Embeddings and Immersions

Translations of

MATHEMATICAL MONOGRAPHS Volume 124

Embeddings and Immersions Masahisa Adachi Translated by Kiki Hudson

American Mathematical Society Providence, Rhode Island

UMEKOMI TO HAMEKOMI (Embeddings and Immersions) by Masahisa Adachi Copyright © 1984 by Masahisa Adachi Originally published in Japanese by Iwanami Shoten. Publishers. Tokyo in 1984 Translated from the Japanese by Kiki Hudson 1991 MatΛewa/ics Subject Classificati^^^ Primary 57R40. 57R42. Secondary 58D10. ABSTRACT. This book provides an introduction to the theory of embeddings and immersions of smooth manifolds and then gives applications of Gromov's theorems to foliations and complex structures on open manifolds.

Library of Congress Cataloging-in-Publication Data Adachi. Masahisa, 1932(Umekomi to hamekomi. English] Embeddings and immersions/Masahisa Adachi: (translated from lhc Japanese by Kiki Hudson). p. cm.—(Translations of mathematical monographs: v. 124) Includes bibliographical references and index. ISBN 0-8218-4612-4 1. Embeddings (Mathematics) 2. Immersions (Mathematics) I. Title. II Series QA564.A3313 1993 9 3-7464 5l63'5-dc20 CIP

Copyright ©1993 by the American Mathematica] Society. All rights reserved. Translation authorized by Iwanami Shoten. Publishers The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America The paper used ίη this book is acid-free and falls within the guidelines established to ensure permanence and durability. © Information on Copying and Reprinting can be found at the back of this volume. This publication was typeset using AMS-TEX. the American Mathematical Society's TEX macro system. 10 9 8

7

65 43 21

98 97 96 95 94 93

Contents Preface to the English Edition

vii

Preface

ix

Chapter 0. Regular Closed Curves in the Plane §1. Regular closed curves §2. Regular homotopies

1 1 4

Chapter I. cr Manifolds, C f Maps, and Fiber Bundles §1. C\n itfy manifolds and C\n itfy maps §2. Fiber bundles §3. Jet bundles §4. Morse functions §5. The transversality theorem of Thom

7 15 32 37 42

Chapter II. Embeddings of C\n itfy Manifolds §1. Embeddings and isotopies §2. Two approximation theorems §3. An immersion theorem §4. Whitney's embedding theorem I: M"\subsetΚ n +1 §5. The theorem of Sard §6. Whitney's theorem on completely regular immersions §7. Special self-intersections §8. The inlersection number of a completely regular immersion §9. Whitney's embedding theorem II: M n\subsetR n

45 45 47 49 53 57 60 62 65 66

Chapter III. I m m e r s i o n s of C\n itfy Manifolds §1. I m m e Γ s i o n s and r e g u l a r h o m o t o p i e s §2. S p a c e s of m a p s : T h e a p p r o x i m a t i o n t h e o r e m §3. c h a r a c t e r i s t i c c l a s s e s §4. I m m e r s i o n s a n d c h a r a c t e r i s t i c c l a s s e s §5. T h e S m a l e - H i r s c h t h e o r e m a n d i t s a p p l i c a t i o n s §6. c r triangulations of a C r manifold §7. Gromov's theorem §8. Submersions: The Phillips theorem V

75

75 79

89 91 92

vi

CONTENTS

§9. Proof of the Smale-Hirsch theorem §10. The Gromov-Phillips theorem §11. Handlebody decompositions of C\n itfy manifolds §12. The proof of Gromov's theorem §13. Further applications of Gromov's theorem

93 94 95 97 108

Chapter IV. The Gromov Convex Integration Theory §1. Fundamental theorem §2. Proofs of the Smale-Hirsch theorem and Feit's theorem §3. Convex hulls in Banach spaces §4. Proof of the fundamental theorem

111 111 114 116 123

Chapter V. Foliations of Open Manifolds § 1. Topological groupoids §2. Γ-structures §3. Vector bundles associated with restructures §4. Homotopies of Γ-structures §5. The construction of the classifying space for Γ-structures §6. Numerable Γ-structures §7. Γ-foIiations §8. The graphs of Γ-structures §9. The Gromov-Phillips transversality theorem §10. The classification theorem for Γ-foliations of open manifolds

127 12 7 128 128 129 129 131 134 135 136 137

Chapter VI. Complex Structures on Open Manifolds §1. Almost complex structures and complex structures §2. Complex structures on open manifolds §3. Holomorphic foliations of complexifications of real analytic manifolds §4. The C-transversality theorem §5. Notes

141 141 142

Chapter VII. Embeddings of C°° Manifolds (continued) §1. Embeddings in Euclidean spaces §2. Embeddings in manifolds §3. The proof of Theorem 7.2 §4. The proof of Theorem 7.3

157 157 160 163 167

Afterword

177

References

179

Subject Index

181

146 1 50 155

Preface to the English Edition For the convenience of readers of this English edition I have replaced the original Japanese references with the appropriate references in English or French. I have also replaced some other references that are hard to obtain with those that are more readily available. 1 wish to sincerely thank Professors Kobayashi and Nomizu for their advice. I am very grateful to Dr. Kiki Hudson, who has provided an excellent translation and pointed out misprints in the original edition. Masahisa Adachi October 28. 1992

Preface Among closed surfaces the torus T 2=S lxS l can be thought of as sitting in three-dimensional Euclidean space R 1 , but the Klein bottle λ'2 cannot be realized there. This observation naturally leads us to the question "can a general «-dimensional manifold M" be smoothly embedded in Euclidean space R p ?'. Further,-it is possible to embed the circle S in three-dimensional Euclidean space R 1 , but there is more than one way to do so. For example, we cannot move one of the two embeddings below to the other via an isotopv; that is, we cannot undo the knot.

This is generalized to the problem'are two given embeddings f . g : M" —· R'' isotopic?' Since the concept of topology wasfirst established, these problems have been in its mainstream, and major contributions to solutions have come from H. Whitney and A. Haefiiger. Still further research and development can be expected in this field. In particular, the problem of classifying embeddings of the circle S1 in three-dimensional Euclidean space R 3 or the three-dimensional sphere S'1 through isotopies—a bit different from the isotopies mentioned in the previous paragraph —forms afield in topology called the theory of knots, which even today generates many research activities. The problem of classifying immersions by regular homotopies is slightly easier than that of classifying embeddings by isotopies. Here is an example. In three-dimensional Euclidean space R 5 , is it possible to turn the sphere .V" inside out smoothly allowing self-intersections? Think about it for a minute. It hardly seems likely, but a classification theorem for immersions shows that it can be done. This classification theorem, the so-called Smale-Hirsch theorem, has been generalized step by step by A. Phillips, M. Gromov. A. Haefiiger. and so on to the present stage where it now offers us a tool for finding solutions

Χ

PREFACE

(or their candidates) to partial differential inequalities or partial differential equations of certain types. It also provides us with a method lor eliminating singularites of certain C°° maps. There are further applications of these methods as well. The aim of this book is to give an introduction to this theory in modern topology and its applications. In accordance with the principle of this series we have tried to make thefirst three chapters easy enough to understand at the level of lower-division mathematics. In this book, unless otherwise stated, embeddings and immersions will be viewed in the C\n itfy category. Wefirst explain in detail the classification of regular closed curves in the plane by regular homotopies; this will serve as an intuitive preparation for the contents of the book. In Chapter I, we give a summary of basic concepts about C' manifolds and C r maps which will be used in Chapter II and beyond. The discussions in Chapter II evolve around Whitney's theorems. This chapter also serves as a prelude to Chapter VII. We develop Chapter III around the Smale-Hirsch theorem which is generalized to Gromov's theorem. In Chapter IV we examine the convex integration theory due to Gromov which is another application of the Smale-Hirsch theorem. In Chapter V we discuss an application of Gromov's theorem, namely, a classification theorem for foliations of open manifolds. In Chapter VI we study complex structures on open manifolds as an application of Gromov's theorem and Gromov's convex integration theory. We study Haefliger's embedding theorem in Chapter VII. which is a continuation of Chapter II. Finally, as references we give a list of books and papers we have either used, adapted, or quoted from directly, and also books and papers basic to embeddings and immersions. The author thanks Kazuhiko Fukui, Shigeo Kawai, and Goo Ishikawa for their valuable help in writing this book. We are deeply indebted to Professor Itiro Tamura who encouraged us to write this book and gave us valuable advice concerning thefirst draft. Last but not least our deepest gratitude goes to Mr. Hideo Arai of Iwanami Shoten Publishers, without whose help this book would never have been realized. Masahisa Adachi May 1983

CHAPTER Ο

Regular Closed Curves in the Plane In this chapter we consider closed curves in the plane R", whose tangent lines move continuously. To each closed curve we assign the "rotation number" )', which is the angle the tangent line makes going around the curve once ( γ = ±2π for a closed curve). Our aim in this chapter is to show the following: Two closed curves with the same rotation number can be deformed from one to the other. This chapter is based on Whitney [C20]. §1. Regular closed curves

Wefirst define closed regular curves. Let / = [0, 1]. Consider / = (/,,/·,), where i\ and f are C 1 functions ( / is called a C 1 map). We say that / is a parametrized regular dosed curve if it satisfies the following: (i) /(0) = / ( l ) , /(0) = / ( l ) , (ii) /(/) ^ 0, for each / e I. The condition (i) shows that the curve is closed and (ii) says that / is regular in some sense with respect to the parameter t. Sec Figure 0.1. To the above / : / —» R" there corresponds a C 1 function / / : (-oc, oc) —· R : . such that (iii) /(0 = /(0 . · ' € / , (iv) / ( / + ! ) = /('),

(v) Conversely to such an f there corresponds an f as above. We say that / is a lift of /. DEFINITION 0.1. Let / and g be parametrized regular closed curves. We say that / and g are equivalent and write f ~ g if there exists a ("' function η : (-oc, oc) —· (-oc, oc) such that >/'(/)> 0,

foreachteR,

η(ΐ + I) = »/(/)+ 1. ι

g(t) = ./' '/(Ο-

2

0. REGULAR CLOSED CURVES IN THE PLANE

ι 0

μ t

Clearly, ~ is an equivalence relation; hence, ~ divides parametrized regular closed curves into equivalence classes which arc called regular closed curves or simply curves. If / ~ g , then we have /(/) = g(l) .

Let C be a regular closed curve. Then there exis g in C such that is constant, where || || is a norm

PROPOSITION 0.1.

element PROOF.

Let f eC and let / be a lift of f . Set L(t)=

f'\\f{s)\\ds, L = L(l ),. Jo Then L = L{C) is the length of the curve C . Since f(t) Φ 0, L(t) is of class c l and monotone increasing. Hence, we can solve s = \/L • L(t) for t, say t = η(ς). η'(5) is continuous and positive. Since / is periodic we have ^ L(t+1)-L(/)

= |' +

\\f(,s)\\ds

= ^ ||/(5)|| ds.

Therefore, »/(j + 1) = t]{s) + 1 . Thus, if we set 1(0 = 7 ° 1(0, we see that g is a lift of some element g of C . Further we have *'(*) =/«>·»(*)· Τ'ΓΎ^' i- W)) So ||g'(0ll ' s constant. •

= ^

PROPOSITION 0.2. Let C be a regular closed curve, and let g be an el of C defined as above. Suppose h is an element of C such that | constant k . Then (i) k = L and (ii) h(t) = g(t + a) for some constant a . In other words, two elements of C with each ||/i'(t)|| cons some rotation of the circle S 1. PROOF. Since h ~ g there exists η : (-oc, oo) — > (-oc, oo) such that ht) = g W ) ) . But h\t) = g'wt))n'(t),

. REGULAR CLOSED CURVES

and so k = L • η' (I) . Hence, 1 =η(\)-η(0)

= J* >l\t)dt = J*

Hence, we get k - L. It follows that η (I) = I and that ;;(/) = I + a . • DEFINITION 0.2. Let f 0 and /, be parametrised regular closed curves. We say that f Q is a deformation of f l or that f Q and f arc regularly homotopic , and write /0 ~ /, if the following holds: For some continuous map F : / χ / —• R2 (i) F(t , 0) = /0 , /·(/, 1) = /,(/), and ( i i ) if we set /„(/) = F(l , it) , then f u : / — R" is a parametrized regular curve for each uel. Here we say that F or the {.l u} is a regular homotopy.

We see that the relation ~ of being regularly homotopic is an equivalence Γ relation. See Figure 0.2. PROPOSITION 0.3. Let C be a regular dosed curve, and let f 0 , J\ Then f 0 is a deformation of f in C ; that is. there e.xists a reg topy f u € C , uel, connecting f 0 and f\ . PROOF. That f 0 and f are equivalent implies that f t(t) = f some function η as in Definition 0.1. Set

'/„(') = «'1(0 + ( 1 - « ) ' , /«(') = /o°'/„(').

0 < it < 1 .

/o is a lift of ,10.

fjr' 7uii)

FIGURE 0.2

0

ο ή(ι)

for

0. REGULAR CLOSED CURVES IN THE PLANE

4

Here we have η0(ή = 1, //,(!) = η(ή . Hence, /j is a lift of ,/j , and we have !?„(/+i) = «[«i(0+ 1] + ( 1 - « ) ( < + l) = '?„(') + 1 ,

%ii)=iAW+(I_w)>0> dt

dt

-

0 < M < 1

-

Therefore, each f u is a parametric regular closed curve, and so we have the proposition. Ο By virtue of Proposition 0.3 the expression " a regular closed curve C is a deformation of a regular closed curve C' " makes sense. §2. Regular homotopies We have the following basic LEMMA 0.1. Let g : I —· R 2 be a continuous for each tel. For ρ e R 2 ,

map and suppose g(t)

At) =p + [ g{s) ds Jo is a parametrized

regular

curve if and only if

S(0) = * ( ! ) ,

f ] g ( s ) d s = 0.

The lemma is obvious. DEFINITION 0.3. For a parametrized closed regular curve / : / define the rotation number y{f) e R of / as follows: the map

/* : I —* 5 c R ,

f[t)

—» R "

we

f(t) 11/(011

defines naturally the continuous map / : S 1 —» S1 . Now define y(f)

— 2π · deg(/*),

where deg(/) is the degree of /* ('). (')In general the degree deg(A) of a continuous mapA : S 1 —> S 1 is an integer which represents the number of the times h(S') wraps around S 1 inclusive of the sign of Λ(-S"1}. The fe. 1' jwing is a more precise definition. Notice that the fundamental group of the circle S [ . , is isomorphic to Z . Let s be the generator of this group. On the other hand h defines the homomorphism

and the image of s by Λ. is n-s, of h to be this π .

« 6 Z . We define the degree or the mapping degree

dcg(li)

52. REGULAR HOMOTOPIES

g be parametrized homotopic then y(f)

PROPOSITION 0.4. LET / ,

and g are regularly

regular closed curves. — y(g) .

PROOF. Let f : I ->(/), fe C. By Proposition 0.3 the above definition does not depend on the choice of

/.

THEOREM 0.1. Regular closed curves if and only if y(C 0 ) = } ' ( c , ) .

C 0 and C, are regularly

This theorem is known as the Whitney-Graustcin

hom

theorem.

COROLLARY 0.1. The family of the regular homotopy classes of r dosed curves in the plane is in one-to-one correspondence with th the integers by the map C y(C)/2n . PROOF OF THEOREM 0.1. The 'only if part is evident by Proposition 0.4. To prove the 'if part set y ( c 0 ) = y ( c , ) = 7 · Choose g 0 e C 0 and /, e c , such that

||^(t)|| = L ( c 0 ) = L 0 .

||./;(t)|| = L ( c | ) = Z,|

(cf. Proposition 0.1). Define g u by g„(t) = *„(0) +U.L·

+ (!-«)

Lo

Then the family {g M } is a homotopy connecting g H and g, . Further as g' u({) φ 0, for each tel. the {g H } is actually a regular homotopy connecting g 0 and g t . Set / 0 = g, . We then have \\.f 0(t)\\ = ||^(ί)|| = L, . We want to show thai f 0 is regularly homotopic to f . Let λ be the circle in R 2 centered at the origin of radius L, . Then f 0 , / : / —· Λ' C R". If , fi : S l — Κ arc the natural maps corresponding to f Q and f . we have deg(jjj) = dcg(/) - y/In. Hence, f 0 and f t are homotopic. Now define 0 : R —- Κ by ()(t) = (L, cos 1, L, sint). (i) For γ φ 0 we have 0(0) = (L,, 0). Without loss of generality we may assume that j^(O) = /(0) = 0(0). As fj(t) e Κ , 1 = 0,1, denoting by F.(/) the argument of /(/), we have the following F : I —> R ,

/,(!) = β oF,(t).

1 = 0,1.

F(0) = 0.

6

0. REGULAR CLOSED CURVES IN THE PLANE

Then by the definition of and the assumption on y we see that F,( 1) = 7,

1 = 0,1.

Now set ' F u{t) = uF 1{t) + (1 ~u)F 0(t),0 < U < 1 . h u{t) = θ Ο F u{t) , Then the {h u } is a homotopy connecting f Q and f t . Set «»„(') = h u(t) - f Jo

l

h„{s)ds.

(0) - /0(0)] + [' v u(s) ds. Jo Evidently /0

) is a c f manifold. A c 0 manifold is a topological manifold. Often a differentiable manifold is understood to be a c l manifold; however, in this book we agree for simplicity that a differentiable manifold is a C\n itfy manifold, which is also called a smooth manifold. Next we discuss orientations of a c\infty manifold (Μ" , 3). Let 3 = [./"]. = {(V-, φj)\ j € J } . For χ e V Γ V let α ·,·(*) be the Jacobian matrix of φ )οφΤ ι at ψ :(χ): a, i(x) = D{

υ = and say that [U , 9f v) is an open submanifold cf (Μ" , 3>). This definition does not depend on the choice of a representative 5 .

10

I. C' MANIFOLDS C' MAS, AND FIBER BUNDLES

(4) Submanifolds. Let (M n ,3) be a C\n itfy manifold, and let A be a subset of M" . Regard R* , 0 < k < η, as a subspace of R" : R* = { (.v x n) e = · · · = x n = 0 } . Now assume that we can choose a representative S" = { ( Κ , φ j) \ j Ε J] of 3 such that for each j with Vj η Λ φ 0 9>jlV.nA

: Vj Π A —> R1, C R"

is a homeomoφhism onto an open subset of R* . Then evidently A is a topological manifold, and - {(VjDA , φl\V J r\A)\j e J } defines an atlas of A. We say that (A , ) is a submanifold of M" . REMARK. A submanifold in Example 4 is different from a "submanifold" as used in differential geometry. Our submanifolds are submanifolds in differential geometry, but the converse is not true. (5) Product manifolds. Let (Μ, 3! ) and (M',3') be C\'n ifty manifolds of dimensions η and η respectively. Set 3 = . 3' = [J?'), •'/ = {(VpfWzJ).^ = {{V k, V' k)\keK). Clearly, Μ χ Μ' is an η + topological manifold. Further, the set S? xS*

= {{Vj

X v

k

,

V j

χ φ[)\ (j,

k) e J Χ Κ \

turns out to be an atlas for Μ χ Μ'. We say that (Μ χ Μ' , [.9 J χ./'']) is the product manifold of (M ,3) and {M 1,3'). When there is no confusion we simply write Μ χ Μ ' . EXAMPLE.

The torus

T 2 = S 1 χ S 1 is the product of two copies of the

S1

circle . (6) The Mobius strip. We obtain a Mobius strip by twisting a strip of a tape and pasting the edges as shown in Figure 1.2. More precisely the Mobius strip M 2 is defined by M 2 = [ 0, 1] χ [0, 1 (0, /) ~ (l, ι - /), t °

2

e [θ, I]. DC

The interior Μ of the Mobius strip is a two-dimensional C^ manifold. This manifold is not orientable. (7) Projective spaces. The n-dimensional real projective space P n(R) S" / ~, χ χ , is an ti-dimensional C\n itfy manifold. We shall give a proof for the case η = 2. We may think of P 2(R) as P 2(R) = { [jc, , a 2 , x 3 ] | not all jc, , x 2 , x^ are zero, ,v( e R, 1 = 1.2.3}.

51. is an embedding, the image ,f(M") is obviously a submanifold of V' . REMARK. An embedding / is an immersion but the converse is not true Even when / is an immersion which is one-to-one into V 1' , it may fail to be an embedding. Consider Figure 1.3. DEFINITION 1.8. Let M " and V'' be c v manifolds of dimensions η and /;, and lei / : A/" —> V 1' be a C\n ifty map. A point r of Γ'' is a regular value of / if the rank of / at each point ,v in f '(y) is /); otherwise, r is a critical value. According to the above definition points not in the image under /' are regular values.

PROPOSITION 1.1. Let M" and V'' be c x mani folds of dimensions and ρ, and let f : Μ" — V 1' be a c x map. If ν is a regular value f . then either f~ ' is the empty set or an n—p dimensional submanifo M" .

The proposition follows easily from the definitions of submanifolds and of regular values. C. Tangent spaces and the differentials of c v maps. DEFINITION 1.9. Let M" be a C \ 'n i fty manifold, and let ν be a point of \f" . A C\n itfy map c : (-ε , r.) —· Μ with c(0) = .v of an open interval (-/:. r.) . ε > 0 (ε is sufficiently small), into M" is called a curve at ν . Suppose
= CJ ~ = { [c] v | f is a curve at ,v }

is the tangent space of A/" at .v .

I. C

MANIFOLDS C

M A S AND FIBER BUNDLES

(-c.r) Vt ' Γ2Ϊ - C, C )

We define the operation of addition on Τ (M") as follows. DEFINITION 1.11. Let [c,]A. and [c2]^ be elements of T X(M"), where c,,c 2 : ( - ε , ε) -» M" are C°° maps with c,(0) = c2(0) = χ . For a chart ((/„,

X by ρ(χ, y) = χ. Then ρ is a continuous map onto X and for each χ in X , p~\x) = Kv is homeomorphic to Y . Fix a point i'0 of Y and define / : X —> Β by f(x) = (,Y,.V0); / is continuous and Ρ 0 f{x) — χ · 2. The Mcbius strip Recall that the Mobius strip is defined as follows: Μ 2 = [0, I] χ [0. I)/ (0, /) ~ (1 , 1 - 0 . 1 € [0, 1 ]. Setting X = S l = [0, 1]/ ~ , / = [0, 1], and Β = Μ 2 we define ρ : Β - X by ί>([(ϊ, 1)]) = [s] e S ] . Then ρ is a continuous map onto A', and for a point χ of X , = Y K is homeomorphic to Y . Further, there exists a neighborhood V(x) of A* such thai p~ ](V(x)) is homeomorphic to V(x) χ Y . In addition the map f : X —> Β defined by f(x) = [(.v, 1/2)] is continuous and satisfies ρ ο f(x) = A' . 3. The Klein bottle The Klein bottle is the surface K~ which we obtain by pasting one pair of facing edges of the rectangular / χ J . I = J = [0. 1 ], in the same direction and the other pair in the opposite direction (Figure 1.5). That is, K 2 = / χ J/

(0, /) ~ ( l , 1 -/), (s,0)~(s,l).

t 6 ./. se/.

Put Β = Κ 2, Χ = S l = 11 ~, and Y = S 1 = .// ~. Define ρ: Β - X by /?([(i, /)]) = [ί] ε Λ', which is continuous onto X. For each .v of .V p~\x) is homeomoφhic to Y = S 1 . For a point .v of A' there exists a

2 FIBER BUNDLES

17

neighborhood V(x) of a* with ρ 1 (Ι-'(λ)) homeomorphic to Γ(.ν)χ)'. The map / : X -» Β defined by f(x) = [(.ν , 0)] is continuous, and pof(x) = χ . 4. Covering spaces Suppose Β is a covering spacc of X and ρ : Β — Λ is its covering map. Then ρ is evidently a continuous map onto X , and for each χ in X the set Kv = x) is discrete. In the case where X is arcwise connected, the )' ( are homeomorphic for all A in X. Further, for each V of X, there exists a neighborhood l'(.v) of χ and a homeomorphism between p~'{V(x)) and K(A") Χ Y y (B is a covering spacc of .V . or (B.p . .V) is a covering space, if (0) A' is arcwise connccted and locally arcwise connected, (i) ρ : Β —> X is a continuous surjcction. (ii) for each ν e A' there exists an arcwise connccted neighborhood Γ of ν such that each connected component is open in X and ρ\1' λ : V. — Γ is a homeomorphism onto v )• 5. The twisted torus Consider [0, 1] χ Λ'1 , and paste {0} χ S1 and { I } χ .V1 with a 180° twist. The resulting surface '/"„ is the twisted torus: r H . = [ο, ι] χ s'/ ~.

(ο. ί·2*'") ~ ( ι , i , 2 " / , " + '")

Define ρ : T w S 1 = [0, 1]/ ~ by p([t , c 2n i"]) = [/]. Then ρ is a continuous map onto X and for each point [1] of S 1 there is a neighborhood Γ of [1] in S 1 such that /> · '(*) is homeomorphic lo Γ χ s ' (Figure 1.6). B. The definition of afiber bundle. DEFINITION 1.13. Let G be a topological group, and let >' be a topological space. Suppose there is a continuous map η : G χ )' — )' ;atisying:

(i) For the unit e of G >/(e . v) = r. (ii) For all g t , g 2 e G and y e Υ , t)(gg' , v) = >i(g . >i(g' . v)) . Then we say that G is a topological transformation group of )' (with respect to η ) and that G acts or operates on Y .

18

I. C' MANIFOLDS C' M A S AND FIBER BUNDLES

We shall write g • y for By Definition 1.1 3 the map g : Υ — Y which associates to each element y of Y the clement g • y is a homeomorphism of Y . Hence, the map η induces a homomorphism ή : G —· H(Y) from G to the group H(Y) of homeomorphisms of Y . DEFINITION 1.14. Let G be a topological transformation gruup of Y. If the homomorphism f) above is injective, that is, g • y = y for all y e Y implies g = e , we say that G is effective . For now unless otherwise stated our topological transformation groups arc assumed effective. DEFINITION 1.15. A coordinate bundle 33 = {Β , ρ, Χ, Υ , G\ is a collection of topological spaces and continuous maps with structures satisfying the following: (1) Β and X are topological spaces; Β is the bundle spacc or the total space and X is the base space, ρ : Β -» X is a continuous map called the projection map of 38 . (2) Y too is a topological space; Y is thefiber of 3! . G is a topological transformation group called the structural group of 33 . (3) The base X has an open covering {Vj\ j e J }, and for each j e ./ there is a homeomorphism φ. :ν.

χ r — p-\Vj)·,

the V. 's are coordinate neighborhoods and the (4) The coordinate functions satisfy the following: (i) ροφ .(χ^) = χ, xeVj, y e Y, j eJ . (ii) The map

t

's are coordinate function

: Y — • p~\x) defined by

gives a homeomorphism of Υ , ΦΪ.,°Φ,.,··Υ

for JC € Κ Π Vj , which agrees with the action of an element g f l(x) of G . (iii) Define a map g/.:KnVj~G

by g i;(x) = Φ7' JI

J ,Λ

ο Φ; l ,Λ

r

• Then g.· is continuous; we say that the g,· are JI

coordinate transformations or a transition Roughly speaking a coordinate bundle is a family

JI

functions of 33 . Κ χ Κ} patchcd by

the { gji}. We write Κ for p~'(x); Y x is thefiber over χ . Let 38 = {Β, ρ , Χ, Υ , C } be a coordinate ordinate transformations {g j :}. Then (i) g k j(x) • gj,W = g k iW, x e v t η Vj η ν' , LEMMA 1.4.

bundle with co

§2. F1BHR BUNDLES

(ii) g u{x) = e, χ e Κ . where e is the unit element of G . and (Hi) £,*(*) = [£,,(Λ-)Γ', -vc Γ η I j..

The lemma follows readily from the definition of coordinate bundles. We next define an equivalence relation in the strict sense between two coordinate bundles. DEFINITION 1.16. We say that bundles 38 - {Β , ρ. X . . (7) and = {β', ρ , X' , )'', G') are equivalent in the strict sense and write '3> ~ 38' i they satisfy the following:

(i) Β = Β', Χ = Χ'. ρ = ρ'. (ii) Υ = Υ', G = G' . (iii) Their coordinate functions {/}, {ij>k) satisfy the conditions thai

=

«/',..,·

rni·'

coincides with the action of an element of G', and that the map

is continuous. It is easy to see that a; is an equivalence relation. DEFINITION 1.17. An equivalence class of coordinate bundles is called a fiber bundle. DEFINITION 1.18. We say that G is a Lie group if (i) G is a topological group. (ii) G is a C\n itfy manifold, and (iii) the group operations on G ψ j : G χ G —· G ,

.

ψ : G —· G , vAg) = g '

arc smooth. G L ( / J , R ) and SO(ti) arc Lie groups. Here SO(n) is the group of «-dimensional orthogonal matrices whose determinants are of the value one, which is called the n-dimensional rotation group. There are natural inclusions SO(/ p~ l (V. ). Further, each φmaps Vj χ Y onto p " ' ( K ) , for if b = {(.ν, y, A)) e />"'(!' ( ) i h e n .γ € I ' n i ; and (x,y,k)~(x, g j k{x)-y , j) ; hence, we can write b = φ,(.χ , g j k (. Y)·.!'). To show that is one-to-one, suppose φ^χ , y) = φ^χ' ,y')\ that is, ( x > y , j ) ~ (x • y , j ) · Then λ- = χ and g tj(x) • y = y • But g n(x) = e , and so y = y . Hence, φ. is one-to-one. We now prove that φ~ ι is continuous. Suppose that H' is an open subset of Vj χ Y ; we want to show that φ t(W) is open in Β . To do so it is enough

2. FIBER BUNDLES

25

to show that q~ is open in T. But '/' is a pairwisc disjoint union of open sets Vk χ Υ χ k; hence, it sulfices to show that the intersection of Q~ {j(I 7)) and χ Υ χ k is an open subset of V k χ Υ χ k . Now the set q~\j(W))n(V k χ Υ χ k) is contained in (l·'. η \' k ) χ Υ χ k , and so we decompose q as follows: (Γ Π v k) χ Y X k C /"\subset,V X }· X ./ ^

\

VjxY-Up

(I·' ) C B =·/·/-

υ

υ

»•'

. k) = (.ν , g j k(x)-y) . Since r is continuous, t' '(M7) is an open set. Thus. "' is continuous. Now the map φ~\.οφ ι v , ν e l^n V , is a homeomorphism of )'. Putting y = Φ~\. ° Φ, ,.(.!'), we get φ^χ,ν') = ·) = g„(x)·y . y e )' We have constructed the desired coordinate bundle whose coordinate transformations arc the -

= {/?,/», V . >', (7}

(ii) Putting A;(.v) - e, .v € I7 in Lemma 1.7. we sec that two coordinate bundles with equal coordinate transformations are equivalent. Thus, the coordinate bundle constructed above is unique up to equivalence classes. • E. Tangent bundles of dilferentiable manifolds. We take an «-dimensional manifold M" for the X, Euclidean «-space R" for the Υ, and the general linear group GL(«, R) for the G in Stcenrod's structure theoremTheorem 1.1. In this case G acts smoothly on >'. Choose a C\n itfy atlas S" = { (t/ .

,., = (ΦΐΐΜ,φ-^ν)). where φ ; : V i χ R" -»

u.v e TJM").

, φ [ V(.v) = φ^χ,ν),

and ( . ) on the right-

hand side of the equation is the usual inner product in R" . The desired

27

2 FIBER BUNDLES

Euclidean metric may be defined by

Λ = Σ ( · * ) ( » . •">,.,.

». " e /,(.»/").

I We can readily check thai { , ) is symmetric and positive definite. • REMARK. WC may also carry out our proof using the reducibilitv of the quotient GL(ti, R)/0(/z). The //-dimensional unitary group U(«) sits naturally in the rotation group SO(2//): consider the map ρ : U(i?) — SO(2n) defined by P(C) C = (c 0),

=

JQ ,

c ( j = a ;j + \T- \ bjj ,

CeU(n). A = (a it).

/* = (/>,).

Then ρ is a continuous isomorphism of U(//) in SO(2//). DEFINITION 1.30. Let M~" be a 2//-dimensional C X manifold. By Theorem 1.2 wc may take 0(2n) for the structural group of the tangent bundle τ(Μ") of M" . A reduction of the structural group 0(2n) of M" to U(//) is called an almost complex structure of M" . An almost complex manifold Μ " is a manifold with an almost complex structure. A complex manifold is almost complex. It is also evident that an almost complex manifold is orientable. Since SO(2) = U(l), a two-dimensional smooth orientable manifold lias an almost complex structure. G. Induced bundles. DEFINITION 1.31. Lei ·1ΰ' - {IS' , // . A"', ) ' . (i\ be a coordinate bundle. Let A" be a topological space, and let ι; : X — .V' be a continuous map. For a system of coordinate neighborhoods { Vj \j'ej'}. the family { i/~ (I'') \ j e j ' } is an open covcr of X . Setting J?„(A-) = jfV(/;(jr)). ve r , n r . we obtain a system of coordinate transformations ({I |.{&' /( }) in A with values in G. We define the pullhack or the induced bundle tf( /> ) of" ./) over X by /; to be the coordinate bundle as constructed in Theorem I.I from { r , ( 7 ; A \ { K } , { g „ } } . The following is an alternative definition of an induced bundle over A Let £6' = {/Λ/Λ X', >', G) be a coordinate bundle and let η : X — V' be a continuous map. Consider the following subspace II of Α χ It : Β = {(χ, b') e Χ χ Β' I //(λ ) = p'(b')}. We then have the commutative diagram Β

B'

X

X'

28

I. C' MANIFOLDS C' MAI'S, AND FIBER BUNDLES

where π, : Χ χ Β 1 —< X and π 2 : Χ χ Β' —· Β' are the projections onto Χ and Bl respectively, ρ = and h = π2|Β. Putting Vj = >f\Vj) and defining φ ] : V χ Υ -> p~'(Vj) by j(x,y) = (x^'jUl(x),y)), we obtain the coordinate bundle {Β , ρ , Χ, Υ, G}, which is equivalent to the pullback //*38 . Induced bundles have the following properties. PROPOSITION 1.2. (i) Let 38[ , 3S 2 be coordinate let η : X —> X' be a continuous map. Then

gg[ ~ gg'

2

bundles over

X' . and

==> η\.„OPn. ] | X f R v OX = λ„ΟΡ„ t t ( /.,. - ι . / > ο / (I an n-simplex. The η is the dimension of the simplex \I' 0I\ • /'J . A zero-simplex |/-"0| is the point P0 , a one-simplex |/'01\ \ is the line 7'()l\ • a two-simplex {PqP^P,] is the triangle with vertices P Q. !\ . I\ . and a threesimplex \PQP\ Ρ-,Ρ^ is the tetrahedron with vertices 1\ . I\ , and l\ See Figure 1.9. /' among the DEFINITION 1.37. Any set of q + I points Ρ ι , P t 0 < q < η . are ... , P of an //-simplex σ = \ l' 1\ •••!'„] n 0 vertices P,0 ' again linearly independent; hencc, they define a (/-simplex •''I

τ

called a q-face

,

= ΙΛ p

· ρ,

I-

of σ . If τ is a face of a . wc write τ χ tr or σ y τ.

(

32

I. C

MANIFOLDS. C

MAI'S, AND FIBER BUNDLES

DEFINITION 1.38. Afinite set Κ of simplices ir. Ν dimensional Euclidean space R v is called a simplicial complex if it satisfies the following: (i) If a e Κ and ay τ , then τ e Κ . (ii) If σ, τ 6 Κ and σ Π τ Φ 0 , then σ Π τ -< σ and σ η τ -< τ .

The dimension of a simplicial complex Κ is the maximum value among the dimensions of simplices belonging to Κ and is denoted by dim Κ . DEFINITION 1.39. Let AT be a simplicial complex. The union of all simplices belonging ίο Κ is a polyhedron of Κ denoted by |A'|:

, //)-matriccs over Κ The set ./'(//, p) is in onc-to-onc correspondence with R' . where Λ' = („//, + · • · + nH r) χ ρ , n Η s = (n + s - I )!/.s!(ti - I)!: we associate to cacli element of J r its partial derivatives of order up to r . This correspondence defines a natural topology in J'(n . p). Now consider C' maps f: (R", 0) — (R', 0), g \ (R'\ 0) —· (R v . 0). We define a map J r(p, q) χ J'(n . p) .l'(n , q) by (g' r ) , / l r | ) i— (g ° f ) ( r > • This definition is independent of the choice of representatives. Further as any partial derivative of go f can be written as a polynomial in partial derivatives of / and g . the above map is algebraic. In particular for // = ρ = q , the above map defines a product in .1' (ii . n) . and the r-jct (l K *) l ' i of the identity map !„., of R" is both the right and left unit of J'(n , n) with respect to this product. Denote by L'(n) a subset of J r(n,n) consisting of all invcrtiblc elements. For 1 < s < r , define a map : J'(" • P) —* •/'("· P) by *,.,(/"') = / 1 " · PROPOSITION 1.4. ( i )

l' (n) = GL(T/ , R ) .

(ii) Z/(//) = (7Tr t)-\L\n)). (iii) L'(n) is a Lie group for each r. I < /· < x . (iv) L r(n) has the homotopv ivpe of GL(ZJ. R) (cf. Thom and Lcvinc [Bll]). B. Singular sets. Wc write L'(n,p) for //(/>) χ //(;/) and define an action of L'(n , p) on J'(n , p) as follows: L r(n , ρ) χ ./'(//, p) —* j'(n , ρ). ,, , ('K),/ /-(—· ,(a-ι ο ,·Job) DEFINITION 1.41. A jet f >r ) e J'(n. p) is regular if its representative J in C'(n, p) has the maximal rank at 0. Wc shall denote by ''.] r(n. p) the set of all regular jets in J'(n . p).

34

I. C' MANIFOLDS. C' MAS, AND FIBER BUNDLES

Let q — min(//, p). We identify j\n , p) with M(p , η ; R) and set S k(n , p) = {A e M(p,

η ; R)|rank of A is q - k },

0 < k < q.

In what follows we simply write S k = S k(n , p). is dense in J'(n,p). (i) S 0 = "j\n,p). (ii) J 1 (//, p) = S Q U S, U · · · U S , pairwise disjoint

PROPOSITION 1.5. ( 0 )

pJ r(n,p)

union.

(iii) S k is the orbit space of /_'(/'. p) • (v) S k is a codimension

(η - q + k)(p - q + k ) submanifold

of ./'(//

PROOF, (i), (ii), and (iii) are immediate. We want to show (iv). We shallfirst prove S k \SUBSETS k . U - • - U S . Let A E S\ . In an arbitrary neighborhood U(A) of A there is a Π whose rank is q - k . Hence, the ."ank of Λ must be less than or equal to q — k . Next show S k U · • · US q C S k . Let A € S k U · · · US q . Then the rank of A is q - k — i, 0 < 1 < q - k . Hence, we can transform A by an element of L l(n , ρ) to

o

"i-k-i

0J ·

where / . stands for the (q - k - /') χ (q - k - 1) identity mat. ix. Hence, it is enough to show that E q_ k_ j is in . But this is obvious, (v) follows readily from the following

LEMMA 1.8. Let M(n,p\ R ) be the set of (ρ, n)-matriccs over R Sinc Μ (ρ , η ; R) is in one-to-one correspondence with R'"'. we give the usu ogy of R^" to M(p , //; R); thus M(p , η ; R) becomes a smooth manifo Let M(p , n \ k) be the set of all (ρ, n)-mat rices of rank k . If k < min(p , η). M(p , η ; k) is a k(p + η - k)-dimensional submanifo of M(p, η• R). PROOF. Let E 0 be an element of M(p,n\k). Without loss of generality we may assume that E 0 = (* £»), A0 is a (k , A:)-matrix and |. l0| # 0. Then there exists an ε > 0 such that \A\ / 0 if the absolute value of each entry of A - A0 is less than ε . Now let U C M(p , η ; R) consist of all [ρ , /z)-matrices of the form Ε = (a o) • where A is a (k , /c)-matrix such that the absolute value of each entry of A - A0 is less than ε . Then we have Ε e M(p , n \ k) if and only if D = CA~ lB.

because the rank of (h

0 \(A

B\ =(

\ A - I p_ k)\C D)

Α

Β

\

νΛ'Λ + C xb + D)

.

BUNDLES

35

is equal t o the rank of Ε for an arbitrary (p - A , k )-matrix X . Now setting X = -C.4 - 1 we see that the above matrix becomes (Α Β \ \0 -CA~ lB + D)' The rank of this matrix is k if D = CA Β. The converse also holds, since if -CA~'B + D Φ 0. then the rank of the above matrix bccomcs greater than k . Let ΙΓ be an open subset of />//-(p-K )(//-A ) - A(/H II - A (-dimensional Euclidean space: "

=

{ (

o)

c

e Λ1{ 1'·

"•

R) H

}

(*): the absolute value of each entry in A - l„ does not exceed v.. Then the correspondence (A B\ \C 0J

(Α \C

Β \ CABJ

defines a dilfcomorphism between ΙΓ and the neighborhood ί.'ηΛ/(/>. //: A ) of E 0 in λί(ρ , η: A). Therefore, Μ (p. ir. A) is a k(p + n- A (-submanifold of M(p. n: R). • C. Jet bundles. Let I'" and M'' be CA manifolds of respective dimensions η and ρ , s > I . For .v e V" and ρ € Λ/'1. I < »' < .ν. set cr

,.(('". M") = (./': I'" - M'\ c'map|/(.v) = r }.

Elements /' and g of

,.(!'", Λ/'') are r-cquiralcni

αι v. / ~ ,tr, if V.τ the partial derivatives of / and g at .v in some local coordinate system agree up to order r. The relation ~ is well defined and is an equivalence relation. Set ./; ,.(!·". A/") = c : ,.('•". M")/ - • We write J[(f) for the cquivalcncc class containing f and we say that J[( f) is the r-jci of f at χ . Set c r

'1

/ ( ! ' " , A/")=

(J

J[

,.(! ". A/").

A€I " .yetf

Using the atlases of I'" and M''. wc turn .!'(Γ",Α/'') into the total space of a bundle over V" χ M n withfiber J'(n.p) and structure group L'(n.p): J[

,.(l'". A/'')\subsetj'(V

n.

A/'') ,— J r(n. p)

i

J

(x.y)

e ν" χ A/''

36

I. C' MANIFOLDS, C' MAPS, AND FlttUR BUNDLES

This is called a jet bundle. An alternative definition of a jet bundle comes from the structure theorem of Steenrod (Theorem 1.1). Set Y = J r(n , p) and G = L r(n,p)\ G acts on Y. Set X = V"xM p . Take a C' atlas ^ = {((/„, φ η) \ a e A ) of V" and an atlas S* = { (W k , ψ λ) \λ e A } of M" . Then with" X t λ = U it χ Η\ , the family {X a e Α, λ e A] is an open cover of X . For X n i Π X fi μ Φ 0 , we define a map Λβ.μ)'· χ„.λ ηΧρ. μ

—

L'("'P)

by ε (.,Α)Αβ. μΜ> y) = ( C · Ό where

(r)

rr

/

^

6

l(0

rr

x /

-l\

°«.β = ^(Λ,^,. ). 6;..;, = ° ψ„ >• Then the { X a λ, g ( ( t ^ j |α, β 6 Λ , Α, μ e Λ } is a system of coordinate transformations in V n χ M p with values in G . Hence, we construct, by Theorem 1.1, afiber bundle, which turns out to be the above jet bundle. The total space J r(V" , M p) may be regarded as a C s~' manifold when r < oo. Let / : V" —» M p be a C s map. Then we call the map J\f):

V" — J'(V"

, M"),

A" — f x(.f) the r-extension of /. The r-extcnsion J r(f) following diagram commute: J'(V"

•/

, M") ( r " , M p) be the set of C* maps from Γ" to M p, and introduce the C r topology in c l ( l " . A'''). I < /· < ,v . DEFINITION 1.42. Let / E C ' ( R " , M p ) . Let A be a compact subsei of V" and let Ο be an open subset of M'' such that f(K) \subsetΟ. For ρ € I'" . Q — f(p) ε Λ-/'', choose a chart ((.', φ) about ν such that A' C U and a chart (U', ψ) about q . Let c > 0 and 0 < / ' < . < . Write φ(/>) = (A, , ... , A'(J) and define the set N'(f : .Ν . Ρ : Κ . ()\r.) as follows: N r(f\

Λ", ρ: Α. Ο; £) = (.?€

c ,

( Γ" , ,1/'')|(i).(ii).(iii) }.

where (i) 8 ( K ) c O , (ii) V, ° f(p) ~ Ψ, ° ·?(/>)! < f- · for each // 6 A'. I

=

'y/ly = 'x/,/iiPx fx t\

= (*

xn)

\ XJ I7

\

2 2 A +X"V+l r J_. "V, ^

2 +--+X., -r

*l τ (8) Finally we show that the index of / at p 0 is r. By differentiating equation (8) twice, we obtain 'ί - 2 , 1=1,. 2, / = /' + ax? I o 2f ϋχβχ;

= 0,

Hence, the Hessian of / at p Q is 1-2

0 =

0 2/ and thus its index is r. We now have proved the theorem. • DEFINITION 1.44. Let Μ be a J°° manifold. We say that a C\n ifty function / : Μ —> R is a Morse function if it satisfies the following: (1) The critical points of / are nondegenerate. (2) If ρ and q are critical points of / such thai ρ # q, then f(p) Φ /(