Characteristic Classes. (AM-76), Volume 76 9781400881826

Table of contents :
Contents
Preface
§1. Smooth Manifolds
§2. Vector Bundles
§3. Constructing New Vector Bundles Out of Old
§4. Stiefel-Whitney Classes
§5. Grassmann Manifolds and Universal Bundles
§6. A Cell Structure for Grassmann Manifolds
§7. The Cohomology Ring H*(Gn; Z/2 )
§8. Existence of Stiefel-Whitney Classes
§9. Oriented Bundles and the Euler Class
§10. The Thom Isomorphism Theorem
§11. Computations in a Smooth Manifold
§12. Obstructions
§13. Complex Vector Bundles and Complex Manifolds
§14. Chern Classes
§15. Pontrjagin Classes
§16. Chern Numbers and Pontrjagin Numbers
§17. The Oriented Cobordism Ring Ω*
§18. Thom Spaces and Transversality
§19. Multiplicative Sequences and the Signature Theorem
§20. Combinatorial Pontrjagin Classes
Epilogue
Appendix A: Singular Homology and Cohomology
Appendix B: Bernoulli Numbers
Appendix C: Connections, Curvature, and Characteristic Classes
Bibliography
Index

Citation preview

Annals of Mathematics Studies Number 76

P R IN C E T O N L A N D M A R K S IN M A T H E M A T IC S A N D P H Y S I C S

Non-standard Analysis, by Abraham Robinson General Theory of Relativity, by P.AM. Dirac Angular Momentum in Quantum Mechanics, by A. R. Edmonds Mathematical Foundations of Quantum Mechanics, by John von Neumann Introduction to Mathematical Logic, by Alonzo Church Convex Analysis, by R. Tyrrell Rockafellar Riemannian Geometry, by Luther Pfahler Eisenhart The Classical Groups: Their Invariants and Representations, by Hermann Weyl Topology from the Differentiable Viewpoint, by John W. Milnor Algebraic Theory of Numbers, by Hermann Weyl Continuous Geometry, by John von Neumann Linear Programming and Extensions, by George B. Dantzig Operator Techniques in Atomic Spectroscopy, by Brian R. Judd The Topology of Fibre Bundles, by Norman Steenrod Mathematical Methods of Statistics, by Harald Cramer Theory of Lie Groups, by Claude Chevalley Homological Algebra, by Henri Cartan and Samuel Eilenberg

CHARACTERISTIC CLASSES BY

JOHN W. MILNOR AND

JAMES D. STASHEFF

PRINCETON UN IVERSITY PRESS AND UN IVERSITY OF TOKYO PRESS

PRINCETON, N EW JE R SE Y 19 7 4

Copyright © 1974 by Princeton University Press ALL RIGHTS RESERVED

Library of Congress Cataloging-in-Publication Data Milnor, Joh n Willard, 1 9 3 1 Characteristic classes. (Annals o f mathematics studies, no. 76) Bibliography: p. 1. Characteristic classes. I. StashefF, James, joint author. II. Title. III. Series. Q A613.618.M 54 5 1 4 ’.7 72-4050 ISBN 0-691-08122-0 Printed in the United States o f America 9 10

Preface The text which follow s is based mostly on lectures at Princeton U niversity in 1957. The senior author w ishes to apologize for the delay in publication. The theory of ch aracteristic c la s s e s began in the year 1935 with almost simultaneous work by HASSLER WHITNEY in the United States and EDUARD STIEFEL in Switzerland. S tie fe l’s th esis, written under the direction of Heinz Hopf, introduced and studied certain “ ch aracteristic” homology c la s s e s determined by the tangent bundle of a smooth manifold. Whitney, then at Harvard U niversity, treated the case of an arbitrary sphere bundle. Somewhat later he invented the language of cohomology theory, hence the concept of a characteristic cohomology c la ss, and proved the basic product theorem. In 1942 LEV PONTRJAGIN of Moscow U niversity began to study the homology of Grassmann manifolds, using a c e ll subdivision due to Charles Ehresmann. This enabled him to construct important new ch aracteristic c la sse s. (Pontrjagin’s many contributions to mathematics are the more remarkable in that he is totally blind, having lo st his eyesight in an acci­ dent at the age of fourteen.) In 1946 SHING-SHEN CHERN, recently arrived at the Institute for Advanced Study from Kunming in southwestern China, defined ch aracteris­ tic c la s s e s for complex vector bundles. In fact he showed that the com­ plex Grassmann manifolds have a cohomology structure which is much easier to understand than that of the real Grassmann manifolds. This has led to a great clarification of the theory of real ch aracteristic c la sse s,

vi

PREFACE

We are happy to report that the four original creators of characteristic c la s s theory a ll remain mathematically active: Whitney at the Institute for Advanced Study in Princeton, S tie fe l as director of the Institute for Applied Mathematics of the Federal Institute of Technology in Zurich, Pontrjagin as director of the Steklov Institute in Moscow, and Chern at the U niversity of C alifornia in Berkeley. This book is dedicated to them.

JOHN MILNOR JAMES STASHEFF

Contents P r e fa c e ....................................................................................................................

v

§1. Smooth Manifolds .......................................................................................

3

§2. Vector Bundles ...........................................................................................

13

§3. Constructing New Vector Bundles Out o f Old ..................................

25

§4. Stiefel-Whitney C la sse s ............................................................................

37

§5. Grassmann Manifolds and U niversal Bundles ....................................

55

§6. A C ell Structure for Grassmann Manifolds .........................................

73

§7. The Cohomology Ring H*(Gn; Z / 2 ) ......................................................

83

Existence of Stiefel-W hitney C la ss e s ..................................................

89

§9. Oriented Bundles and the Euler C lass ...............................................

95

§8.

§10. The Thom Isomorphism Theorem .......................................................... 105 §11. Computations in a Smooth Manifold ...................................................... 115 §12. Obstructions ................................................................................................ 139 §13. Complex Vector Bundles and Complex Manifolds ......................... 149 §14. Chern C la sses ............................................................................................. 155 §15. Pontrjagin C la sses ..................................................................................... 173 §16. Chern Numbers and Pontrjagin Numbers ............................................. 183 §17. The Oriented Cobordism Ring A * ........................................................ 199 §18. Thom Spaces and T ransversality .......................................................... 205 §19. M ultiplicative Sequences and the Signature Theorem ..................... 219 §20. Combinatorial Pontrjagin C la sse s ........................................................ 231 E p ilo g u e................................................................................................................. 249 Appendix A: Singular Homology and Cohomology .................................... 257 Appendix B: Bernoulli Numbers ..................................................................... 281 Appendix C: Connections, Curvature, and C haracteristic C la s s e s

289

Bibliography........................................................................................................... 315 In d ex ........................................................................................................................ 325 v ii

Characteristic Classes

§1. Smooth Manifolds This section contains a brief introduction to the theory of smooth mani­ folds and their tangent spaces. Let Rn denote the coordinate space consisting of all n-tuples x = ( x j , x fl) of real numbers. For the sp ecial case n = 0 it is to be under­ stood that R° co n sists of a single point. The real number them selves w ill be denoted by R. The word “ smooth” w ill be used as a synonym for “ differentiable of c la ss C°°.” Thus a function defined on an open set U C Rn with values in R^ is smooth if its partial d erivatives of all orders exist and are con­ tinuous. For some purposes it is convenient to use a coordinate space R^ which may be infinite dimensional. Let A be any index set and let R^ denote the vector space consisting of all functions* x from A to R. The value of a vector x € R^ on a e A w ill be denoted by xa and called the a-th coordinate of x. Sim ilarly, for any function f : Y -> R^, the a-th coordinate of f(y) w ill be denoted by fa (y)We topologize this space R^ as a cartesian product of copies of R. For any subset M C R^, we give M the relative topology. Thus a func­ tion f : Y - M C R^ is continuous if and only if each of the associated functions fa : Y -> R is continuous. Here Y can be an arbitrary topologi­ cal space.

* Of cou rse our p reviou s co o rd in a te sp a c e Rn can be obtained a s a s p e c ia l c a s e of th is more gen eral con cep t, sim p ly by taking A to be the s e t of in te g e rs b etw een 1 and n.

3

4

CHARACTERISTIC CLASSES

DEFINITION. For U C Rn, a function f : U ->MC RA is said to be smooth if each of the associated functions fQ : U -> R is smooth. If f is smooth, then the partial derivative di/dn• can be defined as the smooth function U

RA whose a-th coordinate is RA defined on an open set U C Rn such that 1) h maps U homeomorphically onto an open neighborhood V of x in M, and 2) for each u e U the matrix [dha (u)/(?Uj] has rank n. (In other words the n vectors dh/duj, ..., N. It is ea sily seen that this definition does not depend on the choice of p, and that Dfx is a linear mapping. In fact, in terms of a local parametrization (U, V, h), one has the explicit formula D fx ^ £ Cj dh/du^j =

Cj N is smooth everywhere. Combining all of the Jacobians Dfx one obtains a function D f: DM -> DN where Df(x, v) = (f(x), Dfx(v)). LEMMA 1.4. D is a functor* from the category of smooth mani­ folds and smooth maps into its e lf.

For the co n c ep ts o f categ ory and fu n ctor, s e e for exam ple [E ilen b erg and Steenrod, C hap ter

iv].

9

§1. SMOOTH MANIFOLDS

In other words: (1) If M is a smooth manifold, then DM is a smooth manifold. (2) If f is a smooth map from M to N then Df is a smooth map from DM to DN. (3) If I is the identity map of M then DI is the identity map of DM; and (4) if the composition f o g

of two smooth

maps is defined, then D(fog) = (Df)o(Dg). The proofs are straightforward. One immediate consequence is the following: // f is a diffeomorphism from M to N then Df is a diffeomorphism from DM to DN. REMARKS. According to our definitions, the tangent space DRX of the coordinate space Rn at x is equal to the vector space Rn itself. In particular, for any real number u the tangent space DRU is equal to R. Thus if f : M -> R is a smooth real valued function, then the derivative Dfx : DMX -> DR^X^ = R can be thought of as an element of the dual vector space HomR(DMx ,R ) . This element Dfx of the dual space, sometimes called the “ total differ­ en tial’ ’ of f at x, is more commonly denoted by df(x). Note that Leib­ niz’s rule is satisfied : D(fg)x = f(x)Dgx + g(x)Dfx , where fg stands for the product function x h- f(x) g(x). For any tangent vector v € DMX the real number Dfx(v) is called the directional d erivative of the real valued function f at x in the di­ rection v. If we keep (x, v) fixed but let f vary over the vector space C°°(M, R) consisting of all smooth real valued functions on M, then a linear differential operator X : C°°(M, R) -> R can be defined by the formula X (f) = Dfx(v). L eib niz’s rule now takes the form X(fg) = f(x) X(g) + X(f ) g(x)

10

CHARACTERISTIC CLASSES

In many expositions of the subject, the tangent vector (x, v) is identified with this linear operator X. One defect of the above presentation is that the “ sm oothness” of a manifold M is made to depend on some particular embedding of M in a coordinate space. It is possible however to canonically embed any smooth manifold M in one preferred coordinate space. Given a smooth manifold M C R^ let F = C°°(M, R) denote the set of a ll smooth functions from M to the real numbers R. Define the embedding i : M -> RF by if(x) = f(x). Let LEMMA 1.5.

denote the image i(M) C R .

This image

F is a smooth manifold in R , and

the canonical map i : M -> Mj is a diffeomorphism, The proof is straightforward. Thus any smooth manifold has a canonical embedding in an associated coordinate space. This suggests the following definition. Let M be a set and let F be a collection of real valued functions on M which separates points. (That is, given x ^ y in M there exists f f F with f(x) ^ f(y).) Then M can be identified with its image under

p

the canonical imbedding i : M -> R . DEFINITION. The collection F is a smoothness structure on M if the subset i(M) C R

p

is a smooth manifold, and if F is precisely the

set of a ll smooth real valued functions on this smooth manifold.* N ote: This definition of “ sm oothness” is sim ilar to that given by [Nomizu]. In the c la s s ic a l point of view the “ smoothness structure” of a manifold is prescribed by the collection of local param etrizations. (See

*

If only the fir s t con d itio n is s a tis fie d , then F m ight be c a lle d a “ b a s is ’ ’ for a sm ooth n ess stru ctu re on M.

11

§1. SMOOTH MANIFOLDS

for example [Steenrod, 19 5 1, p. 21].) In s till another point of view, one uses collections of smooth functions on open subsets. (Compare [de Rham].) A ll of these definitions are equivalent. In conclusion here are three problems for the reader. The first two of these w ill play an important role in later sections. Problem 1-A. Let C R^ and M2 C R® be smooth manifolds. Show A B that Mj x M2 C R x R is a smooth manifold, and that the tangent mani­ fold DCMj x M2) is canonically diffeomorphic to the product DMj x DM2 Note that a function x k (fj (x), f2 (x)) from M to Mj x M2 is smooth if and only if both f^ : M ->

and f2 : M

M2 are smooth.

Problem 1-B. Let P n denote the set of a ll lin es through the origin in the coordinate space Rn+1. Define a function q: Rn+1 - |0!

Pn

by q(x) = Rx = line through x. Let F denote the set of a ll functions f : P n -> R such that f ° q is smooth. a) Show that F is a smoothness structure on P n. The resulting smooth manifold is called the real projective space of dimension n. b) Show that the functions f-j(Rx) = x^Xj/Sx^ define a diffeomorphism between P n and the submanifold of R(n+1)2 consisting of a ll symmetric (n+1) x (n+1) matrices

A

of trace 1 satisfyin g AA = A.

c) Show that P n is compact, and that a subset V C P n is open

if

and only if q- 1 (V) is open. Problem 1-C. For any smooth manifold M show that the collection F = C°°(M, R) of smooth real valued functions on M can be made into a ring, and that every point x e M determines a ring homomorphism F -» R and hence a maximal ideal in F. If M is compact, show that every maxi­ mal ideal in F arises in this way from a point of M. More generally, if there is a countable basis for the topology of M, show that every ring

12

CHARACTERISTIC CLASSES

homomorphism F -> R is obtained in this way. (Make use of an element f> 0

in F such that each f 1 [0, c] is compact.) Thus the smooth

manifold M is com pletely determined by the ring F. For x f M, show that any R -linear mapping X : F -> R satisfyin g X(fg) = X(f)g(x)+ f(x)X(g) is given by X(f) = Dfx(v) for some uniquely determined vector v e DMX.

§2. Vector Bundles Let B denote a fixed topological space, which w ill be called the base space. D E F IN IT IO N .

A real vector bundle £ over B con sists of the follow ­

ing: 1) a topological space E - E(£) called the total space, 2) a (continuous) map n : E

B called the projection map, and

3) for each b e B the structure of a vector space* over the real num­ bers in the set 77_ 1 (b). These must sa tisfy the following restriction: Condition of local triv ia lity . For each point b of B there should exist a neighborhood U C B, an integer n > 0, and a homeomorphism h : U x R n -» n-_ 1 (U )

so that, for each b e U, the correspondence x h- h(b, x) defines an iso ­ morphism between the vector space Rn and the vector space Tr^^b). Such a pair (U, h) w ill be called a lo cal coordinate system for £ about b. If it is possible to choose U equal to the entire base space, then £ w ill be called a trivial bundle. The vector space n~*(b) is called the fiber over b. It may be de­ noted by

or F^(^). Note that F^ is never vacuous, although it may

consist of a single point. The dimension n of F^ is allowed to be a

To be more p re c is e th is v e c to r sp a c e stru ctu re could be s p e c ifie d by g ivin g the su b se t of R x R x E x E x E c o n s istin g of a ll 5-tu p les (ti> t2 ' e l ,e 2 ' e 3^ w ith n t e j) = 7r(e2 ) = n(e 3 )

13

and

e^ = t^e j + t2 e 2 .

14

CHARACTERISTIC CLASSES

(locally constant) function of b; but in most ca ses of interest this func­ tion is constant. One then speaks of an n-plane bundle, or briefly an Rnbundle. The concept of a smooth vector bundle can be defined sim ilarly. One requires that B and E be smooth manifolds, that 77 be a smooth map, and that, for each b e B there ex ist a local coordinate system (U, h) with b e U such that h is a diffeomorphism. REMARK. An Rn-bundle is a very special example of a fiber bundle. (See [Steenrod, 19 5 1, p. 9].) In Steenrod’s terminology an Rn-bundle is a fiber bundle with fiber Rn and with the full linear group GLn(R) in n variables as structural group. Now consider two vector bundles £ and rj over the same base space B. DEFINITION, f

is isomorphic to rj, written M is d e fin e d b y

n (x,v) = x;

a n d th e v e c t o r s p a c e s t r u c t u r e in

77- 1 ( x )

is

d e fin e d b y

tjCxjVj) + t2(x,v2) = ( x^ j Vj + t2v 2) .

The local triviality condition is not difficult to verify. Note that

is

an example of a smooth vector bundle. If 7jyj is a trivial bundle, then the manifold M is called parallelizable. For example suppose that M is an open subset of Rn. Then DM is equal to M x Rn, and M is clearly p arallelizable. The unit 2-sphere S2 C R3 provides an example of a manifold which is not p arallelizable. (Compare Problem 2-B.) In fact we w ill see in §9 that a parallelizable manifold must have Euler ch aracteristic zero, whereas the 2-sphere has Euler characteristic +2. (See Corollary 9.3 and Theorem 1 1 .6 .) Example 3. The normal bundle v of a smooth manifold M C Rn is obtained as follow s. The total space E C M x Rn is the set of a ll pairs (x, v) such that v is orthogonal to the tangent space DMx . The projection map 77: E -> M and the vector space struc­ ture in 77_ 1 (x) are defined, as in Examples 1, 2, by the formulas n(xf v) = x, and tj(x , Vj) + t2(x, v2) = (x, tjV j + t2v2). The proof that v sa tisfie s the lo cal triviality condition w ill be deferred until §3.4. Example 4. The real projective space P n can be defined* as the set of all unordered pairs |x, - x l where x ranges over the unit sphere Sn C Rn+1; and is topologized as a quotient space of Sn. *

A lte rn a tiv e ly

P

n

can be d efin ed a s the s e t of lin e s through the origin in

Rn+1. (Compare Problem 1-B .) T h is am ounts to the sam e thing sin c e e v ery such lin e cu ts S n in two a n tip o d al p oin ts.

CHARACTERISTIC CLASSES

16

Let ECy^) be the subset of P n x Rn+1 consisting of a ll pairs ({±x}, v) such that the vector v is a multiple of x. Define n : E(y*)->Pn by 77(l±xi, v) = i±x|. Thus each fiber ^_ 1 ({±xi) can be identified with the line through x and —x in Rn+1. Each such line is to be given its usual vector space structure. The resulting vector bundle y* w ill be called the canonical line bundle over P n. Proof that y* is lo cally trivial. Let U C Sn be any open set which is small enough so as to contain no pair of antipodal points, and let U1 denote the image of U in P n. Then a homeomorphism h :U x x R -> is defined by the requirement that h(i± x !,t) = (i± x !,tx) for each (x, t) e U x R. Evidently (U j, h) is a local coordinate system ; hence y* is lo cally trivial. THEOREM 2.1.

The bundle y* over P n is not triv ia l, for

n > 1. This w ill be proved by studying cross-section s of y*. D E F IN IT IO N .

A cross-section of a vector bundle £ with base space

B is a continuous function s : B -> E(£) which takes each b e B into the corresponding fiber F^(^). Such a cross-section is nowhere zero if s(b) is a non-zero vector of F j ^ )

for

each b. (A cross-section of the tangent bundle of a smooth manifold M is usually called a vector field on M.) Evidently a trivial R1 -bundle p o sse sse s a cross-section which is no­ where zero. We w ill see that the bundle y* has no such cross-section.

§2. VECTOR BUNDLES

17

Let s : P n ^ E(y* ) be any cross-section, and consider the composition

sn ---, pn

E(y*)

which carries each x e Sn to some pair (i±xi, t(x)x) f E(y*) . Evidently t(x) is a continuous real valued function of x, and t( -x ) = - t ( x ) . Since Sn is connected it follow s from the intermediate value theorem that t(xQ) = 0 for some xQ.

Hence s({±xQ}) = ({± xQ!, 0). This completes

the proof. ■ It is interesting to take a clo ser look at the space E(yJj) for the spe­ cial case n = 1. In this case each point e = (l± xj, v) of E(y*) can be written as e = (|±(cos 0, sin 6)\, t(cos 6 , sin d)) with 0 < 0 E(rj) be a continuous function which maps each v ec­ tor space F ^ ( f ) isom orphically onto the corresponding vector space F^( 77).

Then f is n e ce ssa rily a homeomorphism. Hence

is isomorphic to 77.

§2. VECTOR BUNDLES

19

Proof. Given any point bQ e B, choose local coordinate system s (U, g) for £ and (V, h) for rj, with bQ e U fl V. Then we must show that the composition ( u n v ) x Rn — - - - °-f ° g > ( u n v ) x Rn is a homeomorphism. Setting h- 1 (f(g(b, x » ) = (b, y) it is evident that y = (y1? ..., y n) can be expressed in the form

?i =

X

fi j( t ) x j

j where [fy(b)] denotes a non-singular matrix of real numbers. Further­ more the entries fjj(b) depend continuously on b. Let [F-j(b)] denote the inverse matrix. Evidently g_ 1 o f- 1 oh(b,y) = (b, x) where Xj = 2

F ji (b )yi ' i

Since the numbers Fj-(b) depend continuously on the matrix [fjj(b)], they depend continuously on b. Thus g“ 1 o f” 1 oh is continuous, which completes the proof of 2.3. ■ P roof of Theorem 2.2. Let s 1#. . . , s

be cross-section s of f

are nowhere linearly dependent. Define f : B x Rn -> E by f(b, x) = XjSjCb) + ... + xns n(b)

which

20

CHARACTERISTIC CLASSES

Evidently f is continuous and maps each fiber of the trivial bundle e ” B isom orphically onto the corresponding fiber of Hence f is a bundle isomorphism, and £ is trivial. C onversely suppose that £ is trivial, with coordinate system (B, h). Defining Si(b) = h(b, ( 0 ,..., 0, 1, 0 , . . . , 0)) £ F b( f ) (with the 1 in the i-th place), it is evident that s 1#..., s n are nowhere dependent cross-section s. This completes the proof. ■ As an illustration, the tangent bundle of the circle S1 C R 2 admits one nowhere zero cross-section, as illustrated in Figure 3. (The indicated arrows lead from x e S 1 to x + v, where s(x) = (x, v) = ((x1?x2), (~x2 ,x 1 )).) Hence S 1 is parallelizable. Sim ilarly the 3-sphere S3 C R 4 admits three nowhere dependent vector field s s-(x) = (x, s’-(x)) where Sj (x) = ( - x 2 , x 1 , - x 4 , x 3) s 2 (x) = ( - x 3 , x 4 , x 1 , - x 2) S 3 (x) = ( - x 4 , - x 3 , x 2 , x 1) . Hence S3 is parallelizable. (These formulas come from the quaternion multiplication in R4 . Compare [Steenrod, 19 51, §8.5].)

F ig u re 3.

§2. VECTOR BUNDLES

21

Euclidean Vector Bundles For many purposes it is important to study vector bundles in which each fiber has the structure of a Euclidean vector space. R ecall that a real valued function p on a finite dimensional vector space V is quadratic if p can be expressed in the form

/*(v) = ^

where each £• and each

^i(v) £j(v)

is linear. Each quadratic function determines

a symmetric and bilinear pairing v, w h> v • w from V x V to R, where V • W = i- 0 for v ^ 0. DEFINITION. A Euclidean vector space is a real vector space V together with a p o sitive definite quadratic function p : V -> R . The real number v • w w ill be called the inner product of the vectors v and w. The number v • v = p(y) may also be denoted by |v|2 . DEFINITION. A Euclidean vector bundle is a real vector bundle together with a continuous function li- ECO - R such that the restriction of p to each fiber of £ is p o sitive definite and quadratic. The function p its e lf w ill be called a Euclidean metric on the vector bundle In the case of the tangent bundle dean metric p : DM -> R

of a smooth manifold, a Eucli­

CHARACTERISTIC CLASSES

22

is called a Riemannian metric, and M together with p is called a R iemannian manifold. (In practice one usually requires that ft be a smooth function. The notation p = d s2 is often used for a Riemannian metric.) Note. In Steenrod’s terminology a Euclidean metric on £ gives rise to a reduction of the structural group of f

from the full linear group to

the orthogonal group. Compare [Steenrod, 19 51, §12.9]. Examples. The trivial bundle e ” can be given the Euclidean metric fc>

/x(b, x) = Xj + ... + x£ . Since the tangent bundle of Rn is trivia l it follows that the smooth mani­ fold Rn p o sse sse s a standard Riemannian metric. For any smooth mani­ fold M C Rn the composition DM C DRn - !L R now makes M into a Riemannian manifold. A priori there appear to be two different concepts of triviality for Euclidean vector bundles; however the next lemma shows that these coin­ cide. LEMMA 2.4. L et

be a trivial vector bundle of dimension n

over B, and let p. be any Euclidean metric on £. Then there ex ist n cross-section s s 1 , . . . , s n of £ which are normal and orthogonal in the sen se that s-(b) ■ Sj(b) = S-j

(= Kronecker delta)

for each b f B. Thus £ is trivial also as a Euclidean vector bundle. (Compare Problem 2-E below.)

§2. VECTOR BUNDLES

Proof. Let s ' ^ .- .j S '

23

b e a n y n cross-section s which are nowhere

linearly dependent. Applying the Gram-Schmidt* process to s ' ^ b ) , s ' n(b) we obtain a normal orthogonal basis s^ b ), . . . , s n(b) for the resulting functions S j, ..., s

). Since

are clearly continuous, this completes

the proof. ■ Here are six problems for the reader. Problem 2-A. Show that the unit sphere Sn admits a vector field which is nowhere zero, providing that n is odd. Show that the normal bundle of Sn C Rn+1 is trivial for all n. Problem 2-B. If S n admits a vector field which is nowhere zero, show that the identity map of Sn is homotopic to the antipodal map. For n even show that the antipodal map of Sn is homotopic to the reflection r(xx, ..., xfl+1) = ( - X j, x2 , ..., xn+1) ; and therefore has degree —1. (Compare [Eilenberg and Steenrod, p. 304].) Combining these facts, show that Sn is not parallelizab le for n even, n > 2. Problem 2-C. E xistence theorem for Euclidean metrics. Using a par­ tition of unity, show that any vector bundle over a paracompact base space can be given a Euclidean metric. (See §5.8; or see [K elley, pp. 156 and 171].) Problem 2-D. The Alexandroff line L (sometimes called the “ long lin e ” ) is smooth, connected, 1-dimensional manifold which is not para­ compact. (Reference: [Kneser].) Show that L cannot be given a R ie­ mannian metric.

See any te x t book on lin e a r alg eb ra.

24

CHARACTERISTIC CLASSES

Problem 2-E. Isometry theorem. Let fi and fi' be two different Euclidean metrics on the same vector bundle f . Prove that there exists a homeomorphism f : E(£) -> E(£) which carries each fiber isom orphically onto itse lf, so that the composition /i ° f : E(R is equal to [ i [Hint: Use the fact that every positive definite matrix A can be expressed uniquely as the square of a positive definite matrix y/K. The power series expansion

v/CtTTx) = VF(i + IIt x - Jgt2- x 2 + is valid providing that the characteristic roots of tl + X = A lie between 0 and 2t. This shows that the function A t-» >/A is smooth.] Problem 2-F. As in Problem 1-C, let F denote the algebra of smooth real valued functions on M. For each x e M let I ^ 1 be the ideal con­ sisting of all functions in F whose d erivatives of order < r vanish at x. An element of the quotient algebra F/l£+1 is called an t-jet of a real valued function at x. (Compare [Ehresmann, 1952].) Construct a lo cally trivia l “ bundle of alg eb ras”

over M with typ ical fiber F/l£+1.

§3. Constructing New Vector Bundles Out of Old This section w ill describe a number of basic constructions involving vector bundles. (a) R estricting a bundle to a subset o f the base space. L et £ be a vector bundle with projection n\ E

B and let B be a subset of B.

Setting E = Tr^^B), and letting 77 E -» B be the restriction of n to E, one obtains a new vector bundle which w ill be denoted by f| B , and c a ll the restriction of fiber

B) is equal to the corresponding fiber

to B. Each ), and is to be

given the same vector space structure. As an example if M is a smooth manifold and U is an open subset of M, then the tangent bundle rjj is equal to ^|U . More generally one has the following construction. (b) Induced bundles. Let f

be as above and let B j be an arbitrary

topological space. Given any map f : B j -> B one can construct the in­ duced bundle f*£ over B r

The total space Ej of f*£ is the subset

Ej C B1 x E consisting of a ll pairs (b, e) with f(b) = n(e) . The projection map

77^

: Ej

B ^ is defined by 77^ (b, e) = b. Thus one

has a commutative diagram f

26

CHARACTERISTIC CLASSES

where f(b, e) = e. The vector space structure in

1 (b) is defined by

t x(b, e x) + t2(b ,e 2) = ( b ,t1 e 1 + t2 e2 ) . Thus f carries each vector space F b(f*£) isom orphically onto the vec­ tor space If (U, h) is a lo cal coordinate system for

set Uj = f _ 1 (U) and

define h p U j x R " - ir^ C U j) by hj(b, x) = (b, h(f(b), x)). Then (1^ ,1^ ) is clearly a lo cal coordinate system for f*£. This proves that f*£ is lo cally trivial. (If E( R satisfyin g the symmetry relations: K(v 1 , v 2 , v 3 , v 4) = K(v3 , v 4 , V l, v 2) = - K ( v 1 , v 2 , v 4 , v 3) and K (v l ' v 2 ' V3> V4^ + K ( y l ' V4 ' V2> v 3^ + K ( V1 ' v 3 ' v 4 ' v 2^ = 0 *

(This la st example would be rather far-fetched, were it not important in the theory of Riemannian curvature.) These examples suggest that we consider a general functor of several vector space variables.

S ee for exam ple [Lang, pp. 40 8, 424],

CHARACTERISTIC CLASSES

32 Let 0

denote the category consisting of all finite dimensional real

vector spaces and all isomorphisms between such vector spaces. By a (covariant)* functor T : D x 0 -> 0 1) to each pair V, W e 0

is meant an operation which assigns

of vector spaces a vector space T(V, W ) e ( 3 ;

and 2) to each pair f : V

V', g : W -> W' of isomorphisms an isomorphism

T(f, g) : T(V, W) -> T(V', W) ; so that 3) T (identityy, identity^) = id e n tity ^ y ^

and

4) TCfj o f2 , gl o g2) = T(flf g l) o T(f2 , g'). Such a functor w ill be called continuous if T(f, g) depends continuously on f and g. This makes sense, since the set of a ll isomorphisms from one finite dimensional vector space to another has a natural topology. The concept of a continuous functor T : 0 x . . . x 0 - > 0

in k variables

is defined sim ilarly. Note that examples 1, 2, 3 above are continuous functors of two variables, and that examples 4, 5, 6 are continuous func­ tors of one variable. Let T : 0 x ...x 0 and let

0

be such a continuous functor of k variables,

be vector bundles over a common base space B. Then

a new vector bundle over B is constructed as follows. For each b e B let F b = T(Fb( ^ ) , . . . , F b( 4 ) ) . Let E denote the disjoint union of the vector spaces

and define

77: E -» B by 77(F ^ )= b .

THEOREM 3.6.

There ex ists a canonical topology for E so

that E is the total space of a vector bundle with projection n and with fibers F^.

The d is tin c tio n b etw een c o v a ria n t and c o n tra va ria n t fu n ctors is not im portant h ere, sin ce we are w orking only w ith isom orphism s.

§3. CONSTRUCTING NEW BUNDLES

33

DEFINITION. This bundle w ill be denoted by T ^ , For example starting with the tensor product functor, this construction defines the tensor product £ ® rj of two vector bundles. Starting with the direct sum functor one obtains the Whitney sum £ © rj of two bundles. Starting with the duality functor V t-> Horn (V, R) one obtains the functor f h> H o r n e s 1) which assigns to each vector bundle its dual vector bundle. The proof of 3.6 w ill be indicated only briefly. Let (U ,!^), ...,(U ,hk ) be local coordinate system s for

resp ectively, a ll using the

same open set U. For each b e U define

h ib ; Rn‘ - >Vfi> by hjbOO = hj(b, x). Then the isomorphism T(hl b , . . . , h kb) : T(Rn\ . . . , R nk) - F b is defined. The correspondence (b, x) i-» T(hl b , . . . ,h kb)(x) defines a one-to-one function h : U x T(Rn i , . . . , R nk) -» n-_ 1 (U) . ASSERTION. There is a unique topology on E so that each such h is a homeomorphism, and so that each n~*(11) is an open subset of E. P roof. The uniqueness is clear. To prove existence, it is only neces­ sary to observe that if two such “ coordinate system s” (U, h) and (IT, h') overlap, then the transformation

34

CHARACTERISTIC CLASSES

(u n u o x t c r " 1 , . . . , R °k)--h l o h / >( u n u ' ) x t c r " 1 , . . . , Rnk) is continuous. This follow s from the continuity of T. It is now clear that tt: E -> B is continuous, and that the resulting vector bundle T(£x, REMARK

sa tis fie s the local triviality condition. ■

1. This construction can be translated into Steenrod’s

terminology as follow s. Let GLn = GLn(R) denote the group of automor­ phisms of the vector space Rn. Then T determines a continuous homo­ morphism from the product group GLn^ x ...x automorphisms of the vector space T(Rn i ,

to the group GL' of Rnk). Hence given bundles

over B with structural groups GLf l l , ..., G L ^ resp ectively, there corresponds a bundle T ^ , . . . , ^ ) with structural group GL' and n< n^ with fiber T(R , . . . , R ). For further discussion, see [Hirzebruch, 1966, §3.6]. REM ARK

2. Given bundles

over distinct base spaces, a

sim ilar construction gives rise to a vector bundle T ^ , BC^) x ...x B(^k), with typical fiber T(Fb i ( f j F

over

bk N be a smooth map. Then Horn (r^j,f*r^) is a smooth vector bundle over M. Note that Df gives rise to a smooth cross-section of this vector bundle.

§3. CONSTRUCTING NEW BUNDLES

35

As a second illustration, if M C N with normal bundle v, where N is a smooth Riemannian manifold, then the “ second fundamental form” can be defined as a smooth symmetric cross-section of the bundle Horn (r^j® (Compare [Bishop and Crittenden], as w ell as Problem 5-B.) Here are six problems for the reader. Problem 3-A. A smooth map f : M -» N between smooth manifolds is called a submersion if each Jacobian Dfx : DMX - DNf(x) is su rjective (i.e., is onto). Construct a vector bundle ///£ • Problem 3-C. More generally let P n is clearly covered by a bundle map from y j

to y * . Therefore

j*wi(y„) = Wi(yJ) ^ o . This shows that WjCy*) cannot be zero, hence must be equal to a. Since the remaining Stiefel-W hitney c la s se s of y* are determined by Axiom 1, this completes the proof. ■ Example 3. By its definition, the line bundle y* over P n is con­ tained as a sub-bundle in the trivial bundle s n+1. Let y^ denote the orthogonal complement of y* in e n+1. (Thus the total space E(y^) con sists of a ll pairs ({±x}, v) e P n x R n+1 with v perpendicular to x.) Then w(y^) = 1 + a + a 2 + ... + a 11 . Proof. Since y* © y^ is trivial we have w(y^) = w(y*) = ( l + a )- 1 = l + a + a 2 + ... + an . « This example shows that a ll of the n Stiefel-Whitney c la sse s of an Rn-bundle may be non-zero. Example 4. L et r be the tangent bundle of the projective space P n. LEM M A

4.4.

The tangent bundle

t

of P n is isomorphic to

44

CHARACTERISTIC CLASSES

Proof. Let L be a line through the origin in Rn+1, intersecting Sn -L 114-1 in the points ± x, and let L C R ^ be the complementary n-plane. Let f : Sn -> P n denote the canonical map, f(x) = \±x\. Note that the two tan­ gent vectors (x, v) and (—x, —v) in DSn both have the same image under the map D f: DSn -» DPn which is induced by f. (Compare Figure 5.) Thus the tangent manifold DPn can be identified with the set of all pairs {(x, v), (—x, —v)} sa tis fy ­ ing x • x = 1,

x •v = 0 .

F ig u re 5.

§4. STIEFEL-WHITNEY CLASSES

45

But each such pair determines, and is determined by, a linear mapping £ :L -» L 1 , where £(x) = v . Thus the tangent space of P n at j± xi is canonically isomorphic to the vector space Hom(L, L^).

It follow s that the tangent vector bundle

t

is canonically isomorphic to the bundle Horn (y1 , y^). This completes the proof of 4.4. ■ We cannot compute w(Pn) directly from this lemma since we do not yet have any procedure for relating the Stiefel-W hitney c la sse s of HomCy1 , y^~) to those of y 1 and y k

However the computation can be

carried through as follow s. Let s 1 be a trivial line bundle over P n. TH EO REM

4.5.

The Whitney sum r e s 1 is isomorphic to the

(n+l)-/o/cf Whitney sum y 1 © y 1 © ... © y 1 . Hence the total Stiefel-W hitney c la s s of P n is given by w(Pn) = ( l + a )n+1 = 1 + C11!"1 ) a + (nt } ) a 2 + ... + (n+1 ) a n . 1 Z n Proof. The bundle Hom(y 1 , y 1 ) is trivial since it is a line bundle with a canonical nowhere zero cross-section. Therefore t®

e 1 as HomCyJj, y 1 ) ® HomCy^yJ,) .

This is clearly isomorphic to Horn (y*, y 1 ® y*) ~ Horn {y\, e n+1) , and therefore is isomorphic to the (n+l)-fold sum HomCy1 , s 1 © ... © e 1 ) s HomCy1 , e1 ) © ... © HomCy1 , s 1)

46

CHARACTERISTIC CLASSES

But the bundle HomCy1 , s 1) is isomorphic to y 1 , since y 1 has a Euclidean metric. (Compare Problem 3-D.) This proves that r e s 1 s* y 1 © . .. © y 1 . — /n 'n Now the Whitney product theorem implies that w (r) = w(r © e 1) is equal to w(yj,)

w(y*) = (1 + a ) n+1 .

Expanding by the binomial theorem, this completes the proof of 4.5. ■ Here is a table of the binomial coefficien ts (^ t1 ) modulo 2, for n < 14. 1 1

P1:

1 0

P2:

1

p 3: P4 :

1

p 5:

1 0

p6 :

1

p7:

1

p8:

p9:

1

1

0

1

0

1

1

0

0 0

1 0

1

1

0

1

0

1 1 1

1 0

1

1

1

1

1

1

0

0

0

0

0

1

1

1

0

0

0

0

0

0

1

1

1

1

1

0

0

0

0

1

1

1

1 0 1 0 0 0 0 0 1 0 1

P 10

1 1

p 11 pi 2

1

pi 3 p!4.

1

1 1

0 1

0 1

0 0

1 1

0 0

0 1

1 1

1 1

0 1

0 1

0 0

1 1

0 0

0 1

1 1

1 1

0 1

0 1

0 0

1 1

0 0

0 1

1 1 1 1

1

1 0

1

1 1

The right hand edge of this triangle can be ignored for our purposes since Hn+1 (P n; Z/2) = 0. As examples one has:

1

§4. STIEFEL-WHITNEY CLASSES

47

w(P2) = 1 + a + a 2 w (P3) = 1 and w(P4 ) = 1 + a + a 4 . COROLLARY

4.6 (Stiefel).

The c la ss w(Pn) is equal to 1 if

and only if n + 1 is a power of 2.

Thus the only projective

3 spaces which can be p arallelizab le are P 1 ,P ,P 7 ,P 1 5 ,. . . .

1 o *7 (We w ill see in a moment that P , P , and P actually are p a ralle l­ izable. On the other hand it is known that the higher projective spaces p l 5, P 3 1 , ... are not parallelizable. See [Bott-Milnor], [Kervaire, 1958], [Adams, I960].) 9

Proof. The identity (a + b) = a ( 1 + a)

9r

2

+b

9

= 1 +a

modulo 2 implies that 0r

Therefore if n + 1 = 2 r then w(Pn) = (1 + a )n+1 = 1 + a n+1 = 1 . C onversely if n + 1 = 2 rm with m odd, m > 1, then w(Pn) = ( l + a )n+1 = ( l + a2r)m = 1 + m a2' + !2 (!2 ^pi) a 2 ’ 2r + ...

1 ,

since 2 r < n + 1. This completes the proof. ■ D ivision algebras C losely related is the question of the existence of real division alge­ bras.

48

CHARACTERISTIC CLASSES

THEOREM 4.7 (S tie fe l). Suppose that there exists a bilinear

product operation* p:Rn x R n , R n without zero divisors.

Then the projective space P n _ 1

is

p arallelizable, hence n must be a power of 2 . In fact such division algebras are known to exist for n = 1, 2, 4, 8 : namely the real numbers, the complex numbers, the quaternions, and the C ayley numbers. It follow s that the projective spaces P 1 , 3P and P 7 are p arallelizable. That no such division algebra exists for n > 8 fol­ lows from the references cited above on p arallelizab ility. Proof of 4.7. Let b j, •••>bn be the standard basis for the vector space Rn. Note that the correspondence y ^ pCy.bj) defines an isomorphism of Rn onto itse lf. Hence the formula v ^ p C y,^ )) = p(y, bp defines a linear transformation v. : Rn - Rn . Note that Vj(x), ..., v (x) are linearly independent for x ^ 0, and that v x(x)

= X.

The functions v 2 , •••> v n g*ve fise to n — 1 linearly independent cross-sections of the vector bundle rp n —

1

-

Hom( y L l ’ >/ i> •

In fact for each line L through the origin, a linear transformation

T h is product op eration is n ot req u ired to be a s s o c ia tiv e , or to h a v e an id en tity elem ent.

§4. STIEFEL-WHITNEY CLASSES

49

7 j : L -> L 1 is defined as follow s. For x f L ,

let v^(x) denote the image of v^(x)

under the orthogonal projection Rn -» L 1 . C learly Vj = 0, but v 2 , . . . , v n are everywhere linearly independent. Thus the tangent bundle r n _ 1 is a trivial bundle. This completes the proof of 4.7. ■ Immersions As a final application of 4.5, let us ask which projective spaces can be immersed in the Euclidean space of a given dimension. If a manifold M of dimension n can be immersed in the Euclidean space Rn+k then the Whitney duality theorem w

= Wj(M)

implies that the dual Stiefel-Whitney c la sse s w^(M) are zero for i > k. As a typical example, consider the real projective space P 9 . Since w(P9) = (1 + a ) 1 0 = 1 + a 2 + a 8 we have w(P9) = 1 + a 2 + a 4 + a 6 . Thus if P 9 can be immersed in R9+^, then k must be at least 6 . The most striking results for P n are obtained when n is a power of 2. If n = 2r then w(Pn) = (1 + a) n+1 = 1 + a + a 11 , hence w(Pn) = l + a + a 2 + ... + a 11” 1 . Thus:

CHARACTERISTIC CLASSES

50

THEOREM 4.8. If P 2t can be immersed in R2f+^,

then k

must be at le a st 2 r — 1. On the other hand Whitney has proved that every smooth compact manifold of dimension n > 1 can actually be immersed in R211” 1 . (Reference: [Whitney, 1944].) Thus Theorem 4.8 provides a best possible estim ate. Note that estim ates for other projective spaces follow from 4.8. For example since P 8 cannot be immersed in R1 4 , it follow s a fortiori that P 9 cannot be immersed in R 1 4 . This duplicates the earlier estim ate concerning P 9 . See [Jam es]. An extensive and beautiful theory concerning immersions of manifolds has been developed by S. Smale and M. Hirsch. For further information the reader should consult [Hirsch, 1959] and [Smale, 1959]. Stiefel-W hitney Numbers We w ill now describe a tool which allow s us to compare certain StiefelWhitney c la s se s of two different manifolds. Let M be a closed, possibly disconnected, smooth n-dimensional manifold. Using mod 2 coefficien ts, there is a unique fundamental homology c la s s fiM c Hn(M; Z/2) . (See Appendix A .) Hence for any cohomology c la ss v e Hn(M; Z/2), the Kronecker index 6 Z/2

is defined. We w ill sometimes use the abbreviated notation v[M] for this Kronecker index. Let rl ; ..., r

be non-negative integers with r1 + 2r 2 + ... + nrn = n.

Then corresponding to any vector bundle f

we can form the monomial

W^ / 1 ... wn( f ) rn

§4. STIEFEL-WHITNEY CLASSES

51

in Hn(B (f); Z/2). In particular we can carry out this construction if £ is the tangent bundle of the manifold M. DEFINITION. The corresponding integer mod 2 rl

y n(Rm) it is sufficient to construct a map ?:E («f) - Rm which is linear and injective (i.e ., has kernel zero) on each fiber of The required function f can then be defined by f(e) = (f (fiber The continuity of f is not difficult f

through e), f(e)) . to verify, making use

e fact of that th

is lo cally trivial. Proof of 5.3.

Choose open sets U j, ...,U r covering B so that each

f |Uj is trivial. Since B is normal, there exist open sets

V j , V f

covering B with \L C Uj. (Compare [K elley, p. 171]. Here V- denotes the closure of V-.) Sim ilarly construct Ai :B -> R

with W| C V|. Let

62

CHARACTERISTIC CLASSES

denote a continuous function which takes the value 1 on

and the

value 0 outside of V^. Since £\ U- is trivial there ex ists a map h j : 77—1 Uj - Rn which maps each fiber of f |Uj linearly onto Rn. Define h'• : E (f) -> Rn by h j(e) = 0

for 77(e) / Vj

h j(e) = A|(7r(e))h|(e)

for 77(e) e Uj .

Evidently h^ is continuous, and is linear on each fiber. Now define f :E(£) - Rn ©... © Rn = Rrn by f(e) = (h'^e), h 2 ( e ) ,..., h'r(e)). Then f is also continuous and maps each fiber in jectively. This completes the proof of 5.3. ■ Infinite Grassmann Manifolds A sim ilar argument applies if the base space B is paracompact and finite dimensional.(Compare Problem 5-E.) However inorder to of bundles

take care

over more exotic base spaces it is necessary to allow the di­

mension of Rn+k to tend to infinity, thus yielding an infinite Grassmann “ manifold” Gn(R°°). Let R°° denote the vector space consisting of those infinite s e ­ quences x = (x 1 , x 2 , x 3 , ...) of real numbers for which a ll but a finite number of the x- are zero. (Thus R°° is much sm aller than the infinite coordinate spaces utilized in § 1 .) For fixed k, the subspace consisting of a ll x = (x 1 , x 2 , . . . , x k, 0 , 0 ,. . .) w ill be identified with the coordinate space r K with union R°°.

Thus R 1 C R 2 C R 3 C ...

§5. GRASSMANN MANIFOLDS AND UNIVERSAL BUNDLES

63

The infinite Grassmann manifold

D E F IN IT IO N .

Gn = Gn(R°°) is the set of a ll n-dimensional linear sub-spaces of R°°, topologized as the direct limit* of the sequence Gn(Rn) C Gn(Rn+1) C Gn(Rn+2) C ... . In other words, a subset of Gn is open [or closed] if and only if its inter­ section with Gn(Rn+k) is open [or closed] as a subset of Gn(Rn+^) for each k. This makes sen se since Gn(R°°) is equal to the union of thesubsets Gn(Rn+k). As a sp ecial case, the infinite projective space P°° = GjCR00) is equal to the direct limit of the sequence P 1 C P 2 C P 3 C ... . Sim ilarly R°° its e lf can be topologized as the direct limit of the s e ­ quence R 1 C R 2 C ... . The U niversal Bundle y n A canonical bundle y n over Gn is constructed, just as in the finite dimensional case, as follow s. Let E(yn) C Gn x R°° be the set of a ll pairs (n-plane in R°°, vector in that n-plane) , topologized as a subset of the C artesian product. Define tt: E(yn) -> Gn by n(X, x) = X, and define the vector space structures in the fibers as before.

*

It is cu stom ary in a lg e b ra ic top ology to c a ll th is the “ w eak to p o lo g y ,’ ’ a w eak top ology being one w ith many open s e ts . T h is u sa g e is u nfortunate sin ce a n a ly s ts u se the term w eak top ology w ith p re c is e ly the op p osite m eaning. On the other hand the term s “ fin e to p o lo g y ” or “ larg e to p o lo g y ” or “ W hitehead to p o lo g y” are c e rta in ly a c c e p ta b le .

64

CHARACTERISTIC CLASSES

LEMM A

5.4.

This vector bundle y n sa tisfie s the lo cal triviality

condition. The proof w ill be e sse n tia lly the same as that of 5.2. However the following technical lemma w ill be needed. (Compare [J. H. C. Whitehead, 19 6 1, §18.5].) LEMMA 5.5. Let A ^ C A 2 C ... and B j C B 2 C ... be sequences of lo cally compact spaces with direct limits A and B resp ec­ tively.

Then the C artesian product topology on A x B coincides

with the direct limit topology which is asso ciated with the s e ­ quence A j x B j C A 2 x B 2 C ... . Proof. Let W be open in the direct limit topology, and let (a, b) be any point of W. Suppose that ( a ,b ) f A j x B j .

Choose a compact neigh­

borhood K- of a in A- and a compact neighborhood L^ of b in B^ so that K| x L| C W. It is now possible (with some effort) to choose com­ pact neighborhoods K -+1 of K- in A *+1 and L ^+1 of L^ in B ^+1 so that Kj+1 x L -+1 C W. Continue by induction, constructing neighborhoods C K^+1 C Kj+2 C ... with union U and L- C L ^+1 C ... with union V. Then U and V are open se ts, and (a,b) f U x V C W . Thus W is open in the product topology, which completes the proof of 5.5. ■ Proof of Lemma 5.4. Let XQ C R °° be a fixed n-plane, and let U C Gn be the set of a ll n-planes Y which project onto XQ under the orthogonal projection p : R °° -> XQ. This set U is open since, for each finite k, the intersection

u k = u n G „ (R n + k) is known to be an open set. Defining

65

§5. GRASSMANN MANIFOLDS AND UNIVERSAL BUNDLES

h :U x XQ -> 77- 1 U as in 5.2, it follow s from 5.2 that h |

x XQ is continuous for each k.

Now Lemma 5.5 implies that h its e lf is continuous. As before, the identity h_ 1 (Y ,y ) = (Y, py) implies that h - 1

is con­

tinuous. Thus h is a homeomorphism. This completes the proof that y 11 is lo cally trivial. ■ The following two theorems assert that this bundle y n over Gn is a “ u n iversal” Rn-bundle. THEOREM 5.6. Any Rn-bundle £ over a paracompact base space admits a bundle map

-> y n.

Two bundle maps, f, g : £ -* y n are called bundle-homotopic if there ex ists a one-parameter family of bundle maps ht : £ - y n,

0 Max A_(b) at S “ a lS

§5. GRASSMANN MANIFOLDS AND UNIVERSAL BUNDLES

67

Let Uk be the union of those sets U(S) for which S has precisely k elements. C learly each

is an open set, and b = u 1 u u 2 u u 3 u ... .

For, given b e B, if precisely k of the numbers A.a (b) are positive, then b e U^- If a is any element of the set S, note that U(S) c v a . Since the covering iVa l is lo cally finite, it follow s that {U^l is lo cally finite. Furthermore, since each £| V £ |U(S) is trivial. But the set

is trivial, it follow s that each

is equal to the disjoint union of its

open subsets U(S). Therefore t;\

is also trivial. ■

The bundle map f : £ -> y n can now be constructed just as inthe proof of 5.3. D etails w ill be left to the reader. This proves 5.6. ■ Proof of Theorem 5.7. Any bundle map f :

-> y n determines a map

f:E (£ ) - R°° whose restriction to each fiber of

is linear and injective. C onversely

f determines f by the identity f(e) = (f (fiber through e), f(e)) . Let f, g : f -» y n be any two bundle maps. C ase 1. Suppose that the vector f(e) e R°° is never equal to a nega­ tive multiple of g(e) for e / 0, e 6 E(£). Then the formula ht (e) = (1 —t ) f (e) + tg(e),

0 < t< 1 ,

defines a homotopy between f and g. To prove that h is continuous as a function of both variab les, it is only necessary to prove that the vec­ tor space operations in R°° (i.e ., addition, and m ultiplication by sca la rs)

CHARACTERISTIC CLASSES

68

are continuous. But this follow s e a sily from Lemma 5.5. E vidently hj.(e) ^ 0 if e is a non-zero vector of E(^). Hence we can define h: E(£) x [0, 1] -> E(rj) by ht (e) = (ht(fiber through e), ht(e)) . To prove that h is continuous, it is sufficient to prove that the corre­ sponding function h :B ( £ ) x [0 ,1] - Gn on the base space is continuous. Let U be an open subset of B(£) with f |U trivial, and let s 1? . . . , s n be nowhere dependent cross-section s of f |U. Then h |U x [0 ,1] can be considered as the composition of 1 ) a continuous function b ,t ^ (h^SjCb),..., hts n(b)) from U x [0 , 1 ]

to the “ infinite S tiefel manifold” V n(R°°) C R°° x ... x R°°, and 2) the canonical projection q : Vn(R°°) -> Gn. Using 5.5 it is seen that q is continuous. Therefore h is continuous; hence the bundle-homotopy h between f and g is continuous. General C ase. Let f , g : £ -» y n be arbitrary bundle maps. A bundle map

dx : y n -> y 11 is induced by the linear transformation R°° -> R°° which carries the i-th basis vector of R°° to the (2i-l)-th. Sim ilarly d2 : y n -» y n is induced by the linear transformation which carries the i-th basis vector to the 2i-th. Now note that three bundle-homotopies f ~ dx of -

d2 og ~ g

are given by three applications of C ase 1. Hence f ~ g. ■ Characteristic C la sses of Real n-Plane Bundles Using 5.6 and 5.7, it is possible to give a precise definition of the concept of characteristic c la s s. F irst observe the following.

§5. GRASSMANN MANIFOLDS AND UNIVERSAL BUNDLES

69

COROLLARY 5.10. Any Rn-bundle £ over a paracompact space B determines a unique homotopy cla ss of maps f^r: B -> Gn . Proof. Let f

-*yn be any bundle map, and let

be the induced

map of base spaces. ■ Now let A be a coefficient group or ring and let c f H^G^A)

be any cohomology c la ss. Then £ and c together determine a cohomology c la ss f^*c e H*(B; A) . This c la s s w ill be denoted briefly by c(£).

DEFINITION. c ( f ) is called the ch aracteristic cohomology c la ss of determined by c. Note that the correspondence

c(£)

is natural with respect to

bundle maps. (Compare Axiom 2 in §4). C onversely, given any corre­ spondence H ^ J jA ) which is natural with respect to bundle maps, we have c ( f ) = ^ * c (y n) . Thus the above construction is the most general one. B riefly speaking: The ring consisting of a ll characteristic cohomology c la s s e s for Rnbundles over paracompact base spaces with coefficient ring A is canoni­ c a lly isomorphic to the cohomology ring H*(Gn; A). These constructions emphasize the importance of computing the cohomology of the space Gn< The next two sections w ill give one proce­ dure for computing this cohomology, at lea st modulo 2 .

70

CHARACTERISTIC CLASSES

REMARK. Using the “ covering homotopy theorem” (compare [Dold], [Husemoller]), Corollary 5 .1 0 can be sharpened as follows:

Two Rn-

bundles £ and rj over the paracompact space B are isomorphic if and only if the mapping fg of 5 .1 0 is homotopic to f^. Here are five problems for the reader. Problem 5-A. Show that the Grassmann manifold Gn(Rn+k) can be made into a smooth manifold as follow s: a function f : Gn(Rn+k) -> R be­ longs to the collection F of smooth real valued functions if and only if f °q : V n(Rn+k) -> R is smooth. Problem 5-B. Show that the tangent bundle of G n(R n+k) is isomor­ phic to Horn (yn(Rn+k), y^)-t where y ^ denotes the orthogonal comple­ ment of y n(R n+k) in e n+k. n ow consider a smooth manifold M C R n+k. If g : M -> G n(R n+k) denotes the generalized Gauss map, show that Dg : DM - DGn(Rn+k) gives rise to a cross-section of the bundle Horn (rM, Horn (rM, i/)) s Horn (rM ® rM, v) . (This cross-section is called the “ second fundamental form” of M.) Problem 5-C. Show that Gn(Rm) is diffeomorphic to the smooth mani­ fold consisting of a ll m x m symmetric, idempotent matrices of trace n. A ltern atively show that the map (x 1 , . . . , x n) h>x1 A ...A x n from V n(Rm) to the exterior power An(Rm) gives rise to a smooth em­ bedding of Gn(Rm) in the projective space G 1 (An(Rm)) = p ( n ) - 1

(Com­

pare van der Waerden, Einfiihrung in die algebraische Geometrie, Springer 1939, §7.)

§5. GRASSMAN MANIFOLDS AND UNIVERSAL BUNDLES

71

Problem 5-D. Show that Gn(Rn+^) has the following symmetry prop­ erty. Given any two n-planes X, Y C Rn+^ there exists an orthogonal automorphism of Rn+^ which interchanges X and Y. [Whitehead, 1961] defines the angle a(X, Y) between n-planes as the maximum over a ll unit vectors x e X of the angle between x and Y. Show that a is a metric for the topological space Gn(Rn+^) and show that a(X, Y) = aO H -.X1 ) . Problem 5-E. Let

be an Rn-bundle over B.

1) Show that there ex ists a vector bundle rj over B with £ © rj trivial if and only if there ex ists a bundle map £ - y n(Rn+k) for large k. If such a map e x ists, f

w ill be called a bundle of

finite type. 2) Now assume that B is normal. Show that f

has finite type if

and only if B is covered by finitely many open sets IJ1 , ...,U r with f| U j trivial. 3) If B is paracompact and has finite covering dimension, show (using the argument of 5.9) that every f

over B has finite type.

4) Using Stiefel-Whitney c la s se s, show that the vector bundle y 1 over P°° does not have finite type.

§6. A Cell Structure for Grassmann Manifolds This section w ill describe a canonical c e ll subdivision, due to [Ehresmann], which makes the infinite Grassmann manifold Gn(R°°) into a CW-complex. Each finite Grassmann manifold Gn(Rn+k) appears as a finite subcomplex. This c e ll structure has been used by [Pontrjagin] and by [Chern] as a b asis for the theory of ch aracteristic c la sse s. The reader should consult these sources, as w ell as [Wu] for further information. For a thorough treatment of c e ll complexes in general, consult [Lundell and Weingram]. Grassmann manifolds appear there on p. 17. F irst recall some definitions. Let DP denote the unit disk in Rp, consisting of a ll vectors v with |v| < 1. The interior of DP is defined to be the subset consisting of a ll v with |v| < 1. For the special case p = 0 , both DP and its interior con sist of a single point. Any space homeomorphic to DP is called a clo sed p-cell; and any space homeomorphic to the interior of DP is called an open p-ce//. For example RP is an open p-cell. 6 .1 D e f i n i t i o n [ j. H. C. Whitehead, 1949]. A CW-complex con sists of a Hausdorff space K, called the underlying space, together with a partition of K into a collection {ea ! of disjoint sub sets, such that four conditions are satisfied . 1)

Each ea is topologically an open c e ll of dimension n(a) > 0.

Furthermore for each c e ll ea there exists a continuous map f : Dn(a) -» K which carries the interior of the disk Dn^a ^ homeomorphically onto efl. (This f is called a characteristic map for the c e ll ea .)

73

74

CHARACTERISTIC CLASSES

2) Each point x which belongs to the closure ea , but not to eQ it­ se lf, must lie in a ce ll e^g of lower dimension. If the complex is finite (i.e ., if there are only fin itely many ea ), then these two conditions suffice. However in general two further conditions are needed. A subset of K is called a [finite] subcomplex if it is a closed set and is a union of [finitely many] ea ’s. 3) Closure finiteness. Each point of K is contained in a finite sub­ complex. 4) Whitehead topology. K is topologized as the direct limit of its finite subcomplexes. I.e., a subset of K is closed if and only if its intersection with each finite subcomplex is closed. Note that the closure ea of a c e ll of K need not be a c e ll. For example the sphere Sn can be considered as a CW-complex with one 0-c ell and one n-cell. In this case the closure of the n-cell is equal to the entire sphere. A theorem of [Miyazaki] a sse rts that every CW-complex is paracom­ pact. (Compare [Dugundji, p. 419].) The c e ll structure for the Grassmann manifold Gn(Rm) is obtained as follow s. R ecall that Rm contains subspaces R° C R 1 C R2 C ... C Rm ; where Rk con sists of a ll vectors of the form v = ( v j , . . . , v^, 0 , . . . , 0 ). Any n-plane X C Rm gives rise to a sequence of integers 0 < dim (X n R1 ) < dim (X fl R2) < ... < dim (X fl Rm) = n . Two consecutive integers in this sequence differ by at most 1. This fact is proved by inspecting the exact sequence 0 -> X Cl R k _ 1 - X n Rk k~th coordinate, R Thus the above sequence of integers contains precisely n “ jum ps.”

§6. A CELL STRUCTURE FOR GRASSMANN MANIFOLDS

75

By a Schubert symbol o = (o1 , ..., 0. Note that an n-plane X belongs to

e(a) if and only if it p o sse sse s a basis X j, . . . , x n so that Xj 6 H \ . . . , X n 6 H For if

.

Xp o ssesses such a basis, then the exact sequence above shows

that dim (X fl R *) > dim (X fl R *

)

for i = 1, ...,n , hence X e e(a). The converse is proved sim ilarly. In terms of m atrices, the n-plane X belongs to e(o) if and only if it can be described as the row space of an nxm matrix [x -] of the form

- *...*io...ooo...ooo...o* ... *** ... * 1 0 ... 0 0 0 ... 0

1 00.... ..o 0 J- 11 The c lo su re e(