,R): one uses the same transition functions $!and the preceding action of GL(m,R) on R" x . . . x R". Now let R " x ~ ~ ~ x [ W "be~ the ~ map R m carrying ( x l , . . . , x p ) ~ R" x . . . x R" into the exterior product x 1 A . . . A x p E A"", and let GL(m,R) 5 GL(();, R) be the group homomorphism carrying g E GL(rn,R) I I X " ' X R
1

161
10. Orientable Vector Bundles
into that element of GI,((;), R) with value gx, A * * * A g x p on any x, A . . . A xp E R". The pair (I,@) is a morphism of transformation groups in the sense of Definition 11.2.6, so that by Proposition 11.2.7 one can apply (I, @) to the coordinate bundle E" X to obtain an (;)plane bundle over X , which is independent of the coordinate bundle E 3 X chosen to represent the mplane bundle (.
Ap
10.1 Definition: Given an mplane bundle 5 over X is the preceding (;)plane bundle over X . power
E 3, the
pth exterior
If nr = p, then A" 5 is a line bundle over X , and since gx, A . * * A gx, = (detg)x, A ' . ' A X , for any (xl, . . . , x,) E R" x * * * x R" the group homomorphism GL(m,R) f, GL(1, R) merely carries g E GL(m,R) into its determinant det g, acting via scalar multiplication on A" R". Let (A"Rm)* c A" R" consist of the nonzero elements y E A" R", observe that (A"Rm)* is preserved under the action of GL(1, R), and let be the equivalence relation in (A" Rm)*with y' y if and only if y' = uy for some a > 0. The canonical surjection (A"Rm)*.%(A" R")*/ maps (A" Rm)*onto a space (A" R")*/ with just two elements, and if GL '( 1, R) c GL(1, R) is the subgroup consisting of multiplications by positive real numbers the canonical epimorphism GL(1, R) 5 GL(1, R)/GL+(l,R) has image 2/2, which acts on (A"R")*/ by interchanging the two elements. The pair (I?,@') is another morphism of transformation groups in the sense of Definition 11.2.6, carrying the transformation group GL(1, R) x (A"Rm)*+ (A" Rm)*into the transformation group 2/2 x (A"Rm)*/ + (A" Rim)*/.


10.2 Definition: Let 4 be any real mplane bundle over any X
E
3, and let
(A"'t)*be the fiber bundle with fiber (A" Rm)*and structure group GL(1, R), obtained from the mth exterior power A"( by removing the zerosection. The orientation bundle o(4) over X is induced by applying the preceding morphism (l', 0') of transformation groups to (A" 4)*, as in Definition 11.2.8. There is another way of describing o(t), which we sketch. According to the linear reduction theorem (Theorem 11.6.13) one can reduce the structure group GL(m,R) of ( to the orthogonal subgroup O(m) c GL(m,R), and Proposition 3.4 provides a riemannian metric ( , ) for any coordinate bundle representing 5. One easily alters ( , ), if necessary, in such a way that the action of O(m) on each fiber E x preserves the inner product Ex x E, Iw; in fact, this is automatically the case if one uses ( , ) itself to reduce GL(m,R) to O ( m )c GL(m,R), as suggested at the end of $3. Hence one can replace GL(m,R) by O(m),and E by the subspace of fibers of
162
111. Vector Bundles
unit length, to obtain the sphere bundle associated to ~ 0,, Eand~ R ~ , =fly, . . . t"y, E R". The subspace Em*of nonzero fibers in ,? is then the quotient (RZ)"'* x R m * / z ' ,for the same equivalence relation z'.
six, + . . . + smx,
+
+
2Gm(R") represent the universal 10.9 Proposition : For any m > 1 let oriented mplane bundle y"', and let EU* be the space of nonzero jibers in E x . Then there is a homotopy equivalence @''(R") Em*such that the composition &  ' ( R x ) ; Em*5 @""R") classijies the oriented Whitney sum j j r n  l 0 e 1 ouer @  I ( R 1. ~ PROOF: The proof is virtually identical to that of Proposition 8.10, using the preceding description of E x % @'"'R") in place of the corresponding nr description of E" Gm(RZ). (One also uses the property that if m is even, then c1 0 ym' and y"' 0 1.:' have opposite orientations.)


Let 5 be any real mplane bundle over any X E g,as before. The canonical n(5) involution of the double covering O(5) X is the welldefined homeomorphism O(5) AO( 0 and n > 0 let @'"'R"+") be the total space O(y;) of the orientation bundle o(yr) of the canonical real mplane bundle y r over the Grassmann manifold Gm(Rm+");that is, @'(Rmtn) ,(YIP) Gm(Rm+")represents o(y;). The pullback n(y;)!y; over @'(R'"+") is the

canonical oriented mplane bundle 7;. Since Gm(Rm'") is a smooth closed mndimensional manifold, by Proposition 1.7.3, the same is true of the double covering @""R"""). Furthermore, the bundle over @""Rm'") is canonically oriented by Proposition 10.6.
jjr
10.12 Proposition: Let X E &? be homotopy equivalent either to a compact space or to a finitedimensional simplicia1 space, and let 5 be any real mplane bundle over X . Then 5 iqorientable if and only if 5 = for some finite n > 0 and some map X @""R"""). I f X is a smooth mangold, then 5 is orientable i j and only if a smooth such exists.
f'jjr
7
PROOF: By Propositions 8.12,9.1, or 9.3 one has 5 = f ' y ; for an appropriate map X G"( R"' "), and one substitutes R" " for R" throughout the proof of Proposition 10.8. 10.13 Proposition: 1.f m and n are both odd natural numbers, the canonical involution @'(R'"+tl) A @"''R'''~") is homotopic to the identity.
+
PROOF: Since m n is even, one can substitute Rm+"for R" in the proof of Proposition 10.10.
167
10. Orientable Vector Bundles
10.14 Corollary: For any odd m and any n > 0 the canonical real mplane bundle $' over G'"(R'"+'') is nonorientuble. PROOF: The inclusion Rm+', Rm+ninduces a map Gm(Rm+')5 G'"(R'"+") for which 7'; = f ' y ; , so that nonorientability of 7'; implies nonorientability of yr, by Proposition 10.5. Hence it suffices to consider only the case that n is the odd number 1, for which Proposition 10.13 guarantees that 6'"(Rm+')1,6'"(R'"+') is homotopic to the identity. If 7'; were orientable, then since the double covering Gm(R'"+')of Gm(Rm+l)is the total space O(y;") of the orientation bundle o(y';),it would consist of two disjoint copies of G'"(R'"+ I ) , and z would interchange the two copies of @(R'"+'); however, such a map T would not be homotopic to the identity. '
Since trivial vector bundles are clearly orientable, Corollary 10.14 incidentally guarantees that the bundles 7; are not trivial when m is odd. The simplest case is the canonical real line bundle 7 : over RP', the base space RP' being diffeomorphic to the circle S'. We now develop a more concrete characterization of orientability. Let So be the 0sphere { + 1,  I ) , and let the orthogonal group O(1) act on So as usual, via multiplication by + 1 or  1. Any coordinate bundle E 5 X with fiber So and structure group 0(1)over a space X E 9?is a double covering of X in the sense defined earlier; in fact, since 0(1) c GL(1, R), one easily constructs a real line bundle over X such that E .5 X is precisely the double covering ~ (3 i ) X.
10.15 Lemma: Let E 5 X be u double covering of a space X E 9, a coordinate bundle with respect to some open covering { Uil i E I ) of X . Then E X JI' is trivial $and only fi there is a jumily of transition functions U in U j 4O(1) each of which has the constant value 1 E O(1). PROOF: Clearly E 5 X is trivial if and only if there is a section X 5 E . Let {'PiI i E I ) be a family of local trivializations E I U i5 U ix So. If a section 0 exists, then each composition U i El Lri 3 U i x So maps each x E U iinto (x, 1) E U ix So, and one can alter the local trivializations in such a way that ('Pi 01 U i ) ( x )= (x, + 1) for each i E I and each x E Ui. The transition functions 'Pi are defined by the requirement that the compositions Y , Yx' + U i n U j x So maps (x, I) into (x, $i(x)( f 1)); U i n U j x So hence, using the altered local trivialization 'Pi, one has $i(x) = 1 E O(1) for x E U in U j , as required. Conversely, if the latter conditions are satisfied, then there is a unique section X 5 E such that for each i E I the composition Y i 01 U i carries each x E U iinto (x, + 1)E U ix So.
a
0
168
111. Vector Bundles
The following property of orientable vector bundles is frequently used as the definition of orientability, where GL+(rn,R) c GL(rn,R) is the subgroup of elements with positive determinants, as usual. 10.16 Proposition: A real rnplane bundle 5 over any X E a is orientable
if and only if the structure group GL(m,R) can be reduced to the subgroup GL+(rn,R) c GL(rn,R).
PROOF: Let E 4 X be a coordinate bundle representing 5, with respect to some open covering { Ui I i E I } of X . Then according to the discussion following Definition 11.5.1 we must show that 5 is orientable if and only if there is a family of transition functions U i n U j 3GL(rn,R) whose values all lie in the subgroup GL+(rn,R) c GL(rn,R). The coordinate bundle E 1X induces a double covering E' 5 X that represents the orientation bundle o(5) with respect to the same covering { Uil i E I } . Specifically, for each $! the corresponding transition function U i n U j *," O( 1) for E' 5 X is given by setting $ i j = det ${/ldet $!I. By Lemma 10.15 the orientation bundle o(5) is trivial if and only if there are transition functions $I' for E' 5 X such that $Ij(x) = 1 E O(1) for each x E U i n U,. Hence the bundle 5 is orientable if and only if there are transition functions $i for E 5 X such that det $i(x) > 0 for each x E U i n U,, as claimed.

10.17 Corollary: A real rnplane bundle 5 over any X E G3 is orientable i f and only if the structure group can be reduced to the rotation subgroup O'(rn) c GL(rn, R).
PROOF:By the linear reduction theorem (Theorem 11.6.13) the structure group GLf(rn,R) can always be reduced to the subgroup O'(rn) c GL+(rn,R). 10.18 Corollary: Let 5 be any real mplane bundle over any base space X E 39;then the Whitney sum 5 0 5 is orientable.
PROOF: If 5 is represented by a coordinate bundle using some open covering {U,l i E I } of X and transition functions Ui n U j %GL(rn,R), then 4 0 5 is also represented by a coordinate bundle using the same covering ( U i l i E I } and transition functions U i n U j *:W1 GL(2m,R); however, det(${ 0 $!) = (det $!)' > 0 over Ui n U j .

One can easily strengthen Corollary 10.18:5 0 5 has a natural orientation, which will appear just before Proposition 12.8.
169
11. Complex Vector Bundles
10.19 Corollary: Let 5 and 5‘ he real vector bundles over anj’ base space X E d,and suppose that 5 is orientable; then 5‘ is orientable i f and only i f 5 05’ is orientable.
PROOF: If 5 and 5’ are represented by coordinate bundles using open coverings jUili E I } and ( U , l j ~5 ) of X , respectively, then they can both be represented by coordinate bundles using the common open covering ( U i n U j 1 ( i , j ) E I x 5)ofX;thelattercoveringwillbedenoted ( U i l i € I J f o r convenience. By hypothesis, the transition functions Ui n U j A GL(m,R) for the representation of 5 can be chosen in such a way that det $; > 0 over Ui n U j . If U in U j ‘ ; I GL(n,R) are transition functions for the representation of C’, then U in U j GL(m n, R) are transition functions for a representation of 4 05’; however, det(${ 0 1+9ij)= (det $i)(det $iJ), where det Ic/{ > 0, over Ui n U j . ‘J

+
The Mobius band was introduced for motivation at the beginning of this section. We now sketch the proof that its tangent bundle is indeed nonorientable. First, recall that the Mobius band was described in 811.0 as the total space E of a coordinate bundle E 5 S ’ ( = R P ’ ) whose fiber is the closed interval [  1, + 11 c R and whose structure group is 2/2, acting on [  1, + 11 via multiplication by 1 or  1 ; that is, 2/2 = O(1) c GL(1,R). It is clear that E 5 RP’ is merely a restriction ofa coordinate bundle E‘ 5 RP’ representing the canonical line bundle 1,: over RP’. In any event, one can pull the tangent bundle t ( E ) or r(E’) back along the zerosection RP’ + E c E‘ to obtain a 2plane bundle & ( E ) over RP’, and a direct computation shows that & ( E ) = 7 ; 0 E ’ for the trivial line bundle E ’ . However, y i is nonorientable by Corollary 10.14, so that a!t(E) is nonorientable by Corollary 10.19, so that T ( E )is nonorientable by Proposition 10.5, as claimed.
+
1 1. Complex Vector Bundles One can replace the real field R by the complex field C throughout the development of vector bundles, with virtually no other changes. The replacement is more than an idle exercise, however, even if one is only interested in real vector bundles. For example, “complexifications” of real vector bundles will be used in Volume 2 to compute the real cohomology rings H*(Gm(R’X ); R) of real Grassmann manifolds; Theorem 8.9 suggests
I70
111. Vector Bundles
the real importance of such cohomology rings. Complex vector bundles are also used in the very definition of Ktheory, in Volume 3, which is used to solve the classical problem of real vector fields on spheres. However, complex vector bundles are of geometric interest in their own right, especially if one studies complex manifolds by looking at their tangent bundles. As always, the base space of a fiber bundle belongs to the category 9 of base spaces.
11.1 Definition: A complex vector bundle of rank n, or simply a complex nplane bundle, is any fiber bundle whose fiber is the complex vector space C"and whose structure group is the general linear group GL(n, C)of invertible n x n matrices, acting in the usual way on C".A complex line bundle is a complex vector bundle of rank 1. The obvious complex analog of Definition 2.4 or Proposition 2.5 provides the Whitney sum i0 i'and product i0 i'of complex vector bundles iand [' over the same base space, with the associative, commutative, and distributive properties described for real vector bundles in Proposition 2.6. If one substitutes hermitian inner products C" x @" ( , ) C for real inner products R" x R" R in Definition 3.1, then the result is a hermitian metric for a given complex nplane bundle i.The obvious complex analog of Proposition 3.4 then guarantees that any complex vector bundle (over a base space X E .9d, as always) has a hermitian metric. Consequently there is a complex analog of Proposition 3.6: for any complex subbundle i' of a given complex vector bundle ( there is another subbundle 1' of isuch that [ = i'0 1'. One can also use hermitian metrics, as in the real case, to show that the structure group GL(n, C)of any complex nplane bundle over any X E $28 can be reduced to the unitary subgroup U(n)c GL(n, C);however, since this is a special case of the linear reduction theorem (Theorem II.6.13), the details will be left as an exercise (Exercise 13.19). Definitions 7.2 and 8.2 have obvious complex analogs, which provide the canonical complex mplane bundle y r over the Grassmann manifold Gm(Cm+") and the universal complex mplane bundle y" over the Grassmann manifold Gm(@ a'), respectively; in particular there is a canonical complex line bundle 7 ; over the projective space CP", and a universal complex line bundle y' over the projective space CP'". We already know from Proposition 1.7.5 that Gm(Cmf") E .9d and Cm(Cm)E g.


11.2 Theorem (Homotopy Classification Theorem): Any complex mplane bundle iover a base space X E $28 is a pullback f'y" of the universal complex
171
1 1 . Complex Vector Bundles
mplune bundle 7"' ouer Gm(@" ) up to homotopj,.
E
4, dong a mup X
A G m ( @ "thut ) is unique
PROOF: Substitute C for R throughout the proof of Theorem 8.9.
n"
11.3 Proposition: For any n > 1 let E' G"(C' ) represent the universal complex nplane bundle y", and let E" * he the space of' nonzero Jihers in E x . Then there i s u homotopji equivulence G n  ' ( C z )+! E m* such that the comn= G"(C7) classijes the Whitney sum y"' 0 E' position G"'(@') A E'* over G" I ( C ' ).

PROOF: Substitute C for R (and n for m ) throughout the proof of Proposition 8.10. 11.4 Proposition: I f E" 2CP" represents the universal complex line bundle ill, then the space EL* of nonzero ,fibers in E" is homeomorphic to the contructihle space C"*. If' E : CP" represents the canonical complex line bundle if:,f;w u given n > 0, then the space E* of' nonzero Jibers in E is homeomorphic to @'"+ I ) * , trivially homotopy equivalent to the (2n 1)sphere S2"+
+
'.
PROOF: As in Proposition 8.11, E x * is a quotient C X * x C*/%,with (x, s) % ( J:t ) if and only if sx = t y E @" *; similarly E* is a quotient C("+ I)* x @*/z,with (x,s) = ( y , t ) if and only if sx = t y E V+')*. The proof that @' * is contractible follows the pattern of the corresponding proof for Rz*, given in Proposition 8.1 1. The homotopy classification theorem for complex vector bundles has finite analogs, just as in the real case. Here are some corresponding complex uniqueness and existence results, in that order. 11.5 Proposition (Ersatz Homotopy Uniqueness Theorem): Let i be a complex mplune bundle over X E d such that [ = j'ay'," and i= f'iv',"' jbr mups X 2 G m ( @ m +und " ) X 2Gm(Cm+"'), where r'," and :y are canonicul complex mplane bundles. Then there is an n" >= max(n, n') with Jinite classi~ m ( ~ m + n "Llnd ) ~ m ( c m + n ' ) ~ m ( ~ m + ~ ~ ,jj;llgexterlsiorls ~ m ( , n l + n ) such thut gn,n.. f b and g,I.,n.. j1 ure homotopic maps from X to Gm(Cm+""), fiir which (y,.,.. " j0)'y;. = i= (g,,,,,. ' f J y : . . #:
PROOF: Substitute C for R throughout the proof of Theorem 8.14. 11.6 Proposition: Let X E d be u compact space, a .finitedimensionul simpliciul spucv, or u smooth manijold, and let i be u complex mplane bundle
" )
172
111. Vector Bundles
=
over X . Then there is a natural number n 0 such that [ i s a pullback .f '7: of' the canonical complex mplane bundle j$' over Gm(Cmfn) along a map X Gm(C"'+n).
PROOF: Substitute C for R throughout the proofs of Propositions 8.12,9.1, or 9.3, respectively. 11.7 Corollary: Let X E 98 be a compact space, a ,finitedimensional simplicial sptrce, or u smooth manifold, and let [ be a complex mplane bundle over X . Thenjor some n > 0 there is a complex nplane bundle [' over X such thut the Whitney sum ( 0 [' is the trivial complex bundle d'"'"over X .
PROOF: This is the complex analog of Corollaries 8.13, 9.2, or 9.4, respectively. 11.8 Theorem: Any complex mplane bundle [ over a smooth manfold X can be represented by a smooth coordinate bundle E X ; that is, any complex vector bundle over X is itself smooth.
PROOF: By Proposition 11.6, [ is a pullback f ' ! f of the canonical complex mplane bundle y: along some map X Gm(Cm+"), and by Proposition 1.6.19 one can choose f to be a smooth map. One then substitutes C for R throughout the proofs of Lemma 7.1 and Proposition 5.3 to conclude that 7: and f ' y : are smooth.
12. Realifications and Complexifications For any complex nplane bundle [ over a base space X E 9fthere is a corresponding oriented 2nplane bundle laover X ; one can also ignore the orientation of la and simply regard it as a real 2nplane bundle over X . Similarly, for any real mplane bundle 5 over a base space X E there is a corresponding complex mplane bundle ta:over X . The constructions ( )Iw and ( )a: are studied in this section. Recall from Definition 11.2.6 that a morphism from a transformation of group G x F + F to a transformation group G' x F' + F' is a pair (r,@) 0 maps G I ,G' and F ,F' such that is a group homomorphism and the obvious diagram commutes. For any fiber bundle [ with structure group G and fiber F over a space X , the morphism (r,@) induces a new fiber bundle [' with structure group G' and fiber F' over the same space X , as in Proposition 11.2.7 and Definition 11.2.8.
I73
12. Realifications and Complexifications
Let G = GL(n,C).and let r assign to any (h4, + ic4,) E GL(n, C)the real 2n x 2n matrix consisting of 2 x 2 blocks
(:;3.
n.
where bz and cz are real and i = Similarly let F = C", and let C"5 R2" carry column vectors with pth entry x p + iyp into column vectors whose (2p  l)th and (2p)th entries are x P and y p , respectively, where p = 1, . . . , n. One easily verifies that (F,@)is a morphism from the transformation group GL(n,C)x C"+ @" to the transformation group GL(2n, R) x R2" t R2". 12.1 Definition: For any complex nplane bundle [ over a base space X E J the retilijicution iR is the real 2nplane bundle over X induced by the preceding morphism (r,0). Let I E GL(n, C) be the identity element, so that il E GL(n, C)is scalar multiplication by i = E C. Since (il)' =  I E GL(n, C), the element J = F(i1)E GL(2n, R) satisfies J 2 =  I E GL(2n, R); furthermore J commutes with every element in the image of GL(n, C) 5 GL(2n,R).
J1
12.2 Definition: Let E 5 X be a coordinate bundle that represents a real mplane bundle 5 over X E B. A vector bundle morphism E
J
is a comples structure in E 5 X whenever the restriction E x A E x over each .Y E X satisfies J t =  I for the identity element I E GL(m, R). Clearly a complex structure in one coordinate bundle representing 5 induces a complex structure in any other coordinate bundle representing 5, so that one can regard a complex structure as a structure in 5 itself. Equally clearly, the endomorphism J = T ( i I )E GL(2n, R) induces a complex structure in the realification is of the complex nplane bundle i.We shall show that a real vector bundle has a complex structure if and only if it is of the form iiw for a complex vector bundle i;furthermore, iis unique. 12.3 Lemma: I f J E GL(m, R) suris$es J 2 =  I E GL(m,R), then m is an eiien number 2n und there is a bcisis of R2" of' the form (e,, J e , ; . . . ; e,, Je,).
PROOF: For any nonzero e l E R" suppose there were a linear relation J e , = (A* + l ) r , = ( J 2 + I ) e , = 0. so that I' + 1 = 0, which is
i e , over R; then
1 74
111. Vector Bundles
not possible in R. Hence ( e , , J e , ) spans a 2dimensional subspace I/ c R", and J induces an endomorphism J of Rm/Vsuch that J 2 =  I . The induction on dimension is clear. It follows from Lemma 12.3 that up to a change of basis in Rz" one has J = T ( i I )for the homomorphism GL(n,C)5 GL(2n, R). We shall henceforth use the basis of Rz" described in the proof of Lemma 12.3. 12.4 Lemma: I f A E GL(2n, R) satisfies AJ = J A fix J = T ( i I ) , then A lies in the image o f G L ( n , C)5 GL(2n,R).
PROOF: For the basis ( e , , J e , ; .. . ; e,,Je,) of R2" there are unique real t i x I I matrices B = (b4,) and C = (c4p) such that Ae, = ~ ~ , , ( b 4 , r GJe,,) q for p = 1 , . . . , n, and since AJ = J A , one also has A(Jr,) = J A e , = ,(  tie,, h4,JeJ; hence A = T(B iC).
+
+
+
12.5 Proposition: A real vector bundle 5 has a complex structure J if' urzd only if' it is the reulijication iw of a complex vector bundle [;furthermore iis unique.
cw,
PROOF: If 4 = then T ( i I ) is a complex structure J in 5. Conversely, suppose that J is a complex structure in 4, and let ( U i l i E I ) be an open covering of the base space X E 9 of 4 for which there are trivializations E I U i 3 U i x R" of a representation E 5 X of 4. The restriction of J to E I U iinduces a map U i 2 GL(m,R) such that (Ti J
c
'€';')(x,e) = (x, Ji(.x)e)
E
Ui x
[w"
for every(x, e) E U i x R", and since J 2 =  I , one has J,(x)' =  I E GL(m,R) for every x E U i . It follows as in Lemma 12.3 that m is an even number 2n and that there is a basis (el,Jie,; . . . ; e,,Jie,) of sections of El U i 5 U i , for each i E 1. With respect to any such basis the matrix representation ( J i ( x ) )E GL(2n, R) consists of n blocks
down the main diagonal, with zeros elsewhere; in particular, (Ji)= ( J j ) over any nonvoid intersection U i n U j . The transition function U in U j +*1 GL(2n, R) is given by ( Y j TI: ')(x,e)
for (x, e) E (uin uj)x R2",
= (x, Il/i(x)e)
so that the identity
( T j y; 1 ) (yi,, I>
I,
J
c.
yt: 1)
= yj0 J
n
TI:1
= (yj 'I J
8'
y,: 1) . ( y jr y;
1)
175
12. Realifications and Complexifications
implies ( $ i ) ( J i )= (Jj)($fE GL(2rn,R). Since (Ji)= ( J j ) and (Ji)' =  ( I ) , Lemma 12.4 then implies that ($!) lies in the image of GL(n,C) 5 GL(2n, R). The uniqueness assertion is an easy exercise. Briefly, Proposition 12.5 asserts that any complex vector bundle i can equally well be regarded as a real vector bundle with a complex structure J . By Proposition 11.6.19, GL(n,C) and GL(2n, R) consist of one and two r components, respectively, and since the homomorphism GL(n,C) GL(2n,R) necessarily carries the neutral element into the neutral element, it follows that the image of lies in the component GL+(2n,R) c GL(2n,R). Hence Proposition 10.16 guarantees that any realification irW is orientable. In fact, there is a nuturul orientatiori X 5 O(i,) given by letting the image of each s E X be the equivalence class of e l A J r , A e, A J e , A . . . A e, A Je,, E (l\"'E,)* for any representation E X of ilW and any linearly independent elements r , , . . . , el, E E,. According to the discussion preceding Proposition 10.10,if a real mplane bundle 4 over X E A¶ has an orientation X %O( i R. Thus a ma1 with respect to the inner product (x) x (R" x R") given basis (sl. . . . ,s,) of local sections U j + E provides a new basis (s',, . . . , s;,) of local sections Ui + E , with sb = ;isq for a map U j 4 GL(m,R) carrying each x E U i into a triangular matrix in GL(m, R). If Ui x R" U i x R" carries each (x, e ) E U i x iw" into (x, iLi(x)e) E U j x R", Y A; I the composition El U i4 U ix R" U i x R" is a new local trivialization E I U i 2 Ui x R". There are then new transition functions U in U,i GL(m,R), defined by requiring (Yj '4' ')(x,e ) = (x, $ii(x)e) for every (x, e ) E U in Ui x R", and one easily verifies that $:J(x)E O(m)for every .Y E U j n Ui. Hence the structure group GL(m, R) of 5 can be reduced to the subgroup O(m) c GL(m, R), as claimed.
3
CT=l

r
13.18 Exercise: Carry out the verification required to complete the proof of the preceding reduction theorem. 13.19 Exercise: The linear reduction theorem (Theorem 11.6.13) implies that the structure group GL(n,C)of any complex nplane bundle 5 over any base space X E J can be reduced to the unitary group U ( n ) c GL(n,C). Prove the same result by imposing a hermitian metric on and following the pattern of Remark 13.17 and Exercise 13.18.
13.20 Remark: The C"(X)module €(X) of differentials on a smooth manifold X was described in Definition 6.1 1 as the dual of the C"(X)module 6*(X) of smooth vector fields on X : however, one can also construct & ( X ) directly. The ring Cr>(Xx X ) of smooth functions X x X 5 R is an algebra over the ring C " (X) of smooth functions X A R, with f F E C Y X x X ) defined by setting ( f P ) ( x , y )= f ( x ) F ( x , y )E R for any (x,y) E X x X. Let J c C x ( X x X ) be the ideal of those F E C 7 ( X x X ) that vanish on the diagonal A(X) c X x X , and let €(X) be the quotient C"(X)module J / J 2 . There is a real linear map C m ( X , ) J carrying any f' E Cm(X)into the function with valuef( y )  f ( x ) E R on (x,y ) E X x X , and there is an induced real linear map C" (X) 5 € ( X ) that satisfies the classical product rule d(fy) = f 4 + Ydf.
184
111. Vector Bundles
A smooth map Y X induces an algebra homomorphism C x ( X x X) @* C * ( Y x Y), which in turn induces a module homomorphism A(X)C ’ ( Y), where 8 ( Y ) over the induced ring homomorphism Cz(X) @*f = f @ for any f E P ( X ) , so that @* df = d(@*f). +
0
13.21 Exercise: Verify that the C“(X)module d(X) of Remark 13.20 is canonically isomorphic to the C“(X)module &(X)of Definition 6.1 1. 13.22 Exercise: Verify for any smooth map Y 5 X that the induced homomorphism 8(X) 3 d‘( Y ) of Remark 13.20 agrees up to canonical isomorphism with the homomorphism d‘(X) €( Y) constructed following Proposition 6.14. 13.23 Remark: There is a severe penalty for replacing Cz(X x X) by the subalgebra C r ( X )@ P ( X ) c Cm(Xx X) in the constructions of Remark 13.20. Some of the resulting pathology is described in Osborn [3]. 13.24 Exercise: Let E 5 X and E‘ 5 X represent real mplane bundles over the same X E B, and suppose that they are described by transition functions U in U j + GL(m,R) whose values are transposed inverses of each other. Show that E 5 X and E’ % X represent the same real vector bundle. 13.25 Exercise : Replace “real” by “complex” in the preceding exercise, using conjugate transposed inverses. Show that the given coordinate bundles represent complex conjugate mplane bundles. 13.26 Remark: A smooth ndimensional manifold X is parallelizuble whenever the tangent bundle z(X) is the trivial bundle E” over X. For example, any Lie group G is parallelizable: the leftinvariant vector fields provide a basis of sections of z(G). The spheres S’, S3, and S7 are also parallelizable; in fact S’ and S3 are the underlying manifolds of the Lie groups U(1) and U(2), respectively. However, there are no other parallelizable spheres, a result of Bott and Milnor [l] and Milnor [4], which was later simplified by Atiyah and Hirzebruch [3]; a proof will be given in Volume 3. Products of spheres are better behaved. One result of Kervaire [I] is that S P x Sq is parallelizable whenever at least one of the numbers p > 0 or q > 0 is odd. Staples [ 11 gives a simpler proof of the same result. A classical result of Stiefel [11 asserts that every orientable 3dimensional manifold is parallelizable; more generally, Dupont [3] shows that every orientable (4k + 3)dimensional manifold has at least three linearly inde
185
13. Remarks and Exercises
pendent vector fields. Dediu [ I , 2,3] obtains similar results for (4k + 3)dimensional lens spaces, k 1 0 ; lens spaces are defined in Remark 1.10.20. Since parallelizability of the real projective space RP" would imply parallelizability of the corresponding sphere S", it follows that R P ' , R P 3 , and RP' are the only parallelizable real projective spaces. The question concerning more general real Grassmann manifolds is apparently still open.
+
Recall the identity T ( R P " 0 ) c 1 = (n I)?! of Proposition 7.4, where RP" is the Grassmann manifold G'(R"+l) and y: is the canonical real line bundle over G'(R"+'). Show more generally that 13.27 Exercise:
T ( G " ( R ~ + " )0 ) (1): 01.1:)
= (m
+ n)y:,
where is the canonical mplane bundle over the Grassmann manifold G"( R"' "). :);
+
13.28 Remark: The preceding exercise is not entirely trivial; its solution can be found in Hsiang and Szczarba [11, along with corresponding results for the complex and quaternionic cases. In the latter cases the summand 0j: is replaced by 7; By:, for the conjugate bundle 7;. Different generalizations of these results are given in Bore1 and Hirzebruch [l] and in Lam [4]. ; ! :
13.29 Remark: Whitney sums my! of m copies of the canonical real line bundle 7: over RP" serve other useful purposes. For example, the immersion problem for real projective spaces RP" is equivalent to the problem of finding the largest numbers of linearly independent sections of my: for all m > 0 and I I > 0. The immersion problem for real projective spaces is approached from this point of view in Lam [3] and Yoshida [l, 21. (Recent catalogs of bestpossible immersions RP" + R2"k can be found in Gitler [I], James [ 11, and Berrick [ 11.)
If a complex vector bundle i over a polyhedron IKI has a IKI finite structure group G c GL(n,C),then there is a finite covering X such that the pullback j'! 4 there is a complex 2plane bundle over CP" that cannot be represented by a holomorphic coordinate bun d 1e. Here is an apparently narrower problem: can a (continuous) complex vector bundle over a complex algebraic variety be represented by a coordinate bundle which is algebraic in the obvious sense? A basic result of Serre [3] implies that this question reduces to the preceding one: a complex vector bundle over a complex algebraic variety is algebraic if and only if it is holomorphic. 13.34 Remark: If E 5 X represents a real mplane bundle 5, then n is itself a homotopy equivalence; hence the fiber homotopy equivalence relation of Remark 11.8.21 is of little direct interest. However, if E, 2 X represents the corresponding (m  1)sphere bundle tS,described in Remark 13.3. the projection n, is no longer a homotopy equivalence. Accordingly, two real vector bundles 5 and v] over the same base space are dejined to be jiber homotopy equivalent (or of the same jiber homotopy type) whenever the state
187
13. Remarks and Exercises
ment is true of the corresponding sphere bundles 5, and q s ; fiber homotopy equivalence of complex vector bundles is described in terms of their realifications. It is clear that one can equally well consider fiber homotopy equivalence of vector bundles over different base spaces, provided the base spaces themselves are homotopy equivalent in the usual sense. The first major application of fiber homotopy equivalence coincided with its very definition in Thom [4]: t h e j b e r homotopy type of the tangent bundle 7 ( X ) of a smooth closed manijbld X is independent of the smooth structure assigned to X . A stronger version of Thom's result occurs in Benlian and Wagoner [ 1) : ij' two smooth closed manifolds are homotopy equivalent, then their tangent bundles are j b e r homotopy equivalent; a simplified proof of this statement is given in Dupont [2]. Incidentally, Benlian and Wagoner also show that if the given homotopy equivalent manifolds are ndimensional, and if one of the manifolds has k linearly independent vector fields for some k S ( n  1)/2, then the other manifold also has k linearly independent vector fields. Fiber homotopy equivalence provides a natural setting for other results. Recall that parallelizability of Lie groups was easily established in Remark 13.26, and that Hspaces are natural generalizations of Lie groups. According to Kaminker [ I], the tangent bundle of any (smooth) Hspace is fiber homotopy equivalent to a trivial bundle; this is clearly the natural generalization of parallelizability.
13.35 Remark: In general one cannot strengthen the preceding remark to conclude that a homotopy equivalence X & X ' pulls the tangent bundle T ( X ' ) back to the tangent bundle 7 ( X ) . However, one does have f ! z ( X ' )= 7 ( X )in certain special cases considered in Shiraiwa [l] and Ishimoto [l]. 13.36 Remark: Two vector bundles 4 and q over the same base space are stably equivalent if 4 0E P = q 0 for trivial vector bundles E~ and c4. For example, Proposition 7.4 asserts that the ( n 1)fold Whitney sum (n + I)yi of the real canonical line bundle y,! over RP" is stably equivalent to the tanX ' is a homotopy gent bundle 7 ( R P " ) .In Shiraiwa [2] one learns that if X equivalence of closed evendimensional smooth manifolds such that f ! 7 ( X ' ) is stably equivalent to 7 ( X ) , then the stronger conclusion f ! z ( X ' )= 7 ( X )is valid. There are natural homotopy classification theorems for stable equivalence classes of vector bundles. For each m > 0, let G"(R') + G m + ' ( R ' ) classify the Whitney sum 7" 0I;' of the universal real mplane bundle y"' and the trivial line bundle E ' over Gm(R").In the notation of Remark 13.4, this is a map BO(m) 4 BO(m + l), and one can form the inductive limit
+
188
Ill. Vector Bundles
BO = lim, BO(m). Clearly the stable equivalence classes of real vector bundles over any X E d are classified by homotopy classes of maps X + €30. Similarly, the stable equivalence classes of complex vector bundles over any X E d are classified by homotopy classes of maps X + BU = limffiB U ( m ) , where BU(m) is also defined in Remark 13.4. Stable equivalence classes will be studied in more detail in Volume 3 : they are the very essence of Ktheory. However, representations of such classes are of independent interest. For example, Fossum [13 shows that every real or complex vector bundle over any sphere S" is stably equivalent to a vector bundle represented by an algebraic coordinate bundle. In the special case n = 4k > 16 Barratt and Mahowald [l] show that any real vector bundle over S4kis either stably trivial or stably equivalent to a bundle of rank 2k 1 that is irreducible in the sense that it is not a Whitney sum of nontrivial bundles of lower rank; an alternative proof of the same result appears in Mahowald [l]. Glover, Homer, and Stong [l] show for any k > 0 that the tangent bundle r ( C P Z kof ) the complex projective space C P Z k is similarly irreducible, as a complex vector bundle; this partially strengthens a result of Tango [l], that for unj / I > 2, there is an irreducible complex vector bundle of rank n  1 over CP".
+
13.37 Remark: Surfaces have special properties as base spaces. Cavenaugh [ 13 shows that every nontrivial real 2plane bundle over any orientable surface is irreducible in the sense of the preceding remark. Moore [ 13 sharpens the result of Fossum [ 13 as follows: every vector bundle over the 2sphere Sz can itself be represented by an algebraic coordinate bundle. (The latter result is clearly related to the result of Lernsted [l], cited in Remark 13.6, that any vector bundle over any finite CW space is "algebraic" in a reasonable sense.) 13.38 Exercise: Let X be an rndimensional CW space as in 01.5, so that X has no ncells for any n > rn, and for some fixed n > rn let 5 be a real nplane bundle over X . Use induction on p = 1, . . . , rn and the fact that every map Sp' + S"' is homotopic to a constant map for p < n to show that 5 has a nowherevanishing section. (This is an easy exercise, which is carried out in Husemoller [l, p. 991, for example; it will also be done in detail in Volume 2 of the present work, for any rndimensional weak simplicia1 space
x.1 13.39 Exercise: Use Exercise 13.38 and Proposition 4.8 to conclude that if > H I , then any real nplane bundle 5 over an mdimensional CW space X is of the form q 0 I . : "  ~ for some real mplane bundle q over X . 11
189
13. Remarks and Exercises
13.40 Remark: The geometric dimension of a real vector bundle 5 is the least integer n such that 5 is stably equivalent to a real nplane bundle r ] ; in case 5 is stably trivial its geometric dimension is 0. For example, according to Exercise 13.39 the geometric dimension of any vector bundle over an mdimensional CW space is at most m. However, stronger results are possible in some cases: Remark 13.36 asserts for k > 4 that the geometric dimension of any real vector bundle over the sphere S4kis either 0 or 2k 1. There are other base spaces over which all real vector bundles have severely limited geometric dimensions; several examples can be found in Sjerve [l], Hill [l], and Davis and Mahowald [11.
+
13.41 Remark: Given an mplane bundle 5, a stable inverse of 5 is any nplane bundle r], over the same base space and for some n 2 0, such that 5 0r] = em+''. The existence of stable inverses of real mplane bundles over certain kinds of base spaces was established in Corollaries 8.13,9.2, and 9.4, and there were indications of upper bounds on n. In the following situation one can take n = m.
For any X E W and any point * E X the (reduced) suspension EX is the quotient of the product X x [0,1] by the subspace X x (0) u {*} x [0,1] u X x { 1), in the quotient topology. If X E W is a compact hausdorff space, and if 5 is a complex mplane bundle over EX, then there is a complex mplane bundle r] over EX such that 5 0 r] = cZm. A proof is given in Chan and Hoffman [ 13. 13.42 Remark: The equivalence relations of Remarks 13.34 and 13.36 lead to a useful weaker equivalence relation. Two vector bundles 5 and r] over the same base space are Jequivalent (or stably jiber homotopy equivalent) if there are trivial bundles c p and cq such that 5 0cP and q 0c4 are fiber homotopy equivalent in the sense of Remark 13.34. For example, the theorems of Benlian and Wagoner [l] described in Remark 13.35 first appeared as Jequivalence theorems in Atiyah [13 and Sutherland [11, respectively. The following Jequivalence theorem is due to Atiyah and Todd [l] and to Adams and Walker [l] with a later simplification in Lam [2]: a necessary and sufficient condition that the mfold Whitney sum my,' of the canonical complex line bundle y: over CP" be Jequivalent to a trivial bundle is that m be divisible by an integer i(m,n) defined in Atiyah and Todd [l]. An application of this result will be indicated in Remark VI.9.22. 13.43 Exercise: Let Y 5 E be the zerosection of a smooth coordinate bundle E 5 Y representing a real vector bundle r] over a smooth manifold Y. Show that & ( E ) = r] 0 T( Y ) for the homotopy equivalence r ~ .
190
111. Vector Bundles
13.44 Remark: Let X Y be the restriction of a smooth embedding X + R" of a smooth manifold X to an open tubular neighborhood Y c R" of f ( X ) c R", so that j is a homotopy equivalence. Then any vector bundle 4 over X is of the form f ' q for a vector bundle q over Y, and since T( Y ) = E" over Y the preceding exercise implies ((T f)!z(E) = 5 @ E" over X . Thus, up 0
to the homotopy equivalence (T f, any vector bundle 4 over X is stably equivalent to the tangent bundle of a smooth manifold E. 0
CHAPTER IV
Z/2 Euler Classes
0. Introduction Let H * ( X ; 2/2) be the singular 2/2 cohomology ring of a base space X E 9. In this chapter we assign a cohomology class e(() E H " ( X ; 2/2) to any real nplane bundle 5 over X.Such classes will be used in Chapter V to obtain further 2/2 classes w ( 5 ) = 1 + w l ( l ) + . . . + w,(() E H * ( X ; L/2) for 0. Furthermore, the beginning of the Gysin sequence is Y

HO(RP") 5 H O ( R ~ *A ) HO(RP~)
UP(:l')
.
H I ( R P = )+ 0,
where the initial homomorphism E* is the trivial isomorphism B/2 + B/2, so that we(?') is also an isomorphism for q = 0. Since H o ( R P ' ) = 2/2, this completes the proof.
.. .,
20 1
5. The Splitting Principle
A similar method applies to the 2/2 cohomology ring H*(RP") for any finite n > 0.
4.4 Proposition: For any n > 0 let y j be the canonical line bundle over RP", with 212 Euler cluss e(y:) E H ' ( R P " ) . Then the 212 cohomology ring H * ( R P " ) is a truncuted polynomial ring in a single variable over 212, the generator being e(y,l,)and the relation being e(yf)"+' = 0. PROOF: I f E: RP" represents it!, then, as in Proposition 111.8.1 1, E* is a quotient R'"+ "* x OX*/%, with (x,s) % (y,t ) if and only if sx = ty E R("+I ) * ; that is, E* = R("+')*, which is trivially homotopy equivalent to the sphere s". Since H4(S")= 0 for 0 < 4 < n, the argument used for Proposition 4.3 shows that Ho(RP") = 2 / 2 , that Hq(RP") H 4 + ' ( R P " ) is an isomorphism for 0 S 4 < n  1, and that H"'(RP")H"(RP") is a monomorphism; the latter conclusion implies that H"(RP") contains 2 / 2 as a submodule. Since RP" is an ndimensional CW space, by Proposition 1.5.5, one has HY(RP")= 0 for q > n. In particular, since H " + ' ( R P " ) = 0, the Gysin sequence terminates in the fashion H"(S")2H"(RP") + 0, and since H"(S") = 2/2, the monomorphism H "  ' ( R P " ) H"(RP") is in fact an isomorphism, which completes the proof.


4.5 Corollary: For any n > 0 let RP" A RP" classify the canonical line bundle y: over RP"; then for any p 2 n the induced 212module homomorphism H P ( R P 7 )A H P ( R P " )is an isomorphism carrying e(y')P into e(y:)". PROOF: H P ( R P " )is free on the single generator e(y')", and H P ( R P " )is free on the single generator e(y:)". Since y: = ,f!yl, it follows from the naturality of212 Euler classes (Proposition 3.2) that e(y:) = .f*e(y'), hence that e(y,!)P= j ' * e ( y ' ) P , as required. Observe that Propositions 4.3 and 4.4 establish the existence of nonzero 2/2 Euler classes, hence that the condition of Proposition 3.6 for the existence of nowherevanishing sections is nonvacuous. In particular, y' and y: admit no nowherevanishing sections; a fortiori, they are not trivial line bundles.
5. The Splitting Principle In general, an nplane bundle 5 over a base space X E 93 is not a Whitney sum of n line bundles. Nevertheless, one can always find a map X ' 4 X in the category 9 such that the pullback 9'4 over X' E B is a Whitney sum of n line bundles, and such that the induced homomorphism H * ( X ) 5 H * ( X ' ) of 2/2 cohomology rings is a monomorphism. This result will be used often
202
IV. L / 2 Euler Classes
to verify properties of cohomology classes assigned to arbitrary vector bundles: one need only consider sums of line bundles. The construction of g will itself be used in the next chapter to define further 2/2 characteristic classes. For any nplane bundle over a base space X there is a corresponding fiber bundle P , over X with fiber RP" ',the group being the projective group. To construct P , let E 5 X represent and let E* c E consist of nonzero points, as usual. For each x E X one identifies two points of E: c E x if and only if one point is a real (nonzero) multiple of the other; the total space of P , is the quotient of E* by this equivalence relation, in the quotient topology. There is an obvious induced projection onto X, each fiber is homeomorphic to RP" ',and the action of GL(n, R) on the fibers in E induces an action of the corresponding projective group on the fibers in P , .
* * * 3
tm)*
224
V. StiefelWhitney Classes
2.1 Corollary: Let A,, . . . , A, be m real line bundles over the same base spuce X E 8, and let I , 0 .. ' 0 I , be their Whitney sum. Then .for each q = 1, . . . , m the 4th StiejklWhirney cluss w,,(A, 0 ' ' . 0 I.,) E H q ( X ) is the
qth elementary symmetric junction oq(e(lL1),. . . , e(A,)) in the 212 Euler clusses r(%,),. . . ,e(A,).
PROOF: By the Whitney product formula (Proposition 2.5) and Proposition 1.2 one has 1
+w,(I,0.*.0Am)+"'+
wm(A,0...0A,) = w ( I , 0 ' * ' 0 A,) = w(A,) u . . . u w(L,) = (1 + e(Al))u * . . u (1 + e(Am)).
2.8 Corollary: For any real mplane bundle over any base space X has w,(() = e(p
,
Suppose that u',', . . . , u; are the elementary symmetric functions . . . , t, ,),.. . , ap,(t,, . . . , t, ,) in t , , . . . , t P  , obtained from u ; , . . . , u; by setting t , = 0; clearly u; = t , . . . t,_ . O = 0. Since j ( 0 )= 1, one has a,(t,,
,
c
. f ' ( ~ 1 ~ . ~ . . f ' ( ~ ,  , ) = . f ' ( ~ , ) ~ ' ~ . f ' ( t p ~ , ) . fP"(uy,...,u;) '(0)=
+
c
ojsgp
1
P,(u',', . . . , ui,,O).
n>p I
Since there are no polynomial relations among the elementary symmetric functions, it follows for each n 2 0 that there is a unique P , ( u l , . . . , u,) E 212 [ [ u , , . . . , u,]] of degree n such that P,(u;, . . . , u:) is the term of degree n in .f'(tl) . . . j'(tp),for any p 2 n. 2.9 Definition: For any formal power series f ( t ) E 2/2 [ [ t ] ] with leading term I E 212 the corresponding multiplicative sequence P,(u,, . . . , u,) E 212 [[u1,u2,. . . ] ] is determined by requiring the nth degree term of , j ' ( r l ) . * ..f'(t,) E 2 / 2 [ [ t , , . . . , t,]] to be P,(u;, . . . , ub), where u ; , . . . , u; are the elementary symmetric functions in t , , . . . , t , , .
For example, if , f ( t )= 1 + t then one clearly has P,(u,, . . . , u,) each n > 0.
= u,,
for
c,lzo
2.10 Proposition: Lrr P,,(u,, . . . , u,,) E H / 2 [ [ u , , u 2 , .. . ] ] he rhe multiplicatirc sequence o f uny jormul power series f ' ( t ) E 212 [ [ t ] ] with Ieuding term 1 E U / 2 , und let 5 be any real mplune bundle over a base space X E a. Then the 212 singulur cohomoloyp class ~ ~ (E 5H**(X) ) i s given by
whew w,,( 0 and any q > 0 there is an N(m, q ) > 0 such that lor any n 2 N(n1,y) the 212 cohomology ring H*(C"(R"+")) ugrees in dimensions 0, 1,. . . , y with the polynomial ring Z / 2 [ w l ( y r ) ., . . , w,(yr)] generated over 212 by the StiejelWhitney classes w,(y;) E Hr(G"'(R"'+")) ofthe canonicul real mplane bundle y: over Gm(R"+"). PROOF: Let 7,' be the canonical real line bundle over the projective space R P , and let y,' + * * + y,' be the sum of m copies of y,' over the mfold product
,
231
4. The Rings H" (G" ( R ' ) ; L / 2 )
+
RPY x . . . x RPq = (RP4)";that is, 7; + . . . 7: = pr!'y; 0 . . . 0 prb,y: for thc 111 projections (RP')''' a RP', as usual. Since ( R P ) " is a smooth ymdimensional manifold, by Proposition 1.7.3, it follows from Proposition 111.9.3 that there is an N ( m , q ) > 0 for which there is a "finite classifying map" ( R P ) " 'A G"(R"+") for i Y . . . + 11; whenever n >= N(m,q); that is, i q + . . . + 1 4 = k ' f for ti >= N ( m , q ) . If G"(R"'+'') 5 G"(R") classifies 7: (in the usual sense), then the composition j k classifies y; . . . 7:. Now let R P RP' classify the real line bundle 7 ; over R P , so that the mfold i"' product ( R P ) " +(RP"')" satisfies + . + i q = (i")!(y' + . . . + ; ) I ) ; if (RP')"'+!  G"(R' ) classifies ;I' . . . + 7 ' . then the composition h i'" is another classifying map for 7; . . . + ;!:. By the homotopy classification theorem (Theorem 111.8.9) the two classifying maps j k and h i" for .,I + . . . + .i ,YI are homotopic, so that there is a commutative diagram
+
1'
"1'
1'
+
;I:
+ +
+

*!l
it
0
I Y
H*(G"(Rm))
h.
'*
t
I
J. (Im)*
I
H*(Gm(Rm+" ))
I
H*((RP")")
+
+
k'
H*((RP)")
of Z/Zmodules. We already know for Lemmas 4.1 and 4.2 that h* is a monomorphism whose image H,*,,(( R P " )") c H*( (RP' )"') consists of those elements invariant under the automorphisms induced by permutations of the factors R P " , as in the proof of Proposition 4.3. By Corollary IV.4.5 H P ( R P " ) 2 HP(RPY)is an isomorphism for p 5 q, so that (i")* induces isomorphisms H&,((RP")")5Hr,,((RP)") for p 5 q, for the corresponding subring HZ,((RP)"')c H*((RP"))"). Hence for each p 5 q there is a commutative diagram HP(G"(R~J))
it H P ( G " ( R ~ )) +"
c
H,P,,((RP"1")
HP,,((RP)")
for any I I 2 N ( n i , q ) , with isomorphisms /I* and (i")*. Thus j * is a monomorphism (and k* is an epimorphism). Now let X & G))I(Rm+") be any splitting map for y:, so that H * ( G ( R m + " ) ) is a sum E., 0 . . . 0 E., of J'. H * ( X ) is a monomorphism and line bundles over X ; the composition j / classifies /Ll 0 * . . 0 ,Inl. If X R P " , . . . , X kRP" classify EL,, . . . , j.", respectively, then the map
232
V. StiefelWhitney Classes
. . ., I
X , (RP')" induces a homomorphism H*((RP")") 1:H * ( X ) ; the composition h I also classifies A1 0 * . * 0 A,,,.Since j f ' and h I both classify i , 0 . . . @ in,the homotopy classification theorem (Theorem 111.8.9) provides a homotopy commutative diagram 1=(1,.
)
0
cm(Rm)i
G"( R"
j
T
'1)
T
hl
(RP")"
+
0
I
x,
i
hence a commutative diagram HP(G"( Rm)) j r
h*
HP(
G"( R" ") ) +
I
I
J
J
I
I
' f
HP((RPm)")1 * HP(X), whenever p 5 q and n 2 N(m,q). We already know that h* is a monomorphism with image HE,((RP")"), and that j* and f * are monomorphisms. Consequently the preceding diagram reduces to HP(G"(R"))
* HP(G"(Rm+" ) )
mono jr
I
mono f*
H f J (RP")") [ ' HP(X), where h* is an isomorphism and the compositionf* j * is a monomorphism. It follows that I* is a monomorphism, which becomes an isomorphism in the further reduced diagram 0
mono f*
J
I"
H&,((RP")") ; Im I*. Commutativity of the latter diagram guarantees that the monomorphism f * is also an epimorphism, hence an isomorphism; consequently j* is also
233
5 . Axioms for StiefelLWhitney Classes
an isomorphism, assuming as always that p 2 q and n 2 N(m,q). Since j * w r ( y m )= wr(j!$")= wr(yy)E W ( G m ( R m + "for ) ) r = 1, . . . , m, Proposition
4.3 implies the desired result.
In the preceding proof we appealed to Proposition 111.9.3 for the existence of a "finite classifying map" ( R P ) " 5 G"(R"+") for the bundle y i + . * . y: over ( R P ) " , for sufficiently large n > 0. The following property of such maps is a finitedimensional version of Proposition IV.5.7.
+
4.6 Proposition: Given q > 0,any"jnite classifying map"(RP4)" 5 G"(R"+") f o r the reul mplane bundle yi . ' . y: over (RP)"is also a splitting map for the canonical real mplane bundle yy over G"'(R""), in the restricted sense that H p (G"( R" )) 5 H p (( R P ) " ) is monic for p 5 q and n 2 N(m,q).
+
+
+"
PROOF: The preceding proof contains a commutative diagram H p( Gm(R ) )
j*
* H "( Gm(R" +'))
1
HRv((RP")")
1 (imp
* Hf'tnv((Rf")")
with isomorphisms h* and (i")*, and we later verified that j * is also an isomorphism. Hence k* is an isomorphism onto the submodule H&,((RP4)") c HP( (RPq)"),for p 5 q and n 2 N(m, q).
5 . Axioms for StiefelWhitney Classes We now establish an axiomatic characterization of StiefelWhitney classes of real vector bundles over arbitrary base spaces X E B,following the pattern used in Theorem 1V.6.1 for 2/2 Euler classes. Even if one considers only smooth real vector bundles over smooth manifolds, the corresponding axioms uniquely describe StiefelWhitney classes, as we also show. 5.1 Theorem (Axioms for StiefelWhitney Classes, for the Category 3): For reul vector bundles i: over base spaces X E 9,there are unique inhomogeneous 212 cohomoloyji classes w ( ( )E H * ( X ) which satis#y the following axioms: (0) Dimension: I f ' t i s ( I real mplane bundle over X H o ( X )0 .. . @ H"(X) c H * ( X ) .
E 2,then
w(5)E
234
V. StiefelWhitney Classes
( 1 ) Naturality: lj' X ' 3 X is u mup in L@,und if 5 is u r e d vector bundle over X , then w(g'5) = g*w(() E H*(X'). (2) Whitney product formula: I f ( and q ure reul vector bundles otjer the sume X E B, with Whitney sum 5 0q over X , then w(5) u w ( q ) = w(( 0q ) E H * ( X ) ,for the cup product w(() u w(n). (3) Normalization: If y i is the canonical reul line bundle over RP' ( = S ' ) , then w ( y i ) = 1 e(yi) E H o ( R P ' )0H'(RP'), where e(yf)is the generutor of H * ( R P ' ) .
+
PROOF: The total StiefelWhitney classes of Definition 1.1 trivially satisfy Axiom (0),and they satisfy Axioms (1)(3) by virtue of Propositions 1.3, 2.5, and 1.2, respectively. Conversely, suppose that w( ) satisfies Axioms (0)(3), RP" classify let y' be the universal real line bundle over RP", and let RP' the canonical line bundle 7: over RP'. Then w(y:) E Ho(RP')0 H ' ( R P ' ) by Axiom (0),and Ho(RP") 0 H ' ( R P " ) H o ( R P ' )@ H'(RP') is an isomorphism such that f * ( l e(y')) = 1 e(y:) for the generator e ( y ' ) E H ' ( R P " ) , by Corollary IV.4.5. Since one has f * w ( y ' ) = w ( j ' ! y ' ) = w(y:) = 1 + e(yi) by Axioms (1) and (3), for the same isomorphism j ' * , it follows that w(y') = 1 e(y'). Now let ym be the universal real mplane bundle over Gm(R") and let (RP")" A Gm(Raj)be the splitting map for y" described in Proposition IV.5.7, with
+
+
+
h'y"
= y'
+ . . + y' *
% for the m projections (RP"')'" J
h*w(y")
= pr!,y'
0 .. * 0prLy'
RP". Then by Axioms (1) and (2) one has
w(h!y") = w(pr\y' 0 .. * 0 prky') = w(pr\y') u . . . u w(prky') = pr:w(y') u * . u prEw(y'), =

which is a unique element of H*((RP")") since w(y') is uniquely defined, as we have just learned. Since h* is monic, by Proposition IV.5.7, it follows that w(y'") is a unique element of H*(G"(Rco)). By one final appeal to the homotopy classification theorem (Theorem 111.8.9) any real mplane bundle ( over any X E L@ can be classified by a map X 3 Gm(ROC)which is unique up to homotopy, so that by Axiom (1) one has w ( 5 ) = w(g!y") = g*w(y") E H * ( X ) for a unique homomorphism H*(G"(R")) 3 H * ( X ) and the unique element w(y") E H*(G"(R")). Now let .& denote the category of smooth manifolds and smooth maps, as before. We know from Corollary 1.6.8 that A c .@, and we know from Theorem 111.9.5 that every real vector bundle over any X E .Af is itself smooth, in the sense that it can be represented by a smooth coordinate
0 one easily obtains w(y:) = 1 e(y,') E H o ( R P )0 H ' ( R P ) from Corollary IV.4.5 and Axioms (0),(l), and (3), exactly as in the proof of Theorem 5.1, where y,' is the canonical real line bundle over R P , and where e(y,') E H ' ( R P ) is the uniquely defined generator of H * ( R P ) . Now, for any rn > 0, let q = m, so that there is a finite classifying map (RP")"' Gm(Rm+")for the real mplane bundle y,!, + . . . + y,!, over (RP")" whenever n 2 N(m, 4 ) = N(m, m), as in Proposition 4.5; that is, k'y; = y ,I + . . . + y; = pr\y,!, 0 .. . @ prLy;
+
for the canonical real mplane bundle y; over G"(R"'+"). By Axioms (1) and (2) one has k*w(y;)
= w(pr\y,!, 0 .. prky,!,) w(pr\yi,) u . . . u w(prLy,!,) = pr:w(yA) u . . . u pr:w(y,!,), = w(k!y;) =
and since we already know that w(y,!,) = 1 + e(yf), it follows that k*w(yT) is a uniquely defined element of H*( (RP")"). However, Axiom (0) guarantees that w(y;) vanishes in degrees above m, and since q = m, Proposition 4.6
236
V . StiefelWhitney Classes
guarantees that HP(G"'(R"'+n)) 5 HP((RPm)"')is monic for p 5 rn; hence w(y;) is uniquely defined in H*(G"'(R"'+" )), for any n 2 N(m,m). Finally, if 5 is any smooth real mplane bundle over any X E ~then , Proposition 111.9.3 provides a finite classifying map X 5 G"'(R"'+") for sufficiently large n. By Theorem 1.6.19 one can assume that f is itself smooth; furthermore, one may as well suppose n 2 N(m,m), so that Axiom (1) implies w(5) = w(,f!y;) = f*w(y;) E H * ( X ) for the preceding class w(y;) E H*(Gm(R"'+"1). The final step of the preceding proof clearly does not depend on the choice of the finite classifying map f ;for if w( ) and It() both satisfy the axioms, the earlier steps of the proof give w(y;) = It(yy) E H*(G"(R""")), hence
w(5) = w(f!f) = f*w(y;) = f*It(yY) = I t ( f ! y ; )
= It(S)E
H*(X).
However, there is also an explicit demonstration that the choice o f f is immaterial. Suppose that X % G"'([W"+"') is another smooth finite classifying map for the smooth real rnplane bundle t over X E 4, for some n' 2 N(m, m). The ersatz homotopy uniqueness theorem (Proposition 111.8.14) then provides (smooth) finite classifying extensions gn,n..and gn.,n..for which the compositions Gm(,m+n) g n * * ~ ' ' Gm(,rn+n") + and 3 p(Rrn+n') !/*l'm'' G"'(R"'+"'')
x
x

are homotopic. The axioms then imply f*w(Y;)
= f*w(gb,,*,r;4 = f*g:,,4y;,4
= (gn,,,,
f)*w(y;4
= g*g:,,,..w(y;?.)
= (gn.,d, g)*w(y7,)
= g*w(g~~,,~~y;~.)
= g*w(y3
as expected. Since the Grassmann manifolds Gm(Rrn+")are both closed and smooth (Proposition I.7.3), they are a fortiori compact and triangulable. One therefore has analogs of Theorem 5.2 in which the category 4 of smooth manifolds is replaced by any one of several categories of topological spaces. For example, Theorem 5.2 is valid for real vector bundles over spaces in the category of smooth closed manifolds. Theorem 5.2 is also valid if one replaces A' by the category of those compact spaces which happen to lie in 99,or by the category of finitedimensional simplicia1 spaces (which automatically lie in by Corollary 1.2.2 and Proposition 1.4.6). In each case one of Propositions 111.8.12,III.9.1, or 111.9.3 furnishes the required finite classifying maps.
231
6 . Dual Classes
6 . Dual Classes Since 2/2 is a field, and since we work exclusively with formal power series f ( t )E 2/2 [[t]] with leading term 1 E 2/2, it follows for each such f ( t ) that there is a unique formal power series (l/f)(t) E 2/2 [[t]] with leading term 1 E 2/2 for which , f ( t ) . (l/f)(t) = 1 E 212 [[t]]. For example, if f ( t ) = 1 t , then ( l / . f ) ( t = ) 1 t tZ .. . .
+
+ + +
6.1 Definition: For any formal power series f ( t ) E 2/2[[t]]
with leading term 1 E 2/2, and for any real vector bundle over a base space X €98,the multiplicative 2/2 class u,,,( n ; since the individual Wu classes are polynomials in the StiefelWhitney classes w l ( t ( X ) ) ., . . , w , ( t ( X ) ) , the result can be regarded as a set of polynomial relations among the latter classes. In Brown and Peterson [ 1,2] one finds all the polynomial relations satisfied by w , ( z ( X ) ) ,. . . , w , ( T ( X ) ) for every smooth closed ndimensional manifold X . Incidentally, E. H. Brown [2] and Stong [l] provide independent proofs that if 2p 5 n, then there are no such “universal” polynomial relations P ( w l ( z ( X ) ) ., . . , w,(T(X)) = 0 E H P ( X )in degree p . In sufficiently high degrees there are “universal” polynomial relations satisfied by the dual StiefelWhitney classes W l ( t ( X ) ) ., . . , W,,(t(X))of every smooth closed ndimensional manifold X , where 1 + w l ( t ( X ) )+ . . . + W , ( t ( X ) ) = w ( r ( X ) )= u f ( 7 ( X ) )for f ( t ) = 1 + t + t 2 + * . . E Z/2[[t]]. Such relations were investigated in Massey [l, 31 and in Massey and Peterson [ 11; the latter paper shows that if 0 S p < cc(n), where a(n) is the number of 1’s in the dyadic expansion of n, then W,,,(t(X)) = 0 for every smooth closed ndimensional manifold X . The set of all polynomial relations satisfied by iVl(r(X)),. . . , W,(t(X)) for every smooth closed ndimensional manifold X is implicitly described in Brown and Peterson [l, 21: one merely replaces w by W throughout their original argument (and result); related results are given in Bendersky [l] and in Papastavridis [2]. The result itself is an essential ingredient of our later Remark VI.9.14.
+
+ +
7. Remarks and Exercises
247
7.28 Remark: According to Proposition 3.2 one can reasonably define orientability of a smooth manifold X by the requirement that wl(z(X))= 0 E H ' ( X ) . An orientable manifold which satisfies the additional relation W ~ ( T ( X=) 0) E H 2 ( X )is called a spin munifbld. (Alternative characterizations of spin manifolds are given in Bore1 and Hirzebruch [2, p. 3501, the end of Milnor [ 121, and at the beginning of Milnor [ 161.)The total StiefelWhitney class W ( T ( X ) ) of a spin manifold automatically satisfies further relations, some of which are discussed in Wilson [ 13, for example. 7.29 Remark: Since realifications of complex vector bundles are somewhat specialized, one expects their StiefelWhitney classes to satisfy universal polynomial relations. For example, the realification laof a con3plex vector bundle ( is naturally oriented, so that w , ( i a ) = 0 by Proposition 3.2; other such relations will appear in the second volume of this work. There are also universal polynomial relations satisfied by the StiefelWhitney classes of realifications of "quaternionic bundles," some of which appear in Marchiafava and Romani [I, 2, 31. 7.30 Remark: One of the classical results of differential topology is that S', S3,and S7 are the only standard spheres which are parallelizable in the sense of Remark 111.13.26; that is, if S" has n linearly independent vector fields, then n = 1, 3, or 7. This result was proved in Bott and Milnor [l] and Milnor [4], using the following property of StiefelWhitney classes : If S"is the base space of a real vector bundle 4 with w,(() # 0, then n = 1,2,4, or 8. (Incidentally, the result of Barratt and Mahowald [l] reported in Remark 111.13.36 instantly implies that if n = 4k > 16, then w,(() = 0 for any real vector bundle ( over S",) I n the third volume of this work we shall formulate and prove a result of Adams [l, 21, one of whose principal corollaries is that S', S3, and S7 are the only parallelizable standard spheres. Adams's original proof, which also appears in Cartan [ 5 ] , uses many of the techniques introduced in this chapter, including Steenrod squares; however, the proof given later will be based on an entirely different technique of Adams and Atiyah [ 13. 7.31 Remark: There is a more general result than the one just described: the maximum number of linearly independent vector fields on the standard sphere S" is known for any n > 0. One of the early guidelines for the computation was provided by Stiefel [2], who used StiefelWhitney classes to show that if n = 2 k ~ 1 for an odd number u, then the projective space RP" cannot have 2k linearly independent vector fields. Steenrod and Whitehead [ 13 used Steenrod squares to obtain the same result with S" substituted for RP", and their work eventually led to the complete solution of the problem,
248
V. Stiefel Wh itney Classes
in Adams [4]. (We shall not attempt to present a proof of this result. Adams’s original proof is outlined in Eilenberg [l] and in Husemoller [l], and some simplifications by Karoubi are reported in Gordon [l], for example; Adams himself streamlined the proof in many ways, and a major simplification appeared in Woodward [l]. A complete proof is given in Karoubi [2], incorporating all improvements through 1978; a complete proof is also given in Mahammed et al. [I], incorporating all improvements through 1980.) 7.32 Remark: To any finitedimensional real representation I/ of a finite group G one can associate a real vector bundle 5“ over the classifying space BG of Remark 11.8.18; the total StiefelWhitney class w(tv) E H*(BG) is then the total StieJel Whitney class w(V ) of the representation V . Segal and Stretch [11 present an alternative construction of w(V ) ,which leads to new results about group representations. 7.33 Remark: There is an algebraic construction of Delzant [l], which assigns StiefelWhitney classes to certain “quadratic modules” rather than to real vector bundles; the construction was further developed in Milnor [ 191. A related construction exists for real vector bundles, given in Patterson [l]; however, this variant produces only the terms 1 + wl(t) + w2(t) of the total StiefelWhitney class w(() of a real vector bundle 5. A recent exposition of the constructions of Delzant and Milnor is in Chapter 4 of Micali and Revoy [l]. 7.34 Remark: Lemma 6.3 is a special case of a more general result, with essentially the same proof. Let f ( t ) and g(t) be any formal power series in 212 [[t]] with leading terms 1 E 212, so that the product (.fg)(t)is also such a formal power series. Then for any real vector bundle 5 over a base space X E 2 one has u,( 0 let a(n) > 0 be the number of 1's in the dyadic expansion of n; that is, if n = a02' a121 . . . ar2', where the natural numbers u 0 , a , , . . . , u, are either 0 or 1, then a(n) =
+
+
+
C;=oUj.
4.5 Proposition: For any natural number n > 0 there is a smooth closed l. ndimensional manifdd X that does not immerse in R2na(n)
PROOF: For the set J of those indicesj with aj = 1 in the expansion n = + . . . + ar2', let X be the product nj,,RP2'. Then, for the projections X RP2', the tangent bundle of X is a Whitney sum t ( X ) = @jEJprjs(RP2'), so that the dual StiefelWhitney class of r ( X ) is a cup uo2O
262
VI. Unoriented Manifolds
nJEJ
nJEJ
product w ( r ( X ) )= w(prir(RP2'))= prTW(z(RP2')).The highestorder nonvanishing class in iij(z(RP2'))is m2' l(z(RP2')) = e ( ~ : , ) ~ ' E H 2 '  l ( R P 2 ' ) , as in Proposition 4.3. Since the Kunneth isomorphism OJGJ H*(RP2') H * ( X ) carries the highestorder nonvanishing tensor prj*W2,l ( t ( R P 2 ' ) ) , product @ , E J w 2 J  I ( r ( R P 2 ' ) )into the cup product and since CJEJ(ZJ  1) = n  a(n), it follows that w ~  ~ ( , , ) ( T (#X )0) E H"  a(")( X ) .Hence by Lemma 4.2 the product X cannot immerse in R2""" '.
nJEJ
One cannot improve Proposition 4.5 without further specialization, for the following reason: for any natural number n > 1, every smooth ndimensional munifdd X immerses in R2na(n). This "bestpossible'' immersion theorem of Cohen [l] is briefly discussed in Remark 9.14. Proper embeddings are specialized immersions, for which there are improved versions of Propositions 4.3 and 4.5. Specifically, an immersion X R2"P is an embedding if it is injective ( . f ( x ) = f ( y ) E R 2 n  p implies x = y E X ) , and it is proper if inverse images of compact sets in R 2 "  pare compact in X . For example, any embedding of a compact manifold is necessarily proper. For any proper embedding X 5 R2"P of a smooth ndimensional manifold X, for any E > 0, and for any connected open set V c X whose closure V is diffeomorphic to a closed disk D" c R", let C,( V ) denote the set of points y E R2"p such that l f ( x )  yl < E for some x E V . Since the closure is compact, the inverse image consists of at most finitely many connected components, so that for sufficiently small E > 0 the set j''(W)) contains no points outside the connected component of B c X . In what follows, a cocoon C ( V ) c R2"P of the image j'( V ) R 2 n  pIS any set of the form C,( V ) ,where E > 0 is chosen in such a way that j '  '(C3,(V ) ) contains no points outside the connected component of V c X .
0. Then the 212 Euler class vanishes: e(vs) = 0 E H "  P ( X ) .
FROOF: By definition, e(vs) = cr*j*U,, for the 212 Thom class U , , E H"P(E,E*),where E 5 X represents v,., where E A E, E* is the inclusion, and where X 5 E the zerosection. By the excision axiom one may as well replace E by any open neighborhood of the image of cr, which we also denote E, with a corresponding E* c E. Then Lemma 4.6 provides the isomorphisms f ; ,f : ,and j'* in the accompanying commutative diagram, and the remain
+
j+
H"  "( E, E*)
T
a*
H""(X)
 
2
H" Y( [WZ" P ,

1; =
H"P($E\f(X)) YX
' H"p(E)
'i
I.'*
H"P(i)
)]I
H"P(f(X))
p"P\f(x))j * H" P( R2nP)
ing homomorphisms are induced by the obvious inclusions. The inclusion [ W 2 "  p induces an excision i,i \ f ( X ) 3 ( W 2 n  p , Rznp\ f ( X ) , it follows that gg is also an isomorphism. Since H"P([W2"p) = 0, it then follows that
2%
e(vs) = o*j*U,,
= j'*r*j*(gg) ' ( j ' : ) 
U,,
= 0 E H"p(X),
as asserted.
4.8 Theorem: Let W ( r ( X ) )be the dual StiefelWhitney class of the tangent bundle r ( X ) of a smooth ndimensional manifold X , and suppose that there is a proper embedding X R z "  p ; then m,(r(X)) = 0 E H 4 ( X )for q 2 n  p. PROOF: The normal bundle v f is an ( n  p)plane bundle and one has W ( r ( X ) )= w(v,),where wq(vI)= 0 for q > n  p as in Lemma 4.2. However, according to Corollary V.2.8 w,,(vs) is just the 212 Euler class e ( v f ) E H "  p ( X ) ,which vanishes by Lemma 4.7.
4.9 Proposition: I f n = 2' for r 2 0, then the real projective space RP" does not embed in R2"'. PROOF: Since RP" is closed, a fortiori compact, any embedding is proper. However, W n ,(r(RP")) is the nonzero element e(y,!)" E H"'(RP") as in
265
5 . StiefelWhitney Numbers
Proposition 4.3, so that Theorem 4.8 forbids an embedding of RP" into [W2, I
For any n > 0 the number a(n) > 0 is described in Definition 4.4. 4.10 Proposition: For any natural number n > 0 there is a smooth closed ndiniensional nzun@ld X which does not embed in [W2" '("). PROOF: In the proof of Proposition 4.5 it was shown that iij,m(,J7(X)) # 0 E H""("'(X)for the closed manifold X = RP". Since any embedding of a closed manifold is necessarily proper, Theorem 4.8 therefore forbids any embedding of X into R2na(n'.
n,,,
5. StiefelW hi tney Numbers Let X be any smooth closed ndimensional manifold, and let ( r l , . . . , r,) be any ordered set of natural numbers such that r l + 2r2 . . . nr, = n. We shall use the total StiefelWhitney class w ( t ( X ) )of the tangent bundle r ( X ) to assign an element of 2/2 to each ( r l , . . . , r,). One obtains the value 0 E 2/2 for each ( r l , . . . , r,) if (and only if) X is the (smooth) boundary Y of a smooth compact (n 1)dimensional manifold Y.
+
+
+ 1 + w l ( t ( X ) )+ . . . + w , ( t ( X ) )
If is the total StiefelWhitney class w ( T ( X ) E) H * ( X ) of the tangent bundle T ( X )of the smooth closed ndimensional manifold X , and if(rl, . . . , r,) is an ordered ntuple of natural numbers such that r l Zr, . . . nr,, = ti, the product w , ( ~ ( X ) ) lu' . . . u w,,(T(X)P in H*( X ) belongs to H"( X ) . The boundary X of X is void since X is closed, so that the fundamental 2/2 homology class of Definition 1.4 is an element p x E H , ( X ) . Consequently the Kronecker product H " ( X ) 0 H , ( X ) 2/2 assigns an element of the ground ring 2/2 to the manifold X and the ntuple ( r l , . . . , r,), as follows.
+
+
+

5.1 Definition: Let X be a smooth closed ndimensional manifold, and let ( r l , . . . , r,) be any natural numbers such that r l + 2r2 + * * . + nr, = n. Then the (rl, . . . , r,) th Stieid Whitney number of X is the element (Wl(T(X))ll
U
.
'
*
U
W , , ( T ( X ) P , p x )E 2/2.
5.2 Proposition: Suppose that Y is a smooth compact (n + 1)dimensional manifold with boundary Y , so that Y is a smooth closed ndimensionalmanijold; then all the StiefelWhitney numbers of Y vanish. PROOF: According to Proposition 1.9 the fundamental 272 homology class Y , Y ) under
pu E H,( Y ) is the image of the fundamental 2/2 class pLy E H,,
266
VI. Unoriented Manifolds
the connecting homomorphism d of the exact homology sequence
H , + 1( Y ) L H , ,
1(
Y , Y ) PH,( Y ) i tH,( Y )
for the pair Y , Y ; that is, p p = d p y . Furthermore, the connecting homomorphism S of the corresponding exact cohomology sequence H " ( Y ) ' I , H " ( Y ) "H"+'(Y,Y) j t .n+l(y) is the adjoint of d with respect to the Kronecker product ( , ); that is, (cc,dv) = (6cr,v) for any o! E H"(Y) and any v E H,, l ( Y , Y ) . Consequently the ( r l , . . . , r,)th StiefelWhitney number of Y satisfies (w;l
...w?,py)
= (w;'
. ' . w',",dpy)= ( S ( W ; l . . . w ? ) , p y ) ,
where W?
' ' '
W',"
= W l ( T ( f)rlU *
' *
U W,(5(
Y)y";
We shall show that S(w;l . . . w',")= 0. Let T ( Y ) be the tangent bundle of Y itself. Then for the inclusion Y f Y of the boundary Y , the restriction i ! z ( Y )of t ( Y ) to Y is the Whitney sum z( Y ) 0 c1 of the tangent bundle z( Y ) and the trivial line bundle c 1 spanned by the inwardpointing unit vectors normal to Y , with respect to any riemannian metric on t( Y ) . Since W(T(Y
) )= W ( T ( Y ) @ E l )
= W ( i ! T (Y
) ) = i*W(T( Y ) )
for the total StiefelWhitney classes W ( T ( Y ) )E H*( Y ) and it follows that
W ( T (Y
) )E H*( Y ) ,
6 ( w l ( t ( Y ) ~u1 . . . u w n ( 7 ( Y ) p = ) 6i*(wl(T(Y)ylu * * . u W , ( T ( Y ) P )
=
o
as desired, since Si* annihilates H"( Y ) in the exact cohomology sequence of the pair Y , Y .
5.3 Remark: Proposition 5.2 is due to Pontrjagin [ 5 ] . The converse assertion is also true, due to Thom [3, 681. Thus a smooth closed manifold is a boundary if and only i f all its StiefelWhitney numbers vanish. We shall not prove the converse of Proposition 5.2. The following result depends only on Proposition 5.2. 5.4 Corollary: If n is even, then the real projective space RP" is not the boundary of any ( n 1)dimensional manifold.
+
PROOF: By Proposition 3.1, w ( T ( R P " )= ) (1 + e(y,!))"+',with binomial coefficients computed in 2/2, where according to Proposition IV.4.4, H*(RP")is the polynomial ring over 2/2 generated by the 2/2 Euler class e(y,!), modulo
267
5 . Stiefel Whitney Numbers
("L
the relation e(y,')"+' = 0. In particular, if n is even, then ') = 1 E Z/2, so that w,(z(RP")) is the generator e(y:)" E H"(RP"). Hence the (0,. . . , 0,l)th StiefelWhitney number of RP" satisfies (w,(z(RP")),p R p n )= 1 # 0, so that Proposition 5.2 prevents RP" from being a boundary. 5.5 Proposition: I f n is odd, then all StiefelWhitney numbers of' the real projective space RP" vanish.
PROOF: If n = 2m  1 for m > 0, then Proposition 3.1 and the technique of Lemma 3.3 give w ( t ( R P " ) )= ( 1 + e(~;))~"' = ( 1 + e(+jj,!)2)"'; in particular, all the odd StiefelWhitney classes of RP" vanish. However, ifr, + 2r2 +. . . + nr, is to equal the odd number n, then rp # 0 for at least one odd number y n, so that w,(t(RP"))" u . . . u w,,(r(RP"))'"= 0. 5.6 Corollary : If' n is odd, then the real projective space RP" is the boundary of some ( n
+ 1)dimensionalmanifold.
PROOF: Remark 5.3 and Proposition 5.5. 5.7 Definition: Two smooth closed ndimensional manifolds X and X ' are cobordant if their disjoint union X X ' is the boundary of a smooth compact (n 1)dimensional manifold Y .
+
+
+
If X is any smooth closed ndimensional manifold, then X X is the boundary of X x [0,13, so that cobordism is a reflexive relation. Since the disjoint union X + X ' is assigned no order cobordism is a symmetric relation. Finally, if X + X ' is the boundary of Y and x' x'' is the boundary of Y', then one can identify the common portion X ' of Y and Y' to create a smooth compact (n + 1)dimensional manifold Y" with boundary X X " ; hence cobordism is also a transitive relation. In summary, cobordism is an equivalence relation. Let [ X I and [ X ' ] denote the cobordism classes of smooth closed manifolds X and X ' , respectively. If X and X' are of the same dimension, then one easily verifies that the class [ X + X ' ] of the disjoint union X + X ' depends only on [ X I and [ X ' ] , and for any dimensions one easily verifies that the class [ X x X ' ] of the smooth product X x X ' also depends only on [ X I and [ X ' ] . Hence the set 'Jz of cobordism classes of smooth closed manifolds admits a sum '.Jz x (Jz: 'Jz and a product 'Jz x (Jz i'Jz. Furthermore 'Jz is graded by dimension: if 'Jz, c (Jz consists of classes represented by smooth closed ndimensional manifolds, then (Jz is the disjoint union of the subsets 9tn, and sums and products satisfy 91, x ' J z l l ~ Y 1 ,and , J' z,, x 91, i'ill,,,+,,.
+
+
268
VI. Unoriented Manifolds
5.8 Lemma: The set % of' cobordism classes is a graded commutative ring with respect to the preceding operations
a,, x 3,; a,,
and
%, x %,, i %,+,.
PROOF: This is a trivial verification. In particular, since X + X is the boundary of X x [0,1],it follws that  [ X I = [ X I , hence that every element of % is of order 2 with respect to addition. 5.9 Definition: The preceding graded commutative ring % of cobordism classes of smooth closed manifolds is the unoriented cobordism ring. 5.10 Proposition: If [ X I = [ X ' ] in a,,for two smooth closed ndimensional manifolds X and X ' , then X and X ' have the same StiefelWhitney numbers; conversely, if X and X' have the same StiefelWhitney numbers, then [XI = [X']. PROOF: The first assertion is a rephrasing of Proposition 5.2, and the second assertion is a rephrasing of Remark 5.3. There is an alternative way to construct the unoriented cobordism ring %. Observe that if DP and D4 are disks of dimensions p and 4 with boundaries
+
S p  ' and S q  ' , respectively, then each of the ( p q  1)dimensional manifolds DP x S4' and S P  x Dq has S P  x S4 as its boundary. Suppose that DP x S q  is smoothly embedded in a smooth closed manifold X o of dimension p + 4  1. One can then remove DP x S4' from X , , leaving a boundary S p  x Sq ; since S p  ' x Sq I is also the boundary of S p  I x Dy, one can (smoothly) insert S p  ' x D4 into X,\DP x S  ' , identifying common points on the common boundary S p  ' x S 4  ' , to obtain a new q  1. Two such manismooth closed manifold X , , also of dimension p ifolds X , and X , are surgically related; in general, surgically related manifolds are not diffeomorphic to each other.
'
5.11 Definition: Two smooth closed manifolds X o and X , of the same dimension are surgically equivalent if and if there are smooth closed manifolds X , , . . . , X ,  such that X i  is surgically related to X i for each i = 1, . . . , 4 .
,
Clearly surgical equivalence is an equivalence relation, and the set of equivalence classes [ X I of smooth closed manifolds X forms a commutative semiring %' with respect to disjoint union and smooth product: [ X I + [ X ' ] = [ X + X ' ] and [ X I . [ X ' ] = [ X x X ' ] . The semiring %' is graded by dimension. We shall show that (JL' is canonically isomorphic to the unoriented cobordism ring %. Let 11 ( I p and 11 It, denote the usual euclidean norms on R P and W. The product Rp x Ry then carries a norm 11 I( with Il(x,y)l(= max{IIxllp,llyllp),
2 69
5. Stiefel Whi tney N urn bers
and D'' x S''u S"' x DY is the unit sphere in R p + qwith respect to 11 11. By stretching the unit sphere D p x S Y  ' u S"' x Dq at the "equator" S  I x S Y  ' it follows that D p
x S 4  ' x {O)
u Sp'
x Sq' x [0,1]
u S p  ' x D4 x (1)
is the boundary of a ( p + 61)cell in Rpfy, hence homeomorphic to the sphere SPfq1
c
RP+4.
5.12 Lemma: Any two surgically related smooth closed manifolds are cohordunt : con.sequentlji any two surgically equivalent smooth closed manifolds ure cohordant. PROOF: Suppose that X , and X I are smooth closed manifolds of dimension p 4  1, where one replaces some Dp x S4 c X , by S P  ' x 0 4 to obtain X I ,as before. Then Xo\Dp x S 4  and X \Sp x 0 4 are each diffeomorphic to a smooth compact manifold X with boundary S P  ' x S  ' .Let Y be the union of the product X x [0,1] and the ( p q)cell with boundary
'
+
+
Dp x S4' x {0)
usp'
x SY' x [0,1] u S p  ' x
0 4
x 11)
described earlier, with the obvious boundary identifications: the two boundaries DP x S4I x (0) are identified with each other, the two boundaries Spx S4 I x [0,1] are identified with each other, and the two boundaries Spx Dq x { 1 are identified with each other. Then Y is a smooth compact ( p + 4)dimensional manifold with boundary Xo x (0) + X I x (l}, as required.
' '
The converse of the second assertion of Lemma 5.12 requires some elementary Morse theory, which we sketch. According to Definition 111.6.11 the djflerential of a smooth function Y R on a smooth manifold Y is the C5( Y)linear map €*( Y ) % Cm'( Y ) carrying any smooth vector field L E &*( Y ) into the smooth function Lf' E Cm(Y). A critical point off is any point J' E Y at which df vanishes; that is, ( L f ' ) ( y )= 0 for every L E Q*( Y).
5.13 Lemma (Regular Interval Lemma): Let Y be a smooth compact ndimensional munijdd whose boundary is the disjoint union of two smooth closed ( n  1)dimensional manijolds X , and X , , and let YL [a, b] c R be a smooth ,function with no critical points, such that X , = f  ' ( { a ) ) and X , = j  ' ( { b ) ) . Then there is a diffeomorphism Y 3 X , x [a, h] whose composition with the second projection X , x [a, b] 3 [a, b] is 1' itself'; u jbrtiori X , is difleomorphic to X,.

PROOF: There is a smooth riemannian metric Q*( Y ) x €*( Y ) Cr'(Y ) by Proposition 111.5.8, which induces an isomorphism b(Y ) * €*( Y ) carrying
270
VI. Unoriented Manifolds
a(Y ) into a vector field grad j ' E a*(Y ) such that ( M , grad,/') = M f E Ca( Y ) for any M E a*(Y ) . By hypothesis df is nowherevanishing; hence grad f is also nowherevanishing, so that (grad f , grad f ) is everywhere positive on Y. It follows that there is a unique vector field L E &*( Y ) such that (grad f', grad f ' ) L = grad f ' , for which one trivially has &f = ( L , gradj') = 1 E Cq(Y). The definition of L also yields the identity (grad f , grad f ' ) ( L , L ) = 1, and since Y is compact, it follows that there are positive constants A and B such that 0 c A 5 ( L , L ) B < 00 uniformly on Y . In order to construct Y 3 X u x [a, b] we shall first construct its inverse X u x [a,b]+!  Y by solving ordinary differential equations. In outline, observe that partial differentiation with respect to the coordinate t E [a, b] provides a vector field d / d t on Xu x [u, b]. We shall show that there is a uniquely defined h such that h,(d/dt) = L on Y and h(x,u) = x for each
dj' E
xE
x,.
For any c E [a,b] let X , be the closed set f  '({c}) c Y . Since Y is compact X , is also compact, so that it is contained in the union of finitely many open coordinate neighborhoods Y' in Y. As in Proposition 1.6.10 one can shrink each of the open coordinate neighborhoods Y' to a smaller open coordinate neighborhood Y" whose (necessarily compact) closure satisfies P " c Y ' , in such a way that X, is also contained in the union of the finitely many open sets Y"; one may as well assume that X , actually intersects each Y". If ( y ' , . . . ,y") are coordinate functions on one of the former coordinate neighborhoods Y ' , the restriction of L to Y' is of the form I' d/dy' + . . . . + Ind/dynfor smooth functions I', . . . , I" on Y', as in Lemma 111.6.6.Suppose temporarily that c lies in the interior (a,b) of the closed interval [a, b]. Then according to the classical existence and uniqueness theorem for ordinary differential equations there are positive constants 6 > 0 and E > 0 and a unique ntuple ( h ' , . . . , h") of smooth realvalued functions on the intersection X , n P" x [ c  6, c E ] such that
+
ah' (x; t ) = Il(h*(x;t), . . . , h"(x; t ) ) , at

ah" at
(x; t ) =
l"(h'(x; t), . . . , h"(x; t ) ) ,
such that each x E X , n Y" has coordinates ( h ' ( x ; c), . . . ,h"(x;c)),and such that each ntuple (h'(x;t ) , . . . , h"(x; t ) ) is the coordinate description of a point in the larger neighborhood Y'. The constants 6 > 0 and E > 0 depend on the geometry (in R") of the inclusion 7'' c Y' and on the constants A and
27 I
5. StiefelWhitney Numbers
B described earlier, with 0 < A 5 ( L ,L ) I B < 00; however, A and B themselves depend only on the given function Y s [a,b] and the choice of the riemannian metric ( ,),so that 6 and E effectively depend only on the inclusion Y" c Y'. In the special cases c = a or c = b one has 6 = 0 or E = 0, respectively. One can rewrite the preceding system of differential equations in the form
'
?/I
(7t
s
(7 h " (7
(7
+...+
24'1
d t dy"
=I' +... dyl
a + PF, UY"
or equally well in the coordinatefree form h,(?r/dt) = L. Since X, is contained in the union offinitely many of the coordinate neighborhoods Y", the uniqueness of the ntuples (A', , . . , 11") implies for L' E (a, b) that there are positive h Y with constants 6, > 0 and E , > 0 and a unique map X, x [c  b,, c EJ h,(?/?r) = L over the image of h, and h(x,c ) = s for each s E X , . Since
+

one has f(/z(x, t ) ) = t for each t E [c  b,, c + E,]. Hence for any closed sub) interval [ t , t'] c [c  6,, c 4 there is a bijection from X, (=f' ( { t j , )to X,. ( = j '  ' ({ t i ) which ) identifies the X,endpoint of each integral curve of L with the X,,endpoint of the same integral curve; one easily verifies that the preceding bijection is a diffeomorphism from X, onto X,. . In the special cases = ( I or c = h one has 6, = 0 or cb = 0, respectively. To complete the proof of the regular interval lemma it suffices to note that finitely many of the intervals [c  S,, c + E,] cover [a, b], so that there is a unique diffeomorphism X, x [a, b] 1 :Y such that h,(d/dt) = L on all of Y and h(x,a) = x for x E X,. It follows as in the local case that f ( h ( x ,t ) ) = t for any (x,t ) E Xu x [a, 61. Furthermore, if the inverse diffeomorphism YA X, x [u,b]carriesy~Y i n t o ( . x , t ) ~ Xx, [u,b],thenf(y)= t =pr,(g(y)), so that j ' = pr, y as claimed.
+
(8
19
Now let Y A R be a smooth function on a smooth manifold Y of dimension n = p q, and suppose that y E Y is a critical point o f f . If there are coordinate functions u ' , . . . , u p , o', . . . , ziq on some neighborhood of y, all vanishing at y, such that
+
j ' = f ( y )  [(u')' f . . .
+
(UP),']
+ [(u'),' + . . . + (u",']
in that neighborhood, then y is a nondegenrrate critical point o f f , with criricul ualue ,f(y ) E R. (There is a less restrictiveappearing characterization of nondegenerate critical points; however, for the moment we merely observe that d j ' does indeed vanish at y, so that y is at least a critical point off in the earlier sense.)
272
V1. Unoriented Manifolds
For notational convenience let u = ( u ’ , . . . , u p ) , u = ( u ’ , . . . , u4), lu11’ = + . . . + (up)’, and llull’ = (u’)’ + . . . + (uq)’. For any E > 0 let c Rp+4be the disk of radius 28 about 0 E Rpfq, consisting of those (u, u ) E RP++“ such that IIu11’ + l(u)1’ S ( 2 ~ ) ’and , for any 6 > 0 let S : + q  l c Rpf4 be the sphere of radius 6 about 0 E Rp+4, consisting of those points (u, u) E Rp+q such that lu11’ + lu11’ = 6’. The notations DP, D4, SP S q  without subscripts will denote standard disks and spheres of the indicated dimensions, identified only up to diffeomorphism. (u’)’
’,
+
5.14Lemma: ForanyE > O l e t f = IIuI12 Ilull’onDP,r4;thenj’’((~’)) is dfleomorphic to S p  ’ x 0 4 and f  ’ ( { + E ’ ) ) is diffeomorphic to DP x S Q 
’.
PROOF: If 0 5 6 5 2 ~then , the intersection f  ’ ( {  ~ ’ ) n ) SgfYl consists of those (u,u) E 01;:“ such that IIuII’ + ( 1 011’ = E’ and llu11’ + llu11’ = 6’; that is, l)u11’ = i(6’ E ’ ) and llu11’ = $(a’  c’). Clearlyf’({  8 ’ ) ) n S : + 4   ’ is void for 0 S 6 < E , of the form S P  ’ x (*) for 6 = E , and of the form S p  ’ x S  ’ for E < 6 S 2.5, so that f  ’ ( { e’J) is diffeomorphic to S p  ’ x D4. Similarly f  ’( { + 8 ’ ) ) is diffeomorphic to DP x S 4 
+
’.
5.15 Lemma: Let Y be a smooth compact ndimensional manifold whose boundary is the disjoint union of two smooth closed ( n  1)dimensional manifolds X u and xb,and let Y [a, b] c R be a smooth function with a single critical point y E Y, which is nondegenerate, whose critical wlue c E R satisjies a < c < b, and for which X u = f  ’ ( { a j ) and X b = f  l ( ( b ) ) .Then X , is surgically related to xb.
s
PROOF: One has j’ = c  lu11’ + llull’ in some neighborhood of the critical point y, as in Lemma 5.14, with (u,u) = (0,O) at y itself. One can choose c > 0 sufficiently small that a < c  E’ and c + E’ < b, and also sufficiently small that lies in the preceding neighborhood of y. By the regular interval lemma (Lemma 5.13) X , is diffeomorphic to X C   E and L X C + Eis2 diffeomorphic to X,, so that it remains to show that X c  E 2is surgically related to X,,,,. Let Y, = f  ‘([c  E’, c + E’]) n Dp2:4, and let Y, be the closure in Y ofthe complement fl([c&’, C+E’])\Y~. Then X,,, n Y, and X,+,* n Y, are diffeomorphic to S P  ’ x 0 4 and DP x S 4  respectively, by Lemma 5.14, and since X c   c zn Y2 and X,,,, n Y, are mutually diffeomorphic, by the regular interval lemma, it follows that X, tl is surgically related to X C + E 2as, required.
’,
5.16 Lemma: Let Y be a smooth compact ndimensional manifold whose boundary is the disjoint union of two smooth closed (n  1)dimensional manifolds X , and X , . Then there is a smooth function Y [0,1] with X , = f  ’ ( { O ) ) and X , = f  ’ ( { l ) ) , and with only jnitely many critical points, all
s
273
5. Stiefel ~WhitneyNumbers
qf which are nondegenerate, whose criticul levels ure mutuully distinct real numbers in the open interval(0, I).
PROOF: We omit the proof of this classical result, which is Lemma 1 on pages 4142 of Milnor [9] and Theorem 2.5 on page 9 of Milnor [18]. It also follows from Corollary 6.8 on page 37 of Milnor [131 and from Theorem 1.2 on pages 147148 of M. W. Hirsch [4]. A function Y [0,1] satisfying the conclusion of Lemma 5.16 is an udmissible Morse function. 5.17 Theorem: Two smooth closed manifolds X, and XI ofthe same dimension ure surgicully equivalent if and only if they are cobordant.
PROOF: According to Lemma 5.12 any two surgically equivalent smooth closed manifolds are cobordant. Conversely, if the disjoint union of two smooth closed ( n  1)dimensional manifolds X , and X I is the boundary of a smooth compact ndimensional manifold Y, Lemma 5.16 provides an s [0,1] with critical levels c l , . . . , c, satisfying admissible Morse function Y + 0 < c1 < . . . < c, < 1, where r is the number of (nondegenerate) critical ~ ) 4 = 1 , . . . , r  1, let a, = I, points. Let a, = 0, let uq = +(c, c ~ + for and let X,,q = j  ' ( { a , , ; ) for 4 = 0,. . . , r. Then Lemma 5.15 asserts that X u qis surgically related to X u q +I for 4 = 0, . . . , r  1, so that X , is surgically equivalent to X I , as claimed.
+
5.18 Corollary : The semiring Y1' of surgical equivalence clusses of smooth closed muniJolds is canonically isomorphic to the unoriented cobordism ring fl.
PROOF: Since addition and multiplication in each of 3' and fl are induced by disjoint unions and smooth products of smooth closed manifolds, this follows immediately from Theorem 5.17. 5.19 Corollary: Two smooth closed manifolds represent the same surgical equivcilence cluss (f and only if they have the same StiefelWhitney numbers.
PROOF:This follows immediately from Proposition 5.10 and Theorem 5.17.
One can extend Definition 5.1 in an obvious way without altering the preceding results. Let 2/2 [o1, . . . , a,] be the polynomial ring over 2/2 in ~ assigned degree p , for p = 1,. . . , n; that is, each monomial which each ( T is 0;'. . . T;(: E 2/2 [ol,. . . , (T,,] is assigned degree r 1 + 2r2 + . . . + nr,. A polynomial p(o,,. . . , (T") E 2/2[0,, . . . ,o,,]is homogeneous of' degree n if it is a sum of monomials of degree 17. For any smooth closed ndimensional manifold X , there is then an element p ( w l ( ~ ( X ). ). ,. , w , ( r ( X ) ) )E H " ( X )
274
VI. Unoriented Manifolds
providing a sum ( p ( w l ( r ( X ) ) ., . . , w , ( r ( X ) ) ) , p x )E 2/2 of the StiefelWhitney numbers of Definition 5.1, also called a Stiefftl Wliitney number. For later convenience, we define particular such StiefelWhitney numbers. Let 2/2 [ t , , . . . , t,] be the polynomial ring over 2/2 in which each t, is assigned degree 1, for p = 1,. . . ,n. According to the fundamental property of elementary symmetric polynomials, there is a unique s,(~~,. . . , G,) E 2 / 2 [ G ~ ,. . . , GJ, which becomes t; + . . + t: E 2/2 [ t , , . . . , t,] when one replaces G,,. . . , G, by the elementary symmetric polynomials t l + * * . t , , . . . , t , * * * t , ; clearly s,,(o1, . . . , a,) is homogeneous of degree n in the sense of the preceding paragraph. The following definition will be used in Remark 9.23.
+
5.20 Definition: For any n > 0 and any smooth closed ndimensional manifold X , the number ( s n ( w l ( r ( X ) ) ., .
. ?
w , , ( . ( X ) ) ) , ~ xE) 2/2
is the basic StiefelWhitney number s , ( X ) of X .
6 . StiefelWhitney Genera The set 42 of diffeomorphism classes of smooth, not necessarily orientable, closed manifolds X is a commutative semiring with respect to disjoint unions and smooth products. The subset 9 c 42 of sums of two copies of any X E +2 is an ideal in 42, in the obvious sense, and there is an equivalence relation in % with X , X , ifand only ifthere are elements X , + X , and X , + X , in 9 with X , X , + X , = X 1 X , X , . The algebraic operations in 0, induce corresponding operations in the quotient %/9, and if [ X I E %/Sis the equivalence class of X E 42, then the relation [ X I + [ X I = 0 E %/9 implies that 42/9 is a commutative ring 4 in which every element is of order two. The quotient of 42 by the ideal represented by smooth closed boundaries is precisely the unoriented cobordism ring !Ti of Definition 5.9: the natural epimorphism @ + !Ti factors out boundaries. The purpose of this section is to introduce certain other homomorphisms 42 + 2/2 which will serve as models for later constructions.

+
+
+
For convenience we use the same notation X to denote either a smooth closed manifold in 42 or its equivalence class [ X I E 08. Since every element of 4 is of order two, 4 can be regarded as a (graded) 2/2algebra. For any ndimensional X E % the 2/2 cohomology module H 4 ( X ) vanishes for q > n, so that H * * ( X ) = H * ( X ) . Let r ( X ) be the tangent bundle
275
6 . StiefelWhitney Genera
of X, and for any formal power seriesf(t) E 2/2[[t]] with leading term 1 E Z/2 let u,(z(X)) E H*(X) be the resulting multiplicative Z/2 class, as in Definition V.1.4. Since X is closed, in the usual sense that its boundary X is empty, the fundamental Z/2 homology class of Definition 1.4 is an element p x E H,(X). The Kronecker product (u,(z(X)),pX) E Z/2, in which the homogeneous element px of degree n annihilates all cohomology classes except in degree n, clearly depends only on X as an element of 42. 6.1 Definition: For any formal power series f ( t )E Z/2[[t]] with leading term 1 E 2/2,and for any X E 42, the Kronecker product (uf(r(X)),p x ) E 712 is the flyenus G(f)(X) of X.

6.2 Proposition: For any formal power series f ( t ) E 212 [[t]] with leading W) 212 carrying each term 1 E 212, there is a semiring homomorphism g# X E u& into the flyenus G(f)(X) E 212.
PROOF:Since addition in 42 is induced by disjoint union of manifolds, the additivity of G ( f ) is trivial. For any X, E 42 and X , E 42 the tangent bundle ofthe product X, x X, E 42 is given by z(X, x X,) = pr\t(X,) 0 pr\.r(X,) = t ( X , ) + z(Xz),for the projections X I x X2 % X , and X , x X, X,, so that the crossproduct version of the Whitney product formula (Proposition V.2.11) gives u/(r(X, x XZ)) = U f ( T W 1 ) + G,)) = Uf(.r(Xl)) x UJ(dX2)) E H*(X, x X2). The characterization of fundamental 212 homology classes appearing in Corollary 1.5 immediately implies that the fundamental Z/2 homology class p x I x X 2 ~ H * (xXX,)ofX, 1 ~ X ~ i s t h e c r o s s p r o d u c txpp~x, , ~ H , ( X 1xX,) of the fundamental 212 homology classes p X , E H,(X,) and p x z E H,(X,). Since pxl and p x 2are homogeneous, their degrees being dim X , and dim X,, the classical relation between cohomology crossproducts and homology crossproducts implies G(.f)(Xl x X,)
(u,(.r(X, x X2)L PX] X X J = (U,(W,)) x Uf(.r(X2)),pxl x P x , ) = (Uf(.r(X,))?PX1)(~f(4x2))9 Px,) = G ( f ) ( X l ) .G ( f ) ( X , ) , =
as desired. (See page 217 of Dold [8], e.g., for the crossproduct relation; there are no & signs in the present computation because the coefficient ring is 212.)
276
VI. Unoriented Manifolds
One can equally well apply the term figenus to the 212algebra homomorphism $2 % 212 induced by Proposition 6.2. Any formal power seriesf'(t)E 2/2 [ [ t ] ] with leading term 1 E 2 / 2 induces P,(ul,. . . , u,) E 212 [ [ u , , u 2 , . . .]] as in a multiplicative sequence Definition V.2.9. Furthermore; if X is of dimension n, then Proposition V.2.10 guarantees that G ( , f ) ( X )= ( P n ( w l ( T ( X ) ).,. . , w n ( t ( X ) ) ) , p x )Thus . G ( f ) ( x )is a Z/Zlinear combination of the StiefelWhitney numbers of X ; it is therefore reasonable to call G ( f ) a StiefelWhitney genus, as in the title of this section. According to Proposition 5.2 this remark implies that G ( f ) ( X )= 0 whenever X is the boundary Y of a smooth compact (n 1)dimensional manifold Y . Consequently 42 212 annihilates the kernel of the natural epimorphism % + 3 onto the unoriented cobordism ring \n, so that G ( f ) can equally well be regarded as a HJ2algebra homomorphism 3 + 212, which we shall also denote G ( f ) . However, there are many nonzero elements of 3 which lie in the intersection of the kernels of all Z/2algebra homomorphisms 3 t H/2, as we shall learn in Remark 9.33. The simplest StiefelWhitney genus O 2 212 is induced by the polynomial f ( t ) = 1 t E 2/2 [ t ] (c212 [ [ t ] ] ) .In this case the corresponding multiplicative sequence C n 2 0 ~ n ( u.l.,. , u,) satisfies p,(ul,. . . , u,) = u, for each n > 0, as we observed following Definition V.2.9. Hence if X is a smooth closed ndimensional manifold one has u J , , ( t ( X ) )= w,(T(X))E H " ( X ) , and by Corollary V.2.8 w , ( t ( X ) ) is the 212 Euler class e ( z ( X ) )of z ( X ) . Consequently
+
+
G ( f ) ( X )= ( u f ( G ) ) , P x ) = ( u f , , ( W ) , Px) = ( W f l ( d X ) ) , P x )= ( e ( 4 v ) ? P x ) E 212. 6.3 Definition: For any smooth closed manifold X characteristic x 2 ( X )is the element ( e ( t ( X ) )px) , E 212.
E
42 the 212 Euler
Since we have just observed that the H/2 Euler characteristic is merely a specialized StiefelWhitney genus, Proposition 6.2 guarantees that it too can be regarded as a ZJ2algebra homomorphism @ %212 or as a 212algebra homomorphism 3 3 212; the kernel of the latter homomorphism is geometrically characterized in Remark 9.25.
7. 2/2 Thorn Forms For any smooth closed manifold X E 02 let H * ( X ) 2H J X ) be the 2/2 Poincare duality map npxof Corollary 2.4and let H * ( X ) @ H * ( X ) 212 be the corresponding bilinear symmetric H/2 PoincarP form, with
277
7. 212 Thorn Forms
E 272 for any (a,8)E H * ( X ) x H * ( X ) ;since is nondegenerate. We shall construct a bilinear symmetric 2/2 form H J X ) 0 H J X ) 22/2 which is dual to ( , )p, in the obvious sense, hence a specific inverse H , ( X ) 5 H * ( X ) of the 2/2 Poincare duality map. The form ( ,)T is an element j*T, E H"(X x X ) , where n is the dimension of X ; its behavior will provide the major result of the next section. In this section itself we shall furthermore show that if X 4 X x X is the diagonal map, then A*j*T, is the 2/2 Euler class e ( s ( X ) )E H " ( X ) of the tangent bundle r ( X ) . As in the rest of the chapter, all coefficients lie in 2/2.
(a,/?Ip = ( a . DPP) = ( a
u /?,px)
D, is an isomorphism
(,)p
Let E : X represent the tangent bundle r ( X ) of a smooth closed manifold X E &. One can identify X with the image a ( X ) c E of the zero section X E , and since one can also identify X with the image A ( X ) c X x X of the diagonal embedding X 5 X x X there is a canonical diffeomorphism a ( X ) .+ A ( X ) . One can extend o ( X ) .+ A ( X ) in many ways to a diffeomorphism of open neighborhoods of a ( X ) c E and A ( X ) c X x X , as in the following lemma. Recall that two maps i A X x X and ,6 3X x X are homotopic relative to a subset a ( X ) c E if they are the restrictions to i x {O) and 6, x { 1 respectively, of a map x [0,1] 5 X x X , such that F(x, t )E X x X is independent of t E [0,1] whenever x E a ( X ) . In the following lemma X x X % X and X x X 3 X are the first and second projections of the product X x X .
I,
7.1 Lemma: Let E 1 :X represent the tangent bundle T ( X ) of a smooth closed manijbld X E 4Y. Then there are embeddings E 3 X x X and 2 X x X of' some open neighborhood E c E of the image a ( X ) c E of the zerosection X 5 E, which restrict to the d?fSeomorphism a ( X ) + A ( X ) , and such that each 01' the compositions
X ~ E F " ' X X X  = + X and XE'X
n

F
x XX
is the identity on X . Furthermore, there is a homotopy F ,from F , to F , , relative t o t ( X ) c E, which induces a homotopy from the restriction
, 6 \ a ( X ) A X x X\A(X) to the restriction i\o(X)
AX x
X\A(X).
278
VI. Unoriented Manifolds
PROOF: The initial stage of the construction can be found on pages 108109 of Bishop and Crittenden [13, on pages 3240 of Helgason [13, or on pages 3240 of Helgason [2]. One introduces an afJine connection in T ( X ) ,such as the LeoiCiuita connection associated to a riemannian metric on z ( X ) . This provides an exppential map Ex X for each x E X , defined on an open neighborhood Ex c Ex of 0 E Ex,with the following properties: each exp, is a diffeomorphism such that expxO = x E X , and there is an open neighborhood c E of a ( X ) c E such that each exp, is the restriction to E, c ?I of a map + X . One can therefore define E x [0,1] 5 X x X , as required, by setting
a
F(e9 t ) = (exp,,,,(te), exp,,,,(
 (1 
04 ).
For any X E %! the El2 cohomology module H * ( X x X , X x X \ A ( X ) ) is an H*(X)module with respect to the composition

H*(X)0 H*(X x X , X x X\A(X)) pr: Q id H*(X x X ) 0 H*(X x X , X x X \ A ( X ) ) U H*(X x X , X x X \ A ( X ) ) carrying any o! 0 T E H * ( X ) 0 H*(X x X , X x X \ A ( X ) ) into ( a x 1) LJ T E H*(X x X , X x X \ A ( X ) ) . The 212 cohomology module H * ( E , E * ) is also an H*(X)module, as usual, with respect to the product
H * ( X ) 0 H*(E, E * ) carrying o! 0 U x E X let X , X\{x)

E H * ( X )0
=
n*@id
H * ( E ) 0 H * ( E ,E * )
H*(E, E*) into n*a
LJ
uH * ( E , E * )
U E H*(E, E*). For any
{ x ) x X , { x ) x X \ A ( X ) A X x X , X x X\A(X)
be the lefi inclusion, and let E x , E: 5 E, E* be the inclusion of the fiber pair over x, where E 5 X represents the tangent bundle r ( X )as usual.
7.2 Lemma: For each X E & and each X E X there is an H*(X)module isomorphism G, and a ZI2module isomorphism G,,, such that the diagram H*(X x X , X x X\A(X))
of U/2module homomorphisms commutes.
H*(E,E*)
279
7. HI2 Thom Forms
PROOF: Let i c X x X be the image F , ( i ) of the embedding F , of Lemma 7.1, define a neighborhood Y c X of x E X by setting {x) x Y = ({.XIx X ) n d, and observe that the inclusion I, induces a commutative diagram  * H*(b,D\A(X)) H * ( X x X , X x X\A(X))
H*({x) x
I': I
1
I:
X,{x)x X\A(X)) 5 H * ( ( x ) x Y,{x) x Y\A(X))

=.
H * ( X , X\ ( x )1
* H*( Y, y \ (x)1
with horizontal excision isomorphisms. If Ex = E n E x , = E n E,*, and E* = E n E*, the inclusion E x , E: 5 E, E* also induces a commutative diagram H*(E,E*) H*(&E*)
with horizontal excision isomorphisms. Since the composition
is the identity map, F , induces a commutative diagram
 
E,,E,* A { x ) x YJx) x Y\A(X) = Y, Y \ { x ~
1
1
E,E*
b,b\A(X)
with vertical inclusions, hence a commutative diagram
H*(b,d\A(X))
7 
J
H*( Y, Y\{x) )
* H*(E,E*)
4.
F!x
* H*(E,,
g,*)
280
VI. Unoriented Manifolds
with isomorphisms F: and F:,x. To complete the proof one defines G, and to be the obvious compositions of excision isomorphisms with Fg and F:,x, respectively. The proof that GI is an H*(X)module isomorphism is a direct verification. One can interchange the two factors X x X in Lemma 7.2, in which case H*(X x X, X x X\A(X)) is an H*(X)module with respect to the composition carrying
p 0 T E H*(X) 0 H*(X into (1 x
p) u T E H*(X
x
X, X x X\A(X))
x X, X x X\A(X));
for any x E X the right inclusion is X, X\(X) = X x {x), X x {xf\A(X) % X x X, X x X\A(X). As in Lemma 7.2 one then has an H*(X)module isomorphism G, and Z/2module isomorphisms Gr,xsuch that the diagrams H*(X x X,X x X\A(X))
H*(X,X\{x).)
H*(E,E*)
* 
H*(E,,E,*)
of Z/Zmodule homomorphisms commute.
In the following lemma we temporarily ignore the two H*(X)module structures of H*(X x X,X x X\A(X)). 7.3 Lemma: The isomorphisms GI and G , from H*(X x X, X x X\A(X)) to H*(E, E*) are the same H/2module isomorphisms. PROOF: GI is defined up to excision isomorphisms as the Z/Zrnodule isomorphism H*(b,d\A(X))5 H*(E, B*), and G, is defined up to the same F' excision isomorphisms as the Z/2module isomorphism H * ( b , b\A(X)) A H*($ I?*). By Lemma 7.1 there is a homotopy F from I? 3 b c X x X to k 3 b c X x X relative to 4x1 c i?, which also induces a homotopy from the restriction i \ a ( X ) 2d\A(X) c X x X\A(X)
to the restriction E\a(X) 3d\A(X) c X x X\A(.X). Hence F:
=
FT, so that GI = G,.
28 1
7. 212 Thorn Forms
7.4 Lemma: For uny E H * ( X ) and any T E H*(X x X , X x X \ A ( X ) ) one = T u (1 x p) E H * ( X x X , X x X \ A ( X ) ) ; hence
hu.5 T u ( / I x I )
j*Tu(px l)=j*Tu(l xfl)~H*(XxX)
for the inclusion X x X
X x X , X x X\A(X).
PROOF: Since the Z/Zalgebra H * ( X x X , X x X \ A ( X ) ) is commutative, it suffices to show that ( p x 1) u T = (1 x p) u T. However, Lemma 7.3 gives G , ( ( P x 1) u T )= .*[) u G,T =
.*P
u G,T = G,((l x
for the common 2/2module isomorphism G,
=
p) u T )
G,
Lemmas 7.3 and 7.4 together imply that G, and G, are the same H * ( X ) module isomorphism, which we henceforth denote
H*(X x
x,x x
x\A(x))~H*(E,E*).
7.5 Definition: For any smooth closed ndimensional manifold X E 42 the diagonal 2/2 Thom class Tx E H"(X x X , X x X \ A ( X ) ) is the inverse image G  Ur(,) of the 2/2 Thom class CJrcx, E H"(E, E*) of the tangent bundle z ( X ) ; the 2 / 2 Thom form of X E 42 is the image j*Tx E H"(X x X ) of T, under the inclusioninduced homomorphism H"(X x X , X x X \ A ( X ) ) 5 H"(X x X ) . There are useful alternative characterizations of the diagonal 212 Thom class T,. Recall from Lemma 1.1 and Corollary 1.5 that for each x E X the 2/2 homology module H,(X, X \ ( x } ) is free on the single generator jx,*px, where X 5 X , X \ ( x ) is an inclusion and p, E H,(X) is the fundamental 212 homology class. (The boundary X of any X E 42 is empty, so that H,(X, 8)= H,(X).) Hence the 212 cohomology module H " ( X , X \ { x } ) is also free on a single generator, which we denote a,.
7.6 Proposition: For any smooth closed ndimensional manifold X uny x E x , let
X , X\(x)
= {x)
X , X\[xj
=
x X , { x ) x X\A(X)
AX
E& !
and
x X , X x X\A(X)
and
X x [x), X x ( x ) \ A ( X ) A X x X , X x X\A(X)
be left and right inclusions, respectively. Then the diagonal 212 Thom class Tx E H"(X x X , X x X \ A ( X ) ) is uniquely characterized by the property that l:T, = w, E H " ( X , X\{x}) for each x E X ; similarly, T, is also uniquely characterized by the property that r,*T, = a, E H " ( X , X\{x>) for each x E X .
282
V1. Unoriented Manifolds
PROOF: To prove the first assertion it suffices to reexamine the diagram H*(E, E * )
H*(X x X , X x X \ A ( X ) )

J
GI
J
x
H * ( X ,X\{XS) H * ( L E,*) of Lemma 7.2, where GT, = Ll,(,), and to recall from Definition IV.1.4 that E H"(E, €*) is the unique class such that j,*U,,,, is the 212 Thom class UT(X, the generator of H*(E,, E:) for each x E X . The proof of the second assertion is similar. The following result is a partial rephrasing of Proposition 7.6. 7.7 Lemma: Let p x E H , ( X ) be the fundamental 212 homology class of' a smooth closed ndimensional manifold X E (42, with H/2 Thom form j*Tx E H"(X x X ) ; then ( j * T x ,1 x p x ) = 1 E 212 and similarly ( j * T x , p x x 1) = 1 E 212.
PROOF: One starts with an inclusion diagram X
=
jX
{x}x X
'I
x x x
j
+ {x} x
+
X , { x ) x X\A(X)
=
X,X\(x)
X x X , X x X\A(X).
By applying H*( ) and H,( ) and computing Kronecker products, it follows that ( j * T x ,1 x pX) = (j*Tx71,,,pX)= (l,*j*Tx,px) = ( j X T x 9 p x )= (CTxLix)*px) = (w,,(j,)*px)= 1
for the unique generators w, E H"(X, X\{x} and ( j x ) * p xE H , ( X , X \ { x ) ) . A similar proof yields the second assertion. Here is the main result of this section. 7.8 Proposition : For any smooth closed ndimensional manifold X E 02,let j*Tx E H"(X x X ) he the 212 Thom form of Definition 7.5, and let H * ( X ) H J X ) he the PoincarP duality isomorphism n p , of Corollary 2.4. Then ( j * T , , a x DpP) = ( P , a ) E Z/2 jbr any a E H J X ) and any P E H * ( X ) , and (j * T x ,D,a x b ) = (a, b ) E 212 for any a E H * ( X ) and b E H J X ) .
283
7. L/2 Thom Forms
PROOF: Observe that if u and fl are homogeneous of different degrees then ( j * T x , u x DpP) and ( P , u ) both vanish; hence one may as well assume that u and are homogeneous of the same degree, in which case the cap product [I n u is just (p, a ) 1 E H,(X). Since the coefficient ring is 2/2, cup products commute, and one also has the classical identity (0 x cp) n ( p x v) = (0 n p) x (cp n v) relating cap and cross products (as in Spanier [4,p. 2551, for example). Consequently (j*Tx
x DpB) = ( j * T x , (1 n 4 x (b' n pX)) = ( j * T x , (1 x PI n ( a x pX)> = ( j * T x u (1 x PI, a x p x ) = ( j * T x u ( B x 11, a x px) =
( j * T x , ( P x 1) n (a x PX)) = (.i*Tx, ( B n a) x (1 PX)) px} = ( B , a ) ( j * T x ,1 x pX) = (P,Q>,
= ( j * T x , (B,u>1 x
as claimed, by Lemmas 7.4 and 7.7. Similarly, if a and b are homogeneous of the same degree one has ( j * T x , Dpa x h ) = (j*T,,(a n px) x (1 n b ) ) = ( j * T x , (a x 1) n ( p x x b ) ) = ( j * T , u ( a x l ) , p x ~ h ) = ( j * T X u x( la ) , p , x b ) ( j * T x , (1 x a) n ( p x x b ) ) = ( j * T x , ( 1 n px) x (a n b)) = ( j * T x , p x x ( a , b ) l ) = ( a , b ) ( j * T x , p x x 1) = (Sq A b ) = (a,Sqxa> = ( D T D P w , a> = ( D p w , ~ )= , (j*Tx, Dpw x a ) = (j*Tx, ( w n px) x (1 n a ) ) = (j*T,, ( w x 1) n (p, x a ) = (j*T, u ( w x I), p, x a ) = (Sqj*Tx,Px x a> = (j*TxlSqxPx x = (%X =
sqX a)T = ( D T sqX P X > sqX SqX P X a> E
pX,
( s q DT
7
by Lemmas 8.1 and 8.2. Proposition 8.3 is a variant of the original result of Thom [2] and Wu [3]. Another formulation is given in Corollary 8.5.
For any smooth closed manijold X E 42 and any formal power series f ( t )E 212 [ [ t ] ] with leading term 1 E H/2, the 212 multiplicative class u f ( z ( X ) )E H * ( X ) depends only on the algebraic structure of H , ( X ) and H * ( X ) ; in particular, u f ( r ( X ) )is independent of the smooth structure assigned to X .
8.4 Theorem (Thom and Wu):
PROOF: Since the fundamental H/2 class px E H , ( X ) and the operators Sq,, D, Sq depend only on algebraic properties of H J X ) and H * ( X ) , the identity w ( z ( X ) )= Sq DT Sq, px of Proposition 8.3 implies the result for the total StiefelWhitney class, which is the special case f ( t ) = 1 + t. However, Proposition V.2.10 provides polynomial computations for any multiplicative 212 class u f ( z ( X ) )in terms of w(.(X)), which are also independent of
Go.
Since H,(X) and H * ( X ) depend only on the homotopy type of X E 42, the preceding result can be rephrased as follows: For any X E 42 the 212 multiplicative classes u ,  ( r ( X ) )depend only on the homotopy type of X . Here are some more corollaries of Proposition 8.3. 8.5 Corollary (Wu): For any smooth closed manijold X E 42 the W u class Wu(z(X)) of the tangenr bundle z ( X ) is the unique class v E H * ( X ) such that ( u u a,p,) = (Sq a , ~ , ) for every a E H * ( X ) .
287
9. Remarks and Exercises
PROOF: Since ( Du a,px) = ( 0 , ~ )for ~ the nondegenerate 2/2 Poincare form ( , )p, uniqueness is clear. It remains to observe that (Wu(r(X)) u 4 P x ) = (Sq' W(dX)) u %Px) sqX PX > cl)P = (DT sqX PX Dpa) = (a, sqx Px) = (Sq % Px) = (DT
9
by Remark V.7.20 and the reformulation of Corollary 7.9. 8.6 Corollary: !f X E "11 is of' dimension n, then Wu(r(X))E H * ( X ) vanishes in all dimensions p such that 2p > n.
PROOF: If u E H,(X), then DTu E H""(X) and (Wu(r(X)),u)= (DTSqXPX,a) = (DTa,SqXPX)
=
(SqDTa,PX).
By the dimension axiom for Steenrod squares (Remark V.7.14) SqqD,a = 0 E H"p'4(X) for q > n  p. Hence, if p > n  p one has Sqp DTu = 0 E H " ( X ) , so that ( w u ( ~ ( X ) ) , u=) 0, for every a E H,(X).
8.7 Corollary: For any X E OiY qf'dimension n > 0 one has (Wu(r(X)),px) 0 E 212.
=
PROOF: Since 2n > n, this is a consequence of Corollary 8.6.
9. Remarks and Exercises 9.1 Remark: The 2/2 fundamental class px E H,(X, X ; Z/2) of Definition 1.4 was defined only for smooth ndimensional compact manifolds X with boundary X because the given proof of its existence required results (Propositions 1.8.4 and 1.9.7) which were established only in the triangulable case. However, with additional effort one can extend Definition 1.4 to ndimensional compact topological manifolds. If an ndimensional compact topological manifold X with boundary X is oriented in the sense described in the next volume, then for any commutative ring A with unit there is a unique fundamental class px E H,(X, X ; A); the case A = Z suffices. We shall obtain this result for the smooth case in Volume 2 as an oriented analog of Proposition 1.3. As before, smoothness is not really required; proofs of the topological case can be found in Dold [8, pp. 2592671, Massey [6, pp. 2002051, Milnor and Stasheff [l, pp. 2732741, Spanier [4, pp. 2993061, and (for closed topological manifolds) in Samelson [11.
288
VI. Unoriented Manifolds
9.2 Remark: An alternative proof of the identity d p , = p i of Proposition 1.9 is given in Spanier [4, p. 3041.

9.3 Remark: Here is an alternative proof that the Z/2 Poincare duality Dp'npx H , ( X ) of Corollary 2.4 is an isomorphism. The fact map H * ( X ) that D , is an isomorphism is not used in the proof of Proposition 7.8; furthermore, the Thom form j*T, E H*(X x X ) and induced homomorphism H J X ) 5 H * ( X ) were constructed independently of D,. However, according to Corollary 7.9 the compositions DTDp and D,D, are identity isomorphisms on H * ( X ) and H , ( X ) , respectively; hence D, is an isomorphism with inverse D T .
9.4 Remark: If X is an oriented compact smooth manifold with boundary X , then one can replace the coefficient ring 2/2 of Theorem 2.3 by any commutative ring A with unit to obtain an oriented PoincareLefschetz duality theorem. We shall do so in the next volume, using the obvious analog of Lemma 2.2; a different algebraic format for the proof is given on pages 117 of Browder [l]. PoincarbLefschetz duality is also valid for topological manifolds, using the coefficient ring 2/2 except in the oriented case. One of the standard proofs is similar to the proof of Theorem 2.3; it can be found in Dold [8, pp. 2912981, Greenberg [l, pp. 1621891, Milnor and Stasheff [I, pp. 2762801, and Spanier [4,pp. 296297.1. There are also proofs of the same nature in Bore1 [3] and in Griffiths [I], using AlexanderSpanier and Cech cohomology, respectively, rather than singular cohomology. Given a smooth closed oriented manifold X , one can establish Poincare duality for arbitrary coefficients by a procedure analogous to that of Remark 9.3; the details appear in Milnor [3, pp. 5152] and Milnor and Stasheff [l, pp. 1271281. A similar technique applies to closed topological manifolds; the details appear in Spanier [3] and in Samelson [l]. Finally, there are older proofs of the PoincareLefschetz duality theorem for triangulable manifolds (a fortiori for smooth manifolds) in Mayer [2], Lefschetz [l, pp. 1882041, and Maunder [l, pp. 1701991. A survey of PoincareLefschetz duality and related topics appears in Dold [7], and a 1965 bibliography of proofs of Poincark duality appears in Samelson [l]. 9.5 Remark : StiefelWhitney (2/2 cohomology) classes of tangent bundles of smooth manifolds were first constructed in Stiefel [l] and in Whitney [2] in 1935, using obstruction theory, as noted in Remark V.7.1. Whitney presented the details of his original construction in Whitney [3,4, 61, and some related work appeared in Rohlin [3], also using obstruction theory. In
9. Remarks and Exercises
289
Pontrjagin [ 11 classifying space techniques were applied exclusively to tangent bundles to extend the results of Stiefel [I]; the same techniques were applied in the detailed exposition of Pontrjagin [5]. Chern [ I ] and Pontrjagin [2] used integral formulas to express StiefelWhitney classes of tangent bundles. Classifying space techniques like those of Pontrjagin were then developed in Chern [2] for tangent bundles; they were extended to vector bundles in general in Chern [3] and in Wu [2, 51. 9.6 Remark : The general computation of StiefelWhitney classes via Steenrod squares (Exercise V.7.17) appeared in Thom [l], which concerns arbitrary real vector bundles. However, the method was instantly applied to tangent bundles in Thom [2], Wu [31, and Thom [4]. There are other methods for computing StiefelWhitney classes of tangent bundles of smooth manifolds. For example, Bucur and Lascu [l] compute StiefelWhitney classes of smooth closed manifolds by a method originally applied in B. Segre [l] to algebraic varieties. The alternate computations of Nash [ 11 and of Teleman [131 form part of the next remark. 9.7 Remark: We already know from Remark 111.13.34 that Thom [4] shows that the fiber homotopy type of the tangent bundle z ( X ) of a smooth closed manifold X is independent of the smooth structure assigned to X ; a fortiori the same conclusion applies to the Jequivalence class of z ( X ) . (These results were subsequently strengthened in Atiyah [13 and in Benlian and Wagoner [l].) We also know from Remark V.7.10 that Thom [4] shows that StiefelWhitney classes of vector bundles depend only on the Jequivalence classes of the bundles. Consequently for any smooth (closed) manifold X the StiefelWhitney class w ( z ( X ) )is independent of the smooth structure assigned to X . (The same result was established in Theorem 8.4by methods of Thom [2] and Wu [3].) This is perhaps the best place to recall from the remainder of Remark V.7.10 that one can most easily explain the preceding result by assigning StiefelWhitney classes directly to topological manifolds X in a way which produces w ( t ( X ) )whenever X happens to be smooth. The first such construction is that of Nash [l]. A later construction begins with the microbundles of Milnor [l 1, 141, reinterpreted as tangent topological R" bundles in Kister [ l , 21, to which one assigns total StiefelWhitney classes as in Teleman [I, 2, 31, for example; the latter StiefelWhitney classes agree with those of Nash.
9.8 Remark: The combinatorial construction on page 342 of Stiefel [l] and a later sketch in Whitney [5] assign 2/2 homology classes (rather than 2/2 cohomology classes) to smooth manifolds, using barycentric subdivisions of
290
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given triangulations. These constructions were revitalized by Cheeger [11, Sullivan [13, Halperin and Toledo [11, and Latour [13, with the observation that the H/2 homology classes so assigned to a smooth closed manifold X are the 2/2 Poincare duals of the StiefelWhitney classes wi(z(X)) of the tangent bundle t ( X ) . Halperin and Toledo [l] provided the first complete proof of the duality assertion; later proofs were given by L. R. Taylor [ 11 and Blanton and McCrory [l]. Goldstein and Turner [l] give a variant of Whitney's construction which is explicitly independent of barycentric subdivision, Banchoff [l] gives a visually appealing interpretation of the construction, and McCrory [l] and Porter [l] give an intepretation which is based upon singularities of mappings of the manifold X into euclidean spaces. Related constructions can be found in Akin [l] and in Goldstein and Turner
PI
*
9.9 Remark : The preceding remark suggests that StiefelWhitney cohomology classes of tangent bundles also lead a life of their own, without the machinery of vector bundles in general. In fact, an axiomatic characterization of StiefelWhitney classes of tangent bundles is given in Blanton and Schweitzer [13, and another axiomatic characterization of such classes is given in Stong [7]. The value of the BlantonSchweitzer axioms is demonstrated in L. R. Taylor [l] and in Blanton and McCrory [l], where they are used to prove the duality theorem mentioned in the preceding remark. The key to Blanton and Schweitzer [l] lies in Remark 111.13.44. 9.10 Remark: For any smooth manifold X the classical Whitney duulity theorem in H * ( X ; H/2) is merely the Whitney product formula for the Whitney sum z ( X ) 0 vJ of the tangent bundle z ( X ) and the normal bundle vf of any immersion X R2"k; the result was first announced without proof in Whitney [6], as already noted in Remark V.7.8. An analogous duality theorem in H , ( X ; 2/2) was established in Halperin and Toledo [2], helping to verify that the H / 2 homology classes of Remark 9.8 are Poincark duals of the corresponding StiefelWhitney (cohomology) classes; the homology duality is visually interpreted in Banchoff and McCrory [l]. Finally, the suggestion formulated in Remark 9.9 is further dramatized by a direct combinatorial construction in Banchoff and McCrory [2] of the dual StiefelWhitney (cohomology) classes wi(r(X))of any triangulated manifold X .
29 1
9. Remarks and Exercises
of Exercise 111.1 3.27 to compute the total StiefelWhitney class w ( r ( G " ( R m + " ) ) E H*(G m(Rm+"of )) the tangent bundle T(G"'(R~+"))of the real Grassmann
manifold Gm(R*+").[ H i n t : Do Exercise V.7.9 first.] 9.12 Exercise: For any smooth closed manifold X the tangent bundle T ( X x X ) of the product X x X is trivially the sum t ( X ) t ( X ) . For any riemannian metric ( , ) on t ( X )one can therefore define a riemannian metric ( e ' , f ' ) for elements on t ( X x X ) by setting ( e e',f + f ' ) = ( e , f ) e, e', j ' ,J" in the total space of t ( X ) such that Tce = ~ fand ' ne' = nj', where 7c is the projection onto X . Hence one can define the normal bundle vA of the diagonal map X X x X by requiring t ( X )0 vA = A ! t ( X x X ) as in Definition 4.1. Show that vA is 7 ( X )itself.
+
+
+
>
9.13 Remark:
Some general immersion theorems were briefly described in Remark 1.10.13,and a classical nonimmersion theorem was proved in Proposition 4.5. Here is a very general necessary and sufficient immersion criterion. If X $ ( W 2 n  p is any immersion of a smooth ndimensional manifold X, then the normal bundle vf is of rank n  p and t ( X )0 vf = E ~ over ~ X, as in Definition 4.1. Conversely, according to M. W. Hirsch [l], if there is any real vector bundle 5 of rank n  p over X such that T ( X )@ 4 = . z 2 "  P , then there is an immersion X 5 RZnpwhenever X is compact. The "easy" Whitney immersion theorem of Remark 1.10.13 trivially provides an immersion X 1:R2"+' for any smooth ndimensional manifold X, and according to another result of M. W. Hirsch [l] there is a smooth homotopy x x [O, 11 + R 2 " + 'relating any two such immersions, ho and h such that each X x ( t )+ Rz"+ is also an immersion. It follows that the classifying maps X + G"+' (Ra')for the normal bundles vhuand v h , are homotopic, hence that vho = v h l ; that is, the normal bundle v h is uniquely defined by X itself. (According to Kervaire [2] this result was already known for embeddings X 1:[ W 2 " + l . In fact, for n > 1 Wu [8] had proved an even stronger result: any two smooth embetidings X 2 RZni I and X 2 [W2"+' of a smooth ndimensional manifold X are isotopic in the sense that there is a smooth map x x [O, 13.5 [WZf1' such that each X R2"+' is itself a smooth embedding.) One can always weaken any two immersions X L R 2 n  pand X 3 Rz'8q with normal bundles vf and vg, using inclusion maps R2"p+ R 2 " + ' and [WZnq [ W a n + 1 to obtain immersions X R2"p + R 2 " + 'and X 3 R2"q + R2"+ with normal bundles vf 0 E P + ' and vg 0 .zq+ respectively. According to the preceding paragraph one then has vf 0 E P + = vg 0 E ~ + ; that is, in
',
~
'
', '
'
~
29 2
VI. Unoriented Manifolds
the terminology of Remark 111.13.36,any two normal bundles of X determine the same stable equivalence class, the stable normal bundle of X . By combining the preceding results it follows that the question of a R 2 n  p of a given smooth ndimensional "bestpossible'' immersion X manifold X is equivalent to the determination of the geometric dimension of the stable normal bundle of X , as in Remark 111.13.40. Specifically, if the geometric dimension of the stable normal bundle of X is n  p or less then X immerses in RZnp,at least when X is compact.
s
9.14 Remark : The "bestpossible'' immersion conjecture of Remark 1.1 0.13 was proved by Cohen [I], using results of R. L. W. Brown [2,3] and Brown and Peterson [4,5] : any smooth ndimensional compact manifold X admits a smooth immersion X + [WZn'("), where a(n) is the number of 1's in the dyadic expansion of the dimension n. According to Proposition 4.5 one can do no better. Here is an outline of the proof. Let BO(m) denote the classifying space G"([W") for real mplane bundles, as in Remark 111.13.4, and let BO denote the classifying space lim, BO(m)for stable real vector bundles, as in Remark 111.1 3.36. Since the 212 cohomology rings H*(BO(m))are polynomial rings with one generator w,(y"), . . . , w,(y") in each degree 1, . . . , rn, the identities w i ( y m 0 E ' ) = wi(y") and the naturality of StiefelWhitney classes imply that the 2 / 2 cohomology ring H*(BO) is a polynomial ring with one generator wi in each degree i > 0. Now let X be any smooth ndimensional compact manifold, and let vx be its stable normal bundle; as in Remark 111.13.36, there is a stable homoBO for v x . According to Remark 9.13, one topy classifying map X must show that the geometric dimension of vx is at most n  a(n);that is, one must factor j" in the form X + BO(n  a ( n ) )+ BO, up to homotopy. (According to the result of Wu [8] and M. W. Hirsch [I] cited in Remark 9.13, one can regard v x more concretely as an ( n + 1)plane bundle \I,, whose classifying map X + BO(n + 1) is to be factored in the form X + BO(n  a ( n ) )+ BO(n l), up to homotopy. However, the stable classifying space BO is more convenient than BO(n + 1). Let 1, c H*(BO) be the ideal of all polynomials in w l , w z , , . . lying in the kernel of all the homomorphisms H*(BO) * H * ( X ) induced by the stable homotopy classifying maps ix,for all smooth ndimensional manifolds X . According to Brown and Peterson [4,5] there is a topological space B o i l , and a universal map BO/I, 3 BO with the following two properties: (i) each .fx can be factored in the form X + BO/I, 3 BO, and (ii) the inclusion I , c H*(BO) induces a short exact sequence
*
+
0
 I,
H*(BO)
f
H*(BO/I,)

0.
293
9. Remarks and Exercises
Briefly, BO/I,, is itself a classifying space for stable normal bundles of smooth ndimensional manifolds, and its 2/2 cohomology ring is as small as possible. Brown and Peterson also obtain a weakened form M O I I , + M O ( n  a ( n ) )+ M O of a possible factorization BO/I, + BO(n  a ( n ) )* BO of,j,,. Recall from Remark 1.10.13 that R. L. W. Brown [2,3] showed that every smooth closed ndimensional manifold is cobordant to a manifold that immerses in [W"'(") . Cohen [ 11 uses this result to construct a space X , and maps j;,and g n for which the compositions X , 5BO(n  a ( n ) )+ BO and X I , 3 BO/I,, % BO are homotopic maps from X , to BO; the corresponding weakened compositions T X , 3 MO(n  a($) + M O and T X , 2MOII, % M O are then homotopic maps from T X , to M O . Cohen also constructs a map (T, such that MOII, 3 T X , 3 MOII, is homotopic to the identity and TX,,
Moil,,5TX,,
IJ,,
M O ( H ~ ( n ) )
T g,,. on
is homotopic to Tj". Consequently the composition M O / I , MO(rr  cr(n))+ M O is a new weak version of a possible factorization of j,,, homotopic to the version of the preceding paragraph; moreover, Cohen strengthens the new version toan actual factorization BOII, + BO(na(n))+ BO ofj,. It follows that the stable homotopy classifying map fx of the stable normal bundle v x of any smooth compact ndimensional manifold X E A' can be factored in the form X + BO/I, + BO(n  a ( n ) )+ BO, and hence that \fx is ofgeometric dimension at most n  a(n);consequently X immerses in [WZ" ? ( , I ) as desired, by Remark 9.1 3.
9.15 Remark: Theorem 4.8 first appeared in its present form in Chern and and Spanier [2], following unpublished work of Hopf. The result can also be found in Milnor [3, pp. 4344], Husemoller [I, pp. 2612621, and Milnor and Stasheff [I, p. 1201. There is an important partial converse in Haefliger and Hirsch [I], as follows. Let X be a smooth closed kconnected ndimensional manifold with 0 5 2k < ti  4; then X embeds in R 2 n  k  ' if and only if m n  k  l ( T ( X ) ) = 0E ' ( X ) .(A manifold is kconnected whenever it is connected and the homotopy groups n l ( X ) ,. . . , n,(X) vanish.) Thus, in this special case, the embedding criterion of Theorem 4.8 is both necessary and sufficient. The case k = 0 of the HaefligerHirsch theorem will be used in the next remark. Let X be uny smooth closed manifold of dimension n > 4; then X embeds in R2" if and only if R, ' ( ~ ( x=)0)E H "  ' ( X ) . Observe that the connectedness condition is entirely deleted; one simply applies the k = 0 version of the HaefligerHirsch theorem separately to each connected component of X . '
294
V1. Unoriented Manifolds
9.16 Remark : Some general embedding theorems were briefly suggested
in Remark 1.10.11, and a classical nonembedding theorem was proved in Proposition 4.10. This is an appropriate place for more details about the strongest known general embedding theorem. According to M. W. Hirsch [2], any smooth open ndimensional manifold whatsoever embeds in R2" Accordingly we henceforth consider only smooth closed ndimensional manifolds, for n = 1, 2, 3, . . . . The "hard" Whitney embedding X , Rz"is itselfbestpossible for smooth closed manifolds of dimensions n = 1 and n = 2; for according to Proposition 4.9 the real projective spaces RP' and RP2 do not embed in Iw' and R3, respectively. On the other hand, every orientable closed manifold of dimension n = 2 is one of the familiar orientable surfaces of some genus p , which visibly embed in R3. According to Wall [11 every smooth 3dimensional manifold X embeds in R5. The "hard" Whitney embedding X + R2" is bestpossible for smooth closed manifolds X of dimension n = 4; for according to Proposition 4.9 the projective space RP4 does not embed in R7. However, there is a conjecture that every smooth closed orientable 4dimensional manifold does embed in R7, and many special cases of this conjecture have been verified in M. W. Hirsch [3], Watabe [l, 21, and Boechat and Haefliger [l, 21, for example. At the end of the preceding remark we learned that a smooth closed manifold X of dimension n > 4 embeds in R2"' if and only if the dual StiefelWhitney class W, I(z(X))E H "  ' ( X ) vanishes. However, according to Massey [l, 31 and Massey and Peterson [l] one has W n  ' ( T ( X ) = ) 0 for all orientable manifolds of any dimension n > 1 and for all nonorientable manifolds whose dimension n is not of the form 2'. It follows that every smooth closed orientable manifold of dimension n > 4 embeds in R2"'; the same result is true for smooth closed nonorientable manifolds of dimension n # 2'. Incidentally, the proviso n # 2' is necessary in the nonorientable case; for according to Proposition 4.9 the projective space RP" does not embed in RZn' for n = 2'. The preceding results justify the conclusion of Remark 1.10.11 that, except for easily identifiable exceptions, every smooth ndimensional manifold can be embedded in RZn'. The case of orientable 4dimensional manifolds is probably not an exception to the general rule; however, at the moment it is not known whether every smooth closed orientable 4dimensional manifold embeds in R'. Portions of the preceding results were obtained independently by other authors. S. P. Novikov [l] showed that every simply connected smooth manifold ofodd dimension n > 6 can be embedded in R 2 "  ' . Wu [9] showed
'.
29 5
9. Remarks and Exercises
that every smooth closed orientable manifold of dimension n > 4 can be embedded in Rz" Rohlin [6] showed that every smooth notiorientable 3dimensional manifold embeds in R5. 9.17 Remark: In view of the preceding abundance of smooth embeddings X RzfI 1 of smooth ndimensional manifolds X , it is of interest to examine the normal bundles ti,. of such embeddings. According to Massey [2], vf has a nonvanishing section if and only if w 2 ( v f )u w ,  , ( v f ) = 0 E H " ( X ; 2/2); since w ( v f ) = E(T(X)),as in Lemma 4.2, the latter condition is just W 2 ( . r ( X ) u ) \ t ' f l  2 ( T ( x ) ) = 0 E ff"(X; 2/2). 9.18 Remark: The "bestpossible" embedding conjecture of Remark 1.10.11 states that every smooth ndimensional manifold X embeds in R2"'(")+ I . One justification for the conjecture is the following result of R. L. W. Brown [2,3]: If X is closed, then it is cobordant to a smooth closed ndimensional manifold which does embed in R Z f I  ' ( ' ' ) + l . There is a related result in R. L. W. Brown [4]: a necessary and sufficient condition for X to be cobordant to a smooth closed ndimensional manifold which embeds in S n t k is that all StiefelWhitney numbers involving % i ( ~ ( Xfor ) ) i 2 k vanish. According to M. W. Hirsch [13, if a smooth closed ndimensional manifold X admits a smooth embedding X R 2 n  k t 1for which the normal bundle vf has a nowherevanishing section, then there is an immersion X + R 2 n  k . The condition on vf is necessary since Mahowald and Peterson [l] construct a smooth closed ndimensional manifold X with a smooth R 2 "  k t 1, for which there is no immersion X + R 2 n  k embedding X whatsoever; since k > a(n) in this example, there is no contradiction with the "bestpossible" embedding conjecture. 9.19 Remark: Immersions and embeddings of real projective spaces RP" are of special interest, partly because the nonimmersion and nonembedding techniques of Propositions 4.3 and 4.9 can easily be extended to other dimensions. However, the following Exercise does not in general represent bestpossible results.
+
9.20 Exercise: Show that if n = 2' q for q 2 0, then RP" does not immerse in [ w 2 * * '  2 and does not embed in R2" 
'.
9.21 Remark : Many bestpossible immersions and embeddings are known for real projective spaces RP", and there is now a large literature on the subject. Most of the results known through 1979 are catalogued in Berrick [l]; there are also earlier catalogs and large bibliographies in Gitler [l] and in James [ 1, 21.
296
VI. Unoriented Manifolds
The problem of enumerating immersions and embeddings (up to isotopy) of real projective spaces is considered in Larmore and Rigdon [l, 21 and in Yasui [2, 31, for example.
9.22 Remark : Bestpossible immersions and embeddings of complex projective spaces CP" are also of interest, although most of the known results involve techniques which will be introduced only in later chapters of this work. For example, for any n > 0 Atiyah and Hirzebruch [ 11 show that the complex projective space CP", of real dimension 2n, does not embed in R4, 2a(n). Sanderson and Schwarzenberger [11 use this nonembedding theorem to show for certain values of n that CP" does not immerse in [ W 4 n  2 n ( n ) and Sigrist and Suter [I] find necessary conditions for which CP" does immerse in R4"2a(n)'. A s of 1977 all known nonimmersion results for CP" were consequences of a general technique of Davis and Mahowald [2]; these results depend on the geometric dimensions of Whitney sums my: of the canonical complex line bundle y i over CP", and on the AtiyahTodd number i(m, n), briefly mentioned in Remark 111.13.42. (According to Remark 111.13.29 the corresponding immersion problem for real projective spaces RP" is equivalent to finding the geometric dimensions of Whitney sums my: of the canonical real line bundle y i over RP", for all m > 0 and n > 0.) Yasui [11 enumerates certain embeddings of complex projective spaces CP" (up to isotopy). Oproiu [2] generalizes some of the nonembedding results for real projective spaces RP" ( = G'(R"' ')) to the particular Grassmann manifolds G2(Rn+')and G3([W"+3),and Opriou [3] contains further such generalizations; immersions of Grassmann manifolds are considered in Hiller and Stong [l]. Kobayashi [l] considers certain immersions and embeddings of lens spaces.
',
9.23 Remark: Following some initial work of Rohlin [I, 2,4], the unoriented cobordism ring % was completely computed in Thom [3,681. Specifically, YI is a graded polynomial algebra 2/2 [x2, x4, xs,x6,x 8 , . . .] with one generator x, in each degree n not of the form 24  1. Furthermore, each generator x, can be represented by any smooth closed ndimensional manifold X , such that the basic StiefelWhitney number of Definition 5.20 satisfies s,(X,) = 1 E 2/2. Specific manifolds representing the generators xs and x Z mfor all m > 0 were constructed in Thom [8], and manifolds representing the remaining generators of 91 were constructed in Dold [3]. Alternative sets of manifolds X, representing the generators x, of % are given in Milnor [17], in Stong [ 6 ] , and in Royster [l], for example. Complete expositions of Thom's computation of the unoriented cobordism ring YI are given in Liulevicius [l, 31 and Stong [2]. There are also
9. Remarks and Exercises
297
surveys containing some of the computation in Milnor [lo], Rohlin [S], Poenaru [I], and Gray [ 11. Quillen [1,2] applied jorrnal groups to compute the unoriented cobordism ring 91 via Steenrod squares, and Brown and Peterson [3] used Steenrod squares in an entirely different way to compute '32. Quillen's approach is given in detail in Brocker and tom Dieck [l], and surveys of formal groups and their applications to the computation of '32 appear in Schochet [l], Buhitaber et al. [13, and Karoubi [ 13.
9.24 Remark: Unoriented cobordism is a relatively weak equivalence relation, so that one expects each class in '32 to be represented by at least one manifold with special properties. For example, it has already been noted that according to R. L. W. Brown [2,3] each class in '32 contains at least one representative satisfying the "bestpossible" immersion conjecture and at least one representative satisfying the "bestpossible'' embedding conjecture. According to Stong [4], each class of positive degree in '32 can also be represented by a fibration over the real projective plane R P 2 . 9.25 Remark: Certain classes in 91 contain representatives with further special properties, especially those classes in the kernel of the 2/2 Euler characteristic ' 3 2 3 2/2 of Definition 6.3. According to Conner and Floyd [l], an unoriented cobordism class lies in the kernel of x 2 if and only if it can be represented by the total space of a fiber bundle over S' with structure group 2/2. According to Stong and Winkelnkemper [I], such classes are characterized by the property that there is a representative which admits a locally free action by the product group S' x S'. According to Iberkleid [l], such classes are also characterized by the property that they can be represented by at least one smooth closed manifold X whose tangent bundle s ( X ) splits into a Whitney sum of real line bundles. This implies that every smooth closed manifold is cobordant to a smooth closed manifold X such that the Whitney sum r ( X ) @ c 1 splits into a Whitney sum ofreal line bundles; furthermore, according to Stong [ S ] , if 2 2 2k c n, then every smooth closed ndimensional manifold is cobordant to a smooth closed manifold X such that t ( X ) = 5 @ q for a 2kplane bundle 5 and an (n  2k)plane bundle q. Finally, according to R. L. W. Brown [ 13, every even degree cobordism class in the kernel of !It 52/2 can be represented by a fibration over the 2sphere S2. 9.26 Remark: One can obtain results similar to those of the preceding remark by imposing restrictions on other StiefelWhitney numbers than the one { w " ( r ( X ) )p, x ) which defines the 2/2 Euler characteristic. For example,
29 8
VI. Unoriented Manifolds
the result of R. L. W. Brown [l] continues as follows: If X is a smooth closed manifold of odd dimension n, then X is cobordant to a fibration over S2 ,whenever (w2(z(X))u W ,  ~ ( T ( X ) ) = , ~0. ~) According to Milnor [17] the unoriented cobordism class of a given manifold X contains a complex manifold if and only if all StiefelWhitney numbers constructed from at least one odddimensional StiefelWhitney class w,(t(X)) vanish; this happens if and only if the unoriented cobordism class of X also contains the square Y x Y of a smooth closed manifold Y. Finally, Yoshida [3] shows that if X is a smooth closed ndimensional manifold such that all StiefelWhitney numbers other than those constructed from wl(z(X)), . . . , W , _ ~ ( Z ( Xvanish, )) for some k 5 6,then X is cobordant to a smooth closed manifold with at least k linearly independent vector fields. (The special case k = 1 is an instant corollary of the result of Conner and Floyd [l] described in Remark 9.25.)
9.27 Remark: In some cases the vanishing of certain StiefelWhitney numbers of a smooth closed manifold X implies that X is cobordant to a manifold for which related StiefelWhitney classes vanish. Here are three such results. (i) Recall from Remark V.7.28 that a (smooth closed) spin manifold X is characterized by the conditions wl(7(X))= 0 and w2(z(X))= 0.Suppose that one knows only that X is a smooth closed manifold such that every StiefelWhitney number containing one of the factors wl(z(X))or w2(t(X)) vanishes. Then, according to Anderson, Brown, and Peterson [l, 21, X is cobordant to a spin manifold. (ii) Milnor [17], already cited in Remark 9.26, has the following further consequence. If X is a smooth closed manifold such that every StiefelWhitney number containing any odd factor wZp+,(z(X)) vanishes, then X is cobordant to a smooth closed manifold X' such that w2,,+,(5(X')) = 0 for every p 2 0. (iii) Let X be a smooth closed ndimensional manifold, let i,, . . . , is satisfy 2i, 2 n + 1, . . . , 2i, 2 n 1, and suppose that every StiefelWhitney number containing one of the factors wi,(z(X)),. . . , wi,(t(X))vanishes. Then X is cobordant to a smooth closed manifold X' such that wi,(z(X')) = 0, . . . , wi,(t(X')) = 0.This result was established under the more stringent conditions 2i, > n 1,. . . ,2i, > n 1 by Reed [l], and the same version is in Wall [3, p. 171; the present version is due to Papastavridis [l, 31.
+
+
+
9.28 Exercise: Show for any n > 0 that the complex projective space CP" has the same StiefelWhitney numbers as the square RP" x RP" of the real projective space RP", hence that CP" is cobordant to RP" x RP". Given this
9. Remarks and Exercises
299
result, construct a specific cobordism from CP" to RP" x RP". (The second part of this problem is nontrivial; it is solved in Stong [3].) 9.29 Exercise: According to Corollary 5.6, for each n > 0 the real projective space RP2"' is the boundary of a smooth compact 2ndimensional manifold find such manifolds X 2 " . (See Husemoller [l, pp. 2622631 for example.) 9.30 Remark: The definition of surgical equivalence and the proof of Theorem 5.17, that surgical equivalence is cobordism, appeared independently in Milnor [9] and in Wallace [l41. Wallace uses the terminology "spherical modification" in place of "surgery." 9.31 Remark: The existence of the admissible Morse functions required to complete the proof ofTheorem 5.17 is easily established in Milnor [13, 181 and in M. W. Hirsch [4], as already indicated in Lemma 5.16. Other general accounts of Morse theory are given in Pitcher [11 and Morse and Cairns [I], for example. Admissible Morse functions are also easily constructed for any smooth manifold X by means of the following technique, in Morse [11. Given any embedding X + R", there is a dense set of linear functionals R"' + R whose restrictions to X are smooth functions all of whose critical points are nondegenerate. It remains only to separate critical values, as in Smale [l, 21. 9.32 Remark: Given a smooth manifold X , it is of interest to construct an admissible Morse function X 3 R with as few critical points as possible. According to M. W. Hirsch [2], if X is not closed, then there is a nonconstant smooth function X R with no critical points whatsoever; hence the question centers on closed manifolds. According to Reeb [I] and Milnor [I], any smooth closed ndimensional manifold X with only two critical points is homeomorphic (but not necessarily diffeomorphic) to the sphere s". A related set of conditions, involving an entire family of Morse functions, is used in Rayner [l] to characterize those manifolds X which are djfeomorphic to S". One easily constructs an admissible Morse function RP2 5 R with only three critical points. Eells and Kuiper [l, 21 and Banchoff and Takens [11 study other manifolds with the same property. Specific admissible Morse functions are constructed for Grassmann manifolds in Hangdn [I] and in Alexander [I], for lens spaces in Vrhceanu [11, and for other special cases in Vrgnceanu [2] and Masuda [11, for example.
300
VI. Unoriented Manifolds
Aside from the intrinsic interest of constructing admissible Morse functions with a minimal number of critical points on a given smooth manifold X , one is interested in relating the latter number to the LjusternikSchnirelmann category of X (Remark 1.10.2),a relation which is studied in Threlfall [ 11 and Takens [ 11, for example. A similar question for cobordism classes of smooth closed manifolds is studied in Mielke [l41. 9.33 Remark: Since any Z/Zalgebra homomorphism '31 % 2/2 whatsoever can only assume one of the values 0 E 2/2 or 1 E 2/2 on each of the generators x2 E '31 and x4 E '31, for example, it follows that the element (x2)3x4+ xZ(x4)' E '31 of degree 10 lies in the kernel of every such cp. Consequently any manifold X E 02 representing the element (X$X4 + x2(x4)2E % lies in the kernel of every StiefelWhitney genus 42 3 2/2. Thus one cannot compute all StiefelWhitney numbers in terms of StiefelWhitney genera: there are smooth closed manifolds that are not boundaries all of whose StiefelWhitney genera vanish. It is probably not difficult to compute the ideal n,kerG(f) c of those unoriented cobordism classes lying in the kernel of every StiefelWhitney genus '31 % 2/2, or to compute the quotient 91/(), ker C ( f ) . However, the geometric significance of such computations is not clear to the author.
9.34 Exercise: Observe that the structure theorem '31 = Z/2[x2, x4, x5,x 6 , x8,.. .] of Remark 9.23 implies that every smooth closed 3dimensional manifold is the boundary of a smooth compact 4dimensional manifold, a result first announced in Rohlin [l]. It follows from Proposition 5.2 that all the StiefelWhitney numbers of any smooth closed 3dimensional manifold vanish. Prove the latter statement directly, without using Remark 9.23. 9.35 Exercise: Show that the 2/2 Thom formj*T,,,, E H"(RP" x RP") of any real projective space RP" is the sum x p + q = n e ( y ! )xP e(~!)~,for the generator e(y,') E H'(RP"). 9.36 Exercise: Show that the 272 Thom form j*T,, E H"(S" x S")of any sphere s" is the sum 1 x w(Sn) w(S") x 1, for the generator w(Sn)E H"(S").
+
9.37 Remark: The conclusion of Theorem 8.4 can be reformulated as follows: For any smooth closed ndimensional manijold X the Stiejhl Whitney classes w ~ ( T ( X ).) ., . , w , ( T ( X ) ) ure independent uj' the smooth structure assigned to X . This result was also formulated in Remark V.7.10, with indications of the alternative proof given in Thom [4].
30 1
9. Remarks and Exercises
9.38 Remark: According to Remark V.7.27 there are no “universal” polynomial relations P ( w l ( t ( X ) )., . . , w , ( z ( X ) ) )= 0 E H P ( X )of weighted degree p which are valid for every smooth closed manifold X of dimension n 2 2p; this was proved independently in E. H. Brown [2] and in Stong [l]. However, the total W u class W u ( z ( X ) )= 1 + W u , ( . c ( X ) )+ . . . + W u , ( z ( X ) ) of the tangent bundle T ( X ) of an ndimensional manifold X is of the form P p ( w l ( z ( X ) ). ,. . , w,,(T(X)), Pp(ul,. . . , up) being the multiplicative sequence associated to the “ W u series” of Exercises V.7.21 and V.7.22, where each P p ( u l ,. . . , up) is a nontrivial polynomial of weighted degree p . Since W U , ( T ( X ) )= 0 whenever 2p > n, by Corollary 8.6, the Wu polynomials P p ( u l , .. . , up) do provide nontrivial relations P p ( w l ( z ( X ) )., . . , w , ( z ( X ) ) ) = 0 E H P ( X ) which are valid for every smooth closed manifold X of given dimension n < 2p. It follows that the ideal generated by the Wu polynomials P p ( u l ,. . . , up)such that 2p > n is contained in the ideal of all polynomials P such that P ( w , ( T ( X ) ) ., . . , w , ( z ( X ) ) )= 0 for every smooth closed manifold X of dimension n. The latter ideal was completely determined in Brown and Peterson [ l , 21, as indicated in Remark V.7.27.
cp20
9.39 Remark: For any given n > 0 let P ( u , , . . . , u,) be a polynomial over 2/2 of weighted degree n such that P ( w l ( z ( X ) )., . . , w,(.c(X))) = 0 E H ” ( X ) for every smooth closed manifold X of dimension n. According to Dold [2], in this special case the polynomial P(ul, . . . , u,) then lies in the ideal generated by the W u polynomials P,(u,, . . . ,up)for 2p > n. This result does not generalize to polynomials of weighted degree less than n. 9.40 Remark: Let X Y be any map of smooth closed manifolds X , Y of dimensions m and n, respectively, and let H J X ) H,( Y) be the induced 2/2 homology homomorphism. Since X and Y are closed, one also has Poincare duality isomorphisms H * ( X ) %H J X ) and H * ( Y ) % H,( Y), as in Corollary 2.4 and Remark 9.3, with specific inverses H J X ) 3 H * ( X ) and H,( Y ) 3 H*( Y ), as in Corollary 7.9. The composition H*(X)
aH * ( X )
H*(Y)
*
DT=
Dp”
H*(Y)
is the Gysin homomorphism H * ( X ) H*( Y). Since fH proceeds in the reverse direction ( = Umkehr) of the usual cohomology homomorphism H*( Y ) I*,H * ( X ) , the Gysin homomorphism is an example of an Umkehr homomorphism; more general Umkehr homomorphisms will be constructed in the third volume of this work, along with an explanation of the notation f H and a proof of a general result which includes the following special case.
302
VI. Unoriented Manifolds
Recall from Remark V.7.14 that the Steenrod square Sq is natural in the sense that H*(Y) I
I
*
* H*(Y) I
I
J.
J.
H*(X)
f*
sq H * ( X )
s
commutes for any map X Y , and that Sq is multiplicative in the sense that it satisfies the Cartan formula Sq(a u p) = Sq a u Sq p. It is reasonable to ask whether Sq is in some sense natural with respect to the Gysin homomorphism H * ( X ) & H*( Y ) induced by any map X Y . The answer is affirmative only if one introduces a correction consisting of cup products by the Wu classes Wu(z(X))E H * ( X ) and Wu(z(Y ) )E H*( Y ) of the tangent bundles z(X) and z( Y ) . Specifically, the 2 / 2 RiemannRoch theorem asserts that
s
H*(X)
j*l
H*(Y)
*
b
H*(X)
uWu(r(XII
ss H * ( Y )  H * ( Y )
uWu(r(Y1)
H*(X)
I*
commutes. This result underscores the importance of Wu classes. The preceding 2/2 RiemannRoch theorem is a slightly specialized version of the main result of Atiyah and Hirzebruch [2], later generalized by Spanier [2]. However, it is also a special case of a much broader generalized RiemannRoch theorem, which will be proved in Volume 3. In the most useful special case the Steenrod square H*(X; Z/2) 5 H*(X; 2/2) will be replaced by the Chern character K @ ( X ) H*(X; Q) and the correction factor Wu(z(X)) E H * ( X ; 2/2) will be replaced by the Todd class td(r(X))E H * ( X ; Q), along with corresponding changes for Y .
a
9.41 Exercise: Let X be a smooth compact ndimensional manifold with boundary X and 2/2 fundamental class px E H,(X, X ) , Use couariant MayerVietoris functors on L?(@to show for each q E 2 that the cap product H"4i(X,X)HI(X) is a 2/2module isomorphism. (This is also a 2/2 PoincareLefschetz duality theorem.)
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Glossary of Notation
L
C" C P" CP'
C ( V )c R2"p CW complex CW space C W structure CW structure of projective spaces @'
C"(X)modules .F C"(X)modules .F*
any element of the complex general linear group the adjoint of A E GL(n,C) the modulus of A E GL(n,C) the category of 72graded abelian groups classifying space for a topological group G barycenter of a simplex I category of base spaces complex field, in its usual topology standard ndimensional complex vector space, in its usual topology complex projective space ( = Gl(C"+1 ) ) complex projective space (=Gl(C")) cocoon of V c X
87 87 89
45 101 9
2, Sff, 7 34 23, 34 23.34
262 20 19ff, 20 20 22.23 the complex vector space limn@" 34 in the weak topology the ring of continuous functions 116 X+R 117, 118, 119 1I9
35 1
352
Glossary of Notation the ring of X+Iw
C'(X)module d ( X ) of smooth differentials on X C'(X)module 6 * ( X ) of smooth vector fields on X C"(X)module 9 of smooth sections X + E C'(X)module 9* C"(X)module .F** M(G, F )
EG
9
.F
Y* Y** fgenus
f!5 G G x F+F
I I9 I 34, I 35. in3 127, 129, 130, 131 I22
category of families of fibers, with respect to a group action G x F+F Poincare duality map n p x inverse Poincart duality map D; ' the closed unit ndisk the differential of g E C r ( X ) the total space of a family E X of fibers (especially a coordinate bundle E : x) the total space of a universal Gbundle ya the fiber n  ' ( ( x } ) over X E X of a fiber bundle represented by E:X the exponential of A E End C" the C'(X)module of smooth differentials on X the C" (X)module of smooth vector fields o n X the 2/2 Euler class of a real mplane bundle over X E S? the fiber of any family of fibers (especially a coordinate bundle), also applied to fibre bundles (and especially to fiber bundles) a C"(X)module of (continuous) sections X + E a C"(X)module ofsmooth sections X+E the first conjugate of 9 the second conjugate of 9 a homomorphism 4 %2/2 or 'JL G U I 7 / 2 assigned to a forma1 power series f(t) E 2 / 2 [ [ t ] ] pullback of 5 along X X ' structure group action of a transformation group G (structure group) on a fiber F
r
F
smooth functions

I23 I33 61
250, 258,216 217,283 19 134, 183, 184 57, 61, 61
I02 67
87 134, 135, 183 127, 129, 130. 131 196 3, 51, 60
117, 118, 119
122 118, 119. 123 I33 214tT, 275,216
3, 63 3, 58, 60
2, 60,65
Glossary of Notation
Grelated isomorphisms Grelated set of homomorphisms G(j) GIK G U n , C) G U m , R) G L + ( m R) , G"(C" ") +
G"(Q3") G*( R" '" ) G"(R') G"(R") GI(@"+I )
GI(@') G1(R"' ')
G'(k8")
s'5 H*(Gm(R"'" ); 6/21 H*(G"(R"); Zj2) H*(RP"; Z/2) H*(RP', 212) I I , , . . . , I , and I
i(m,n ) J j* T ,
K K'
IKl*
353
Stiefel Whitney jgenus homogeneous space complex general linear group real general linear group the component of the identity in G L h k8) complex Grassmann manifold complex Grassmann manifold real Grassmann manifold real Grassmann manifold the total space O(ym)of o(y") complex projective space CP" complex projective space CP" real projective space RP" real projective space RP" pullback of ( along X 5 X'
a twosided ideal in @ V with @ V / I = AV simplexes in a simplicial complex K . which also constitute the vertices of the barycentric subdivision K' AtiyahTodd number a complex structure in a real vector bundle a Thom form ( , )T an abstract simplicial complex the first barycentric subdivision of K the vertex set of K a family of functions K + [O, I ] a family of functions KO+ [O, I] the simplicial space of K the simplicial space of K' metric simplicial space of K weak simplicial space of K the metric telescope of IKI
Ktheory
x lim,, X , , (in the weak topology) In A E GL(n,C)
Steenrod's convenient category of compactly generated spaces the logarithm of a positive element A
E
60 60 2 1 4 6 275.276 83 59.87 I, 5n 59.90 34 34 34 34 I66 34 34 34 34 63 227tT, 230 221fT, 229 199ff. 201 199fr, 200 35 9
I 89,296 I13
n 9
n n 9 17, in 17, i n 12 170 48 19.20 n7
GL(n,C)
left inclusion
278.281
Glossary of Notation any real or complex vector bundle of rank m the category of smooth manifolds and smooth maps a category of modules over a fixed commutative ring the category of Zgraded modules over the commutative ring R the natural numbers (0. I. 2,. . .;
mplane bundle
w1: N ncell (for n = 0. I, 2, . . . ) ndisk D" (for n = I , 2, 3 , . . . ) nplane bundle
2, 23ff, 25 41 45
19 19
any real or complex vector bundle of rank n
( n  1)sphere s"' (for ii = I , 2, 3 , . . .) nth type of topological space (for n = 0, I , 2,. . . ) nskeleton (for n = 0, 1, 2 , . . . ) nsphere S" (for 11 = 0, 1,2, . . .) 91
a'
O(1) c GL(1,R) O(0
psimplex (for p = 0, 1,2,. . . ) qdimensional abstract simplicia1 complex K (for q = 0, I , 2,. . . ) ydimensional simplicia1 space K (for y = 0, 1. 2,. . .)
106. 110 19
6 20 19 250. 268.213 268, 273
the unoriented cobordism ring the unoriented surgical equivalence ring the orthogonal group, usually act 3, 59, 81. 86,90 ing on [w" or on S"I the rotation group, usually acting 59, 81, 86. 90 on [w" or on S" I the orthogonal group (acting on S o ) 160 the total space of the unique coordi 162, 163 nate bundle representing an orientation bundle o ( 5 ) [i? particular, O ( y r ) is denoted Gm([wm+") and O(y") is denoted (7'""R")] the orientation bundle of a real 160, 161, 162, 163 vector bundle 5 over X E &3 the category of open sets in V c X 43 for X E the category of open sets in X E D 41 Poincare form 276, 217,283,284 the projective bundle of a real vec 3,202 tor bundle 5. or the total space of the projective bundle 8 9
1 I
v
2JV) t e(v)
a specific family of open sets in
2(k,
a category of open sets in the interior X of a manifold X real field, in its usual topology
R
1, 106, 170
43
vcx 46, 252tf. 256f
355
Glossary of Notation R"
R P" RP" Iw"
T x E H"(X x X , X x X \ A ( X ) ) ( 9 )T
u1
,(O
standard ndimensional real vector space, in its usual topology real projective space ( =GI(&!"' I ) ) real projective space ( =G1(R')) the real vector space limnR" in the weak topology right inclusion the Steenrod square the(n I)sphere(forn= 1,2,3,. . .) the nsphere (for n = 0, I , 2, . . . ) the 0sphere (consisting of two points) the diagonal Thom class of X E .A Thom form j*Tx the unitary group(usually acting on @" or on S 2 "  ' ) L/2 Thom class of a real nplane bundle 5 multiplicative Z/2 class of a real vector bundle 5 with respect to a formal power series f ( t ) E 212 "[I1 dual of a 2/2 multiplicative class
34 22. 23. 34 22, 23, 34 22 280,281 243R, 285ff 19
19 19, 162, 167
281tl' 216K 211,283, 284 59, 81, 86, 89 3. 191, 192K 194ff, 21 1 2, 3, 215tl', 216
231,238
U,(O
Y Y
V
the semiring of smooth closed unoriented manifolds the category of spaces homotopy equivalent to spaces in Steenrods category x' any of the real or complex vector spaces R" ",R" , C" ",@ (used to generate the tensor algebra @V and the exterior algebra AV = @V/I 2/2 Thom class of a real nplane bundle 5 2/2 Thom class of a real nplane bundle 5 the qth StiefelWhitney class of a real vector bundle 5 the total StiefelWhitney class of a real vector bundle 5 the dual StiefelWhitney class of a real vector bundle 5 the category of spaces homotopy equivalent to metric simplicia1 spaces the category of spaces homotopy equivalent to countable metric simplicial spaces +
#a
214 49
34
+
20tl', 21 I 21 I 4,216 2, 4, 215, 216 2,231 2, 13, 21
14
356
Glossary of Notation
x
x Y
e
H 212
the boundary of a manifold X the interior of a manifold X the boundary of a manifold Y the interior of a manifold Y the ring of integers the field of integers modulo 2
24 25 24 25
number of 1's in the dyadic expansion of n real line bundle group over X E
2, 50, 52, 246, 261.292 206lT, 207,208, 213,214 70, 71, 72, 81, 109 110,111,115, 161, 162, 172, 173. 176. 177, 178 170 3, 149 164
2
Greek iilphabet
u(n)
morphisms groups
of
transformation
universal complex mplane bundle universal real mplane bundle universal oriented mplane bundle
n(v")!r"
.. .. (and also 5,