Job #: 111368
Author Name: Conlon
Title of Book: Differentiable Manifolds
ISBN #: 9780817647667
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Differentiable Manifolds Second Edition
Lawrence Conlon
R e p rint of the 2001 S e c o n d Edition
Birkh~iuser Boston 9 Basel ~ Berlin
Lawrence Conlon Department of Mathematics W a s h i n g t o n University St. Louis, M O 6 3 1 3 0  4 8 9 9 U.S.A.
Originally p u b l i s h e d in the series B i r k h d u s e r A d v a n c e d Texts
ISBN13:9780817647667 DOI: 10.1007/9780817647674
eISBN13:9780817647674
Library of Congress Control Number: 2007940493 Mathematics Subject Classification (2000): 57R19, 57R22, 57R25, 57R30, 57R45, 57R35, 57R55, 53A05, 53B05, 53B20, 53C05, 53C10, 53C15, 53C22, 53C29, 22E15 9 Birkh~iuser Boston All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkh~iuser Boston, c/o Springer Science+Business Media LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Cover design by Alex Gerasev. Printed on acidfree paper. 987654321 www. birkhauser, com
Lawrence Conlon
Differentiable Manifolds Second Edition
Birkh~iuser
Boston
9 Basel ~ Berlin
Lawrence Conlon Department of Mathematics Washington University St. Louis, MO 631304899 U.S.A.
Library of Congress CataloginginPublication Data Conlon, Lawrence, 1933Differentiable manifolds / Lawrence Conlon.2nd ed. p. cm. (Birkh~iuseradvanced texts) Includes bibliographical references and index. ISBN 0817641343 (alk. paper)ISBN 3764341343 (alk. paper) 1. Differentiable manifolds. I. Title. II. Series. QA614.3.C66 2001 516.'6dc21 2001025140
AMS Subject Classifications: 57R19, 57R22, 57R25, 57R30, 57R35, 57R45, 57R50, 57R55, 53A05, 53B05, 53B20, 53C05, 53C10, 53C15, 53C22, 53C29, 22E15 Printed on acidfree paper. @2001 Birkh~iuserBoston, 2nd Edition 9 Birkh~iuser Boston, 1st Edition
Birkh~user ~ 
All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkh~iuser Boston, c/o SpringerVerlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. ISBN 0817641343 ISBN 3764341343
SPIN 10722989
Typeset by the author in lATEX. Printed and bound by Hamilton Printing Company, Rensselaer, NY. Printed in the United States of America. 987654321
This book is dedicated to my wife Jackie, with much love
Contents Preface to the Second Edition Acknowledgments
xi xiii
Chapter 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7.
1. Topological Manifolds Locally Euclidean Spaces Topological Manifolds Quotient Constructions and 2Manifolds Partitions of Unity Imbeddings and Immersions Manifolds with Boundary Covering Spaces and the Fundamental Group
17 20 22 26
Chapter 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 2.9.
2. The Local Theory of Smooth Functions Differentiability Classes Tangent Vectors Smooth Maps and their Differentials Diffeomorphisms and Maps of Constant Rank Smooth Submanifolds of Euclidean Space Constructions of Smooth Functions Smooth Vector Fields Local Flows Critical Points and Critical Values
41 41 42 50 54 58 62 65 71 80
1 1
3 6
Chapter 3. The Global Theory of Smooth Functions 3.1. Smooth Manifolds and Mappings 3.2. Diffeomorphic Structures 3.3. The Tangent Bundle 3.4. Cocycles and Geometric Structures 3.5. Global Constructions of Smooth Functions 3.6. Smooth Manifolds with Boundary 3.7. Smooth Submanifolds 3.8. Smooth Homotopy and Smooth Approximations 3.9. Degree Theory Modulo 2* 3.10.Morse Functions*
87 87 93 94 98 104 107 110 116 119 124
Chapter 4.1. 4.2. 4.3. 4.4. 4.5.
131 131 136 142 145 150
4. Flows and Foliations Complete Vector Fields The Gradient Flow and Morse Functions* The Lie Bracket Commuting Flows Foliations
viii
CONTENTS
Chapter 5.1. 5.2. 5.3. 5.4.
5. Lie Groups and Lie Algebras Basic Definitions and Facts Lie Subgroups and Subalgebras Closed Subgroups* Homogeneous Spaces*
161 161 170 173 178
Chapter 6.1. 6.2. 6.3. 6.4. 6.5.
6. Covectors and 1Forms Dual Bundles The space of 1forms Line Integrals The First Cohomology Space Degree Theory on S 1.
183 183 185 190 195 202
Chapter 7.1. 7.2. 7.3. 7.4. 7.5.
7. Multilinear Algebra and Tensors Tensor Algebra Exterior Algebra Symmetric Algebra Multilinear Bundle Theory The Module of Sections
209 209 217 226 227 230
Chapter 8.1. 8.2. 8.3. 8.4. 8.5. 8.6. 8.7. 8.8. 8.9.
8. Integration of Forms and de Rham Cohomology The Exterior Derivative Stokes' Theorem and Singular Homology The Poincar6 Lemma Exact Sequences MayerVietoris Sequences Computations of Cohomology Degree Theory* Poincar5 Duality* The de Rham Theorem*
239 239 245 258 264 267 271 274 276 281
Chapter 9.1. 9.2. 9.3.
9. Forms and Foliations The Frobenius Theorem Revisited The Normal Bundle and Transversality Closed, Nonsingular 1forms*
289 289 293 296
Chapter 10.1. 10.2. 10.3. 10.4. 10.5. 10.6. 10.7.
10. Riemannian Geometry Connections Riemannian Manifolds Gauss Curvature Complete Riemannian Manifolds Geodesic Convexity The Cartan Structure Equations Riemannian Homogeneous Spaces*
303 304 311 315 322 334 337 342
Chapter 11.1. 11.2. 11.3. 11.4.
11. Principal Bundles* The Frame Bundle Principal GBundles Cocycles and Reductions Frame Bundles and the Equations of Structure
347 347 351 354 357
CONTENTS
ix
Appendix A.
Construction of the Universal Covering
369
Appendix B.
The Inverse Function Theorem
373
Appendix C. Ordinary Differential Equations C.1. Existence and uniqueness of solutions C.2. A digression concerning Banach spaces C.3. Smooth dependence on initial conditions C.4. The Linear Case
379 379 382 383 385
Appendix D. The de Rham Cohomology Theorem D.1. Cech cohomology D.2. The de RhamCech complex D.3. Singular Cohomology
387 387 391 397
Bibliography
403
Index
405
P r e f a c e to t h e S e c o n d E d i t i o n In revising this book for a second edition, I have added a significant amount of new material, dropping the subtitle "A first course". It is hoped that this will make the book more useful as a reference while still allowing it to be used as the basis of a first course on differentiable manifolds. In such a course, one should omit some or all of the material marked with an asterisk. More information about these optional topics will be given below. Presupposed is a good grounding in general topology and modern algebra, especially linear algebra and the analogous theory of modules over a commutative, unitary ring. Mastery of the central topics of this book should prepare students for advanced courses and seminars in differential topology and geometry. There are certain basic themes of which the student should be aware. The first concerns the role of differentiation as a process of linear approximation of nonlinear problems. The wellunderstood methods of linear algebra are then applied to the resulting linear problem and, where possible, tile results are reinterpreted in terms of the original nonlinear problem. The process of solving differential equations (i.e., integration) is the reverse of differentiation. It reassembles an infinite array of linear approximations, resulting from differentiation, into the original nonlinear data. This is the principal tool for the reinterpretation of the linear algebra results referred to above. It is expected that the student has been exposed to the above processes in the setting of Euclidean spaces, at least in low dimensions. This is what, we will refer to as local calculus, characterized by explicit computations in a fixed coordinate system. The concept, of a "differentiable manifold" provides the setting for global calculus, characterized (where possible) by coordinatefree procedures. Where (as is often the case) coordinatefree procedures are not feasible, we will be forced to use local coordinates t h a t vary from region to region of the manifold. When theorems are proven in this way, it becomes necessary to show independence of the choice of coordinates. The way in which these local reference frames fit together globally can be extremely complicated, giving rise to problems of a topological nature. In the global theory, geometric topology and, sometimes, algebraic topology become essential features. These themes of linearization, (re)integration , and global versus local will be emphasized repeatedly. Although a certain familiarity with the local theory is presupposed, we will try to reformulate t h a t theory in a more organized and conceptual way that will make it easier to treat the global theory. Thus, this book will incorporate a modern treatment of the elements of multivariable calculus. Fundamental to the global theory of differentiable manifolds is the concept of a vector bundle. As the global theory is developed, the tangent bundle, the cotangent
xii
PREFACE TO THE SECOND EDITION
bundle and various tensor bundles will play increasingly important roles, as will the related notions of infinitesimal Gstructures and integrable Gstructures. For conceptual simplicity, all manifolds, functions, bundles, vector fields, Lie groups, homogeneous spaces, etc., will be smooth of class C ~ It is possible to adapt the treatment to smoothness of class C k, 1 < k < oc, but the technical problems that arise are distracting and the usefulness of this level of generality is limited. On the other hand, in much of the literature, the study of Lie groups and homogeneous spaces is carried out in the real analytic (C ~) category. In these treatments, it is customary to note that C a groups can be proven to be analytic, hence that no generality is lost. It seems to the author, however, that nothing would be gained by this approach and that the ideal of keeping this book as selfcontained as possible would be compromised. The optional topics (sections, subsections and one chapter, with titles terminating in an asterisk) can safely be omitted without creating serious gaps in the overall presentation. One topic that is new to this edition, covering spaces and the fundamental group, is not starred and should not be omitted unless the students have seen it in some prior course. Some of the optional topics fall into subgroupings, any one of which can be included without dependence on the others. Thus, Subsection 2.9.B and Sections 3.10 and 4.2 constitute a brief introduction to Morse theory, one of the most useful tools in differential topology. Similarly, Sections 3.9, 6.5, and 8.7 constitute an introduction to degree theory, together with some classical topological applications, but in this case any one of these three sections can be treated without serious logical dependence on the others. Apart from minor revisions, this treatment of degree theory is not new to this edition. In Subsection 1.6.B, we classify 1manifolds. This intuitively plausible result needed is only in the optional Section 3.9. Also easily omitted is the brief Subsection 1.6.A, this being an extended remark on cobordism theory. New to this edition is an optional introductory treatment of Whitney's imbedding theorems (Subsection 3.7.C). We prove only the "easy" Whitney theorem, while stating carefully the general theorem. Imbeddings of manifolds in Euclidean space will be used only in treating some other optional topics, namely, the smoothing of continuous maps and homotopies (Subsection 3.8.B) and the existence of Morse functions (Section 3.10). In Chapter 5, an introduction to Lie theory, adequate for a first course on manifolds, requires only the first two sections. Accordingly, Sections 5.3 (the closed subgroup theorem and related topics) and 5.4 (homogeneous spaces) are optional. Certain topics in de Rham theory, Sections 8.8 (Poincar5 duality) and 8.9 (a version of the de Rham theorem), can be omitted, as can the treatment of foliations defined by closed 1forms (Section 9.3). Also easily omitted is the brief treatment of Riemannian homogeneous and symmetric spaces (Section 10.7). Finally, Chapter 11, on principal bundles and their role in geometry, gathers together and slightly expands on topics treated in various parts of the first edition and can be reserved to introduce a more advanced course or seminar. There are some significant changes in the appendices also. The original Appendix A has been replaced by one that gives the construction of the universal covering space. The former Appendix D (Sard's theorem) has been moved to the main body of the text. The current Appendix D (formerly Appendix E) has been expanded to include a proof of the de Rham theorem for singular as well as Cech cohomology.
Acknowledgments I am grateful to the late Robby Gardner and his students at Chapel Hill who "beta tested" eight chapters of a preliminary version of the first edition of my book in an intensive, onesemester graduate course. Their many suggestions were most helpful in the final revisions. Others whose input was helpful include Geoffrey Mess, Gary Jensen, Alberto Candel, Nicola Arcozzi and Tony Nielsen. I particularly want to thank Filippo De Mari, whose beautiful class notes, written when he was one of my students in an earlier version of this course, were immensely useful in subsequent revisions and first suggested to me the idea of writing a book. Finally, my students in the academic year 19992000 have offered much helpful input toward the final version of this edition.
CHAPTER 1
Topological Manifolds This chapter pertains to the global theory of manifolds. See also [3, Chapter I] and [41, Chapter 1].
1.1. Locally Euclidean Spaces Classical analysis is carried out in Euclidean space, the operations being defined by local formulas. One might hope, therefore, to extend this classical theory to all topological spaces that are locally Euclidean. While this is not generally possible without further restrictions on the spaces, the locally Euclidean condition is fundamental.
Definition 1.1.1. A topological space X is locally Euclidean if, for every x E X, 3 n > 0 (an integer), an open neighborhood U C X of x, an open subset W C ]I~n and a homeomorphism ~ : U ~ W. If we can show that n is uniquely determined by x, we will write n = d(x) and call this the local dimension of X at x.
Example 1.1.2. Any open subset X c_ ~ n is a locally Euclidean space that is also Hausdorff and 2nd countable. We will see that the local dimension is n at every xCX.
Example 1.1.3. Let X = ll~ U (*}, where * is a single point and sqcup denotes disjoint union. Topologize this set so that a basis of open subsets V C X consists of the following: 9 I f * ~ V, then V is open as a subset of ~. 9 If * E V, then 0 ~ V and there is an open neighborhood W C R of 0 such that V = (W \ {0}) U {*}. This space is locally Euclidean and 2nd countable. It is not Hausdorff since every open neighborhood of * meets every open neighborhood of 0. In this case, the local dimension is everywhere 1, even at . and at 0. E x a m p l e 1.1.4. In [41, Appendix A], there is described a bizarre space called the long line. It is connected, Hausdorff, and locally Euclidean with d(x) = 1, but it is not 2nd countable. In fact, the long line contains an uncountable family of disjoint, open intervals.
Exercise 1.1.5. Prove that each connected component of a locally Euclidean space X is an open subset of X. E x e r c i s e 1.1.6. Prove that a connected, locally Euclidean space X is path connected.
2
1. T O P O L O G I C A L
MANIFOLDS
E x e r c i s e 1.1.7. Give an example of a connected, 2nd countable, Hausdorff space t h a t is not path connected. Recall t h a t a regular space X is one in which any proper closed subset C C X and point x C X \ C can be separated by disjoint, open neighborhoods of each. E x e r c i s e 1.1.8. Prove that a locally compact (in particular, a locally Euclidean), Hausdorff space must be regular. By a theorem of Urysohn [8, p. 195], 2nd countable regular spaces are metrizable. Thus, manifolds are metrizable and questions of continuity, closure, compactness etc. involving manifolds can be reduced to corresponding questions of sequential continuity, sequential closure, sequential compactness, etc. The following difficult result, known as L. E. J. Brouwer's theorem on invariance of domain, is needed in order to show that local dimension is always well defined on locally Euclidean spaces. The proof will not be given. It is best carried out by the methods of algebraic topology [10, p. 303], [39, p. 199], [13, p. 110]. For a more classical proof, see [20, pp. 9596]. In the theory of smooth manifolds, differential calculus reduces the appropriate analogue of this theorem to elementary linear algebra. T h e o r e m 1.1.9 (Invariance of domain). I f U C_ IRn is open and f : U ~ ~'~ is continuous and onetoone, then f ( U ) is open in IRn. C o r o l l a r y 1.1.10. I f U C_ ~ n and V C_ I~m are open subsets such that U is horneornorphic to V , then n = m. Proof. Assume that rn ~ n, say, m < n. Define i : R m ~ lI~~ by i(x 1 . . . . ,x m)
:
(X 1 , . . . , x m , 0 , . . . , 0 ) . nm
This m a p is continuous and onetoone and i(N m) is not open in N ~ and does not even contain a subset that is open in N n. By assumption, there is a homeomorphism ~o : U ~ V, so the composition
f :U ~
V ~d~U~n
is continuous and onetoone. Also, while U C_ R ~ is open, we see t h a t f ( U ) = i ( V ) C i(N m) cannot be open in R n. This contradicts Theorem 1.1.9. [] C o r o l l a r y 1.1.11. I f X is locally Euclidean, then the local dimension is a welldefined, locally constant function d : X * Z + . Proof. Let x C X and suppose that there are open neighborhoods U and V of x in X, together with open subsets U C_ R n, V C N m, and homeomorphisms
~:U*U
r Since V R U is open in X, it follows that
~ ( v n g ) c_ ~c_
Rn
are inclusions of open subsets and, similarly, t h a t r or
:r
n U) is open in R m. But
n v ) ~ ~ ( u n v )
is a homeomorphism, so m = n by the previous corollary. A l l assertions follow.
[]
1.2. T O P O L O G I C A L MANIFOLDS
3
C o r o l l a r y 1.1.12. If X is a connected, locally Euclidean space, then the local dimension d : X ~ Z + is a constant called the dimension of X . E x e r c i s e 1.1.13. Let X and Y be connected, locally Euclidean spaces of the same dimension. If f : X ~ Y is bijective and continuous, prove that f is a homeomorphism. 1.2. T o p o l o g i c a l M a n i f o l d s Some authors designate by the term "manifold" an arbitrary locally Euclidean space. It is more common, however, to require more. D e f i n i t i o n 1.2.1. A topological space X is a manifold of dimension n (an nmanifold) if (1) X is locally Euclidean and d(x)  n = dim X; (2) X is Hausdorff; (3) X is 2nd countable. Of the three examples in the previous section, only the open subsets of Nn were manifolds. L e m m a 1.2.2. I f X is a compact, connected, metrizable space that is locally Euclidean, then X is an nmanifold, for some n c Z +. Proof. Indeed, X is Hausdorff because it is metrizable. It is 2nd countable because it is locally Euclidean and compact. Being locally Euclidean and connected, X has constant local dimension. [] Here are some examples of manifolds. E x a m p l e 1.2.3. The nsphere S n = {v E IR~+1 I Ilvll = 1} is an nmanifold. One way to see that it is locally Euclidean is by stereographic projection. Let p+ = ( 0 , . . . , 0 , 1 ) , p_ = ( 0 , . . . , 0 ,  1 )
be the north and south poles of S n, respectively. Then the stereographic projections 7r+ : S ~ \ {p+ } + IR~' 7r_ : S n \
{p_}
+ II{ n
onto the subspace R~= {(xl,...,xn,0)} are homeomorphisms and {S ~ \ {p_}, S ~ \ {p+}} is an open cover of S n. (For a pictorial definition of ~+, see Figure 1.2.1.) Since S ~ is compact and metrizable, it is an nmanifold. E x e r c i s e 1.2.4. In Example 1.2.3, write down formulas for the stereographic projections 7r• and prove carefully that they are homeomorphisms. E x a m p l e 1.2.5. If N is an nmanifold and M is an mmanifold, then N x M is an (n + m)manifold. Indeed, if (x,y) E N x M, let U be a neighborhood of x in N homeomorphic to an open subset of IRn and V a neighborhood of y in M homeomorphie to an open subset of R m. Then, the neighborhood U x V C_ N x M of (x, y) is homeomorphic to an open subset of IRn x R m = R n+m. Since M and N are Hausdorff and 2nd countable, so is N x M.
4
1. TOPOLOGICAL MANIFOLDS
F i g u r e 1.2.1. Stereographic projection from p+ E x a m p l e 1.2.6. The ntorus T n S 1
X
S1
X
...
X
S1
n factors
is an ndimensional manifold. Indeed, by Example 1.2.3, S 1 is a 1manifold and Example 1.2.5, applied successively, then implies that T n is an nmanifold. E x a m p l e 1.2.'7. A vector w E ~ + 1 is defined to be tangent to S n at v E S n if w J_ v. This conforms to naive geometric intuition and can be seen to conform to the general definition of tangent vectors to differentiable manifolds that we will give later. In order to keep track of the point of tangency, we will denote this tangent vector by (v, w) E R ~+1 x •n+l. Thus, the set of all tangent vectors to S ~ is T ( S n) : { ( v , w ) e ]1:~n + l >< ]l~n + l I IlVll ~ 1 , W J V}.
This space is topologized as a subspace of I~~+1 • R ~+1 . We also define the continuous map p : T ( S n) + S n by p ( v , w ) = v. Thus, p assigns to each tangent vector its point of tangency. For each v0 E S ~, consider
Tvo(SD = {(v0,w) 9 T ( S D } = p  l ( v 0 ) , the set of all vectors tangent to S ~ at v0. This is an ndimensional vector space under the operations r . (vo, w) = (vo, r " w)
(vo,w,) + (vo,w2) = (vo,Wl + w2). This structure, p : T ( S ~) * S '~, is called the tangent bundle of S n. The space T ( S ~) is the total space of the bundle, the space S ~ is the base space of the bundle, and p is the bundle projection. By a common abuse of terminology, the total space is often referred to as the tangent bundle itself. In Exercises 1.2.10 and 1.2.11, you are going to prove that T ( S '~) is a 2nmanifold. A couple of definitions are needed first. D e f i n i t i o n 1.2.8. If U c S n is an open subset, then T ( S n ) I U = T ( U ) is the space p  l ( U ) . The tangent bundle of U is given by Pu : T ( U ) + U, where Pu denotes the restriction pIT(U).
1.2. T O P O L O G I C A L MANIFOLDS
5
D e f i n i t i o n 1.2.9. If U C_ S n is open, a vector field on U is a continuous m a p s : U + T ( U ) such t h a t Pu o s = idg. E x e r c i s e 1.2.10. Given v0 E S n, show t h a t there is an open neighborhood U C S n of v0 and vector fields si : U ~ T ( U ) , 1 < i < n, such t h a t
{Sl(V), s2(v),., sn(v)} is a basis of the vector space Tv(Sn), V v C U. E x e r c i s e 1.2.11. Let U C S n be as in the previous exercise. Using t h a t exercise, c o n s t r u c t a continuous bijection ~o : U x IR~ + T ( U ) and prove t h a t ~ is a homeom o r p h i s m . (This is not very deep. You do not, for instance, need T h e o r e m 1.1.9.) Using this, prove t h a t T ( S n) is a 2nmanifold. Prove also that, for each v E U, the formula qov(w) = ~ ( v , w ) defines an isomorphism ~Ov : IR'~ * T v ( S n) of vector spaces. A t h o r o u g h u n d e r s t a n d i n g of the tangent bundle of S n eluded topologists for several decades. For instance, it was long unknown w h a t is the m a x i m u m number r(n) of vector fields si : S ~ ~ T(S'~), 1 < i < r(n), t h a t are everywhere linearly independent. T h a t is, we require that, for each v C S n, the vectors { s l ( v ) , . . . ,sr(~)(v)} be linearly independent in T , ( S n) and t h a t no set of r(n) + 1 fields has this property. T h e problem of c o m p u t i n g r(n) was known as the "vector field problem for spheres". Definition
1.2.12. T h e sphere S ~ is parallelizable if r(n) = n.
This brings us to a striking example of global versus local properties. If S ~ is parallelizable, Exercise 1.2.11 implies t h a t T ( S n) ~ S n x IRn. For general n, this s a m e exercise implies t h a t the tangent bundle T ( S ~) is locally a Cartesian p r o d u c t of an open set U C S ~ w i t h IR~, but it is only 91obally such a p r o d u c t when S ~ is parallelizable. Not every sphere is parallelizable. For instance, it has long been known t h a t r(2n) = 0. This means t h a t every vector field on S 2n is somewhere zero. In the case of S 2, this is s o m e t i m e s stated facetiously as "you c a n ' t comb the hair on a c o c o n u t " . It was also known for some t i m e t h a t S 1, S 3, and S 7 are parallelizable. T h e following was finally proven in the late 1950s [4], [27]. Theorem
1 . 2 . 1 3 (R. B o t t and J. Milnor, M. Kervaire). The sphere S ~ is paral
lelizable if and only if n = O, 1, 3, or 7. T h e case n = 0 is tile trivial fact t h a t the 0sphere S o = {41} c IR a d m i t s 0 i n d e p e n d e n t fields. T h e r e is an interesting relationship between T h e o r e m 1.2.13 and the problem of defining a bilinear multiplication on IR'~ w i t h o u t divisors of zero. Such a multiplication is a bilinear m a p # : IR~ x R ~ + IR~, w r i t t e n #(v, w) = vw, such t h a t v w = 0 ~ v = 0 or w = 0. Theorem
1.2.14.
and only if n = 0 , 1 , 3 ,
There is a multiplication on IRn+l without divisors of zero if or 7.
Indeed, R 1 = IR, R 2 = C, and IR4 = ]HI (the quaternions). T h e nmltiplication on ]Rs is given by the Cayley numbers, a nonassociative division algebra whose elements are ordered pairs (x,y) of quaternions [43, pp. 108109]. This proves the "if" in T h e o r e m 1.2.14.
6
1. T O P O L O G I C A L M A N I F O L D S
E x e r c i s e 1.2.15. If ]~n+l admits a multiplication without divisors of zero, prove that S n is parallelizable. In light of Theorem 1.2.13, this gives the "only if" part of Theorem 1.2.14. The full solution to the vector field problem for spheres was given by F. Adams in the early 1960s [1], culminating a long history of research on that problem by several algebraic topologists. We state Adams' result. Define the function p(n), n > 1, by requiring that S n1 admit p(n)  1 everywhere linearly independent vector fields, but not p(n) such fields (thus, r(n) = p(n + 1)  1). Write each natural number n uniquely as n = (2r + 1)2 c+4d, where r, c, d are nonnegative integers and c n and an imbedding i : M ~+ IRk.
Theorem
Proof. Since M is compact, there is a finite open cover II = {Uj}}'=I of M and a collection of h o m e o m o r p h i s m s ~oj : Uj * Wj C_ ]Rn, 1 < j < r. Let ~ = {AJ}5=l be a p a r t i t i o n of unity subordinate to ~d. We will take k = r(n + 1) and c o n s t r u c t an i m b e d d i n g i : M ~ IRk. Define i:M+iRn• ... x ]~n XIRX ... X IR r fact . . . . .
f•tors
by
i(~) = ( ~ ( ~ ) ~ 1 ( ~ ) , . . . , ~(x)~,.(x), ~1(~),..., ~,.(x)). Here we m a k e the convention t h a t 0.p3 (x) = () 9 IR~, even when ~ j (x) is undefined. Since supp(Aj) C Uj and d o m ( ~ j ) = Uj, the expression )~j(x)~oj(x) is identically (~ n e a r the settheoretic b o u n d a r y of Uj and on all of M \ Uj. This implies t h a t t h e m a p i : M + IRa is continuous. Since M is c o m p a c t and i(M) is Hausdorff, we only need to prove t h a t i is onetoone in order to prove t h a t i is a h o m e o m o r p h i s m onto its image. Let x , y 9 M and suppose t h a t i(x) = i(y). Since ~ is a p a r t i t i o n of unity, there is a value of j such t h a t h i (x) ~ 0. B u t the (nr + j ) t h coordinates of i(x) and i(y) are Ay(x) = Aj(y), so x , y 9 supp(/~i) C Uj. Also, )U(x)~oj(x) = )U(y)~oj(y), so ~j (x) = ~j (y). Since ~oj : Uj ~ IRn is onetoone, it follows t h a t x = y. [] T h e i m b e d d i n g dimension k = r(n + 1) given by this t h e o r e m for c o m p a c t nmanifolds is often much too generous. For example, Exercise 1.3.26 gives a covering of p 2 by r = 3 open sets h o m e o m o r p h i c to ]R2, so the theorem guarantees only t h a t P~ can be i m b e d d e d in ]R9. In fact, it is possible to imbed p2 into ]R4, as you showed in Exercise 1.5.5. Generally, if an nmanifold is differentiable (Definition 3.1.6), it can be proven t h a t M imbeds in ]R2~+1 ( T h e o r e m 3.7.12). This result, due to H. W h i t n e y , is best possible in the sense t h a t there exist nmanifolds t h a t cannot be i m b e d d e d in IR2n.
1.6. Manifolds with B o u n d a r y Manifolds are m o d e l e d locally on Euclidean nspace. S o m e t h i n g like the closed nball D n = {v E IR~ I I[v[I < 1} fails to be a manifold because a point on the b o u n d a r y 0 D n = S ~1 does not have a neighborhood h o m e o m o r p h i c to an open subset of IR~. It does, however, have a neighborhood h o m e o m o r p h i c to an open subset of E u c l i d e a n halfspace. Definition
1.6.1. T h e Euclidean halfspace of dimension n is ]HIn =
x n) e R n l x 1 0, from which it follows that J is open in the topology of N. By Theorem 1.1.9, g(0, 1) is also open in N, so g(0, 1] is open. Similarly, if x_ c ON, g[0, 1) is an open subset of N. Thus, if x+ and x_ both belong to ON, the image of g is open and, being compact, is also closed in N. By connectivity, this image is all of N, proving t h a t N is homeomorphic to [0, 1]. Suppose, therefore, that x+ c int(N) and deduce a contradiction. The same argument will show that x_ r int(N). Let h : V + IR be a homeomorphism of an open neighborhood V of x+ in N onto an open subset of 1R and let J denote the connected component of 9(0, 1] a V containing x+. Since x+ is the limit of a sequence in g(0, 1), J does not degenerate to a single point, hence h(J) is a halfopen interval with h(x+) as its endpoint. Rechoosing h, if necessary, assume t h a t h(J) = (  1 , @ For a small value of e > 0, there is an open set W C V such t h a t h(W) = (  1 , e) and J = g(0,1] n W . It follows easily that W Ug(0,1) = W U U is homeomorphic to an open interval in R, contradicting the maximality of U. [] The following innocuous corollary will be quite important for our treatment of rood 2 degree theory in Section 3.9. C o r o l l a r y 1.6.15. Every compact, 1dimensional manifold A~ has OM equal to a finite set of points with an even number of elements. 1.7. C o v e r i n g S p a c e s a n d t h e F u n d a m e n t a l
Group
Covering spaces play a fundamental role, not only in manifold theory, b u t throughout topology. In this section, there will be no reason to restrict our attention to manifolds, everything being true for a quite large class of topological spaces. Accordingly, we fix only the following hypothesis for the entire section. Hypothesis.
All spaces are locally pathconnected.
Note t h a t local pathconnectedness implies that connected spaces are pathconnected. Note also that we do not require the Hausdorff property. There are actually useful applications of covering space theory to nonHausdorff 1manifolds. 1.7.A. The basics of covering spaces. D e f i n i t i o n 1.7.1. Let p : Y + X be a continuous map. An open, connected subspace U C_ X is said to be evenly covered by p, if each connected component of p1 (U) is carried homeomorphically by p onto U.
1.7. C O V E R I N G
SPACES AND THE FUNDAMENTAL
GROUP
27
D e f i n i t i o n 1.7.2. A continuous m a p p : Y ~ X is a covering map if X is connected a n d each point x C X has a connected neighborhood t h a t is evenly covered by p. T h e triple (Y, p, X ) is called a covering space of X. In practice, one usually abuses this terminology, referring to Y itself as the covering space. Note t h a t we do not require Y to be connected. However, we will be mostly interested in connected covering spaces. Of considerable i m p o r t a n c e are automorphisms of covering spaces, defined precisely as follows. D e f i n i t i o n 1.7.3. Let p : Y + X be a covering map. A covering transformation, also k n o w n as a deck transformation or an automorphism, is a h o m e o m o r p h i s m h : Y + Y such t h a t p o h = p. T h a t is, the following diagram commutes: Y
h
~ Y
X L e m m a 1.7.4. The set F of covering transformations associated to a covering map p : Y +X forms a group under composition, called the covering group.
Proof. Indeed, if hi and h2 are covering transformations, p o ( h i o h 2 ) = (p o h i ) o h2 = p o h~ = p
Furthermore,
poh =pop=
( p o h ) o h 1 = p o h 1.
Finally, it is clear t h a t idv is a covering transformation, so F is a group. Example
[]
1.7.5. T h e m a p p : IR + S 1, defined by p(t) = e 27tit,
is a covering map. Indeed, if z0 = e 27tit~ C S 1, p carries the compact interval [to  1/4, t0 + 1/4] onetoone, hence homeomorphieally, onto a compact arc in S 1 c o n t a i n i n g z0 in its interior U. T h e n p  l ( U ) is the disjoint union of open intervals (to + n  1/4, t0 + n + 1/4), as n ranges over the set Z of all integers. Evidently, each of these intervals is a connected c o m p o n e n t of p1 (U) and is carried by p homeomorphically onto U. Finally, p(t) = p(s) if and only if s = t + m, for some m E Z. Thus, h : IR ~ II{ is a covering t r a n s f o r m a t i o n if and only if h(t) = t + m t where rat E 2~,  o o < t < oo. By continuity, mt depends continuously on t. B u t Z is a discrete space a n d R is connected, so ra t = ra is constant. The group of covering transformations is the group of translations by integers, hence is canonically isomorphic to the additive group Z. E x e r c i s e 1.7.6. Let G be a connected, locally pathconnected, topological group. Let H C G be a closed, discrete subgroup. (Recall that a subspace is discrete if, in the relative topology, each of its points is open.) Prove t h a t the coset projection
p: G ~ G/H is a covering space (where G / H has the quotient topology). Show that the group F of covering t r a n s f o r m a t i o n s consists of right translations
g E G~~ gh
28
1. T O P O L O G I C A L M A N I F O L D S
by elements h E H. More precisely, prove that
~o : h E H+ qOh E F, ~h(g) =
gh 1,
defines an isomorphism of tile group H to the group P. (The need for h 1 rather than h is due to the possible noncommutativity of these groups. W i t h this definition, one has that ~ghlh2 ~hl o ~9h2. For commutative groups, left and right translation are equivalent and the inverse could be omitted.) :
Note that Example 1.7.5 is a special case of Exercise 1.7.6. More generally, the projection p : R ~~
Rn/Z ~ = T n
is a covering map with covering group isomorphic to the integer lattice Z n. E x a m p l e 1.7.7. The quotient map p : S ~ ~ P~, defined as in Exercise 1.3.26, is a covering map. The group of covering transformations is generated by the antipodal interchange map, hence is Z2. Definition
1.7.8.
If
y,
X'
f
f
~y
~X
is a commutative diagram of continuous maps, where p~ and p are covering maps, we say t h a t f i s a lift of f to the covering spaces. In the case that X ~ = X and f = i d x , such a lift is called a homomorphism of covering spaces. A homomorphism of covering spaces that is also a homeomorphism is called an isomorphism of covering spaces. Of course, an automorphism of covering spaces, as defined earlier, is an isomorphism. L e m m a 1.7.9. If f is a lift o f f , as in the preceding definition, and if the covering
space Y~ is connected, then "f is completely determined by f and by the value of f at a single point. Proof, Let f and f" be lifts of f which agree at some point y C Y'. Let x = p'(y) and note t h a t f ( x ) = p('f(y)) = p('f(y)). Denote this point by z. Let V' be an evenly covered neighborhood of x and let U be an evenly covered neighborhood of z such that f ( U ' ) cc_ U. This is possible since f is continuous. Let V ~ be the component of (p')l(U') that contains y and V the component of p  l ( u ) that contains f ( y ) = f'(y). By the definition of "lift", the diagrams V' ~P'I U'
Y )V Ip , U
1.7. COVERING SPACES AND THE FUNDAMENTALGROUP
29
and V'
r
P'I
,V 1~
U'
~U f exist and commute. But p~ and p are onetoone on the components V I and V, so the maps y and f must agree on the open subset V ~ C_ Yq This proves that the set of points o n which f = f is open in Y~. On the other hand, the set of points on which f and f do not agree is also open. Indeed, if f ( y ) r f ( y ) , then the component V1 of p  l ( u ) containing f ( y ) is disjoint from the component V2 containing ]'(y). Since V' is connected and y E V', we conclude that f ( V ' ) C 1/1 and "f(V') C V2. T h a t is, f a n d f'disagree at every point of the open set V'. The fact t h a t Y~ is connected and that the functions agree at some point implies that
1=_?.
[]
C o r o l l a r y 1.7.10. Let p : Y ~ X be a coverin 9 map, assume that Y is connected, and let y E Y . Then a coverin9 transformation h is uniquely determined by the point h(y). Indeed, a covering transformation is a lift of id : X ~ X, where we take Y~ = Y and p~ = p. Another important type of lift is one for which the covering p~ : Y~ ~ X ~ is the trivial covering id : X ~ ~ X ~. In this case, the continuous lift fits into a commutative triangle Y
X'
~
, X f
L e m m a 1.7.11 (Pathlifting property). Let p : Y ~ X be a coverin9 space, let x c X and y C p  l ( x ) , and let cr : [a,b] * X be a continuous path with ~(a) = x. Then there is a unique lift ~ : [a, b] ~ Y such that ~(a) = y. Proof. For each t C [0, 1], let Ut denote an evenly covered neighborhood of or(t). Then {~rl(Ut)}o 0), w r i t t e n 8(~;) :
(X 1 ( ~ ) , . . . , x n ( t ) ) ,
such t h a t each xi(t) is of class at least C 1 a n d s(O) = p, ~(o) = ~.
T h a t is, ~i(0) = a i, for 1 < i < n. By s t a n d a r d calculus,
D~(f) = lira f ( s ( h ) )  f(p) h~O h ' an e q u a t i o n t h a t makes sense without explicit reference to coordinates. In other words, although the directional derivative was defined as differentiation at t = 0 along a straight line curve
g(t)=p+t~,
~ 0 d e p e n d on .s) such t h a t s(0) = p is denoted by S(U,p).
44
2. LOCALTHEORY
F i g u r e 2.2.1. Some infinitesimally equivalent curves at p D e f i n i t i o n 2.2.3. If sl,s2 C S(U,p), we say that 81 and s2 are infinitesimally equivalent at p and we write sl ~p s2 if and only if
~f(sl(~)) t = 0 : df(s2(t))t=0' for all
f C C~176
It is easy to check that ~p is an equivalence relation on the set S(U, p). Following Isaac Newton, we think of each equivalence class as an "infinitely short curve", but not as a single point. In fact, we are simply lumping together all curves sharing the same position p and velocity vector ~ at time t : 0 (see Figure 2.2.1). But, while the notion of "velocity vector" will not have an obvious meaning for curves in a manifold, the definition of "infinitesimal equivalence" will be meaningful in that context, allowing us to define the velocity vector as an "infinitesimal curve". D e f i n i t i o n 2.2.4. The infinitesimal equivalence class of s in S(U,p) is denoted by (S)p and is called an infinitesimal curve at p. An infinitesimal curve at p is also called a tangent vector to U at p and the set
Tp(Y) : S(U,p)/~p of all tangent vectors at p is called the tangent space to U at p.
Remark. Once we are sufficiently familiar with these notions, we will replace the notation (S)p with the more standard i(0) and call this the velocity vector of s at time t = 0. The following is immediate by the definition of infinitesimal equivalence. L e m m a 2.2.5.
['or each (S)p E Tp(U), the operator
D(~>.: C~(U,p)
~ ]I{
2.2. T A N G E N T V E C T O R S
45
is welldefined by choosing any representative s E (S)p and setting D(s),,(f) = d f ( s ( t ) ) t = 0 '
for all f E C~(U,p). Conversely, (S}p is uniquely determined by the operator D(s),. Note t h a t D<s),, is a linear operator. That is,
D<s>,"(af + bg) = aD<s>,,(f) + bD(s>,,(g), Vf, gECoo(U,p),
Va, b E R .
Since the whole reason for introducing tangent vectors is to produce linear approximations to nonlinear problems, it will be necessary to exhibit a natural vector space structure on Tp(U). In order to carry this structure over to manifolds, we do not want it to be dependent on the coordinates of ~n. The key lemma for this follows. L e m m a 2.2.6. Let (Sl)p,(S2)p E Tp(U) and a,b E JR. Then there is a unique
infinitesimal curve {S}p such that the associated operators on Coo(U,p) satisfy D,, is the desired operator. By Lemma 2.2.5, {s}~ is uniquely determined by
D,,.
[]
D e f i n i t i o n 2.2.7. Let { S i p , {S2}p E Tp(U) and a, b E R. Then a {Sl)p + b {.s2)p E Tp(U) is defined to be the unique infinitesimal curve {s}p given by Lemma 2.2.6. E x e r c i s e 2.2.8. Prove that the operation of linear combination, as in Definition 2.2.7, makes Tp(U) into an ndimensional vector space over R. The zero vector is the infinitesimal curve represented by the constant p. If (s)~ E Tp(U), then  (S)p = (S}p where s  ( t ) = s (  t ) , defined for all sufficiently small values of t. The operator D is defined by
D(s>,[f]p
:
~f(s(t))lt:o.
Remark. The discussion so far would have worked equally well if we had fixed an integer k k 1, replaced C~(U,p) with the set Ck(U,p) of C k functions defined in neighborhoods of p, and taken {~p : {~pktO be the germs of these functions. This remark is crucial if one wants to formulate the theory of C k manifolds. There is a purely algebraic characterization of Tp(U) that, though admittedly more formalistic than the one we have given, has its charms. This definition of the tangent space is valid only for the C 0r case (the default). First, recall that an algebra ~ over R is a vector space over ]i{, together with a bilinear map
called multiplication. If (r = r for all (, r E ~, the algebra is said to be commutative. If, for all ~, C,X E ~, (~r : ~(r the algebra is associative. If there is e ~ such that ~( = (~ : (, for all ( 6 ~, then L is called a unity. Now, define algebraic operations on germs as follows. * Scalar multiplication: t[f]p : [tf]p, Vt 9 11{,V [f]p 9 OF. . Addition: [f]p + [g]p = [f[W + g[W]p, V [f]p, [g]p 9 ~gp, where W is an open neighborhood of p in dora(f) A dora(g). 9 Multiplication: [f]p[g]p : [(f]W)(g]W)]p, V If]p, [g]p 9 r where W is again as above. L e m m a 2.2.12. The above operations are well defined and make r
a commutative
and associative algebra over ]~ with unity. The elementary proof of Lemma 2.2.12 is left to the reader. The unique unity, of course, is the germ of the constant function 1. D e f i n i t i o n 2.2.13. The evaluation map ep : {~p ~ ]~ is defined by
ep[f]p
=
f(p).
The following is immediate. L e m m a 2.2.14. The evaluation map ep : ~)p + ]~ is a welldefined homomorphism
of algebras. D e f i n i t i o n 2.2.15. A derivative operator (or, simply, a derivative) on ~hp is an ]l{linear map D : {~p ~ If{ such that
D(ab) : D(a)ep(b) + ep(a)D(b), for all a, b 9 ~p. Temporarily, we will denote the set of all derivatives on Op by T(~p). However, we will see shortly that it is a vector space that is canonically isomorphic to Tp(U). We define algebraic operations on T(C3p).
2 , 2 . TANGENT
VECTORS
47
9 scalar multiplication: (tD)(a) = t(D(a)), Vt C IR and V D C T(Op), Va C Or; 9 addition: (D1 + D2)(a) = Dl(a) + D2(a), VDa,D2 E T(Op), Va C Or. Lemma
2.2.16.
The space T(~Sv) is a vector space over R under the above oper
ations. Again, the proof will be left to the reader. Example
2 . 2 . 1 7 . Define Di,p : Op ~ IR by
of
Di,p[f]p = ~xi (P). This is a welldefined, Rlinear map. Furthermore, by the Leibnitz rule for partial derivatives,
Di,p([f]p[g]p) = ~~(f g)(p) = ~fxi(p)g(p)
09 + f(P)~~x~(P)
= Di,p([f]p)ep([g}p) + ep([f]p)Di,p([g]p). T h u s Di,p is a derivative, 1 < i < n. E x a m p l e 2 . 2 . 1 8 . If (S>p is an infinitesimal curve, then D<s>, : 0 v ~ derivative. Indeed,
]R is a
n
D<s>~ [f]p = E aiDi,P[f]P' i=1
where
~(o)
=
; a
so the assertion follows from the previous example. It is obvious that
(s>p ~ D,, defines a linear m a p from the space of infinitesimal curves into the space of derivatives of Op. It is also clear t h a t this linear m a p is injective. The fact t h a t it is an isomorphism of vector spaces (Corollary 2.2.22) requires proof. Lemma
2 . 2 . 1 9 . If c is a constant function on U and D E T(Ov), then
D[c]v = O. Proof. Consider first the case c = 1. T h e n D[1}p = D([1]p[1]p)
= D([1]p)%([1]p) + %([1]p)D([1]p) = 2D[1]p, from which it follows t h a t D[1]p = 0. For an arbitrary constant c, D[c]p = ~D[1]p
by linearity.
= 0,
[]
48
X=
2. LOCAL THEORY
In order to get more information on T(qSp), we need a technical lemma. Let ( x l , . . . , X n) a n d p = (xl(p),...,xn(p)).
L e m m a 2.2.20. Let f E C~~
Then there exist functions g l , . . . , g n C C~176
and a neighborhood W c dora(f) A dom(gt) A . . . N dom(gn) of p such that (1) f ( x ) = f(p) + E n = l ( X i  xi(p))gi(x), V x e W ; (2) g~(p) = ~ ( p ) , 1 < i < ~ Proof. Define gi(x) =
/01
( t ( x  p ) +p)dt.
This is clearly a smooth function defined at all points x sufficiently near p. In order to prove (2), consider g~(P) =
J l I Of 7x~ (p) dt
Of fo 1 = Ox i (p) dt
of = Ox~ (p)"
In order to prove (1), consider
f ( x )  f(p) = fro1
d (f(t(x  p) + p) dt
dt =fo 1 { ~~"~fxi(txp)+p)(xixi(p)i=l {~ol ~fxi(t(x p) +p)dt} (xix~(P))
i=l g~(x)(z ~  ~(p)). i=1
[] T h e o r e m 2.2.21. The set { D l , p , . . . , Dn,p} is a basis of the vector space T(C3p),
Proof. Suppose that n
E
aiDi,p = O.
i=l For the coordinate functions x j, 1 < j O on A and gl(II~n \ A) = O. Lemma
Proof. T h e definition of k gives functions ki, by taking a = ai and b = bi in t h a t definition, 1 < i < n. T h e n g(Xl,:~2,...
,Z n) ~ ]~l(xl)k2(x2)... ]~n(x n)
is as desired.
[]
Next we define a s m o o t h function g : R ~ [0, 1] by
e(t)  f'a k(X) dx
dx' where a a n d b are the n u m b e r s in the definition of k. This function is weakly m o n o t o n i c increasing, t~(t)  0 for t < a a n d g(t) = 1 for t >_ b. The graph is depicted in Figure 2.6.5.
F i g u r e 2.6.5. The graph of
2.7. S M O O T H
VECTOR
FIELDS
65
Proof of Theorem 2.6.1. Let K and U be as in the s t a t e m e n t of the theorem. For each x E K , let A~ be an open, bounded, n  d i m e n s i o n a l interval, centered at x a n d having A~ C U. Apply L e m m a 2.6.3 to o b t a i n a smooth function 9~ : IRn * [0, 1), strictly positive on A~ a n d vanishing identically outside of Ax. Since K is compact, it is covered by finitely m a n y A x l , . . . , Axq. The function G = g X l ~ " ' " ~  g X q i s C o o on IRn, strictly positive on K , and has supp(G) = A~, U . . . U A~, C U. Since K is compact, we can also find m i n ( G I K ) = ~ > 0. In the definition of g : N   * [0,1], take a = 0 and b = ~. T h e n f = C o G : R ~ ~ [0,1] is smooth, s u p p ( f ) C U, a n d f I K ~ 1. [] E x e r c i s e 2.6.4. Let U c_ R ~ be open, f : U ~ IR smooth, a n d p E U. Prove t h a t there is a Coo function f : R n ~ R such t h a t [f]p = [f]p in Op. Conclude that, for a r b i t r a r y p E/R ~, @p can be identified canonically with the set of germs at p of globally defined, smooth, real valued functions on R n. E x e r c i s e 2.6.5. Let C C tR~ be a closed subset, U C_ R ~ an open neighborhood of C. Show t h a t there is a smooth, nonnegative function f : IR~ ~ IR such t h a t flC > 0 a n d s u p p ( f ) C U'. 2.7. S m o o t h V e c t o r F i e l d s Let U _C Nn be an open subset. In particular, this is a s m o o t h submanifold and we consider the set of smooth vector fields on U (Definition 2.5.13). This set is c o m m o n l y denoted by ~ ( U ) . It is a vector space over R under the pointwise operations. If X E X(U), its value at a point x E U is c o m m o n l y denoted by X~. T h r o u g h o u t this section, the preferred way to think of Xx E Tx(U) will be as a derivative of the algebra ~!ix of germs (Definition 2.2.15). For intuition, however, it is helpful to identify Tx(U) = ]R~, viewing Xx as a column vector. Thus,
[
fl(~) 1
Lf i )j 9
i=1
where Di,~ = O/Ox~I~ is the i t h partial derivative at x. As x varies, each i f ( x ) varies smoothly, so
11] f2
X =
n
= ~
n
ffDi,
i=1
where Di = O/Ox ~ a n d fi E C~176 1 < i < n. Using the column vector interpretation, we can picture X as a smoothly varying field of directed line segments (arrows), issuing from points of U and possibly degenerating to zero length segments somewhere (Figure 2.7.1). While this makes sense only in Euclidean space, the second i n t e r p r e t a t i o n makes X a first order differential operator on the space Coo (U) a n d will continue to make sense on manifolds.
66
2. LOCAL
THEORY
F i g u r e 2.7.1. T h e vector field X can be pictured as a s m o o t h field of arrows on U. I n t e r p r e t i n g X E 3r
as an operator, we write
X(g)
= "  . ' f bi Og 7'
Vg~Coo(u).
i=1
We will give an abstract, purely algebraic definition of the t e r m "first order differential o p e r a t o r " and t h e n show t h a t such operators are exactly the elements of
~:(u).
We view Coo(U) as an algebra over R ( c o m m u t a t i v e and associative, with unity the constant function 1). While ~ ( U ) is a vector space over IR, it has more algebraic s t r u c t u r e t h a n that. For example, it is a module over the algebra Coo(U) under pointwise scalar multiplication:
C~176 • X(U) ~ X(U), (f, X ) ~+ f X , where ( f X ) x = f ( x ) X ~ , for each x C U. T h e formal definition of a m o d u l e over an algebra is as follows. D e f i n i t i o n 2.7.1. Let F be an algebra over ]R, ~ a vector space over IR. Suppose t h a t there is an 1Rbilinear m a p F x ~   ~ JV[,
such t h a t
Vp,,~CF, V~C:~.
(po).~=p.(o.~),
T h e n 3V[ is said to be a module over F. If the algebra F has a unity ~ E F , it is further required t h a t ~.#=>, V # C :IV[. T h e m o d u l e is free if there is a subset {#~}~e~ C ~V[, called a basis of :M over F , such t h a t each # E :M has a unique representation r
tt = ~_~Pi " ftc~i, i=1
2.7. S M O O T H V E C T O R F I E L D S
67
with coefficients Pi E F, 1 < i < r. E x a m p l e 2.7.2. It should be clear that ~(U) is a module over C~176 Because U is an open subset of R n, this module is free with canonical basis {D1, D 2 , . . . , D , } . While 3~(U) will continue to be a module over Coo(U) when U is an open subset of a manifold, it will not be true, generally, that this module is free. Thus, while a module over an algebra is analogous to a vector space over a field, one must not press this analogy too far. We are going to give a deeper algebraic interpretation of ~(U). For this, we need some definitions. D e f i n i t i o n 2.7.3. Let F be an (associative) Ralgebra with unity. A linear map A : F ~ F such that A ( f 9 ) = A ( f ) 9 + fA(9), V f, 9 E F, is called a derivation of F. Derivations of Coo(U) are also called first order differential operators. The set of derivations of F will be denoted by 9 L e m m a 2.7.4. The set of der'ivations 9
is a vector space over R under the
linear operations (aA1 + bA2)(f) = a A l ( f ) + bA2(f), Va, bE R, VA1, A2 E fD(F), V f E F. The proof of this is completely elementary and is left to the reader. If the algebra F is commutative as well as associative, define an operation of "scalar" multiplication by
(fA)(g)=f(A(g)),
V f, 9 E F ,
VAE 9
E x e r c i s e 2.7.5. If F is commutative, prove that f A is an element of ~ ( F ) , V f E F , VA E 9 This makes 9 a module over the algebra F. L e m m a 2.7.6. The space )~(U) is a Coo(U)submodule of 9
[email protected] Indeed, given X E ~(U), write it as X = ~   ~ f i D i, i=1
where fi E C~(U), 1 < i < n. The Leibnitz rule for each partial derivative Di implies that
X(hg) = X(h)g + hX(9),
Vh, g ~ Coo(U).
[]
The main goal in this section is to prove that, all derivations of C~(U) are vector fields. T h e o r e m 2.7.7. The inclusion map )C(U) ~ [D(Coo(V)) is surjective. Before comiRencing the proof, we consider another algebraic structure on the module of derivations of an algebra.
68
2. LOCAL
THEORY
D e f i n i t i o n 2.7.8. If A1, As E ~D(F), then the Lie bracket [A1, A2] : F ~ F
is the operator defined by [A1, A2](f) = A I ( A 2 ( f ) )  A 2 ( A I ( f ) ) , V f E F . This is also called the commutator of A1 and As. E x e r c i s e 2.7.9. Prove that the Lie bracket satisfies the following properties: 1. [A,, A2] 9 9 V / ~ I , A 2 9 ~)(F); 2. the operation [., . ] : 9 x 9 * 9 is R bilinear; 3. [A1, A2] : [A2, A1] , V A i , As 9 D ( F ) (anticommutativity);
4. [AI,[A2, A3]] : [[A1,A2],A3] ~[A2,[A1,A3]], VA1,A2, A3 9 ~)(F) (the Jacobi identity). Thus, we can think of the operation [.,.]: 9
x 9
~ 9
as a bilinear multiplication making 9 into a kind of Ralgebra. This algebra is nonassociative, however, with the Jacobi identity replacing the associative law. The algebra is also anticommutative and does not have a unity.
Remark. One way to remember the Jacobi identity is to notice that, by this identity, the operator [A, 9] : 9 ~ 9 is a derivation of the (nonassociative) algebra 9 VA 9 9 D e f i n i t i o n 2.7.10. A nonassociative algebra having the properties in Exercise 2.7.9 is called a Lie algebra. E x e r c i s e 2.7.11. It will follow from Theorem 2.7.7 that :~(U) is a Lie algebra. Here, you are to prove directly that iX, Z] 9 •(U),
VX, Y 9 )C(U).
For this, note that the composed operators X o Y and Y o X are second order differential operators, but show by direct calculation that the commutator X o Y Y o X is first order. L e m m a 2.7.12. If t E F is the unity, then A(ct) = 0, VA E 9
and Vc E R.
The proof is exactly like that of Lemma 2.2.19. In order to prove Theorem 2.7.7, we must establish the reverse inclusion
~D(C~(U)) c :~(u). For this, we need to show that a derivation of C ~ (U) can be localized to a derivative of the germ algebra ~5~, at each x E U. This is by no means evident. L e m m a 2.7.13 (Key Lemma). Let A E 9
f 9 C a ( U ) , and suppose
that V c U is an o;en set such that f t V  O. Then A ( f ) I V ~ O.
Proof. Let x E V. By Theorem 2.6.1, we find ~ E C a ( U ) such that
~(~) =
0,
~l(U \ v) 
1.
2.7. SMOOTH
VECTOR
FIELDS
69
Indeed, since {z} is compact, w e find ~ E C a ( U ) such that g)(x) = 1 and supp(~b) C V. Then ~ = 1  ~ is as desired. Since f l V = O, we see that ~ f = f. Thus, A ( f ) = A((pf) = A ( ~ ) f + ~ A ( f ) . Hence A ( f ) ( x ) = A(~p)(x)f(x) + ~ ( x ) A ( f ) ( z ) . But f ( x ) = 0 = ~(x), and so A ( f ) ( z ) = O. Since x E V is arbitrary, A ( f ) I V = O. [] C o r o l l a r y 2.7.14. Let A C 9
f E C a ( U ) , and z E U. Then A ( f ) ( z ) depends only on A and the germ [fix E @z.
Proof. Let f , g E C~176 have the same germ If], = [g],. Choose an open neighborhood W C U o f x such that fl W = gl W. By the Key Lemma 2.7.13, ( A ( f )  A(g))IW = A ( I  g)l W  O, and so A ( f ) ( x )
=
A(g)(x).
[]
Given aJ E @~, x E U, there exists f E C a ( U ) such that aJ = [f]x. This is by Exercise 2.6.4. This allows us to make the following definition. D e f i n i t i o n 2.7.15. Given A E 9176
and x E U, Ax : @~ ~ IR is given by
Ax(aj ) = A ( f ) ( x ) ,
V~ = [f]x E 
where f E C a ( U ) . By tim above discussion, it is clear that Ax is welldefined, Vx E U, VA E P r o p o s i t i o n 2.7.16. I r A E 9
and x E U, then A , E T,(U).
Proof. Let [f]~, [g]x E @x, where f , 9 E C a ( U ) . Then, A~([f]~[g]x ) = A~[fg]~
= A(fg)(x) = (A(f)g + fA(g))(x) = A(f)(x)g(x) + f(x)A(g)(x) = Ax[f]~ex[g]~ + e~[f]~Ax[g]z. It is clear that A~ :
@ ~ +
IR is linear; so Ax E T~(U).
[]
Given A C 9 define 2x : U , T(U) by A(x) = A~ E T~(U). This function satisfies p o A = idu. (Maps with this property are called sections of the tangent bundle.) If this section is smooth, then A E :E(U). Write
7, =
]+D+, i=1
relnarking that the smoothness of A is equivalent to ]+ E C ~ ( U ) ,
l 0 so t h a t #Ptk~s(q) a n d q2sg2t(q) are defined, ($q < s,t < 5q. As q E U varies, these b o u n d s 5q will also vary. T h e local flows vary too, b u t they agree on overlaps by T h e o r e m 2.8.4. D e f i n i t i o n 2.8.19. The local flows of X a n d Y c o m m u t e on U if
62tq2s(q) = qJsOt(q),
5q < s,t < ~Sq, Vq E U.
T h e vector fields themselves c o m m u t e on U if [X, Y] ~ 0 on U. 2 . 8 . 2 0 . The vector fields X and Y commute on U if and only if their local flows commute on U.
Theorem
Proof. If the local flows c o m m u t e on U, then, for  a x < t < 5,, ~ t carries any flow line { ~ s ( x ) [  ~ < s < 6~} of 9 onto another flow line of ~. By t a k i n g the infinitesimal curve point of view, we see immediately t h a t (~t).:~(Yx) = Y~,(:~),  5 ~ < t < ($x, V x E U. T h a t is,
tArv,~1~ = lira 4)t. (Y)  Y _ lira Y   Y t0 t t~0 t
I
0
t h r o u g h o u t U. For the converse, we assume t h a t [X, Y] = 0 on U and deduce t h a t the local flows commute. Let q E U, fix s E (5q,($q), and let q' = ~s(q). Define v : (5q, 5q) + Tq, (U) by the formula v(t) = 4)_t.(Ye~(q,)). (Here a n d elsewhere, in an a t t e m p t to streamline notation, we drop the subscript { on differentials f.~.) T h e n
2.8. L O C A L F L O W S
79
v(t) is a differentiable curve in the vector space Tq, (U) = R ~, and dv = lira (~th),(Y,L+,(q,)) q~t,(Y~dq,))
dt
h~o
h
= lim q)t. h~0
~h.(Y~+,xq'))
= ~  t * lira h~0

Y~(q')
h
d~h*(Y~h(~t(q')))  Y~L(q') h
C r,i,t(q,)(g) = ~  t * [X, Y]e,(q') = 0,
 6 q < t < 6q. It follows t h a t v(t) is constant o n (6q, 6q); so
9 _t.(Yeodq,)) : Yq,,
(~q < t < (~q.
B u t q' ranges over a(s) = 9~(q), 6q < s < 6q, an integral curve to Y. Thus, ~(s) = Y~(~) a n d 4)t.(~(s)) = Y~d~(4) as s and t range i n d e p e n d e n t l y over (6q, 6q). Therefore, ~ t o a is also an integral curve to Y with initial condition (I)t(a(0)) : 'I~t(q). B u t ~ t g ~ ( q ) = 9 ~ t ( q ) ,  6 q < s , t < 6q, by the uniqueness part of Theorem 2.8.4. [] E x e r c i s e 2 . 8 . 2 1 . Given A c glI(n), view the right translation operation R A : Gl(n) ~ ~ ( n ) as a vector field RA E Z ( G I ( n ) ) . (1) For
show t h a t the local flow generated by RA has the formula
Vt C N, VQ C Gl(2). In particular, we o b t a i n a 9lobal flow 9 : ~{ • Gl(2) ~ Gl(2). (2) Note t h a t the formal definition
e~ A = [ + t A +
t2A2 tnAn 2! + " " + n ! +""
yields
C o m p u t e e tB for the m a t r i x
a n d make a n educated guess of a flow on Gl(2) generated by RB. Prove t h a t your guess is correct.
80
2. LOCAL
THEORY
2.9. C r i t i c a l P o i n t s a n d C r i t i c a l V a l u e s Let U C_ ll~n a n d V C ]l~m be open and let 9 : U ~ V be smooth.
D e f i n i t i o n 2.9.1. A point x E U is a regular point of (I) if 9 .~
: T x ( U ) ~ T~(~)(W)
is surjective. Otherwise, x is a critical point. Thus, s m o o t h maps from lower dimensions to higher dimensions have only critical points. At the other extreme, if 9 has only regular points, it is a submersion.
D e f i n i t i o n 2.9.2. A point y E V is a critical value of 4) if (I)l(y) contains at least one critical point of 4). Otherwise, y is a regular value of ~. You must take care with these terms. If ~  l ( y ) = 0, t h e n this set contains no critical points. T h a t is, a point y E V t h a t is not a value of 9 at all is a regular value of (I)! 2 . 9 . A . S a r d i s t h e o r e m . In this subsection, we prove the following fundam e n t a l result of Sard. It has i m p o r t a n t applications in topology, some of which will be t r e a t e d in the next chapter.
T h e o r e m 2.9.3. I f ~ : U ~ V is a smooth map, then the set of critical values has Lebesgue measure zero. We will u n d e r s t a n d the term "almost every" to m e a n "Lebesgue almost every". Thus, almost every point of V is a regular value. Our proof will follow closely t h a t given by J. Milnor [30]. First, however, we consider some examples a n d corollaries. E x a m p l e 2.9.4. It is well known t h a t one can construct a continuous surjection s : ~ ~ I~2 (a "space filling curve"). However, if s is smooth, every true value of s is a critical value; so s(R) C R 2 has measure zero. Smooth curves c a n n o t be space filling. More generally, smooth maps from lower to higher dimensions always have images of measure zero.
C o r o l l a r y 2.9.5. Let { ~ : Ui ~ V}i=l N , 1 1, Ck C_ C is the set of points x E U such that all mixed partials of dPi of order < k vanish at x, 1 < i < m. It is clear that we obtain a nest C _D C1 _DC2 _D ... _D Ck _D ... of closed subsets of U. The basic estimates behind Sard's theorem are contained in the following proof. P r o p o s i t i o n 2.9.8. For ]~ >_ 1 sufficiently large, the set ~(Ck) has Lebesgue measure zero.
[email protected] Let Q C U be a compact cube. Since Ck n U is covered by countably many such cubes, it will be enough to find a value of k, depending only on m and n (not on Q) such that ~(Ck a Q) has Lebesgue measure zero. Let ~ denote the edge length of Q. Let p range over Ck a Q. The kth order Taylor series, expanded about p, takes the form 9 (p + v)  ~(p) = n(p, v), where the remainder term satisfies a uniform estimate
NJ~(P,V)II ~ CIIV]Ik+l, for all p C Ck n Q and all v E R n such that p + v E Q. Subdivide Q into r n subcubes of edge length 5 / r and let Q~ be one of these subcubes containing a point p E Ck. Every point of Q~ has the form p + v, where Ilvll
_ 1, the set q~(Ck \ Ck+l) has Lebesgue
measure zero. Proof. Again, it will be enough to show that, for each p E Ck \ Ck+l, there is a neighborhood Wp of p in U such that ~(Ck n Wp) has measure zero. Let ~ : U * R be a kth order partial of a coordinate function ~ such t h a t some first order partial of ~ fails to vanish at p. Without loss of generality, we assume that O~ OX 1 (p) ys O.
Of course, ~ vanishes identically on Ck. The inverse function theorem again gives a change of coordinates
( x l , x 2 , . . . , X n) b+ ( ~ ( x l , . . . , x n ) , x 2 , . . . , X
n) ~_ ( y l , y 2 , . . . , y n ) ,
defined on some neighborhood Wp of p, thereby coordinatizing Ck n Wp as a subset of the hyperplane {0} x ]Rn1. Every point in this set is a critical point of the restriction q~0 of q5 to U n ({0} x R~~), so the inductive hypothesis implies t h a t ~(Ck n Wp) = 02o(Ck n Wp) has Lebesgue measure zero. [] C o r o l l a r y 2.9.12. For each integer k >_ 1, the set ~(C'..Ck) has Lebesgue measure zero.
Proof. Indeed, C \ Ck = (C \ C1) U (C1 \ C2) U . . . U (Ck1 \ Ck).
[]
By this corollary and Proposition 2.9.8, the proof of the inductive step is complete.
2.9. C R I T I C A L P O I N T S
83
2.9.B. N o n d e g e n e r a t e c r i t i c a l p o i n t s * . Of special note are the critical points of a smooth, realvalued function. Suppose that f : U + R is such a map, where U C_ R n is open. We present some facts that are the beginnings of "Morse theory", a remarkable application of critical point theory to topology due to M. Morse. This introduction to Morse theory will be continued in Sections 3.10 and 4.2. E x e r c i s e 2.9.13. Let p E U be a critical point of f.
If X , Y C X(U), then
X p ( Y ( f ) ) = Y p ( X ( f ) ) , and this number depends only on the tangent vectors Xp, Yp, not on their extensions to fields on U. Each element Yp C Tp(U) can be extended to a vector field Y E ~(U). Indeed, if Yp = ~ n = l aiDi,p, we can define Y = ~ i ~ 1 aiDi" This remark, together with the exercise, insures that the following definition makes sense. D e f i n i t i o n 2.9.14. If p C U is a critical point of f, the Hessian of f at p is the symmetric, bilinear form Hp(f) : Tp(U) • Tp(U) ~ ]~ defined by
Hp(f)(Xp, Yp) = X p ( Y ( f ) ) . D e f i n i t i o n 2.9.15. The critical point p E U of f is nondegenerate if the symmetric matrix representing the Hessian Hp(f), relative to some choice of basis, is nonsingular. The (Morse) index A of the critical point is the number of negative eigenvalues of this matrix. E x e r c i s e 2.9.16. Show that, relative to the standard basis of Tp(U), the matrix representing the Hessian is the matrix c~2f of 2nd partials of f at p. It is straightforward to check that a matrix representing the Hessian with respect to some coordinate system is nondegenerate of index )~ if and only if this is true relative to every coordinate system. E x a m p l e 2.9.17. Let U = 1Rn and let f : R n ~ Ii~ have the formula (*)
f ( z l , . . . , z n) = f(0)  ~](zi) 2 + i=l
(zi) 2, i=A+I
relative to suitable coordinates. Then 0 is the only critical point and the Hessian at 0 is represented by the matrix In),
"
It is evident that this matrix is nondegenerate of index A. The following, due to M. Morse, asserts that this is essentially the only example. T h e o r e m 2.9.18 (The Morse Lemma). Let p E U be a nondegenerate criticaI point
of index )~ of the smooth function f : U ~ R. Then there is an open neighborhood Up of p in U and a smooth change of coordinates (i.e., diffeomorphism) z : Up + W onto an open neighborhood W of 0 in ]~n such that, relative to the new coordinates z=(zl,...,zn),
84
2. L O C A L T H E O R Y
(1) p = 0;
(2) f has the formula (,). We need a preliminary lemma. By a translation, we assume that the critical point p = 0. L e m m a 2.9.19. In a suitable neighborhood V of O in U, there arc defined smooth,
realvalued functions his = hsi, 1 < i , j < n, such that f ( x l , . . . ,x ~) = f ( O ) + ~ , xixJhij(xl,... ,X n)
i,j=l and such that the matrix 2[hij(0)] represents the Hessian Ho(f). Proof. Choose V to be the open eball centered at 0, where e > 0 is small enough t h a t V C_ U. Write f ( x l , . . . ,x n)  f(O) = =
(tx~,... , t x n ) d t
~fa liOf'l
x f~xiitx , . . . , t x ~ ) d t .
z=l
0
Thus, setting
gi(xl,... ,X n) ~
~ x i ( t x , ' " ,txn)dt,
we write n f(xl,...,X
n)
 
f(O) : ~~X i gi( X 1 ,...,xn) 9 i=1
By differentiating the formula for gi under the integral sign, we compute
OV(o ) ~o~soz* "
~(o)Note t h a t
g i ( 0 ) = ~~(0) = 0 ,
lr for suitable s m o o t h functions H i j . Symmetrizing these coefficients by
Hij(v)
= Hij(v) + Hji(v) 2
completes the inductive step.
[]
C o r o l l a r y 2 . 9 . 2 0 . Nondegenerate critical points are isolated. By contrast, degenerate critical points m a y easily fail to be isolated. For example, the function f ( x , y) = x 2 has the entire yaxis as its set of critical points, while f ( x , y) = x~y ~ , n, m > 1, has the union of b o t h axes as its critical set. Examples of isolated, b u t degenerate, critical points include f ( x ) = x 3, having 0 as its sole critical point, and the "monkey saddle" f ( x , y) = x a  3xy 2. This latter has only the origin as critical point a n d the Hessian there is the zero matrix.
CHAPTER 3
The Global Theory of Smooth
Functions
Our present goal is to extend the theory of smooth functions, developed on open subsets of IRn in Chapter 2, to arbitrary differentiable manifolds. Geometric topology becomes an essential feature.
3.1. S m o o t h Manifolds and Mappings Let M be a topological manifold of dimension n. Tile locally Euclidean property allows us to choose local coordinates in any small region of M. D e f i n i t i o n 3.1.1. A coordinate chart on M is a pair (U, g)), where U is an open subset of M and qo : U + R n is a homeomorphism onto an open subset of IRn. One often writes ~o(p) = (zl(p),... ,zn(p)), viewing this as the coordinate ntuple of the point p E U. Relative to such a coordinatization, one can do calculus in the region U of M. The problem is that the point p will generally belong to infinitely many different coordinate charts and calculus in one of these coordinatizations about p might not agree with calculus in another. One needs tile coordinate systems to be smoothly compatible in tile following sense. D e f i n i t i o n 3.1.2. Two coordinate charts, (U, V)) and (V, ~b) on M are said to be C ~  r e l a t e d if either U N V = (~ or o~1 :W(UnV)~(UNV) is a diffeomorphism (between open subsets of IRn). This is illustrated in Figure 3.1.1. We think of ~ o ~  l as a smooth change of coordinates on U N V. Thus, on U n V, functions are smooth relative to one coordinate system if and only if they are smooth relative to the other. Indeed, differential calculus carried out in UNV via the coordinates of ~(UNV) is equivalent to the calculus carried out via the coordinates of ~b(U n V). The explicit formulas will, of course, change from the one coordinate system to tile other. Furthermore, piecing together these local calculi produces a global calculus on M. The concept that allows us to make these remarks precise is that of a smooth atlas. D e f i n i t i o n a . l . a , a C ~ atlas on M is a collection r = {(U~, p~)}~e~ of coordinate charts such that 1. ( U ~ , ~ ) is C~176
s. M = U ~
to (U~,F~), Va,/3 E 9.1;
u~.
D e f i n i t i o n 3.1.4. Two C ~176 atlases J[ and A t on M are equivalent if JtUCV is also a C ~ atlas on M. It will be seen that global calculus carried out relative to A will be identical to global calculus carried out relative to the equivalent atlas A'.
88
3. G L O B A L T H E O R Y
F i g u r e 3.1.1. T h e s m o o t h coordinate change ~ o ~b1 E x e r c i s e 3.1.5. Equivalence of Coo atlases is an equivalence relation. Each 6"~176 atlas on M is equivalent to a unique m a x i m a l 6 ,0o atlas on M . D e f i n i t i o n 3.1.6. A m a x i m a l C ~176 atlas .4 on M is called a s m o o t h s t r u c t u r e on M (also called a differentiable s t r u c t u r e or a Coo structure). T h e pair (M,.4.) is called a s m o o t h (or differentiable or Coo) nmanifold. By a typical abuse of notation, we usually write M for t h e s m o o t h manifold, the presence of t h e differentiable s t r u c t u r e A being understood. By Exercise 3.1.5, any 6 ,0o atlas (not necessarily maximal) on M completely determines the differentiable structure. N o t e t h a t the dimension n of a s m o o t h nmanifold is welldefined by P r o p o s i t i o n 2.3.12. E x a m p l e 3.1.7. T h e manifold IRn has a canonical s m o o t h structure, n a m e l y the set Ytn of all pairs (U, ~) where U C_ ]Rn is open and ~ : U * IRn is a diffeomorphism onto an open set ~o(U) _C 1R~. E x a m p l e 3.1.8. If M and N are s m o o t h manifolds, d i m M = m and d i m N = n, w i t h respective s m o o t h structures A = { ( U ~ , ~ a ) } ~ and $ = {(V~,r t h e n M x N is canonically a s m o o t h (ra + n)manifold. Indeed,
is a O0o atlas, d e t e r m i n i n g uniquely a m a x i m a l one, called tile Cartesian p r o d u c t of the two s m o o t h structures. E x a m p l e 3.1.9. If W C_ M is an open subset of a s m o o t h nmanifold, then W is a s m o o t h nmanifold in a natural way. Details are left as an easy exercise.
3.1. S M O O T H M A N I F O L D S
89
Remark. By substituting C k for C ~176 in the above discussion, one obtains the notion of a C k manifold, 1 < k < co. Similarly, one defines the notion of a real analytic (C ~) manifold. The reader should have no trouble in adapting the following discussion to these cases. Let M be a smooth nmanifold with a smooth atlas A = {(Ua, ~)}aE~*. We do not require this atlas to be maximal. Set
gaf~ = (~a o ~ 1 : ~ ( V a ~ V/3) ~ ~a(Va n Vf~). These local diffeomorphisms in Rn satisfy the coeycle conditions (1) gaf~ o gz'Y = g~'~ on ~g.y(Ua n U, n Uv) , (2) gaa = i d ~ ( u ~ ) , (3)
=
It should be noted that properties (2) and (3) follow from property (1). D e f i n i t i o n 3.1.10. The system {9aZ}a,Ze~ is called a structure cocycle for the smooth manifold M. The term "cocycle" is borrowed from algebraic topology due to certain formal similarities to cocycles in Cech cohomology.
Remark. It will be useful to see how to "reassemble" M out of the d a t a {Ua = Pa(Ua),ga~}a,ZE~. On the disjoint union aEP2
define the relation x~yc:~3a,~E91suchthatxEU~,
yCUz andy=gz~(x).
By properties (1), (2), and (3), this is an equivalence relation, so we form the topological quotient space M / ~ . We will show that this space is homeomorphic to M and exhibit a natural smooth structure on it. Let [z] E M / ~ denote the equivalence class of z E M. Define
~:M* M / ~ by setting ~(x) = [ ~ ( x ) ] if x C Ua. If x C UZ also, then =
so ~ is well defined. It is also continuous. The map from M to M that takes z E U~ to ~ l ( z ) respects the equivalence relation, hence passes to a continuous map
r It is easy to see that ~ and ~p are mutually inverse, so M and M / ~ are canonically homeomorphic. Each U~ imbeds canonically in ff~'~/~ as an open subset and ida : Ua * U~ c_ ]R~ defines a coordinate chart (Ua, ida) on M / ~ . These charts are C~176 via the cocycle {gaz}a,ZE~, SO M / ~ is canonically identified with M as a smooth manifold via the mutually inverse diffeomorphisms ~ and r (see Definition 3.1.18). E x e r c i s e 3.1.11. Prove that the topological nmanifold P~ (see Exercise 1.3.26) is a smooth nmanifold.
90
3. GLOBAL
Exercise 3.1.12. Show with just two charts. We
THEORY
that the manifold
turn to the smooth
maps
S n can be assembled
from a C a atlas
defined on a manifold.
Definition 3.1.13. A function f : M * ~ is said to be smooth there is a chart (U, ~) E .4 such that x E U and
if, for each x E M,
is smooth.
will be denoted
The
set of all smooth,
real valued functions on M
by
C~176 The definition only requires us to be able to find some such chart about each point x E M, but the following assures us t h a t all charts will then work. L e m m a 3.1.14. The function f : M * N is smooth if and only if
f o ~21 : ~ ( U ~ ) ~ R is smooth, V ( U ~ , ~ a ) E A. Proof. Clearly this condition implies that f is smooth. For the converse, suppose that f is smooth and let z E U~ where ( U ~ , ~ ) E A. By Definition 3.1.13, choose (U~, ~o~) E A such t h a t x E U~ and f o (fl~l : ~ ( U ~ )
+ ]t~
is smooth. Then, f o W21 : ~ ( U ~ N UZ) ~ IR is given by the composition ~(U,
n U,) ~
~ z ( U , n U,) s ~
1 a.
As a composition of smooth maps, this is smooth. T h a t is, f o ~  1 : ~a(gc~ ) _.+ is smooth on some neighborhood of the point qo~(x). But x E U~ is arbitrary, so f o ~ 1 is smooth on all of ~ ( U ~ ) . [] We think of f o ~21 as a formula for flU~ relative to the coordinate system ~a = ( x ~ , . . . , x~). We generally write (Ua, x ~ , . . . , x n) or (Ua, xa) for (Ua, ~a). D e f i n i t i o n 3.1.15. Let M and N be C a manifolds with respective smooth structures .4 and ~B. A map f : M ~ N is said to be smooth if, for each x E M, there are ( U ~ , ~ a ) E A and (Vz,r E N such that x E U~, f(U~) C_ V~, and r
o f o ~21 : ~ ( g ~ )
~ ~b~(V~3)
is smooth. L e r n m a 3.1.16. The map f : M ~ N is smooth if and only if, for all choices of
(u., ~ ) E ~ and (V~, r
E ~3 such that f(U.) C_ V,, the map ~3 o f o ~ 1 : qo~(U~) ~ ~b~(V~)
is smooth.
3.1. SMOOTH MANIFOLDS
91
The proof is similar to the previous one and is left to the reader. Again, we think of CZ o f o g)~l as a local coordinate formula for f. These two lemmas give an important part of the content of our remark that differential calculus in one coordinate chart is equivalent, in overlaps, to differential calculus in Coorelated neighboring charts. L e m m a 3.1.17. I f f : M ~ N and g : N , P are smooth maps between manifolds, then g o f : M + P is smooth. This is also elementary and is left to the reader. D e f i n i t i o n 3.1.18. A map f : M + N between smooth manifolds is a diffeomorphism if it is smooth and there is a smooth map g : N + M such that f o g = idN and 9 o f = idM. E x a m p l e 3.1.19. The maps : M/~.
M,
~o:M+M/~ are mutually inverse diffeomorphisms (see the remark following Definition 3.1.0). L e m m a 3.1.20. I f (M,A) is a smooth nmanifold, U C_ M an open subset, and : U ~ R n a diffeornorphism of U onto an open subset ~o(U) of R n, then (U, qo) C A. Proof. By the definition of diffeomorphism, (U, qo) is C~176 A, so (U, ~) C A by the maximality of this atlas.
to every (Us, qoa) C []
If we write (U, ~o) as (U, x 1, x 2 , . . . , xn), the above discussion allows us to write f ( x l , . . . , x n) for f l U , whenever f : 5,I + N is smooth. This is logically a bit sloppy, but it is psychologically helpful. E x e r c i s e 3.1.21. Suppose that 54 is a smooth nmanifold and that r~:MI~
M
is a covering space. Prove that M ' has a unique smooth structure relative to which the projection ~r is locally a diffeomorphism. If p E M, it makes good sense to talk about germs at p of real valued C ~ functions defined on open neighborhoods of p. As before, these form an associative algebra ~Sp over R. The evaluation map ep : ~Sp ~ R is defined exactly as before. D e f i n i t i o n 3.1.22. A derivative of q3p is an Rlinear map D : ~Sp, R such that D(~r
= D(r162
+ ep(~)D(r
V~, ~ C I~ip. This operator D is also called a tangent vector to M at p and the vector space T p ( M ) of all derivatives of Op is called the tangent space to M at p. Definition 3.1.23. If f : M ~ N is a smooth map between manifolds and if p E M, the differential
f.p = dfp : T p ( M ) ~ Tf(p)(N)
92
3. G L O B A L T H E O R Y
is the linear map
defined by
(f,p(D))[g]/(p) = D[g o f]~, for all D E T p ( M ) and all [g]f(p) 6 ~)f(p). L e m m a 3.1.24 (Global chain rule). If f : M ~ N and g : N ~ maps between manifolds and x E M , then d(g o f ) x = dg/(z) o dfx,
P are smooth
Proof. Consider ((g o f),p(D))[h]g(f(p)) = D[h o g o f]p = (f,p(D)[h o glfo)) = 0,S(~)(Lp(D)))[h]~(S(,)
)
Since [h]g(f(p)) ~ Og(f(p)) and D E TB(M) are arbitrary, the assertion follows.
[]
It is clear that id,p = id : T p ( M ) ~ Tp(M), so the chain rule has the following consequence. C o r o l l a r y 3.1.25. I f f : M ~ N is a diffeomorphism, then f,p : T p ( M ) ~ T f o ) ( N ) is an isomorphism of real vector spaces, V p 6 M . C o r o l l a r y 3.1.26. I f M is a smooth manifold of dimension n, then T z ( M ) is a real vector space of dimension n, V x 6 M . Proof. Let (U,~) be a coordinate patch on M with x E V. Then, T~(U) = T ~ ( M ) and, by the previous corollary, ~ , x : Tx(U) * T~(~)(~(U)) is an ~linear isomorphism. Since p(U) C_ R ~ is open, we know t h a t
T~(x)(~o(U) ) : ~n. [] The same kind of argument gives the following. C o r o l l a r y 3.1.27. I f f : M ~ N is a diffeomorphism, then d i m M = d i m N . Remark. We do not have a canonical basis for T ~ ( M ) since there is no preferred choice of local coordinates about x. Thus, we cannot write T x ( M ) = •n. The notion of infinitesimal curve (S)p makes sense in our context, as does the derivative D(s)p 6 T p ( M ) . Viewing tangent vectors as infinitesimal curves is preferable from an intuitive point of view, making the differential of a map "visible" and making the chain rule evident. If one is developing a theory of C k manifolds, defining tangent vectors to be infinitesimal curves rather than first order differential operators is essential [33]. The following is evident via local coordinates and the corresponding facts in ~n. L e m m a 3.1.28. The correspondence (S)p ~ DO) ~ between the set of infinitesimal curves at p C M f : M ~ N is a smooth mapping between manifolds, T / ( p ) ( N ) is given, in terms of infinitesimal curves,
is a onetoone correspondence and T p ( M ) . Furthermore, if the differential f,p : T p ( M ) by
f,p( (S}p) = ( f o s) / o ) .
3.2. DIFFEOMORPHIC
93
STRUCTURES
Finally, for s m o o t h m a p s f : M + N, we define the notions of regular point, critical point, critical value, and regular value exactly as before.
3.2. Diffeomorphie Structures T h i s section is really an extended remark on some very deep theorems, the point of which can now be easily appreciated. Let M be a differentiable manifold with s m o o t h structure .q = { ( u ~ , ~ ) } ~ .
Let 9 : M ~ M be any homeomorphism. Set
aL[~ = {((!D1 (Ucr), qPa o (]))}c~EN. P r o p o s i t i o n 3.2.1. The set Am is a Coo structure on M having the same structure cocycle as A. Proof. Indeed, g~
_ ~,~ o ~ F ~ _ ( ~
o ,~) o ( ~
o ,~)_ O. Define D : r ~) ~ IR by D[g]x = lira g(s(t))  g(x) t~O+
t
It is elementary that D 9 T~(IHI~) and that p*(D) = D n.
Theorem
In fact, the following much more general theorem, due to H. W h i t n e y [50], is known. We will prove it for c o m p a c t manifolds. For a proof of the general case, see [2, C h a p t e r 6]. 3 . 7 . 1 2 ( W h i t n e y imbedding theorem). If M is an arbitrary differentiable nmanifold, then there is a smooth, proper imbedding of M into H 2n+l.
Theorem
Proof for M compact and OM = 0. Since OM = ~, we imbed in IR2'~+1. By Theorem 3.7.11, we choose a s m o o t h imbedding M C R k for a suitably large value of k >_ 2 n + 1 . I f k = 2 n + l , we are done. We assume t h a t k > 2 n + l a n d s h o w t h a t M imbeds s m o o t h l y in IRk1. Finite repetition of this a r g u m e n t t h e n yields t h e t h e o r e m for the c o m p a c t case. View IRk1 C IRk as the subspace x k = 0 and let p : R k + ]R be projection onto t h e kth component. For each unit vector v E S k1 \ ]Rk  I , define pv : Rk +]Rt~1 by
v(w)~ p~(w) = w  ~ T h a t is, p~ is the linear projection of IRk onto 1Rk1 along v. T h e idea will be to choose v so t h a t p~]M actually imbeds M in IRk1. To begin with, we choose v so t h a t p~]M is injective. Consider the diagonal
A
=
{(x,x) tx e M}
C
M x M
and the m a p
f:MxM..,A~S
a1
defined by
xy f ( x , y )  iix  YlI' where Itwl[ denotes the usual Euclidean n o r m of w r IRk . Since k  1 > 2n, Sard's t h e o r e m guarantees t h a t f is not surjective (cf.Example 2.9.4), so we choose v C S k1 not in the image of f . T h a t is, v is not a scalar multiple of x  y, for any two distinct points x , y C M. Thus, pv(X  y) ~ O, whenever x and y are distinct points of M , proving t h a t pv[M is injective.
114
3. G L O B A L T H E O R Y
If we can prove t h a t Pvl M is an immersion, then compactness of M and injectivity of p r i M implies t h a t this m a p is an imbedding. Equivalently, we must find v as above such that, for every nonzero tangent vector w E T ( M ) , v r w/[Iwll. L e t
S ( M ) = {w E Z ( M ) l liwll = 1}, the socalled unit tangent bundle of M . It is convenient to view this as a subset of M x S k1 and, in Exercise 3.7.13, you will show t h a t it is a smooth, c o m p a c t submanifold of dimension 2n  1. T h e canonical projection of M x S k1 onto S k1 restricts to a s m o o t h m a p
g : S ( M ) * S k1 t h a t can be viewed as parallel translation in iRk x iRk of unit t a n g e n t vectors to M to vectors issuing from the origin. Again, by the dimension hypothesis and Sard's t h e o r e m , there is v E i m g U i m f . Since Pv is linear, pv. = Pv at each point of M , so p , is b o t h onetoone and an immersion. [] E x e r c i s e 3 . 7 . 1 3 . Prove t h a t S ( M ) is a s m o o t h submanifold of
T ( M ) C M x iRk. (Hint. F i n d a suitable m a p v : T ( M ) ~ iR having 1 as a regular value.) E x e r c i s e 3.7.14. If M is a compact nmanifold w i t h o u t boundary, show t h a t it a d m i t s a s m o o t h immersion into iR2n.
Remark. T h e existence of proper imbeddings M ~~ ]HI2n+1 of c o m p a c t nmanifolds w i t h b o u n d a r y is proven by modifying carefully the above p r o o f (cf. [16, Theorem 4.3 on page 31], where proper imbeddings are called "neat imbeddings"). T h e n o n c o m p a c t case is proven by suitable modifications of the t r e a t m e n t in [2, C h a p ter 6]. E x e r c i s e 3 . 7 . 1 5 . Let M C ]HIk be a properly imbedded nmanifold w i t h b o u n d a r y and prove t h a t there is a R i e m a n n i a n metric on iRk agreeing with the s t a n d a r d E u c l i d e a n metric outside of a neighborhood of ON k and such that, at each point x of OM, the orthogonal complement of T~(M) in T~(IR k) lies in T~(OEIk). We say t h a t , relative to this metric, M meets 0]HIk orthogonally along its boundary. E x a m p l e 3 . 7 . 1 6 . If M C ]HIn is the hemisphere S n1 N ]HIn, t h e n M is properly i m b e d d e d in this halfspace and meets OIEn orthogonally along its boundary. Here it is not necessary to change the s t a n d a r d Euclidean metric on R ~ near 0IN~. If i : M ~ ]HIk is a smooth, proper imbedding of an nmanifold, we routinely identify M w i t h i(M), as above, realizing T ( M ) c M x R k as a s u b b u n d l e in the usual way. Via the inclusion II{k C irk, we view M C R k wherever convenient. M o d i f y i n g t h e Euclidean metric in N k as in Exercise 3.7.15, we define ~ ( M ) = {(x,v) E M x iRk I v • T~(M)}. If 0 M = ~, we view M C iRk and define v ( M ) via the s t a n d a r d Euclidean metric. E x e r c i s e 3 . 7 . 1 7 . Prove t h a t the p r o d u c t projection M x iRk ~ M restricts to a m a p 7c : ~ ( M ) , M t h a t is the projection m a p of a vector bundle of fiber dimension k  n. T h i s is called the normal bundle of M in ]HIk.
3.7. S M O O T H S U B M A N I F O L D S
115
Remark that the normal bundle u(M) is a manifold of dimension k, generally with boundary u(M)IOM, and that this boundary is exactly the normal bundle of OM in 0/IIIk = IRk1. Define a smooth map : v ( M )  , H k, ~ ( x , v) = z + v,
using the additive structure of IRk. Note that we can identify the (image of) the zero section {(x,0)lx E M} C u(M) with M and that, under this identification, v I M = idM. P r o p o s i t i o n 3.7.18. If M C N k is a smooth, properly imbedded submanifold of
dimension n, there is an open neighborhood U of M in ~(M) that is carried diffeomorphically by ~ onto an open neighborhood ~(U) of M in N k. If M is compact, this neighborhood cart be taken to be of the form U(e) = {(x,v) c ~(M) I Ilvll < e}, for suitably small e > O. Pro@ We give the proof for the case that M is compact, leaving the general case as Exercise 3.7.19. Let (x, 0) E M C u(M) and remark that there is a natural identification T(x,O)(~'(M)) = Tx(M)  ~,x(M). Relative to this identification, we can write ~.(~,0) = idT~(M)  id,x (M), so the inverse function theorem (if x C OM, use Exercise 3.7.6) guarantees that there is a neighborhood U~ of (x, 0) in ~(M) carried diffeomorphically by qz onto a neighborhood of x in H k. This neighborhood can be taken to be of the form
u~(~4 = {(y,v) e . ( M ) l y e W~, Ilvll < ~x}, where Wx is an open neighborhood of x in M and e~ > 0 is small enough. Cover M with finitely many sets of the form U~(e~), let e be the smallest e, that occurs, and consider the union of the corresponding neighborhoods U~(e). This is an open neighborhood of M in u(M) of the form
g ( 6 = {(x,v) ~ ~(M) I IIvll < d . Although ~ is locally a diffeon~orphism on U(e), it might fail to be globally onetoone. We claim that, by choosing e > 0 smaller, if necessary, we can make sure that ~ is onetoone on U(e). If not, we could choose sequences (x~, v~) r (Yn, wn) in U(e) such that
~(xn, ~n) = ~(y~, ~ ) while IIv~H < 1In and IIw~ll 0 and 5 > 0. Use this to prove t h a t homotopy (respectively, isotopy) is an equivalence relation on C~176 N) (respectively, on Diff(M)). The equivalence classes for these relations will be called, respectively, homotopy classes and isotopy classes. D e f i n i t i o n 3.8.4. A diffeomorphism f E Diff(M) is compactly supported if there is a compact subset K C_ M such that f l ( M \ K ) = idM.K. The set of all compactly supported diffeomorphisms is denoted Diff,(M). A compactly supported isotopy
3.8. H O M O T O P Y
117"
between fo, fl E Diffc(M) is an isotopy such that there is a compact subset C C_ M with f t i ( M \ C) = i d M \ c , 0 < t < 1. E x e r c i s e 3.8.5. Prove that the set Diffc(M) is a group under composition. E x e r c i s e 3.8.6. Prove that compactly supported isotopy is an equivalence relation on the group Diffc(M).
Remark. For a compactly supported isotopy, ft belongs to Diffc(/1,J), 0 < t < 1. T h e o r e m 3.8.7 (Homogeneity lemma). If N is connected, boundaryless, and x, y E N , then there is f E Diffr and a compactly supported isotopy ft such that f ( x ) = y and f0 = idN, fl = f. The proof of this theorem uses flows and will be deferred until the next chapter. C o r o l l a r y 3.8.8. If g E C ~ ( M , N), y E N, and N is connected and boundaryless,
then g ~ ~ such that y is a regular value of'g. Proof. By Lemma 3.7.5, we choose a regular value x E N of g. Let f and H be as in Theorem 3.8.7. Since f ( x ) = y and f is a diffeomorphism, it follows that y is a regular value of ~ = f o g. But fi o g is a homotopy of g = f0 o g with ~ = fl o g. [] The definition of smooth homotopy and isotopy that we have given does not adapt nicely to manifolds with boundary. The problem is that [0, 1] is itself a manifold with boundary, hence M x [0, 1] will be a manifold with corners when OM ~ ~. This minor difficulty can be overcome by slightly modifying our definitions. D e f i n i t i o n 3.8.9. If f0, fl E Coo(M, N), these maps are (smoothly) homotopic if there is a smooth map H : M x R * N such that (1) H(x,O) = fo(x), g x E M; (2) H(x, 1) = f l ( x ) , V x E M. As usual, we set
f~(x) = H ( x , t ) and say that f0, fl E Diff(M) are isotopic if there is a homotopy between them such that ft E Diff(M), Vt E ]R. E x e r c i s e 3.8.10. Prove that, under the second definition of homotopy and isotopy, these continue to be equivalence relations. If 0 M = 0, prove that the second definition of homotopy and isotopy is equivalent to the first. D e f i n i t i o n 3.8.11. A map f c C~176 N) is a (smooth) homotopy equivalence if there is g E C ~ 1 7 6 such that f o g ~ idN and g o f ,.~ idM. In this case, we say that M and N are homotopy equivalent manifolds and that f and g are homotopy inverses of one another. E x a m p l e 3.8.12. Let M c H ~ be a compact, proper submanifold with normal bundle rr : ~(M) ~ M, and let U(e) be the open neighborhood of the zero section M C ~(M) as in Proposition 3.7.18. Then ~lU(e) e C ~ ( U ( e ) , M) is a homotopy equivalence with the inclusion i : M ~ U(e) as a homotopy inverse. Indeed, 7rlU(e ) o i = idM, so we investigate i o 7rlU(e ) : U(e) + U(e). But g :
u ( 6 x i  . u ( 6 , defined by
H((z,~),t)
=
(~,t~)
=
~(~,~),
118
3. G L O B A L T H E O R Y
is clearly a homotopy ioTrlU(e ) = 7r0 ~ ~rl = idu(6. Notice that each stage 7rt of this homotopy fixes M pointwise. In this situation, M is called a deformation retract of U(e). This is a very special type of homotopy equivalence. Since the normal neighborhood U of M in IHIk is just the image ~(U(e)) under the diffeomorphism of Proposition 3.7.18, we have shown that M is a deformation retract of its normal neighborhood. Intuitively, we can "shrink" the normal neighborhood U down to M while fixing IV/itself pointwise.
3.8.B. Smooth approximations*. Topologists usually formulate homotopy theory purely in the topological category. Differential topologists, on the other hand, like to take advantage of the differentiable structure of manifolds in using homotopy theory. The fact that smooth homotopy theory is equivalent (on manifolds) to the purely topological version is due to the approximation theory t h a t we now develop. The key to this is the following classical result. 3.8.13 (StoneWeierstrass Theorem). Let X be a locally compact topological space, C ( X ) the algebra of realvalued, continuous functions on X . Let A C_ C ( X ) be a subalgebra containing the constant functions and separating points. That is, for arbitrary x , y E X such that x ~ y, there is f E A such that f ( x ) 7~ f ( y ) . Then, for each f E C ( X ) , each compact subset K C_ X , and each e > O, there exists g E A such that If(x)  g(x)] < e, for all x E K . Theorem
For a proof, see [8]. C o r o l l a r y 3.8.14. If W and V are open, relatively compact subsets of a manifold N such t h a t W C V, i r e > 0 and i f f E C~ Hk), there is "fC C~ Hk),
uniformly eclose to f , smooth on W and equal to f on the complement of V. Proof. Consider first the case k = 1. By the StoneWeierstrass theorem, we can find f ' E C ~ ( M ) t h a t is eclose to f on V. Let O = N \ W and let {Av, Ao} be a smooth partition of unity subordinate to the open cover {V, O} of N. Then ? =
has the desired properties. For the general case, apply this argument to each of the coordinate functions of f = ( f l , . . . , fk), replacing e by e/x/~. [] We consider maps f E C~ M), appeal to Theorem 3.7.12, we imbed this does not use the full force of the enough dimension k will do. Define a Euclidean metric of INk. That is,
where N and M are both manifolds. By an M as a proper submanifold of INk. In fact, Whitney imbedding theorem since any large topological metric p on M by restricting the
p(x, y) = IIx  yll. By an esmall perturbation of f , we will mean a C O homotopy ft such t h a t f0 = f and p ( f ( x ) , ft(x)) < e, uniformly for x E N and 0 < t < 1. When we say "there is an arbitrarily small perturbation such that ...", this should be read: "for each e > 0, there is an esmall perturbation such t h a t ...". We also fix a choice of normal neighborhood 7r : U ~ M for the imbedded submanifold M. 3.8.15. If W and V are open, relatively compact subsets of N such that W C V and if f E C ~ there is an arbitrarily small perturbation ft
Proposition
3.9. D E G R E E T H E O R Y M O D U L O 2*
119
of f such that fl i8 smooth on W and ft agrees with f on the complement of V, 0 O. We claim that, for r sufficiently large, ft has no zeros on OW~, 0 < t < 1. Indeed,
ft(Z)z "~  l + t
(~
a2 +7+...+77
am)
,
and the term in the parentheses converges to 0 as z + oo. Thus, for r > 0 sufficiently large, we define
G : OW~ x [0, 1] + S ~ by
H(z,t)
a(z,t) qH(z,t)l This is a homotopy between
f(z)
Cl(z)
If(z)l
and
Co(re ~~ = e~'~o, so degz(G0 ) = deg2(G1 ). But G o l ( y ) contains exactly m points, Vy E S 1, so deg2(G1 ) = 1 since m is odd. It follows, by Example 3.9.13, that f has a zero in Wr. [] There is an integer valued degree for maps f C C~ N ) when M and N are both orientable. This can be used to give a proof of the full fundamental theorem of algebra. We will take this up in Chapter 8 when we study differential forms and de Rham cohomology. D e f i n i t i o n 3.9.15. Let W be a compact manifold, possibly with boundary, and let X c_ W. A retraction of W to X is a s m o o t h map f : W   , X such that f] x = idx. T h e o r e m 3.9.16. I f W is a compact manifold with O W # 0 connected, then there is no retraction f : W * OW.
Proof. Indeed, d e g 2 ( f l O W ) = 1, since f l O W = idow, and this contradicts the existence of the extension f : W + O W by Theorem 3.9.12. [] C o r o l l a r y 3.9.17 (Brouwer fixed point theorem). If f : D~ ~ D~
is smooth, there is a point x G D n such that f ( x ) = x. Proof. Suppose f has no fixed point. Define g : D n + OD n = S n1 as follows. For each x E D ~, construct the ray Rx starting at f ( x ) and passing through x ~ f ( x ) . Let g(x) be the unique point Rx N S ~1. If x C O D n, it is clear that g(x) = x, so if g is smooth, we have contradicted Theorem 3.9.16. The smoothness of g is left for the reader to check. []
3.9. DEGREE
THEORY
MODULO
2*
123
E x e r c i s e 3.9.18. Prove that Corollary 3.9.17 holds when f is assumed only to be continuous. Another famous theorem that can be proven using mod 2 degree is the J o r d a n Brouwer separation theorem (smooth version). We introduce the key idea, that of "winding number", and then, in a series of exercises, lead you through a proof of the separation theorem in the plane (the smooth version of the Jordan curve theorem). D e f i n i t i o n 3.9.19. Let f : S 1 ~ IR2 be a smooth map and let p C IR2 \ f(S1). Define
fp : S 1 + S ~ by the formula
fp(Z) 
f(z)
i]f(z)
p
P[t'
where [l" I[ denotes the usual Euclidean norm. Then the (mod 2) winding number of the closed curve f around p is
w2(f,p) = deg2(L). Remark that the winding number is defined for an arbitrary smooth closed curve f. It is not required that f be an imbedding or even an immersion. In the case that f is a diffeomorphic imbedding (a smooth Jordan curve), you will show in the exercises that the open set ]Re \ f ( S 1) has exactly two components, distinguished from one another by the fact that wz(f,p) = O, for every point p in one component, and w2(f,p) = 1, for every point p in the other. The component in which w2(f,p) = 0 is unbounded (and called the "outside" of f(S1)), while the other component is bounded (the "inside"). E x e r c i s e 3.9.20. Let f be a smooth Jordan curve, let U be a connected component of JR2 \ f(S~), and let p,q 9 U. Prove that w2(f,p) = w2(f,q). (Hint: The mod 2 degree is a homotopy invariant.) D e f i n i t i o n 3.9.21. If p 9 IR2, the ray in ]R2 out of p and having direction given by the unit vector v 9 S 1 will be denoted by Rp(V). E x e r c i s e 3 . 9 . 2 2 . If f : S 1 + ]R2 is a smooth Jordan curve and p 9 ]R2 \ f(S1), prove that v 9 S 1 is a critical value of fp : S 1 +S 1 if and only if the ray Rp(v) is somewhere tangent to the Jordan curve f.
E x e r c i s e 3.9.23. Prove the smooth Jordan curve theorem: I f f is a smooth Jordan
curve, then ]R2 ".. f ( S 1) has exactly two components, one of which (called the inside of f) is bounded (i.e., has compact closure) and the other of which (the outside of f) is unbounded. For every point p in the outside o f f , w2(f,p) = O, and for every p on the inside, w2(f,p) = 1. Finally, f ( S ~) is the settheoretic boundary of each of these components. Proceed as follows. (1) If p 9 IR2 \ f(S1), prove that w2(f,p) is the number of points rood 2 in Rp(V)Af(S1), for v 9 S a any regular value of fp. (Hint: Use Exercise 3.9.22.) (2) Use (1) to prove that there are points p, q 9 IR2 ",, f ( S 1) such that w2(f,p) w2(f,q). By Exercise 3.9.20, conclude that IR2 \ f ( S 1) has at least two components. Also remark that the winding number is 0 about points in at least one of these components and is 1 about points in at least one of the other components.
124
3. G L O B A L T H E O R Y
(3) Using the fact that f is a smooth imbedding, choose a coordinate chart (U, u, w) about any point f(z) in which g = ((~,w)l
2 < ~ < 2,2
< w < 2}
and f ( S 1) A U is just a horizontal line segment w = 0. Show t h a t every point of ~2 \ f(S1) can be connected by a continuous path in R 2 \ f ( S 1) either to the point (0, 1) C U or (0,  1 ) E U. This proves that R 2 \ f ( S 1) has at most two connected components. (4) Prove t h a t one of these components is bounded, that the other is not, and t h a t the winding number of f is 0 about the points in the unbounded component. (5) Show that each point of f ( S 1) lies on an arbitrarily short arc that meets both components. Conclude that f ( S 1) is the common settheoretic boundary of these components. 3.10. M o r s e F u n c t i o n s * In Subsection 2.9.B, we defined the notion of a nondegenerate critical point of a function f c C ~ ( U ) , where U is an open subset of Euclidean space. Via local coordinates, this notion carries over to functions f E C ~ ( M ) on a r b i t r a r y manifolds. Since the definition of the Hessian (Definition 2.9.14) is coordinatefree, the actual choice of coordinates about the critical point is immaterial. D e f i n i t i o n 3.10.1. If f E C ~ ( M ) and a E N,
M~ = {, e M f f ( x ) 0. Prove t h a t there is a n u m b e r co > 0 and a s m o o t h i m b e d d i n g such t h a t In this case, the flow line Rx is the imbedded circle ~ ( S 1) and we say t h a t x is a periodic point of the flow of period c0. Not every vector field is complete. For example, Exercise 2.8.13 showed t h a t
e t ~ C ~ ( N ) is not complete. Lemma
4.1.10.
If the maximal local flow of X E ~ ( M ) contains an element of
the form 4~ : (  e , e ) x M  + M
with e > O, then X is complete. Proof. Let t 9 ~. T h e n one can find k 9 Z and r 9 (  e / 2 , e/2) such t h a t t = r + k . e/2. Given x 9 M , define
[~r(x),
k = 0.
If 02t(x) is well defined by this formula,  o c < t < oc, t h e n it will be an integral curve to X . To see this, remark t h a t ~T(x),  ~ < r < ( / 2 , is integral to X and use t h e fact t h a t (~• = X (Exercise 2.8.14). We show t h a t ~ t ( x ) is well defined. For simplicity, let t > 0. Obvious modifications of the a r g u m e n t give the general case. Suppose t h a t r + k . e/2 = t = .s + q . ~/2, where s, r 9 e  e / 2 , e/2) and k, q 9 Z. It follows t h a t r  s 9 (  ~ , e), hence t h a t q  k = 0, 1, or  1. If q  k = 0, then r  s = 0 and we are done. Assume, therefore, that q  k = • W i t h o u t loss of generality, take q  k = 1. Then, r  s = e/2, so
,~/~
o...o
~/,~ o~,.(~) = ,~/~
o...o
,~/~o,~+~/~(~)
= ,'F~/2 o ' " o 'F~/~o~(x).
k•
134
4. F L O W S A N D F O L I A T I O N S
Thus, (Pt(x) is well defined,  o o < t < oo, for each x E M , and is an integral curve to X . We must show t h a t d) : R x M ~ M is smooth. Let (to, x0) E ]R x M . For small enough z / > 0, we fix k0 E Z such t h a t t = r + k0 9 c/2, for each t E (to  r/, to + 7) and suitable r E (  c / 2 , c/2). Then, (to  r], to + r]) x M is an open n e i g h b o r h o o d of (to, x0) in IR x M on which
ko
=
o .2.0
k0 < 0,
I ~0
[~r(X),
k0 = 0.
This is a s m o o t h function of ( r , x ) = (t  k0e/2, x), hence a s m o o t h function of [] Theorem
4.1.11. If X E :E(M) has compact support, then X is complete.
Pro@ Since s u p p ( X ) is compact, cover it with finitely many open subsets U 1 , . . . , U~ of M such t h a t the local flow of X contains elements o~ : (  c i , ~i) x Ui ~ M, 1 < i < r. Let U0 = M \ s u p p ( X ) , an open set with XIUo ==O. Define ~5~ : II~ x U0 + M r by qst~ )  x, Vx E Uo, Vt E R. Since { U~i}i=o covers M and the 4)i agree on overlaps, we have a local flow on M generated by X. Let e = minl 2q it vanishes outside the ellipsoidal ball E C B defined by x + 2y < 2e and the local definitions o f / 7 fit together smoothly. The solid E is indicated in Figure 4.2.2 by a dashed ellipse, in which the handle is shown to be inscribed. C l a i m 1. M} = M~.
4.2. T H E G R A D I E N T F L O W A N D M O R S E F U N C T I O N S *
141
Pro@ Everywhere, F _< f , and so M ] C_ M g, for every value of a. Since F and f are equal outside the ellipsoidal ball E, we have (,)
M~ \ E = M~, ../~.
Inside E, F < f _< (x + 2y)/2 _< ~, and so E g M~ n M~. Thus,
M r ( ~ .. E) u E_c (Mr . E)uMr _c ML Mb c_ (Mb . E) u E c (Mb \ E) u M~ c M~. These inclusions must all be equalities and, by (*), M} = M b.
[]
C l a i m 2. The function F has exactly the same critical points as f.
Pro@ Since F and f coincide outside of U, it is enough to show that FlU has p as its sole critical point. Write F =  z + y  #(x + 2y) in U and compute v r = ~ 9F v x' + N OFv~y' this being obvious since, in U, the formula for V F is the classical one. Purtherinore, the partials
OF Oz OF

0y
are never 0, so origin p.
1#'(z+2y)
'(z+2y)_>
1+0=

1,
Vf[U vanishes only where both V z and Vy vanish, namely, at the []
One notes that F ( 0 ) =  p ( 0 ) <  e , hence that there are no critical points of F in Fl[e,e]. By Theorem 4.2.3, applied to the function F , we obtain C l a i m 3. M } ~ M ~ ~. Evidently, we can write
MF ~ = MT~ u H, where H is a compact set inscribed in E as in Figure 4.2.2. Thus, the final step in the demonstration of Theorem 4.2.6 would be to prove the following intuitively plausible fact (cf. [38]). We omit this step.
C l a i m 4. The space H is a Ahandle of dimension n.
Remark. For compact manifolds, Morse theory gives a description of M as a space obtained by gluing together finitely many handles of dimension n. Each handle is attached only along tile part (ON~) x B ~~ of its boundary, hence it is possible to "flatten" it appropriately, proving that M} has the homotopy type of the space M~~UwB"x, obtained by attaching the Acell B "x by an imbedding ~ : OB~ ~ OMj7~. This leads to a description of M, up to homotopy, as a finite "cell complex". For many applications of algebraic topology, this is more useful than the precise decomposition of M into handles. For more details, see [28].
142
4. FLOWS AND FOLIATIONS
4.3. T h e Lie B r a c k e t
As we have already remarked, 2r is a Lie algebra over IR under the Lie bracket. Vector fields D, when viewed as derivations of the function algebra C~176 localize to open subsets U C M. This localization is equivalent to the restriction DIU of D as a smooth section of the tangent bundle, so it is elementary to check the following. L e m m a 4.3.1. ff X, Y E X(M) and ifU C_M is open, then
[xlu,
YlU] = [x, Y]lU.
In particular, properties of the bracket that were proven with coordinates can be extended to global properties on M. We apply this remark to Lie derivatives. Since every vector field X E ~ ( M ) generates a local flow q) on M, the definition of the Lie derivative
L x ( Y ) = lira ePt*(Y)  Y t+O
t
makes sense pointwise on M and defines a new field L x ( Y ) E :~(M). The proof that L x ( Y ) = IX, Y] (Theorem 2.8.16) that was given in R n can be carried out in local coordinate charts hence, by Lemma 4.3.1, globalizes. T h e o r e m 4.3.2. If X , Y G X(M), the Lie derivative of Y by X is defined and
smooth throughout M and L x (Y) = [X, r ] . Similarly, Theorem 2.8.20 globalizes. Here, commutativity of local flows r = {~a}~e~ and g~ = {~e}eem means that q*t5 o r = ~5~ o q~t~ wherever both sides are defined. Commutativity of vector fields X, Y E X(M) means that [X, Y]  0 on M. T h e o r e m 4.3.3. Vector fields X, Y on M commute if and only if the local flows
they generate on M commute. C o r o l l a r y 4.3.4. Complete vector fields X, Y e 2E(M) commute if and only if the flows that they generate commute. We are going to be interested in Lie subalgebras of X(M). Of course, ;~ C_ X(M) is a Lie subalgebra if it is closed under the vector space operations and the bracket. If F C_ T ( M ) is a kplane subbundle (which we also refer to as a kplane distribution on M), P(V) c ~(M) is a Coo(M)submodule and a real vector subspace. It is not generally a Lie subalgebra but, when it is, there are important geometric consequences.
D e f i n i t i o n 4.3.5. The kplane distribution F C_T(M) is a Frobenius distribution (or an involutive distribution) if P(F) is a Lie subalgebra of X(M).
Remark. If f,g r Coo(M) and X, Y E 2~(M), it is easy to verify the identity [fX, gY] = f X ( g ) Y  g Y ( f ) X + fg[X, Y]. Consequently, if F C T(M) is a kplane distribution and if the fields X1, X 2 , . . . , X,. e p(F) span r ( f ) over Coo(M), then F will be a Frobenius distribution if and only if [Xi, Xj] C F(F), 1 _< i,j < r.
4.3. LIE BRACKET
143
E x a m p l e 4.3.6. On the group manifold Gl(n), we will define a couple of interesting Frobenius distributions. First recall (Example 2.7.18) that the space of leftinvariant vector fields g[(n) C 3r is a finite dimensional Lie subalgebra, canonically identified with tile Lie algebra 9Jr(n) of n x n real matrices under tile commutator bracket. Let sl(n) C g[(n) be tile set of matrices of trace 0 and let o(n) C g[(n) be the subset of skew symmetric matrices. These are clearly vector subspaces and the reader should have little diMcnlty in computing dims[(n) = n 2  1, ~ ( n  1) dim o(n) 
2
Since t r ( A B ) = t r ( B A ) , we see that tr[A, B] = 0, hence si(n) is a Lie subalgebra of l~[(n). If A, t3 c o(n), then [A,B] T = B T A T   A T B
T = [B,A] =  [ A , B ] ,
so o(n) C l~[(n) is also a Lie subalgebra. If { S ~ , . . . , S ~ _ ~ } is a basis of s[(n), extend it to a basis of g[(n) by adjoining a n2 suitable vector Sn2 and view {S~}~=~ as a set of leftinvariant vector fields on Gl(n). It is clear that, at each point P E Gl(n), these fields give a basis { P S I , . . . , PSn2} of TF(GI(n)). Let Sp C Tp(Gl(n)) be the (n ')  1)dimensional subspace spanned by ;l p .~qJ .i =pl ~  I and remark that this subspace does not depend on the choice of basis of s[(n). Let
S=
[_J
Sp.
PEGI(n) n2
This is an (n ~  1)plane distribution on GI(n). Indeed, the vector fields {S~}~=1 define an explicit triviMization of T(GI(n)) ~ Gl(n) x g[(n) relative to which S becomes the trivial subbundle Gl(n) x s[(n). (Warning: We are not using the standard trivialization of T(GI(n)) = Gl(n) x ffJt(n). This would give an imbedding GI(n) x M(n) C T(GI(n)) (tiff)rent from the one we have defined and not very interesting.) Similarly, extending a basis of 0(n) to one of g[(n) and viewing these as leftinvariant vector fields on Gl(n), we obtain a distribution O C T(GI(n)) of fiber dimension n ( n  1)/2 and independent of the choices. By tile remark preceding this example, the fact thai; s[(n) and 0(n) are closed under the bracket implies that S and O are Frobenius distributions on Gl(n). Also, by our construction, if P C Gl(n), then L p , (Sl) = Sp,
Lp,(Oi) = Op. Recall t h a t the special linear group is the subgroup Sl(n) C Gl(n) consisting of the matrices of determinant 1 (Example 2.5.5). In Example 2.5.8, we showed that Ti(Sl(n)) is the subspace of TI(GI(n)) = gJl(n) consisting of the matrices of trace 0. For each Q E Sl(n), LQ carries Sl(n) onto itself', so it follows from our construction that TQ(SI(n)) = S 0. Also, for arbitrary P C Gl(n),
TpQ( P . Sl(n)) = Lp,:ib(Sl(n) ) = Lp,( SQ) = Spq. We say t h a t tile left cosets P . Sl(n) are integral submanifoIds to tile distribution S. In exactly the same way, using Exercise 2.5.9, we see that the left cosets P . O(n) of tile orthogonal group are integral submanifolds to tile distribution O C T(GI(n)).
144
4. F L O W S A N D F O L I A T I O N S
These are the first examples of foliations in this book. The connected components of the left cosets of these groups are the leaves of the foliation integral to the respective distributions. A distribution F on M with (onetoone immersed) integral submanifolds through each point of M is said to be integrable. We will see that the Frobenius property is precisely the integrability condition (ef. Example 3.4.20). This is the theorem of Clebsch, Deahna, and Frobenius, commonly called the Frobenius theorem. Generally speaking, a smooth map f : M ~ N does not push a vector field X C X(M) forward to a vector field f . ( X ) E X ( N ) . There are two problems. I f f is not surjective, there would be points of N where f . (X) would not even be defined. If f is not injective, there could be points of N where f . ( X ) would be multiply defined. Nevertheless, there are situations in which f fails to be bijective, b u t the following concept makes sense. D e f i n i t i o n 4 . 3 . 7 . If f : M ~ N is a smooth map between manifolds (possibly with boundary), vector fields X E ~ ( M ) and Y E ~ ( N ) are said to be frelated if, for each q C M , f . q ( X q ) = Yf(q)"
E x a m p l e 4.3.8. It is possible that X E 3~(M) is not frelated to any Y E 3r For example, let f : IR ~ S 1 be the map f ( t ) = e 27tit. Then
f,t
t~
= tf, t
= 27rite 2~it e T / ( t ) ( S 1) C Ty(t)(C ) = C.
There is clearly no vector field on S 1 satisfying this. E x a m p l e 4.3.9. It is possible that X E X(M) may be frelated to many vector fields in ~ ( N ) . For example, let f : S 1 ~ C be the inclusion map. Let U C C be an open neighborhood of S 1 such that U ~ C, Let ~ : C * [0, 1] be smooth such that supp(~) C U and ~IS 1 = 1. Define the vector field Y~ = iz on C. Since z Z iz, we obtain X E X(S 1) by setting Xz = iz, V z C S 1. Clearly, X is frelated to both Y and ,kY and these are distinct fields on C. P r o p o s i t i o n 4.3.10. Let f : M ~ N be smooth and let X , Y C ~ ( M ) be frelated to X , Y C 3~(N), respectively. Then [X,Y] is frelated to [ X , Y ] . Equivalently, L x ( Y ) is frelated to L ~ ( Y ) .
P r o @ For each h E C ~ (N) and for each q C M, Y ( h o f ) ( q ) = Yq(h o f ) = f,q(Yq)(h) = Y/(q)(h) = ( Y ( h ) o f ) ( q ) . T h a t is,
Y(hof)=Y(h)of,
VhEC~176
There is a similar relation between X and X. Thus,
[2, Yls(
)(h) =

= f , q ( X q ) ( Y ( h ) )  f,q(Yq)(_X(h)) = X q ( Y ( h ) o f )  Yq(X,(h) o f ) = Xq(Y(h o f))  Yq(X(h o f)) = [X,Y]q(ho f) = f.q([X,Y]q)(h),
4.4. COMMUTING FLOWS Vh E C~176
145
T h a t is, [X, Y]/(q) = f.q([X, Y]q),
Vq 9 M.
This is the assertion of the proposition.
[]
E x e r c i s e 4 . 3 . 1 1 . Let f : M , N be a onetoone immersion and let Y 9 X(N). Prove t h a t there exists a field X 9 ~ ( M ) t h a t is frelated to Y if a n d only if Yf(q) 9 f.q(Tq(M)), Vq 9 M. In this case, prove t h a t X is mfique (we call X the restriction of Y to the immersed submanifold). E x e r c i s e 4 . 3 . 1 2 . Let f : S 2~1 ~~ R 2n be the inclusion (an imbedding, hence a onetoone immersion). F i n d Y 9 fl~(IR2~) t h a t restricts, as above, to a nowhere zero vector field X 9 X(S2n1). (By Theorem 1.2.16, this is false for the inclusion f : S 2~ ~~/R2~+1. This fact is more elementary t h a n Theorem 1.2.16, however, a n d will be proven later (Theorem 8.7.5).)
4.4. C o m m u t i n g Flows A n IRachart (U, x i, ... , x n ) about q C M determines n c o m m u t i n g vector fields
(0
Oxl,...
0}
,Ox n
c X.(U).
The corresponding local flows about q are of the form
(~(xl,...
,xi,...
,X n) =
(xl,...
, X i ~ t , . . . , x n ) .
Conversely, the following implies t h a t commuting, linearly i n d e p e n d e n t vector fields correspond to a coordinate chart. 4.4.1. Let M be an nmanifold without boundary, let q E M , let U be an open neighborhood of q, and let X 1 , . . . , X k E :E(U). If these vector fields commute and if { Xlq,... , Xkq } is linearly independent, then there is a local coordinate chart (W, ~) about q such that
Theorem
0 qo,(Xi]W) = 0x7, 1 < i < k. Proof. Making U smMler, if necessary, we assume t h a t it is a coordinate chart, hence view it as an open subset of [R~. We can do this so t h a t q becomes the origin 0 a n d so t h a t the vectors
xl'
.
.
' x~. .Ox~k+l 0' . .
0
~ Oxn
0
form a basis of To(U). Let
qsi
: (  e , e) x W * U
be a local flow a b o u t q = 0 generated by X i, 1 < i < k. We can choose W to be of the form (  e , e) n with e > 0 so small that the formula o(xl,...,
x n) =
~
~
(o,...,
o, x
+z , . . .
is defined, V ( x l , . . . , x ~) C W. This defines a smooth m a p
O:W~U.
,
146
4. FLOWS AND FOLIATIONS
Since [Xi, X j]  0 on U, 1 r, h~ and hn+lN lie in a common plaque P of W. Similarly, h~+l and h~+2 lie in a common plaque Pq Since hn+l E P N P ~, it follows that P = P ' . Proceeding in this way, we see t h a t / 5 contains hm, V m >_ r. [] This proposition has a surprisingly strong converse. T h e o r e m 5.3.2 (Closed subgroup theorem). If G is a Lie group and H C G is a
closed subset that is also an abstract subgroup, then H is a properly imbedded Lie subgroup. E x e r c i s e 5.3.3. Let G be a topological group whose underlying space is a topological manifold. Use Theorem 5.3.2 to prove that there is at most one differentiable structure on the topological manifold G making G into a Lie group. (The positive solution to Hilbert's fifth problem guarantees that a topological groupmanifold does have a smooth (in fact, analytic) structure making it into a Lie group.)
174
5. L I E G R O U P S
Our proof will be modeled, in certain important ways, on the proof given in [14, pp. 105106]. The problem with that proof is that it is based on a lemma [14, Lemma 1.8, p. 96] that assumes that Lie groups are real analytic groups. The fact t h a t C ~~ groups are, in fact, real analytic, will not be used in our proof. For a somewhat different presentation, also carried out in the smooth category, the reader can consult [49, pp. 110 112]. F i x the assumptions in Theorem 5.3.2. Define
b = { X e L(G) l e x p ( t X ) 9 H ,  o o < t < oo}. This contains 0 9 L(G) and is closed under scalar multiplication. It is not evident t h a t I? is a vector subspace, let alone a Lie subalgebra. It is also unclear t h a t l) ~ 0 if H is not discrete. In fact, [1 will turn out to be the Lie algebra of an open Lie subgroup of H. Let V C L(G) be the vector space spanned by O. In the following proof, it will be convenient to use the notation a X = L a . ( X ) and X a = R a . ( X ) (where Ra denotes right translation by a), a 9 G, X 9 2E(G). Thinking of vectors as infinitesimal curves makes this notation particularly natural. L e m m a 5.3.4. The vector space V is a Lie subalgebra of L(G).
Proof. Since the Lie bracket is bilinear and V is spanned by I1, it will be enough to prove t h a t IX, Y] 9 V, VX, Y 9 I1. By Theorem 4.3.2 and Proposition 5.1.23, [X, r ] = lim Y e x p (  t X )  r t~0
t
Since Y is a leftinvariant field, Y e x p (  t X ) = exp(tX) Y e x p (  t X ) , and, for a fixed t, this is a leftinvariant field whose corresponding 1parameter group is c~(r) = exp(tX) e x p ( r Y ) e x p (  t X ) . Since X, Y 9 ~, a(r) is a product of elements of the subgroup H, hence a ( r ) 9 H,  o o < r < co. It follows that, for each value of t, Y e x p (  t X ) 9 0 _c V. Thus, IX, Y] 9 V, as desired. [] Let H0 C G be the connected Lie subgroup with L(Ho) = V. L e m m a 5.3.5. There is an open neighborhood U ore in Ho (in the manifold topol
ogy of rio) that is contained in H. Proof. Let {Y1,... ,Yq} C I) be a basis of V. The map qo : V + H0, defined by qa
tiYi
= exp(tlY1) exp(t2Y2)...exp(tqYq),
\i=1
satisfies ~o,0(Yi) = Yi, 1 < i < q, so the inverse function theorem implies t h a t ~o carries some neighborhood U0 g V of 0 diffeomorphically onto a neighborhood U c_ H0 of e. But exp(tiY/) 9 H, 1 _< i < q, and H is a subgroup, so U C_ H. [] C o r o l l a r y 5.3.6. The Lie group Ho is a subgroup of H.
Proof. By Proposition 5.1.13, H0 is generated by U C H, and so H0 C H.
[]
5.3. C L O S E D S U B G R O U P S *
175
Remark that, at this point, we know that I) = V, hence I? is the Lie algebra of Ho.
L e m m a 5.3.7. The subgroup Ho C_ H, with its manifold topology, is open in the
relative topology of H. Proof. (Compare [14, p. 106].) It will be enough to prove that some open neighborhood U of e, in the manifold topology of H0, is a neighborhood of e in the relative topology of H. Tile problem is that, for each such U, there might be a sequence o~ {X k}k=l C H \ U such that xk ~ e in the topology of G. Assuming that this is so, we deduce a contradiction. Find a direct sum decomposition L(G) = b O W and remark that, by the inverse function theorem, the map p : 0  W + G, defined by ~(v, w) = exp(v) exp(w), carries some neighborhood N of 0 in L(G) diffeomorphically onto a neighborhood of e in G. Choose U = exp(% N N). Thus, for k sumciently large, we can write xk = exp(vk)exp(wk) e ~o(N), where vk C 0/q N and wk E I/V N N. Since xk ~ U, it is clear that wk ~ 0, for all large values of k. Select a bounded neighborhood TWoC W of 0 and positive integers nk such that, for k sufficiently large, n,:wk E Wo, but (nk + 1)wk g W0. Since W0 is bounded, we can assume that nkwk +w C W. Since wk + 0 and (nk + 1)wk ~ W0, we must have w ~ 0. For arbitrary t C IR, we will show that exp(tw) C H. That is, w G 0, hence 0 5r w C b ~ W, the desired contradiction. Write t n k = sk + tk, where sk E Z and Itkl < 1. Thus, tkwk ~ 0 and exp(tw)
=
lim exp(tnkwk) /c~oo
:
lim exp(skwk) exp(tkw/~) k~oo
= =
lira exp(skwk)
k~oo
lira exp(wk) ~k k~oo
=
lira ( e x p (  v ~ ) . ~ ) s~ /c~oo
Since H is closed in G, it follows that exp(tw) E H.
[]
Proof of theorem 5.3.2. Let i : Ho ~+ H be the inclusion map. We have proven that i carries Ho, with its manifold topology, homeomorphically onto an open subset of H in the relative topology. In particular, the manifold topology of H0 coincides with its relative topology, so H0 is an imbedded Lie subgroup of G. By Proposition 5.3.1, H0 is closed in G, so H0 = i(Ho) is a connected, openclosed subset of H. Thus, H0 coincides with the component of the identity in H. The other components La(Ho), a E H, of H are also properly imbedded submanifolds of G, so H is a Lie subgroup. Since H has the relative topology, each of its components is relatively open in H, so H is a properly imbedded Lie subgroup. [] C o r o l l a r y 5.3.8. Let F C G be an abstract subgroup Then the closure F in G is
a properly imbedded, Lie subgroup of G.
176
5. LIE GROUPS
E x a m p l e 5.3.9. Let v = ( a l , . . . , a n) C ]Rn be a point such that, when N is viewed as a vector space over the rational number field Q, the subset { a l , . . . ,a n} c ]R is linearly independent. Let p : 1Rn ~ T n be the standard projection and let g C ]Rn be the line through v and 0. A classical theorem of Kronecker asserts that this line projects onetoone to a 1parameter subgroup p(g) C T n that is everywhere dense in T n (for the case n = 2, cf. Example 4.1.8 and Exercise 4.4.12). It is now fairly easy to prove Kronecker's theorem. Indeed, one proves (Exercise 5.3.10) that, if v l , . . . ,vk E Z n and k V = ~2
xivi
i
i=1
for suitable coefficients x i E N, then k = n. By Corollary 5.3.8, the closure ~ C T n is a compact, connected, abelian Lie subgroup, hence a toroidal subgroup of dimension r < n (Exercise 5.1.30). It follows that v E g C V where V C N n is a subspace and = p ( V ) . In particular, V is spanned by V (3 Z n, so Exercise 5.3.10 implies that d i m V = n and ~ = T n. E x e r c i s e 5.3.10. Prove the assertion in Example 5.3.9 that the vector v E ]Rn, with rationally independent coefficients, cannot be expressed as a real linear combination of fewer than n elements of the integer lattice 7/.n. E x e r c i s e 5.3.11. Let v = ( a l , . . . , a n) E R n be a point such that the set of coefficients {1, a l , . . . , a n} is linearly independent over Q. Prove that the subgroup A C T n, generated by a = p ( v ) , is everywhere dense. (Hint: Every point in the coset qv + Z n has rationally independent coefficients, Vq C Z. Prove this and use it to show that every 1parameter subgroup of T n meeting a nontrivial element of A is dense in Tn.) Following Helgason [14, pp. 107108], we deduce the following classical result as another corollary of Theorem 5.3.2. For a proof that does not depend on that theorem, see [49, p. 109]. T h e o r e m 5.3.12. I f ~ : G ~ H is a c o n t i n u o u s group h o m o m o r p h i s m
between
L i e groups, t h e n qp is s m o o t h . Proof. The product G x H is a Lie group and the projections 7ra
GxH
~
G,
~rH
G x H
*
H
are smooth group homomorphisms. Let F C G x H be the graph of ~. T h a t is, P = {(x, ~(x)) ] x C G}, clearly a closed subgroup of G x H. Thus, F is a properly imbedded Lie subgroup. Also, 7ral F = ~ : F * G is a smooth group homomorphism and is bijective. If it can be shown that r : G * F is smooth, then ~ = ~rHor 1 will be smooth and the assertion will be proven. By the inverse function theorem, it will be enough to show that r is bijective, Vy C F. Since P is a Lie group and ~ is a smooth homomorphism, it is enough to prove this at y = (e, e). Remark that the exponential maps for the groups G x H, G, and H are related
by eXpaxH = expa x exPH.
5.3. C L O S E D S U B G R O U P S *
177
This, together with the proof of Theorem 5.3.2, implies that L(F) = { ( X , Y ) r L(G) x L(H) [ (expGtX, e x P H t Y ) 9 F, Vt 9 ~}. Since r162
Y) = X, we must show that, for each X 9 L(G), there is a unique
Y 9 L ( H ) such that (X, Y) 9 L(F). We first show uniqueness of Y. If (X, Y) 9 L(F) and (X, Z) 9 L ( r ) , then the difference is (0, Y  Z) 9 L (F), implying that (e, eXpg t ( Y  Z)) 9 F, V t 9 N. Thus, e x P H t ( Y   Z) = ~(e) = e, Vt 9 R, and Y  Z = 0. Choose open neighborhoods Uo C_ L(G) and Vo C L(H) of the origin and Ue C G and Ve G H of the identity such that 1. exp C : U0 + U~ is a diffeomorphism onto; 2. expH : Vo ~ Ve is a diffeornorphism onto;
3. ~(u~) c v~; 4. e x p a •
carries (U0 x V0)n
L(r)
diffeomorphically onto (U~ • V~)N F.
Let X
9 L(G) and choose an integer r > 0 such that ( 1 / r ) X 9 Uo. Thus, ~ ( e x p a ( 1 / r ) X ) 9 Ve and there is a unique Yr 9 V0 such that e x p g Yr = ~ ( e x p a ( l / r ) X ). There is also a unique Z,. 9 (Go x Vo) N L(F) such t h a t exPaxH Z~ = ( e x p a ( 1 / r ) X , exPH Y~). Since e x p c •
is onetoone on Uo x Vo, this implies that
((1/r)X, Y~) = Z,. 9 L(P). Take Y = rY.~, obtaining (X, Y) = rZ,. r L(F).
[]
Corollary 5.3.13.
L e t ~ : C, + H be a continuous homomorphism of Lie groups and let K = ker(~). Then K is a properly imbedded, normal Lie subgroup of G, G / K is canonically a Lie group, and the induced map ~ : G / I ( +H is a onetoone immersion of this Lie group as a Lie subgroup of H.
Proof. Indeed, K is a normal subgroup by standard group theory and K = ~  1 ( c ) is a closed subset of G, so Theorem 5.3.2 guarantees that K is a properly imbedded Lie subgroup of G. By Theorem 5.3.12, ~ is a smooth homomorphism, so Exercise 5.2.14 guarantees that ~(G) is a Lie subgroup of H. Obviously, ~ is an isomorphism of the group G / K onto ~(G), so ~ (:an be used to transfer the Lie structure of qo(G) back to G / K . [] E x e r c i s e 5.3.14. Maximal abelian subalgebras of Lie algebras play an important role in Lie theory, as do the maximal abelian subgroups of Lie groups. Prove the following. (1) Show t h a t every finite dimensional Lie algebra contains a nontrivial abelian subalgebra that is not itself contained properly in another such subalgebra. Similarly, show that every compact Lie group G contains a maximal subgroup that is Lie isomorphic to T k, some k >_ 1. This is called a maximal torus of G. (2) W h e n G is compact, prove that the correspondence between Lie subalgebras and connected Lie subgroups sets up a onetoone correspondence between the maximal abelian subalgebras of L(G) and the maximal tori in G. (3) Let G be compact and connected, T C_ G a maximal torus. Prove that T is a maximal abelian subgroup. (4) There are maximal abelian subgroups of a connected Lie group G t h a t are not maximal tori. Find a finite subgroup of SO(3) that is maximal abelian.
178
5. LIE G R O U P S
E x e r c i s e 5.3.15. Let G be an ndimensional Lie group and g = L(G) its Lie algebra. Let Gl(g) denote the group of nonsingular linear transformations of the vector space g and let Ant(g) C Gl(g) be the subgroup of Lie algebra automorphisms of g. Prove the following. (1) Gl(g) has a canonical Lie group structure under which it is (noncanonically) isomorphic to Gl(n). Also, for use in Exercise 5.3.16, show that L(GI(g)) is canonically the space End(g) of linear endomorphisms of the vector space g, the bracket in End(g) being the commutator product of endomorphisms. (2) Ant(g) is a closed subgroup of Gl(g), hence a properly imbedded Lie subgroup. (3) Assume that G is connected and let C C_ G denote the center of G, clearly a closed subgroup. Each element a 9 G determines an inner automorphism of G, denoted by Ad(a) and defined by Ad(a)(g) = aga 1,
Vg 9 G.
Prove t h a t {Ad(a)}aec is canonically a Lie subgroup of Ant(g), isomorphic as a group to G/C. This subgroup is denoted by Ad(G) and called the adjoint group of G. E x e r c i s e 5.3.16. Let G be a Lie group and again denote its Lie algebra by g. A derivation D : g + g is a linear transformation such that
D[X, Y] = [DX, Y] + [X, DY], VX, Y 9 g. Let 9
be the space of derivations of g.
(1) Prove that, under the commutator product, 9 is naturally identified as a Lie subalgebra of L(Gl(g)) (cf. Exercise 5.3.15, part (a)). (2) For each X 9 g, define a d ( X ) : g + g by ad(X)Y=[X,Y],
VY 9
and prove that ad(X) 9 9 (3) Prove t h a t ad : g + L(GI(g)) is a homomorphism of Lie algebras. Thus, ad(g) C_ L(GI(9)) is a Lie subalgebra. (4) Assume that G is connected and prove t h a t the connected Lie subgroup of Gl(g) corresponding to the Lie subalgebra ad(g) is exactly the adjoint group Ad(G). (Hint: Prove that a d ( e x p ( t X ) ) = exp(t ad(X)), V X 9 g.)
5.4. Homogeneous Spaces* Lie groups arise in many natural ways as transformation groups of differentiable manifolds. When the group action is transitive, the manifold is called a homogeneous space and one has considerable control over its structure. D e f i n i t i o n 5.4.1. Let M be a smooth manifold and G a Lie group. A smooth map
#:G•
M,
written p(g, x) = gx, is said to be an action of G (from the left) on M, and G is called a Lie transformation group on M, if (1) gl(g2x) = (9192)x, Vgl,g2 E G and Vx E M; (2) e x = x , V x 9
5.4. HOMOGENEOUS SPACES*
179
Remark. One can also define a right action #:MxG~M by making the obvious changes in the above definition. D e f i n i t i o n 5.4.2. An orbit of the action
GxM~M is a set of points of the form {gxo I g C G}, where x0 E M. The action is transitive if M itself is an orbit, in which case M is said to be a homogeneous space of G.
Remark. It is elementary that the orbits of a group action are equivalence classes, two points x, y E M being equivalent under the action if 3 g C G such that g x = y. E x a m p l e 5.4.3. The orthogonat group O(n) acts on R n in the usual way, leaving invariant the unit sphere S n  : . Note that, if el C S n is the column vector with first entry 1 and remaining entries 0, then Ael is the first column of A C O(n). Every unit vector appears as the first column of suitable orthogonal matrices, so the action O ( n ) X S n  1 > S n  1
is transitive and S ~1 is a homogeneous space of O(n). In a completely similar way, there is a transitive action U ( n ) x S 2 n  1 + S 2 n  l ,
where S 2n: C C n is the unit sphere in the standard Hermitian metric. D e f i n i t i o n 5.4.4. Let M be a homogeneous space of G and let x0 C M. isotropy group of z0 is the set Gxo = {g C G I gxo = z0}.
The
L e m m a 5.4.5. The isotropy 9roup Gzo as above is a properly imbedded Lie sub9roup of G.
Pro@ It is obvious that Gxo is an abstract subgroup of G. If {gn}n~=: is a sequence in Gxo converging to g C G, then, by the continuity of the group action, gxo=
lim g ~ x o =
n~oo
lim x 0 = x 0 .
n~oo
Thus, G~o is a closed subset of G. By Theorem 5.3.2, Gxo is a properly imbedded Lie subgroup. [] E x a m p l e 5.4.6. If el G S ~  : is as in Example 5.4.3, the isotropy group O(n)~: is the set of matrices
where A r O(n  1). Similarly, for e: E S 2n1, U(n)~: is the set
where A C U(n  1). We are going to show how to put a smooth structure on the quotient space G/G~o and prove that this manifold is diffeomorphic to the homogeneous space M. Under the identification M = G/Gxo, the Gaction on M becomes the action
G x G/G~o ~ G/G~o , g(ha~o ) = (gh)G~o.
180
5. L I E G R O U P S
In what follows, we consider an arbitrary properly imbedded Lie subgroup H C_ G, put the quotient topology on G / H , and construct a natural smooth structure on this space. Throughout this discussion, we set t} = L ( H ) . Decompose L(G) = moil, where m is any fixed choice of complementary subspace. Let
r :m247 be the map r
B) = exp(A) exp(B)
and choose a neighborhood V of 0 in il and a neighborhood W of 0 in m such that r sends W x V diffeomorphically onto a neighborhood U of e in G. Choose a compact neighborhood C C W of 0 with the property that  C = C and exp(C) exp(C) C U. We can assmne that coordinates x l , . . . ,x k in il define V by the inequalities  1 < x i < 1, 1 < i < k. Similarly, coordinates y l , . . . ,yq for m define C by  1 0. Let s : [e,e] * M be smooth such t h a t s(0) = z a n d ~(0) = v. Choosing e > 0 smaller, if necessary, we can assume t h a t ~ s ( , ) ( ~ ( t ) ) > 0,
~ < t
0,
contradicting the hypothesis.
[]
D e f i n i t i o n 6.3.8. We say t h a t w E A I ( M ) has p a t h  i n d e p e n d e n t line integrals if, for every piecewise smooth path s : [a, b] ~ M , L ~ depends only on s(a) a n d s(b) a n d n o t otherwise on s. D e f i n i t i o n 6.3.9. A piecewise s m o o t h p a t h s : [a, b] * M is a loop if s(a) = s(b). Theorem
6 . 3 . 1 0 . For aJ E A I ( M ) ,
the following are equivalent.
(1) aJ is an exact f o r m . (2) fs w = 0, f o r all piecewise smooth loops s. (3) w has pathindependent line integrals. Proof. We prove t h a t (1) ~ (2). If a = df is exact and s : [a, b] * M is a piecewise s m o o t h loop, s(a) = q = s(b), then Corollary 6.3.6 implies t h a t
where q denotes the constant path q(t) = q, a < t < b We prove t h a t (2) =~ (3). Let sl a n d s2 be piecewise smooth curves s t a r t i n g at the same point x and ending at the same point y. W i t h o u t loss of generality, assume t h a t Sl is parametrized on [1, 0] and s2 on [0, 1]. Let u : [0, 11 ~ [0, 1] be defined by u(t) = 1  t. T h e n s2 o u starts at y a n d ends at x and Sl + s 2 o u
= s : [1,1]~ M
is a piecewise smooth loop. By our assumption,
1
where
we have used part (2) of Lemma
2
6.3.2 to write
2 ou
2
We prove t h a t (3) =~ (1) by using (3) to construct f C C ~ ( M ) such t h a t = dr. W i t h o u t loss of generality, we assume t h a t M is connected (otherwise, carry out the construction of f on each c o m p o n e n t individually). Fix a basepoint x0 c M. Given any point x C M , use connectivity to find a piecewise s m o o t h p a t h s : [a, b] ~ M such t h a t s(a) = xo and s(b) = x. (In fact, the homogeneity lemma, T h e o r e m 3.8.7, implies the existence of a s m o o t h path, b u t the present claim is more elementary and is left to the reader.) Set
f(~) = f ~ .
6.3. L I N E I N T E G R A L S
193
By the a s s u m p t i o n of pathindependence, this is independent of the choice of piecewise s m o o t h p a t h s from x0 to x. R e m a r k t h a t f(xo) = O. We prove first t h a t f : M + IR is smooth. Let q C M be arbitrary and choose a c o o r d i n a t e chart (U, x l , . . . , x n) a b o u t q in which q is the origin and U = i n t D n, where D n is the unit ball in R ~. In these coordinates, we can write
colU = ~
gi dxi.
i=l
For each x C U, let Sx : [0, 1] + U be defined by sx (t) = tx, 0 < t < 1, and express flU by the formula
f(x) = f(q) + /
CO.
Js x
Thus, on U,
I ( x 1, .
,x. '~) .
f(O) . .+
.
. gi(tx 1,
z=l
,tx'~)~d (txi)d t
0
~ i f01 9i(txl,...
= f(O)+Ex
,tx~)dt.
i=1
T h i s is clearly smooth. Since q E M is arbitrary, f E C ~ ( M ) . Next, we prove t h a t co = df. Let s : [a, b] ~ M be an arbitrary piecewise s m o o t h path. Let c < a and let so : [e,a] + M be piecewise s m o o t h such t h a t so(c) = xo and so(a) = s(a). T h e n
O+S
o
T h a t is
It follows t h a t tile form c~ = w  df satisfies
f
c5 = 0,
for all piecewise s m o o t h paths s. By L e m m a 6.3.7, c~ = 0, so co = df.
[]
D e f i n i t i o n 6 . 3 . 1 1 . A 1form co E A I ( M ) is locally exact if, for each x C M , there is an open n e i g h b o r h o o d U of x such t h a t colU E AI(U) is exact. Example
6 . 3 . 1 2 . On the manifold M = IR2 \ {(0, 0)}, define the 1form
 Y dx + x  x 2 + y2 ~
dy.
We claim t h a t ~ is locally exact. Indeed, if q C M is not on the yaxis, a branch of 0 = arctan(y/x) is defined and s m o o t h on a neighborhood of q. A direct comp u t a t i o n gives dO = ~]. Similarly, if q C M is not on the xaxis, select a branch of 0 =  arctan(x/y) and check t h a t dO = r/. Since no point of M is on b o t h axes, this proves t h a t r/ is iocally exact. We claim, however, t h a t r] is not exact. Indeed, consider the s m o o t h loop s : [ 0 , 1] + M defined by s(t) = (cos2r4, sin 2rrt). Clearly,
rl4t) =  sin 2rrt dx4t ) + cos 2rrt dys(t),
194
6. C O V E C T O R S
AND
1FORMS
and
s* (dx) = 27r sin 2~rt dr, s* (dy) = 27r cos 27rt dt, SO
s* (rj) = 27r(sin 2 2~rt + cos 2 2~t) dt = 27r dt. Thus
= 27r
/0
dt = 27r r O.
R e m a r k t h a t the form r/ in the above example cannot be extended to a 1form on N 2. We are going to see shortly (Corollary 6.3.15) that, on N 2, every locally exact 1form is, in fact, exact. The above example reflects a topological feature of R 2 , {(0, 0)}, the missing point, t h a t distinguishes t h a t space from R 2. T h e n o t i o n of smooth homotopy extends to a notion of piecewise smooth hom o t o p y in a fairly obvious way. Here is the formal definition. D e f i n i t i o n 6.3.13. Let s0, sl : [a,b] ~ M be piecewise smooth loops. We say t h a t so is (piecewise smoothly) homotopic to sl, and write so ~ sl, if there is a continuous m a p H : [a, b] x [0, 1] ~ M a n d a p a r t i t i o n a = to < tl < . . . < tr = b such t h a t (1) Hl([ti_l,t~ ] x [0, 1]) is smooth, 1 < i < r; (2) H ( t , O ) = so(t) a n d H(t, 1) = sl(t), a < t < b; (3) H ( a , T ) = H(b, 7), 0 < T < 1. As usual, (piecewise smooth) homotopy is an equivalence relation. E x e r c i s e 6 . 3 . 1 4 . If w c= A I ( M ) is locally exact and if the loops Sl a n d s2 are piecewise s m o o t h and homotopic, show t h a t
1
2
Proceed as follows. (1) Let U C R 2 be open, let R = [a,b] x [c,d] C U, a n d let w E A I ( U ) be locally exact. For e > 0, let R~ = (a  e, b + c) x (c  e, d + c). Show t h a t there is e > 0 such t h a t R~ C_ U and co[R~ is exact. (2) Let ~ : M * N be a smooth m a p between manifolds a n d let cv E A 1 (N) be locally exact. Prove t h a t ~* (w) E A 1 (M) is locally exact. (3) Use these two results to prove the proposition. C o r o l l a r y 6 . 3 . 1 5 . Every locally exact 1form on R n is exact.
P r o @ Let s : [a, b] ~ R n be a piecewise smooth loop and define H : [a, b] • [0, 11 ~ ]~n by H ( t , T) = Ts(t). T h e n H is a homotopy of the constant loop 0 to the loop s. If cv is locally exact, Exercise 6.3.14 implies t h a t
Since the loop s is arbitrary, Theorem 6.3.10 implies t h a t w is a n exact form.
[]
6.4. FIRST COHOMOLOGY
195
Corollary 6.3.15 is a special case of one version of the Poincar~ Lemma, to be treated later.
6.4. The First Cohomology Space In Example 6.3.12, we saw that a locally exact form on a manifold can fail to be exact and that this seems to be related to the topology of the manifold. This insight is formalized and exploited by the de Rham cohomology H 1(M), a vector space associated to the manifold M which measures, in some sense, how much the notions of "locally exact" and "exact" differ on M. D e f i n i t i o n 6.4.1. The space of (de Rham) lcocycles on M is
Z 1(M) = {co E A 1(M) lco is locally exact}. The space of (de Rham) 1coboundaries is B I ( M ) = {co E AI(M) lw is exact}.
Remark that, if we regard A 1(M) as a vector space over JR, then Z I(M) and It is also clear that B I(M) C_ Z I(M).
B I ( M ) are vector subspaces. They are not C~176
D e f i n i t i o n 6.4.2. The vector space
HI(M) = ZI(M)/BI(M) is called the first (de Rham) cohomology space of the manifold M. If co is a 1eoeycle, its cohomology class is [co] = co + B 1(M) C H I(M). Although, whenever d i m M > 0, the vector spaces Z I ( M ) and B I ( M ) are infinite dimensional, it frequently happens that H I ( M ) is finite dimensional. We will see, for instance, that this is the case whenever M is compact. Cohomology is a contravariant functor from the category of differentiable manifolds (smooth maps are the morphisms) to the category of real vector spaces (and linear maps). Indeed, by Exercise 6.2.16 an arbitrary smooth map ~ : M + N induces a linear map g)* : Z I ( N ) ~ Z I ( M ) and g)*(BI(N)) C B I ( M ) , so g)* passes to a welldefined linear map (of the same name) ~* : H 1(N) + H 1(M). It is trivial to check that (~0 o r
= ~b* o ~* and (idM)* = idH*(M).
Proposition
6 . 4 . 3 . Let co, ga C 2 l ( a ) . Then [co] z [~] C H I ( a ) /f and only if f~ co = f~ ~ as s varies over all pieeewise smooth loops in M. These numbers are called the periods of co and of the cohomology class [co].
Proof. The locally exact forms co, ~ E Z 1(M) have the same periods f~ co = fs ~, for every piecewise smooth loop s, if and only if fs (co  ~) = 0 for all such loops. By Theorem 6.3.10, this holds precisely when co  ~ belongs to B I(M). Equivalently, [co] = p]. [] E x a m p l e 6.4.4. Consider the sphere S n, n >_ 2. By stereographic projection, we know that the complement of a point in S n is diffeomorphic to IRn, so every piecewise smooth loop in S n that misses a point is homotopic to a constant loop. By Sard's theorem, no piecewise smooth curve in S ~ can be spacefilling if n > 2 (see
196
6. COVECTORS AND 1FORMS
E x a m p l e 2.9.4), so all piecewise smooth loops a on this sphere are homotopically trivial. By Exercise 6.3.14, jfa aJ = 0 , for all locally exact 1forms ~o. By Proposition 6.4.3, we conclude t h a t
H I ( S n) = 0, Proposition
whenever n _> 2.
6.4.5. If fo, f l : M ~ N are smooth and homotopic, then
f~ = f~ : H I ( N ) + H I ( M ) . Proof. Let [aJ] 9 H I ( N ) . If s : [a,b] > M is a piecewise s m o o t h loop, t h e n si = f i o s : [a,b] ~ N is also a piecewise s m o o t h loop, i = 0, 1. Let H : M x N + M be a h o m o t o p y of f0 to fa. Then, the composition
[a, b] x [0, 1]
sxid> M x If{ ~
N
is a h o m o t o p y of so to sl, so
~ f~(o:)= f bs*f3(o~) b f
(fo o S)*(w)
'1W (Exercise 6.3.14)
Since s is an arbitrary piecewise smooth loop, Proposition 6.4.3 implies t h a t
f;[w] = [f;(w)] = [f~(w)] = f~[w]. Since [co] C H 1 (M) is arbitrary, f~ = f~ at the cohomology level, as desired.
[]
D e f i n i t i o n 6.4.6. A smooth m a p f : M + N is a homotopy equivalence if there exists a s m o o t h m a p g : N + M such t h a t f o g ~ i d g and g o f ~ idM. C o r o l l a r y 6.4.7. A homotopy equivalence f : M ~ N induces a linear isomorphism f * : g 1 (N) + H 1 (M).
Proof. Since f o g ~ i d g , it follows by the (contravariant) functoriality of cohomology a n d Proposition 6.4.5 t h a t g* o f* = ( f o g ) * = id~v = i d H l ( y ) . Similarly, f* o g* = idHl(M), so f* and g* are m u t u a l l y inverse isomorphisms on cohomology. [] E x a m p l e 6 . 4 . 8 . Let f : {0} ~ D ~ be the inclusion. Let g : D ~ + {0} be the only map. These maps are smooth a n d g o f = id{0}. Consider the m a p f o g : D n ~ D n having image {0}. We claim t h a t this is homotopic to idD,,.
6.4. FIRST
197
COHOMOLOGY
Indeed, let ~) : IR + [0,1] be s m o o t h such t h a t ~(0) = 0 and qo(1) = 1. Define H : D ~ x IR + D ~ by
H(x, t) = ~(t)x. Then
H(x,O)=O=f(g(x)),
gxeD
~,
H(x, 1 ) = x = i d D , ~ ( x ) ,
V x E D n.
and T h i s establishes the desired h o m o t o p y and completes the proof t h a t f is a h o m o t o p y equivalence, so f* : H I ( D n) + H i ( { 0 } ) = 0 is an isomorphism. T h a t is, H 1(D '~) = 0 or, equivalently, every locally exact 1form on D n is exact. A similar proof shows t h a t IRn is homotopically equivalent to a point, and we recover Corollary 6.3.15. Example
6.4.9. Let i : S n  1 c__+ R n \
{0}
be t h e inclusion. Let g :Rn
\
{ 0 } + S n  1
be the m a p defined by ?2
These m a p s are smooth, and g o i = idsn1 . We claim t h a t i o g ~ id~.,..{o}, hence t h a t i is a h o m o t o p y equivalence. Indeed, define H : (R n \ {0}) • [0, 1] ~ ~ n \ {0) by the formula
H(v, t) 
V
t + (1  t ) l l ~ l [
This is s m o o t h since II~ll > 0 implies t h a t t + (1  t)>ll > 0, 0 < t < 1. Then H ( ~ , 1) = v,
W e ~
\ {0},
and
It follows t h a t H I ( R ~ \ {0}) = Hl(Sr~l). In particular, t o g e t h e r With E x a m p l e 6.4.4, this proves t h a t H 1(N ~ \ {0}) is trivial, whenever n _> 3. Proposition
6.4.10.
There is a canonical isomorphism H I ( S 1)
11{.
198
6. COVECTORS AND 1FORMS
We will prove this via three lemmas. Recall the universal covering m a p
p : ]~ * S 1, p(t) = (cos 2~t, sin 27a). This is the m a p t h a t induces the s t a n d a r d diffeomorphism R / Z = S 1. Define O~
: Z l ( S 1 ) +]1~
by
a(~)
=
a linear map. Lemma
6.4.11.
So
p*(~),
The linear map a passes to a welldefined linear map a : H l ( S 1) + R.
Proof. Indeed, a = p[[O, 1] is a smooth loop and w C B I ( S 1) implies t h a t
a(~) = / ~ = 0, by T h e o r e m 6.3.10. Lemma
6.4.12.
[]
The linear map a : H I ( S 1) * 1~ is injective.
Proof. Let co E ZI(S 1) be such that a(w) = 0. We must prove t h a t co E BI(S1). For n E Z, let rn : IR ~ R be the t r a n s l a t i o n rn(t) = t + n . Then, porn = p, so r* op* = p * : AI(S 1) +AI(]R). This a n d the change of variable formula for the integral gives
f t + n p * ( ~ ) = fot ~ ; ( p * ( ~ ) ) = ~ot p*(~),
VncZ,
o_ 2, H 1 (S n) = 0 a n d H 1(T n) = ]R~, proving t h a t S ~ a n d T n are not homotopically equivalent. Of course, S 1 = T 1. E x e r c i s e 6 . 4 . 1 8 . We will say t h a t a locally exact form w E Z I ( M ) is integral if all of its periods are integers. For example, ~ / 2 ~ C Z I ( S 1) is integral. By P r o p o s i t i o n 6.4.3, ~ is integral if a n d only if every a / E [w] is integral, in which case we say t h a t [~] is an integral cohomology class. We denote by H i ( M ; Z) C H~(M) the subset of integral cohomology classes. (1) Prove t h a t H~(M; Z) is a subgroup of the additive group of the vector space H i ( M ) . We call H I ( M ; Z ) the integral cohomology of M.
6.4. F I R S T C O H O M O L O G Y
201
F i g u r e 6.4.1. The pair of pants P
(2) If f : M ~ N is smooth and w E Z I ( N ) is integral, prove that f*(w) C Z I ( M ) is also integral. Using this, show that integral cohomology is a contravariant functor from the category ~ of smooth manifolds and smooth maps to the category 9 of abelian groups and group homomorphisms. (3) Referring to Exercise 6.4.16, prove that H 1(Tn; E) is canonically the integer lattice ~ n C ]~n : H I ( T n ) .
(4) To each smooth map f : T ~ ~ T ~, show how to assign canonically an n • n matrix A / o f integers, depending only on the homotopy class of f , such that Afog = A g A / (matrix multiplication). If f is a diffeomorphism of T n onto itself, prove that A / is unimodular (i.e., has determinant • (5) Prove t h a t every n • n unimodular matrix of integers occurs as the matrix A f assigned to some diffeomorphism f : T n ~ T n. E x e r c i s e 6.4.19. If the vector space H i ( M ) has finite dimension k, prove that there is a set of piecewise smooth loops { a l , . . . , ak} on M such that the map
H i ( M ) ~ ll~k defined by
is an isomorphism of vector spaces. In light of Exercise 6.4.19, one might expect to generalize part (3) of Exercise 6.4.18 to all manifolds with finite dimensional first cohomology. T h a t is, one asks whether the loops in Exercise 6.4.19 can be chosen so as to carry H i ( M ; Z) isomorphically onto Z k. In fact, this can be done if M is compact, but we sketch an example t h a t shows what can go wrong in general.
202
6. C O V E C T O R S
AND
IFORMS
E x a m p l e 6.4.20. Let P denote the 2manifold with boundary obtained by removing two small, disjoint, open disks from the interior of D 2. The boundaries cl and c~ of these disks should be disjoint, each from the other and from co = OD2. The resulting surface, called by topologists a "pair of pants", is pictured in Figure 6.4.1, together with a dotted loop a that is homotopic to the outer boundary circle. In Figure 6.4.2, we cross this manifold with a closed interval and identify opposite ends with a twist through ~ radians. The result is a solid torus with a "wormhole" drilled out that winds around twice longitudinally. Denote this 3manifold by V0. Note that this manifold is a kind of bundle with fibers diffeomorphic to P. A meridian on the outer boundary corresponds to the boundary curve co of P in Figure 6.4.1. By Exercise 6.3.14, the integral around Co of any locally exact form w is equal to the integral of w around the loop a in Figure 6.4.1 and this, in turn, is equal to the sum of the integrals around cl and c~. In V0, the loops cl and c~ are homotopic along the boundary of the wormhole, so we obtain
o
1
We now glue another copy of V0 (longitudinally) into the wormhole, obtaining a manifold V1 containing a loop c2 such that ff~ c o = 2 ~ 0
co=4~ i
co. 2
Inductively, a manifold V~ is obtained by gluing a copy of V0 longitudinally into the wormhole of Vn1 and V~ contains a new loop cn such that /coco=2nfc
co"
Proceeding ad infinitum, we obtain a limit manifold Vcr This is the complement in the solid torus of a very complicated compact subspace E called the solenoid. If one first imbeds the solid torus in S 3 in the standard unknotted fashion and then removes the solenoid, the noncompact manifold M = S 3 \ E that results can be shown to have first de Rham cohomology H 1(M) = N and the set of loops chosen in Exercise 6.4.19 can be taken to be the singleton {co}. In fact, one can show that all periods of any locally exact form co are sums of periods of co corresponding to loops ci, i >_ 0. If co is a locally exact lform that is integral, we obtain integers
~ co=hi,
i>O,
i
and no = 2nl . . . . . 2ini . . . . . This can only happen if no = 0, in which case every ni = 0 and co has all periods 0. That is, the isomorphism in Exercise 6.4.19 identifies Hi(M; Z) = 0. 6.5. D e g r e e T h e o r y o n S 1. Recall from Example 1.7.33 that ~1(S 1, 1) = Z. This is a canonical isomorphism, produced by lifting a loop a based at 1 in S 1 to a path ~ in the universal cover ~ starting at 0. The endpoint of this path is the integer corresponding to [a]. By Exercise 5.1.37 and the fact that S 1 is a Lie group, the group 71[S1, S 1] is canonically isomorphic to 7rl (S 1, 1). This isomorphism, denoted by deg : 7r[S 1, S 1] + Z,
6.5. D E G R E E T H E O R Y ON S 1.
203
F i g u r e 6.4.2. Forming the manifold Vo
is called the degree map and deg([f]) is also called the degree of any f E [f] and denoted by deg(f). We are going to give two equivalent definitions of this degree, one using cohomology and one in terms of regular values. The first remark is that, since H 1(S 1) = R, f induces a linear map f* :IR ~ I R depending only on the homotopy class of f . Thus, f* is just multiplication by a certain constant a / E 1R and a / depends only on the homotopy class of f . By part (4) of Exercise 6.4.18, a I is an integer, being the sole entry in the 1 x 1 integer matrix A f . We are going to give another way to see that a / E Z. By an application of Theorem 1.7.39, if f : S 1 ~ S 1 is smooth, we can lift the m a p f o p : R + S 1 to a smooth map 37: N ~ N. T h a t is, the diagram
R
?,R
S1
, S1 I
is commutative. Also, since the group of covering transformations consists of translations by integers, f" : IR * 1R will be a lift of f o p if and only if ]" = y + k, for some integer k.
Proposition 6.5.1. If f : S 1 ~
S 1 is smooth, then
a / = 37(1)  37(0) = deg(f) e •, where 37 is any lift of f o p.
204
6. COVECTORSAND 1FORMS
Proof. R e m a r k t h a t p ( f ( 1 ) ) = f(p(1)) = f(p(O)) = p ( f ( 0 ) ) ,
SO ?(1)
 ?(0)
= m 9 Z
Thus, let ~ 9 Z I ( S 1) be as in the c o m m e n t a r y following the proof of L e m m a 6.4.12 and c o m p u t e 2~ra/= ai~[~
=
~1 ( f o p ) * ( ~
~01( p o f ) * ( ~
=
f0
1 f* (2~ dt)
: 2~(f(1)
fo ep * ( f * ( ~ )
= c~(a/[~) = (~(f*[~) =
 f(0))
27r
f01
:
2~m.
f* (dt)
=
~01 f * ( p * ( ~ )
27T
j l?(
t) dt
T h a t is, a I = m E Z. The fact t h a t this integer is d e g ( f ) as defined above is e l e m e n t a r y and left to the reader. []
Remark. In Section 3.9, we defined deg2(f) e Z2 for s m o o t h m a p s between manifolds (without b o u n d a r y ) of the same dimension. We will see t h a t , for s m o o t h m a p s of the circle to itself, deg2(f) is just the residue class modulo 2 of d e g ( f ) (Corollary 6.5.4). 6.5.2. If f : S 1 * S 1 is smooth, if f is a lift of f o p, and if t E ]R is arbitrary, then d e g ( f ) = f ( t + 1)  ]'(t).
Corollary
Proof. View p : ~ * S 1 C C as a group homomorphism. Then
p(f(t +
1)  f(t))

p(y(t
+ 1))
p(i(t))
f(p(t + 1)) f(p(t)) so f ( t + 1)  f ( t ) E Z, Vt E ~. This function of t, being continuous and integervalued, is constant on N, hence equal to f ( 1 )  f(0) = d e g ( f ) . [] We turn to the description of d e g ( f ) in terms of regular values. Let z0 E S 1 be a regular value of f : S 1 * S 1. Then f  l ( z o ) = { Z l , . . . ,Zr}. Here, if f  l ( z o ) = ~), we take r = 0. Recall t h a t deg2(f) = r (rood 2). Choose ~/ c N such t h a t P(ii) = zi, 1 < i < r. The s m o o t h m a p f : S 1 ~ S 1 preserves orientation at zi if f~(~i) > 0 and reverses orientation at zi if ~ ( ~ / ) < 0. Let ei = ? ( ~ ) / I ] ) ( ~ i ) l e {  1 , 1} and r e m a r k t h a t this depends only on f and zi. Proposition
6.5.3.
r
With the above conventions, d e g ( f ) = ~ = 1 ~i.
Proof. Choose a 9 R \ { p  l { z l , . . . ,z,.}}. T h e n pl(z~) C~(a,a + 1) is a singleton and we choose this point as our zi, 1 _< i _< r. We will also use the fact t h a t f ( a + 1) = f(a) + d e g ( f ) . Let pl(zo) = {b + k}kez and consider the graph of s • f ( t ) over the open interval (a, a + 1), together with the horizontal lines s = b + k, k 9 Z. Each time
6.5. DEGREE
Figure
THEORY
ON
S 1.
205
6 . 5 . 1 . G r a p h of f
t h e g r a p h crosses a line s = b + k, t h e p a r a m e t e r t is equal to one of t h e Ei a n d ei records w h e t h e r t h e g r a p h crosses this line while increasing (ei = 1) or d e c r e a s i n g (ei =  1 ) . F i g u r e 6.5.1 i l l u s t r a t e s a case in w h i c h r = 7, q = e2 = ea = e4 = e7 = 1, a n d e5 = e6 =  1 . T h e s u m of t h e Qs p e r t a i n i n g to a single line s = b + k is 1,  1 , or 0, t h e n e t n u m b e r of d i r e c t e d crossings. Clearly, t h e s u m of all t h e s e n e t n u m b e r s is r
E e i = "f(a + 1)
 f(a) =
deg(f).
i=1
(In F i g u r e 6.5.1, t h e degree is 3.) Corollary Example
6.5.4.
deg2(f)
[]
= d e g ( f ) ( m o d 2).
6 . 5 . 5 . For each n C Z, define f n : S 1 ~ S 1, A(z) = ~n
Here, of course, we view S 1 C C.
5(t)
W e c a n choose t h e liR f ~ : IR ~ R to b e
= he, so
deg(A) =/.(1)  into) : If z 9 S 1 is a regular value of fn, then
f~l(z)
= { P l , . . . ,PinS},
n.
206
6. C O V E C T O R S A N D 1  F O R M S
where P l , . . . ,Plnl are the distinct n t h roots of z. Of course, if n = 0, t h e n f0 is constant and f o l ( Z ) = 0. I f n > 0, all r = +1 and, i f n < 0, all r =  1 . Thus, M
n=E(i i=1
in all cases. T h e o r e m 6.5.6. A smooth map f : S 1 ~ S 1 extends to a smooth map F : D 2 + S 1 if and only if d e g ( f ) = 0. Proof. First suppose t h a t the s m o o t h extension F exists. T h a t is, f = F o i where i : S 1 ~ D 2 is the inclusion. T h e n f* = i* o F* and F * : H I ( s 1) + H I ( D 2) = 0, implying t h a t f* = 0. Therefore, d e g ( f ) = 0. For the converse, suppose t h a t d e g ( f ) = 0. Since the degree is a complete invariant for homotopy, it follows t h a t f ~ f0  1. By the C ~ U r y s o h n trick, choose t h e h o m o t o p y H:S 1 x[0,1]~S 1 so t h a t H(z, 1)= f(z), H ( z , t ) = 1,
V z C S 1, 0 0. But g,. extends smoothly to F~ : D 2 ~ S 1, a contradiction to Theorem 6.5.6. [] L e m m a 6.5.8. If f, g : S 1 ~ :{1 are smooth, then
deg(f
o
9) = d e g ( f ) deg(g ).
Indeed, functoriality of cohomology implies t h a t aio 9 = afag , so the lemma is immediate. Corollary
6.5.9. I f f , 9 : S 1 * S 1 are smooth, then f o 9 and g o f are homotopic.
By Exercise 5.1.37, the Lie group structure on S 1 makes ~r[M, S 1] into an abelian group and one obtains the following. Theorem
6.5.10.
The m a p x: ~[M,S
1] ~
Hi(M; Z),
defined by x[f] = f*[g/27r], is an i s o m o r p h i s m of groups.
Here, the integral cohomology H i ( M ; Z) is defined as in Exercise 6.4.18. The fact t h a t [~/27r] is an integral class implies t h a t f*[~/27r] is also integral by t h a t same exercise. E x e r c i s e 6 . 5 . 1 1 . Prove Theorem 6.5.10. Proceed as follows. (1) Show t h a t X is a group homomorphism. (2) Let aJ C Z I ( M ) be an integral form. You are going to define a s m o o t h m a p f~ : M * S 1 such t h a t f~(~/2~r) = aJ. For this, no generality will be lost in assuming t h a t M is connected (why?), so make t h a t assumption and fix a basepoint x0 C M . For each x E M , choose any piecewise s m o o t h p a t h s : [a, b] ~ M such t h a t s(a) = xo and s(b) = x, and show t h a t f~o(x)= ( / w
(modE))
elR/Z=S
1
208
6. C O V E C T O R S
AND 1FORMS
depends only on x (and xo), not on the choice of p a t h s. (3) Prove t h a t fw : M + S 1 is s m o o t h and t h a t its h o m o t o p y class is independent of the choice of basepoint x0. (4) Prove t h a t f~(~/2rr) = co. In particular, conclude t h a t ;g is surjective. (5) Let f : M + S 1 be such t h a t co = f* (~/2rr) is an exact form. You are to prove t h a t f ~ 1, so note that, again, no generality is lost in assuming t h a t M is connected. In this case, show t h a t fo0, as defined in step (b), is actually well defined as a m a p f~ : M + IR and t h a t there is a constant e such t h a t f = p o (f~ + c). Conclude t h a t f ~ 1, hence t h a t X is onetoone. E x e r c i s e 6.5.12. Use degree theory to show t h a t the group Diff(S 1) has exactly two isotopy classes. (Hint. An easy application of degree theory will show t h a t there are at least two isotopy classes. The hard step is to show t h a t , if f E Diff(S 1) and d e g ( f ) = 1, then f is isotopic to fl. It then follows fairly easily t h a t there are at m o s t two isotopy classes.)
CHAPTER 7
Multilinear Algebra and Tensors Smooth functions, vector fields and 1forms are tensors of fairly simple types. In order to handle higher order tensors, we will need some rather sophisticated multilinear algebra. The reader who is well grounded in the multilinear algebra of Rmodules can skip ahead to Section 7.4, referring to the first three sections only as needed. 7.1. T e n s o r A l g e b r a We will be working in the category ?V[(R) of Rmodules and Rlinear maps, where R is a fixed commutative ring with unity 1. In order to study Rmultilinear maps, we build a universal model of multilinear objects called the tensor algebra over R. In the typical applications in this book, R will be either the real field N or the ring C a (M). D e f i n i t i o n 7.1.1. An Rmodule V is free if there is a subset B C V such that every nonzero element v E V can be written mfiquely as a finite Rlinear combination of elements of B (terms with coefficient 0 being suppressed). The set B will be called a (free) basis of V. If R is a field, every R module is free. Another example is the integer lattice Z k, a free Zmodule. At the other extreme, the abelian group Z2, when viewed as a Zmodule, is not free. A basis would }lave to contain 1 C Z2, but 0 C Z2 would then have infinitely many representations a. 1, a E 2Z. The following example will be very important. E x a m p l e 7.1.2. Let 7r : E  M be an nplane bundle. Then F(E) is a free Coo(M)module on a basis of n elements if E is trivial. Indeed, if E ~ M • N n, let { e l , . . . , en} be the standard basis of ]Rn, and define si C F(E) by the formula si(x) = (x, ei), 1 < i < n. An arbitrary section s(x) = (x, f l ( x ) , . . . , f n ( x ) ) has the n unique expression s = ~ i = 1 fisi. E x e r c i s e 7.1.3. Suppose that rr : E + M is an nplane bundle and that F(E) is a free C ~ (M)module with basis B. One easily checks that B must contain at least n elements. Using local triviality and the Coo Urysohn lemma, show that B has exactly n elements 81,... , 8n and that 81(X),..., 8n(X) form a basis of E , , for each x E M. Thus, E must be trivial.
Remarks. There are strong but limited analogies between vector spaces over a field and free Rmodules. Here are some of the facts. (1) If V is free on the basis B, then Rlinear maps g) : V ~ W into arbitrary Rmodules W correspond onetoone to maps ~ : B + W of sets, the correspondence being ~ = ~IB.
210
7. M U L T I L I N E A R A L G E B R A
(2) If V is a free Rmodule, it can be shown that any two bases of V have the same cardinality, called direr V. For example, dimz Z k = k. (3) On the other hand, there are important dissimilarities. A submodule W C V of a free Rmodule can fail to be free and, even when the submodule W is free, it may have no basis that extends to a basis of V. We give two examples illustrating Remark (3). E x a m p l e 7.1.4. If M C IRn is a nonparallelizable submanifold, then we have the canonical inclusion T ( M ) ~~ M x ~ n of the tangent bundle as a subbundle of the trivial bundle. Thus, •(M) C F ( M x ~ n ) is a C ~ ( M )  s u b m o d u l e . By Exercise 7.1.3, X(M) is not free, but by Example 7.1.2, F ( M x ~ n ) is free. E x a m p l e 7.1.5. The submodule 2Z C Z is a free Zsubmodule of a free Zmodule. But there are two bases {2} and {  2 } of 2Z, neither of which extends to a basis of Z. Modules will not be assumed free unless that is explicitly stated. D e f i n i t i o n 7.1.6. If V1,1/2, 1/3 are objects in ?d(R), a map ~ : Vi x V2 ~ V3
is Rbilinear if
~(, v2): 1/1 ~ v3 ~(Vl, .): v2 ~ V3 are Rlinear, Yvi E V/, i = 1,2.
Remark. For fixed choices of V1,1/2, 1/3 E :SI(R), the set of Rbilinear maps q : 1/1 • 1/2 ~ 1/3 is itself an Rmodule under the pointwise operations. D e f i n i t i o n 7.1.7. If V1 and 1/2 are Rmodules, their tensor product is an R  m o d ule V1  V2, together with an Rbilinear map  : Vix
V2 .> V1 
2
with the following "universal property": given any Rmodule 1/3 and any Rbilinear map p : V1 x 1/2 * 1/3, there is a unique Rlinear map ~ such that the diagram  VIXV 2 , Yl 2
Va commutes. We write 
w) = v  w.
Thus, the Rmodule of Rbilinear maps V1 x V2 ~ Va is canonically isomorphic to the Rmodule HomR(V1  V2, 1/3) of Rlinear maps Vi  V2 ~ V3,
T h e o r e m 7.1.8. Given V1,V2 a$ above, a tensor product Yl  V2 exists and is
unique up to a unique isomorphism. That is, if 
iV1 x V2 .~ Y l 
6 : Vl x V2 ~ Vl 6 V2
7.1. T E N S O R A L G E B R A
211
are two such tensor products, there is a unique isomorphism O : Vl O 72 ~ 7 1 ~ 72 of Rmodules such that the diagram

VIXV2
,
71072
Ul g W2
commutes.
Pro@ First we prove uniqueness. If 0 and O are two tensor products, the universal property gives unique Rlinear maps 01 and 02 making the following diagrams commute:  71x72 v10v2 ,
VI @V2 71x 7~
, 71~ 72
710V2
Then the diagram 0
Vl x g2
. Vl O V2
~
[02o01 V1072
also commutes, as does  V~ x 72
tid
. 71o72
71072 By the universal property, we conclude that 02 o01 = id and, similarly, that 01002 = id, so 01 and 02 are mutually inverse Rlinear isomorphisms. Since 01 is unique, we are done. The existence proof, though elementary, is a bit more long winded. Let W be the free Rmodule spanned by the set V1 x 72. The module W is just the set of all formal linear combinations k
E ai(vi, wi) i=1
212
7. M U L T I L I N E A R A L G E B R A
where ai E R and (vi,wi) E Vi x 8 9 This is an Rmodule under the obvious operations and each element 0 r w C W is uniquely expressed as an Rlinear combination of finitely many members of the basis 171 x 1/2. Any linear combination with all coefficients 0 is equal to the 0 E W. Let [R C_ W be the submodule spanned by all elements of the form
(av + bu, w)  a(v, w)  b(u, w), (v, aw + bu)  a(v, w)  b(v, u) where a, b E R and u, v, w are in 171 or 89 appropriately. We think of :R as the submodule of bilinear relations and set
Vl  89 = w/:~. The cosets of the elements of the basis 1/1 x 89 will be denoted by
(v,w)+ J~=vQw, and we define 
: 1/1 x 89 , V1  1 8 9
by

= v  w.
Bilinearity follows immediately from tile definition of [R. For example,
(av + bu)  w = (av + bu, w) + R = a(v, w) + b(u, w) + :R = a(v  ~ ) + b(,,  ~). Note that, as a special case of bilinearity, (av) @ w = a(v  w) = v  (aw) and, in particular, v @ 0 = 0 = 0 @ v. We establish the universal property. Let ~o : V1 x 89 + V3 be an Rbilinear map. Since V1 x 89 is a free basis of W, there is a unique Rlinear map
7:W~V3 such that ~(v,w) = ~ ( v , w ) , V(v,w) E 171 x 172. Since qDis bilinear, it follows that vanishes on the generators of :R, hence that ~IIR = 0. Consequently, ~ passes to a welldefined Rlinear map
? : W/:R= VI  89 , 89 such that the diagram 
V~ x V2
, w~189
va commutes. Since 1/1  89 is spanned by elements of the form v @ w, ~ is unique.
[]
7.1. TENSOR ALGEBRA
213
In a completely parallel way, one can consider Rtrilinear m a p s and prove t h e existence and uniqueness of a universal Rtrilinear m a p
Vlxv2xvs2s174174 sending (Vl, ~32, V3) H Vl @ V2 @ V3.
It is a trivial exercise to check t h a t tile composition
(Vl X 72) X V3 
(V1 @ V2) x V.3 ~
(V1 @ V2) @V3
also has the universal property, as does V1 x (V,2 • V3) idvl •174 V1 • (V2 @V3) ~
V1 @ (V2 @ V3).
7.1.9. If Vi is an Rmodule, i = 1, 2, 3, there are unique Rlinear iso(V1 @V2)@V3VI@V2@V,3 identifyingvl174 (V 1  l)2)  V3 = V1  V2  V3, V V i E Vi, i = 1, 2, 3.
Corollary
morphisms V1  1 7 4
More generally, for each integer k >_ 2, there is a unique universal, klinear m a p (over R)
vlXv2x...vk ~~v~ov20...evk and canonical identifications "Vl e ( V2 e . . . e Vk ) = ( E l 
. . . e Vk_ l ) 
v k = v1 
v2 
...
tk .
A n obvious induction shows t h a t all groupings by parentheses are equivalent, so parentheses can be d r o p p e d or used selectively as desired. D e f i n i t i o n 7.1.10. An element v c !/1  ...  Vk is decomposable if it can be w r i t t e n as a m o n o m i a l v = vl  999  vk, for suitable elements vi E V~, 1 < i < k. Otherwise, v is indecomposable. By the construction of the tensor p r o d u c t in the proof of T h e o r e m 7.1.8, the decomposable elements span. 7.1.11. If V and W are flee Rmodules with respective bases A and B, then V  W is free with basis C = { a  b l a E A, b c B }.
Lemma
Proof. An a r b i t r a r y element v E A  B can be written as a linear c o m b i n a t i o n of decomposables. A decomposable element v  w can be expanded, via the multilinearity of tensor product, to a linear combination of elements of C, proving t h a t C spans V  W. It remains for us to show that, if P,q
E
P,q
cijai  by = E
i,j=l
dijai  bj,
i,j=l
where ai C A and bj E B, 1 < i < p, 1 _ 0, the r t h tensor power of V is
7~(v) =
v,
174
r = 1,
~>2.
R e m a r k . By Lemma 7.1.17, ~Y~  7~(V) = Y~(V) = 7 n ( v ) @ 70(V). When n and m are both positive, the identity ~ ( V ) @ ~Ym(V) = 9~+'~(V) is given by the associativity of the tensor product.
Set 9"(V) = {T'(V)}~= o and note that  defines an Rbilinear map
9 ~(v) x ~ ( v ) This makes IT(V) into a
~ ~(v) 
= ~+~(v).
graded algebra over R in the following sense.
D e f i n i t i o n 7.1.19. A graded (associative) algebra A over R is a sequence {Ar}~=0 of Rmodules, together with Rbilinear maps (multiplication) An x A ~ Z ~ A ~ + ' ~ , V n , m Z O , (written (a, b) ~+ a . b or, sometimes, (a, b) ~+ ab) that is associative in the sense that the compositions ( A ~ x A m ) x A k .xid An+,~ x A k & A ~+'~+k
216
7. MULTILINEAR ALGEBRA A n x (Am x A k) id x . A n x A 'n+k ~ A n+'~+k
are equal, Vn, m , k _> 0. D e f i n i t i o n 7.1.20. The graded algebra A is connected if A ~ = R and A ~ x A m  4 A m .2 A m x A ~ are equal to scalar multiplication, V m _> 0. R e m a r k that a connected graded algebra has unity 1 r R = A ~ D e f i n i t i o n 7.1.21. If V is an Rmodule, then ~T(V), with multiplication •, called the tensor algebra of V.
is
It is clear t h a t the tensor algebra ~Y(V) is connected. D e f i n i t i o n 7.1.22. A homomorphism ~ : A * B of graded R  a l g e b r a s is a collection of Rlinear maps 9~n : A n ~ B n, V n >_ 0, such t h a t the diagrams A n x A "~
~ A n+m
,~x~m I B n X B m
~,~+" ~ B n+m
commute, V n , m > O. The homomorphism ~ is an isomorphism if 9 n is bijective, Vn>0. T h e o r e m 7.1.23. [f )~ : V * W is an Rlinear map, then there is a unique induced h o m o m o r p h i s m ~Y()~) : ~Y(V) * ~Y(W) of graded Ralgebras such that T~ = idR and 9q ( ~ ) = ;~. This homomorphism satisfies ~ ( ~ ) ( ~ ,  ~2 
 ~n) = ~(vl)  ~ ( ~ ) 
 ~(~n),
V n >_ 2, Vvi E V , 1 < i < n. Finally, this induced homomorphism makes 9~ a covariant functor from the category of Rmodules and Rlinear maps to the category of graded algebras over R and graded algebra homomorphisms. Proof. T h e formula on decomposable tensors is imposed by the requirement t h a t 9"(,k) be a homomorphism of graded algebras, together with the stipulation t h a t 9"1 (,k) = ,k. Existence and uniqueness of the linear maps 7n(,~) are given by Proposition 7.1.13. T h e fact that 9"(~) preserves  multiplication is immediate. T h e final assertion a m o u n t s to the obvious identities
~'(~, o #) = ~(~) o g'(~), 9 (idv) = id~(v) 9 [] T h e following is an elementary consequence of Corollary 7.1.12 a n d Proposition 7.1.16. Theorem
7.1.24. If V is a free Rmodule with basis { e l , . . . , era} then {eil @ ' ' ' @ e i k } l < _ i 1 ..... i k < r n
is a free basis ofiTk(V) and ITk(V *) = 9"k(V) *. Remark. In particular, if V is a free Rmodule with d i r e r V = m, then dimR ~Tk(V) = d i r e r iTk(V *) = m k.
7.2. E X T E R I O R A L G E B R A
217
Terminology. The established terminology a b o u t covariance a n d contravariance of tensors in geometry is inconsistent with the usage of "covariant" a n d "contravariant" in category theory. For later reference, here are the geometer's definitions. Let V be a finite dimensional vector space over the field F.
D e f i n i t i o n 7 . 1 . 2 5 . For each integer r > 0, 7r(V*), viewed as the space of rlinear m a p s V ~  " IF, is called the space of covariant tensors on V of degree r and is denoted by T~(V). D e f i n i t i o n 7.1.26. For each integer s >_ 0, 7~(V), viewed as the space of slinear m a p s (V*) ~  " IF, is called the space of contravariant tensors on V of degree s and is denoted by T~s(V ) D e f i n i t i o n 7.1.27. T h e space of tensors on V of type (r, s) is the tensor p r o d u c t
%(V) = %'(V)0~(V). A tensor c~ E '3,"(V) is said to have covariant degree r and contravariant degree s. Obviously, 7~(V) is the space of (r + s)linear maps V • ... x VxV*
x ... x V*.
IF.
Y 7"
s
E x e r c i s e 7.1.28. Let V be a n Rmodule, V* its dual. (1) E x h i b i t a canonical Rlinear m a p a : V*  V  . R. (2) If V is free, prove t h a t c~ is a surjection. If, in addition, V has a basis with one element, prove t h a t a is a bijection. (3) If V a n d W are Rmodules (not necessarily free), exhibit a canonical Rlinear m a p fl : V* @ W  . HomR(V, W). (4) If R is a field, prove t h a t / 3 is injective. Do not assume t h a t V a n d W are finite dimensional. (5) If R is a field, prove t h a t fl is surjective if a n d only if either d i r e r V < oo or dimR W < oc. E x e r c i s e 7 . 1 . 2 9 . Let A be a connected, graded Ralgebra. (1) Show t h a t there is a unique homomorphism ~ : 7 ( A 1)  . A of graded algebras such that ~0 = idR and ~i = idAa. (2) Define a suitable notion of graded 2sided ideal I C A so t h a t A / I = {An/[~}~=o is again a graded Ralgebra. (3) If A is generated, as a graded algebra, by A 1, show t h a t there is a canonical ideal I C 'y(A1), with I ~ = {0} = 11, and a canonical isomorphism ,~ : r ( A ~ ) / I
~ A
such t h a t 7 o = idR and 71 = idm*.
7.2. Exterior A l g e b r a Let R be any c o m m u t a t i v e ring with unity 1 such that ~1 E R. T h a t is, if 2 = 1 + 1 E R, t h e n ~i E R has the property t h a t 1 . 2 = 1 . I n t t l e c a s e t h a t R = F is a field, this means t h a t the characteristic of F is not 2. Lemma
7.2.1. L e t V be a n R  m o d u l e ,
v E V.
Then
v = v
~=> v = O.
218
7. M U L T I L I N E A R A L G E B R A
Proof. Evidently, v = 0 ~ v =  v . For the converse,
v=v~2v=0~v=l(2v)=l(0)= 2
0.
2
[] Let Ek be the group of permutations of {1, 2 , . . . , k}, a group of order k!. D e f i n i t i o n 7.2.2. The sign of a E Ek is
(1)~ =
1, 1,
a an even permutation, cr an odd permutation.
D e f i n i t i o n 7.2.3. Let V and W be Rmodules. An antisymmetric klinear map o2 : V k + W is a klinear map such that ~ ( v o o ) , . . . , va(k)) = (  1 ) ~ o ( v l , . 99 , vk), V v l , . . . ,vk E V, Va E Ek. Remark that this definition will be useful only because 1 ~  1 in R. As in the definition of tensor product, for each k > 2, define a universal antisymmetric klinear map V x . . . x V 2+ Ak(V), written A ( v l , . . . ,vk) = vl A . . . A v k . Here, the subspace ~ of relations is generated by the klinear relations and all elements ( v l , . . . ,vk)  (  0 ~ , ~ ( k ) ) , ~ ~ ~k. Existence and uniqueness, up to unique isomorphism, are established exactly as for tensor product. One sets A~ = R and AI(V) = V. D e f i n i t i o n 7.2.4. The Rmodule Ak(V) is called the kth exterior power of V. We set A ( 7 ) = {Aa(V)}~~ . D e f i n i t i o n 7.2.5. An element w E Ak(V) that can be expressed in the form vl A vs A 9.. Avk, where vi E V , 1 < i < k, is said to be decomposable. Otherwise, w is indecomposable. It is clear that Ak(V) is spanned by decomposable elements, but generally there are plenty of indecomposable elements as well. As before, Rlinear maps ~ : V ~ W will induce canonical Rlinear maps Ak(~) : Ak(V) ~ Ak(W) such that A~
= idm AI(~) = ~, and
Ak(A)(vl A . . . Avk) = /~(Vl) A ' "
A
~(Vk).
We are going to define a bilinear, associative multiplication AP(V) • Aq(v) ~ A P + q ( V ) that will satisfy (Vl A " " A Vp) A (Vp+l A ' ' " A Vp+q) = Vl A " " A Vp+q.
7.2. E X T E R I O R A L G E B R A
219
By the above remarks, this will make A a covariant functor from the category of Rmodules and Rlinear maps to the category of graded algebras over R and graded algebra homomorphisms. To define the algebra structure on A(V), we must relate A(V) more directly to
O. Thus, A(V) is a graded algebra over R (the exterior algebra of V) and A is a covariant functor from Rmodules to graded Ralgebras9 Pro@ For k = 0,1, it is clear that Ak(V) = 7k(V)/P2k(V). consider the klinear map
For each k >_ 2,
v k ~ ~k(v) & ~k(V)/~k(V), where rr is the quotient projection 9 The reader will check easily that this is antisymmetric. Given an arbitrary antisymmetric klinear map ~ : V k ~ W , we obtain the commutative triangle 
Vk
, 7k(v)
W
220
7. MULTILINEAR ALGEBRA
and ~lP.lk(V) = 0. Thus, c~ induces ~ : O ' k ( V ) / P l k ( V ) + W making the following diagram
commutative: V k
W

)
ok(v)
)
W
id
~
, ~k(V)/~k(V)
,
id
W
T h a t is, the triangle ~oO v ~
~k(V)/~k(V)
.
W
commutes. Since g'k(V)/P.l;v(V) is spanned by
D(vl 
o vk) I vl,... ,~k c v}
and commutativity of the diagrams forces ~(Tr(Vl ~ ' ' ' O V k ) )
=
~O(Vl,... ,Vk),
we see t h a t ~ is the only linear map making the triangle commute. T h a t is, ~ o  : V k ~ g ' k ( V ) / P A k ( V ) has the universal property for antisymmetric, klinear maps, hence is uniquely identified with A : V k ~ Ak(V). Finally, this identification makes vl A 9 .. A vk : vl  " "  vk + P2k(V), and all assertions follow. [] D e f i n i t i o n 7.2.9. A graded algebra A is anticommutative if a E A k and/3 E A r => a/3 = (1)kr/3a.
C o r o l l a r y 7.2.10. T h e graded algebra A(V) is a n t i e o m m u t a t i v e . Proof. It is enough to verify Definition 7.2.9 for decomposable elements of Ak(V) and At(V). But that ease is an elementary consequence of the case k = r = 1, and this latter case is given by
v n w = vOw
+ ~12(V)
= w  v + ~2(V) =
Vv,w
W
A V,
E V.
[]
C o r o l l a r y 7.2.11. I f w 6 A2r+I(V), then w A w = O. Proof. Indeed, w A w :
By Lemma
7.2.1, w A w = 0.
Corollary
7.2.12.
For the remainder
(I)(2"+I)(2~+I)w
If w 6 Ak(V)
A w =
w
A w.
[] /s decomposable,
then w Aw
= O.
of this section, we specialize to the case in which V is a free
Rmodule on a basis { e l , . . . ,era}.
7.2. EXTERIOR ALGEBRA
Lemma
7.2.13.
221
I f V is as above, then { e i 1 A el2 A . . . A e i k } l < < i l < i 2 < , . . < i k < m
is a free basis of A k ( V ) , k > 2. In particular, dimRAk(V) =
and, i f k
> .~, A k ( V ) =
 kT(m  k)!
(0}.
P r o @ T h e basis {%174
..... j~2.
Show that, if {el, , era} is a basis of V, this identifies the m o n o m i a l e* e* ... e~ w i t h (ei, ei2 " " eit:)*. D e f i n i t i o n 7.3.6. Let V be a finite dimensional vector space over a field F of characteristic zero. A function f : V + F is a homogeneous polynomial of degree k on V if, relative to some (hence, every) basis { e l , . . . , era} of V,
I
}~,e~
= P(~l . . . . . ~m)
i=1
is a homogeneous polynomial of degree k in the variables x l , . 99 , x ~ . The vector space of all homogeneous polynomials of degree k on V will be denoted by Tk(V). Exercise
7.3.7. For all k _> 0, establish a canonical isomorphism
0: sk(v *) ~ ~k(v) of vector spaces. For the case k = 2, construct 0 5 explicitly. (~P2(V) is called the space of quadratic forms on V and the process 0 1 of recovering the s y m m e t r i c bilinear form from its associated q u a d r a t i c form is called polarization.)
7.4. Multilinear B u n d l e T h e o r y J u s t as the linear construction of dualizing a vector space passes to the construction of dualizing a vector bundle, so the multilinear constructions of the previous three sections pass to corresponding constructions on vector bundles. Let 7ri : E.i ~ M be a kiplane bundle, 1 < i < m. We want to define a bundle
7r : E1 
"" Q Em + M
228
7. M U L T I L I N E A R A L G E B R A
with fiber over x C M canonically equal to E 1 x  9 ' '  E m x. The fiber dimension will be klk2 .." km. Let {Us, r be the maximal family of simultaneous trivializations Edg~
r
, Us x R k,
Us
,
U~.
id
Note that, if Aj C Gl(kj) is viewed as a nonsingular linear transformation of IRkj , 1 _ j _< m, then A1  A2  9 9 Am is a nonsingular linear transformation of R kl  N k2  999 N k" , hence A1  A2 
 Am C G l ( k l k 2 . . . k m ) .
Here, we identify N kl  9  1Rk,,` = ]Rklk2'"k,,~ by lexicographic order on the basis l<j<m
where {e~,... , e~; } is the standard basis of IRk;, 1 _< j _< m. The system {Us, r i gives rise to a Gl(k~)cocycle { 7i~ } ~ e ~ , for each i = 1 , . . . , k , with the aid of which the bundle Ei can be assembled from the products Us x R k~, a E 92. We try to define a cocycle for assembling the tensor product of these bundles by setting % ~ : Us n U~ ~ G l ( k l k 2 . . . kin),
L e m m a 7.4.1. The map ~/aZ is smooth and ~/= {3'~}~,Ze~ is a cocycle. Proof. The cocycle property is rather obvious. For smoothness, it is enough to show that the vectorvalued function x ~ ~.Z(x)(e~ 

is smooth on Us n Uz. But kj
s~(x)(%) ~=1
and the realvalued functions a~ (x) are smooth. Expanding ~o~/3(x)(e~l 
" .  eim) "
1 = %1A X )(%)
"
kl
=
m km
~ (=)e 
m
174
ae (x)e= \t'=l
, "
we obtain a linear combination of the elements of the basis B with smooth functions of x as coefficients. []
7.4. M U L T I L I N E A R
BUNDLE THEORY
229
E x e r c i s e 7.4.2. Let rr : E * M be the bundle determined by the cocycle { U ~ , % ~ } ~ and produce a canonical isomorphism
Ex = E l x  1 7 4 1 7 4 for each x E M. In particular, given an nplane bundle rr : E + M, we can form the tensor powers 7r :Irk(E) ~ M, the fiber over x E M being, canonically, Irk(E~). If E has an associated cocycle {%Z}~,~ 1, then r~ = da for some a E A ~  I ( M ) , hence wA~/ = w A d ( ~ = d~ A (  1 ) P ~ + (  1 F w A d((1)P~) = d(~ f (  1 ) ' ~ )
Since r] A aJ = (1)PqoJ A r/, it follows that B*(M) is a 2sided ideal in Z*(M).
[]
Since smooth maps ~ : M ~ N preserve exterior multiplication and pass to welldefined maps in cohomology, we see that
y)* : H*(N) ~ H*(M) is a homomorphism of graded algebras. We have completed the proof of the following. T h e o r e m 8.1.16. The graded cohomology construction defines a contravariant fune
tot H* from the category of differentiable manifolds and smooth maps to the category of anticommutative graded algebras overN and graded algebra homomorphisms. The graded algebra H* (M) is connected if and only if M is connected. The graded algebra H* (M) is called the (de Rham) cohomology algebra of M. Whether or not it is connected, H* (M) has a unity, namely the constant function
1 e Z~
= H~
244
8. I N T E G R A T I O N A N D C O H O M O L O G Y
D e f i n i t i o n 8.1.17. The space of compactly supported pforms on M is denoted by APc(M). It is clear that the exterior pactly supported. Indeed, it is Thus, each APc(M) is a module gebra A*(M) over C~176 It our purpose. Furthermore,
product of two compactly supported forms is comenough that one of them be compactly supported. over C~176 and these assemble into a graded alis also a graded algebra over IR, which is more to
d(A~(M)) C_ A~+I(M), so one can define the space
ZP(M) = {~ e AP(M) I &z = 0} and the vector subspace
B~(M) = {~ = da I a e A~I(M)}. If we use the common convention that A 1 (M) = A~1 (M) = 0, the above definition of BP(M) includes the case p = 0. As before, Z~(M) is a graded subalgebra of A*~(M) and B c ( M ) C Z*(M) is a 2sided ideal. D e f i n i t i o n 8.1.18. The (de Rham) cohomology algebra with compact support is
H* (M) : Z c ( M ) / B c (M). Remark that H*(M) = H*(M) if M is compact. At the other extreme, if M has no compact component, the space Z~ of compactly supported, locally constant functions on M is trivial, so H~ = 0. In any event, each element of Z~ will vanish on all but finitely many components of M. These observations establish the following. L e m m a 8.1.19. The vector space H~
is isomorphic to a direct sum of copies of ~, one for each compact component of M.
Note the different conclusions in Lemma 8.1.19 and Corollary 8.1.13. Each element of a direct sum has terms from only finitely many summands, while elements of a direct product are allowed to have terms from infinitely many of the factors.
Remark. The graded algebra H i (M) generally does not have a unity unless M is compact. Recall that a smooth map F : M ~ N is said to be proper if, for each compact set C C_ N, the set T  I ( C ) is also compact. For example, id : M + M is always proper. If M is compact, ~ is always proper. In any event, the composition of proper maps is proper, so the class of differentiable manifolds and smooth, proper maps between them is a category. If qp : M * N is proper and if co E AP(N), then F*(w) E AP(M). As usual, 9)* o d = d o ~*, so we get an induced homomorphism of graded algebras ~ * : H c(N) ~ H 2(M). T h e o r e m 8.1.20. Cohomology with compact supports is a contravariant functor
H~ from the category of differentiable manifolds and smooth, proper maps to the category of anticommutative graded algebras over ~ and graded algebra homomorphisms. E x e r c i s e 8.1.21. Prove that Hcl(~) ~ R. This is the onedimensional case of the Poincar~ lemma for compactly supported cohomology.
8.2. STOKES' THEOREM
245
8.2. S t o k e s ' T h e o r e m a n d S i n g u l a r H o m o l o g y
In this section, we define integration of forms and give a detailed treatment of two versions of Stokes' theorem. As an application of the second (combinatorial) version, we define the singular homology of a manifold and relate it to de Rham cohomology, stating the celebrated de Rham theorem. A detailed sketch of the proof of this theorem will appear in Section 8.9. Throughout what follows, we assume that dim M = n and that M is oriented. We also allow 0 M ~ 0. T h e o r e m 8.2.1. For each oriented nmanifold M , there is a unique Rlinear func
tional ~
: A2(M ) ~
R,
called the integral and having the following property: if (U, ~) is an orientationrespecting coordinate chart, if co E A ~ ( M ) has supp(a~) C U, and if ~l*(w) = g d x 1 A . . . A dx ~ e A2(p(U)), then /MCZ=~
(v)
g
(the Riemann integral).
Proof. First we prove uniqueness. Let {(Us, ~ ) } ~ e ~ be a smooth Hnatlas on M respecting the orientation. Let {A~}~e~ be a smooth partition of unity subordinate to the atlas. If ~ E A~(M), then A ~ e A ~ ( M ) and A ~ r 0 for only a finite number of ~ C 92. This is because supp(w) is compact and the partition of unity is locally finite. Thus, = E AaaJ ~692
and this sum is actually finite. Then, if fM exists, linearity gives
and s u p p ( A ~ ) = supp(A~) N supp(~) is a compact subset of Us. By the local property of fM, each fM A ~ is uniquely given as
where g~ dx I /k. .. A dx, n : ~ l * ( ~ l V a ) . We give one way to define fM, establishing existence. This will depend on a choice of orientationrespecting coordinate atlas {U~, ~ } ~ e ~ and of subordinate partition of unity {A~}~e~. (One could remove some of this arbitrariness by requiring the atlas to be the maximal one, but the choice of partition of unity still could not be made canonical.) We appeal to the uniqueness already proven to show independence of the choices. If aJ C A~(M), only finitely many A ~ are not identically 0. Define =
~
~(u~)
)g~,
246
8. I N T E G R A T I O N A N D C O H O M O L O G Y
where g~ d x I A . . . A d x n
= ~jl*(o2lUc~).
Then
define
a finite sum. Defined in this way,
is an JRlinear map.
We must check that, if supp(w) C U, where (U, ~) is an arbitrary orientationrespecting coordinate chart (not necessarily in our atlas), and if cfli*(w) = g d x I A . . . A d z n,
then M w = L ( u ) g' First remark that
= E
~
U~m U)
c~e~ ~ (
(,,~ 0 (~a,1)go~,
since supp(w) C U. By Exercise 7.5.22, (Aa~ = L (Ac~o~l)g, ~,(s~nu) (u~nu) for each a C 92. The fact that the charts are compatibly oriented is essential. Thus, aEPa
(U~nU)
=
(a~ c~C92
J1 =L
(u)
9. []
Remark. By Exercise 7.5.22 and the above proof, _MO2 =  / M ~
The orientation of M induces an orientation of O M in the following way. Let {Us, x ~1 , . . . , x n~ } ~ be an IHI~atlas on M respecting the orientation. By the deftnition of H ~, x~i < _ 0, wherever defined, and x~1 = 0 exactly on Us n OM, V a E 92. Let 9,1' = { a C 92 I Us N O M r 0} and consider the N~latlas
{us n OM,xL...
8.2. S T O K E S ' T H E O R E M
247
of O/V/. Let g~z and gaz~ denote the respective changes of coordinates for these atlases. At x E U~ n UZ n O M , rOx~
0 gc~
] z
Here, since x~ decreases with x~, the upper lefthand entry in this matrix is strictly positive. Since d e t ( J g ~ ) x > 0, it follows that d e t ( J g ~ > 0. Thus, this IRn  I atlas on 0 M defines an orientation of 0 M . D e f i n i t i o n 8.2.2. The orientation of O M , produced as above, is said to be induced by the given orientation of M. We always assume that, when M is oriented, O M has the induced orientation. The following fundamental result asserts that exterior differentiation is the "adjoint" of passing to the boundary. For this reason, d is often referred to as the "coboundary operator". T h e o r e m 8.2.3 (Stokes' Theorem). L e t M be an oriented n  m a n i f o l d a n d let i : O M ~ M be the inclusion. T h e n , i f w E A ~  I ( M ) ,
where, i f O M = ~, the r i g h t  h a n d side is interpreted as O. Proof. First we prove the local case. That is, M = H n and O M = IR~1. Any a~ C A ~  I ( H n) can be written as n
a) = E (  1 ) J  l f j d x
1 a'.'AdxJ
A".Adx
n,
j=l
where f j has compact support, 1 < j < n, and d x j indicates that this term is omitted. Then i*(co) = (fl
o
i) d x 2 A . . .
A
d x n E AnI(cO]H[n)
and dw =
_
OxJ J d x l A . . . A d x n C A~c~(tHIn).
By the fundamental theorem of calculus and the compactness of s u p p ( f j ) , for j = 2 , . . . , ?z,
/? 2o, oo""
oo ~ x J d x l "'" d x n =
~ '"
~ \J_~
OxJ d x j
d x l "" " d x j "" " d x n = O.
248
8. I N T E G R A T I O N
AND
COHOMOLOGY
Therefore,
L
dco = n
F
/cx~; OO .
=
.
Co .
Ofl " 1
dx n
. Oo . ~ x l a.X
l i l?; "
l(O, x 2 , . . .
,z ~)dx 2"''dx ~
oo
9
fl oi
: L N ~ i*w.
Now we can prove the global case. Let i M : O M '* M ,
iH~ : OH n ~ H n
denote the respective boundary inclusions and let
{u~,~ = (~L... , ~n ) } ~ , { U~, M O M , ~o~ = ( x 2 , . . .
,x2)}~,
be the orientationrespecting atlases on M and O M , respectively, as chosen above. If { A ~ } ~ is a smooth partition of unity subordinate to the atlas on M, then {ha o iM}aE~l' is a smooth partition of unity subordinate to the atlas on O M . Note that ~ 1 oiH~ = i M o (~0)1, E For w E A n  I ( M~ ~ j, the fact that w = ~ A~w is a finite sum gives finite sums
VO~ ~[/
d~ = ~ d(a~), aE~
*
% M a) =
E ~;~(~).
aE~'
Therefore,
L
&o=
Y~ fuo d ( ~ )
c~E9.1
E L (r aE~l o(U~)
= ~ f~
aE~ ~ (U~)
=
EL,,
d(~21"(~))
~Hn~ _1,
c~6 9 / '
(d()~aca))
(~).
H~
The last equality is by the local version proven above. If O M = O, then supp(w) A O M = O,
8.2. STOKES' THEOREM
249
and this integral vanishes. Otherwise, we get dw =
r /o
=
a ~ j~ ,
(~)
~M(;~o)
( ~O ) 1..,~ M ( ~ ) ,,(U~nOM)
o~EgX'
~nOM
= F. /
(x~ o i,~,)iS(~)
o~Ega' U ~ n O M =
7,A/IOO. hi
[] Theorem
8.2.4. Let M be an oriented nmanifold with O M = ~). Then f
: H ~ ( M ) ,
is a welldefined, Nlinear surjeetion. Proof. Since A ~ + I ( M ) = 0, we have Z ~ ( M ) = A ~ ( M ) . I f w = dr] E Bn~ ( M ) , then
Stokes' theorem and the fact that cgM = {3 imply that
Thus, the linear map
~ : Zg(M) ~ r
induces a welldefined linear map M : H n ( M ) ' N.
To prove surjectivity, we only need prove that this map is nontrivial. Let (U, x 1, ... , x n) be a compatibly oriented chart and let A G C ~ 1 7 6 have compact support contained in U, with A _> 0 everywhere and ,~ > 0 somewhere. Thus co = A dx 1 A ... A dx '~ can be interpreted as an element of Z 2 ( M ) and of A~(IR~), so /Me~ = s
A>0' []
A deeper fact, to be proven later (Theorem 8.6.4), is that, if M is both oriented and connected, then fM is a bijection from H ~ ( M ) to IR. In order to integrate pforms, where p < dim M, it is necessary to define suitable pdimensional domains of integration. For the case p = 1, we have already studied lille integrals, the domain of integration being a (piecewise) smooth curve in M. In general, it is convenient to use singular psimplices (defined below) as domains for integrating pforms. A singular 1simplex is simply a smooth curve. Recall t h a t a subset A C ]Rp is convex if, for each pair of points v, w G A, the straight line segment joining v and w lies entirely in A. If C C IRv is an arbitrary
250
8. I N T E G R A T I O N A N D C O H O M O L O G Y
subset, the c o n v e x h u l l of C is defined to be the smallest convex set C containing C. Since an arbitrary intersection of convex sets is convex, and NP is itself convex, is just the intersection of all convex sets containing C. D e f i n i t i o n 8.2.5. The standard psimplex /~p (~ ]I~p is the convex hull of the set { e 0 , e l , . . . ,ep}, where ei is the ith standard basis vector, 1 < i < p, and e0 = 0. Thus, A0 = {0}, a single point, and A1 = [0, 1]. The cases p = 2 and p = 3 are pictured in Figures 8.2.1 and 8.2.2, respectively.
F i g u r e 8.2.1. The standard 2simplex
F i g u r e 8.2.2. The standard 3simplex A more explicit definition of the standard psimplex is Ap =
( x l , . . . , x P ) leach x i > 0 and ~ x
i _< 1 .
i=1
It is sometimes convenient to set A_I = 0. D e f i n i t i o n 8.2.6. A (smooth) singular psimplex in a manifold M is a smooth map s : Ap ~ M. Thus, each point of M can be thought of as a singular Osimplex and smooth curves, up to parametrization, are singular 1simplices. One could also define piecewise smooth singular psimplices, but we will not do so.
8.2. S T O K E S '
THEOREM
251
D e f i n i t i o n 8.2.7. For 0 < i < p, the ith face of the standard psimplex Ap is the singular ( p  1)simplex Fi : Ap1 * Ap defined by Fi(xl'""xP1)
l , . . . ,xil,0, x i , . . . ,3::p  l ) = [(1x 1 ..... xPI,Xl,...,X pl)
f(x
i f / > 0, if/=0.
The 0th face of A0 is considered to be defined but empty. If s : Ap ~ M is a singular psimplex, the ith face of s is tile singular (p  1)simplex Ois = s o Fi. It is clear t h a t Fi : Ap_I * Ap is a topological imbedding and t h a t the image of Fi is exactly the subset ordinarily thought of as the "face" of Ap opposite the vertex el. The ith face Ois of a singular psimplex s is essentially the restriction of s to the ith face of Ap, but parametrized on the standard Ap_I. 8 . 2 . 8 . If s : Ap + M is a singular psimplex and co E AP(M), then s*(co) has the form g d x I A . . . A dx p and we set
Definition
p
where the righthand side is tile Rietnann integral. If s : {0} ~ M is a singular 0simplex and co = f E A ~ the integral is interpreted to mean f f = f(s(O)). There is a combinatorial version of Stokes' theorem, according to which the integral of an exact pform dr/over a singular psimplex s is equal to the integral of r/over the "boundary" of s. T h e o r e m 8.2.9 (Combinatorial Stokes' Theorem). I f s : Ap ~ M is a singular psimplex and r~ C A P  I ( M ) , then P i=0
is
Remarks. The signs in the combinatorial Stokes' theorem are dictated by comparing the s t a n d a r d orientation of Ap_ 1 with the induced orientation of Fi (A v_ 1 ) &q a part of the b o u n d a r y of Ap. We write the fornml combinatorial expression P
08 = y ~ (   1 ) % S /=1
and express Stokes' formula as fdr/=
f0s r/.
This highlights the analogy with Theorem 8.2.3 and agrees with established usage in algebraic topology. For f E A ~ and a smooth curve s : [0, 1] * M, Theorem 8.2.9 asserts that s d f = f(s(1))  f(s(0)), which is just Lemma 6.3.3. T h a t lemma was a thinly disguised version of the fundamental theorem of calculus and Theorem 8.2.9 is a somewhat less thinly disguised version of the same fundamental theorem.
252
8. I N T E G R A T I O N
AND
COHOMOLOGY
Proof of theorem 8. 2.9. It is clearly sufficient to prove that
/ dv=~,(1)is n;(V), P
P
i=O
p1
where ~/is a (p  1)form defined on an open neighborhood of Ap in N p. We can write P
=
~.s dx 1 A . . . A~XJ
A...
A dx v
j=l
and, by the linearity of the integral, prove the formula for each term of the sum. That is, without loss of generality, we assume that 1 _< j _< p and = f dx I A 999 A dxJ A 999A dx p. By the local formula for exterior differentiation,
of
@ = (1) jl~dx
1 A...Ad2,
and we are reduced to proving the formula (8.1)
fAp(1)Jl~dxl
A...AdxP
= E(1)
i
F * ( f d x 1 A . . . A dxJ
i=0
A...
A
dxP).
v 1
The righthand side of equation (8.1) can be simplified. For this, it will be helpful to let x i denote the coordinates in N p and z i the coordinates in N p1. Remark t h a t fdz j1 ~  E P  : dz i
FG (dx~ )
i f j > 1,
if j = l ,
and, if i > 0, zj
if j < i, i f j = i,
( dz j  1
ifj >i.
I~
F [ ( d x j) = One obtains the formula
F ~ ( f dx 1 A ... A dz"~ A ... A dx v) = (  1 ) J  l ( f o Fo)dz 1 A . . . A d z p1 and, if i > 0, F*(fdxl
A A'"AdxJ
A'"AdxP)=
l0 ~,(foFj)dzl
A...Adz
p1
if/~j, ifi=j.
Substituting these terms in the equation (8.1) and multiplying both sides by (  1 ) j  1 reduces us to proving /A
O f dxl A . . . A dxP = / A ( f o Fo) dzl A . . . A dz pI r, OxJ p1

f J~
( f o iaj) dz 1 A ' " p1
Adz p  l ,
8.2. STOKES' THEOREM
253
which, rewritten in terms of the Riemann integral, becomes (8"2)
/ ~ p OxJ 0 f = //x ,,1 f(1  Z 1 . . . . .
Zp  l , Z 1 , . . . , Zp  l )  f
f(zl, "'',zjI,0,zj,''',zp1)" p 1
The linear change of coordinates in ]Rvl, defined by f lz 1 ..... w i = J z i1 (z i
if i = 1, if2 0. These vectors must be linearly independent, so we can find a local coordinate chart (U, x l , . . . , x ~) about x in which vi is the value of the ith coordinate field ~i = O/Ox i at x, 1 < i _< p + 1. By making this chart sutficiently small, we can guarantee that r/(~l A ... A ~;+1) > 0 on all of U. Let s : Ap ~ U be any orientationpreserving, smooth imbedding into the coordinate (p + 1)plane { ( x l , . . . , x n) E U [ x p+2 = . . . . x ~ = 0}. It follows that
[] Singular simplices are used to detect topological features of a manifold. For instance, a piecewise smooth, closed curve s = Sl + 999+ Sq in the punctured plane IR2 \ {(0, 0)} can detect the missing point, provided that s has nonzero "winding number" w ( s ) about the origin. The closed curve s is assembled from the singular 1simplices s l , . . . , Sq which join together, end to end, to form a "icycle" and the winding number itself is defined by integrating the locally exact (hence, closed) 1form r/ of Example 6.3.12 over s. Piecewise smooth closed loops s in R 3 \ {(0,0,0)} do not detect the missing point (Example 6.4.9). However, a map s : S 2 ~ IR3 \ {(0,0,0)} can snag the missing point. One effective way to use this observation is to triangulate S 2 (Section 1.3) and form singular 2simplices Sl,. 9 Sq by restricting s to these triangles. It is necessary to assume only that each si is smooth, so we get a piecewise smooth map s: s 2 ~ R ~ \
{(0,0,0)}
and write s = Sl + "" + Sq by analogy with the case of loops. We call this a "singular 2cycle". One should test whether or not the singular 2cycle has snagged the missing point x by integrating a suitable closed 2form co over this cycle: q
The possibilities for singular 2cycles are richer than for 1cycles. For instance, triangulations of T 2 and corresponding piecewise smooth maps s of T 2 into M define "toroidal" singular 2cycles s = sl + "" + Sq in the manifold M and such a cycle might well detect a topological feature that would be missed by a "spherical" cycle. Again, a test of what this cycle detects is made by integrating closed 2forms over the cycle. These remarks are extended and made precise by defining the singular homology of a manifold, a covariant functor H , from the category of smooth manifolds to the category of graded vector spaces over IR. The celebrated de Rham theorem asserts that this functor is dual to de Rham cohomology. We sketch the main
8.2. S T O K E S '
THEOREM
255
facts, illustrating the importance of the combinatorial Stokes' theorem for algebraic topology. D e f i n i t i o n 8.2.11. The set of all singular psimplices in M, p > 0, is denoted by A v ( M ). The space C v ( M ) of singular pchains on M is the free Nmodule (real vector space) generated by the set A v ( M ). By convention, if p < 0, A v ( M ) = and Cp(M) = O. Each pform co E AP(M) can be viewed as a linear functional
co: Cp(M) ~ R as follows. An arbitrary pchain c C Cp(M) can be written uniquely (up to terms with coefficient 0) as a linear combination m
C = Eajsj~ j=l
where sj E A p ( M ) , 1 0, is the linear map P
o =
Z(~)'o~. i=0
For co E A p  I ( M ) and c ~ @ ( M ) , the combinatorial Stokes' theorem asserts that cz(0c)=/acW=fcdW=da~(c)' T h a t is, the operators d and 0 are adjoint to one another. The boundary operator is an algebraic analogue of the geometric notion of a boundary. The following crucial property can be viewed as the algebraic analogue of the fact that the boundary of the boundary of a manifold is empty. E x e r c i s e 8.2.13. Prove that the composition
Cp(M) o Cp_~(M) a Cp_2(M) is trivial (0 2 = 0). D e f i n i t i o n 8.2.14. The subspace Zv(M ) C_ C v ( M ) of all (singular) >cycles is the kernel of the boundary operator cq: Cp(kJ) ~ Cp_I(M). The subspace Bp(M) C_ Cp(M) of all (singular) pboundaries is the image of the boundary operator 0 :
Cp+I(M) +C,,(M). An immediate corollary of Lemma 8.2.13 is that Bp(M) C Z v ( M ).
256
8. I N T E G R A T I O N A N D C O H O M O L O G Y
D e f i n i t i o n 8.2.15. T h e pth singular homology of M is the vector space
Hp(M) = Z p ( M ) / B p ( M ) . If z e Zp(M), the homology class of z is the coset [z] E Hp(M) represented by the cycle z. E x a m p l e 8.2.16. Since C1 (M) = 0, the b o u n d a r y operator vanishes identically on Co(M). T h a t is, Zo(M) = Co(M) is the real vector space with basis the set of points of M . Define c : Zo(M) ~ N by aixi \i=1
=
ai, i=1
where a l l a i E ]R and all xi E M. I f s E A I ( M ) , e(Os) = e ( s ( 1 )  s ( 0 ) ) = 0, so B o ( M ) C_ ker(c) a n d ~ passes to a welldefined linear m a p g : Ho(M) ~ R. We assume t h a t M ~ !~, so there is a point x E M a n d ~([x]) = 1, proving t h a t ~ is a surjection a n d Ix] 7~ 0, Yx c M. If M is connected, then every two points x , y E M can be joined by a piecewise smooth p a t h s = sl + ... + sT a n d Os = y  x. This implies t h a t [x] = [y], hence that H0 (M) has basis consisting of a single element [x]. We have proven t h a t the 0th singular homology of a nonempty, connected manifold is canonically isomorphic to JR. E x a m p l e 8 . 2 . 1 7 . Let M be contractible (cf. Exercise 8.1.9) with contraction ~t : M ~ M. This is a homotopy of P0 = idM to a constant m a p ~1. Using this contraction, we are going to define linear maps
Lp : Cp(hl) ~ Cp+I(M), Vp > 0, with a remarkable property. T h e s t a n d a r d inclusion R T ~~ Rp+I restricts to an inclusion A T ~~ Ap+l which is j u s t the face m a p Fp+l. For each point v E Ap+l \ {eT+l}, there is a u n i q u e point v' E A T and a unique n u m b e r t C [0, 1] such t h a t
v = teT+l + (1  t)v', a n d every point oflR p+I of such a form is a point in Ap+l. (For v = ep+t, v' is not unique, b u t t = 1, so this will cause no problem in what follows.) If s : Ap ~ M is smooth, define a s m o o t h m a p Lp(s) : Ap+l ~ M by the formula
Lp(s)(tep+l + (1  t)v') = ~t(s(v') ). T h e fact t h a t this is well defined when t = 1 is due to ~1 being a c o n s t a n t map. We view s H Lp(s) as a set m a p
Lp : A p ( M ) ~ Cp+l(M) a n d take the linear m a p L T to be the unique linear extension of this set m a p to all of CT(M ). In Exercise 8.2.18, you are invited to check t h a t
(,)
0 o Lp = Lp1 o 0 k (  1 ) p+I idcp(M),
provided t h a t p _> 1. This is the remarkable property promised above. If z E Z T ( M ) a n d p > 1, it follows t h a t
O(LT(z)) = Lp_l(OZ ) + (1)P+lz = (1)T+lz, hence t h a t Z v ( M ) C Bp(M). The reverse inclusion also holds, so we have the result t h a t the singular pcycles and the singular pboundaries in a contractible space are exactly the same, Vp >_ 1. T h a t is, HT(M ) = 0 in all degrees p > 0.
8.2. S T O K E S ' T H E O R E M
257
E x e r c i s e 8.2.18. Prove the identity (*) in Example 8.2.17. The above two examples give T h e o r e m 8.2.19. I f M is a contractible nmanifold, then Hp(M)
= ~,
p = o,
[ 0,
p>0.
In particular, this is true for M = ]R~. P r o p o s i t i o n 8.2.20. I f co C ZP(M) and z ~ Zp(M), then the real number fzco depends only on the cohomology class [co] C H P ( M ) and the homology class [z] C
H~(M). Proof. Indeed, [co] is the set of all closed pforms co + dTi, where ~ C AP~(M). We have
by Stokes' theorem and the fact that z is a cycle, so
Similarly, [z] is the set of all pcycles of the form z + Oc, where c E Cp+~ (M). Since
we obtain +0c
c
[] Thus, we can define an Rlinear map D R : H P ( M ) + ~ , ( M ) * , by (DR[co])([z]) = [
[co]. )
T h e o r e m 8.2.21 (The de Rham Theorem). The linear map DR is a canonical isomorphism of vector spaces. This is a deep result. For the case in which M is compact, the proof will be discussed in some detail in Section 8.9. In that case, the vector spaces are finite dimensional (Theorem 8.5.8), so we also get H p ( M ) = H P ( M ) *. The following corollary generalizes Proposition 6.4.3. C o r o l l a r y 8.2.22. Let co,~ C ZP(M). Then [w] = [c5] if and only if fz w = fz ~ as z ranges over all singular pcycles in M . These numbers are called the periods of w and of the cohornology class [co]. In particular, co is an exact form if and only if all of its periods are 0, which generalizes the equivalence of properties (1) and (2) in Theorem 6.3.10. E x e r c i s e 8.2.23. Let M be an nmanifold and let z E Z~+I(M). Assuming the de Rham theorem, prove that there is a chain c E C , + 2 ( M ) such that z = Oc. E x e r c i s e 8.2.24. Show that singular homology is a covariant functor, proceeding as follows.
258
8. I N T E G R A T I O N A N D C O H O M O L O G Y
(1) If f : M ~ N is a smooth map between manifolds, exhibit a canonical way to induce a linear map f # : Cp(M) ~ Cp(N), Vp >_O. (2) Prove that the diagram
Cp+l(M)
f#
~
Cp+l(N)
o~ Cp(M)
1o f#
,
Cp(N)
commutes, Vp _> 0. Conclude that the linear map f # passes to a linear map f , : H , (M) ~ / 4 , (N) of graded vector spaces. (3) Verify the properties ( f o g), = f , o g, and id, = id. (4) Under the de Rham isomorphism of g k ( M ) with Hk(M)*, show t h a t
f* : Hk(N) ~ Hk(M), f, : Hk(M) + Hk(N) are adjoint to each other. E x e r c i s e 8.2.25. W i t h o u t appealing to the de Rham theorem, extend the argument in Example 8.2.16 to show that Ho(M) is a direct sum of copies of N, one for each connected component of M. 8.3. T h e P o i n c a r 6 L e m m a
In the following discussion, R will stand for any nondegenerate, compact interval [a, b] or for JR. We consider an arbitrary nmanifold M, not necessarily orientable. If M has nonempty boundary, M x R will always denote M x ~, thereby avoiding manifolds with corners. Homotopies, therefore, will be understood in the sense of Definition 3.8.9 whenever convenient. We will agree to denote the s t a n d a r d projections by
lr : M x R* M and
p:MxR~R. The coordinate of R will be denoted by t and p* (dt) E A 1(M x R) will be denoted by dt (an abuse). A locally finite atlas {(W~,x~)}~e~ on M determines such an atlas {(W~ • R , ( x ~ , t ) ) } ~ e ~ on M x R. Here, i f 0 M = ~ and R = [a,b], we model ( n + l ) manifolds with boundary on Rn x [a, b] instead of on IHIn+l. A smooth partition of unity {A~}~e~ , subordinate to {W~}~e~, determines a smooth partition of unity
on M x R subordinate to {W~ x R}~e~. Here,
~ (x, t) = As (x). If w C A k ( M x R), let
~ = ~ l ( W ~ x R) and write
aJ~ = E f~ dxI A dt + E g~ dxJ' I J
8.3. POINCARE LEMMA
259
where we use the conventions I=
i1,i2,...
J=jl,j2,...
,/kl,
l H . + I (C*, (5)
H*(L*,5)
, H*+I(J*,5) 5*
also commutes.
266
8. I N T E G R A T I O N
AND
COHOMOLOGY
In the case of a short exact sequence of chain complexes, the connecting homom o r p h i s m has degree  1 . We show how to find 6*[e] E Hk+I(C*,5), where [e] E Hk(E*,5). Consider the c o m m u t a t i v e diagram 0
Ck
~
Dk
J
Ek
ck+l
i
)
Dk+l
j
)
Ek+l
) 0
ck+2
i
)
Dk+2
j
)
Ek+2
) 0
and choose e E E k representing [el. In particular, 6(e) = 0. Since j is surjective, choose e' E D k such t h a t j(e') = e. T h e n j(5(e')) = 6(j(e')) = 6(e) = 0 and exactness of the middle row implies t h a t there is a unique c E C k+l such t h a t i(c) = 6(e'). T h e n i(5(e)) = 5(i(e)) = 6(5(e')) = 0. Since i is onetoone, it follows t h a t 6(c) = 0, so we define 5* [el = [c] E H k+l(C*, 5). More d i a g r a m chasing proves t h a t [c] is i n d e p e n d e n t of the choices of e E [e] and of e' E D k such t h a t j(e') = e. E x e r c i s e 8.4.7. P r o v e t h a t the connecting h o m o m o r p h i s m 5* is natural as defined in t h e s t a t e m e n t of L e m m a 8.4.6. E x e r c i s e 8.4.8. Prove t h a t a short exact sequence
0 ~ (c*, 5) L (D*, 5) ~ (E*, 5) ~ 0 of cochain complexes induces a long exact sequence
... ~ Hk(C.,5 ) i~ Hk(D.,5) i_~ Hk(E.,6) ~
Hk+I(C.,5) ~ . . .
in cohomology. O n e s o m e t i m e s writes the long exact sequence more c o m p a c t l y as an exact triangle: H* (C*, 6)
i*
, H* (D*, 5)
H*(E*,5) E x e r c i s e 8.4.9. Let R be a field. If
ALBJ~C is an exact sequence of vector spaces over R, prove t h a t the dual sequence A* ~ i*
B* ~ J*
C*,
where i* and j* are the respective adjoints, is also exact. F i n d an e x a m p l e showing t h a t this m a y fail for modules over a c o m m u t a t i v e ring.
8.5. MAYERVIETORIS SEQUENCES
267
8.5. M a y e r  V i e t o r i s Sequences Let U1 and U2 be open subsets of the nmanifold M and consider the inclusions
Jl
U1N U2 ~+U1,
j:
U l n U: ~ U~,
and
il : UI,~~ UI U U2 i2 : U2 ~~ UI U U2 . L e m m a 8.5.1. The above inclusions give rise to a short exact sequence
0 ~ (A*(U1 u U2), d) ~ (A*(U1)  A*(U2), d  d) ~ (A* (U1 vi U2), d) * 0 of cochain complexes, where i ( w ) = (i~(w),i~(w)),
V w e A*(Vl UU2),
and j(wi,w2) = Y ~ ( w l )  i i ( w 2 ) ,
Vwe ~ A*(Ue), e = 1,2.
Proof. Indeed, a nontrivial form on U1 U U2 must be nontrivial on either U1 or U2, so i is onetoone. Since j~" o i~ = j~ o i~, it is clear that ira(i) c_ ker(j). For the reverse inclusion, let (Wl,W2) E ker(j). Then Wl](U1 n U2) = w2](U1 N U2), so these forms fit together smoothly to define a form w on U~ U U2 and (~1,w2) = i(~). Finally we must prove that j is surjective. Let co be a form on U1 N U2. Let {A1, A2} be a partition of unity on U~ U U2 subordinate to {U1, U2} and set ~1 = A2~, ~2 = AlW. (Note that, since A2 is supported in /]2, A2~ extends smoothly by 0 to all of U1. Similarly, AlW is a form on/72.) Then, j(a~l, w2) = a;l + w2 = oz. [] T h e o r e m 8.5.2. There is a long exact sequence
... d~ H q ( u ~ U U~) ~
Hq(u~) e H q ( U 2 ) AL ~ " ( U i n u : ) ~L H , + I ( u 1 u U:) ~A~ ...
called the MayerVietoris sequence. Indeed, the cohomology of the cochain complex (A*(U1) A*(U2),d  d) is clearly H*(U1) 9 H*(U2), so we apply Proposition 8.4.8. We turn to the Mayer Vietoris sequence for compactly supported cohomology. Again, U1 and U2 are open subsets of some nmanifold M. Clearly, there are inclusions ae : A*~(U1 N U2) ~ A*~(Ue), g = 1,2, and ~e : Ac(Ue) ~ Ac(U1 U U2), g = 1, 2. It is evident that these inclusions commute with exterior differentiation, hence induce linear maps a~, t3~ in compact cohomology, g = 1, 2. L e m m a 8.5.3. The above inclusions induce a short exact sequence
0 ~ (A*~(U~ n U2),d) ~ (A~(U~) A*~(U2), d  d) A (A~(U1 u U2),d) ~ 0 of cochain complexes, where ~(~) = ( ~ ( . ~ ) ,  ~ 2 ( ~ ) ) ,
w
~ A;(UI n U~),
and 9(Wl,W2) = g x ( w i ) + Z 2 ( w 2 ) ,
Wgc
A*~(Ue), g = 1,2.
268
8. INTEGRATION AND COHOMOLOGY
Pro@ Everything is clear except, perhaps, the fact that fi is a surjection. If co is a compactly supported form on U1 U U2 and {A1,A2} is a partition of unity on U1 UU2 subordinate to {UI, U2}, then AlaJ has compact support in U1 and A2w has compact support in U2 (note the switch from the proof of Lemma 8.5.1). Then, /3(Alco, A2co) = w and/3 is surjective. [] Remark. We have chosen the signs differently than in Lemma 8.5.1. This is not necessary for our present needs, but will be useful in our treatment of Poincar~ duality. Theorem
8.5.4. There is a long ezact sequence
d* Hkc(U1 fl W2) ~
Mck(U1) @ Hck(W2)
o/3"Hkc(U1U U2)
d ~ IILrk+l/rT c ktJl n U2) cz*' ""
called the M a y e ~ Vietoris sequence for compactly supported cohomology. D e f i n i t i o n 8.5.5. Let M be an nmanifold without boundary. An open cover {U~}~e~ of M is said to be simple if it is locally finite and every nonempty, finite intersection
U=U~onU~, n...nv~, is contractible and has H~ (U) = H~ (R~). By Theorem 8.2.19 and Theorem 8.3.8, simple covers have the following property. L e m m a 8.5.6. If ~[ is a simple cover of M and U is any nonempty, finite intersection of elements of ~[, then
H*(U) = H*(Nn), H , ( U ) = H,(Nn). T h e o r e m 8.5.7. If M is a manifold with OM = O, then every open cover of M admits a simple refinement. We will postpone the proof of this theorem to Section 10.5, since it requires methods from Riemannian geometry. The idea is to produce a locally finite refinement by geodesically convex open sets and to prove that a geodesically convex open set U has the property in Definition 8.5.5. Since finite, nonempty intersections of geodesically convex sets are geodesically convex, we obtain a simple refinement. Using Theorem 8.5.7 and MayerVietoris sequences, we obtain the following interesting result. T h e o r e m 8.5.8. If M admits a finite simple cover, then H * ( M ) and H * ( M ) are finite dimensional. In particular, if M is a compact manifold without boundary, then H * ( M ) is finite dimensional.
Proof. Select a finite simple cover {Ui}i~=l of M. We proceed by induction on r. If r = 1, then M = U1 has the ordinary and compact cohomology of R ~. Thus, H* (U 1) H* (singleton), hence is finite dimensional. For H2 (U 1), the assertion is given by Corollary 8.3.17. Suppose, then, that it has been shown that H~ (N) and H* (N) are finite dimensional whenever N has a simple cover by r  1 elements, r r1 ui and remark some r > 2. Let M have the simple cover { U i}i=l, let U = Ui=l t h a t {U1 N (Jr, U2 n U~,... , U~I n U~} is a simple cover of U N U~.. We consider the =
8.5. MAYERVIETORIS SEQUENCES
269
compactly supported case. By the inductive hypothesis, Hc(U ) and H*(U n U,.) are b o t h finite dimensional as, of course, is H i (U~). Since M = U U U,, the M a y e r Vietoris sequence gives an exact sequence
H*(U)  H2(U~ ) ~
H2(M ) d~+ H . + I ( u n [fir).
By s t a n d a r d linear algebra,
H2(M ) ~ ker(d*)  im(d*) = im(/3*)  ira(d*). Since/3* has finite dimensional domain and d* has finite dimensional range, the assertion for H i (M) follows. The proof for ordinary cohomology uses the appropriate MayerVietoris sequence in the same way. []
Remark. Even if the compact manifold M has boundary, it is true that H* (M) is finite dimensional. One way to prove this is to show that int(M) has a finite simple cover and that M and int(M) are homotopically equivalent. E x e r c i s e 8.5.9. If the manifold M is connected, but not necessarily compact, prove t h a t the real vector spaces H$ (M) and H . (M) have dimension at most countably infinite. You may use the de Rham theorem. E x e r c i s e 8.5.10. Prove that
H~(S~) =
R, 0,
k = 0, n, otherwise,
for all n _> 1. There is also a MayerVietoris sequence for singular homology. The proof is similar to those for cohomology except for one technical point, the proof of which is very tedious and would take us too far afield. Since we will need this sequence for the proof of the de R h a m theorem, we derive it here, referring the reader to standard references in algebraic topology for the bothersome technicality. The inclusions j,
Ul n U2 ~ U1,
j2
Ul n U2 ~ U2
il
Ul ~ U~ u U2,
and
i2 U2 ~,Ul U U2 induce an exact sequence
(c.(ul) e (c.(u2), o e o) 2~ i(c. (Ul u u2), o), where j(c) = ( j l # ( c ) ,  j 2 # ( c ) ) and i(cl,c2) = il#(Cl) + i2#(e2) (and the induced homomorphisms i1#, jl#, etc., are as in Exercise 8.2.24. The exactness is imme(S.4)
0 + ( C , ( U 1 I"l U2) , 0) 5_+
diate. If (*) were a short exact sequence, the MayerVietoris sequence for singular homology would follow immediately, but it is generally false that i is a surjection. This brings us to the technical point. D e f i n i t i o n 8.5.11. Let Ii = {Uo}~c~ be an open cover of tile manifold M. A singular psimplex s : a p ~ M is said to be llsmall if, for some a C 91, s(Ap) C_ U~. Tile set of llsmall singular psimplices is denoted by A ~ ( M ) . The vector subspace of Cp(M) spanned by A ~ ( M ) is denoted by C~(M).
270
S. INTEGRATION AND COHOMOLOGY
By the definition of the singular boundary operator, it is immediate t h a t
O: CUp(M) ~ C~_I(M ). D e f i n i t i o n 8.5.12. The chain complex (C,~(M),O) is called the complex of ~ small chains. The homology of this complex is H~,(M), the llsmall homology of M. It is clear that the natural inclusion of the space of ~lsmall chains into the space of all chains is a homomorphism of chain complexes
zu : (C~, (M),O) ~ (C,(M),O), so there is induced a canonical homomorphism in homology Z.u : H~,(M) __~H,(M).
We arrive at the technical result. P r o p o s i t i o n 8.5.13. The homomou)hism zu is a canonical isomorphism
Hr.~(M) = H, (M). Proofs of Proposition 8.5.13 will be found in the standard references in algebraic topology, such as [la, pp. 8588] and [46, pp. 207208]. The idea is to subdivide the singular simplices in each cycle z until all simplices in the subdivision are llsmall. W i t h appropriate choices of signs, there results a Rsmall cycle z' with [z'] = [z]. Thus, homology can be computed using llsmall chains, for any open cover II of M. In our situation, II = {U1, [72} is an open cover of the manifold U1 U Uu and we replace the sequence (*) with the short exact sequence
0 e (C'. (U1 n U2), oq) & (C. (Ul) (~) (6'. (U2), O @ O) ~e (C.II (U1 u U2), O) ~ O. 8.5.14. There is a long exact sequence
Theorem
... o_:,HAUl n []2) At, H,(U1) r Hp(U2) ~ HAUl VU2) 2 V HpI(U1 nU~) 2 : + . called the Mayer Vietoris homology sequence. Using the result of Exercise 8.4.9, we obtain dual MayerVietoris sequences. 8.5.15. The Mayer Vietoris sequences dualize to exact sequences
Theorem
i' "'" + H q ( U l ) * ([]3Hq(U2) * ~
9 '
H q ( U l n g2)* d.~ H q + l ( u 1 U U2)* ~ "
Hg(U1)* eH:(U2)* ~A+gg(g, uGh)* ~ H ~+1~" riG2)*
"
c
j,
\~'1
j'
Hp(UI)* r Hp(U2)* ~+ Hp(Ux UU2)* a~+ HpI(U1 rid2)* +"
where i' is the adjoint of i* (respectively, of i,), etc.
8.6. C O M P U T A T I O N S
OF COHOMOLOGY
271
8.6. Computations of Cohomology In this section, we compute the top dimensional cohomology of connected m a n ifolds. 8.6.1. If the open subsets U1, U:, and U1 A U2 of an nmanifold M all have the same compact cohomology as IRn and are coherently oriented, then
Lemma
s
H:(U1o U2) +R 1UU2
is an isomorphism. Proof. Consider the diagram H~(U1 N U2)
oz
*
)
,fUlnU2I IR
Hn(u1)  H~(U2)
~* ~ Hn(u1 U U2)
ful ~,fu2 I A
~
IR 
IR
5
d*
)
0
,fu1uu2~
id l
IR
~ 0
>
where A(t) = ( t ,  t ) and 5(s,t) = s + t, Vs, t C JR. C o m m u t a t i v i t y of the diagram is obvious (coherency of orientations is essential), exactness of the b o t t o m row is obvious, a n d the top row is exact by Theorem 8.5.4. The m a p fu~nu~ is a n isomorphism by the hypothesis t h a t H'~(U1 A U2) = R (as in Corollary 8.3.18). Similarly, fu~ Q fu2 is an isomorphism. Since the diagram can be extended harmlessly by a c o m m u t a t i v e square of 0s on the right, it follows from the Five L e m m a that feluu~ is a n isomorphism. [] 8.6.2. Let {U~}sc~ be a simple open cover of a connected, oriented nmanifold M without boundary, each Us being oriented coherently with the orientation of ll~[. Let w s , w z C A n ( M ) have respective supports in Us and Ufl, a, fl C 92. Then [w~] = [wfi] if and only if fM ~ = fM ~Z"
Lemma
Proof. If [cz~] : [czZ], t h e n we know that fM a& : IM aJZ. For the converse, assume equality of the integrals. Since M is connected, we can find a sequence of indices a = a 0 , a l , . . . , a ~ . = fl in 92 sucll that U~{_ 1 rhU~{ ~ ~, 1 < i < r. Choose a~, E A ~ ( M ) such t h a t supp(c~s{) C U~{, 1 < i < r, and such that aJ~o = a&, eva,. : :vfl, and
This is clearly possible. By L e m m a 8.6.1,
v
H n ( G '  ~ U G~) ~ N
is an i s o m o r p h i s m , 1 < i < r, so
[(A). . . . ] = [(A)Si] ~ Hn(U C \ O~i1 U Usi ) ,
T h a t is, there is a form t] E A nc  l f \U s i  1 U Ua~) such that ws, = ws,_~ + &]. These forms all live in A~(M), so [w~,] = [w. . . . ] E H 2 ( M ) , 1 < i < r. In particular, [~sl = [czZ] as desired. []
272
8. I N T E G R A T I O N
AND
COHOMOLOGY
L e m m a 8.6.3. If {U~}~e~ is a simple cover of the connected, oriented nmanifold M with OM = O, then, for each a o r 92, the natural inclusion e : A'~(U~o) ~ A'~(M)
induces an isomorphism e. : H2(U~o ) ~ H : ( M ) . Proof. Indeed, since fUoo : Hn(u~o) ~ R is an isomorphism, [co] 9 H'~(U~o) is nontrivial if and only if fM co = fu,, o co # O. Since fM vanishes on B 2 ( M ) , it follows t h a t e.[co] 7~ 0, so e. is injective. We prove surjectivity. Let co 9 A'~(M) and use a partition of unity {/~}~e~, subordinate to the simple cover, to write
aEg.l
where )~,co 9
A2(U~,), 1
1, the Gl(n)cocycles do not form a group of any kind because of the n o n c o m m u t a t i v i t y of Gl(n). 8.9.7. If each Us E Ii is connected, the space I:I~ is canonically isomorphic to the space of locally constant, real valued functions on M. In particular, [t0(11) = HO( M). Lemma
Proof. R e m a r k that/2/0(11) = ~0(11) and t h a t this is the space of 0cochains 0 such t h a t 0s = 0 3 whenever Us N U3 7~ 0. T h i n k i n g of 0s as a constant function on Us, g ~ E 91, we see t h a t these constant functions agree on overlaps of their domains, hence unite to form a coherent locally constant function 0 on M. Conversely, if 0 : M + I~ is a locally constant function, its restriction 0~ = OIUs is constant by the connectivity of Us, V ~ E ~. [] R e m a r k t h a t simple covers 11 satisfy L e m m a 8.9.7. 8.9.8. If 11 is a simple cover, there is a canonical linear isomorphism l~ 1(11) = H a(M).
Lemma
Remark. It is an i m m e d i a t e consequence of Lemmas 8.9.7 and 8.9.8 t h a t [tP(11) does not d e p e n d on the choice of simple cover, hence this vector space can be denoted b y / ~ / ~ ( M ) , p = 0, 1. For the c ~ e p = 0, the purely topological condition on 11 in L e m m a 8.9.7 proves that HO(M) and /:/~ are topological invariants. However, the definition of a simple cover requires a difl>rentiable structure, so we c a n n o t conclude from L e n m m 8.9.8 that H i ( M ) and t715(M) are topological invariants of M. T h e proper definition of Cech cohomology involves passing to an algebraic limit over the directed set of all open covers of M, thus o b t a i n i n g a true topological invariant. In Section 10.5, we will show that every open cover has a simple refinement and, in A p p e n d i x D, use this fact to prove Theorem 8.9.6. We sketch the construction of the isomorphism in L e m m a 8.9.8 a n d leave verification of several details to the exercises. Fix the choice of simple cover We define a linear m a p ~ : S 1 ( M ) + Iv/1 (11).
Given [co] E H I ( M ) , select a representative w E [co]. By simplicity of the cover, H i ( u s ) = 0, so the restriction cos = colU~ of the closed 1form co is exact, V(~ C 9.1. Thus, we can choose f s E A~ such that co~ = dfs, Vc~ E 91. On Us0 NU~x # (~, d(f~ o  f ~ ) = w  co  0, so f~o  f ~ is locally constant on Uso A Us,. The cover
286
8. INTEGRATION AND COHOMOLOGY
being simple, this set is connected, so fao  f~l = cao~l E IR is a constant. This defines a Cecil 1cochain c E ~,1(ll). But (5c)~o~,~
2 = c~m:

Oil UO:0 [~ Uoq N U0r we call set
c,~o~ 2 + c . . . .
SO C 9 21(r
=
(f~

f,~2) 
(f,~o 
f~2)
+ (f~o

f~l)
 0
If [c] 9 /;/1(1/) depends only on [co] 9 H I ( M ) ,
~([co]) = [ 4 E x e r c i s e 8.9.9. Prove t h a t the class [c] defined above is independent of the choice of representative co 9 [co] and of the choices of f~ 9 A~ such t h a t df~ = cox as an infinitesimal curve, where s : (  e , e) + M is smooth and s(O) = z. Then,
(D29)~
Dx,.Y
=
~(P~(t) d )~=0
=
d(y,(,)) ~=0
and this depends only on Y[(U N M), not on the choice of extension !/. Therefore, D x Y is a welldefined element of F(T(Rm)IM).
306
10. R I E M A N N I A N
GEOMETRY
D e f i n i t i o n 10.1.2. If X, Y C 32(M), t h e n the operator V : X(M) • X(M) ~ X ( M ) , defined by
VxY = p(DxY) is called the LeviCivita connection on M C IRm. T h e following is totally elementary, as the reader can check. Lemma
10.1.3.
The LeviCivita connection is a connection on M.
If V is a connection on M and (U, x l , . . . ,x ~) is a coordinate chart, set ~i =
O/Ox i a n d write
v~j = ~ r~& k=l
D e f i n i t i o n 10.1.4. The functions F ~ E C~176 are called the Christoffel symbols of V in the given local coordinates. D e f i n i t i o n 10.1.5. Let V be any connection on a manifold M. T h e torsion of V is the IRbilinear m a p T : ~ ( M ) x ~ ( M ) + ~ ( M ) defined by the formula
T ( X , Y) = V x Y  V z X  [X, Y]. If T _= 0, t h e n V is said to be torsion free or symmetric. E x e r c i s e 10.1.6. Prove t h a t the torsion T of a connection V on M is Coo(M)bilinear. T h a t is, T C 'J~I(M) and T ( v , w ) E Tx(M) is defined, Vv, w C T z ( M ) , Vx E M . Torsion is a tensor. E x e r c i s e 10.1.7. Prove that the connection V is torsion free if a n d only if, in every local coordinate chart (U, x l , . . . , xn), the Christoffel symbols have the s y m m e t r y k k Fij = Fji ,
1 < i , j , k < n.
This is the reason that torsion free connections are also said to be symmetric. E x e r c i s e 10.1.8. If M C IRm is a smoothly imbedded submanifold, prove t h a t the LeviCivita connection is symmetric. We use connections to define a way of differentiating vector fields along a curve. Indeed, if X E 2E(M) a n d s : [a, b] * M is a s m o o t h curve, then, at each point s(t), one can c o m p u t e
X's(t) = V~(t)X e Ts(t)(M). B u t we will also be interested in differentiating vector fields along s t h a t are only defined along s. In fact, it is often n a t u r a l to consider fields Xs(t) along s t h a t are also p a r a m e t r i z e d by the p a r a m e t e r t, allowing Xs(tl) ~ Xs(t2) even if s (t~) = s (t2), t l r t2. For instance, Xs(t) = i(t), the velocity field, may exhibit such behavior. In these cases, it is not immediately clear how to use a connection to produce the desired derivative. D e f i n i t i o n 10.1.9. Let s : [a,b] ~ M be smooth. s m o o t h m a p v : [a, b] ~ T ( M ) such t h a t the diagram
A vector field along s is a
10.I. C O N N E C T I O N S
307
T(M)
[a, b]
s
9M
commutes. The set of vector fields along s is denoted by Y(s). We have already seen two examples, namely, the restriction (VIs)(t) = Ys(t) to s of a vector field Y E X(M) and the velocity field i(t). Via pointwise operations, it is evident that Y(s) is a real vector space and, indeed, a C~[a, b]module. D e f i n i t i o n 10.1.10. Let V be a connection on M. An associated covariant derivative is an operator V 
dt
: X(s)
~ X(s),
defined for every smooth curve s on M, and having the following properties:
1. V/dt is Nlinear; 2, (V/dt)(fv) = (df/dt)v + f V v / d t , V f E C~[a,b], Vv E X(s); 3. if Y E ~ ( M ) , then
V (yls)(t) = V~(t)(Y) E Ts(t)(M), a < t < b. Remark. By property (3), V/dt is associated only to the one connection V. T h e o r e m 10.1.11. To each connection V on AJ, there is associated a unique co
variant derivative V/dt. Proof. We prove uniqueness first. For this, it is enough to work in an arbitrary coordinate chart (U, x l , . . . ,x~). Let F~j be the Christoffel symbols for V, set ~i = O/Ox i, consider a smooth curve s : [a, b] ~ U, and let v c ~(s). Write
v(t) = ~ v~(t)~ ~(,), i=1
~(t) = ~ ~J (t)~j ~(~). j=l
Then any associated covariant derivative must satisfy
Vv

=
dt
~
k,/dv~
~
+ v
i V~i J
i=1 n
k
dv = ~ j~k + ~~.v~V~ k=l
i=1
dv
=
k=t


viuJF~i~k
+
i,j,k=l
n
i,j=l
)
308
10. R I E M A N N I A N
GEOMETRY
evaluated along s(t). This is an explicit local formula in terms of the connection, proving uniqueness. We turn to existence. In any coordinate chart (U, x l , . . . ,zn), use the above formula to define V / d t for curves lying in the chart. The reader can easily check t h a t the three properties in the definition of covariant derivative are satisfied, where M = U. Thus, in U, the connection VI U has an associated covariant derivative and, by the preceding paragraph, this covariant derivative is unique. Consequently, on overlaps U N V of charts, the two sets of Christoffel symbols must define the same covariant derivative for VIU Cl V. (Classical geometers and physicists defined connections and covariant derivatives by Christoffel symbols and they checked this invariance via explicit change of coordinate formulas.) Thus, along any smooth curve s : [a, b] + M, these local definitions of V / d t can be pieced together to give a global definition. [] In particular, for the Euclidean connection D in ]R"~, all Fi~  0 and we get the expected formula: m
i=1

Dv dt 
m
= ~  dv~
i
i ,(t).
For M C N "~, the covariant derivative associated to the LeviCivita connection U on M is dt
P
the orthogonal projection into T ( M ) of the usual Euclidean covariant derivative. Convention. From now on, we adopt tile "summation convention" of Einstein. According to this convention, the summation symbol is omitted and it is understood that any expression is summed over all repeating indices. For example, at +
=~=1 \ dt + ~
i,j=l
It is necessary that the indices repeat for terms in a product, not just in a sum, and it is customary t h a t the repeated index occur once as a superscript and once as a subscript (a custom we will usually honor). Thus, dvk + viuJP~i  dv k ~, 9 dt  dt+ ~ v*uaP~i' i,j=l
the index k being repeated only in terms separated by +. D e f i n i t i o n 10.1.12. Let M be a manifold with a connection V. Let v E 2E(s) for a smooth p a t h s : [a, b] ~ M. If V v / d t ~ 0 on s, then v is said to be parallel along s (relative to the given connection). T h e o r e m 10.1.13. Let V be a connection on M , s : [a,b] * M a smooth path, c C [a,b], and vo E T~(~)(M). Then there is a unique parallelfield v C ~ ( s ) such that v(e) = vo. This field is called the parallel transport of vo along s.
10.1. C O N N E C T I O N S
309
Pro@ In local coordinates, write
~(t) = ~J(t)~ 5 s(~), ~(t) = ~ ( t ) ~ s ( ~ ) , VO = ai~is(c).
Here, as promised, we are using the summation convention. The condition that v be parallel along s becomes the equation
~Z(t). + v~(t)uJ ( t ) r ~ ( s ( t )
o = / dv k
)~
or, equivalently, the linear O.D.E. system (10.1)
dv k dt 
viuJF~i, 1 < k < n,
with initial conditions
vk(c) = a k, l < k < n. By the existence and uniqueness of solutions of O.D.E., there is e > 0 such that the solutions vk(t) exist and are unique for c  e < t < c+e. In fact, these equations being linear in the vks, it is standard in O.D.E. theory (Appendix C, Theorem C.4.1) that there is no restriction on e, so the unique solutions vk(t) are defined on all of [a, b], l 0. Let
hij : (xi, xj>, fij = (Z,, Zj>. Then, [h~j] = [Tki]T[fke][Tej] and it follows that = (act 7 ) ~ / 7 . Also, (det7')~ 1 A . . . At/n = a~1 A . . . AaJ n, where 7 ~ = (7 T )  I . P u t t i n g this information together, we obtain v / h ~ 1 A . . . AcJ ~ = (det 7)X/f(det 7') r/I A . . . Arl n = v / f r l I A .  . Ar] n. Thus, the locally defined volume forms fit together coherently to define f~ as desired. [] D e f i n i t i o n 10.2.7. Let (U, z 1, ... , z ~) be a coordinate chart with coordinate fields
~i = O/Oz i. Then the functions 9ij = ([i,~j>, 1 0 for which U ~ y and ~(U, y) < ~ ~
d'ff(y) dr2
A(g(U)) A(U) < e.
E x e r c i s e 10.a.7. For each y E M, prove that this derivative exists and that d~(y) _ ~(y). df~ E x e r c i s e 10.3.8. Let M C ]Ra be the graph of the equation z = x 2  y2. Prove that ~(x, y, z) < 0, g (x, y, z) E M. In the above discussion, the normal field g played a central role. The curvature of M was seen as a measure of how much this field "spreads" infinitesimally at a point. Thus, curvature appears to be an extrinsic property of the surface. But Gauss proved a remarkable theorem (he called it his "Theorema Egregium") that showed the Gauss curvature to be intrinsic. A twodimensional inhabitant of the surface can take measurements leading to the computation of curvature. We turn to this theorem. Let X, 17, Z E 5~(U) and extend these to fields X, Y, Z E 2E(U), where U C IRa is an open set such that U n M = U. L e m m a 10.3.9. For fields chosen as above,
D ~ Y = V x Y  (L(X), Y} r~ along U. Proof. Along U, D 2 Y depends only on X and Y. As in the proof of Lemma 10.3.4, (DxZ, ~) =  ( n ( x ) , Y) , so the component of D x Y perpendicular to M is  (L(X), Y) g. By the definition of the LeviCivita connection, V x Y is the component of D x Y tangent to M. []
320
10. R I E M A N N I A N G E O M E T R Y
The Euclidean connection D satisfies a simple commutator relation:
[D~, Dr] = DE~,~1 This is because D ~ and D~p operate on a vector field Z by applying .~ and respectively to the individual components of Z. It turns out that, on M, curvature is an obstruction to this commutator relation for V. D e f i n l t i o n 10.3.10. The curvature operator n ( x , v ) : ~(M) ~
X(M)
is defined, for arbitrary X, Y E ~(M), by R(X,Y)Z = VxVyZ  VyVxZ
 V[x,y]Z,
v z E X(M). The fact that this operator is related to curvature is far from obvious. It is the content of the Theorema Egregium. By Lemma 10.3.9, at every point of U we have D ~ ( D ~ 2 ) = D ~ ( V y Z  (L(Y), Z) ~) = D x ( V y Z )  (L(Y), Z) L ( X )  X (L(Y), Z) = Vx(VyZ)
 (L(Y), Z) L ( X )  X (L(Y), Z)
 (L(X),VyZ)~.
Similarly, at every point of U,  D~(D~2) = Vy(VxZ)
+ (L(X),Z) L ( Y ) + Y (L(X), Z) ~ + (L(Y), V x Z ) ~.
Finally, at every point of U,  D [ ~ , 9 1 2 :  V [ x , y ] Z + 0 is sufficiently small, then exp~ : Bx(ex) ~ M,
called the exponential map at x, is a diffeomorphism onto an open neighborhood of x inM. Proof. Write ~ = exp~ : B~(e) ~ M. Clearly, V(0x) = x. We c o m p u t e ~.o~ : To~(B~(e)) : T~(M) ~ Tx(M). For 0 # v 9 To:r(B~(e)) = Tx(M), set s(t) = tv, e/llvlt < t < e/llvll , a s m o o t h curve on B~(e) w i t h s(0) = 0~, i(0) = v. T h e n r / o s defines a curve on M , r/(s(O)) = r/(0z) = x and r/(s(t)) = expx(tv ) = vv(t). Thus,
~7,0~(~(0)) = , , 0 ~ ( v ) = %(0) = v T h a t is, r/.0x is the identity under the identification To~ (B~(e)) = T~(M). By the inverse function theorem, 7/ will be a diffeomorphism for e = e~ > 0 sufficiently small. [] R e m a r k t h a t , if the manifold is compact, the n u m b e r e = e~ > 0 can be chosen to be i n d e p e n d e n t of x. D e f i n i t i o n 1 0 . 4 . 1 1 . T h e R i e m a n n i a n manifold M is geodesically complete if exPx(V ) is defined for all x 9 M and for all v 9 Tx(M). Equivalently, every geodesic segment extends (uniquely) to a geodesic 7(t),  o o < t < oo. D e f i n i t i o n 1 0 . 4 . 1 2 . A p a t h a : [a, b] ~ M is regular if it is a s m o o t h immersion. If there is a partition a = to < tl < ... < tT = b such t h a t al[ti_l,ti I is regular, 1 < t < r, then a is piecewise regular. N o n d e g e n e r a t e geodesics cr are regular. Indeed, if d~(t0) = 0, for some to, then t h e fact t h a t a is evenly p a r a m e t r i z e d implies t h a t #  0 and cr degenerates to a c o n s t a n t path.
326
10. RIEMANNIAN GEOMETRY
D e f i n i t i o n 10.4.13. Let M be a connected Riemannian manifold. Then the Riem a n n (or Riemannian) distance function p : M x M + [0, oc) is defined by
p(x,y) = inf M, where a ranges over all piecewise regular paths from x to y in M. Fixing the hypothesis that M is a connected, Riemannian nmanifold with OM = 0, we are going to prove the following results. P r o p o s i t i o n 10.4.14. The Riemann distance function is a topological metric on the Riemannian manifold M and the metric space topology is the same as the manifold topology. T h e o r e m 10.4.15 (Hopf Rinow I). I f the Riemannian manifold M is geodesicalIy complete, x , y C M , then there is a geodesic "7 from x to y such that [7[ = p(x, y). In particular, expx : Tx (M) ~ M is surjective. T h e o r e m 10.4.16 (HopfRinow II). The Riemannian manifold M is a complete metric space in the metric p if and only it is geodesieally complete. C o r o l l a r y 10.4.17. If M is compact, then M is geodesically complete in any Riemann metric. Because of these results, it is standard to use the term "complete Riemannian manifold" when either geodesic completeness or metric completeness is intended. A number of preliminary considerations are necessary for the proofs of Proposition 10.4.14 and the HopfRinow theorems. To begin with, remark that the distance function p has the following properties:
1. p ( x , x ) = O, V x c M; 2. p(x,~) = p(y,x), w , y 9 M; 3. p(x,y) < p(x,z) + p(z,y), Vx, y,~ 9 M. Thus, to prove that p is a topological metric, it is only necessary to prove that
p(x,y)
=
0 ~
x = y.
Let xo 9 M C T ( M ) , choose an open neighborhood U C_ M of x0, and let e > O. Then W = {v~ 9 T~(M) I x 9 U and IIv~JI < e} is an open neighborhood of x0 = 0~0 in T ( M ) . If U and ~ are chosen small enough,
c(v) = (~(~), exp~(v)(v)) is defined and smooth as a function O : W  ~ M x M. We can take U to be a coordinate neighborhood, with coordinates x l , . . . ,x n. We can also assume that there is a smooth, orthonormal frame field Z 1 , . . . , Zn defined on U. We obtain, thereby, coordinates
(xl,... ,xn,yl,..
,y~) ~, y~Z~(~:~...... ,,)
on 7rl(U) such that n
W:
{ ( x l , . . . , x n , y l , . . . , y n)
(xl,...
,X n) C U, E ( y i ) 2 < i~1
= U x B(Q.
e 2}
10.4. C O M P L E T E
RIEMANNIAN
MANIFOLDS
327
L e m m a 10.4.18. Given xo C M , the neighborhood W of xo in T ( M ) can be chosen, as above, so small that G : W 4 M x M carries W diffeomorphieally onto an open neighborhood of (xo,xo) in M x M .
Proof. Indeed, let (i represent the basic coordinate fields for x i and (j those for yJ, 1 < i , j >
II~(t)ll dt dt i b dr dt
= It(b)  r(a)t. If equality holds, n o t only is v(t) = v constant, b u t dr~dE c a n n o t change sign and, since r(t)v = 5(t) is piecewise regular, dr/dt can never be O. T h a t is, r(t) is strictly m o n o t o n i c a n d piecewise regular. []
330
10, R I E M A N N I A N G E O M E T R Y
1 0 . 4 . 2 3 . For V and r > 0 as above, let V : [0, 1] + M be the unique geodesic of length < e joining two points x, y E V. Let a : [0, 1] +M be an arbitrary piecewise regular path joining the same two points. Then IV] r  5 . L e t t i n g 5 ~ 0, we conclude t h a t lal _> r = 171. I f l a l = r, t h e n the segment from each $5 to S, must be a reparametrization of (the same) radial geodesic. Since a is piecewise regular, the reparametrization r(t) is continuous and § has only j m n p discontinuities, occurring only finitely often as 5 I 0, so a itself is a piecewise regular reparametrization of V. Conversely, if a is a piecewise regular r e p a r a m e t r i z a t i o n of V, then 1~7]= IV]. [] C o r o l l a r y 1 0 . 4 . 2 4 . Let a : [a,b] + M be piecewise regular and have minimal
length for any piecewise regular path from a(a) to a(b). Then a is obtained from a geodesic by piecewise regular reparametrization. If a is regular, it is a regular reparametrization of a geodesic. If ]]~]] is constant, a is a geodesic. Proof. Consider any segment of a lying in a n open set V as above and having length < r By the above, this segment must be a piecewise regular r e p a r a m e t r i z a t i o n of a geodesic. Since every interior point of a lies in the interior of such a segment a n d a is m a d e up of finitely m a n y such segments, a must be a piecewise regular r e p a r a m e t r i z a t i o n v ( r ( t ) ) of a geodesic V. If cr is regular, then
dr
0 # ~(t) = 7{a/(r(t)) a n d this implies t h a t r(t) is a regular change of parameter. Since ]]~(r(t))l] is constant, dr/dr will be constant if IIa(t)II is. In this case, r(t) = ct + e for suitable constants c ~ 0 a n d e, so a(t) = v(ct + e) is a geodesic. [] T h e next two results complete the proof of Proposition 10.4.14. Proposition
1 0 . 4 . 2 5 . If x , y C M and p(x,y) = O, then x = y.
Proof. Suppose x # y. Choose V and e > 0 as usual, b u t such that x E V, y r V. For a suitable choice of r] E (0, e), expx(B,(r/)) C V. Then, every piecewise regular p a t h a from x to y must meet the spherical r/shell centered at x, so ]or] k r/ and p(x, y) >_ rl > o.
[]
Thus, the R i e m a n n distance function is a topological metric on M. Proposition
1 0 . 4 . 2 6 . The topology induced on M by the metric p coincides with
the manifold topology. Proof. Let x C M a n d choose e > 0 so small t h a t exp~ : B~ (e) + M is a diffeomorphism onto an open neighborhood U~ (e) of x in the manifold topology. T h e set of all such U~ (e) is a base for the manifold topology of M. But
(10.6)
v~(~) = {v c M I P(~, v) < ~}.
Indeed, if y E Ux(e), then p(x,y) < e. Furthermore, if z C M \ U~(e), every piecewise regular a from x to z must meet every ~?shell, 0 < ~] < e, centered at x, so Ia] > e and, consequently, p(x, z) >_ e. This proves the assertion (10.6), showing
10.4. C O M P L E T E R I E M A N N I A N M A N I F O L D S
331
F i g u r e 10.4.1. The HopfRinow setup
t h a t {Ux(e) [ x E M, e > 0} is also a base for the topology induced by the metric p. [] We note also the following useful fact. 1 0 . 4 . 2 7 . If C C M is compact, there exists 5 > 0 such that any two points x, y E C with p(x, y) < 5 are joined by a unique geodesic (parametrized on [0, 1]) of length < 5. This geodesic is the shortest piecewise regular path from x to y and depends smoothly on its endpoints. In particular, if M is compact, 5 can be chosen uniformly for all of M . Proposition
Proof. Cover C by open sets V~ with corresponding e~ as in the above discussion. Select a finite subcover { 1/.~ } i r= 1 ' Let 5 > 0 be a Lebesgue n u m b e r for this cover (i.e., i f p ( x , y ) < 5 a n d x , y E C, t h e n x and y lie in a c o m m o n V~). We can also d e m a n d t h a t 5 < minl_ 0 was arbitrarily small, there is a halfopen interval [to, ~'), r / < r, on which Ft holds. The union of all such intervals produces the maximal one [to, rl). (c) Since F~ holds, let [5, r]) be the maximal halfopen interval on which Ft holds. But the truth of Ft on [5, rl) implies Fv, by continuity. Thus, if r / < r, we could apply (b) to obtain a contradiction to the maximality of [5, 7). Consequently, r / = r and Fr holds. []
Proof of theorem 10.3.16. We first assume that M is geodesically complete and we prove that M, as a metric space under p, is complete. For this, it will be enough
10.4. COMPLETE
RIEMANNIAN
MANIFOLDS
333
to prove that, whenever B c_ M is a p  b o u n d e d subset, the closure B is compact. Choose any x C B a n d consider the continuous map exp x : T x ( M ) + M, defined because M is geodesically complete. Since B is bounded, there is a n u m b e r r > s u p y e u p(x,y). If D C T x ( M ) is the closed ball of radius r, t h e n B C_ e x p , ( D ) (Theorem 10.4.15) a n d expx(D) is compact. Thus, B C_ expx(D ) is compact. Next, a s s u m i n g t h a t M is complete in the metric p, we prove t h a t M is geodesically complete. Let x C M and let v C T ~ ( M ) have u n i t norm. Let (a,b) denote the m a x i m a l open interval a b o u t 0 in R such that exp~(tv) is defined, Vt E (a, b). We must show t h a t a =  o o and b = oo. If b < oo, choose {ta:}~=x C (a, b) such t h a t tk T b strictly. This is a Cauchy sequence. Set xk = exp~(tkv) and remark t h a t p(xe,xk) 0 such that Ilvxll = 1 and x = exPxo(txvz ). Then, F : X x [0,1] + X, defined by
F ( z , r) = expx o (rtxvx), is the desired contraction.
[]
It seems that open, star shaped sets U C M are always diffeomorphic to IRn, but this is extremely difficult to prove. The problem is that the set theoretic boundary OU may be very badly behaved. For instance, the "radius function" r : S ~1 zo
,
[0, oc],
even if it takes only finite values, may not be continuous, let alone smooth. This function is defined on the sphere of unit vectors in Tx0 (M) and assigns to v C S xo n1 the supremum r(v) of the numbers r > 0 such that expx o(tv) CU,
O O, flUa=O. Let Z E Y(U \ {x0}) be nowhere 0, tangent to the radial geodesics out of x0, and everywhere pointing toward x0. Let Ft : U ~ U denote the flow, defined for all time t, generated by the compactly supported vector field f Z E Y.(U). Then, since C ".. Upo is contained in the interior of the support of f Z , as is 0U;o , there is a value 7 > 0 such that F~(C) C U;o. Set r = F_~. Since ~L~ is a compactly supported diffeomorphism of U onto itself, isotopic through such diffeomorphisms Ft to F0 = idu, it follows that r : H~(U) ~ H~(U) is the identity. Let aJ E Z~(U) and let C = supp(a@ By the previous paragraph, we obtain r U ~ U such that r E Z~(Upo ) and r = [a~] E H~(U). It follows that [aJ] E im(i.), hence that i. carries H;(Upo ) onto H~(U). Suppose that a~ E ZP(Upo) and that i.[~] = 0. Choose a > 0 as above such that supp(cJ) C U~. Viewing '~ = i.(w) in ZP(U), we find r~ E AP~I(U) such that w = dr/. Let C = supp(r/) and obtain r : U + U, as above, so that
~*(~]) = r/o e ApI(Upo), but dr/o = d r
= ~*(dr/) = r
= ~.
That is, [w] = 0 in H*(Upo), proving that i. is onetoone.
[]
C o r o l l a r y 10.5.8. Every open cover of M admits a simple refinement.
Proof. By Theorem 10.5.4, each open cover admits a refinement by open, geodesically convex sets. With a little care, one chooses this refinement to be locally finite (Exercise 10.5.9). By Lemma 10.5.3, any finite intersection of elements of this refinement is also an open, geodesically convex set, hence star shaped. By Lemma 10.5.5 and Proposition 10.5.7, this refinement is simple. [] E x e r c i s e 10.5.9. Check the assertion in the proof of Corollary 10.5.8 that the refinement by open, geodesicafly convex sets can be chosen to be locally finite.
336
10. R I E M A N N I A N
GEOMETRY
E x e r c i s e 1 0 . 5 . 1 0 . Let x c U C M where U is open in M and star shaped w i t h respect to x. Let r : S~ 1 + [0, oo] be tile radius function for U. T h a t is, $2 1 C T x ( M ) is the unit sphere and U = {exp~(tv) I v 9 $2 1 and 0 _< t < r(v)}. If r is finitevalued of class C ~ , prove t h a t U is diffeomorphic to ]Rn. E x e r c i s e 1 0 . 5 . 1 1 . Let x 9 U C_ M and r : S~ ~ + [0, oo] be as in the preceding exercise, but do not assume t h a t r is s m o o t h or even continuous. Prove t h a t r is lower semicontinuous. T h a t is, r  l ( a , oo] is open in S zn1 , V a E R . E x e r c i s e 1 0 . 5 . 1 2 . Let x 9 U C_ M and r : S~ 1 * [0, oo] be as in t h e preceding exercises. C o n s t r u c t an example in which r is finitevalued everywhere and discontinuous on a dense subset of S Sn   1 . We t u r n to the proof of T h e o r e m 10.5.4. Let x 9 W, as in the s t a t e m e n t of the theorem. As in Section 10.4, choose a neighborhood V of x in W and a n u m b e r e > 0 such t h a t any two points y, z 9 V can be joined by a unique geodesic ay,z in kcr of length < e. As usual, ay,z is p a r a m e t r i z e d on [0, 1] and depends s m o o t h l y on ( y , z ) 9 V • V. Choose 5 > 0 such t h a t the open ball Bx(a) c T x ( M ) of radius a is carried diffeomorphicaliy by expx onto a neighborhood Ux C_ V of x. Let ( v z , . . . ,v~) be an o r t h o n o r m a l frame for T s ( M ) and coordinatize this vector space by
(xl,... ,xD
U n d e r the diffeomorphism exp~T : Us + B~(a), these become coordinates on U~ (called a normal coordinate system on U~). The corresponding coordinate fields are {i 9 :~(Us), 1 < i < n. If y 9 Us has coordinates (bz,... , b~), t h e n 7Z
y)" = Z
i=1
If 0 < 5. < 5, if $5. C U~ is the spherical shell of radius 5., centered at x, if y = ( b l , . . . ,b,~) 9 $5., and i f v = aJ~j 9 r y ( S 5 . ) , then
bia i = O. T h e key l e m m a for the proof of T h e o r e m 10.5.4 is the following. Lemma
10.5.13.
I f & 9 (0,5) is small enough, then every geodesic "y : (  rl , rl) , M ,
such that 7(0) = y C $5. and "9(0) E T y ( S & ) , has the property that p(x, v(t)) > & , for all sufficiently small values of ]tlr o. Proof. As above, denote the normal coordinates of y by ( b l , . . . , bn). Let 5. be so n small that, for ~ i = 1 b~ _< 5., the s y m m e t r i c m a t r i x O = 215ke  biFkg(bl,... i ,bn)] is so close to [25ke] as to be positive definite. Let "y(t) =
(xl(t),...
,xn(t)),
7"] < t < l],
be a geodesic in M , tangent to $5, at 7(0) = y = ( b l , . . . , bn), and let ~(0) = ai{i. Define n
F ( t ) = p(x,'~(t) ) 2  52. = ~~ xi(t) 2 i=i
52,.
10.6. C A R T A N S T R U C T U R E E Q U A T I O N S
337
For small values of Ill, this is a smooth function and V(0) = 0,
r ' ( o ) = 2~(o>~(o) = 2bia i
~0, F"(t) = 2(2i(t)2i(t) + xi(t)~i(t)). Since 7(t) is a geodesic, it satisfies
~i
=
_~k:bgpi he, 1 < i < n,
giving r"(o)
= 2(x~(O) ~  ~ q o > k ( o ) S ( o ) p ~ e ( x ~ ( O ) , . . .
,.'~(o)))
= 2((ai) 2  a~ae(biFike(bl,... , b~))) _ [ a l
...
, a~]Q
,
the value of a positive definite quadratic form on the vector "~(0) r 0. T h a t is, F"(O) > 0. Plugging this d a t a into the 2nd order Taylor expansion of F(t) about t = 0, we see that
F(t) = ~ F ' ( O ) + O(t a) > 0, for small enough values of Itl r 0.
[]
Proof of theorem 10.5.3. Choose N , = exp~(Bx(5.)), where 5, is chosen by the above lemma. Let R C_ N : • N : be the subset of all (y,z) such that (r>z lies entirely in N : . By the smooth dependence of this geodesic on its endpoints, R is an open subset. It is also clear t h a t / g ~ 0. If we prove that R is also a closed subset, then, by the connectivity of N : x N~, R = N: • N: and N~ is geodesically convex. Let {(Yk,Zk)}~=l C R with limkoo(yk,zk) = (Yo, Zo) in Nx x X~. If (yo, zo) r then Cr~o,zo meets ON= = Sa.. If ayo,~o is tangent to the spherical shell at some point of intersection, an application of the lemma shows that ayo,zo contains points in Ux ".. Nx. But smooth dependence on (Yo, zo) implies that this remains true for all values of (y, z) sufficiently near (Y0, z0), hence for (Yk, zk), k sufficiently large. This contradicts the fact that (Yk, zk) C R. But if an intersection point of ayo,zo with the shell is not a point of tangency, it is clear that a:o,yo contains points in U~ \ N , , leading to the same contradiction. Thus, (xo,Yo) C R, proving that R is closed in N : . [] 10.6. T h e C a r t a n S t r u c t u r e E q u a t i o n s We return to the torsion and curvature tensors that were introduced earlier for a connection V. The key to understanding the geometric significance of these tensors is a pair of equations, written in terms of differential forms, called the equations of structure. In tiffs section, we derive these structure equations in local coordinate charts. (In tile next ct~apter, where we treat principal bundles, we will be able to obtain global, coordinatefree versions of these equations by lifting them
338
10. R I E M A N N I A N G E O M E T R Y
to the frame bundle.) As an application, we will prove that the Riemann tensor is exactly the obstruction to the integrability of the Riemannian structure. T h a t is, the vanishing of curvature is equivalent to the existence of a coordinate atlas {U~,x~,l.. . ,x~}~e~ such that the coordinate fields O/Ox~, 1 < i < n, form an orthonormal frame field on Us, for each ~ C 92. Equivalently, the Riemannian manifold is locally isometric to Euclidean space. In what follows, V is a general connection on the nmanifold M, n > 2. To begin with, we will work in an open, trivializing neighborhood U for T ( M ) and fix the trivialization by a choice of a smooth frame ( X 1 , . . . ,X~) on U. Define 0 i r A I ( U ) by Oi(Xj) = 6~, 1 < i , j < n. Then, each X E X(U) can be written
X = Oi(X)Xi. Remark. Elie Cartan called the frame field a "moving frame". The discussion in this section concerns his "method of moving flames". In the next chapter, we will use principal bundles to globalize this method and give some geometric applications. Define forms wji E A I(U), 1_ 0 smaller, if necessary, we see t h a t integral curves to X , s t a r t i n g in B~(xo) a n d p a r a m e t r i z e d on (  c , c), must stay in the region W where X has not been altered.
C.1. Existence and uniqueness of solutions Since we will n o t be t h i n k i n g of the vector field X as a differential operator, we will write X ( x ) for Xx. A curve s : (  5 , c) ~ R n is integral to X if a n d only if
s(t) = s(0) +
f
x(s(u)) du, ~ < t
0 is an integer, a (Cech) psimplex of l[ is an ordered (p + 1)tuple (U~o, U~I,... , U~p) of elements of U such that U~o A ... n U~p is nonempty. D e f i n i t i o n D.1.2. If p > 0 is an integer, an Rvalued (Cech) pcochain on ~/is a function ~ that, to each psimplex (U~0, U~I,... , U~.), assigns an element w(U,o,U~l,...
, u ~ , ) = w , 0 , 1 .... . ~ R.
The set of all Rvalued pcochains on ~/is denoted by CP(I[; R). Evidently, the operations of "simplexwise addition" of cochains and "simplexwise scalar multiplication" make C'B(I[; R) into an Rmodule. More precisely,
(~+r
....p = ~ 0 ~
.... ~ + r
.... ~
and (a~) . . . . ...~p = a(~ . . . . ...ap), Ya E R and V ~ , r C CP(~[; R). D e f i n i t i o n D.1.3. The ((~eeh) coboundary operator
is the Rlinear map defined by the formula p+l
i=0
388
D. DE
RHAM
THEOREM
Thus, we obtain a sequence r
~ dP+~(11;R) •
dP+2(11;R) ~ . . . .
The following is a routine computation. L e m m a D.1.4. The sequence of coboundary operators satisfies 52 = O. D e f i n i t i o n D.1.5. The module of ((~ech) pcocycles is 2P(U; R) = ker(5) N C'P(11; R) and the module of (Cech) pcoboundaries is /)P(11; R) = im(5) A alP(U; R).
As usual,/)P(ll; R) C_2~(11; R). D e f i n i t i o n D.1.6. The pth (2ech cohomology of 11, with coefficients in the ring R, is /:/~(11; R) =/)P(U; R)/2P(U; R). We have defined /2/* (11; R) as a graded Rmodule. It is made into a graded algebra by the "cup product". D e f i n i t i o n D.1.7. If ~ E C'P(11;R) and r E Cq(ll;R), the cup product ~or E 6"P+q(11; R) of these cochains is defined by (~r
= ~ o .... p C ~ p . . . ~ + q .
This makes C*(ll; R) into a graded algebra. Another straightforward computation proves the following. L e m m a D.1.8. / f ~ E OP(~/.;R) and r E 0q(~/.;R), then ~(~r
= (6~)r + (1)P~(5r
This is formally the same as the formula for the exterior derivative of the wedge product of a pform and a qform. As in that case, we get the following consequence. L e m m a D.1.9. The graded module Z*(~I; R) of Cech cocycles is a graded subalgebra of C*(~; R) and B*(~A;R) C_ 2* (]g; R) is a 2sided ideal. Consequently, cup
product is well defined on/2/*(ll; R), making that graded Rmodule into a graded algebra over R. Finally, we will define the Cech cohomology algebra of the space X to be the "direct limit" /2/* (X; R) = li__.m/2/.(11; R) over finer and finer open covers. We make this precise. Let { V * } ~ be a family of graded Ralgebras, indexed by a partially ordered set 92. That is, there is a partial ordering c~ ~ ~ on ~ such that, whenever a, ~ E 92, 3 3' E 91 with a 21_3' and ~ _~ ~/. Assume also that, whenever c~ _~/3, there is given a homomorphism ~ : V* * V~ of graded algebras and that, whenever ~ ~ ~ ~ 3", then ~o~ o ~a = ~ ' * We say that {V2, ~}~,~ is a directed system of graded Ralgebras.
D.1. C E C H C O H O M O L O G Y
389
On the disjoint union
v*= Hvd, c~e~
define the equivalence relation ~ generated by ~
where v E Vc~ and c~ ~ ft. Then V * / ~ has a natural graded Ralgebra structure. Indeed, scalar multiplication a[v] = [av] is clearly well defined. As for addition, if [v], [w] E V * / ~ are represented by v c Vd and w E V~, find 7 c 92, c~ ~ V, fl < 7, and set Iv] + [w] = [ G ( v ) + ~7~(w)]. It is trivial to check that this is well defined and that these operations make V * / ~ into a graded Rmodule. Similarly, the algebra multiplication passes to a well defined multiplication making V*/~ into a graded Ralgebra. D e f i n i t i o n D . I . 1 0 . In the above situation, we set lim Vd = V*/~ and call this graded Ralgebra the direct limit of the directed system of graded Ralgebras. E x a m p l e D . I . l l . For a differentiable manifold M, let ~lx denote the set of open neighborhoods U of x E M. This is a directed system under the partial order U _~ V ev U _D V. The graded algebras {A*(U)}ueux form a directed system under the restriction homomorphisms
= lV, where a~ E A*(U) and U _D V. Then = lira A * ( U )
is just the graded algebra of germs at x C M of smooth forms. Let D ( X ) denote the set of open covers of X. This is partially ordered by: ~1 ~ V e$ V is a refinement of ~. Since any two open covers of X have a common refinement, this makes O ( X ) into a directed system. If U = {U~}~ev~, V = {Vfl}Ze~ , and ~1 _~ V, then there is a choice function i : ~3 + 92 such that VZ C_ Ui(~), Vfl C lB. This induces a homomorphism i # : O * ( ~ ; R ) * C'*(V; R) of graded algebras, where i # (~)Zo&..G = ~(Zo)~(;h)..4(~,J" The following is trivial. Lemma D.1.12. ~ o i #.
The homomorphism i # : 0"(~1; R) ~ C*(V; R) satisfies i # o (~ =
We cannot use i # as a homomorphism 9~v C'* (L[; R) C* (V; R) for a directed system of algebras. The problem is that i # depends on arbitrary choices, so we could never guarantee that ~ w o ~vu= ~wu. But the above lemma implies t h a t i # induces i* :/:/* (]~; R) ~/:/* (V; R) and it turns out that, at the level of cohomology, the arbitrariness disappears.
390
D. DE
Definition D.I.13. If11 ~ V in above, and if p E Z, define
RHAM
D(X),
THEOREM
ifi, j : ~ * 9/are two choice functions, as
S : CP(U; R) ~ C p  I ( v ; R) by the formula p1
~=0
As usual, if p  1 < 0, we understand that CPI(1L;R) = 0 and S = 0. The following is checked by a routine (if somewhat tedious) computation, left to the reader. L e m m a D.1.14. If i, j, and S are as above, then
S o 5 + 6 o S = j #  i #.
Consequently, i* = j* :/:/* (ll; R) */:/* (V; R). By this lemma, whenever ~A__ V in D(X), we define a homomorphism ~
= i * : / r * ( U ; R) ~ H*(V; R)
of graded algebras that is independent of the (allowed) choice of i : fl~ * 9/. L e m m a D.1.15. IfU < V~ W in D(X), then ~oW oqo~
~w
Pro@ Indeed, set u = {u~}~, V = {v~}~, w = {w~},~,
and let i : f13 * 92, j : ~ . ~ be suitable choice functions. Then i o j : E ~ 9/is an allowed choice function relative to the refinement 11 ~ W. But qow = ( i o j ) * = j * o i *
= ~ W o q o uv. []
v,
V
Thus, we get a directed system {H (11;R), ~u}u,ve~(z) of graded algebras over R. D e f i n i t i o n D . l . 1 6 . The Cech cohomology of the space X with coefficients in R is the direct limit = hm H (11;R), taken over the directed system D(X). Let f : X * Y be a continuous map between spaces. Given 11 E D(Y), define f  l ( ~ l ) E D(X) by the usual pullback construction. If we are given a psimplex ( f  i ( U ~ 0 ) , . . . , f  i ( U a , ) ) of f  l ( l l ) , then it is clear that (U~o,... ,Uap) is a psimplex of ~1. Consequently, each Cech cochain 0 E CP(11; R) has a natural pullback f#(O) E ~ p ( f  i (ll); R). This defines a homomorphism f # : C*(ll;R) ~ C * ( /  I ( l l ) ; R ) of graded algebras.
D.2. DE RHAM(~ECHCOMPLEX LemmaD.l.17. If f : X + Y, as above, then f # o($ = 5 o f # canonically defined an induced homomorphism
391 and there is
f* :/:/*(Y; R) +/:/*(X; R) of graded algebras over R. This makes Ceeh eohomology into a contravariant functor on the category of topological spaces and continuous maps.
D e f i n i t i o n D . l . 1 8 . If P/ is a directed system, a cofinal subsystem ~ _C 92 is a directed subsystem with the property that, whenever a C 9/, ~7 E ~ such that a_~7. Finally, the proof of the following lemma is a straightforward application of definitions9 L e m m a D.1.19. Let {Vo:,F~}~,~e~ be a directed system of graded Ralgebras. If C_ P2 is a cofinal subset, then there is a canonical isomorphism hm V~ = hm V~ of graded Ralgebras, where the first limit is taken over all c~ C P2 and the second is taken over all 7 c ~.
By Corollary 10.5.8, the family of simple covers (Definition 8.59 tiable manifold is a cofinal subset of D ( M ) .
of a differen
C o r o l l a r y D.1.20. The Cech cohomology H * ( M ; R ) of a differentiable manifold can be computed by taking the limit only over the directed system of simple covers. D.2. T h e de R h a m  ( ~ e c h c o m p l e x The proof we will give of the de RhamCech theorem is essentially that of Andr@ Weft [48]. The main step is to prove the following. T h e o r e m D.2.1. If11 is a simple cover of M, there is a canonical isomorphism e u : / : / * ( U ; JR) + H~)R(M ) of graded algebras and, if II ~_ V, where both are simple covers, then the diagram II
/7/*(11;]I~) / ~ 1 " ~
. ~ / / ) ~ 2 ' /7/*(~; R)
HSR(M) is commutative.
The equivalence of de Rham theory and Cech theory follows easily. Indeed, the Cech cohomology can be computed by passing to the limit over the simple covers only (Corollary D.1.20), so the isomorphisms e u induce a welldefined homomorphism ~b:/:/*(M;]R) + H~)R(M ). The fact that each e u is an isomorphism implies the same for ~b.
392
D. D E R H A M T H E O R E M
T h e o r e m D . 2 . 2 (de Rham). There is a canonical isomorphism H~)R(M ) = / : / * ( M ; R )
of graded ]~algebras. For use in the following section, we record the following corollary, implicit in the above argument. C o r o l l a r y D . 2 . 3 . If]1 is a simple cover, the natural homomorphism of/:/*(]1;N)
into the limit/:/*(M;N) is an isomorphism of graded algebras. In order to prove Theorem D.2.1, we will build an enormous cochain complex of graded algebras that includes both (A* (M), d) and (C* (]1; R), 5) as subcomplexes. If ]1 is simple, we will prove that the inclusions of these subcomplexes induce isomorphisms in cohomology. F i x an open cover ]1 = {U~}~e~ of M. For the following definitions, it is not necessary that ]1 be simple. D e f i n i t i o n D . 2 . 4 . A Cech pcochain on ]1 with values in A q is a function qo that, to each psimplex (U~o, U ~ I , . . . , U~p) of ]1 assigns
~c*oal .... v C Aq(Uao N Uc~l N . . . N Uav ). The set of all such cochains will be denoted EP'q(]1) = CP(~/; Aq). Although the coefficient ring Aq(u~ 0 A U~I A . . . N U~,) changes with each simplex, one can still add cochains simplexwise and multiply them by real scalars. These operations make EP'q(]1) = CP(I/; A q) into a real vector space. There is also a bigraded multiplication
EP,q(]1)  E',S(]1) ~ EP+~,q+s(]1). In defining this and other operations, we make the notation less bewildering to the eye by abusing it (the notation, that is). Whenever respective forms have been defined on respective open sets with common, nonempty intersection, addition of such forms and exterior products of such forms are understood to be defined on their common domain. For instance, if (Uao,U~,Ua2) is a 2simplex of ]1 and COalo~~ E Aq(Uai CI Uo~j), then
a;c~1o~2 Wc~oa24 a;o~oo~l C Aq(U,~o N Uoq N Uo~2). Similarly, if ~oaoa~ C Aq(U~o N U~,~) and r
C As(Uc,~ M U,~2), then
W i t h these conventions understood, we define the bigraded multiplication as follows. If ~ C EP'q(]1) and r C E~'~(]1), then ~ r 9 EV+~,q+~(]1) is defined on a (p + r)simplex (U~o,... , U ~ , , . . . , U~,+~) by (~r
.... ,+~ = (  1 ) q ~ , o .... p Ar
.... ,+~ 9 Aq+s(U~o n . . . n U ~ + ~ ) .
Often we suppress explicit reference to the simplex on which this formula is being evaluated and write
~r = (  1 ) q ~
A
r
We say t h a t E** (~) = {E p'q (~A)}~,~=0 is a bigraded algebra under this multiplication. Note how this operation combines the cup product from Cech theory with the exterior multiplication from de Rham theory. The strange sign (  1 ) q~ will be needed in the proof of Lemma D.2.8.
D.2. D E R H A M  ~ E C H
COMPLEX
D e f i n i t i o n D . 2 . 5 . The de Rham operator e : EP'q('~) setting
(c~) . . . .
= (1)Pd(~aoal...a,) E Aq+l(Uao
...c~,,
393 *
n
EP'q§
Ual
is defined by
n . . . lq
for arbitrary ~ C CP(II; A q) and for every psimplex (U~o, U ~ , . . .
Uc~r,),
, U~,) of II.
D e f i n i t i o n D . 2 . 6 . The Cech operator 0 : EP'q(ll) * EP+~'q(~) is defined by setting p+l
(0~:')~o~
.... ,+,
= E(1)i(Fo~o...a,
.... p+, C A q ( U ~ o
n U,~, n . . .
r~ U , ~ p + , ) ,
i=0
V~ C (~'P(~[; A q) and for every psimplex (Uao, U ~ , , . . . , U ~ ) of II. The following are evident: 9 E2=0, 9
02 =
9
~o0=0oe.
O,
Remark t h a t the sign (  1 ) p in the definition of e is responsible for the anticommutativity of e and 0.
/~0,2(~[)
5
, E1,2(~
E0,~(lt)
~ , Eu(lt)
Eo,o(~)
~
, E1,~
)
5
~
, E2,2(~ )
5
, E2,1(~)
5 ) 9
, E2,~
5
) .
F i g u r e D . 2 . 1 . The de Rham(~ech complex
D e f i n i t i o n D . 2 . 7 . For each integer rn >_ O, Em(~i) = (~p+q=m EP'q(~'i) and the total differential operator D : Em(~l) ,Em+l(ll) is D = c + 5. It is a good idea to picture E**(~i) laid out as a first quadrant array in the (p,q)plane, having EP'q(~I) at the point (p,q) of the integer lattice as in Figure D.2.1, with the de R h a m operators e as vertical arrows and the Cech operators 5 as horizontal arrows. This array is called the de R h a m  C e c h complex. The total degree of an element ~ C EP'q(~i) is p + q and Em(~l) is spanned by the elements of total degree m. One can view E'~(~2) in this diagram as lying along the diagonal p + q = rn. If ~ E EP'q(~I), where p + q = m, then
D(~) : ~(~) + 0(~) c E~'~+I(U) 9 E~+I,~(u) c Em+~(u).
394
D. DE
RHAM
THEOREM
L e m m a D.2.8. The pair (E*(1/),D) is a cochain complex in which E*(1/) is a graded algebra over ]R and D(~r
= D(~)r + (1)m~D(r
where ~ E Era(t[). Proof. Indeed, E*(1/) = {Em(1/)}~m=o is a graded vector space and it is clear that the bigraded multiplication in E** (11) induces a graded algebra structure on E* (1/). Because of the anticommutativity of r and 5, D 2 =r162
=0.
Finally, it is only necessary to verify the Leibnitz formula for ~p E EP'q(1/) and r E E r's (1/), p + q = m. Suppress reference to the (p + r)simplex (U~o,... , Ua,+~) and compute
r162
= (1)P+~d((1)'q~ = (1)P+r+rq(d(~)
A
r
A r } (1)q~P A d(~)))
~ (1)P+r+rq+P+(q+l)r(E((D)r
~ (1)P+r+rq+q+r+rq(~g(~)))
= ~(~)r + (1)P+q~s(r That is, (D.1)
~(~r
= E(~)r + (1)m~zs(r
Similarly, suppress reference to the (p + r + 1)simplex (U~o,... ,Uap+,.+l ) and compute 5(~r
= (1)ra(5(9~) A r + (  1 ) P ~ A 8(r = (1)'q(1)rqs(~)r
+
(1)P+'q(1)q(r+l)~5(r
= 5(~)r + (  1 ) ' + q ~ 5 ( r That is, (D.2)
5(Vr
= 5(~)r + (1)ra~5(r
By adding equation (D.1) and equation (D.2), we obtain the desired Leibnitz rule for D. [] L e m m a D.2.9. There are canonical inclusions
(A*(M),d)
i
(E*(1/),D),
(C* (ll; R), 8) ~ (E* (1/), D)
of subcomplexes, respecting the graded algebra structures. Proof. Indeed, ifw 6 Aq(M), i(w) E ~0(1/; A q) = E0,q(1/) assigns to each 0simplex (U~o) the element i(~J)~ o = wlU~ o. Since (5(i(~))).o~,
= ~ l g ~ o n u s ,  wlg~o n u ~ l = 0,
we see that
n(i(w) ) = E(i(w) ) = i(dw). Likewise, if ~ E CP (1/; R), j (~) E CP (1/; A ~ assigns to each psimplex (U~o,... , Us,) the Oform on the open set UaoN...NUa, that is the constant function ~ o . . . a , E ~.
D.2. D E R H A M  C E C H
COMPLEX
395
Clearly, e(~O)so...s, = O, and so j also commutes with the coboundary operators of these complexes. It is clear from these definitions t h a t
i(~ A ~?) = i(aJ)i(~), whenever w 9 AV(M) and r / 9 A~(M), and that
j(go%b) = j(~o)j(r whenever g) 9 OP(11; 1R) and %b9 O~(ll; ]R). Corollary D.2.10.
[]
There are canonical homomorphisms i * : H~)R(M ) ~ H*(E*(~I),D)
and
/*:/~*(u; R) ~ H* (E* (U), D) of graded algebras. We augment the rows of the diagram in Figure D.2.1 by i. T h a t is, the new rows are
Aq(M) & E~
a+ EI,q(U) A .
Similarly, we augment the columns by j :
Op(u;~) & Ep,~ Lemma D.2.11.
& E~,I(U) &
The augmented diagram has exact rows.
Proof. If w e Aq(M), we have seen in the proof of Lemma D.2.9 that 5(i(w)) = 0. Conversely, if ~ C d'~ A q) and 5(~) = 0, then the forms ~ o C Aq(U~o) and ~sl E Aq(u~I) must agree on Uso n U~I, if this intersection is nonempty. Hence, the forms p~ E Aq(u~) piece together smoothly to give a form co C Aq(M) such t h a t i(cz) = ~. This proves exactness at E~ We prove exactness at EP'q(~l) = d'P(ll; A q) when p > 1. Let {As}~ea be a smooth partition of unity subordinate to ~i. Define
as follows. Given ~ E OP(~I; A q) = EP'q(~I), define A(~) E 0Pl("d; A q) to be the element, the value of which on the (p  1)simplex (U~o,... , Us,, 1) is
a(~)oo...~_l : ~ ~s~sso .... .1, s
where each term of this locally finite sum is interpreted, in the obvious way, as a qform on Uso N . . . N U~,_I. If 5(qD) = O, the reader can check that 9~ = a(A(w)), proving exactness at EP'q(~). [] L e m m a D . 2 . 1 2 . If the cover ~i is simple, the augmented diagram has exact columns.
Proof. If ( E CP(ll; R), we have seen in the proof of Lemma D.2.9 t h a t e ( j ( ( ) ) = 0. Conversely, it is clear that, if ~ C CP(U, A ~ and c(~) = 0, then, d ( ~ o.... ,,) = 0, for each psimplex ( U s o , . . . , Us,). The fact that U~ o n ... N U~,, is connected implies t h a t ~o.s,, e A~ N . . . n Uo,) is constant. Thus, we can define ( C CP(~I;R) by (~o...s~, = the constant ~so...~, and j ( ( ) = ~. This proves exactness at E p'~
396
D. DE RHAM T H E O R E M
If q _> 1, exactness at F,P'q(~[) follows from the Poincar~ l e m m a a n d the fact t h a t U~ o N 9. N U ~ , if not empty, has the de R h a m cohomology of ~ n . [] Lemma
D . 2 . 1 3 . If the cover ~1 is simple, the homomorphisms i* and j* are sur
jective. Proof. Let ~ E E m ( l l ) be a Dcocycle. We show t h a t the element [~] E H*(E*(II), D) is in the image b o t h of i* and j*. If m = 0, t h e n ~ = 0 = c~ and the assertions follow from L e m m a s D.2.11 and D.2.12, respectively. Assume, therefore, t h a t m > 1. T h e cocycle ~ lies along the diagonal p + q = m, so we write
r = }~r p=0
where (~ E E P ' m  ~ ( l l ) , 0 _< p _< m. The equation D(~) = 0 implies t h a t e(@) = 0, so L e m m a D.2.12 allows us to find {0 E E~ with e({0) = @. T h e n  D ( { 0 ) has 0 as its c o m p o n e n t in E~ and is Dcohomologous to ~. Suppose, inductively, t h a t {k E E m  l ( ~ ) has been found so t h a t ~  D({k) has 0 as its c o m p o n e n t in EP'mP(~[), 0 _ 1, t h e n i(co) E E~ is of the form is, co = 0, so [co] = 0 E H ~ i(co) = D(~), ~ E E m  l ( l l ) . Write m1
~= ~ , p=0
where ~p E EP"~lP(ll), 0 _< p _< m  1. Since the component of i(co) in E m ' ~ is 0, 5(~,~1) = 0 and L e m m a D.2.11 implies t h a t ~m1 = 6(0), 0 E E'~2'~ Thus, ~' = ~  D(O) has component 0 in E ' ~  I ' ~ a n d D(~') = i(co). Again using L e m m a D.2.11 a n d finite induction, we see t h a t no generality is lost in assuming t h a t ~ is concentrated in E~ a n d t h a t 5(~) = 0. By one more appeal to L e m m a D.2.11, we find a unique r/ E A m  I ( M ) such t h a t i(r/) = (. T h e n , i(dr/) = e(i(r/)) = i(w) and, i being injective, w = dr/. T h a t is, [col = 0 as desired. A completely parallel argument, using L e m m a D.2.12, proves t h a t j* is injective. []
D.3. SINGULAR COHOMOLOGY
397
If the cover 11 is simple, we define Cu = (i.)1 o j* :/:/*(l~; N) , H ~ a ( M ) , an isomorphism of graded Ralgebras by Lemmas D.2.13 and D.2.14. The following completes the proof of Theorem D.2.1. L e m m a D.2.15. If11 ~ V are simple covers of M, then the diagram ~oIIv
H~)R(M) is commutative. Proof. Indeed, if ~ = {U~}~e~ and V = {V~}ze ~ recall that Fv is induced by a choice function g : ~3 + 91 such that V/~ C_ U~(~), V~ E ~ . The same choice function g defines a homomorphism ~
: H* (E* (U), D) + H* (E* (V), D)
and the diagram H*(M)
i* , H*(E*(ll),D) ~ j"
~i
idl H;R(M)
/:/*(~A;R)
~~
i* , H * ( E * ( V ) , D ) ,
y"
commutes.
/:/*(V;N) []
Remark. In fact, the de RhamCech isomorphism r
N) + H ~ a ( M )
is functoriaL That is, whenever f : M ~ N is a smooth map between manifolds, the diagram
H*(N;N)
s"
, ZP(M;N)
H;R(N )
> H;R(M ) S* is commutative. That is, on smooth manifolds, Cech theory and de Rham theory are equivalent as functors. Checking this functoriality is straightforward.
D.3. Singular Cohomology We will define the graded singular cohomology algebra H* (M; N) and prove the following. T h e o r e m D.3.1. There is a canonical isomorphism /:/* (M; N) = H * ( M ; R ) of graded Ralgebras.
398
D. DE
RHAM
THEOREM
We note that with little extra effort, the proof of this theorem can be carried out with N replaced by an arbitrary commutative ring with unity. It can also be generalized to a larger class of topological spaces than manifolds. The proof of Theorem D.3.1 will be analogous to that of Theorem D.2.2. Coupled with Theorem D.2.2, this will prove the de Rham theorem for singular cohomology. T h e o r e m D.3.2 (de Rham). There is a canonical isomorphism
H~)R(M) = H*(M;N) of graded Nalgebras. Again, these isomorphisms are easily checked to be functorial. The singular cohomology should be defined via duality at the chain level. Recall (Definition 8.2.11) that Cp(M;R) denotes the real vector space with basis the set Ap(M) of smooth (respectively, continuous) singular psimplices in M. This is called the space of singular pchains. Set
C'(M; N) = Home(Cp(M; R), N), the space of singular peochains. The boundary operator 0 : Cp+,(M; N) + C,(M; N) has adjoint 0* : CP(M;N) * Cv+I(M;N), called the singular coboundary operator and the identity 02 = 0 dualizes to 0 *2 = 0. In the standard fashion, this gives rise to the vector spaces Z* (M; N) and B* (M; N), called the spaces of singular cocycles and singular coboundaries, respectively. The singular cohomology theory is then the quotient
Z*(M;N)/B*(M;N). The above construction is functorial. That is, smooth (respectively, continuous) H*(M;N)
=
maps f : N ~ M induce graded, Nlinear maps f# : C, (N; N) ~ C, (M; N), f # : C*(M;N) * C*(N;N). These commute, respectively, with 0 and 0", hence pass to graded, Rlinear maps, f, and f* on homology and cohomology, respectively. The usual functorial properties are satisfied, making singular homology into a covariant functor, singular cohomology into a contravariant functor. For homology, this is the content of Exercise 8.2.24. Dualizing this gives the corresponding result for cohomology. It remains that we define the graded algebra structure on H* (M; N). Multiplication in this algebra is called the singular cup product and will be denoted by a dot "." to distinguish it from the Cech cup product. For this, let n = p+q, p, q >_O, and consider the maps O'p : A p ~ A n , 6rq : Aq ~
/~n~
defined by
o,(xi,...,xp)
= (x~,...,xp,o,...,o),
~q(xl,...,xq)
= (0,...,0,x~,...,xq).
D.3. S I N G U L A R C O H O M O L O G Y
399
One calls Op the front pface operator and a q the back qface operator. Given c C P ( M ; N ) and r C Cq(M;N), the cup product ~ . r C C n ( M ; N ) will be completely determined by its values on the set A n ( M ) of singular nsimplices on M. This is because this set is a basis for the vector space Ca(M; R). If s E A n ( M ) , define ~. r = ~(s o ~ ) r o ~q). A little combinatorics gives the following expected relation. L e m m a D.3.3. [f ~ E CP(]ll;]I~) and r E Cq(M;N), then 0"(~o. r
= (0*p). r + (  1 ) v ~ 9 (0"r
As usual, we get the following consequence. L e m m a D . 3 . 4 . The graded vector space Z*(M;]R) of singular cocycles is a graded subalgebra o f t * (M; ]R) and B* (M; JR) C Z* (M; ~) is a 2sided ideal. Consequently, cup product is well defined on H*(M;I~), making that graded vector space into a
graded algebra over JR. Finally, if f : N ~ M is smooth, the induced map f* : H * ( M ; ~ ) * H * ( N ; N )
is a homomorphism of graded algebras. Remark. In the above discussion, we allowed Aq(M) to be either the set of smooth singular simplices in M or the set of continuous ones. This yields two possibly different singular cohomologies, the smooth and the continuous. The proof that we will give of Theorem D.3.1 works equally well in either case, hence both theories, being canonically isomorphic to Cech cohomology, are identical. Also, since continuous maps between manifolds are homotopic to smooth ones (Subsection 3.8.B), the homotopy invariance of singular theory (not proven here, but cf. [13]) implies t h a t these theories are canonically isomorphic as functors. C o r o l l a r y D . 3 . 5 . The singular cohomology algebra, computed by using smooth
singular simplices, is canonically and functorialIy equivalent to that computed by using continuous singular simplices. Let ~ = {Us}ae~ be an open cover. We are going to mimic the construction of the de Rham(~ech complex to produce a singularCech complex E**(~). Ultimately, we will need ~1 to be simple. The proof of Theorem D.3.1 will then proceed almost exactly as that of Theorem D.2.1. D e f i n i t i o n D . 3 . 6 . A (~ech pcochain on 11 with values in C q is a function ~ that, to each psimplex (U~o , U s ~ , . . . , Uo~,) of II assigns ~sos~ .... , ~ c q ( U s o n us~ n . . .
n
Us,;~).
The set of all such cochains will be denoted F,P'q(]~) = CP(~; cq). Once again, this is naturally a real vector space. We have simply replaced
A q(Uso (3 Us~ N . . . N Uo,,) with C q(Uso n Us~ N . . . n Us,,; R) in the earlier definition. Similarly, we get a bigraded multiplication on the resulting double complex and, setting c = (  1 ) P 0 * : EP+q(~) + E p'q+l and defining 5 in complete analogy with Definition D.2.6, we complete the definition of the double complex. The total differential is D = e + 5. The first significant difference between the current and former construction is t h a t there is no natural inclusion of (C*(M;R),O*) into (E*(11),D). If ~ C
400
D. D E R H A M T H E O R E M
Cq(M;~), go~ E ~1,, and if i~ : U~ ~* M is the inclusion, then we can define the restriction of ~o to U~ by ~IU~ = i~(~). This defines a homomorphism i : (Cq(M; R), 0") ~ (C~
cq), ~),
i(~)~ = wlu~ of cochain complexes. However, it is quite possible that ~ ~ 0, but that ~IU~ = 0, for every ~ E 91, and so i will not be injective. The solution to this is the delicate subdivision process for singular homology (see commentary following Proposition 8.5.13) and cohomology that allows computation of these theories using only "11small" singular simplices (Definition 8.5.11). The sketch for homology accompanying Definitions 8.5.11 and 8.5.12 dualizes to a similar procedure for cohomology and the references for details are the same. We note that the definition of cup product also works for this ~[small theory and record the following. T h e o r e m D.3.7. The inclusions AUq(M) ~~ Aq(M), q >_ O, induce canonical isomorphisras of graded vector spaces H.U(M;N) = H . ( M ; R ) , H ~ ( M ; R ) = H*(M;]R).
In the case of cohomology, this is an isomorphism of graded algebras. One
now
notes that
i: (C~(M;N),O*) ~ (d~
Cq),e)
is injective. Lemma D.3.8.
There are canonical inclusions (Cu(M,N),O*) ~ (E*(U),D), (d'*(li;]R), 5) ~ (E*(II),D)
of subcomplexes, respecting the graded algebra structures. Proof. Indeed, for each Cech 1simplex (Uao , U~I ) and each cochain ~o E C~(M; JR), we see that (~i(~))~o~1 = ~]V~o n U ~  ~lV~o n V~, = 0 for the above definition of i. Thus, D(i(~)) = ~(i(~)) = i(0*~). For the definition of j, note that, for each Cech psimplex, (U~o,...,U~,), C~ ~ ... N Uc~p) is just the set of arbitrary JRvalued functions on the open set U~ o N ... Cl U~p (the singular 0simplices in a space are just the points). A Cech 0cochain r E ~,0(~; JR) assigns to each psimplex (U~o,... , U ~ ) an element r .... ~, E IR. We define j ( r E d'P(U; C ~ by letting J(r .... ~ denote the constant function on U~o N ... N Ua~ with value r .... p. Evidently, j ( r = 0 if and only if = 0, and so j is injective, By the definition of the singular coboundary operator, the coboundary of a constant 0cochain vanishes, so
D(j(r
= 5(j(r
 j(5r
D.3. SINGULAR COHOMOLOGY
401
The fact that i and j respect the graded algebra structures can be checked by the reader. [] Again we augment the rows of E** (11) by i, obtaining the sequence i C q ( M ; IR) c__+ E~
~
5 E l ' q ( 1 1 ) +.
,
and we augment the columns by j, obtaining the sequence Ov(11;R) & EP'~
& EP'l(11) ~ . .
L e m r n a D.3.9. If the cover 11 is locally finite (in particular, if II is simple) the
augmented diagram has ezact rows. Pro@ We have seen that a o i = 0. Conversely, if ~ E 6'~ see that, for each 1simplex (U~o, U~I) ,
C q) and a~ = 0, we
~olS~o n g~l = ~ , l g ~ o n U~,, and so the ~b,s patch together to define ~b' E C~(M) such that i(~b') = ~b. This proves exactness at E ~ (11). We prove exactness at E;'q(~) = CP(11; C q) when p _> 1. For this, we construct A : Ev,q(11) ~ SP1'q(11), p > 1, such that ~
= 0 ~
~ = ~A(~).
Since a 2 = 0, this will prove the lemma. Let ~ : AqU(M) , Z + be defined by ~(a) = cardinality of {Ua E 11[ o(~Xq) C Ua}. The hypothesis that 11 is locally finite guarantees that ~(a) is finite. If U~ E 1{ and (Uao,..., U~,_,) is a Cech (p  1)simplex, define
ia o.... ~~ : c ~ ( e ~ n U ~ o n . . . n u ~ , _ , ) ~ c ~ ( U ~ o e .
.nV~,_,)
by i~o .... v*(~9)(~)= ~ ( a ) '
ifcr(Aq) C U a n U a ~
nUa, ....
otherwise.
to Finally, for each ~o E EP'q(~A), define 1(~) E Ep1'q(11) by a(~)