Some Applications of Topological K-Theory (North-Holland Mathematics Studies, 45)

SOME APPLICATIONS OF TOPOLOGICAL K-THEORY

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NORTH-HOLLAND MATHEMATICS STUDIES

45

Notas de Matematica (74) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester

Some Applications of Topolo@al K-Theory N. MAHAMMED University of Sciencesand TechnicalStudies Lille, France

R. PKXININI Memorial University of Newfoundland Newfoundland, Canada

U. S U E R University of Neuchhtel Newhatel, Switzerland

1980

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

NEW YORK

0

OXFORD

0 Notth-Holland

Publirtu'ng Company, I980

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted,in any form or by any means, electmnic, mechanical.photocopying, recording or otherwise, without theprior permission of the copyright owner,

ISBN:0444861130

Pirhlishiw: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAMONEW YORK*OXFORD Sok cli.~trihirtorsfi,rrhc U.S.A.c r i d ('titicitlit:

ELSEVIER NORTH-HOLLAND. INC. 5 2 VANDERBILT AVENUE. NEW YORK. N.Y. 10017

Llbrary of Congress Catdoglng la Publlentloa Data

Mahamd, N 1944Some applicatlons o f topologlcal K-theory. (Notas de materngtlca ; 74) (North-Holland mathematics studies ; 45) Blbllooraphy: p. Includes Index. 1. K-theory. I . P l c d n l n i , Renzo A., 1933j o l n t author. j o l n t author, 11. Suter, U., 1935111. T l t l e . I V . Series. QA1.NM no. 74 BA612.331 510s @14'.23 I S B N 0-444-86113-0 80-23219

PRINTED IN THE NETHERLANDS

PREFACE

In the intervening years since its conception and early development by Grothendieck, Atiyah and Hirzebruch there has been a tremendous expansion in the knowledge and applications of Topological K - Theory. Among its impressive applications we cite Adam's solution of the famous (real) Vector Field problem on spheres and the considerable simplification of the solution of the Hopf Invariant One problem. Perhaps the real power of Topological K-Theory lies in the depth and diversity of its applications. In this book we attempt to present systematically some applications which are more or less accessible to a graduate student or the non-specialist in Algebraic Topology who has some feeling for the concepts and techniques of this branch of Mathematics. With the above philosophy in mind we have presented the material in Chapters one to seven in such a way that it is almost self-contained. We have also included a descriptive Chapter (Chapter 8) about the Atiyah-Singer Index Theorem, a theorem rieavily based in K-Theory and undoubtedly, one of the most important mathematical results of the last two decades. Again, in the interest of self-sufficiency, we have included an introductory Chapter (Chapter 0) on the results of Topological K-Theory which are pertinent to the development of our book. This Chapter is descriptive in nature: it is not intended to be an exhaustive study but rather that it may, together with the indicated literature, form the basis of, and motivation for, later detailed study of this subject. The important results within each Chapter (with the exception of Chapter 0 ) are identified by a.pa1r of positive integers, the first of which refers to the Section in which V

vi

Preface

the result is contained; if a reference is given by a triple, the first number indicates the Chapter. Chapter 0 is not divided up into Sections and all references to it are given by a pair whose first entry is 0 Blaise Pascal once observed that no book is actually written by a single author, since every book contains ideas and results of others. The large bibliography should convince the reader that this is particularly true in the book she or he is presently reading. However, Pascal's observation is true in our case also for other reasons: firstly, because we are three authors and secondly, because our book would not have been written were it not for the help, encouragement and suggestions of many friends among whom we wish to note B.Eckmann, R.Fritsch, H.Glover, D.Gottlieb, P.Heath, P.Hilton, D.Lehmann, G.Mislin, D.Rideout, F.Sigrist and A.Zabrodsky. To all of them our heartily thanks. We wish further to express our gratitude to the three Universities with which we are connected, to the Natural Science and Engineering Research Council of Canada and to the Mathematics Research Institute of the Swiss Federal Institute of Technology in Zurich for the encouragement and material support. Many thanks are due to Frau H.Jordan of the University of Konstanz for the excellent typing work.

.

Lastly, we wish to thank L.Nachbin for accepting our book into his prestigious collection.

.

N Mahammed R.Piccinini U. Suter

TABLE OF CONTENTS

PREFACE

V

CHAPTER 0 : A REVIEW OF

K - THEORY

CHAPTER 1 : THE HOPF INVARIANT 1. Introduction 2. The Hopf Invariant of Maps from S3 onto S 3. The Hopf Invariant of Maps f : S2n-1+Sn 4. Cohomological Interpretation of the Hopf Invariant 5. K - Theoretical Solution of the Hopf Invariant One Problem and Applications CHAPTER 2

CHAPTER 3

:

:

TORSION FREE H - SPACES OF RANK TWO 1. Introduction 2. Hopf Construction, projective Plane and Type of Torsion free Rank two H - spaces 3. Torsion free H - spaces of Type (3,7) 4 . The Homotopy Type Classification 5. K - Theoretical Proof of the Type Classification Theorem HOMOTOPY AND STABLY COMPLEX STRUCTURE 1. The Question of Complex Structure 2. Almost Complex Manifolds and Stably Complex Manifolds 3. The Homotopy Type of M and H 4 . The manifold is not stably complex

CHAPTER 4 : VECTOR FIELDS ON SPHERES 1. Introduction 2. Vector Fields and Sphere Bundles over Projective Spaces vii

1 13

13 14

25 30

36 43 43 46

54 63 72 81 81 83 87 90 93 93 97

Table of Contents

viii

The K - Theory of the projective Spaces 4 . Real Vector Fields on Spheres 5. Cross-Sections of Complex Stiefel Fibrations 6 . Cross-Sections of Quaternionic Stiefel Fibrations

106

CHAPTER 5 : SPAN OF SPHERICAL FORMS 1. Introduction and Generalities about Spherical Forms 2 . Vector Fields on Spherical Forms 3 . G - Fibre Homotopy J - Equivalence 4 . G - (Co) Reducibility 5. Span of Spherical Forms of Cyclic Type 6 . Span of Spherical Forms of Quaternionic Type

139

CHAPTER 6 : IMMERSIONS AND EMBEDDINGS OF MANIFOLDS 1. Background 2 . A brief Historical Survey 3. Atiyah’s Criterion 4 . About Immersions and Embeddings of Lens Spaces 5. The Case of the Qm Spherical Forms 6. Parallelizability of the Spherical Forms 7. Immersions of Complex Projective Spaces

177

3.

-

CHAPTER 7

:

GROUP HOMOMORPHISMS AND MAPS BETWEEN CLASSIFYING SPACES; VECTOR BUNDLES OVER SUSPENSIONS 1. Generalities Spaces 2. Cartan-Serre-Whitehead Towers and H 3. Remarks about the KU - Theory of certain Classifying Spaces 4 . A Theorem of Non-Surjectivity for a GrH 5. Vector Bundles over Suspensions

-

CHAPTER 8 : ON THE INDEX THEOREM OF ELLIPTIC OPERATORS 1. Introduction 2 . The Index of an Elliptic Differential Operator 3. Four Standard Complexes

119 123 130

139 146 152 160 168 173

177 182 188 198 206 212 217

225 225 226 229 232 234 239 239 244 254

Table of Contents

4. The Index Theorem 5. The Generalized Lefschetz Fixed-Point Formula

ix 267 281

BIBLIOGRAPHY

29 5

INDEX

315

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CHAPTER 0 A

REVIEW OF

K

-

THEORY

In what follows, IF will represent the field IR of real numbers, the field C of complex numbers or the skew-field H of quaternions. We use CW to denote the category of finite CW - complexes and continuous functions (maps). (0.1)- For a given X E obj C W , let K IF ( X ) be the group obtained by symmetrization of the semi-group VectIF(X) of all isomorphic classes of IF - vector bundles over X , with addition induced by the Whitney sum. It follows that all the elements of K I F ( X ) are of the type “1 - [ q l , where 5 and rl are IF - vector bundles over X ( * ) . Notice that since there is, for every vector bundle 6 over X , a vector bundle 5 ’ such that 5 @ 5 ’ is trivial, the elements of K F ( X ) can be written 151 - n , where n is an integer. On the other hand, it also follows from the appropriate universal property that KIF(X) is unique up to isomorphism. Pull-back of vector bundles shows that K l F is a contravariant functor from CW to the category of abelian groups. (0.2) - If G is a compact Lie group and X E obj CW is a G - space, considering G - vector bundles over X we obtain, Hence, all construcanalogously to K I F , a functor K I F G tions and properties relative to K I F which respect G - actions can be transported to K I F G (0.3) - If X is based, say with base point x0 , define K F(X) as the kernel of the homomorphism

.

.

N

(*)On several occasions we shall identify a vector bundle with the element 1 5 1 in K I F - theory. 1

6

2

review of

A

induced by the inclusion map Ixo)

sing map X -> KIF(X)

K-Theory

i

{xo} ->

:

shows that there is a spl:

gIF(X) 8 KIF({xo})

which is natural; in this splitting, if identify x = 151 - n E K IF (X) with ([El

-

x ;

dim 5 , dim 5

-

is con1

X

n ) = (151

-

dim

5,1

Furthermore, if for every (unbased) object X we c as the union of X and some point * - which will as the base point of X+ - then, KIF(X)

EIF(X+)

.

Given a pair of finite CW - complexes (X,Y), def: as K IF(X/Y) ; the isomorphism K IF(X) IF ( X ' ) i! X/@ = X+ by writing (0.4) - The cofunctors KIF will be denoted by 1 KSp for IF = IR, C and M , respectively; likewise, corresponding reduced functors. The functors K IF are corepresentable; in fact, N

.

KO= [ r Z KU = ,z KSp= [ , z

X

x x

BOI

r

BU] , BSp] ;

the reduced functors are given by the based homotol maps. This is a consequence of the C l a s s i f i c a t i o n which, in particular, shows that for every connect plex such that dim X 5 d(n+l) - 2 (d = dim IF a vector space) N

[X,Gn( IF2n) 1 ,

K IF(X)

Gn( IF2n) is the appropriate Grassmann mani ( 0 . 5 ) For n _r 0 , define KIF-"(X) = aIF(ZnX) fo where EnX is the n fold suspension of X ; fo KIF-"(X) = FIF(ZnX+) and for pairs, KIF-"(X,Y) = Then, according to Atiyah-Hirzebruch [281, we hav sequences -n -> KIF-"(X,Y) -> KF (XI -> KF-"

where

-

...

... ->

KFo(X,Y) --->

KF

0

(X) ---->

KF

review of

A

CW - complexes

for any pair of

...

... ->

K IFo(X,Y)

3

(X,Y), and N

KIF-"(X,Y) ->

->

K-Theory

N

KIF-"(X) -> K IF-" (Y) ->

->

N

KIFo(X)

~

>

EIFO(Y)

if X,Y are based CW - complexes, Y c X and have the same base point. (0.6) Actually the previous exact sequences can be extended infinitely to the right; the argument goes as follows. The Bott Periodicity Theorem ([46],[134]) states that

Z L

x

x

Z x

--

BU RLBU , BO R 4BSp , B S p a Q 4BO ;

the first homotopy equivalence gives rise to a 52 - spectrum KU2n = Z x BU, KUZn+, = RBU, n E Z ; the other two homotopy equivalences define the R - spectra KO = {KO,} and k k KSp = {KSpn}, with KOn = R BO, XSpn = Q BSp, n Z 8 - k (mod 81, 0 < k c 7 , and with the convention that ROB0 = 2 x BO, ROBSp = Z x BSp . These spectra define "generalized cohomology theories" 12521 and thus, the claim made at the beginning of the paragraph. Notice that the coefficient groups of KO and KSp are the same but with a shift of 4 in dimension. The previous observations give readily the KIF - theory of the spheres, which we can summarize in the following table:

-

N

N

n mod 8

0

Z

1

2

z2

z2 E

H

0

Z

0

-- KO sn+41 ) N

0

3

4

5

6

7

0

2

0

0

0

0

h

0

Z

0

G

2

22

z2

0

.

The theories KO and KSp differ also in another significant way: if IF =IR or B: , the tensor product of vector bundles over X induces a ring structure on KO and KU; since the tensor product of two quaternionic vector spaces i s a real vector space, we do not obtain a multiplicative structure in KSp. We also observe that the theories KO and KSp

A review of

4

are related to KU(X)

KU

K-Theory

by the Bott Homomorphisms

>

KU (XI

I491

>

KU(X)

KSp(X) >

KU(X)

KO(X)

;

these satisfy the relations rc=2=qh,

c r = h q = l + t ,

where 2 is multiplication by 2 in KO or KSp and t is the homomorphism induced in KU by conjugation of complex 4n vector bundles. In this context we notice that if X = S then c (resp. r) is an automorphism of Z whenever n is even (resp. odd) and multiplication by 2 whenever n is odd (resp. even). (0.7) - If 5 is a real vector n - bundle over X E Obj C W , the S t i e f e l - W h i t n e y c l a s s e s of 5 are elements wi(6) E Hi ( X ; Z 2 ) , 1 < i < n ; if 6 is a complex vector n - bundle over X , the Chern c l a s s e s of 5 are elements ci( X , there is a space F(E) and a map i : F(E) -> X such that the induced bundle over F(E) splits off as a Whitney sum of line bundles and such that i induces a monomorphism in cohomology with the appropriate coefficients (i.e., Z in the complex case and Z2 in the real one) 1491, [134].

(0.8) A real vector n - bundle is said to be o r i e n t a b l e if its structural group reduces from O(n) to SO(n) ; notice that a complex bundle can be viewed as a real bundle and as such it is orientable since U(n) c SO(2n). Now let 5 be a real vector bundle E -> X; for a riemannian metric on E let D(E) -> X and S(E) -> X be respectively the associated disc and sphere bundles. Set T( 0, a $ 0, there is a z € KU*(X) such that ch(z) = a + higher terms. Finally, the Integrality Theorem shows that under the stated hypothesis on X, ch(X) is a ring monomorphism. (0.12) - If 5 is a real or complex vector bundle E + X, define AiS (i = non-negative integer) to be the vector bundle whose fibre at x E X is the ith exterior power AiEx The exterior power operations on bundles are used to define operations in KIF, that is to say, natural transformations of the functor K l F on itself (see [17,Chapter 1111). To this end, let

.

1 + KIF(X)[[t]] be the multiplicative group of formal power series in t with coefficients in KIF'(X) and starting with 1. For every vector bundle as before, let At( J'(X) 1 4 , Part 11, tors through an epimorphism J(X) Theorem 6.11. On the other hand, J"(X) is an upper bound for J(X) in the sense that W(X) 5 T(X) and the quotient map KO(X) -> J(X) factors through an epimorphism J"(X) + J(X); the reader should notice that these results put together imply that J(X) = J' (X) = J" (X). The fact that J" (X) is an upper bound for J(X) is a consequence of the Adams C o n j e c t u r e which asserts that for any k E Z , X a finite CW - complex and y E KO(X) , there is a non-negative integer e = e(k,y) such that ke ($k-l 1 y maps to zero in J(X). This conjecture was proved by Quillen in 11991; more recently, a proof based on the notion of transfer was obtained by Becker and Gottlieb [411. i > x -> X/A is a cofibration of finite If A -

A review of

connected

-

J - sequence is X in general. However, the following special (see [ 4 , Part 11, 9 3 I ) :

CW

not exact at result holds "if then

I

K-Theory

complexes, the corresponding

-

i' : KO(X)

-->

KO(A)

is an e p i r n o r p h i s r n ,

11

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CHAPTER 1 THE HOPF INVARIANT

1.

Introduction Let f be a map of the sphere S 3 onto the sphere. S 2 and let us suppose it to be simplicia1 relative to some triangulations of S2 and S 3 If q and r are any two points in the interior of a 2-simplex of S 2 , f-' (9) and f-l (r) are 1-cycles of the complex S 3 (see 9 2 1 . Furthermore, as shown by Hopf in a paper of 1931 [121], the linking number of these two cycles depends only on the homotopy class of f We call this linking number the H o p f i n v u r i u n t of f and denote it by y ( f ) . Hopf also proved that the function

.

.

y : n3(sL)

z

->

defined by Y[fl = Y(f) for every class [f] E n3(S 2 ) is a group epimorphism. These ideas can be generalized to maps from S2n- 1 onto Sn, n 2 2 ; as proved by Hopf in a successive paper 11221, one obtains a group homomorphism y : n

2n- 1 (S")

->

z

which is trivial if n is odd (because of the anti-commutativity of the linking numbers); in addition to that, Hopf proved that y is epic if n = 4 or 8 and that there are always maps of S2n-1 onto S" of ~ o p finvariant 2 for n even (this, in particular, shows that n (s") is never trivial 2n- 1 for n even). The obvious problem, left unsolved by Hopf, was to find out for which values on n there were maps f : S 2n-1 --> Sn with y(f) = 1 . A lot of effort was devoted to the solution of this question, known as the "Hopf invariant one problem"; the keen interest of mathematicians on it was also due to the fact that the Hopf invariant one problem has 13

14

The Hopf Invariant

deep connections to other important problems like the determination of the real division algebras and the parallelizability of the spheres. The solution came about only in 1960 by work of Adams, who proved that the only values of n €or which there are maps f : S2n-1 -> Sn with y(f) = 1 are n = 2, 4 or 8 We wish to quote here two important partial results obtained before Adams' conclusive theorem. The first, proved by Ad&m in 1951 [9], says that there is no map f : S2n-1 -> sn of Hopf invariant equal to 1 if n is not a power of 2 ; its proof was established via a delicate study of the Steenrod operations. The second result, due to Toda [2411, showed that (thus, not there is no map f : S 31 -> S16 with y(f) = 1 all powers of 2 could give maps of Hopf invariant 1 ) . The Chapter is organized as follows. In sections 2 and 3 we present an account of Hopf's beautiful work in [121] and [122] Since the homology approach devised by Hopf is not suitable to the development and ultimate solution of the Hopf invariant one problem, we devote 5 4 to the study of a cohomology version of the Hopf invariant. The idea of using cohomology for the definition of such invariant was first described by Steenrod [230]; our development follows that of [71]. Finally, we use 5 5 to give a K - theoretical proof of Adams' Theorem and to discuss some of its consequences.

.

.

.

The Hopf Invariant of Maps from S 3 onto S2 We recall thatapolyhedron is a topological space M together with an abstract simplicia1 complex and a homeomorphism between M and the geometric realization of d ; we shall admit that all simplices of %T had been coherently oriented by a local ordering of the vertices. In order to simplify the notation we shall use the same letter for both the topological space and the subjacent simplicia1 complex of a polyhedron: furthermore, throughout this chapter, whenever we refer to a simplex, we normally understand that we are dealing with the closed (or geometric) simplex. Let M and N be given polyhedra, with M of dimension > 2 and N of dimension 2 ; we also are given a simplicial map f : M + N. Let q be an interior point of a fixed 2.

Maps from

2

M

onto

S3

15

S2

-

simplex T of N : then, for every 2 which is properly mapped onto T~ by

-

simplex a2 of f , there is a unique 2

p E int a2 such that f (p) = q. We write cpu2(q) = + p (or - p) according to that f(u2) has the same (or opposite) orientation as T 2 . If c2 = Z ai u2i is a 2 - chain of M with integral coefficients, define ( q ) = c ai cp i(q) , with Ucpc2 L the proviso that

i a2

( q ) = 0 whenever

cpui

is not mapped prop-

2

erly onto

T~

by

f.

-

(q) is a 0 chain of a convenient cpc2 subdivision of M [ 2 2 5 ; 3.31 Let now u 3 be a tetrahedron of M which is mapped onto T~ by f . Observe that u3 has exactly two faces, say 1 2 a2 and a2 , which are mapped properly onto T~ ; moreover, the set of points of a3 which are taken into q by f is a line segment with end-points p1 E int a21 and p2 E int u22 1 If f (0,) and -t2 have opposite orientations, then f has the same orientation as T2 ; in this case we orient the segment (pl,p2) from p to p2 (otherwise, we orient it from p2 to p,). It is immediate to verify that if acp, ( q ) Notice that

.

.

3

(respectively, au3) of

a,)

then

acp,

to 3 - chains of of M , we define moreover,

is the boundary of

(q) =

cp

. The

(q)

(respectively,

(q) cp,3

previous ideas generalize

3 aa3 M : if c3 = C ai is an integral ( q ) by setting it equal to C ai (OC3

05

(2.1)

One should also notice that cp

2(q) = c3 + c 3

(4

C

,(q) + 3

cp 2 ( q ) C

3

and that these equalities allow us to write that

I

3-chain (9) ;

cp U

3

The Hopf Invariant

16

We now recall, for the sake of completeness, that a c l o s e d , o r i e n t e d , t o p o l o g i c a l n - m a n i f o l d is a finite, connected, n - dimensional (oriented) polyhedron M such that, given any point p E M , 2,if

i = n ,

We also recall that each simplex of M is a face of an n-simp l e x of M , each (n-I) simplex is face of exactly two n simplices and finally, given any two n - simplices a and T of M , there is a finite sequence of n simplices of M , 1 2 i say a = a, a , ,aq = T such that 0 and a have an (n-1) face in common, i = 1 q - 1 [212; 5681 For the remainder of this section we shall assume that M is a closed, oriented, topological 3 manifold; indeed, most of the time M will be the unit 3 - sphere of IR4 If we (according regard M as a 3 - chain with coefficients * 1 to orientation), (2.1) and (2.2) show that cpM(q) is a 1 cycle of a convenient subdivision of M. Let a2 be a triangle of M which has one point in common with cpM(q) ; this intersection point is endowed with a sign, according to the following rule: u2 lies in exactly two tetrahedra u 31 and u 23 of M , s o if we take a2 with the orientation induced by a31 , cpM(q) n a2 will be given a

-

-

-

.. .

'+'

,...,

-

.

.

-

positive (negative) sign according to that spectively, enters) a 31 The number

wM(q)

leaves (re-

.

obtained is the i n t e r s e c t i o n number of

cpM(q) with

a2. If

cpM(q) n a2 = 0 , we define the intersection number of qM(q) i is a with a2 to be zero. More generally, if c2 = C a. a2 1 2 chain of M such that vM(q) has at most one point in common with each a; , we define the intersection number of cpM(q) with c2 by

-

Maps from

onto

S3

17

S2

This number has an interpretation which is directly related to f and T~ : if u 2 is mapped properly onto T 2 by f, define the d e g r e e of f on u 2 (denoted by def flu2) or - 1 according to that f (a,) has the as the number + 1 or opposite orientation. If u 2 is same orientation as T~ not mapped properly onto T~ by f , deg flu2 = 0 Then, i and the previous oriendeg flc2 is defined by C ai deg flu2 tation convention show that

.

I(c2r wM(q))

(2.3)

=

deg f / c 2

.

We next specialize M to be S3 and N to be S 2 ; also, to simplify the notation, we shall write Q ( q ) for cp,(q) Because H1 ( S J ; Z ) = 0 , the 1 - cycle cp(q) of a subdivision of S 3 is homologous to 0 , that is to say, there is a 2chain c2 such that 3c2 = @ ( q ) Of course, c2 is not uniquely determined; we shall however, make a choice of c2 which will prove itself to be very useful. If a 3 is a tetrahedron of S 3 whose image by f covers r 2 properly, then a3 contains a 1 - simplex (arb) of cp(q). Let e be one of the two vertices that the triangle containing a has in common with the triangle containing b : let us replace (arb) by the pair of segments (are) and (e,b). If we do this with every tetrahedron which contains a segment of w(q) , we end up by replacing ~ ( q )with a 1 - cycle w, which runs over triangles of S 3 : note that w1 together with w(q) bounds a 2chain determined by triangles of type (a,e,b). Next, we replace in each triangle (e',a,e), (e,b,e"),.. the pairs of edges ( (el,a), (are) , ((e,b) , (b,e") ,.. by the 1 - simplices (e',e) , (ere") respectively. The edges (el ,e) , (ere''),.. define a 1 - cycle v, of S 3 , which together with w(q) bounds a 2 - chain ci determined by the triangles (e',are), (a,e,b) On the other hand, vl being homologous to 0 and being formed by 1 - simplices of S3 , it is the boundary constructed with triangles of S 3 . Thus, of a 2 - chain c2 2 2 take c2 to be ci + c2 Y

.

.

,.. .

.

.

,... .

.

We discuss now why the previous choice of c2 is usebelong to the original trianful. Since the triangles of c2 2 gulation of S 3 , they will be taken by f simplicially into

The Hopf Invariant

18

.

triangles, edges or vertices of the triangulation of S 2 As 1 for c, , its triangles of the type (a,e,b) are taken by f into segments (q,r), where r = f(e) is a vertex of T ~ the ; triangles of type (elrare) are taken by f into triangles (r',q,r), where r'= f(e') is also a vertex of T, (notice that (r',q,r) may be degenerated). If we subdivide T~ s o that q is connected to the vertices and we subdivide M to include a r b as vertices, then with respect to such triangulations of S 3 and S 2 , f is simplicia1 on c,. Recall that the boundary operator commutes with a simplic.ia1 map; this implies that the (geometric) boundary of f(c2) reduces to just the vertex q and hence, f(c2) is a 2 - cycle of S 2 . We conclude that f(c2) is homologous to an integral multiple of the fundamental cycle z 2 of S 2 ; in other words, if we represent the homology class of a cycle z by { z ) , {f(c2)) = y(f){z2), with y(f) E Z One should also notice that {f(c2)) is equal to deg flc, {z,) and so, y (f) = deg f I c2. The integer y (f) is known as the Hopf i n v a r We give next two properties of i a n t of the simplicia1 map f

,...

. -

.

y(f).

-

.

y(f) d o e s n o t d e p e n d o n t h e c h o i c e of c2 In fact, if c, is another 2 chain such that 3c2 = q(q), c 2 - c2 is a 2 - cycle of S3 and so, is homois also homologous to 0 and logous to 0 ; then, f (c,- c,) hence, (f(c2)l = {f(c2)l = y(f){z23 1

The number

-

.

-

.

y(f) i s i n d e p e n d e n t of t h e p o i n t q Suppose first that r is an interior point of T; , a triangle of S2 .different from T~ Let c, and d2 be 2 - chains of convenient subdivisions of S3 such that ac, = q(q), ad2 = q(r); notice that q(q) r l q(r) = 0 . Let yq (respectively y r ) be the value of y(f) relative to q (respectively r). Hence yq = deg flc, = I(c,,q(r)) and qr = I(dzrq(q)). Now let c2 f l q(r) (respectively, d2 n co(q)) be the 0 - chain intersection of c, and q(r) (respectively, of d2 and q(q)) ; if E : C o ( S 3 ) + Z is the augmentation homomorphism, €(c2 n q(r)) = E(d2 n q ( q ) ) , because c2 n q(r) d2 flcp(q) % 0 But €(c2 n qfr)) = I(c2"P(r) 1 r € ( d 2 il cp(q) 1 = I1

The v a l u e

.

.

Maps from

S3

onto

19

S2

It is clear that if s is any yq - yrpoint in the interior of T; , different from r , then y r - y-s .

= I(d2rW(q) 1

and

Remark. In the literature the number I (c2,cp(r)1 = I (d2,cp(q) is also called the l i n k i n g number of cp(q) and cp(r) and is indicated by L ( c p (q),cp (r) The simplicial approximation theorem suggests that we could possibly define the Hopf invariant of any c o n t i n u o u s function f : s 3 + s2 ; for a given simplicia1 approximation f' to f , set y(f) = y(f'). It is clear that if we are to obtain a meaningful definition, the number y(f) must be independent of the simplicial approximation chosen. This is the substance of the following result.

.

Lemma

2.4.

the integer

FOP

any c o n t i n u o u s f u n c t i o n

of

f

s3

into

i s w e l l d e f i n e d and i s i n d e p e n d e n t o f

y(f)

s2 , f

w i t h i n i t s homotopy c l a s s .

Proof. The proof of this result is rather long and will be given in two steps. STEP 1. Suppose that f is itself simplicial (for some basic triangulation of S 3 and S 2 ) and let f be anWe shall show other simplicial map which approximates f that y(f) = Let q be an internal point of a 2 - simplex T 2 of S2 and let r be interior to T; T ~ .Construct the 1 - cycles cp(q) ,cp(r) (respectively, cp(q) ,G(r)) relative to f (respectively, f) ; we must show the equality of the linking numbers L(cp(q) ,cp(r)) and L(G(q) , cp(r)). To this end we construct the 2 - chains c2 and d2 so that:

.

y(r).

*

(i)

c2

cp(r), cp(r)

is disjoint from

cp(q), cp(q) ;

(ii)

d2

(iii)

ac2 = cp(q)

(iv)

ad2

-

is disjoint from

=

cp(r)

-

;

-

cpw

-

cp(r)

;

.

These conditions on c2 and d2 show immediately the desired equality of the linking numbers; notice that due to symmetry, it is enough to prove the existence of c2 with properties (i) and (iii). We observe first that since f is a simpli-

20

The Hopf Invariant

cia1 approximation to f , if u2 is a 2 - simplex of s3 such that f(a2) is non-degenerate, then f(02) is also nondegenerate and indeed, f(a2) = T ( a 2 ) . Let c3 be the 3 - chain formed by all tetrahedra of S 3 mapped onto T~ by f and f ; this 3 - chain has no tetrahedra in common with the 3 chain c; formed by all tetrahedra of S3 which are mapped onto T; by f and Since w(r) and w(r) lie in c' - 3' the problem will be solved if we show that w ( q ) and w ( q ) together bound a 2 - chain c2 in c3 Let a2 be a face of a tetrahedron a; of such c3 that f(02) is not degenerate. Since I(a2,w(q)) = I ( a 2 , w ( q ) ), the 1 - cycle w ( q ) - w(q) has intersection number 0 with u2 , and indeed, with any face of a; We are going to show that every 1 - cycle z1 of a convenient subdivision of S3 which has intersection number 0 with every triangle in c3 is homologous to 0 in c3 ; this shows, in particular, that there exists a 2 - chain c2 in c3 such that ac, = w(q) Suppose that z 1 intersects a2 at a point a , which can becounted positively by a suitable choice of orientation for a2 ; since I(a2,z1) = 0 , z , must intersect a 2 at another point b , which must be counted negatively. Now besides being a face of a; I the triangle o2 is a face of just another tetrahedron a'; Let al,b, € u; and a2,b2 € u'; be points of z 1 so that a, ,a2 are close to a and bl ,b2 are close to b ; we assume that al a a2 is the positive direction of z , around a and thus, b2 b b, will be the positive direction of z, around b (see figure).

.

.

.

~ ( 4.)

.

Maps from

S3

onto

S2

21

Connect al to bl by a segment in u; and connect a 2 to b2 by a segment in 0 ; and denote the closed, oriented polyNotice that P, is gonal line al a a2 b2 b bl al by PI and hence, in c3 ; consequently, homologous to 0 in u; + u: 1 z 1 = z 1 - PI and z 1 are homologous in c3. The 1 - cycle does not contain the points a and b and moreover, its intersection with u 2 has two points less than the intersection of z 1 and a 2 ; finally, the intersection number of z 11 with any triangle of c3 ist still 0 In this way we obtain a 1 - cycle z n 1 which is homologous to z 1 in c3 and which is disjoint from all triangles of c3 ; one should notice that n z 1 consists of a certain number of mutually disjoint cycles, each one contained in the interior of a tetrahedron of c3. Hence, is homologous to 0 in c3. This completes the proof of Step I.

.

.

STEP 2 . We show next that if fl and f 2 are two simplicial maps from S 3 to S2 which belong to the same homotopy class, then y(fl) = y ( f 2 ) . Let F : S ~ ~ I + be S the homotopy F : f l = f2. Assume f l and to be simplicial with respect to the same triangulation of f;3 ; this gives rise to a triangulation o f S 3 x I . Let F' be a simplicial approximation to F . Then F' is a simplicial map from a convenient barycentric subdivision of S 3 x I into a triangulation of S 2 . This new triangulation of S3 x I induces a triangulation on S 3 x CO} = S i and on S 3 x ( 1 ) = S: and these triangulations are subdivisions of the original triangulation of S 3 . Let f ; = F ' I So3 and f; = F'IS: i because F' = F, f ; is a simplicial approximation to fl and is a simplicial approximation to f 2 . By the argument developed in Step I , y ( f l ) = y ( f ; ) and y ( f 2 ) = y ( f ; ) i we have to show that y(f;) = y(f;). Let q be a point in the interior of a triangle u 2 * of 'S and let cp (q),q ,(q) and cp 3(q) be the chains of

fi

SXI S3 x

by

I

and

F', f;

S3

and

s1

whose carriers are the inverse images of q , respectively. Since a ( S 3 x I) = S 13 - S o3 ,

fi

22

The Hopf Invariant

From this we conclude that if S i

and

S:

c;

respectively, such that 1 then, c2

-

(9)

cp

1 c2

and

ac;

- c2 0

--

are

2-chains of and

= cp 3 ( q )

s,v

z2

is a

2-cycle of a

S3XI

triangulation of S 3 x I. Notice that z2 like any cycle of 3 x I, is homologous to a cycle of S1, namely the projection 1 z2 of z2 onto S: Hence, F'(z2) is homologous to 1 ) = f;(z2) and thus, is homologous to 0 . This implies F'(z 2 1 0 1 that f ; (c;) f; (c,) , thus deg fi Ic2 = deg f;lc2 and thereS3

.

-

fore, y(f;) = y(f;) Theorem

2.5.

which completes the proof of the lemma.

The f u n c t i o n y

which t a k e s any

,

: n3(S

[fl E

2

.+ z

a3(SL)

into

y(f)

i s a g r o u p homomor-

phism.

Proof. Let [f], [g] E n3(S2,qo) be taken arbitrarily: we may assume that these elements are represented by maps f,g : S 3 -* S2 so that: ( i ) f and g are the geometric realizations of simplicia1 functions (which we represent by the same letters) relative to some triangulation of S2 and S 3 ; 3 3 3 3 (ii) f (E-1 = g(E+) = q,, where E+, E- are respectively the northern and southern hemispheres of S3 Recall that [f] + [g] can be represented by h : S3 + S2 defined by hlEl = f, hlE-3 = g; of course, we may view h as a simplicia1 function. Take q in the interior of some triangle of S2; let cph (9), cpf (q) and cpg(q) be the 1-cycles of a subdivision of .S3 whose carriers are the inverse images of q by h,f and g, respectively. Let C s c E+3 3 (c; c E-) be the 2 - chains of S 3 such that acz = cpf (9)

.

(respectively, ac;

= cp

9

( 9 ) ) . Then,

-

h

+

c2 + c;

of s3 such that a(cz + c,) = (4 ( 9 ) that if z2 is the fundamental cycle of

+ Ch(c2

+ ci)) =

+ {f(c2)} +

{g(c;)l

is a 2 - chain

. . All S2,

this shows y(h)'{z2) =

= (y(f) + y(g)){z21

.

Maps from

S3

onto

23

S2

We conclued this section by showing that the homomorphism y is indeed an epimorphism; we shall achieve this goal by first exhibiting a map f : S3 + S 2 with y(f) = i 1 and then, by discussing the Hopf invariant of the composition of any map from S3 to S 2 with a map of degree c. In what follows, the points of IR3 will be given 4 coordinates (x1,x2,x3) and those of IR , the coordinates (Y1 rY2rY31Yq) Define g :IR4 +IR 3

-

by the formulas =

2 ( ~ 1 ~+ 3~ 2 ~ 4 )

x2 = 2 ( ~ 2 ~- 3~ 1 ~ 4 ) 2 2 2 2 x3 = Y1 + Y2 - Y3 - Y4 : a simple computation shows that 2 2 2 x 1 + x2 + x2 3 = (y: + y; + y3 + Yi>’

I

which means that the unit sphere S 3 of IR4 is mapped by g 3 onto the unit sphere S2 of IR . Using the stereographic projection (from the north pole) of S2 onto the plane (xl,x2) , 2 we identify the points (xl,x2,x3) E S (x3 9 1 ) with the complex number x1 + ix2 q = 1 - x 3 and identify

(0,081)with

Observe that if q = all (yl,y2,y3,y4)E

EO

q

= a.

Then, if

f = glS

S3;

,

, the carrier of

S3

such that

V(q) is the set Of y3 - y4 = 0, which is a

x1 great circle of

3

for any other point

q

=

+ ix -

, the

3 carrier of y1 that

Y3

+ +

V(q)

iy2 iY4

-

is the set of all x1 + ix 1 - x3

and so

( Y 1 ~ Y 2 ~ Y 3 ~2‘Y 4s ) 3 such

24

The Hopf Invariant

(l-.x3)y1- x1y3 + x2y4 = 0 (l-x3)y1 - x1y4 - x2y3 =

o

.

These equations show that, again, the carrier of cp(q) is a great circle. The following result Droves that y(f) = 1. Lemma 2.6. L e t f : S 3 + S2 be a map s u c h t h a t t h e c a r r i e r of cp(q) i s a g r e a t c i r c l e of S 3 f o r a n y q E S2 ; t h e n 3 y ( f ) = * 1 , t h e s i g n d e p e n d i n g o n t h e o r i e n t a t i o n of S . Proof. We develop only the geometric argument leaving the simnlicial reasoninq to the reader. A

3

- dimensional subswace of IR4

and a

2

-

dimen-

sional subsDace of IR4 intersect on a straight line if the Dlane is not contained in the 3 - space: hence, a great sphere and a great circle of S 3 intersect in two diametrically opwosite Doints, if the circle is not contained in the sphere. Hence, if H is a hemisDhere of a great sphere of S 3 , any great circle which is disjoint from a H intersects H in exactly one Doint. This indicates that any two great circles of S 3 which are disjoint to each other have linking number equal to ? I : thus, for any two distinct points of S2 , say and

q

r

Theorem

, L(@(q),cp(r)) 2.7.

maps of d e g r e e

c

-

y(f)

and

= * 1

I > e t g : S:

+

and so, the desired result. S3 and h : S 2 + S ::, b e g i v e n

c . T h e n , f o r a n y map y(hf) = c2 y(f).

-

f

: S3

+

S

, y(fg)

=

Proof. A s usual, we may assume f,g and h to be simolicial. For any q interior to a triangle of S2 , let cpfg(q) be the 1 - cycle of S 3 obtained from the inverse image of 7 by fg. Let u 3 be a tetrahedron of contains a 1 - s i m ~ l e x (a,b) of v(q).

S3

which

Suppose next that n tetrahedra of S 31 are mapped by g onto o3 in the positive fashion and that n - c tetrahedra of S: are mapped negatively by g onto u 3 : then, n respectively , n - c) segments of cpfg(q) will be mapped positively (respectively, negatively) onto (a,b) by g. This shows 1 that 9(cpfg(q)) = c c p ( q ) . Let now c2 and c2 be two 2 chains, the first of S 3 and the second of S: , such that

-

Hopf Invariant of

f

:

S2n-1 +

sn

25

1 = cpfg(q). It follows that ag(c2) 1 =c . ac2 = cp(q) and ac2 ac2 1 - cf ( c 2 ) 0 in S 2 because g ( c12 )- c c 2 is and so, fg(c2) 1 a 2 - cycle of S 3 . This implies that deg fglc2 = c deg f Ic2 y(f). and hence y(fg) = c To show the second part of the theorem, we begin by taking a Point q interior to a triangle o 21 of S: ;. let

-

-

-

-

{ql,...,qn ; rl,...,rn-c } be the inverse image of q by h. We observe that each point of this set is interior to a triangle of S 3 ; we give a positive orient.ation to the triangles of S2 which contain a point q 1. in their interior and a negative orientation to these triangles which contain points r it j’ then follwos that

-

Let

i i = l,...,n c2,

the

2 - chains of

S3

(respectively, di, j = l,...,n - c) be i = w(qi) (respectively, such that ac2

ad; = cpfr.)).By the previous characterization of qhf (9) , 7 1 1 is the fundamental {hf(Zci - Ed;) 1 = y(hf){z2} , where z2

On the other hand, €or all i = 1,. . .,n and i l,...,n - c, {f(c2)} = y(f){z21, If(d2)l = y ( f ) I z 2 ) and so,

cycle of S : j

=

.

if(2c; - L‘d;)} = c * y ( f ) t z 2 1 . Our result is obtained from the 1 last equality and the fact that {h(z2)} = c({z2)

.

Remark. The existence of a map f : S 3 S2 of Nopf invariant 1 and ( 2 . 7 ) show that the homomorphism Y of ( 2 . 5 ) is an 2 0 An alternative way to prove epimorphism and thus, n j ( S 1 that n 3 ( S 2 ) 0 is to show that the map f : S 3 + S 2 which we proved to have Hopf invariant 1 is in fact a fibration 1 with fibre S ; then, the exact homotopy sequence of this fi2 E . Of course, these ideas were not bration shows that IT 3 ( S ) avaible to Hopf when he wrote his paper. +

*

.

The Hopf Invariant of Maps f : S2n-1 -P sn. In the previous section we described the construction of the Hopf invariant of a map from S 3 to S 2 , along the lines traced by Hopf in [1211. In a subsequent paper [1221, Hopf generalized this construction to maps from S2n-1 into 3.

26

The Hopf Invariant

Sn, for n > 2 . Such a generalization is readily obtained: given S2n-1 -. S" (relative to triangulaa simplicia1 function f : tions of Sn and S2n-1) and a point q interior to some nsimplex of Sn, let (o(q) be the (n-1) -'chain of a convens2n-1 whose carrier is the inverse ient triangulation of image of q by f ; notice that since S2n-1 is a closed manifold, (o(q) is indeed a cycle. This cycle was called by Hopf the o r i g i n a l c y c l e of q , relative to f. Now the Hopf invariant y(f) is defined as the linking number L((o(q) ,cp(r)) , where r is an arbitrary point in the interior of an n - simplex of Sn, possibly different from that containing q. This number is independent of the points q and r, and as before, the Simplicia1 Approximation Theorem can be used to define the Hopf invariant of an arbitrary map f of S 2n-1 onto S" -

-

_.

-

f f f by y(f) = L((o (9),cpf (r)) , where cp (q) and (o (r) are the original cycles of q and r respectively, relative to a simplicial approximation f of f; again, y ( f ) is independant of f within its homotopy class. This approach gives immediately an interesting result. In fact, it is known that if and 'n-k are non-intersecting cycles of dimensions Z k- 1 k - 1 and n - k respectively, belonging to a subdivision of an orientable n - dimensional topological manifold, then the linking numbers L ( z ~ - , , z ~ - ~and ) L( Z, -~ , Z~ -~ )can be de= fined and they are related by the formula: (k-I) (n-k) + 1. L ( z ~ - ~ , z ~ I211; - ~ ) $ 7 7 1 . In our case, (-1)

L(cp(q) ,(o(r)) = (-1)( n - 1 ) 2 + 1 L(cp(r) ,cp(q)) thus, if n i s o d d , L.(cp(q),(o(r)) = 0 and so, the Hopf invariant of any map S2n-1 sn f : with n odd is always trivial. Hence, from now on, we shall restrict n to be an e v e n integer. ~

Recall that for n = 2 we have proved the existence of maps with Hopf invariant 1 (and indeed, for that value of n, there are maps S 3 -. S 2 with any preassigned integer'as Hopf invariant); for higher values of n, Hopf has shown the existence of maps S2n-1 -. Sn with Hopf invariant 1 , if n = 4 and 8 [122]. We shall reproduce here these results but first, we prove the following

Hopf Invariant of

Theorem 3 . 1 . Tf w i t h y(f) = 2.

n

f

:

S2n-1

-,

s"

is e v e n , t h e r e i s a nap

27

f

: S2n-1 +

s"

Proof. - Let E+" be the northern hemisphere of S" n sn-l and let g : ( E + , 1 -, ( S n l q o ) (9, = (l,O,...,O)) be any map whose restriction to EY - sn-1 is a homeomorphism (an example of such a map is the following: first take h : Ef -, S" given by h(xl,...,xn+l) = ( ~ X ~ , . . . , ~ X -~ 1) ,~X where ~+~

next, take the rotation p of S" given by p (xl,.. . , x ~ + ~= ) ( - x ~ + ~ ~ x ~ , . . . , x ~The , x ~map ) . ph has the desired property S2n-1 as 1 1 0 4 1 ) . Suppose now that g is simplicial and view the boundary of Dn x D" (where D" is the unit n - ball of IR"); notice that the (simplicial) boundary of D" x D" is also given by aDn x D" + D" x aDn, because n is even. Hence define f : aD" x D" + D" x aDn -D Sn by taking 1-

(x, 2 +

9(Yl,""Ynl

(y,( x , ,

. . . ,xn)

...

+ x 2 1 ) and n J L y ; +

... +

E aDn

x

D"

into

g(xl ,.. . ,xn,

((yl,... ,yn),x) E D"

x

aDn

into

y ,2) ) ' .

Let q be an interior point of some n - simplex of S"; the carrier of the (n-1) - cycle p(q) determined by f and q is aDn x p U p x aDn, where p is the unique point in the interior of Dn such that f(p) = q. Take next p' E aD" and form the 0 - cycle (p) - (PI): let c1 be a 1 - chain of D" such that ac, = (p) - ( p ' ) . Then a(Dn x (p') aDn x c 1 + (p') x D" + c1 x aDn ) = c p ( q ) . Since f(aDn x cl), f(cl x aDn) are homologous to 0 and f(Dn x (p')) ,f((p')x D") are equal to the fundamental cycle of S", it follows that deg f i D n x ( p ' ) - a D " x c , + ( p ' ) x D n + c , x a D n = i.e.,

2

,

y(f) = 2. An immediate consequence of the previous theorem is that €or n even, TI (S") 9 0 . 2n- 1

The Hopf Invariant

28

Now we go on to the existence of maps of Hopf invariant 1 . We begin by observing that a map f : Sn-l x Sn-l Sn-l defines a map H(f) : s2n-1 + S n . In fact, let EY and E'f be the northern and southern hemispheres of S", respectively. n Since E: n E- is the equator Sn-l , we can view EY and Ef +

as cones over Sn-l; we shall write E: = c+sn-l and E: = c-sn- 1 , €or the sake of precision. On the other hand, the unit ball D" is homeomorphic t.o CS"-', so we view S2n-1 as being sn-l x csn-1 U CSnml x Sn-l. The map H(f) is then trivially defined: it is the map taking any (x,(x',t)) E Sn-l x CSn-l + n-1 and any ((x,t),x')E CSn-l x sn-1 into (f(x,x'),t) E C S - n-1 . into (f(x,x'),t) E C S The construction of H ( f ) we just described is known as the Hopf c o n s t r u c t i o n : we shall consider it again in Chapter 2, in connection with the study of H spaces. We also recall that a map f : Sn-' x Sn-l - r s n- 1

-

is of t y p e

(c1,c2) if

deg f/S"-l

q2

x

c1 and

=

sn-1 - c2, where ( q l l q 2 ) is a base point for de9 f l q l sn- 1 x Sn-l. We are now ready to prove the crucial result of this section. Theorem y(H(f))

._ -

3.2. .Tf = i c c 1 2'

f : Sn-'

x

Sn-l

-+

is

Sn-l

Proof. We begin by decomposing union of the sets

v,

=

{(X1,...rx2n) E

2 s2n-1 Ixl

+

...

v2

=

{ (x,

s 2n-: 1. 1

+

... +

,X2J

E

type

(c,,c2),

S2n-1 into the

+ x2

and I . ' .

Of

2'1

2

5 Xn+l+ . . . + x 2ni

xi 2

Xn+l 2

+

... +x;,}

,

D" x Sn- 1 and and by observing that V1 Sn-l x Dn , V 2 sn- 1 x sn-1. v1 n v2 For each real number c such that c2 - 1 / 2 we consider the (n-1) - spheres

sn-l 1 ,c

=

{ ( x l,. .. ,X2J

E s2n-1 l x 1 = c,x2=

. . . = xn

Xn 2+ l +

*

-. +

= 0, X2n=

1-2)

5 v1

Hopf Invariant of

f

- r sn

S2n-1

:

29

and sn-1 2,c -

{ (X1,...,X2n) X

Let

2 s2n-1IXl + . . . +

E

z~~~

n+ 1 -

...

C,Xn+2 -

x2 n

=

-

1

= 0) - X2n

c

2 I

.

5 v2

Sn-' l I c , i = 1,2 : we

be the fundamental cycle of

should notice that the homology classes

(znLl '/vZ}

and

--

(z211/v21 form a basis for the homology of n- 1 furthermore, when viewed as and

zn2'1'Q 1

V1

(n-1) - cycles of

are homologous to

.

0

n

and

V2

S2n-1

1

Actually, these cycles are

boundaries: for example, we can see geometrically that

n - chain

bounds the

{

(Xl,...,Xn)

2 X n+l any sphere

E

c2 n

s 2n-1 IXl

of

V2

,l/fi

"n-1

z1 , I / f i n- 1

given by the set

l/fl, x2 -

... =

x

=o,

n Now the intersection of

2 with . cn X2n < - l/@] Sn-' is the point (x, = dl-c2,x2 = . = xn = 0 ; 2,c X = x = 0 ) and so, the intersection nurnn+ 1 = c, xn+ 2 2n Z11c in v1 , it f o l ber 1(cnt z2rc) n-l = * 1 Since z'"'11 n- 1 n- 1 lows then that +

+

..

...

-

.

(3.3)

L(Zn-1' 1,c 22,c) n-1 =

f

1

.

Following our costumary procedure, we assume that H(f) is simplicial; we also take q+ and q- to be interior , respectively. Let points to n - simplices of EY and E! and q- , recp(q+) and cp(q-) be the original cycles of q, lative to H ( f ) . The Hopf construction shows that co(q+) is an (n-1) - cycle of a subdivision of V1 and that W(q-1 is an (n-I) - cycle of a subdivision of V2. Now Q ( q + ) and (p(q-) are homologous to integral multiples of zltc and n- 1 Z2lc in V 1 and V2, respectively: say that V ( q + ) - b l z1n-l tc n- 1

is identified to

Sn-l

x

q2)

is mapped by

H(f)

into

Sn-l

30

The Hopf Invariant

with degree

c l , deg H (f)I :c

I ( ~2 ~ ~ q ( q -=1 i) c 1

and hence, L(znml ltl/d',q(q-))

l,l'o ,z2") n-l other hand, L(Z,-~ L(~~'~'~,q(q-)) =

that say in

cp(q-) V1

.

-

i

Using

c1 , which implies that

=

c 1 z2rc n-1

*

in

= i 1

and

cp(q-)

-

b2. Therefore, b2 =

v2

;

similarly, q(q,)

(3.3) , we conclude that

*

= i

c l . On the

b2z:17

imply

c, that is to

-*

c2zn-l 1I C

L(cp(q+) , q ( q - ) ) =

i-

c 1 c 2'

Remark. The previous theorem shows that a map f : sn-' x sn-l + sn-l of type (cl,c2) determines a map n g : S2n-1 -+ S of Hopf invariant c c In fact, if y(H(f)) = 252n-1 S2n-1 is a map -c1c2, take g = H(f). f', where f' : of degree -1 . -+

Corollary 3.4. Tf n = 2,4 or 8 t h e r e u r e maps from S2n- 1 o n t o sn o f ~ o p fi n u a r I i a n t 1

.

Proof. According to the theorem and the previous remark, it is enough to show the existence of maps sn-l ,~Sn-l -+ sn-l of type ( 1 , l ) for the values of n given f : in the stctemenk. Now, this is trivial: f is just the map induced on the corresponding product of spheres by the multiplication of the complex, quaternionic and Cayley numbers. In closing this section we want to observe that (3.4) provides a new proof €or the existence of maps S3 -+ S 2 of Hopf invariant 1 (see 5 2 ) ; furthermore, Hopf used (3.2) 2n- 1 + sn of Hopf to show that for n even, there are maps invariant 2 . The proof we gave here to (3.2) is based on [104; Theorem 1 . 6 . , Ch. IV]

.

4.

Cohomological Intgretation of the Hopf Invariant. In order to describe the purely combinatorial defini-

tion of the Hopf invariant in t.erms of ordinary cohomology theory, we shall analyse more closely the structure of the original cycles c p ( q ) ; to this end, we shall also review some basic ideas. Given a (closed, oriented) topological n - manifold n ; M , denote its ordered i - simplices by n i l i = 0 ,

...,

Cohomological Interpretation of

31

y(f)

...,ai .

Select a point in the interior of 0 : , for each i and j as before; this point will be called, by abuse of language, the b n r y c e n t e r of u!1 Now we construct a new complex M' by defining its abstract simplices to be ordered i0 sequences 0 ; = (pj ,...,pj where n > i > i l > > i > 0 - 0 k0 j = 1,

.

...

and

uijr

is a face of

r is a subdivision of

u

Jr-1 ir- 1

, r

=

l,...,k

.

Notice that

M'

M in the sense of [225; 3 . 3 1 , that M' is ordered and that M' is also a triangulation of M. The complexes M and M' are related by a simplicia1 map a:M'+M which gives rise to a chain equivalence : a is defined by associating to each vertex pi the first vertex of the simplex u: ; following [110;3.5.6] one shows that a# : C# (MI)+ C # (M) is a chain equivalence with inverse chain map p : C#(M) + C # ( M ' ) . 4.1. L e t M and N b e t o p o l o g i c a l m a n i f o l d s and l e t N be a s i m p l i c i a 2 map. S u p p o s e t h a t N i s s u b d i v i d e d l b a r y c e n t r i c a Z 2 y l b y N' t h e n , t h e r e i s a s u b d i v i s i o n MI of M s u c h t h a t f i s s i m p l i c i a 2 w i t h r e s p e c t t o M' a n d N'. Lemma f : M

+

that the

Proof. By induction on the skeletons of M. Suppose ith skeleton of M is subdivided so that the re-

striction of f to it is simplicial. Let u be an ( i + 1) simplex of M; because f is simplicial, f(0) = 7 is a simplex of N. Let q be the barycenter of 7 ; we are going to prove that there is a point p in the interior of u such that f(p) = q . In fact, if dim T = i + l , this is so because < i, there are at least of the simpliciality of f . If dim T 1 2 and u 2 , such that f(u ) = f(u ) =I-. two faces of u , say 0' 2 By induction, there are points p1 E int 0' and p2 E int u , such that f(p1 ) = f(p2) = q . By linearity, the segment p1p2 is entirely taken onto q; then choose p interior to the segment

PIP2. Given a topological

n - manifold

M

as before, we

define for each i - simplex 0 ; the c l o s e d d u a l (n-i)-ceZl bA-i to be the union of all simplices of M' which have as last vertex. The cell bi-i is an (n-i)-dimensional subcomplex of M. The set

The Hopf Invariant

32

n- i li = 0 , . ..,n : j = 1,... According to [71;3] is called the d u a l c e l l s u b d i v i s i o n of M and [ I l O ; 3.8.21, k is a b l o c k d i s s e c t i o n of M ; moreover, [110; 3.8.81. H,(k) y- H,(M)

M

= {bj

.

Lemma

4.2.

Let

M

and

N

be t o p o l o g i c a l m a n i f o l d s s u b d i -

v i d e d s o t h a t a g i v e n s i m p l i c i a 2 map

M onto N i s also simplicia1 with respect t o these subdivisions. I f m = dim M > dim N = n and q is t h e b a r y c e n t e r of a n n - s i m p l e x T o f N , t h e n t h e c a r r i e r of t h e o r i g i n a l c y c l e ~ ( q ) is an (m-n)-dimensional subcomplex o f M

f

of

.

Proof. Let a be an n-simplex of M such that f(u) = T . If p is any simplex of M I of which 6 is a face, f ( p ) = T and the barycenter of p is mapped into q by f. It follows that the entire dual cell bm-n of u is mapped into q by f . Conversely, let p be an arbitrary point of the carrier of

rp(q).

.

p E u ' = (pjior...rpji") If 0 k O,...,k, then q 6 f ( u ' ) and p

Suppose that

q for every r = is ):]p(f not in the carrier of ~ ( p,)contradiction. Hence, let a ' ( + @ I be the simplex of M' generated by all vertices of a ' which are taken into q by f and let uI2 be the face of a ' generated by the remaining vertices. There are 'points p1 E U ' I , p2 c a t 2 and integers a r b such that

'

0< a, b < 1,

a + b

= 1

and

P = aP,

+

bP2

*

Since f(p) = f(pl) = q and f(pZ) q , it follows that b = 0, a = 1 and p = p, E 0'1. Since f is simplicial, the vertices of 0'' are barycenters of simplices of dimension > n. Now, if p is a simplex of M such that dim p > n and f (barycenter of p ) = q , then f ( p ) = T and hence, p has at least one face, say u , such that f ( a ) = T . This shows that a' is contained in the dual (m-n)-cell of u We restrictnext M and N to be respectively

.

Cohomological Interpretation of

y(f)

33

S2n- 1

and Sn. We want to study the Hopf invariant of a simplicial map f : s*~-'+ Sn; we shall assume that both spheres are subdivided so that f is still simplicia1 with respect to these subdivisions (see ( 4 . 1 ) ) . Let q be interior to an nsimplex T Of Sn; since y(f) is independent of q , there is no loss of generality in assuming that q is indeed the barycenter of T Let T ' be the unique n-simplex of the subdivision (S") ' such that a h ' ) = T and let un be the nn cocycle of (S")' defined by (u~J') = 1 and (.u r p ' ) = 0, for every n-simplex p ' of (S")' different from T ' (here ( , ) is the Kronecker index). We shall also consider the fundamental cycles zn and z ~ of ~ Sn - and ~ S2n-1, where zn is taken so that appears in it with coefficient + 1 , Finally, to conclude this list of generalities we observe that the cohomology class {un) is a generator of Hn(Sn) (the Kronecker index induces an isomorphism Hn(Sn) + Hom(Hn(Sn) , E ) ). Theorem 4 . 3 . Let f# : Cn((Sn)') --* C # ( ( S 2n-1 ) ' ) be the co-

.

chain map induced by f. The carrier of the (n-1)-cycle u = f# (un ) n Bz2n-, coincides with the carrier of (o(q). i i Proof. Suppose that (p 0 ,...,p n-l) is an (n-1)simplex in the carrier of u. From the definition of the cap product, we can assume the existence of a (2n-l)-simplex (p2n-1, ,p0 ) of (SZn-' ) ' such that # 0 (f (u"), (pn,.. Ip 1 ) 9 0 and i0 = (p2n-l, . . . , p n 1 . (P r - - - iin-l) P

. ..

.

# 0 On the other hand, (f (un), (p",.. .,p 1 ) = n (u ,f (p", ,PO)) * 0 implies that f (p", . , P O ) coincides, up to orientation, with T ' ; this fact together with the definition of T ' shows that f (p") = q . Let u be the n-simplex of S2n-1 of which pn is the barycenter; then f ( a ) = T and S2n-1 which contains indeed, the barycenter of any simplex of 0 as a face is mapped onto q by f Hence, .f (pi,,. . , pin-^ = n f(p ,...,p 0) = q and therefore, (pio,...,pin-l) is an (n-1)-

. ..

..

.

simplex of the carrier of

Ip(q).

.

34

The Hopf Invariant

p be an arbitrary point of the carrier of (~(q).By ( 4 . 2 ) p belongs to the dual (n-1)-cell of an nsimplex o of s ~ . Let ~ -p" ~be the barycenter of o : then f(pn) = q and f ( o ) = T I . Hence, t.here is a (2n-l)-simplex of (S2n-11 ' , say (p2n-1, .. ,pn , .. ,p0) such that n 0 2n- 1 n 2n- 1 0 and p E (p ,...,p ) . Since (p ,...,p ) f(p ,...,p ) = T I is a (2n-l)-simplex which appears in the cycle B(z2n-l)n and n 0 (p2n1 r...,p 1 is (f # (u") , (p , . . . , p 1) = 1, it follows that an (n-1)-simplex of the carrier of u and therefore, p belongs to the carrier of u . 2n- 1 Because H,-,(S ) = 0, there is an n-chain v of (S2n-1) I such that av = u. Now, applying ( 4 . 3 ) we conclude that f#(v) is an n-cycle and thus, there is an integer d such that f (v) = d-B(zn). # p - 1 ( (s2n-1 On the other hand, there is vn-l 1 such that fH(un) = : notice that vn- 1 U f#(un) is a Let now

.

.

I )

(2n-1)-cocycle. Theorem 4 . 4 . Let 5 H2n-1((S2n-1)') Z . equal to y(f) 5

.

Proof.

(v"-l,av)

be t h e f u n d a m e n t a l c Z a s s of The c h o m o l o g y c l a s s

U

f#(un)}

is

We begin by observing that

=

(6vn-',v)

=

(f # (U"),V)

= (U

n

,d-B(z,))

= d

.

and hence, {vn-' U f# (u") } = d. 5 We are going to prove that y(f) = d. To this end, we take the barycenter q* on an nsimplex T * of Snr T * * T , and we construct un* and u* n following the same argument used in the construction of u and u, given q Because of ( 4 . 3 ) , yCf) = L(u,u*); the theorem is then proved if we show that L(u,u*) = d. By definition, L(u,u*) = I(v,u*) : on the other hand, since I(B(zn),q*) = 1 and d.(3(zn) = f++(v),it follows that d = I(f (v), q * ) . We are going to show that I(v,u*)'=I(f#(v),q*). #

.

Cohomological Interpretation of

Observe that

y(f)

35

I ( v , u * ) is defined: for this, we recall that the

carrier of u* is an (n-1)-subcomplex of S2n-1 , that the intersection number is bilinear and finally, that I(x,y) * 0 only whenever x and y are dual 1211; 5 6 8 1 . Suppose that n 0 2n- 1 n 2n- 1 x = (p ,...,P ) and y = (p ,...,P 1 , with ( p ,..., pn,.. . , P O ) a (2n-l)-simplex of ( ~ 2 n - 11). NOW (p2n-1 ,.. ,p") = tT*l ( E = ? I ) , in appears in u* only if (f(p", ...,p which case it appears with coefficient (f#(un*) , (pn,.. . ,p0 ) ) -(u~*,ET*')= E . Hence, I ( x , u * ) = E : on the other hand, I(f(X),q*) = I(€T*',q*) = E .

.

For a c o n t i n u o u s f u n c t i o n , f : S2n-1 + Sn we have defined its Hopf invariant via the simplical approcimation theowe attempt to define rem (see 5 3). In view of theorem ( 4 . 4 ) y(f) for the continuous case directly in terms of singular cohomology. Let r! and F; be the fundamental classes of Hn(Sn) 2n-1 (S2n-1) , respectively. Suppose that r) is represented and H by an n - cocycle u : the cocycle f# ( u ) is a coboundary, i.e., there is an (n-1)- singular cochain v of S2n-1 such that f# (u) = 6(v). The obstruction to proceed with a parallel argument to that developped for the simplical case is now given by the fact that v U f#(u) is not necessarily a cocycle; however, note that u u u = 6 ( w ) , for a certain (2n-1)- singular cochain, and that v U f#(u) - f# ( 0 ) is a (2n-1) - cocycle. Let ys(f) be the integer defined by

Iv

u ftt (u) -

f# ( w ) l

=

ys(f)

*

5

.

Clearly, if f is simplicial, y,(f) = y(f). We will now give another characterization of y,(f), showing its homotopy invariance. From the simplical approximation theorem it then follows that y s ( f ) = y ( f ) for all continuous functions f. Let Mf be the mapping cylinder of f ; the exact 2n- 1 ) shows singular cohomology sequence of the pair (Mf,S is free abelian of rank 2 with generathat H*(Mf,S2n-l ) N

tors

q

Theorem

in dimension

4.5. (Steenrod). Proof:

a

:

n

-

5

and q

U

in dimension =

-ys(f)

-

5

2n. Then,

.

We claim that the connecting homomorphism "2n-1 (s2n-1) -+ H2n ( M f , S 2n-1 ) maps the cohomology class

The Hopf Invariant

36

{V

U f# (u)

-

f# ( 0 1 )

-! U {

onto

(4.5).

this will imply

;

Consider the canonical maps j : S 2n- 1 +Mfi p : Mf + S n for which f = p . j, and choose a singular (n-1)- cochain w of Mf such that j# (w) = v The homomorphism j# maps the

.

cochain c we compute

=

w

U

p# (u)

-

p#

6c = 6w u p#. (u) Now, since

-

(w)

onto

p#(6w)

=

v

u

f# (u)

(6w-p# (u))

-

f# ( w )

and

.

p# (u)

u

6w - p# (u) is a cocycle which goes to zero under

, thus representing an element of Hn(Mf ,S 2n- 1 , and since # p (u) represents a generator of Hn(Mf) , the claim follows. Remark: The homotopy type of (Mf,S2n- 1 ) depends only on the j#

homotopy class of

f

;

hence

ys(f)

is a homotopy invariant.

-

Theoretical Solution of the Hopf Invariant one Problem _ and _ Applications. We have seen in ( 3 . 4 ) that for n = 2, 4 or 8 The obthere are maps € : S2n-1 + Sn such that y ( f ) = 1 5.

K

.

vious question to ask is then: are there any other values of n for which there exist functions f : S2n-1 + Sn with Hopf invariant 1 ? Much research has been done on the question but a conclusive result was published only in 1 9 6 0 by Adams : Theorem 5.1:. f : S 2n- 1 +

The o n l y v a l u e s o f

n S

with

y(f) = 1

are

n

for w h i c h t h e r e a r e maps,

2,4

or

8

.

Adams' proof was established with the aid of secondary cohomology operations in singular cohomology; we present here a much simpler and shorter proof, obtained using K - theory and its primary operations. It is due to Eckmann 1 8 6 1 ; a very similar proof was given independently by Adams-Atiyah 161. Proof of (5.1). To start with, we observe that in view of theorem 4.5 we have to search for all integers n = 2m n such that, if for which there are (based) maps f : S2n-1 -B S = , where rl is Cf is the mapping cone of f , then rl U and 5 is a generator of H2n(Cf ,Z 1 . a generator of Hn(Cf , Z ) In other words, we are asking €or what values of n there ex3 ists a finite CW - complex X such that H*(X;Z) zz [ x ] / ( X ) ,

37

Solution of the Hopf Invariant one Problem

with dim x = n. As a consequence of the Integrality Theorem (see (0.11)) , there exists z E KUo(X) such that ch(z) = x + qx2, q E (9 ; in addition, as seen at the end of (0.12), ch(Qkz) = kmx + qk 2mx2 and thus,

-

ch($k(z)

kmz)

q(k2m

=

-

km)x2.

Again, from the Integrality Theorem it follows that pk = qkm(km-l)

is an integer. Also, because ch is a monomorphism (see (O.ll)), $k( z ) -m k z =k p2 ~

.

2

For k = 2, the previous equality and the definiton of show that z2 ( p 2 - 1) = -2(X 2 ( 2 ) + 2 m- 1 2 ) . Now

2

does not divide

otherwise, it would divide

z2

and so, p 2

the truncated polynomial ring Z[x]/(x3) This means that

-- -

p2

q =

2m(2m-1) tegers. Hence, the integer as

and SO, necessarily possible only if m

9

U

with

u

and

v

x2

in

is odd. odd in-

2m.v

-

p3 = q

2m divides = 1,2 or 4

3m(3m-1)

can be written

3 m - 1 . But this division is (for a proof see Chapter 2 ,

5).

The first application of the preceeding theorem is the determination of the dimension of the spheres which possess a m u l t i p l i c a t i o n ; we recall that a m u l t i p l i c a t i o n on a sphere Sr (with base point e) is a continuous function :

such that

p(e,x)

=

x

sr =

x

sr

+

p(x,e) ,

sr for all

x E Sr.

Corollary 5.2. The o n l y s p h e r e s w i t h a m u l t i p l i c a t i o n a r e t h o s e o f d i m e n s i o n s 1,3 a n d 7

.

Proof. Clearly S1 ,S 3 and S 7 have a multiplication induced by the multiplication of the complex numbers C

,

The Hopf Invariant

38

the quaternions El Suppose now that:

and the Cayley numbers

u : sn- 1

x

s n-l

-3

Ul

, respectively.

sn- 1

is a multiplication, with n > 1. The Hopf construction gives n a map H ( p ) : S 2n-1 -D S and since p is of type ( 1 , l ) y(H(u)) = 1

(see (3.2)). Once the dimension of the spheres with a multiplica-

tion has been established, one would like to find out when two multiplications on the same sphere are homotopic and how many homotopy classes there are in each case. This problem can be solved completely using some results due to James. We begin by recalling the notion of S e p a r a t i o n e l e m e n t of two maps

u,v : K -D X which coincide on a subspace r and E r L c K such that K - L = e , an open r-cell. Let :E be the northern and southern hemispheres of the sphere Sr; the Sr- 1 r is the bounds a cell Vr and suppose that e equator image of the interior of Vr by a map f, with f(Sr-l) C L . r r be the orthogoFinally, let p+ : Vr --t E, and p- : Vr .+ Enal projections. The element d(u,v) = [g] E nr(X) defined by the map g : sr+x gp, = uf and gp- = vf is called the S e p u r u t i o n u and v The following properties hold [136]: (5.3) Let u,v : K .+ X be as before; then u y v, relative to L if, and only if, d(u,v) = 0 r ( 5 . 4 ) Given that K - L = e and u,v,w : K + X are such that u/L = v/L = w/L then, d(u,v) = d(u,w) + d(w,v). (5.5) Given 6 E nr(X) and u : K + X with r K - L = e , there is a map v : K -D X with v/L = u/L and such that

e l e m e n t of

.

.

d ( u v) = 6. In particular, any two multiplica ions fltf2 of define a separation element d(fl,f2) € n2n-2(Sn- ) beSnsn- 1 cause f l and f2 coincide on v Sn-1 and the complement of sn-l sn-l in sn-l x sn-1 is an open (2n-21-cell. NOW, if f : sn-l x s n-l -+ s n- 1 is a fixed multiplication, we associate to the class [f'] of any multiplication f' on Sn- 1 the element d(f,f'); notice that this cor-

39

Solution of the Hopf Invariant one Problem

respondence is well-defined because of the other hand, if 6 E n2n-2 (Sn-l) is is a multiplication f' on sn- 1 such previous observations and theorem 5.2 Proposition

5.6.

For

n

-

1 = 1,3

OP

(5.3) and ( 5 . 4 ) . On given, by ( 5 . 5 ) there that d(f,f') = 6 . The show that: 7 , t h e r e i s a one-to-

one c o r r e s p o n d e n c e b e t w e e n t h e homotopy c l a s s e s ( r e l a t i v e t o

sn-I

sn-I) o f m u l t i p l i c a t i o n s o n sn-I and t h e e l e m e n t s o f (Sn-' "2n-2 1 3 Z I 2 and Observe that since n2(S ) - 0, n 6 ( S l l 1 4 ( S ' ) = z120 there are, respectively, one homotopy class of multiplications on S 1 (all are homotopic to the multiplica3 tion induced from C ) , 12 classes of multiplications on S 7 and 120 classes of multiplications on S . Theorem 5.2 has several interest ng consequences v

.

-

-

which will be discussed next. n L e t p : IRn x IRn + IR be a c o n t i n u o u s f u n c s a t i s f y i n g t h e ' n o r m p r o d u c t r u l e ' l l p x,y) II = I I X I I Ilyll

Corollary tion

5.7.

-

and h a v i n g a t w o - s i d e d u n i t : t h e n

Proof.

n

=

1,2,4 o r

The multiplications on IR,b: , M

a. and

4,

show

that such a map exists. Suppose now that we are given a function p : IRn x IRn + lRn with the properties stated. Then, )1 I sn-l x s n- 1 is a multiplication on Sn-'; the result now follows from ( 5 . 2 ) . n is said to be p a r a l l e l i z a b l e if A sphere Sn-' S I R the bundle of (n-1) - frames on Sn-' has a cross-section or, in other words, if there are n - 1 tangent vector fields on Sn- 1 which are linearly independent at each point. Corollary 7 and

S

5.8.

The o n l y p a r a l l e l i z a b l e s p h e r e s a r e

1 3 S ,S

.

Proof. We begin by observing that if there exists a continuous function n p :IRnxIRn+lR

which is bilinear, norm preserving and with a two-sided unit, then Sn-' is parallelizable. In fact, we may assume that el is the unit and that {el,...,en} is an orthonormal basis of IR". Then, for every x E S"-', {p(k,el)= ~ , p ~ ~ , e ~ ~ , . . . , ~

40

The Hopf Invariant

is an orthonormal system of independent vectors which varies In particular, the usual multiplications continuously with x of C , M and d, show that S1,S3 and S7 are parallelizable.

.

Suppose now that Sn-' cIRn is parallelizable. For each x E Sn-l let fi(x), i = l,...,n-1, be the independent vectors at x , which we suppose to be orthonormal. Hence, the (n x n) - matrix

-(XI1 having the columns x,f,(x), ...,fn-,(x) is o r t h o n o r m a 2 depends continuously on x . Let M(x)

e

= (Xrfl

=

(XIr

.

Ifn-,

(l,O,...,O) E S

and

n- 1

and define

v :

sn- 1

x

->

sn-1

sn- 1

by P(Xry)

=

M(x)

.

M

-1

(el

.y

(here y is viewed as an (n x 1) -matrix). Clearly p is continuous and one checks readily that e is a two-sided unit. The assertion follows now from ( 5 . 2 ) . Remark. The statement ( 5 . 8 ) above was first proved by Kervaire; his proof is based on previous partial knowledge on the parallelizability of the sphere and the results of Bott on the periodicity of the stable homotopy groups of SO(n) and U(n) [ 1 4 9 1 . We say that IRn there is a b i l i n e a r map

is a d i v i s i o n a 2 g e b r a over IR

u : I R n x I R n

if

-> IRn

which has no z e r o d i v i s o r s . Corollary 5.9. The onZy finite dimensional division algebras o v e r IR a r e t h o s e of d i m e n s i o n s 1,2,4 o r 8 . Proof.

The usual multiplications on IR, C , M and 0 show that IR, l R 2 , IR4 and IR8 are division algebras over IR. Now assume that there exist a v : IR" x IRn + Rn which is bilinear (hence continuous) and without zero divisors. To sirn-

Solution of the Hopf Invariant one Problem

41

-

plify the notation we shall write ~ ( x , y )= x y . Let ,en} be a basis of IR". The conditions on p imply Ie, , that for every x * 0 the sets {x * el,...,x en] and {el * x,... ,en * x) are bases of lRn. Indeed, if Cxi(x * ei) = 0 we get x 1 A . e . = 0 and s o , xi = 0 In particular, 1 1 since {el e l , e2 el,... ,en * el} is a basis, we conclude that for all y E Sn-1 there is a u n i q u e element x E IRn such that y = x * el : moreover, x depends continuously on e2,...,x en onto the tangent y. Projecting the vectors x plane of sn-' at the point y = x we obtain n 1 el linearly independent vector fields on Snql Hence, ( 5 . 9 ) follows from (5.8). An a l m o s t - c o m p l e x structure on Sn-l is a continuous function with domain Sn-l which associates to each x E Sn-l an endomorphism Jx of the tangent space to Sn-l d C x , such that J2 = -1 X

...

-

-

-

.

-

-

-

-

1

.

-

.

Corollary ture are

The o n l y s p h e r e s w i t h an a Z m o s t - c o m p l e x

5-10. S*

and

Proof. (see [231: 41.16

~6

struc-

.

and S6 have almost complex structures and 41.211).

S2

Suppose that Sm (m 1) has an almost complex structure. Let x,y E Sm with x and y perpendicular: then the vectors Y and Jx (y) in the tangent plane to Sm at x are linearly independent (this follows from :J = -1). We define V ' (x,y) to be the unit vector orthogonal to y in the plane spanned by y and Jx(y) , i.e.

Clearly, v ' (v'(x,y)1x1

=

is continuous in both variables and 0 = ( v ' (x,y)ly) We extend V ' to a function

.

u :

,m+l

&m+l

>

mm+ 1

as follows. For two linearly independent vectors x,y E IRm+ 1 let x',y' be the two orthanormal vectors obtained by a fixed

42

The Hopf Invariant

orthogonalization process, i.e.

r

x'

=

X -

I1 xll

and

, if x and y are linearly dependent

0

otherwise. The map v perties:

is cont (V(XrYI

(5.11) (v

for every

(XIY)

x,y E IRm+l

We complete the proof of the Corollary, showing that v gives rise to a continuous multiplication on 7Rm+2 which satisfies the norm product rule and has a two-sided unit, and then applying (5.7). m+ 1 Let {el,...,em+,} be an orthonormal basis of IR lRm+2 and complete it to an orthoviewed as a subspace of normal basis {eorel, If X,Y are elements em+1 1 of nrn+*. of IRm+2 write them as

...

J

Define

p

:

IRm+2

IRm+2

~

*m+2

bY

The map p is continuous, and using the properties one verifies readily that

(5.11)

x = p(Xreo) llxll . IIYII

p(eo,X) = U(XrY) for all

=

1

.

X,Y E I R ~ + ~

( * I Eckmann calls such a function v a c o n t i n u o u s v e c t o r IRm+1 p r o d u c t of t w o v e c t o r s on [861, [871

.

CHAPTER 2 TORSION FREE

H

-

SPACES OF RANK TWO

Introduction A f i n i t e d i m e n s i o n a l H o p f s p a c e or, a f i n i t e h'-space for short, is a finite, based CW-complex (X,e) endowed with a continuous m u Z t i p Z i c a t i o n , that is, a map 1.

m : X x X + X such that the maps m(e,-) and m(-,el from X into X are homotopic to the identity of X (in other words, e acts as a two-sided homotopy unit). This notion obviously generalizes the concept of compact Lie group; notice that we do not require the CW-complex X to have a manifold structure, nor do we ask the multiplication to satisfy all group axioms, with the exception of the existence of a unit element (and this, only up to homotopy) . 7 The sphere S 7 and the real projective space IRP , with multiplication induced by the Cayley product of lR8 , are examples of connected finite H-spaces which are not groups. For some time it was thought that any 1-connected finite H7 space was of the homotopy type of a product G x S7 x x S , where G is a compact Lie group. Then, in 1969 Hilton and Roitberg 11081 produced an example of an H-space not homotopy equivalent to a Lie group nor a product with S ' ; their example was given by the total space of a principal S3 -bundle over S 7 . The discovery of this Hilton-Roitberg H-space greatly stimulated the study of finite dimensional Hop€ spaces; other interesting H-spaces were soon found, bringing along the problem of their classification (see [681, [1091, [1821, In this chapter we shall study the 12281, 12681 and [269] classification problem for a special family of H-spaces, the

.. .

7

.

43

44

Torsion free

H

-

spaces of rank two

so called "torsion free rank two

H-spaces". In [I241 Hopf proved that the rational cohomology ring of a finite, connected H-space X is an exterior algebra withodd dimensional generators,

,.. . ,xr)

H* ( x ; Q ) = A (x,

r

dim xi = 2ni - 1. Hopf also pointed out that if X is a compact, connected Lie group, the number r of generators is equal to the Lie group rank of X . Thus, for any finite Hspace X as before, the number r is called the rank of X. The t y p e of X is the sequence (2nl - l , . . . , 2nr - 1). A connected H-space of rank one and having no torsion in integral homology is a sphere. By a theorem of Adams (Chapter 1 , Theorem 5.1 ) , such a space has type ( 1 ) , (3) or (7) (thus, in Chapter 1 we solved the classification problem for the torsion free rank one H-spaces). The case of the torsion free, rank two H-spaces was discussed by Adams [31, Douglas-Sigrist 1811, Hosli [128] and Hubbuck [1291; they proved that the torsion free, connected, finite H-spaces of rank 2 have the following types: (Ir1)r (Ir3)r (Ir7)r (3r3), (3,5), ( 3 r 7 ) or (7,7) (see Theorem 2.10). The types we just listed are realized by S1 x Sl, S1 x S 3 , S1 x S7, S 3 x S 3 , SU(3) , S 3 x S7 and S 7 x S 7 , respectively. It is not hard to prove that, except €or the case (3,7), the previous is a complete list of homotopy types for the torsion free, rank 2 H-spaces (see § 4 1 . For the type (3,7) the situation is more complicated; for example, the Lie group Sp(2) and the product S 3 x S 7 are two spaces of this type and they are not homotopically equivalent. Actually, a torsion free, connected, finite H-space of type (3,7) is, up to homotopy, a 3-cell complex of the form S 3 u e7 u elo; moreover as shown by Hilton and Roitberg, such an H-space has the homotopy type of the total space of a principal S 3-bundle over S 7 . We now observe that these bundles are classified by the elements of n6 ( S3) ; Z 12 and that the latter group has a canonical generator w characterized by Sp(2) = S U w e 7 u e lo (see (3.3)). Let En be 3 the total space of the principal S -bundle over S7 which corresponds to n u ; then, En = S 3 Unue U elo. Actually, the

Introduction

45

spaces En and Eqn are homeomorphic and thus, for the type (3r7) there are only seven possible homotopy types, namely 7 Eo, E,, E2, E 3 , E4, E5 and E6. Of these, Eo = S 3 x S , E l = Sp(2) and E5 (the Hilton-Roitberg H-space) are Hspaces: using Zabrodsky's method of "mixing homotopy types" [267], Stasheff [228] and Curtis-Mislin [68] proved that also E j and E4 are H-spaces. So, it only remains to decide whether or not E2 and E6 can carry an H-space structure. It was Zabrodsky who first proved that both E2 and E6 are not H-spaces [268], by showing that if the CWcomplex s3unW e u e l o is an H-space then n + 2 (mod 4) (see Theorem 3.4). In order to prove this result, Zabrodsky had to recur to tertiary cohomology operations; we shall present a K-theoretical proof of Zabrodsky's Theorem [2181 The results quoted before, Zabrodsky's method of "mixing homotopy types" and some classical Algebraic Topology lead to the proof of a Classification Theorem (Theorem 4 . 1 ) due to Hilton-Roitberg [lo91 and independently to Zabrodsky [268]. (At this point we mention that the classification problan is also solved for rank two H-spaces having torsion in integral cohomology. According to Browder - [ 5 6 1 and [ 5 7 1 Hspaces of that sort have at most 2-torsion. Hubbuck (mimeographed notes) then proved, that an H-space of rank 2 and with 2-torsion has the same cohomology as the Lie group G 2 ; in particular, it is of type ( 3 , l l ) and primitively generated. Finally, Mimura - Nishida - Toda [ 1 8 2 ] proved there are exactly four distinct homotopy types of primitively generated Hspaces of type ( 3 , l l ) ) . The chapter is organized as follows. In §2 we define the "Hopf construction" and the "projective plane" PX of an H-space X ; we then compute the KU-theory of QX - the (4n+l) - skeleton of PX - and give the type classification of torsion free rank 2 H-spaces, modulo several number-theoretical results (introduced by the KU - theory of Q X ) , which shall be discussed in Section 5. The rationale behind this move is that although such results are directly related to Ktheory, their manipulation is long and hard enough to make one loose track of the global picture, at least on a first reading.

.

-

Torsion free

46

In

5

H

-

spaces of rank two

we investigate the H-spaces of type ( 3 , 7 ) and in 9 4 , we prove the Classification Theorem for the homotopy type of torsion free H-spaces of rank 2. Throughout this chapter we assume (without further mentioning) that our H-spaces are connected, f i n i t e CW - comp l e x e s together with a multiplication with strict unit (this assumption is no real restriction: since the pair '(Xx X , X v X) has the homotopy extension property, any multiplication on X is homotopic to one with strict unit). We close this introduction noting that K-theory has been used to solve other H-space problems. Refining and expanding the methods described in this chapter Hubbuck has greatly restricted the possible types of finite H-spaces having no 2-torsion in integral homology (see [129] and [1301). Furthermore, he proved (again with K-theory) that a homotopy commutative finite H-space is of the homotopy type of a product of circles 11311. (For another application of K-theory to Hspaces see also [132]). Finally, Wilkerson has applied K-theory in his investigation of spheres which are loop spaces mod p (see 12571 and [2581). 3

2.

Hopf Construction, projective Plane and Type of Torsion free Rank two H - Spaces In chapter 1 ( 9 9 3 and 5 ) we answered the question of whether or not a sphere Sn" admits an H-multiplication using the Hopf construction H(f) : sn- 1 * sn-' ZSn-' and investigating the Ku-theory of the mapping cone of H(f). To study our H-space problem we shall proceed in a similar way. Assuming that X carries an H-multiplication m , we define a map H(m) : X * X + SX and then study the KU-theory of its mapping cone, the so-called projective plane PX (actually, it will be enough to determine the KU-theory of a convenient subspace of PX). The cohomology of PX has a richer structure than that of X, reflecting the combination of algebra and topology involved in the definition of H-space. In this chapter we indicate the integral nth cohomology group of X simply by Hn(X). We begin by putting together some facts about the

-

Hopf Construction and projective Plane

47

(non-reduced) join X * Y of two connected CW - complexes and Y. For any space Z, let CZ = (I x Z)/(l x Z) be the c o n e over Z; then, the join X * Y is the subspace of CY x CY defined by

x *

Y

=

x

CY

x

u cx

X

Y .

x

We are also interested in the (non-reduced) s u s p e n s i o n SZ of Z; this is the union of the cones C+Z and C-Z glued together along the base 0 x Z, i.e., C

SZ =

+Z u

+ n

with

C-Z

C Z

C-Z =

.

Z

There is a canonical map

->

j : X * Y

S(X x Y)

defined by taking any ([t,z],y) E CX x Y into [t,x,y] E C+(X x Y), and any (x,[s,y]) E X x CY into [s,x,y] E C-(XxY). Now consider the diagram

sx X

*

Y

S(X

X

Y)

\

S(X

A

Y)

SY

where p1,p2 and q are the obvious projection and quotient maps. As one can check in [103], the composition Sq o j is a homotopy equivalence, while Spl 0 j and Sp2 0 j are nullhomotopic. Note that there is a canonical isomorphism cp

:

H"(x)

3

H"(x,,

Y)

c+

H"(Y)

H"(X x Y) , n

1. 1 ,

given by cp(a,b,c) = py(a) + q*(b) + p;(c); similarly for the suspended spaces and maps. Hence, the kernel of j * is equal to

ap; (E* ( X )

)

apz (;*(Y) )

morphically onto Hk+l



X X Y

i2 > C X X Y r2 > -

XXCY

y

Torsion free

48

H - spaces of rank two

and the commutative diagram H"(C(X1.Y) ,X x CY)

Hm(C(X+YhCX

@I

x

"

y) ->

Hn+m(C(X*Y) ,X4Y) 2 76 Hn+m- 1

( X * Y)

T6B 6

1.

r, 8 r:

>

Hn-' (X) % Hm-' (Y)

Hn+m-2

X

(X x Y )

(here 6 stands for the coboundary operator of the exact cohomology sequence of the appropriate pair of spaces); we wish to observe that the commutativity of (2.1) is readily established working with the exact triad (C(X * Y), X x CY, CX x Y), and using the basic properties of the cup product (see [79,V11,§8]) plus the fact that the composition Hk (X x Y) -> Hk+'(Xx CY, # x

Y)

Hk+'(X+Y,CXxY) ->

x * x

->

X

H (m) ~

>

and the compo-

sx

:

let RX = SX U C ( Z * Z) be the mapping cone of g which, in 4 this context will be called r e d u c e d p r o j e c t i v e p l a n e of X. Proposition 3.1. (i) H*(RX) = z[a,bI/ (a3,a2b,ba2 lb3 ) ,

dim a

dim b

= 4,

8.

=

(ii) RX c o n t a i n s sz = s 4 u e8 u s a s u b c o m p l e x . ~f i : SX -. RX i s t h e i n c l u s i o n map, i*(a) and i*(b) g e n e r a t e 4 8 t h e g r o u p s H ( S Z ) - Z and H ( S Z ) Z , respectively. Proof.

(i) The commutative diagram

induces a homomorphism of the exact cohomology sequence of into that of g I

-> (3.2)

H

r- 1

f* (SX)--->

I=

V

Hr-' (X * X)

---->

;

r H (PX) ---> f

r H (SX) --->

(k * k) * V

f

V!

=

Torsion free

H - spaces of Type

(3.7)

55

Since the of H*(X) whenever establish cribed in

subcomplex Z carries the generators x3 and x7 it follows easily that (k * k)* is an isomorphism Hr(Z * Z ) * 0. Using (3.2) and (2.4) we readily that h*IA : A y- H*(RX), where A is the ring desTheorem 2.4. Sk (ii) The composition SZ -> sx '1 > RX embeds SZ into RX. In dimensions 4 and 8 the homomorphisms g;o h*, h* and (Sk)* are isomorphisms thus, part (ii) follows. From now on we shall be more specific about the attaching maps of the CW-complex X = S3 U e7 U elo ; although we shall deliberately confuse maps and homotopy classes (thus, S 3 U, e7 U B elo with a E n 6 ( S 3 1 , B E n g ( S 3 U, e7) is really a homotopy type). 3 (3.3) Recall that the homotopy group n 6 ( S ) is isomorphic to Z,2 1 2 4 2 1 . We claim that the attaching map w of the 7cell in the Lie group Sp(2) = S 3 uw u elo, w E n 6 ( s 3) r e presents Q generator of n 6 ( S 3) . This is seen as follows. Take zw = s3 U w e7 and the sequence

3

part of the exact homotopy sequence of the pair (Zw,S ) . The characteristic map u : (D7 , S 6 ) ---> (Zw,S3) for the 7-cell in Zw generates the cyclic group n7(Zw,S3) and thus,aa = w generates the image of a . Since n 6 ( Z w ) n 6 (Sp(2)) 2 n 6 (Sp)=O 3 (see ( 0 . 6 ) ) , a is onto and thus, w generates n 6 ( S 1 . Our aim is to prove the following result. Theorem

3.4.

Let

S3 Unw e7

u

el o

be an

H-space. Then

n

2

(mod 4 ) This theorem shows that of the list E O , E 1 , E2' E 3 ,E4,E5 and E6 of possible torsion free rank 2 H-spaces of type (3,7), presented in the Introduction, the spaces E2 and E 6 certainly are not H-spaces. To achieve our goal we shall work with the functor L(Y) = KO(Y)@

KSp(Y)

and so, we now put together the basic properties of

L(Y)

(see

Torsion free H - spaces of rank two

56

[ll] and [2181). Observe that the tensor product of real and quaternionic vector bundles induces a Z 2 - graded ring strucare ture on L(Y). (If 5 is a real vector bundle and q , q ' quaternionic ones, 6 @ q is a quaternionic and r- Q r l ' is a real vector bundle). Thus, we have an isomorphism of r i n g s 1471 L(Y)

=

KOo(Y) @ KO 4 (Y)

KO(Y) @ KSp(Y)

compatible with the usual filtration (see (0.6)). For a based space (Y,*), there is a canonical decomposition L(*)@

L(Y)

GY)

.

The free abelian "subgroup" L(*) is generated by 1 € KO(Y) and an element [El E KSp(Y), represented by the trivial quaternionic line bundle, satisfying [El2 = 4 € KO(Y) If v is 8k 8k a generator of KO(S ) then {v,[Elv) is a basis of L(S (i.e. , [ E I ~generates KS~(S'~)). S-imilarly,if u generates K S P ( S ~ ~ + ~ {u,[E]u) ), is a basis of L(S8k+4). The exterior powers of real and quaternionic vector bundles define a Z 2 graded X - ring structure on L(Y). The X - operations give rise to Z 2 - graded Adams operations Qk, k € N, on the ring L(Y). These coincide on the subring KO(Y) with the usual real Adams operations. For a quaternionic bundle q over Y, the element J l k ( [ q ] ) i s reaZ if k i s e v e n and q u a t e r n i o n i c if k is odd.The operations Jlk are ring homomorphisms and satisfy

.

-

-

-

N

N

Furthermore, J,k([€l) =

{

2,

k is even

if

[€I, if

k is odd.

The following two commutative diagrams give the relations between the J, - operations on KSp(Y) and the real J, - operations on KO(S4 A Y) (we write 3 for the Bott isomorphism KS~(Y) ;0(s4 A Y)). N

N

N

H - spaces of Type

Torsion free k

57

(3,7)

even:

N

N

KSp ( Y )

>

N

-

.

KO(S~ A Y )

Recall now that the reduced projective plane RXn of 3 an H-space of the form Xn - S Unw e7 U elo contains the (reCZn -

duced) suspension pute the ring

-_P >

L(CZn)

S4

Unzw e8

To com-

as a subcomplex.

consider the cofibration

4 j > S UnEwe8

S4

s 8 , giving rise to exact sequences

-KO(CZn

I

(3.5)

"

8

KO(S

0 ->

)

P' - Z ->

j! -

) ->

4

Z ->

KO(S

0

N

The ring

is therefore torsion free of rank

L(CZn)

4.

._

u € KSp(CZn) be s u c h t h a t j!(u) i s a g e n e r a t o r o f KSp(S ) and l e t v E FO(IZn) be t h e p!- i m a g e of a g e n e r a t o r o f -KO(S8 ) . T h e n ,

Theorem

3.7.

Let

-

4

N

(i)

{u,[Elv} KO(CZn).

i s a basis for

KSp(CZn)

a n d {[€]u,V}

N

i s a basis of

(ii) 2

2

k (k - 1 )

1-

2

k 2u + q(n)

2 ,(2k

-l)[€lv,

k

even

k

odd

58

Torsion free H - spaces of rank two

prime t o

12.

4

k

I) (v) = k v .

(iii)

Proof. Let u and v be as in the statement. Then, I 4 if j ! ( u ) is a generator of KSp(S ) , the element [Elj'(u) = 4 j!([€]u) generates KO(S 1 ; similarly, [€]v is the p!-image " 8 of a generator of KSp(S ) . Part (i) now follows from the exact sequences (3.5) and (3.6). In order to prove (ii), consider the KO - theory of L> 8 us ae 12 8 the cofibre sequence S l2 s , a E nl1(s '24 We have KO(S8 U, el2) 2 Z i%> Z and so, choose generators w and z such that j!(w) generates zO(S8) and z is the p! - image of a generator of K0(Sl2). According to Adams [4, Part IV], the JI - operations on w are given by k 4 2 JI,(w) = k4w + X(a)k ( k -1)z N

-

N

N

where X ( a ) is a rational number that does not depend on in fact, X ( a ) (mod 1) is the so-called e - invariant of

k; a

[4, Part IV, 5 31. The homomorphism 8 e& : n l , ( S 1 -> Q/Z which takes Q into X ( a ) (mod 1 ) is injective (see [ 4 , Part 5 8 IV, 7.171) and the element C w E n l l ( S ) has order 12, according to [242; (5.5), (13.2) and (13.6)1 . We conclude that m+ C 5 w ) = n -12 t X(n where m,t f Z and m is prime to 12.

-

m + 12t, we conclude that for the e l 2 ) one has nC w 4 2 k k (k -1) IJIIR(w) = k4w + q(n) 12

Setting q(n) = n rations of x~(s'

JI

- ope-

u

.

Consider the Bott isomorphism B : KSp(S 4 UnCwe8) 4 4 8 el2) and set w = B ( u ) , KO(S A ( s unCwe 1 ) K O ( S ~u nC w z = B([E]v). Recalling the properties of I)K listed before, we obtain, f o r k even: N

N

1-

Torsion free

=

H - spaces of Type

+ q(n)

B(k4u

4

(3,7)

2

1[:

.

-l)[Elv)

Hence, [El$ k (u) = 2k 2u + q(n) k2(k2-1) [Elv

and multiplying by

6

[El

59

I

2

k

$ (u) =

2 2 - [ E I +~ q(n) k (k6 - 1 )

If k is odd we proceed in a similar way. Finally, part (iii) follows using the $-operations on Theorem

3.8.

Let

be a f i n i t e

R,

CW-compZex c o n t a i n i n g t h e

subcomplex

S4 UnIue8 and s a t i . s f y i n g c o n d i t i o n s o f P r o p o s i t i o n 3.1 . T h e n , (i)

L(Rn)

(x 3 ,x3yIxy2 ,y3 )

L ( * ) @ E[x,yl/

N

KO(S8 ).

(i)

and

(ii)

, where

N

x E K S ~ ( R ~ and )

In particular,

y E KO(R,). N

is a b a s i s o f b a s i s of KSp(Rn). y2)

KO(Rn)

~[~~x,y,x~,[~~xy

and ~x,[Ely,[EIx2 ,xy,[EIy*)

is a

N

2 (ii)

k2

+ q(n)

f,[Elx

2

y + a x2+bk[E1xy+cky 2 , k

(k -’) 6

q(n) = n - m + 12t

where

akl bkl ck a r e i n t e g e r s and m, t E Z and (m,12) = 1 . k (iii) J, (y)

=

4

k y + dk[Elxy

Proof. tral sequence of

+

eky2

,

where

dk,ek

(i) The E 2 - term of the Rn is given by

KO

-

with

are i n t e g e r s . theory spec-

E 2 = H * ( R n ) g~ KO*(*) =Z[a,b]/ (a3,a2b,ab2,b3 ) Q KO*(*)

dim a

= 4,

even,

k

,

dim b = 8. Using the derivation property of the dif-

ferentials, it is easy to show that the previous spectral sequence collapses. (In fact, assuming that dr = 0, r = 2 , . . . , m - 1 , we obtain for example, dm(a Q 1) E H 4+m (Rn)@ (*) : H4+m but this tensor product is always trivial since (R,) = 0 if

m

0

(mod 4 )

and

(*)

=

0

if

m = 0

(mod 4 ) ,

and

Torsion free

60

H - spaces of rank two

therefore, dm(a B 1 ) = 0).We conclude that and so, for the subring (3.9)

QL(R,)

zH*(Rn)

E2 2 E,

BKO* (R,)

L(Rn) c KO*(Rn)

we obtain 2 3 L(*) TZ[a,bl/(a3,a2b,ab ,b ) C3 L(*)

@

.

It follows that L(Rn) is torsion free. Now choose x E -KO4 (R,) - KSp(Rn) and y E EOo(Rn) representing a 8 1 and b @ 1 , respectively. Because of (3.9) we conclude that 2 2 the elements x,y[El, x [ E l , xy, y [El E kSp(Rn) and ~ " E l r Y r N

x

2

, xy[E], y 2

-

N

KO(Rn) form a basis of L(R,). 2 2 3 It remains to prove that x3 = x y = xy = y = 0 . First observe that the composition €

N

is an injection of rings, since KO(Rn) has no torsion ( 0 . 1 1 ) Observe that according to (3.1) all 3-fold products of the latter ring vanish, and so, the same can be said about KO(Rn). Next, notice that in L(Rn) the product of a non-zero element N

with [ E l is always non-trivial. We can now prove that x3 = 0 as follows. The element x[E] E EO(Rn) is such that 0 = 3 (x[E]l3 = 4x [ E I : since L(R,) is torsion free x3[€] = o 3 and so x = 0 . The remaining relations are established in a similar way and part (i) of (3.8) is established. Let us now prove (ii). Consider the inclusion 4 8 i : S unCUe -> Rn : from the previous discussion it is clear that the elements _I

u = i!(x) E v = i!(y) E

KO& 4 unCue8 -

satisfy the hypothesis of Theorem 3.7 (Observe the homomorphism E2(i) : E2(Rn) - qKO*(Rn) - E2(S 4 UnCUe ) = $?KO* 4 8 (S Unxue ) ) . We obtain the JI - operations on x using

-

'

Qki! = i!Qk, (3.7) (ii)

(the space

S4 unCoe

and

'

i!(x2) = i!(xy) = i.(y

2

= 0

is a suspension).

Finally we prove (iii). The complexification homoKU(Rn) is injective and does not morphism c : KO(Rn) -> increase filtration because KO(Rn) is torsion free. From

H - spaces of Type

Torsion free

61

(3,7)

[14, Proprosition

5.61 we infer that 4 ~ , ~ ( c ( y= ) k) c(y) + elements of filtration > 8

(this consequence of Atiyah's work will also be heavily exploited in § 5 ; the reader could therefore consult (5.1) which describes more precisely the property of Jlk we just used). But c commutes with the Adams operations; thus, k 4 Q (y) = k y + elements of filtration > 8

.

This concludes the proof of the Theorem.

-

Proof of Theorem 3.4. The L theory of a CW complex Rn satisfying (3.1) is given by the previous Theorem; this Theorem, together with the properties of L - theory described after the statement of (3.4), shows that the generators [E], x, y E L(Rn) satisfy.

J,2([E1)

= 2

J,~(Y)=

4 2 2 y + d2[€1xy + e2y

J,2

(XI = 2[ ~ ] x+ 2q(n)y + a2x2 + b2[E]xy + c2y

J,3([€l) J,

3

J,3

[El 4 2 3 y + d3[E1XY + e3Y

=

(y)

=

(x)

= 9x

+ 6q(n)[E]y + a3[Elx2 + b3xy + c3[E1y

Recalling that [El2 = 4 , we have: J, 2J, 3 (y) = 2434y + (34d2 + 2

+

(34e2

+ 26q(n)d3 + 28e3)y2

3 2 4 4 4 6 Q J, (y) = 2 3 y + (2 dj + 3 d,)[~lxy

4 + ( 2 e3 Since

J12J,3

=

2

$3J12

2d3

=

+

+

+ 38e2)y 2

.

4 5 10-3 -e2 + 3 q(n)d2

-

3 5 2 3 q(n)d2

, 3

3 d2

3 30e3 + 2 q(n)d3 Eliminating d 2

r

=

in the second equation we obtain 3e3 = 3 4e 2 + q(n)dj

2

.

Torsion free

62

H - spaces of rank two

-

and since q(n) = n m + 12t = n (m,12) = 1) , we conclude e3

(3.10)

For the element J,

x

I-

(mod

e2 + nd3

(mod

2)

(recall that

.

2)

we have:

2 3

IJJ (x) = 18[E]x + 210 q(n)y + Ax2 + B[E]xy +

+

(32c2 + 2 23q(n)e2 + 23q(n) 2a3+ 2 5q(n)b3+ 2 c

$ 3 ~ 2 ( x )= 18[E]x

+

210 q(n)y + A'x2 + B'[EIxy +

+ (Z3c3+ 2q(n)e3+233 3q(n) 2a2+ where

A,B,A',B'

3 5

2 3 q(n)b2+ 3

are expression whose explicit form is not

needed in our argument. Equating the coefficients of dulo 8 , we obtain: 2

2 3q(n)e2 = 2q(n)e3

8 2 ( 3 -3 )c2 = 2q(n)e3

+

y 2 , mo-

(mod

8).

Thus 6q(n)e2 = q(n)e3 since either

q(n) = n

(mod 4 )

(mod or

-

:

4)

q(n)

3n

(mod 4 )

, we

see that 2ne2 = ne3

(3.11)

(mod

4)

.

NOW, the relation

IJJ 2 ( y ) holds in the

h

Y2 - 2h2(Y)

- ring L(Rn). Hence,

IJJ 2 ( y ) and thus, e 2 = 1 (mod and ( 3.1 1 ) become e3

and we infer

=

y2

(mod

2)

2). With this, the congruences

+

nd3 = 1

(mod

2)

ne3 = 2n

(mod

4)

(mod

4).

n + 2

(3.10)

The Homotopy Type Classification

63

4.

The Homotopy Type Classification In this section we give a complete list of homotopy types of torsion free rank 2 H-spaces. More precisely, we shall prove the following. Theorem

4.1.

torsion free f i n i t e

A

H-space

o f rank

2

i s

h o m o t o p y e q u i v a l e n t t o o n e of t h e f o l l o w i n g e l e v e n s p a c e s :

s1

S l , S1

x

x

S 3 , S1 x S7 , S 3

E l = Sp(2.1, E j r E41 E g A22

and

SU(3), Eo = S 3

S3,

x

7

S

S7 x

x

S7 I

.

t h e s e s p a c e s r e p r e s e n t m u t u a l l y d i s t i n c t homotopy t p p c s .

We begin by recalling, once more, that the integral cohomology of a torsion free H-space X of type (2q-lI2n-1) is an exterior algebra

We first dispose of the cases

(2q-1,

2n-1) = ( 1 , 1 ) 1

and ( 1 , 7 ) by means of the following result (recall that the fundamental group of an H-space is abelian). (1,3)

Theorem

Let

4.3.

be a c o n n e c t e d

X

complex. I f t h e fundamental groiip o f summand o f r a n k (S1)k

X1

x

,

k

where

,

H-space X

t h a t i s t o say, i f

n1 (X1)

n

w h i c h is a

CW-

contains a free direct nl(X)

.

- Z

k

@ n 1 then

Proof. Assume first that k = 1 . Let f : S’ + X be a representative of a generator for the subgroup Z 5 n l ( X ) and let p : X 1 -, X be the covering corresponding to the sub- n. The composition group n 5 n, ( X ) , so that n 1 ( X 1 ) :

s1

x

x1

fXP

m >

x

x

x

___

> x

induces an isomorphism between the homotopy groups of

S

1

x

X,

1

hence X N S x X1 . The case k > 1 follows easily by induction since the space X1 is again an H-space (cf. Chapter 7, Lemma 2.2).

and

X

I

Corollary 4.4. Tf X i s a t o r s i o r . f r e e H-space of t y p e 2n- 1 (1,2n-1), t h e n x = s1 x s Proof. Follows immediately from (4.2) and (4.3).

Torsion free H - spaces of rank two

64

We turn next to the case

q = n.

Theorem 4.5. If X is a torsion free H-space of type 2n-1. (2n-lI2n-1), then X = S 2n-1 Proof. The Theorem holds for n = 1 ; let us suppose n 2. The Hurewicz Theorem, the Universal Coefficient Theorem and (4.2) imply that 2n- 1 H (XI ~ Z O :Z n 2n-1") 2 H2n-l (XI thus, let fi : S2n-1 + X I i = 1,2, be maps representing a basis of n2n-1(X). The composition 2n- 1

m

S2n-1

> x

induces an isomorphism between the integral cohomology groups in dimension 2n - 1 and hence, it induces an isomorphism beS2n-1 tween the exterior algebras H*(X) and H* (S2n-1 ) . BY Whitehead's Theorem, S2n-1 x s2n-1 CI! x

.

Now we consider the type (3.5). A torsion free H-space of type (3,5) is hoTheorem 4 . 6 . motopicaZZy equivaZent to the Lie group S u ( 3 ) . Proof. An H-space of the kind given in the statement is simply connected and therefore belongs to the homotopy 3 3 3 5 Z2, B E n 7 ( S Ua e 1 (see type s U, e5 u B e8, a E n4(s (4.2)). We discuss first the case a = 0. Consider the space X 0 = ( S 3 v S 5) U e8 ; the inclusions i* : ) ' s ( H S the proof of

i3

: S

3

+

Xo, i5 : S5

H'(X~) , s = 3'5. If (4.5) we see that

s3

x

s5

i3

i5

>

+

Xo

xo

Xo

induce isomorphisms

were an

x

xo ->

H-space, as in

xo would induce an isomorphism between the integral cohomology groups and thus, S3 x S5 = Xo. But a retract of an H-space is still an H-space; thus, the last homotopy equivalence would imply that S 5 is an H-space, contradicting Corollary 5.2 3 8 of Chapter 1. Hence, a complex of the form ( S v S5) U e cannot carry a Hopf multiplication. So, let now

65

The Homotopy Type Classification

X = S3 u q e 5 u

e8

D E n 4 ( S3 Z2, rl 0, and B E n 7 ( S 3 U,, e5 ) , be an H-space. Now, it is known that the Lie group SU(3) has a cellular 3 e8 and hence, X and SU(3) structure of the form S u,, e "Y have the same 7 - skeleton. Since n7(SU(3)) = 0 12431, the inclusion of S 3 u rl e5 into SU(3) extends to a map f : X + SU(3), inducing isomorphisms between the integral cohomology groups in dimensions 3 and 5. But H * ( X ) and H*(SU(3)) are both exterior algebras with generators in dimensions 3 and 5, thus, f* is an isomorphism and therefore, x 2,+ SU(3). Finally, we investigate the most interesting case, namely the type (3'7). Classical examples of H-spaces having this type are S3 x S7 and the Lie group S p ( 2 ) . Both spaces are the total spaces of principal S3 - bundles over the sphere 7 S ; we search for other bundles of this type, whose total spaces rl

3

carry an H-multiplication. Recall that the principal S -bun3 dles over S 7 are classified by the elements of n 7 ( B S ) 3 ns(S 2 1 2 and that Z 1 2 has a canonical generator w characterized by Sp(2) = S 3 U, e7 U elo. Let En be the total space of the S3-principal bundle over S 7 induced from the 7 3 > s 7 of > S 7 by a map fn : S S -bundle Sp(2) degree

n

P1

:

gn

>

En

Sp(2)

(4.7)

BS (Here w '

corresponds to

a base point S7

-

{*}

(En,S3 )

*

E S7.

The

w

3) Z n 6 ( S 3 1 ) . Choose IT~(BS bundle En + S 7 is trivial over

under

S3 -

and there is a map of pairs which is a homeomorphism of

-1 7 (S - { * I ) ; pn

(here D7

is the closed

h

:

(D7 x S3,S6

(D7- S 6 )

x

S3

x

S3)

onto

7 - ball). Hence we

+

66

Torsion free

H - spaces of rank two

obtain a CW- decompositon En = S3 U (e7 X ( * ~ e3 ) ) = 3 7 U e ) U elo. The bundle map gn : En S p ( 2 ) gives rise to (S a commutative triangle .+

Since the homomorphism (gn)* is multiplication by n and a(11 = w , we conclude that the attaching map of the 7-cell in En is equal to nu ; thus (4.8)

=

(s3 uno

e7)

u

e 10

En

En and hornotopy e q u i v a l e n t if, and o n l y i f , n

Proposition

4.9.

The s p a c e s

. Em, -6 5 n, m 5 6 a r e = i m .

Proof. It is clear that En and E-n are homotopy equivalent (even homeomorphic). Conversely, let h : En Em be a homotopy equivalence, which we can assume to be cellular. Since S3 carries the 3-dimensional homology of En and Em, h' = hlS3 must be of degree i 1 on S 3 . A comparison of the homotopy sequences of the pairs (En,S3) and (Em,S3 ) shows that 3 3 h* : m7(En,S 1 n7(E 1s ) : m notice that these groups are cyclic infinite and moreover, the following diagram commutes: .+

I n addition, observe that a map of degree -1 on S3 induces the isomorphism -1 on the homotopy groups of S 3 because S 3 is an H-space. Thus, starting with a generator I n € n7(En,S 3 122

The Homotopy Type Classification

67

we compute h$an(ln) = h$(nw) = * nu and a mh,(ln) = am(*lm) = f mu and conclude that n = m . Now we restrict our attention to the seven spaces Eo, E,, E 2 ’ EjI E q I E 5 and E g I which by (4.9) represent mutually distinct homotopy types. Of these, two are not Hspaces, namely E 2 and E6 (this follows immediately from (3.4) and (4.8)); we claim that the remainder carry an Hmultiplication. More precisely, Theorem and E5

4.10. are

The spaces

Eo = S 3

x

H-spaces.

We shall give a proof of th s Theorem relying on Localization Theory of 1 - connected CW-complexes. We shall review here the basic results of Localization Theory needed in the proof of (4.10); the reader who desires to become more familiar with such theory is advised to consult [107], where a proof of Theorem 4.10 can also be found. We denote the set of all prime numbers by TI . For any subset P 5 TI I we say that a group G is P - l o c a l if n x , x E G I is a bijection for all the function x b> n E n - P. For any ’I-connected CW-complex X we write Xp for a P - Zocalization of X (Xp is a I-connected CW-complex with P - local homotopy groups and there is a map ep : X + Xp Inducing isomorphisms (ep), Q Z p : n,(X) C3 Z,

-

N

where z p 5 Q is the ring of integers localized at P). For I-connected CW-complexes P - localizations always exist and are unique up to homotopy type. A map f : X + Y gives rise to a map fp : Xp + Yp ; if f, : n , ( X ) + n,(Y) is a P - isomorphism, then fp is a ho> n,(Xp) @

Z,

I

motopy equivalence. The space X# corresponding to the empty subset # c n is a rationaZization of X. For any P 5 TT one has (XP)@ = X# and there is a factorization eg:

x-

eP

>

xp > e

X#

I

p I@

.

A rationaliwhere e is the rationalization map Xp + X# PI# 2q+ 1 is given by the Eilenberg-MacLane zation of the sphere S space K(Q,2q+l). Finally, we note that if X is an H-space,

Torsion free

68

H - spaces of rank two

any localization Xp inherits an ep : X + Xp is an H-map. Proof of Theorem

s7

>

~

i4

4.10.

s7

H-space structure such that

Consider the diagram

>

~

s7

f3

(fr is a map of degree r, r = 3 , 4 ) . The sphere S 7 is an H-space and so, using the left-distributive law, we conclude that (f3)* : n,(S 7 ) + n , ( S 7 ) is just multiplication by 3 . The bundle morphism (grf3) induces a morphism between the appropriate exact homotopy sequences; we conclude that g is a ( n - ( 3 1 ) - equivalence and thus, (E3)p ( S P ( ~ ) ), ~where P = n - { 3 1 . Similarly, h is a ( n - (2)) - equivalence and 3 7 so, (E3lpl = ( S x S , with PI = { 3 3 Now, since Sp(2) and S 3 x S 7 are H-spaces, it follows that (E3)p and (E3)p i are H-spaces. Therefore, the multiplications on the latter two spaces give rise to two H-space structures on 7 = K ( Q , 3 ) x K ( Q , 7 ) . These structures must (E3)@ = (S3 x S coincide, since (up to homotopy) there is a unique Hopf-multiplication on the product K ( Q , 3 ) x K ( Q , 7 ) of Eilenberg-MacLane spaces. (In fact, two multiplications on an H-space X "differ'

.

(E3)p by an element of [ X A X I X I ) . Now, we turn e into a fibration and consider the pull-back diagram

-+

(E3)g

V

Since P ' = n - p , the basic pull-back theorem of Localization Theory (see [107, Theorem 11.5.11) shows that E is homotopy equivalent to E 3 . Furthermorei, e and epl ,@ being H p ,0 maps, the space E carries an H-space structure and thus, E 3 is an H-space. Interchanging the roles of 3 and 4 in the

The Homotopy Type Classification proof, we see that E 4 is an before, kut with the diagram

H-space. Finally, proceeding as

f7 one shows that

E5

Theorem (mod 4 )

Let

En

-

f5

H-space.

+

3

Xn = S Unw e7 U D el0, -6 < n < 6 , n 2 H-space. Then Xn is hom ot op y e q u i v a l e n t t o

4.11.

,

is an

69

be an

In order to prove the Theorem we need the next three

lemmas. Lemma

The g r o u p

4.12.

ng(En)

is f i n i t e and its order d i -

v i d e s t h e g r e a t e s t common d i v i s o r

Proof.

(n,6).

The exact homotopy sequence of the fibration

and the finiteness of the groups n k ( S 2 r + 1 ) , k * 2 r + 1 ; show that n k ( E n ) is finite for k 3 , 7 . In addition, since 7 3 L 3 (generated by ng(S 1 z 2 ( 2 4 2 1 and n g ( S 3 &>

A> S3)

,

the exact homotopy sequence proves the existence of an exact sequence S9

S6

[ 2141

The left-distributive law valid for the H - space S 3 im3 3 plies that n(w 6 C W ) = (nw)oE W . If n 0 (mod 3) the ele3 3 ment n(w o C w ) generates n g ( S ) and i,(n(w a C 3 w ) ) = (io nw)o C 3 w = 0 ; therefore,

+

(4.14)

is exact for

O - > n ( E )

+

n 8

9

n

->

z2

(mod 3). We recall now that

0

2

3 nS(S )

z2,

generated by S S6 > S3 [ 2 4 2 , Prop. 5 . 9 1 . If n isodd,the left-distributive law shows that i*(wT)2 ) =

70

H - spaces of rank two

Torsion free

2

i,(n(wn

) )

=

(4.15)

i,((nw) o n

2

)

23

( i o n w )o n 2

=

= 0

>

n (En)

>

and thus,

9

is exact. The statement follows from

o

(4.13), (4.14) and

(4.15). 4.16. The o r d e r of t h e t o r s i o n s u b g r o u p of n ( S 9 divides (n,6).

Lemma

Proof.

3

7 Unwe )

Consider the following part of the exact ho7 (En, S3 Unu e ) :

motopy sequence of the pair n

10

(E n

--->

n

( E ,S

10

n

3

Unwe7

a

->

n (S 9

7

nw

e ) -->n

(E I n

9

3 7 The group n,O(En) is finite and nlO(En,S Unw e ) : - E 2 is injective and the result follows from (4.12).

;

.

thus,

g : S 3 v S7 + S 3 Unw e7 w h i c h i s 7 t h e c a n o n i c a l i n c l u s i o n o n S3 a n d s u c h t h a t (glS ) * : H7(S ) 3 7 -+ H7(S Unw e ) i s m u l t i p l i c a t i o n b y k , w h e r e k is p r i m e t o n. Proof. Let k be an integer such that (n,k) = 1 and kn E 0 (mod 12). Consider Lemma

4.17.

T h e r e i s a map

The restriction g quired properties.

=

hklS3

v S7

provides a map with the re-

and fix Proof of Theorem 4.11. Let Zn - S3 Uno e 7 a map g : s3 v s + zn satisfying the properties stated in 3 7 10 (4.17). Assume now that Xn = ( S Unwe ) U B el0 = Zn U B e p E no(Zn), is an 11-space with multiplication 2 unit). The composition glS3xglS7

h : S 3 x S in an extension of

>

xnx

g. The homomorphism

xn

m

(and strict m >

n'

The Homotopy Type Classification

71

is such that h*(x7) = ky7 (see (4.17)) and thus, h has degree 2 k on the top cell. Consider the following commutative diagram induced by h :

,=

(hlS3 v S7)

h* Z

g,

V

at

nlO(XnrZn)

>

ng(Zn)

and represent the generators of the two infinite cyclic groups on the left by the characteristic maps y 1 : (D10,S9) + ( S 3 x S 7, S 3 v S7) and y 2 : (D10,S9) (xnrZn)(i.e., ayl = [ i 3 , i , 1 and a'y2 = 6). Since h has degree f k o n the top cell, h,yl +

.

For the H +kY2 and g,a(yl) = a1h,(y2) = f kal(y2 ) = f kB space En - Zn U o l elor 8 ' E n ( 2 ) , the same map g : S 3 v S7 9 n --. Zn implies that g,a(yl) = ? r kB' and thus, kf3 = kB' or kB = -kB', i.e., k(B-B')

= 0

or

k(B+B') = 0

.

Since (k,n) = 1 , Lemma 4.16 implies that either B = B ' or 8 = - B ' , thus proving that Xn and En have the same homotopy ' '-8 elo). type. (It is trivial that Zn U B el0 N n Theorem

4.18.

A torsion free

H-space

of type

(3,7) i s h o -

motopy e q u i v a l e n t t o one o f t h e f o l l o u i n g s p a c e s :

A 1 2 t h e s e s p a c e s r e p r e s e n t m u t u a l l y d i s t i n c t homotopy t y p e s .

Proof. An H-space satisfying the hypothesis of (4.18) is homotopy equivalent to a 3-cell complex of the form 3 e7 u e lo (see 5 3). The result now follows from 3.9) 'nu (4.9), ( 4 . 1 0 ) and (4.11). Proof of Theorem (4.5), ( 4 . 6 )

and

(4.18).

4.1.

Combine

(2.101, (4.4)

I

Torsion free H - spaces of rank two

72

K - Theoretical Proof of the Type Classification Theorem We have seen in 9 2 that our proof of the Type Classification Theorem for torsion free H-spaces of rank 2 (Theorem 2.10) depends directly on Theorem 2.9. In this section we give a proof of the latter theorem. 5.

(5.1) In what follows we shall make constant use of the f o l lowing properties of the Adams Operations Q k in K U - theory (see (0.12) and [14;(5.6)1) : (i)

The operations Jlk

(ii)

~

l

= Q~

(iii) Q 2 (z) (iv)

if

I

~~ ; Jl

are ring homomorphisms;

~~

~

(mod 2) , for every

z2

is a torsion free finite

Y

and z E KU(Y) has exact filtration Theorem 2 . 8 ) then, for 0 5 r < q , 9 k (z)

=

z E KU(X) ;

kq-ryl + ~2

in addition, k Q (z) = kqZ + y2

r

r

2q

CW-complex

(see footnote for

~1 E KU(Y) r ~2 E f2q+2r+2KU(Y)

y2 E f

2q+2 KU(Y)

;

.

The following result of elementary number theory (already used in Chapter l , Theorem 5.1) will also be needed: For a n y i n t e g e r k Z e t v2(k) be t h e e z p o n e n t of t h e h i g h e s t power of 2 d i v i d i n g k !*I Then

Lemma

5.2.

(i)

v 2 (3m-1)

=

{

1 , i f m i s odd ; 2 + v2(m) , i f m i s e v e n ;

1,2 o r

ii)

i f

m

9

iii)

i f

q

2r

then

4, ~ ~ ( 3 ~ -< 1m ) ; v2(3q

+' -1)
l

+ 1 = 4

(mod81

(mod 4)

.

,

,

and v

-

1

I

2

Part (ii) observe its veracity 8 and let that r But a 1 4 because 5 2

+

follows easily from (i). A s for (iii), for r 5 7 (by direct inspection). Assume a be the integer such that ' 2 5 r < 2'. r 8 : using (i),

v (q+r) 5 2 2

+ a

< 2a- 1

5 r

.

Proof of Theorem 2 . 9 . We shall distinguish two ca< n < 2q and n 1. 2q. ses, namely, q Case

1.

q < n < 2q.

Write n = q + r, 0 5 r < q and consider a CW-complex Q satisfying the hypothesis of ( 2 . 9 ) . The elements x , 2 y,x ,xy and y2 of KU(Q) have filtration 2q, 2(q+r) , 4q, 2(2q+r) and 4(q+r), respectively. From (5.1) (iv) we conclude that $

k

(XI

= kqx

k Ji (y) =

where

+ kq-raky + bkx2 + ckxy + dky 2 kq+ry

+

kZrekx2 + krfkxy

ak,bk,ck,dk,ek,€k and

$mJik(x) = mqkqx

+

(kqbm + m2rkq're

are integers. Hence,

gk

(mq-rkqam

+ gky 2 ,

+

mq+rkq-ra')k

m ak + m2qbk)x2

+

+

(kqcm + mrkq-rfmak + 2m2q-rambk + m 2q+rck)xy

+

(kqdm + kq-rgmak + m 2q-2ra2b m k + mzqamck + m 2q+2rdk)y2

,

H - spaces of rank two

Torsion free

74

m k (y) = mq+rkq+ry

+ Bx2 + Cxy +

J, $

+ (kq+rgm + m 2q-2r

2r arnek + m2qkramfk+m 2q+2rgk)y2

I

where B and C are expressions whose explicit form is not 2 xy needed in the sequel. Equating the coefficients of y I x and y2 in the equality 2 k k 2 J, (x) = J, (XI I I

*

*

and those of

yL in 2 k J, ic) (Y)

=

J,k$21Y)

I

we obtain the following conditions: 2r(2r-l)ak = kr(kr-l)a2

+

2q(2q-1)bk

;

2q ( 2q+r-1 ) ck + 2rkq-rakf

kq(kq+'-l

)

c

2

+'-'2

2q ( 2q+2r-l ) d k

bka2

+

+

-

kq+r ( kq+r- 1 ) g 2 + 2 2 r k 2 q - 2re2ai

(R6)

Lemma

(mod 2 )

,

b2

d2

0

(mod 2 )

I

g2

5.3.

If

2q,

=

22qkra2fk =

+

2rk2qakf2

5

1

(mod 2 )

1

(mod 2 )

.

.

r = 0 a n d q P 1,2 or 4 (q > (R1) to ( R 6 ) a r e s a t i s f i e d .

Proof. filtration

+ 22qcka2

(5.1) (iii), we obtain the congruences

c2 = 0

the relations

;

k2q-2rb a2 + k2qc2ak : 2 k

+

2q+r ( 2q+r- 1 ) gk + 2 2q-2rk2',ka2

=

k2q-rb2ak

22q-2rbka2 2

+

Zq-'a2gk

;

22q-r+l

+

k a2fk + 2

+ kq-'akg2

kq(kq+2r-1)d2

In addition, using

+ 2 q-rk 2ra2ek

22rkq-rake2 = kq(kq-l)b2

Since

r

= 0, both

x

(5.1) (iv) implies that

and ak

y

O),

not QZI

have exact

= 0.

Then, ( R 2 )

Proof of the Type Classification Theorem

with

k

=

3

75

becomes 2’(2’-l)b3

.

3’(3’-1)b2

=

since b2 is odd (see (R6)), 2q divides 3‘ - 1. By (ii) this is possible only for q = 1,2 or 4. Lemma 5.4. If q is odd , q 2 3 and q > r r e Z a t i o n s (RI) t o (R6) a r e s a t i s f i e d .

1

(5.2)

n o t a11 t h e

Suppose first that q - r ? 2, q 2 3, r 2 1. (R2) with k = 3, implies

Proof. In this case,

0

Since

q

is odd, 3q 0

(mod 4).

3q(3q-l)b2

I

I

-

2b

1

I

2

(mod 4 1 , by

(5.2) (i), and hence,

,

(mod 4)

.

contradicting (R6) If q - r = 1 , q 2 3, then

r 2 2

0 = 32q-1 (329-1-,)92 = 2g2 again contradicting Lemma

5.5.

rezations

and

(mod 4 )

(R5) implies

,

(R6).

If q 2 3, r is odd and q > r 2 1, n o t a21 t h e (Rl) t o (R6) a r e s a t i s f i e d .

Proof. By the previous Lemma it suffices to prove the claim for q even. If q > r 2 2 , equation (R5) ‘with k = 3 , implies: 0

I

.

(mod 4)

3q+r(39+’-l)g2

Since q + r is odd, 3q+r - 1 P 2 (mod 4) and the former congruence implies that 0 = 2g2 (mod 4) contradicting (R6). If q > r = 1, we infer from (R4) and d 2 I 0 (mod 2) that kq-’akg2 P k 2q-2akb2 2 + k2qakc2 (mod 4). Since c2 is even and both conclude that (5.6) Relation

ak

+

(R5) with

2

b2

(mod 4) k = 3

and

,k

g2

are odd (see (R6)) we

odd.

and reduced

mod 4

gives

Torsion free

76

H - spaces of rank two

2g2u + 2a3f2

0

(mod 4 )

,

u

odd.

Since 9 2 = 1 (mod 2), we conclude that sider now (RI) with r = 1 : 2ak

must be odd. Con-

.

k(k-l)a2

=

a3

Since a3 is odd, the same holds for a2. But this implies that a5 I 2 (mod 4 1 , contradicting ( 5 . 6 ) .

z

Lemma 5 . 7 . If q 3 and q > r (R1) t o (R6) a r e s a t i s f i e d .

,

1

not a l l the reZations

Proof. By Lemmas ( 5 . 4 ) and ( 5 . 5 ) it suffices to prove the statement for q and r even, q > r >- 2 . Using ( 5 . 2 ) (ii) and (RI) we conclude that (5.8)

= v 2( a2) + l , i f 5 v2(a2) , if

v2 (a,)

r = 2 r > 2.

Suppose first that q > - 6. From (5.2) (if and q and r interchanged, we have

(iii) with

2 l + )v2(q+r). q > ~ ~ ( 3 ~ + =~ -

Reducing that c2

(R3) (with k = 3) mod 2 3+v2 (q+r), and recalling is even, we obtain that 2r3q-rf2a3 - 2q-r3rf3a2

a

3+v2 (q+r) 2

*

32q-rb2a3

(mod

2

)

-

Since b2 is odd, (5.8) shows that the 2-divisibility of the terms appearing on the left handside of the last congruence is strictly greater than the one of the terms on the right handside ( q 6, ~ r -> 2, q-r ? 2 ) This implies 2+v2 (q+r) (5.9) a3 5 0 ( mod 2 1

.

and because of

(5.8), l+v2(q+r)

a2

5

0

(mod

2

a2

I

0

(mod

2

(5.10)

)

,if

r = 2

,if

r > 2

2+v2 (q+r) )

.

77

Proof of the Type Classification Theorem 3+v2 (q+r) Reducing (5.9)

(R5) mod

and

(5.10)

2

1 +v2 (3'+'-1) = 2

( q _ > 6 ,r 2 , 2 r q-r?2)

This contradicts g2 = 1

(mod 2 )

I

and using we obtain

and the Lemma is proved for

q 2 6.

Finally, consider the case q = 4, .r = 2 . By ( 5 . 8 ) the integer a3 is even. Reducing (R4) mod 4 we obtain 32a3g2

I

34b 2a3 2 + 3 8c2a3

(mod 4)

.

Since c2 = 0 (mod 2 ) and g2 E 1 (mod 2) we conclude that a3 2 (mod 4) thusr a3 = 0 (mod 4). Reducing (R5) mod 16 we obtain that

+

0

3

36(36-1)g2

contradicting g 2 Remark then

5.11.

+

2 2 3 8 a3f2

3

8g2

(mod 2).

1

Lemmas

5.3

and

5.7

show that, if

(q,n) = ( l r l ) r (2,2), ( 2 , 3 ) Case

2.

n

(mod 16)

or

(4,4)

q

5 n < 2q,

.

2q.

Write n = q + r r r _> q and consider a CW-complex Q satisfying the hypothesis of (2.9). The elements x, x ~ ,y r xy, y2 E KU(Q) have filtration 2qr 4q, 2 ( q + r ) , 2(2q+r), 4(q+r) respectively. For the J, - operations we have: k 2 J, (x) = kqx + aky + bkx2 + ckxy + dky J,k (y) =

+ krfkxy + gky2

kq+ry

and hence', m k + (kqam + mq+'a k)y J, J, (x) = mqkqx + (kqbm + mZqbk)x2 (kqcm+ 2mqa b +mrfmak+m2q+rc )xy + A ' y 2 m k k J,mJ,k(y)= m

q+r q+r k y

+

(mrkq+rfm

(kq+rgm + mq+rkramfk

+ m2q+rkrfk

)xy +

+

m

+

Torsion free

78

H - spaces of rank two 2 x , y and 2 3

Equating the coefficients of and those of

xy

and

(R' 1)

2'(2'-1)b3

(R'3)

2q(2q+'-1)c3

y2 =

2q+r (29+'-1)

(R'5)

3q(3q-1)b2

;

+

+ 2'f2a3

2q+1,2b3

g3

+

+

2

+

(mod 2) ,

a2 = 0

lations

3qa3b2 + 3rf3a2

Consider

;

=

2r3q+ra3f2

;

(mod 2 )

,

= 1

(mod 2 )

.

5.12. If q = 1, r 1. 1 a n d r $ 1,3 (R'l) t o (R'6) a r e s a t i s f i e d . Proof.

2r

=

b2 = 1 g2

Lemma

-

29+r 3r a2f3

3q+r ( 3q+r-1)g 2 (R'6)

J, J, ( y ) =

in

3q ( 3q+r-1 c2

2 3 xy in Q Q ( X ) = $ J ~ $ ~ ( X ) 3 2 J, Q (y), we obtain:

(R'5) for

q = 1

,

not a l l the re-

and reduce modulo

:

0

Since

g2

I

1

31+r(31+r-l)g2

( m o d 2r)

.

(mod 2) , it follows that ' 2

divides

3l+r - 1 ;

using Lemma 5 . 2 , (iii), we conclude that r = 1,3, contradiction. Lemma 5.13. If r 2 q 2 2 and r > 2 , n o t a l l t h e r e l a t i o n s (R'l) t o (R'6) a r e s a t i s f i e d . Proof. 0

Reducing I

(R'5) mod '2

(3q+r-l)g2

(mod 2r)

one gets

.

According to the hypothesis and (5.2) (iii), the previous congruence is possible only if (q,r) = (3,3), (3,5) or ( 4 , 4 ) . The first two cases can be excluded working with (R'l), which, for q = 3 gives (33-l)b2 = 0 (mod 8 ) , and thus, contradict(mod 2 ) . Now we investigate the case q = r = 4 . i n g b2 = 1

Proof of the Type Classification Theorem

With these values of a3

=

27a2

q

and and

r

(R'2) and

f3

=

27f2

79

(R'4) become

,

respectively. Hence a3f2 Since

a 2 -= 0

a2f3

=

(mod 2 )

it a l s o follows that

a3 = 0

Reduce

(R'3) modulo 0

9

Now consider 0

4

.

(mod 2) to obtain

2a3b2 + a2f3 = a2f3 = ajf2 (R'5) mod 8 8

26

5.14.

3

1

Lemmas

(mod 4)

.

(mod

26)

and get

= 3 ( 3 - l ) g 2 + 243 8a3f2

contradicting g 2 Remark

.

E

25g2

,

(mod 2). 5.12

(qrn) = ( 1 r 2 ) r (1,4) or Remarks 5 . 1 1 and 5 . 1 4

and

5.13

show that if

2q 5 n r

(2r4). conclude the proof of Theorem

2.9.

This Page Intentionally Left Blank

CHAPTER 3 HOMOTOPY AND STABLY COMPLEX STRUCTURE

In this chapter we discuss examples of manifolds constructed by W.A.Sutherland, showing that the property of admitting a stably complex structure is not homotopy invariant. The Question of Complex Structure. Let M be a complex manifold of complex dimension n. The admissible charts of M define on the same underlying topological space the structure of a real differentiable manifold of dimension 2n. It is well known that this manifold, which is called realification of the original complex manifold, is orientable. Conversely, given a real orientable differentiable manifold of even dimension 2n, the question arises whether it admits a complex structure compatible with the given real manifold structure. If n = 1, that is to say if we are dealing with orientable surfaces, such a complex structure always exists. The situation is different when n > - 2; for example it was shown by Hopf [125] and Ehresmann [ 8 8 ] that the sphere cannot be given a complex structure. S4 Examples of complex manifolds are the complex projective spaces 6: Pn and non-singular algebraic varieties. The latter objects are compact complex manifolds which can be analytically embedded into some complex projective space (see [66]). The notion of complex manifold is more general than the notion of algebraic manifold. In [125] Hopf exhibited a complex structure on S 1 x S 3 ; this example was generalized by Calabi 2q+l and Eckmann [61] giving a complex structure on S2‘+’ x S The study of complex manifolds has been pursued in different ways. One line’of development was concentrating on special families of manifolds, satisfying certain additional 1.

81

82

Homotopy and Stably Complex Structure

properties. In this context we mention the so called KahZer Manifolds. These are complex hermitian manifolds the metric ds2 = Cgik(z,z)dzJdzk of which is kahlerian, i.e. such that the associated exterior differential form w

=

Cg ik ( z ,z)dzJ A dzk

is closed. Iiahler manifolds have been investigated extensively (Hodge theory) and have interesting properties (see [249] and [1161). Non-singular algebraic varieties are kahlerian; but there are Kahler manifolds which are not algebraic. Furthermore, not any complex manifold admits a Kahler metric; we quote once again the examples of Calabi and Eckmann. Another approach in the study of complex structures consists in viewing a weaker situation: the almost complex structure. The following observation leads to the definition of this notion. The tangent bundle of a complex manifold M admits a complex vector bundle structure. Multiplication by defines at each point x E M a linear transformation 2, of the tangent space TMx with 1, c 1, = -Id(TMx) and such that 1, depends continously on x If on a real differentiable manifold M a continuous field 1, of linear transformations with 1,2 = -Identity is given (without reference to any complex structure), M is called an almost complex manifoZd with almost compZex structure 1 It is easy to see that an almost complex structure defines a canonical orientation on the manifold. Some necessary conditions of existence of almost complex structures on M can be stated in terms of characteristic classes. Historically, several results on 4 - dimensional manifolds have been obtained as early as 1948, by Wentsiin Wu [262]. Since then several papers had been written following the same lines of thought; in particular, we wish to bring the attention of the reader to the paper of W.S.Massey [1721. Using different methods, Kirchhoff has shown that, if the sphere S2n admits an almost complex structure, then S2n+l is parallelizable [151]. At the end of chapter 1 , we have already pointed out that Bore1 and Serre proved back in 1953 that the only spheres carrying an almost complex structure are S 2 and S6 .

.

.

Almost Complex and Stably Complex Manifolds

83

The study and construction of almost complex manifolds have revealed some interesting phenomena. For instance, the property of admitting an almost complex structure is not a homotopy invariant (cf. [125]). Also, this property depends on the orientation of the manifold, in a sense explained by the following example. The complex structure on cP2 induces a canonical orientation on this manifold. It can be proved that there is no almost complex structure on CP2 which induces the opposite orientation (see [145]). Almost Complex Manifolds and Stably Complex Manifolds. Let M be a 2n-dimensional orientable manifold and let T ( M ) be its tangent bundle. After reduction, we may suppose that the structure group of the 2n-dimensional real vector bundle T (M) is SO(2n) ; let us denote by T' (M) the associated bundle with fibre rn = S0(2n)/U(n). 2.

Definition

2.1.

The m a n i f o l d

complex s t r u c t u r e i f t h e

rn-

M

i s s a i d t o a d m i t an a l m o s t

bundle

T'(M)

has a c r o s s - s e c -

tion.

Hence, the existence of an almost complex structure on M amounts to the possibility of reducing the structure group SO(2n) to U(n). It is well known that such a reduction requires, among other conditions, the nullity of each odddimensional Stiefel-Whitney class [231; cor. 4.1.91. Note that the preceding situation may be viewed as a particular case within the more general context of X - structures (see [176; p.181). Before we formaly establish the notion of stably complex structure on a manifold, let us describe a particularly interesting(*) standard construction of an almost complex manifold. Let x be a closed differentiable manifold (i.e., compact and without boundary) of dimension n. Then, t h e t o t a l s p a c e M = T*X of t h e c o t a n g e n t b u n d t e T * ( X ) = (T*X,n,X) h a s a n a t u r a l s t r u c t u r e o f a l m o s t c o m p l e x m a n i f o l d . In order to prove this statement we have to determine clearly the na-

(*I

For its usefulness, see section

4

of chapter

8.

84

Homotopy and Stably Complex Structure

ture of the tangent bundle T*(M) = T(T*X) , whose base space is, of course, a 2n-manifold. First of all, let us observe that €or every (differentiable) real vector bundle 5 = (E,p,X), the differential dp : TE > TX induces a bundle epimorphism (dp), :

>-

T(E)

p*T(X)

whose kernel is the bundle of vectors tangent to the fibres of 5 . We also observe that ker(dp), is canonically isomorphic to p*S since, for every x E X, the space tangent to 5, at any point of that fibre is isomorphic to 5,. It follows that the seauence (dP)* (2.2) 0 -> P*S -> T ( E ) > p*T(X) -> o is exact. Taking

T*(X)

for

5 , we have .

is exact. NOW, since the manifold X is riemannian, T(X) and T* (X) are isomorphic and moreover, the last sequence splits. Hence, n*'r*(X) @ IT*T(X) T(M) = T(T*X) T*(T(X) @ T(X))

.

But T(X) 8 T(X) is just the real vector bundle subjacent to the complexification T (X) = T(X) 2' 3 C of r(X). This shows 6: inherits the complex structure defined (**I by that T ( M ) d. = ~ * ( T ~ ( X ) )In . other words, for any v E M = T*X, let us write the elements u E TV(M) in the form u = (arb)E n*(TV(X) @ TV(X)) Then, the automorphism

.

.

= (-bra) Finally, let us give is given by ],(arb) clearly the orientation of the manifold M. Let x,,...,~n be the local coordinates of X; then, every v E TC(X) (where n x E X) is of the form vkdxk : the 2n-tuple k= 1 (**) This ci-structure depends on the riemannian structure of X

Almost Complex and Stably Complex Manifolds

85

(X1rvlr**-rxnrvn) defines the orientation of M = T*X induced In order to avoid confusion, we shall call this orienby a tation the % - o r i e n t a t i o n of M. If X is itself oriented (in the preceding construction we did not make any requirements about the orientability or not of X), this orientation induces a natural orientation on T*X. With the previous notations, one can see that the natural orientation corresponds to that given by the 2n-tuple (xl,...,xn,vl,...,vn). Hence, the %-orientation of M differs from the natural orientation by the signature of the permutation (x,,...,xn,vl,...,vn) +

.

(x~,v~,...,x~,v~ that ) , is to say, by a factor

(-1)(n-I)n/2

The notion of almost stably complex manifold consists in replacing the space rn (see Definition 2.1) by the homogeneous space r = SO/U. More precisely, let 8 ( M ) be the r - bundle associated to T ( M ) : then, we give the following: M admits a stabZy c o m p Z e x structure if r Definition 2.3. bundZe e(M) has a cross-section. For the manifold M , to be stably complex means that any stabilization of T ( M ) with even-dimensional fibre may be endowed with a complex vector bundle structure. This is possible if and only if the element [ T (M)3 - dim M of ? O ( M ) belongs to the image of the realification homomorphism N

KU(M) + *iO(M). (If 5 is a stably complex structure on a connected and closed manifold M , we can show that it reduces to an almost complex structure if and only if its nth Chern 1: :

cn(c) is the Euler characteristic of M : a proof, K-theory, is given in [145; p . 3 4 7 1 . ) The 71 -manifolds provide an interesting family of stably complex manifolds. By definition, the Whitney sum of the trivial line bundle and the tangent bundle of such a manifold is trivial. This property enables us to associate, in a stand-

class using

ard way, a stably complex manifold to any orientable k-manifold X Let f be a differentiable embedding of X into Rd (this is always possible when d is large enough) and let us write v = ( E , p , X ) for the normal bundle of this embedding. We have the following split exact sequence of vector bundles (see chapter 6)

.

Homotopy and Stably Complex Structure

86

where

sd(X)

is the trivial

d - dimensional bundle over

X.

Let S(v) be the (d-k-I)-sphere bundle associated to v We shall denote its total space by M and its projection by ?T , i.e.

.

S ( v ) = (M,v,X)

.

Since the submanifold M of E normal vectors in E , we get

admits a field of non-trivial

Restricting the vector bundles of 0 -

>

p*v

(see 2.2) to M bundles over M 0

-

>

,

?T*v

->

T(E)

->

p*T(X)

---->

0

we obtain the following exact sequence of

->

?T*T(X) ->

->

T ( E ) IM

The previous remarks imply that

0.

'

in other words, the total space M of S(v) is a (d-I) - dimensional IT -manifold. The procedure just described was used by Sutherland to build his examples of manifolds mentioned in the introduction. More precisely, in the following let M be t h e r - m a n i 2 f o l d o b t a i n e d from a n emb ed din g of the manifold X = S1 x @P Rn+l I , where we suppose that n > 18. Here, @P2 denotes into the CayZey p r o j e c t i v e p l a n e with its canonical C W - structure, obtained by attaching a 16 - cell to S8 via the Hopf map a : sI5 + s8, i.e. @P 2 = S 8 u e 'I 6 (I

.

Let g : X = S1 x @P2 + SI7 be the canonical projection map collapsing the 16- skeleton of X , and let c1 be a n o n - t r i v i a l vector bundle over S l 7 of dimension n + 17, where the integer n is the same as before. Since n > dim X there is an n-dimensional differentiable vector bundle

The Homotopy Type of

6

over

= (%,&X)

5 Let

%

M

M

X I such that

0T (X) =

.

g*a

be the totaZ s p a c e of t h e s p h e r e bundZe S(5)

=

(Fi,Y,X)

I

associated to 5 . As for the bundle lowing exact sequence over Z : >

N

N

T*


Since again

(%)

@E

(2.41

@E

0-

and

T(Z)

2

T(E)

*;

2 T

N

->

IM

(E) ;1

there is the fol-

S(v)

Y*T(X)

->

0 .

, we obtain

(5 @ T ( X ) )

'2 ?*g*Cr

.

Thus, the bundles T ( z ) and ;*g*a are stably equivalent. With this stable equivalence we will prove in 9 4 that does not admit any stably complex structure. N

3.

.

The Homotopy Type of M and M To compare the homotopy types of M and % we will work with the group KH(Y) of stable fibre homotopy classes of sphere bundles over the space Y. This is no surprise for the reader, since both M and M are defined as total spaces of certain sphere bundles. Let us briefly recall the definition of KH. The semigroup H(n) of all homotopy equivalences of S"-', endowed with the compact - open topology, is an associative H - space; let BH(n) be its classifying space. According to Dold-Lashoff [ 8 0 ] , the set of fibre homotopy classes of Sn-'fibrations over a finite CW-complex Y is in one-to-one correspondence with the set [Y,BH(n)]. Under suspension we have a natural inclusion H(n) + H(n+l) which gives rise to a map BH(n) + BH(n+l). We put BH = Lfm BH(n); this space admits an H-space structure. For a finite C W - complex Y we define N

N

N

N

KH(Y) = [Y,BH]

.

If n > 1 + dim Y , the canonical map BH(n) + BH induces a bi-jection [Y,BH(n)] 2 [Y,BH]. (For further details, see D o l d Lashoff [ 8 0 ] and Stasheff [ 2 2 7 1 . ) Restriction of orthogonal maps of iRn to s"-' provides an inclusion O(n) + H(n) , which induces a map BO(n) + BH(n). These maps give rise to an H - m a p J : BO + BH. The

88

Homotopy and Stably Complex Structure N

N

image of the induced homomorphism J : KO(Y) = [Y,BOI + EH(Y) [Y,BHI may be identified with the group ?(Y) defined in (0.14) (see Atiyah [12]). If Y is the r - sphere we have

According to [12], the latter group isomorphic to IT^-^+^ (sn-') ; hence

is for

2

=

2 r5 n - 2

-

(Tr-1 is the so called stable (r-1) stem). In the following we will identify ?H(Sr) and IT^-^ The homomorphism 5 : FO(Sr) + FH(Sr) may now be interpreted as a map

.

N

It can be proved that J coincides with the classical stable J-homomorphism of G.W.Whitehead (see [134, 15.61). After this brief sketch on the relations among the functors FO, FH and 3 , we provide the following lemma which is the key to all our investigation concerning the manifolds M and ii

.

Lemma

3.1.

C

Let

the H o p f map

ci

.

2

ci

: SI7 +

S1'

The non-trivial element a € KO(S

satisfies :

(i)

be t h e double-suspension of

(c

a 6 im (EO(S'O)

2 0 )!

>

(C2a)* >

(ii) J(a) € im (FH(Sl0)

17) Z J ( S 1 7 ) g'e2

.i('o(sl7))

?H(S17))

.

Proof. Let q E IT; be the non-trivial element. ACcording to 1242, prop. 3.11, the 8 - fold suspension of the diagram 2 C 0 > slo

s1

4

In

V

V

S1

Cci

>

s9

N

The Homotopy Type of

and

M

8 9.

M

commutes (up to sign), thus giving rise to a commutative dia9 ) z ? O ( S 10) gram in KO- theory. Since r ) '' : KO(S (see [4,Iv, (* 1 1.2 I ) and because (Xu) is obviously zero, we conclude that (C2u)! is trivial and (i) is established. s16 To prove (ii) we first note that the map S17 induces an epimorphism in KO-theory (see [4,IV,1.2 I) and therefore also an epimorphism for the J - groups; hence ?H(S 17) ) . The only non-trivial eleJ(S 17) C, im (?H(S 16 ) ->rl* ment in the latter sub-group of FH(S 17) is r) o p , as one N

1

N

N

-

may check, consulting Toda's tables [ 2 4 2 , p.189 and Th.14.11 (w are now following Toda's notation). One concludes that J(a)

phic to invoking

= rl

0

p

.

According to [242], the group FH(Sl0) @ Z 2 @, Z 2 generated by v 3 , p and [ 2 4 2 , Th. 14.11, we obtain

Z2

(C

2

a)* ( p ) = p

o

is isomorE Again

.

ou = crop = rlop

and (ii) is proved. (In the latter relation we use the same symbol €or the map CT : SI5 + S8 and the class represented by S this map in n7.) Corollary

S17 be the canonicaZ p r o jection. F o r the non-trivial element a € .i(O(S17) one has (i)

g!(a)

(ii)

duct

S

3.2.

1

g*J(a))

g : S1

Let

$

Q

= 0

in

FO(S'

x

@p2)

in

'iTH(S1

x

@P

Proof. The map 2 A (DP ,

g : S 1 x @ P2

x @P2 +

->

g

2

)

;

.

factors through the smash pro-

u C U e 17 q > 517

.

It is well known that the projection p onto the smash induces a monomorphism in any cohomology theory. Thus, the corollary will be proved if we can show that q!(a) These -

9

0 and

q*(J(a)) = 0

.

two relations follow readily from Lemma 3.1 and the induces an epimorphism of (*)The composition SIO 2 S 9 -? S8 KO-groups. N

Homotopy and Stably Complex Structure

90

N

N

exact sequences in KO- and KH-theory associated to the relevant part of the Puppe sequence of u , namely to

s 9 u c 0 e1 7

q > s1 7

CLu

s 10

>

We are now ready to prove the main result of this section. N

Theorem 3.3.

The m a n i f o l d s

M

and

are homotopically equi-

M

valent.

Proof. According to 9 2 , M is the total space of the sphere bundle associated to the normal bundle v of an Rn+17 , n > 1 8 , whereas arises as toembedding X -> tal space of a sphere bundle S ( t ) , with ,€ satisfying

5 Here, a

@ T ( X )

z g*a , dim €, = dim v

=

n

.

is a (non-trivial) vector bundle over

S1 7

.

Part (ii) of corollary 3.2 implies that €, f+' ?(I dim x + 1 the bundles S(u) and S ( < ) are fibre homotopy equivalent and the theorem is proved. The manifold k is not stably complex. In this last section of chapter 3 we show that the manifold does not admit a stably complex structure. By (2.4) the tangent bundle ? (i) is stably equivalent to n*g*u and it would therefore be sufficient to prove that the latter bundle is not stably complex. With this in mind we first establish the following result. 4.

N

Proposition

4.1.

Let

g : X

= S1 x

@P2

+

S17

c a l p r o j e c t i o n . Then, f o r any v e c t o r b u n d l e w h i c h is s t a b l y n o n - t r i v i a l ,

t h e bundle

g*a

a

be t h e c a n o n i 17 over S does n o t admit a

s t a b l y complex s t r u c t u r e . 17 Proof. Our hypothesis on a implies that in KO(S 1 we have [ a ] = a + dim a , with a $ 0 Accord ng to the redoes not bem a r k s following ( 2 . 8 ) we must show that g ! (a long to .

.

~(Ku(x)

Is

a Stably Complex Manifold?

91

The

KU-theory of X is torsion free and r : XU(X) + KO(X) is injective. (The projection X + @P2 in2 duces an isomorphism KU(0P ) KU(X) ; for the two-cell com8 plex OP2 = S U el6 it is easy to see. that the realification is injective). According to (3.2) (i), the element g! (a) is nonzero of order 2 and hence g!(a) B r(?U(X)) , proving the proposition. N

N

N

N

Theorem

4.2.

The m a n i f o l d

M

d o e s not a d m i t a s t a b l y c o m p l e x

structure. N

Proof.

The

(n-1) -sphere bundle

S([)

=

N

(M,n,X)

N

has a cross-section s : X dim X < n - 1. In view of

+

M , since by hypothesis we obtain that

(2.4)

By Proposition 4.1, the bundle g*a does not admit a stably complex structure and we conclude with ( 4 . 3 ) that the same (i) holds for

.

This Page Intentionally Left Blank

CHAPTER 4 VECTOR FIELDS ON SPHERES

Introduction Let sn-l be the unit sphere in the euclidean n space IR". A I n o n - z e r o ) v e c t o r f i e l d on Sn-l is a continuous function v : sn-1 > IRn - lo1 1.

which associates to each x E Sn-l a vector v(x) orthogonal to x , that is to say, a vector "tangent" to Sn-l at x. For example, if n is even, say n = 2m, then

. -

v(x) = (-x2,x1,-X4,X3,.. , X2m,x2m-1 ) defines a non-zero vector field on S2m-1 It was PoincarC who, in one of his celebrated papers inspired by what he called "le problsme astronomique de la stabilitg du syst6me solaire" [ 1 9 7 ] , first observed that the sphere S 2 does not have a vector field. Later on, Brouwer proved that the spheres S2n , n 2 1 , do not have a vector field [ 5 5 ] . Actually, the non-existence of vector fields on even dimensional spheres is a very simple question from nowadays standpoint. In fact, suppose that a Sn- 1 sphere possesses a vector field v and let w be the normalized field given by w(x) for every

=

.

-v(x) ,

Ilv(x) II

x E Sn-l. Then,

h(t,x)

=

(cos nt)x + (sin nt)w(x)

,

O Sn-l

~

t

~

l

defines a homotopy between the identity of and the antipodal map. Since the degree of the former map is 1 and that of the latter is (-l)n, one concludes that n must be even. The natural question now is to determine which spheres 93

,

94

Vector Fields on Spheres

admit several vector fields that arc linearly independent at each point. More generally, we shall study the following problem. Let IF be one of the classical division algebras over IR, that is, IF = IR, C or M , and let S IFn be the unit sphere of Fn with respect to the standard inner product ( 1 1. A non-zero F - v e c t o r f i e l d on s IF" is a map v : SIF"

-> IFn -

(0)

v ( x ) with (xlv(x)) = 0. We say that r such fields v (1),...,v (r) are Z i n e n r Z y i n d e p e n d e n t when, for every x E S IFn , the vectors

which assigns to each

x E

V(l) (XI I

S

. .

F"

a vector

. Jr)

(XI

are linearly independent o v e r IF. The "vector field problem" (for the spheres) then reads as follows: find t h e maximal numb e r o f Z i n e a r Z y i n d e p e n d e n t IF - v e c t o r f i e l d s o n t h e s p h e r e s IF". There are two reasonable ways of attacking this problem. On the one hand, one might try to exhibit as many vector fields as possible on a given sphere. This, of course, should be done in the most simple way and is the sort of problem one tries to solve by giving explicit constructions using linear algebra methods. On the other hand, one might try to show that a given sphere does not admit a certain number of linearly independent vector fields. This is the kind of problem one usually approaches with Algebraic Topology methods. As it turned out, the vector field problem for the reals was solved precisely along the previous lines. The strongest positive result derives from the works of Hurwitz [ 1 3 3 1 , Radon 1 2 0 0 1 and Eckmann [ 8 5 ] in Linear Algebra and Representation Theory of Finite Groups (the first two papers were already published in 1923). It may be stated as follows. Write n = 2 4d+c (2a+1), where a,d,c are integers with 0 5 c 5 3 and let p(n) be the integer defined by p(n)

= 2'

+

8d

;

then, the sphere Sn-' admits p(n) - 1 linearly independent vector fields. The negative part of the vector field problem

Introduction

95

over lR was solved by Adams who, inventing powerful methods in Topologica'l K - theory, was able to prove that the number of vector fields given by Hurwitz and Radon cannot be improved. This result is contained in Adams' famous paper "Vector Fields on Spheres" 1 2 1 , published about 40 years after the papers of Hurwitz and Radon. The situation is quite different in the complex and quaternionic cases. Indeed, no explicit construction giving two or more linearly independent vector fields on SCn is known, up to the present moment. As for the quaternions, one does not even have a single non-zero vector field on SIHn which is given by a formula "qui peut Qtre montree en public". Nevertheless, it is a remarkable fact that the vector field problem €or C and M could be "solved" by the following result. The sphere SCn (resp. SMn) admits k 1 1 linearly independent complex (resp. quaternionic) vector fields if, and only if, n is a multiple of an explicitly given integer bk (resp. c,) , called the kth complex (resp. quaternionic) J a m e s Number (see Theorems 5.9 and 6.6)!*)It should be noted that these numbers can be explicitly computed. We now enlarge upon the historical remarks already made, to give a short history of the complete solution of the Vector Field Problem. The first important result after those obtained by Poincarg, Brouwer, Hurwitz and Radon was the f o l lowing Theorem of Alexandroff and Hopf: "A differentiable manifold M has a (continuous) vector field if, and only if, its Euler characteristic is zero" [ l o ] . As a trivial consequence, we note that all odd dimensional differentiable manifolds have a vector field. Actually, Alexandroff-Hopf's Theorem is contained in the results of Stiefel [ 2 3 3 1 , who had addressed himself to the more general question of finding the maximal number of linearly independent vector fields on a differentiable manifold: it is on that work that Stiefel introduced his widely known class of differentiable manifolds, called the Stiefel manifoIds. The next key result was obtained by Eckmann [ 8 3 1 (and independently by G-Whitehead 125011, who proved that on Our indexing of these numbers differs by a shift of 1 from that one adopted in [ 1 3 9 1 , [ 1 2 1 and [ 8 1

.

Vector Fields on Spheres

96

a sphere S4n+' there is no pair of linearly independent vector fields. Theimportance of this paper lies not only in the fact that it exhibited a class of spheres for which the number of linearly independent vector fields would not go over p(n) - 1 (actually, it should be noted that Eckmann had shown before that S 5 cannot have two linearly independent vector fields [821), but also, because the powerful ideas and techniques of homotopy groups were used for the first time. In this same order of ideas, the next important theorem we should quote is that announced by Steenrod and J.H.C.Whitehead in 1951 [232] : "Let n be an integer and let 2k be the highest power of 2 dividing n + 1; then Sn does not have 2k linearly independent vector fields." Their proof relies on cohomology properties of the real projective spaces (Steenrod Operations). Around that time there was also a lot of work done on the question of parallelizability of the spheres; the reader is invited to consult Chapter l for the relevant results. The next generation of results came about with the papers of Ioan James on Stiefel manifolds [138], [I391 and [140]. In them James proved that for each positive integer k , the sphere SIF", with IF = C or M , admits k linearly independent vector fields if, and only if, n is a multiple of a certain number bk (if IF = C ) or ck (if IF = M ) I without being able to determine explicitly these numbers. Making use of his notions of S - reducibility and S - coreduclbility, he then reduced the vector field problem to a problem concerning truncated projective spaces. Later on, Atiyah proved (see [ 121 ) that the numbers bk and ck are the J - orders of the canonical line bundle over the projective space F P k . Next, Atiyah and Todd showed [37], with the aid of complex K - theory, that if SCn ha5 k linearly independent vector fields, then n must be a multiple of an explicitly determined number mk This condition was shown to be sufficient by Adams and Walker in 1965 [ a ] , who proved that bk = mk. In the meantime, as we observed before, Adams had solved the vector field problem for SIR" using real and complex K - theory, together with James' result on S coreducibility of truncated projective spaces.

.

-

Vector Fields and Sphere Bundles over lFPm

97

Finally, in 1973, Sigrist and Suter published a paper in which they solved the vector field problem for the quaternionic sphere S M n [ 2191- The method used was similar to that of [ 81 , but it required the use of the "Adams Conjecture", then already proved by Quillen. The Chapter is organized as follows. In 5 2 we first formulate the vector field problem in terms of crosssections of Stiefel fibrations. We then relate the cross-section problem to a question concerning the J - order of the canonical Hopf bundle over an appropriate projective space. In 5 3 we compute the KO and KU - theory of the projective spaces. The vector field problem for the reals is treated in section 4. We follow the method of [49] and also sketch a proof of the Hurwitz Radon - Eckmann Theorem. In 9 S we solve the cross-section problem for complex Stiefel fibrations [160] and working primarily with following and idea of K.Lam the functor J' of [4]. Section 6 is devoted to the quaternionic case. We close the chapter with a comparison between the complex and quaternionic cases.

-

Vector Fields and Sphere Bundles over Projective Spaces. In this section we first interpret vector fields on spheres as cross-sections of the so called Stiefel Fibrings. Following Woodward [261] and James [141], [143] we then relate such cross-sections to homotopy properties of certain sphere bundles over a projective space lFPk-l , reducing the vector field problem to the computation of the J - order of the canonical line bundle over IFPk- 1 2.

.

Let IF be one of the classical real division algebras IR,C or M , with the usual norm and conjugation operation. Let lFn be the euclidean n - space corresponding to IF. We regard IFn as a right IF - module and consider the standard inner product ( I ) so that if x = (x,, Xn) are elements of IF", then = ~ 1 + ~ 1 + xnyn r IIxII = (X x1 + ... + nxn ) 1/2 (XI~) (2.1)

...

x

...

An ordered set

{

x (1),..., x k))

of

k

vectors in

.

Vector Fields on Spheres

98

IFn is called an o r t h o n o r m a l k - f r a m e if (x(i)Ix(1)) = 6ij (i,j=l,...,k). Such a frame can be identified to the n x k matrix having the orthonormal columns x ( 1 ) ,...,x (k), moreover, with the linear norm preserving map IFk + IFn given by (ylr ryk) -B x(')yl + ... + x (k)yk. The set of all orthonorma1 k - frames in IFn,

...

pn,k

=

{{x ( 1 ) r . - * r X (k) Ix(~)€ IFn, ( x ( ~ lx(1) ) )

=

bij

, i,j = 1 , .. .,k,1

...

x IFn (n 1 k) , corresponds to a subset of the space IF" x (k factors) and hence, it inherits a canonical topology. Actually, IFnrk is a smooth (real) manifold, the so called S t i e f e l M a n i f o l d . We shall frequently identify IF with the n,k to IF". space of all IF - linear norm preserving maps from The space IF,, coincides with the sphere SIF" formed by the vectors of norm 1 in IFn ( SIF" is a sphere of dimension dn - 1, where d = dimm@')). The projection map

IF^

p :IF

nrk

->

SIF"

= x(l), is a fibration with defined by p({x(l)r...rx(k)~) fibre IFn-lrk-l (see 231; § 7 . 8 ) . Clearly, a cross-section of p gives rise to k - 1 orthonormal IF - vector fields on the sphere SIF" : introducing some notation, if s is a crosssection of p and x € SF", s(x) is an orthonormal k - frame 1 k- 1 (x)1 and the vectors s1 (x),.. . , s k- 1 (x) are {x,s (x),.. . , S tangent to SIF" and perpendicular to each other. Conversely, suppose that we are given k - 1 linearly independent IF-vector fields on SF". Then the Gram-Schmidt orthogonalization process provides us with k - 1 orthonormal vector fields 1 k- 1 v (x),.. ,v (x) (the process is continuous) and one gets a cross-section of p defined by s(x) = {x,v'(x) ,...,vk- 1 (x)1 . Hence : (2.2) The s p h e r e SIF" a d m i t s k - 1 Z i n e a r Z y i n d e p e n d e n t IF-

.

v e c t o r f i e l d s if, and o n l y i f , t h e S t i e f e l f i b r a t i o n n p : IFnrk +SIF has a c r o s s - s e c t i o n .

The vector field problem formulated in the introduction is therefore equivalent to the following problem: f o r what n pairs ( n , k ) d o e s t h e f i b r a t i o n IFnrk -+ SIF a d m i t a c ~ o s s -

Vector Fields an Sphere Bundles over IFPm

99

section?

The sphere SIF is a Lie subgroup of the multiplicative group IF* =IF - { O ) (that is, S I R = S0 , SC = S1 and S M = S 3 ). We consider the canonical right action of SIF on the sphere SF^ ; factoring out this action, one obtains the projective space IFP"-' associated to IFm Let < be the canonical IF(2.3)

.

line bundle associated to the principal SF-bundle S F m of this line bundle is given by The total space E F,

(2.4)

E

5

= ( S I F ~x

IF)/(v,Y)

-

(VZ,YZ)

(z E SIF)

-+

IFPm-'.

*

It follows that S(nE), the sphere bundle associated to the vector bundle nc, is obtained by factoring out the diagonal action of SIF in S I F ~x SIF", i.e., s(n (SFk x SIF") / (v,x) -

-

-

vz

= IFP

(VZfXZ)

k- 1

-

that is to sayf o is a map between sphere bundles over IFPk-l, o : S(nE) S(n6) (here E is the trivial IF - line bundle) On the other hand, (ii) shows that u is the identity on the fibre over [e") 1 € FPk-' and so, it follows from a wellknown theorem of Dold [ 7 7 1 that u is a fibre homotopy equivalence. At this point we introduce a notation which will be used in the remainder of the Chapter: if a is a real vector bundle over X , we denote by J(a) the element of J ( X ) represented by the stable class of a , that is to sayf J ( a ) = J ( x ) , where x = [ a ] - dim a E KO(X). With this and the preceeding observations, we can state the following -

.

+

N

N

N

N

N

Fn,k -rSIFn h a s a c p o s s - s e e t i o n , t h e n t h e s p h e r e b u n d l e S ( n < ) o v e r IFP k-' i s f i b r e homotopically t r i v i a l . In particuZar, n J ( [ ) = 0 i n t h e group J ( F P k - ' ) , t h a t i s t o s a y , n i s a m u l t i p Z e o f t h e J - o r d e r of Theorem

2.5.

If the fibration

-

-

N

N

5 The next Theorem is a partial converse of will be proved in (2.12).

(2.5) and

Theorem 2.6. S u p p o s e t h a t n > 2k. If t h e s p h e r e b u n d l e S(nS) o v e r E'Pk-' h a s s t a b l y t r i v i a l f i b r e h om o t opy t y p e , t h a t i s t o s a y , i f n is a m u Z t i p Z e o f t h e o r d e r o f J ( < )E 7 ( F P k - ' ) , then the fibration +SFn has a c r o s s -

N

section.

(2.7) In an attempt to prove Theorem 2.6., the considerations preceeding ( 2 . 5 ) . Let u topy equivalence between S(nc) and S(n6). elementary principles in bundle theory that to a map u over SIFk that is, to a map

we try t o reverse be a fibre homoIt follows from u can be lifted

Vector Fields and Sphere Bundles over IFPm

a : SIFk x SIF"

I

(vrx)

~

>

SIFk XSIF"

>

(vral (v,x)1 r

101

which is compatible with the appropriate S I F - action, i.e., such that, for every v E S F k , x E SIF" and z E SIF, cs(v-z,x)= u(v,x) z . Now we focus on the map

.

a1

:

slFkxSIFn >

SIF"

obtained by composing cs with the projection onto the second factor. The map k -> SIF" al( ,XI : SIF

C---> al(vrx)

v

in general, will not be the restriction of a linear, norm preserving map from IFk into IF", but is compatible with the S P action on the two spheres, i.e., u (v*z,x) = u l (v,x) z . The 1 adjoint of al provides, therefore, a continuous map

.

defined by s ( x ) = a l ( ,x) , for every x E SF", where X (IF) n,k is the s p a c e o f a l l SIF - e q u i v a r i a n t maps f r o m SIFk i n t o SF", endowed with the compact-open topology. We choose e ( 1 ) = (l,O,...,O) as base point of SIFk . Evaluation at e (1) defines a continuous map

we shall prove in (2.11) that Observe that the map 01

(e(1) I

)

:

e

is a Hurewicz fibration.

sp" -> sIFn

has degree one, since we may presuppose that, the fibre homotopy equivalence a restricted to fibres is of degree one. Hence, the composition e c s is homotopic to the identity map 1 , and because e is a Hurewicz fibration ( a s we SIF" shall see), s can be deformed to a cross-section of e. SO far, we have proved the following result. Proposition 2.8. If t h e s p h e r e b u n d l e S(n6) o u e r IFP k- 1 i s f i b r e hornotopioaZly

t r i v i a l , the projection

e

:X

n,k

(IF)+

SIF"

Vector Fields on Spheres

102

admits a cross-section.

(2.9) The Stiefel manifold IF may be identified with a n,k subspace of X (F), namely, with the subspace of those SIFnfk equivariant maps that are given by linear norm preserving maps from IFk into IF". We have a commutative diagram

SF"

The following Lemma implies that a cross-section of e can be deformed into a cross-section of p, provided n is sufficiently large when compared to k (see [lo21 and [2421). Lemma

2.11.

Let

d

be t h e d i m e n s i o n o f

IF

as a real vector

space. Then, t h e h o m o m o r p h i s m : nj(FnIk) ->

induced by the incZusion p h i s m if

ik

n.(x I n,k O F ) ) :

+

xnlk@)

is an i s o m o r -

j < 2d(n-k+1)-3.

Proof. Choose the standard inclusion fk : SIFk +SF", which takes any k-tuple (x,,...,x,) into (xll...,xkl O,...,O) as the base-point of Fn,k and X (F). nfk We prove the Lemma by induction on k. If k = 1, we have IF = X (F) = SF", and the result is true. Let then n,1 n, 1 k > 1 and assume by induction that (ik-l)* is an isomorphism for j < 2d(n-k+2)-3. Let us identify SIFk-' with the k subspace { x = (X,,...,X~-~,~)I l l x l l = 1 1 of SIF Restriction of linear and SIF - equivariant maps SIFk + SIF" to SIFk-' defines maps

.

q :lF

>

n,k

and

-

q : Xn,k(F) > -

IFn,k-l 'n,k-l

(F)

I

respectively. Notice that q ik = i c q. It is well known k- 1 that q is the projection of a fibre bundle with fibre homeo=n- k+1 morphic to (see [231; 9 7.81). Furthermore, we have

Vector Fields and Sphere Bundles over F P m

103

the following two facts: -

the map q is a Hurewicz fibration with fibres homotopically equivalent to the iterated loop space )SF"; (A)

(B) let

j,

=

-1 iklq (fk-l) be the inclusion of the

fibre SIFn-k+l into the fibre

-

Y

n,k Then the induced homomorphism n-k+l (jk)* : n . l S I F ) I is the

d(k-I)

-

-1

)

'j+d(k-I) CSIF")

n.(Y

-->

I

Qd(k-l)SIFn

(fk-l

=

nrk

fold suspension.

Statements ( A ) and ( B ) will be proved shortly. Consider now the following homomorphism of exact homotopy sequences: + n j+l @n,k-l )-->

n-k+l

n 1. (SIF

I->

q*

n I. aFnrk)--> n j aFn,k-l)-'

...

Assertion (B) and the Freudenthal Suspension Theorem show that the above suspension is an isomorphism for j < 2d(n-k+1)-3; it follows inductively that (ik)* is an isomorphism in the relevant range. For the sake of completeness we give next the proofs of ( A ) and (B). Proof of ( A ) . Consider the subspace Ek- 1

=

{(x,

....

Xk- 1 (l-ilxl12)1'2~ I

Ii=

1

of S F k . This subspace is homeomorphic to a d(k-1) - ball and has the canonically embedded sphere SFk-' as its boundary. The following two facts are readily checked: ( A 1 ) - any SIF - equivariant map defined on SIFk is uniquely determined by its restriction to the subspace Ek"

-

(A2) any continuous map

l ' k E

+

SIF"

.

which is SIF -

104

Vector Fields on Spheres

can be extended equivariant on the boundary aEk-l = S Fk - l continuously (and uniquely) to an SIF- equivariant map s F k +SF”. -> Ek- 1 is a cofiObserve that SIFk-’ = aEk” bration and that both spaces are compact; hence, by 1225; Theorem 2.6.21, the map

9

:

Map(Ek-’,SIFn)

->

Map(SFk-’ SF”)

defined by @(f : Ek-’ + S F ” ) = f l S Fk-’ is a Hurewicz fibration. By Dold’s Theorem, the fibres are of the homotopy type of 4 - l (fo) = Qd(k-l)SIFn , where fo E Map(SFk-l , SF”) is the constant map onto (O,...,O,l) E SFn. N o w form the commutative diagram X

nrk

Map(Ek-’ , SF”)

(F)

where j is defined by restriction on Ek-’ ; as the reader can easily check, it turns out that this is a pull-back and hence, claim ( A ) is verified. Remark.

The statement that

is a Hurewicz fibration, made earlier, follows trivially from (A).

Proof of

(B).

Direct inspection shows that the in-

clu s ion

is the adjoint of the map n-k+ 1 g :BIFk-l x S F given by

g ( x , y ) = fx, I

...,(l-llxll here

2 1/2

. . .

>

SP”

,xk~l,(l-llxl12~”2-yl,..

-Y,-~+~),for every

x E B P k-l

and y E S F n-k+l. I

Vector Fields and Sphere Bundles over lFPm BIFk-’

is the closed unit ball of IFk-’. Write

105

(x~~...~x~-~

..

. , u ~ - 1~ =+ (x,u) ~ E SIF” and consider the following homotopy of the identity map 1 SZ?”

U, r

-1

The map h, Qd(k-l)SIFn

N

gives rise to a map from Y into @ (fo) = n,k which in turn, induces an isomorphism between the corresponding homotopy groups. The reader can now check that the composition

is the adjoint to the map B F ~ - ’ / s F ~ - ’x SF n-k+l

__ >

which sends an arbitrary element into [

SFn

x SF n-k+l (x,y) E BF~-’/sF~-’

(I-llxll)(x,(1-11x112)l/2.y)

+

(0,.

. . ,Or1)1

The latter map is readily seen to be homotopic to the canonical map BFk-l,sFk-l

SFn-k+l

-->

BFk-l/SFk-lASFn-k+l 2 SF” r

whose adjoint is the d(k-1) -fold suspension [ 2 2 5 ; 0 8.51. It follows that h, t j, is homotopic to this suspension; this concludes the proof of ( B ) . (2.12) Proof of Theorem 2 . 6 . Suppose that n > 2k and that nS(5) = 0 in :(IFPk-’). Since the dimension of the CWcomplex FPk-’ is smaller than the dimension of the fibre of the sphere bundle S(nS)f it follows that S ( n E ) is fibre homotopically trivial ( * ) . Hence ( 2 . 8 ) implies that the fi-

(*I According to Atiyah [ 1 2 1 r in the stable range (m-2 dim x) we may identify J(X) with the set of fibre homotopy types of orthogonal Sm-’ -bundles over X N

.

106

Vector Fields on Spheres

.

bration e : X (F) + SFn admits a cross-section s Since nrk n > 2k, it follows from (2.11 that (ik)* : ndn-l(Fn,k) Z ndnel (Xnrk(F) and so, there exists s ' : SFn+Fn,k such that (i,),[s'] = [ s ] . Now (2.10) shows that s ' is a cross-section of p. Remark 2.13. In sections 4 and 5 we shall prove that if nJ(

-

-

N

N

II 0

If

t

E

o or Z 2

Z 0 (mod.2) we have N

KO(Y~)

zO(S4t-2) = E O ( S ~ ~ )z

0

.

iol(s45 II 0 .

, hence

N

If t P 1 (m0d.2)~then KO(S4t-2) 2 z 2 ; in this case, we map the previous sequence by complexification into the corresponding KU-theory sequence and obtain a commutative diagram with exact rows >

0-

zV

Since

c

i!

>

KO(Yt) V

IR>

>

o

V

is contained in

is trivial, the image of

. s 4t-2 ker ii , which is generated by a2t

Cyt

I

=

c (a). Hence, Yt

K-Theory of projective Spaces

image c Yt

ker it

=

=

image jl

109

.

Z

NOW, the homomorphism

c is multiplication by 2 (see S4t (0.6)) and we conclude that the upper row does not split. This implies that KO(Yt) , and s o , (3.5) is proved. N

Proof of Proposition 3.2. We first establish parts (i) and (ii) €or m even, say, m = 2t : this is done by induction on t , starting with the trivial case CPo = point. Consider the following commutative diagram with exact rows, induced by

C'

C

and therefore,

c

is injective. Observing that

+7

c(y) = v (cor(c) = 5 obtain that

+

c(y

r) t )

=

C"

=

v2

+

higher terms,

and because of

v

2t

= c$

1

c.

c'(a)

(3.1), (3.3) and

=

c

-

(3.51, we

1

c

q;R (a).

We conclude that qk (a) = yt and that yt+' = 0. So far, we have proved (i) and (ii) of (3.2) for m even.

Vector Fields on Spheres

110

Suppose now that m = 4s the cofibration 4s+3 CP 4s+2 i > CP we get readily that i! : x0(cp4s+3) R Finally, let

m

4s + 1

=

i >

cp4s+1

+

0). Working with

(s

3

> s8s+6

- z[YI/(Y 2s+2

XOCCP4s+2

)

=

0). The cofibration

(s

>

CP 4s+2

s8s+4

j

gives rise to the following commutative diagram with exact rows:

z

= kJ(s8s+4) I> ih(6:P4s+2

I > iU(CP 4s+l ) i’ C

)

1;

(the vertical arrows are given by the map c is injective and the map c’ I 2 (see (0.6)); the homomorphism j; onto v 4s+2 = C(Y~~+’).It

jk

is generated by conclude that KO(CP4s’1)

KO(CP

2y2’+’.

o

complexification). The is just multiplication by maps a generator of

follows that the image of i h is an epimorphism, we

Since

4s+2 )/(2y2’+’

->

)

2‘ Z[y]/(2y

2s+l

rY

2s+2

)

*

Parts (i) and (ii) of (3.2) are now established for all cases. In order to determine the JI - operations on y , we again map KO (C Pm) into KU (CPm) by complexification. We have c(y)

=

c

(here 5 = cm(C)) homomorphism,

o

r(c) - 2 = 5

+

5-l

-

2 E KU(CP”)

and because complexification is a -1

=

Tp([ + 5-l-2)

=

T (c(y)) P

=

c

0

Tp(y).

, JI - ring

K-Theory of projective Spaces

If

111

+

m 1 (mod.4), then c is injective and we infer that ;$ (y) = Tp(y). For m = 1 (mod.4) we get the same result via the epimorphism KO(CP 4s+2 ) + KO(CP4s+’) This completes the proof of (3.2). In 9 4 we shall make use of the following properties of the polynomials Tp(x) , that is, of $‘(y). Lemma

(i) T2(x)

3.6.

(iil

4x + x 2

=

;

p

f a r any odd i n t e g e r

= 2q

+

,

1

Proof. We shall prove formally only (ii), since the first part of the Lemma is easily verified. In the ring ~ [ z , z - I’ we compute T2q+l

(z+z

-1

-2)

= z 2q+’

2q-1

-2q-1

+

+ ...+

-

2 =

( 22q+l+

z+l)z-q}2(z-lP

2 z-2q-1 =

2-1

Using the identity

one proves inductively that there exists a unique polynomial Sq(x) of degree q , with integral coefficients, such that

4

k=-q

Zk =

s

9

(z+z

-1

-2)

,

q =

0 1 1 ,

...

(In fact, one has the recursion formula

s9 (XI

= (X+2)S

q-1

(x)

- s9-2 ( X I ,

q

2

I

with SO(X) = 1

S1(X) = 3

+ x

;

notice also that these formulas imply, by induction, that the coefficient co of Sq(x) is equal to 2q + 1). Hence,

Vector Fields on Spheres

112

T2q+l (XI = {Sq(X)}2

-

x

.

In order to determine the coefficients of

we write down the relation 4 z-q(z2q+1-1) = (2-1)s (Z+z-l-2) = (2-1) 1 C.(z+z-l-2)j q j=o 3 and in it, replace z by e2u , that is to say, replace z + z-l - 2 by (eu-e‘u)2. This implies .U(e(’q+’)U,

e-(’q+’)U)

=

eu(eu-e-u)

9

1

j=o

c .(eu-e-u) 2j 3

and hence, that 9

sinh ((2q+l)u) =

1

j=o

22jc sinh 2J+1u j

.

By differentiating twice both sides of the last relation we obtain (set c ~ =+ 0) ~: (2q+l) 9

1

sinh ((2q+l)u ) = 22JC. 3 (2]+1) {2J-Sinh2j-1U + (2J+l) sinh2j+’u} =

j=o

replacing the left hand side by (2q+l) times the expression of sinh ((2q+l)u) obtained before, we conclude that (2q+112 z2jc = 22j(2j+1)2c + 22j+2 (2j+2) (2j+3) cj+l j j A

.

simple induction procedure now shows that

and s o , the proof of Lemma 3.6 is complete. Next we investigate the K theory of the spaces MPm. The canonical projection S4m+3 + MPm factors through CP2m+1 giving rise to an S2 - fibre bundle

-

K - Theory of projective Spaces

113

The map g induces injections in ordinary cohomology as well as in K - theory. So we might regard KU@iPm) as subring of the corresponding ring of C P2m+’ C Let 5,QH) be the 2 - dimensional complex vector bundle underlying the quaternionic line bundle $,,(El). Using (2.4) we readily establish the following bundle isomorphism over C P ~ ~ +: ’

.

(3.7)

(see also

[ 4 4 ; 9.61).

Proposition

Let

3.8.

w

(i) The r i n g

w

to t h e r e l a t i o n

[(,(H) c

=

KUWPm)

= 0

,

I-

2 E kJWPm)

i s g e n e r a t e d by

. w

subject

t h a t i s t o say,

KU(IHPm) = ~[W]/(W~+~);

-

(ii) t h e homomorphism g ! : K U Q H P ~ ) KU(CP 2m+1 ) i s i n j e c t i v e and g i u e n by g ! ( w ) = v + v , whe_ne v = 5 2m+1 ( C ) - 1 . Proof. The cofibration sequence Zipm-’ + MPm + S 4m shows, by induction, that KU@iPm) is a free abelian group of rank m + 1. By ( 3 . 7 ) we have g!(w)

v + higher powers of v ; (w),.. . , g ! ( w ) ~therefore generate a

= v 1

+

the elements 1 , g ’ summand of rank m in position is proved.

direct KU(CP2m+1) 2 E[V]/(V~~+~) and the Pro-

Proposition 3.9. (i) The c o m p l e x i f i c u t i o n homomorphism c : K O Q H P ~ )-. K U O H P ~ )i s i n j e c t i v e ; (ii) t h e s u b r i n g

~ ( K o ( E ~ P ~of )

KUQHP~) i s generated (as a e.wJ, j = I,...,m, 3 i s e v e n and e q u a l t o 2 i f

f r e e a b e Z i a n g r o u p ) b y 1 and t h e e Z e m e n t s where

j

e

j i s odd.

i s e q u a l to

1

if

j

Proof. Again, we make use of the cofibration HPm-’ +lHPm + S4m and proceed by induction on m , starting with the case M P0 = point. Consider the exact sequence

114

Vector Fields on Spheres

inductively we conclude that the short exact sequence

o

X O ( S ~ ~->)

-->

Z0-l WPm)

KOWP

m

is finite. This gives

KOOH~-') ->

-->

o

and by induction one proves that KOWPm) is torsion free. Thus, the relation r 0 c = 2 (see (0.6)) implies that c is injective. To prove part (ii) we first observe that for every integer q 2 1 , 2w and w2 are elements of c ( ~ 0 W P q ) ) .This and g C = gc O H ) . The is seen as follows. Write 6 = 5,W) 4c bundle '5 is self-conjugate, that I s to say, 5 = 5 , = c 0 r(w) E 2w c o r(CC) = 2s' ; this shows that and hence 2 c(~O(lKPq)). As €or w , notice that by definition is equal to c'c, where c' stands for the map which associates to a quaternionic vector bundle its underlying complex vector bundle. The tensor product 5 8 M 5 is a real (4-dimensional) vector c't This gives w2 E bundle and we have c(c @ m 5 ) = c ' c C c(KOQHPq)). (We quote 1 4 7 1 as a reference to the preceeding remarks). The second part of (3.9) now follows easily by induction on m from the commutative diagram

cc

.

o

-->

Z O ( S ~ ~--> )

Ic

Ic

V

V

0 ->

K O W P ~ )-->

KU(S 4m)

-->

K U [ M P m ) --->

KOOKPm-1) ->

0

Ic

V

K U W Pm- 1 ) ->

0

and noting that c : 'ijO(S4m) -B %l(S4m) is an isomorphism if m is even and multiplication by 2 if m is odd (see (0.6)). The canonical projection g : CP2m+1 +ZIPm induces, by restriction, a map glCp2m = h : cP2m

->

MPrn

.

K-Theory of projective Spaces

Proposition h!

115

( i ) The homomorphism

3. l o .

: KOQHPm)

->

KO(CP2m)

E[y]/(y"+')

i s a monomorphism. I t s image i s t h e f r e e a b e l i a n g r o u p w i t h 2 j m base {1,2yry , e j y , ..., emy } , # h e r e

...,

1 ,

R 5,QH)

(ii) Let

be t h e r e a l

m

euen

m

odd.

4

,

- dimensional bundle underly-

i n g t h e c o n o n i c a l q u a t e r n i o n i c l i n e b u n d l e and l e t

5R, ~ )

z =

-

4 E

E O Q H P ~;) t h e n

1

h' ( z ) = 2y. P a r t ( i ) f o l l o w s f r o m (3.21, ( 3 . 8 )

Proof:

,( 3 . 9 )

and

t h e f o l l o w i n g commutative diagram: KO'(b:P2m+') > -

KOQHPm)

Ic

V

KO(GP2m)

Ic

Ic

V

V

I

KU(CP2m+ 1 )

KU(IHPm) >:9

>

KU(CP2m)

.

c : KO(CP2m) + KU(CP2m) i s m o n i c ) The s e c o n d p a r t of ( 3 . 1 0 ) stems f r o m (3.7) W e f i n a l l y c o m p u t e t h e K - t h e o r y of t h e r e a l p r o (3.2)

(In particular,

( i i ) shows t h a t

.

.

jective spaces I R P ~ . Proposition

Let

3.11.

u = C~,W)

-

I E

KUWP~).

N

( i ) The r i n g

KUWPm)

i s generated by

subject t o the

u

relations u2

+

2u = 0

--

and

u [m/21+1 = o

2k+1) i n p a r t i c u l a r , KUQRP2k ) = KUWP N

der

i s a c y c l i c g r o u p of o r -

2k ( i i ) The o p e r a t i o n s

9 P (u) =

{

are g i v e n by

QP

0

, i f

p

i s even,

u

,

p

i s odd.

i f

;

116

Vector Fields on Spheres

If the relation u 2 + 2u if, and only if, 2ku = 0 .

Remark.

uk+l

holds, then

= 0

= 0

Proof of (3.11). We begin by observing that tm(IR)Q tm(IR) = E ~ In . fact, for any C W - complex X the multiplicative group of line bundles over X is isomorphic to 1 H (X;Z2). Since complexification is compatible with tensor products we obtain ( 1 + ~ =) 1~ and hence, 2u + u2 = 0. Next we show that u is a non-zero element of filtration 2 in KU@Pm) , m ? 2 , that is to say, if i : IRP2 --. lRPm is the inI clusion map, then i‘(u) 0. The total Stiefel-Whitney class of tmQR) is given by

N

*

w(5,D))

= 1 + b

r

;z2)=

where b E H’ DPm;Z2) is the generator of H* (IRPm) Z2[bl/(bm+’) [ 7 9 ; VII, 9.41. Hence, = w(2Cm(IR)) = (l+bI2= 1 + b 2 1 w(rec(5,W)))

*

,

if m 2 . It follows that c(cmaR)) and its restriction to RP2 are stably non-trivial, proving that u = c(Sm[IR)) - 1 is a non-zero element of filtration 2 (m 2 2 ) . The ring H*(LRP2k;Z) is generated by an element 2 a E H (IRP2k;z) subject to the relations 2a = 0 and

ak+’ = 0

(see 1 7 9 1 ) . Since there is no odd-dimensional integral cohomology, the Atiyah-Hirzebruch spectral sequence of IRP2k collapses and we infer that the associated graded ring $KU(lRP2k) is isomorphic to H*(IRP2k;Z) (see ( 0 . 1 0 ) ) . By the above remarks, the element u represents a E 42%J(IRP2k) and we conclude that uJ P 0, j = l,...,k, and uk + l = 0 . N

Part (i) of (3.11) is now proved for m = 2k 2 0 (the case k = 0 being trivial). With the cofibration IRP2k one gets readily that Part

i

>

IRP

2k+l

i’I : -KU@RP2k+1)

(ii) follows from

S2k+l kJ(IRP2k )

.

117

K - Theory of projective Spaces

and the fact that complexification commutes with the rations. Proposition

Let

3.12.

x

=

-

ERI,)

II,

-

ope-

1 E ZO(IRPm).

N

The r i n g KO@Pm) i s g e n e r a t e d by x (i) relations , x2 + 2x = 0 a n d x f(m)+l =

subject t o the

f(m) is t h e number of i n t e g e r s q w i t h q = 0,1,2 o r 4 (mod.81 and 0 < q 5 m. In p a r t i c u l a r , t h e g r o u p KOmPm) i s cycZic of o r d e r 2f ( m ) where

N

JIp

(ii) The o p e r a t i o n s

a r e g i v e n by

o, $P(xl Remark f(m+8)

=

{

even

if’ p

x , i f

odd.

p

3.13. =

The integer valued function f ( m ) + 4 and

In particular, f(8q)

, f(m)

is given by

4q.

=

Proof of (3.12). We begin by showing that the order IEO(RPm) I of KO(RPm) divides IKO(RPm-’) I em , where N

N

(3.14)

m =

Recall that lRPm projection

{

2

,

1

, if m

if

m- 1

and so, the Puppe sequence of exact sequence 2k, except for a small number of cases, namely for n = 2f(k-1), with k = 1 ,8 (see (3.13)) Hence, for all but the latter cases, it follows from (2.2) and (2.61, that sn- 1 admits k - 1 linearly independent vector fields. For the exceptional cases one can explicitely construct the vector fields (see (4.4)). We illustrate this for k - 1 = 8, n = 2f(8) = 16. Let lH(2) be the R - algebra of the 2 x 2-matrices over the quaternions and let i,j,k be the usual generators of M We set in M(2)

.

,.. .

.

One checks r,eadily that e e = -e e if p + q , and e2 = P 9 9 P ’ P oper-lH(2) (P=l,...,4). The lR-algebra A = M ( 2 ) QmlH(2) ates on IR 16 M(2) by linear transformations as follows:

fx, Q x2) (x) (in fact

= x1

-

x

-

x2

zlR(16)). For the elements

A

a,

,...,a8

E A

defined

by a

P

=

ep@ 1, ap+4 = ep ~

we show that a a

P 9

=-aa q P

if

e1 2e3 e

p

9

q

and

(p=l,2,3,4)

L

a P

=

-lA

-

(pi9 = 1i*--i8) We choose an inner product ( I ) in lR16 which is invariant under the action of the finite (multiplicative) group generated by all...,a8 and compute (xla (XI 1 P

=

(a (x)l a 2 (XI)

P

P

=

(a (x)I-x)

P

=

-(xlap(x)).

Real Vector Fields on Spheres

Thus (xla (x)) = 0. Similarly, one obtains P if p q. Hence, the frame

*

{x,a,(x)

I . . . I

a8(x))

121

(a (x)la (x)) = 0, P q

I

S1 5 lR1 , is orthonormal, providing 8 linearly independent vector fields on S15 Working with quaternions and Cayley numbers we exhibit in the same way k - 1 vector fields on the sphere of dimension 2f(k-1) - 1, k = 3,...,8. The case k = 2 was treated in the Introduction. Theorem (4.2) is now proved.

x

E

.

Corollary

The maximum number o f L i n e a r l y i n d e p e n d e n t

4.3.

v e c t o r f i e Z d s on Sn-l is e q u a l t o p(n) - 1 , w h e r e is d e f i n e d a s f o l l o w s : w r i t e n = (2*a(n)+1)2b (n)I b(n) = c(n) + 4 d(n), where a(n), b(n), c(n), d(n)

-

t e g e r s and

0

5 c(n) < p(n)

=

p(n) are in-

3 ; then,

2c(n)

+

8d(n)

.

Proof. According to (4.2) we have to find for a given n the greatest m such that Zf (m) divides n. Hence, we must look for the greatest m such that f(m) 5 b(n). By (3.13) this is obviously p(n) - 1. Remark. As €or k - 1 = 8, n = 16, it is in fact possible to construct explicitely the vector fields in all cases. We shall sketch a proof of the following so-called Hurwitz-Radon-Eckmann Theorem (see [851, [133] and [2001). sn- 1 t h e r e e x i s t p(n) - 1 l i Theorem 4 . 4 . On t h e sphere nearly independent vector f i e l d s

v(q) (XI = (+) q

=

near -

l,.. .,p(n) in

x

=

-

(x)1 . .

. ,VA@

(x)1

1, w h i c h a r e g i v e n b y f u n c t i o n s

Zivi( ~ (x) ) , -

(xlI...,xn).

Proof. We shall show that Sn-l admits m vector fields which are linear in x if n is a multiple of 2f (m) This is equivalent to ( 4 . 4 ) as the proof of (4.3) shows. To this end we shall apply the theory of real Clifford algebras (see [261). Let us recall that a Clifford algebra Cm is an generated associative algebra with a unit element 1 over IR I

122

Vector Fields on Spheres

(as an IR- algebra) by elements el ,e*,...,em subject to the relations e2 = -1, e.e = -e e j (i,j=l,...,m). i 1 1 j i if i

*

Let V be a real vector space of dimension n which is a left Cm - module. Thus, Cm acts on V by linear transformations. If a E Cm and x E V , we write a(x) f o r the action of a on x. Choose now an inner product ( 1 ) on V which is invariant under the action of the multiplicative group generated by e l , e2,...,em. Let S(V) be the unit sphere in V with respect to this inner product. For any x E S ( V ) the elements x,e,(x),

...

em(x)

form an orthonormal (m+l)-frame in V , providing us with m linearly independent vector fields on the (n-1)-sphere S(V). (Using the fact that the selected inner product is invariant under the action of the multiplicative group described, 2 it follows that if i j, (e,(v) I e .(v) = (ei(v) I e . e .(v) = 3 1 7 -(vle.e.(v)) = (v1e.e (v)) = -(e.(v)le.(v)) and hence, 1 3 7 i 3 1 (ei(v)I e.(v)1 = 0. Similarly, (v I ei (v) = 0 and (ei(v)lei(vl)=1).

+

3

In view of the preceeding remarks, ( 4 . 4 ) will be proved if IRn can be given the structure of a Cm-module, whenever n is a multiple of 2f(m). Let F ( q ) denote the IR - algebra of q x q - matrices with entries in IF (IF = I R , C or M ) . The algebras C ,m=l,. . . , 4 m can be identified with the following IR- algebras:

,---.

1--

-

-

5

4

7

6

8

1

Furthermore, Cm+8 Cm BIRIR (16). From these statements, we see that Cm, m = 1,...,8, acts by linear transformations on XIam where am is given in the following table: m

1

2

3

4

5

6

7

8

z2

z2

23

z3

23

z3

z4

(4.5)

am

Complex Stiefel Fibrations

123

(The algebra M O M acts on IR4 via the algebra homomorphism H @ M + H ,defined by the projection on the first factor; similarly for I R ( 8 ) O I R ( 8 ) ) .Notice that if IRs is a C m - module, 4

'm+ 8 - module; this, together with (4.5) implies that IR is a C, - module (see (3.13)). Hence, for any multiple of 2f(n11 say n , the vector space IRn admits a C m - action and ( 4 . 4 ) is established. then IRs 8 IR16 = I R S ' *

5.

is a 2f (n)

Cross-Sections of Complex Stiefel Fibrations. The cross-section problem for the fibrations

was solved by Atiyah and Todd [37], and by Adams and Walker [ 8 ] ; the former authors gave necessary conditions, the latter proved that these conditions were also sufficient. Referring to ( 2 . 5 ) and (2.6) the task is to compute the J - order of the real bundle underlying the can- F;k-l(C) over CPk- 1 We shall work onical line bundle with the functor 3' defined in Chapter 0. The groups J ( X ) and r'(X) are naturally isomorphic for any finite C W - complex X : we also notice that Adams and Walker use a different J' in N

.

ck-,

[8], defined in terms of the classes bh method invented by K.Lam [160], we first the order of J' (y,) , y, = r (6,) - 2, for show that y2t+l and y2t have the same

and sh. Using a determine explicitely

N

In the sequel we shall write

5

even m : then, we J-order (t > 1). N

and

y

for

5,

and

ym respectively. According to the definition of the functor J' the order b2t of J ' ( y ) E 5' (CP2t) is the smallest positive integer such that there exists an element w E ?O(2:P2t) with (5.1)

for all primes

p

(see (0.13) and ( 0 . 1 4 ) ) . For the ring

124

Vector Fields on Spheres

(see (3.2)), we have the canonical embeddings KO(6P2t) cKO(CP 2t) @ QpCKO(CP2t) 8 Q = Q[y]/(yt+') and we extend the $ - operations to the latter rings in the obvious way, namely, by taking 8 1 and JIp @ l Q The QP multiplicative groups of polynomials with constant term 1 are denoted by

.

1 + :[YI/(Y~+~)

and

1 + 6[yl/(ytf1)

,

respectively. The next result is readily established using the formula $ ~ ~ ( y=~p2m ) ym + higher terms, proved in

(3.6).

1 + u be a n e l e m e n t o f 1 + c[y]/(y t+l ) . T h e n , t h e polynomial $(1+u) h a s i n t e g r a l c o e f f i c i e n t s f o r a l l p r i m e s 1 +u p i f , and o n l y if, u E ~[yl/(y~+') . t+l ) We also note that every element 1 + u € l+a[yl/(y has an nth - root given by

(5.2) L e t

is a ring homomorphism, J I ( ~(l+u)'In) = ($' (l+u)) 'In . NOW, for any integer b , the element e (by) = ( 8 (y))b P P According to corresponds to a polynomial in 1 + Q[y]/(yt+l). (5.1) and (5.2) this polynomial has integral coefficients for all primes p , if b is a multiple of the order of J'(y). We show that the converse also holds. Since $ p

N

N

is t h e s m a l l e n t p o s i t i v e i n t e g e r b, s u c h t h a t f o r a l l p r i m e s p t h e e l e b ment (ep(x)) E 1 + Q[y]/(yt+l) is a p o l y n o m i a l w i t h i n t e g r a l Lemma

5.3.

The o r d e r

b2t

of

J'(y) E z'(CP2t)

coefficients.

n

=

b2t)

primes

p

Proof. There exists and integer n (for example, and an element 2 E i[y]/(yt+l) such that for all one has

(ep(y))n = $p(l+z) l + z

. Hence, for every prime

Complex Stiefel Fibrations

, where (y) = P integer b we obtain

p

,

125

1 + u = (l+z)l/”. For any positive

8

1 + w = (1+~)~ Thus, . if for all primes p the polynohas integral coefficients, from (5.2) we mial (OP(y)) and hence, b J ’ (y) = 0, provconclude that w E Z[yl/(yt+’)

with

-

N

ing the Lemma. We now determine the polynomial Lemma

5.4.

(i) B2(y) = (l+y/4)1 /2

(ii) F o r a n y o d d i n t e g e r

p

q-1 j=l (iii) If

p = 2q

,):;q( 2j1+ ring

+

=

2q + 1

1 2j + 1

N

BP(y) E 1 +E[yl/(y t+l ).

. one h a s

1 2q

+

1 Yq

*

is a p r i m e , t h e c o e f f i c i e n t s

1 j

= 0

,...,q -

1

are integers.

Proof. Complexification injects the torsion free KO(CP2t) Z[y]/(yt+l) into the ring KU(CP 2t) =

-

Z[V]/(V~~+’) . Referring to (3.2 (ii)) this embedding is -1 given by c(y) = v + 7 = 5 + 5 - 2 (note that -5 = 5 -1 ) .

-1 Consider now the complex 4 - dimensional bundle T J =25+ 25 4 From a wellOne has h C (n) = 1 and r(n) = 4r(c) = 8 + 4y known property of the Bott operations (see (0.13)) we have

.

Hence ,

.

126

Vector Fields on Spheres

implying that

(see (3.2 (iii))). The proof of the Lemma is now concluded using the explicit formula for T (y) given in ( 3 . 6 ) . (If P 2q + 1 is a prime, then the i n t e g e r s

are obviously divisible by 2q + 1. ) We now recall that given any integer n 0 and for any prime number p I the p - a d i c v a l u a t i o n vp(n) of n is the exponent of p in the prime power decomposition of n ; we set v ( 0 ) = 0 for any prime p(?)Moreover, the p - adic valP uation function can be extended trivially to rational numbers.

*

Lemma

5.5.

Let

b e a p o s i t i v e i n t e g e r and

b

p

be a prime.

The f o l l o w i n g t h r e e c o n d i t i o n s a r e e q u i v a l e n t .

(ilp

: The p o l y n o m i a l

(ep(y)))

E

a[y1/(yt+')

has i n t e g r a l

coefficients.

(ii) : For a12 P

with

s

the binomial c o e f f i c i e n t

(iii) P

:

v (b) P

2 s 5

0

b

P

t h e power

ps

divides

.

(s)

max{s+v ( s )

[3]

I

O

~

s

~ } [

.~

]

Proof. ( A ) (i), w (ii) for odd primes p = 2q + 1 . P According to ( 5 . 4 ) the polynomial 8 (y) is of the form P (y) = 1 + w + - 1y q e2q+l P w E

(+)

[y]/(yt+')

.

Hence,

This definition is convenient for our purpose.

Complex Stiefel Fibrations

127

[$]

(l+w)b-s yq s , 0 5 s 5 , form a partial (note that yqs = 0 basis for the abelian group z[y]/(y t+ 1 ) if qs > t) , we conclude that (eP(y)) is integral if , and Since the elements

only if, pS (B)

Let

(:)

(ii)

P

(:)

divides

Y

(iii)

,

0

5

=

for all primes

P

[s] .

[i]

5

s

.

p

m be a non-negative integer. We show that ps for all s with 0 5 s 5 m if, and only if, v (b)

P

(;)

.

. max \I s + v ( s ) 10 5 s 5 m 1 P J

This claim holds trivially for m we write b b-I . b-2 = m 1 2

.

= 0

.-.

and observe that if

vp(b)

z

m

-

divides

m = 1 . For m

and

2,

b- (m-1 ) m- 1

1 , then

v (b) = vp(b) - v (m) : P m P

-

this remark implies that, in case v

(”)2 m

P m

P

v (b) 2 m + v (m)

P

P

The claim now follows by induction on (C)

(i), * (iiiI2

m

v (b)

m

-

1 , one has

. .

.

Both conditions are satisfied only if say b = 2b’. The polynomial

b

i

an even integer,

(l+y/4)(1/2)b - (l+y/4Ib’E Q[yl/(yt+ has integral coefficients if, and only if, s ’ = 0 , ..., t A s in (B) one shows that the latter‘holds if, and only if, v2(b’) 2 rnax (2s’ + v 2 ( s 1 )Is’ = O,...,t}

.

.

Since

v2(b)

=

1

+ v2(b1) and

Vector Fields on Spheres

128

1

+ max

1 2 s 1 +v2(sI) Is = max { 2 s 1 + v 2

..., 2t)

,

max {s+v2(s)Is = 0,

=

the proof is complete. Proposition given by

The o r d e r

5.6.

b2t

J(y)

of

in

J(CP2t)

I,

v (b2t) = max {S+V ( s )I 0 5 s 5 2t P P I,-1 f o r e v e r y prime

is

r

-

p

Proof. Since J(y) and (y) have the same order, the Proposition follows from ( 5 . 3 ) and ( 5 . 7 ) . We show next that the J - order of the canonical line bundles over CP 2t+1 and CP2t are equal.

-

N

Proposition

For

5.7.

t

1

;(CP2t+1) Proof.

the order

b2t+l

is e q u a l t o

b2t

1

The homomorphism

i’

by the inclusion i : CP2t + CP2t+’ Hence, b2t divides b2t+, If t

.

is an isomorphism. Thus, ?(CP2t+1) Now suppose that

t

E

0

3(CP2t)

(mod.2), say, t

in

KO-theory

52t+l onto S2t (mod.2)

1

J(Y)

.

induced in

maps 5

of

=

the map

I

i‘

b2t+l - b2t.

and

2q. The exact se-

quence 0 ->

‘iio(s8q+2) ->

KO(CP

z 2 and is generated by (see ( 3 . 2 ) ) shows that ker(i!) y2q+1. Because of the final remarks in (0.14) , we conclude that the kernel of J(i) : y(CP 4q+1) + 3(cp4q) is at most and is generated by z(y2q+1). The homomorphism J(i) maps the element b ? ( y ) E 7(CP4q+1) onto zero, that is to say 4q T =

b

N

N

J(y) E ker J(i) = 4q

*

2q+ 1

z2

or

0

z2

.

We are going to show that T z(y : this implies that T = 0 and the proposition will be proved. Consider the canoni-

Complex Stiefel Fibrations

129

cal projection g

> CP44+1

: IRP 8q+3

,

By (3.15) the homomorphism J(g) maps ?(y) onto 2?(x), hence J(g1-r = 2b4q?(~) E ?(IRPRq+3). Using (5.61, (4.1) and (3.13) it is easy to see that N

v2(b4q)

2

4q + v2(4q)

.

N

which implies that J(g)r = 0 and (4.1) we conclude that

Thus

T

*

Theorem CPm

?(y2'+l)

and

(5.7)

N

The

5.8.

2 4q + 2

J-order

I?mP 8q+3) I

=

,

On the other hand, from

(3.15)

is proved.

of t h e c a n o n i c a l l i n e b u n d l e o v e r

is e q u a l t o t h e i n t e g e r

bm

given by

[*I>

v (bm) = max {s+v ( s ) 10 5 s 5 P P f o r e v e r y p r i m e p . I n p a r t i c u l a r , v (b ) P m a n d v (b,) 1 1 if p 5 m + 1.

if

= 0

p > m

+

1

P

Proof.

P

r

For any positive integer m

and any prime

let N(m,p)

=

Observing that for

max {s+v t 1 1

P

( s ) 10

5 s 5

and for every prime

p

one has

N(2trP) = N(2t+lrp) the theorem follows for

m 1 2

from

(5.6)

and

(5.7).

2 Let us assume that m = 1. Then, J" (CP ) = ( S ) =Z2, generated by J " ( y ) (see (3.2) and ( 3 . 6 ) ) . Hence l ? ( y ) I = 2. Since N(1,2) = 1 and N(1,p) = 0 for p odd, the theorem holds for m = 1 1

N

?It

N

.

Remark. b4 = b5

The first few values of bm 2880, b6 = b, = 362 880.

=

are

bl

=

2, b2

=

b3 = 24,

Vector Fields on Spheres

130

Theorem

$he compZex S t i e f e Z f i b r a t i o n

5.9.

h a s a c r o s s - s e c t i o n if, und o n l y if, integer

‘n,k

+

bk-l

’

C

nik

+S2n-1

k>2,

is a m u l t i p l e of t h e

n

caZZed Complex James number, d e f i n e d i n (5.8).

and (5.8) , if is a multiple of

Proof. According to (2.5) 2n - l has a cross-section, n Conversely, let

n

be a multiple of

v2(n) 2 V ~ ( b k - ~ ) k

-

1

bk-l.

bk-,. Then

I

.

that is to say, n 2 2k-l If k 1 5 then, n > 2k; if k = 3 or 4, then bk-l = 24 and again, n > 2k. Hence, €or k 2 3, Theorem 2.6 implies the existence of a cross-section of S2n-1 ‘n,k If k = 2 we have that bk-l = bl = 2 and hence, n is even, say n = 2q. In this case, we give an explicit crosssection s of the Stiefel fibration: for every +

Remark 5.10. (1) In view of (2.2), Theorem lent to the following statement. “The s p h e r e

S

admits

m

5.9

l i n e a r l y independent

c o m p l e x v e c t o r f i e l d s i f , a n d onZy if, n of

is equiva-

i s a muZtiple

bm”.

(2) Contrary to the situation in the real case there are no known formulas which give explicitly the complex vector fields on the spheres (except for m = 1)

.

6.

Cross-Sections of Quaternionic Stiefel Fibrations.

In this last section of the Chapter we shall determine the J-order of the canonical line bundle over the quaternionic projective spaces. We shall use methods, results and notation of the preceeding section. IR Let cmW) be the real 4 - dimensional vector bundle N

Quaternionic Stiefel Fibrations

131

underlying the quaternionic line bundle

z [y]/(ym+l)

induced by the canonical map h : CP2m -D M P m is injective and identifies the ring gO(lHPm) with the J1 - subring Am(y) generated (as a ring) by 2y and y 2 Furthermore, h!(z) = 2y. Since the classes 0 are natural, we infer that the order of P ?I ( 2 ) E @Pm) is the smallest positive integer cm such that there exists an element u E Am(y) with

.

?I

for all primes u E 5[y]/(ymt1) ment

1 + u @(l+u)/(l+u)

.

It is easy to see that a polynomial belongs to Am(y) if, and only if, the ele-

p

belongs to =

1 + u

P

1

.

(If

for all primes

p,

+ Am(y) , for all primes p

lies in

1

+ A,(Y)

then according to (5.2) we have that u E ;[y]/(ym+’). Since 2 2 2 il, (y) = 4y + y is in Am(y) , one obtains that J1 (l+u) E 1 + Am(y)

for every

u E y[y]/(ym+’).

This implies that

1

+ u

=

2

(l+u2)/il, ( l + u ) E 1 + Am(y)). With the preceeding remark, the following Lemma is now proved in exactly the same way as (5.3).

J’(z)

J’@IPm) i s t h e s m a l l e s t 2c positive integer c m s u c h t h a t t h e p o l y n o m i a l (ep(y)) m E 1 + %[y1/(ym+’) b e l o n g s to 1 + ~,(y), f o r ~ Z pZ r i m e n u m b e r s p. Lemma

6.2.

T h e o r d e r of

in

In the following two Lemmas we determine the valuation of the integer cm

p - adic

.

Lemma

6.3.

Let

d

and

m

be i n t e g e r s

2 1

a n d Let

p

be

an odd p r i m e . The f o l l o w i n g a r e e q u i v a l e n t .

(i)

:

The p o l y n o m i a l

( ~ ~ ( y ) ) b’e ~ longs t o

1 + Am(y)

.

Vector Fields on Spheres

132

(ii)

The p o l y n o m i a l

:

(iii) : v (d)

max {s+v

-

P

belongs t o

(ep(y))d P

(s)

lo 5 s 5

1 +z[y1/(ym+').

[*]I P-1 1 .

Proof. (i) (ii) : If (Op(y))d is integral, then (Bp(y) 2d E 1 + Am(y) , since the subring Z @ Am(y) generated by 1 , 2y, y2 contains all squares of z[y]/(ym+l). To prove the converse, we note that by definition the polynomial BP(y) has its coefficients in Qp (see Chapter 0 and ( 5 . 4 ) ) . Therefore, if (Bp(y))d = 1 + w does not one of its coefficients has a belong to 1 + z[y]/ (ym+') power of p in its denominator and we conclude that the same 2d must be true for the square (Bp(y)) . (ii) y (iii) : Lemma

This follows from

Let

6.4.

d

and

(5.5).

be i n t e g e r s

m

2 1.

The f o Z i ! o # i n g

are equivalent.

(i)

The p o l y n o m i a l

(ii) F o r a l l i n t e g e r s ficient

(;)

e v e n and

e

j (iii) v2(d)

j

with

i s d i v i s i b l e by = 2

i f

belongs t o

(e2(y))2d

j

1 + A,(y).

5 j 5 m, t h e b i n o m i a l c o e f e.41 , where e = 1 i f j i s 3 j 0

i s odd.

max (2j+v2(j), 2m+110 5 j 5 m}

ProoE.

According to

(5.4)

we have

. B2(y)

=

(1+y/4)1/2

and hence ,

2 j m Since 1,2y,y ,...,ejy ,...,emy form a basis of the free abelian group Z @Am(y) , we conclude that (i) and (ii) are equivalent. Note that condition (ii) can be reformulated as follows: 2j + e - I , 0 5 j 2 m (ii)' : v2(4) 1

.

As in the proof of (5.5) (B) we show that if v2(d) > m - 1, then v2 1. 2m + em- 1 M v 2 (d) 2m+ v2(m) + e m - 1

(:)

.

2 2(m-1)

Quaternionic Stiefel Fibrations

133

Starting with the case m = 1 we conclude now by induction on m that (ii)' holds if, and o n l y if, v2(d) 2 max C2j+v2(j) + e - 1 1 0 < - j - 2j+l, if 2j+l 2j+v2(j) + e - 1 = , if j

{

the equivalence proved. Theorem

Y

is even, is odd,

(iii) is established and

(6.4)

is

=

max

v (c,) P

=

max {s+v ( s ) l o 5 s 5

+ 1.

P

j 2 m1

{ 2j +v2(j) , 2m+ 110 5

v2(cm)

P

I n p a r t i c u l a r , v (c ) = 0 p < - 2m

j j

6.5. The 3 - o r d e r o f t h e c a n o n i c a l l i n e b u n d l e o v e r 1 , i s equal t o t h e i n t e g e r c m g i v e n by

z

HPm, m

(ii)'

.

p > 2m

i f

m

+

*I,1,

[P-l 1 and

I

odd.

p

v (c P

m

)

z

1

i f

Proof. Since 3 ( z ) and J ' ( z ) have the same order, the Theorem follows from (6.2), (6.3) and (6.4). Remark. and c3

The first three values of c are: c1 = 24, c2=1440 m 362 880. We are now in a position to give a solution of the cross-section problem for the quaternionic Stiefel fibrations.

Theorem

=

6.6. HI n,k

The q u a t e r n i o n i c S t i e f e l f i b r a t i o n

,

S4n-1

, n > k > 2

I

n is a m u l t i p l e o f t h e ck-1' c a l l e d t h e Q u a t e r n i o n i c James number, d e f i n e d (6.5). Proof. If MnIk S4n- 1 has a cross-section, then ( 2 . 5 ) and (6.5) the integer n is a multiple of c ~ - ~ . Conversely, let n be a multiple of c ~ - ~ Then, .

has a c r o s s - s e c t i o n i f , and o n l y i f , integer in

by

-.

v2(n) ~ v ~ ( c ~2 -2k~ - ) 1 Hence, for all

and

k > - 2, one has

(2.6) we conclude that M

n,k

+

v3(n) >v3(ck-,)

.

n 1 22k-1 3 > 2k S4n-'

z

1

.

and with

has a cross-section.

Vector Fields on Spheres

134

Remarks. (1) In view of lowing result:

s

"The s p h e r e

~

(2.2), Theorem 6.6

~a d m -i t s ~m

gives the fol-

l i n e a r l y independent

q u a t e r n i o n i c v e c t o r f i e l d s i f , and onZy if,

cm

multiple of

n

is a

.I1

(2) As for the complex case, no explicit formulas are known s4n- 1 (even for giving the quaternionic vector fields over the case m = 1 ) . We conclude this section investigating some relations between the complex and the quaternionic James numbers b2k-l and c ~ - ~

.

Proposition

T h e q u a t e r n i o n i c J a m e s number

6.7.

e i t h e r equaZ t o

b2k-l

or t o

(1/2)b2k-l

ck-,

is

.

Proof. The Proposition follows at once from ( 5 . 8 ) and (6.5). However, we give a second proof which does not depend on the explicit knowledge of c ~ -and~ b2k-l

.

The homomorphism ? ( g ) : ?(HPk-') + J(cP~~-') induced by the canonical map g : CP2k-1 + MP k-l sends J ( z ) onto ?(2y) (see ( 3 . 7 ) ) . Hence (1/2)b2k-l divides ck-I ' According to

(3.10)

the homomorphism

h!

=

(glCP2k-2)!

iden-

tifies ?0(HPk-l) with the $ - subring Ak-l ( y ) of z0(CP2k-2) 2 Z[y]/(y k) ; moreover, h!(z) = 2y. Since b2k-2 is equal to the order of ? ' ( y ) , there is a polynomial w E F[y]/(yk) such that

for all primes

p , and hence,

.

The subring Ak-,(y) contains 2w + W ' for all primes p for any w E z[y]/(yk) and we conclude that c ~ -divides ~ b2k-2

(compare with (6.2)). But b2k-2 divides b2k-l (ac- b2k-,) and it follows that ck-, = (1/2)b2k-, tually, b2k-2 Or 'k-1 - b2k-l

.

Quaternionic Stiefel Fibrations

135

Remark. Geometrically the equality c = (1/2)b2k-1 means k- 1 that if 2n = 0 (mod. b2k-,) , the complex vector fields on s4n- 1 can be chosen "quaternionic", that is to say, S4n-1 admits 2k - 1 complex vector fields that s.tem from k - 1 quaternionic vector fields. (The identification of the quaternions with the complex 2 x 2-matrices of the form u v

(-v

ii

)

provides a canonical embedding of M into C2n,2k compatn,k ible with the projections onto S4n-1. A cross-section of H + S 4n-1 therefore gives rise to a cross-section of n,k '2n12k -+ s4n-1 ) . We now give a description of the integers k for which c ~ -- (1/2)b2k-l ~

-

Theorem 6.8. L e t k > - 2. T h e n , c k-1 onZy if, k b e Z o n g s to t h e s e t A =

(Here IN

= ( 1/

2 b2k- 1

if, a n d

{s+r.2 2s- 1 Is,r E IN1

is the set of inteqers 2 1).

Before giving the proof, let us draw some consequences which ilustrate the result. ~ (a) If k is odd, c ~ =- (1/2)b2k-l tegers belong to A .

.

Indeed, all odd in-

(b) The first even integer k for which is 10 , that is, c9 = (1/2)b19 .

(6.5)

c ~ - (1/2)b2k-l ~ =

Proof of (6.8). We first recall from (5.8) that ~ ~ ( b =~max ~ {-j +~ v2(j)10 ) 5 j 5 2k - 1 )

and

and v ~ ( c ~ -= ~max(2t ) + v2(t), 2k =

max{ j +v2(j) - 1,2k- 1

We now show that for any integer properties are equivalent. (i)

Ck- 1 =

-

(1/2)b2k-l

.

k -> 2

1 I 0 5 t 5 k - 1)

I

0

5 j 5 2k - 1

the following four

}

136

Vector Fields on Spheres

(ii)

-

V2(b2k-l) > 2k

1

.

(iii) There exists an integer 2s-I k = s (mod. 2 ) . (iv)

such that

s

0 < s c k

and

k E A .

-

The equivalence (i) Y (ii) follows immediately from the compaand v2 (b2k-1) The equivalence (iii) (iv) rison of v 2 ( c ~) - ~ is trivial. (ii) (iii) : If ~ ~ ( b >~2k~ - -1 ~there ) exists an even integer j such that 1 5 j < 2k - 1 and j + v2(j) > 2k 1. Setting j = 2k 2 s we obtain a number s . such that 0 < s < k and 2k - 2 s + v2(2k-2s) > 2k - 1. This latter 2s-1 condition is equivalent to k - s = 0 (mod. 2 1. (iii) * (ii) : If j = 2k - 2s then

.

I,

-

-

~ ~ ( b> ~ - ~+ ~) ~ ( 2 k - 2 ~(2k-2s) - ~ (2k-2s) ) + 2s > 2k

-

1

.

The theorem is now proved. Finally, we turn to an evaluation of the density of Any integer of A belongs to exactly one the set A in N arithmetic progression A = {q + r-22q"1 with q Ct A Moreq over the set A is the disjoint union

.

and its density

a

.

is equal to

2- (2m-I )

1 mct A

a

=

1

2-(2m-1)

mCA -

-2 -

l n$A

1

= m IE N

mEAn

2- ( 2m- 1 )

2- ( 2m- 1 )

-

. We l mE A

then write 2-(2m-1)

137

Quaternionic Stiefel Fibrations

The series converges in an extraordinarily rapid fashion. Using the fact that

A

contains sequences of con-

secutive integers of arbitrary length, one deduces easily from Liouville’s criterion that approximate value of a is all cases, the James number

a

is a transcendental number The 0.63. Hence, in about 6 3 % of ck-, is equal to (1/2)b2k-

.

This Page Intentionally Left Blank

CHAPTER 5 SPAN OF SPHERICAL FORMS

Introduction and Generalities about Spherical Forms In this chapter we will study the vector field problem for the quotient manifolds associated to free actions of a finite group G on the sphere Sn. Among the manifolds Sn/G to be considered, we find an important class of riemannian manifolds with constant curvature, namely the S p h e r i c a l S p a c e Forms which, for reasons of simplicity, we shall just call S p h e r i c a l Forms. The present chapter aims at determining the span of the spherical forms, that is to say, the maximum number of linearly independent vector fields on these manifolds. Except for the dimension 7, the span of an n - dimensional spherical form depends only on the dyadic valuation of n + 1 and on the order IGI of its defining group G. Complete results have been given by J.C.Becker [40] under the assumption that whenever the set of the 2-Sylow subgroups of G contains a generalized quaternion group Qm then G is a metabelian group of rank 2 . These results, stated in 55 5 . and 6 , are independent of the choice of the free action of G on Sn This is not the case in dimension 7 , where the action must be taken into account: we shall see this in the following chapter, in which we discuss the problem of the parallelizability of the spherical forms. The methods used to compute the span of spherical forms are just extensions towards the G - equivariant side of methods developped by J.F.Adams in [2]. Basically the ideas we present here are very close to those explained in the previous chapter. Thus, after a brief preliminary on spherical forms we shall recall some of the properties concerning the notion of G - fibre homotopy type ( 5 3) ; then, we 1.

.

139

140

Span of Spherical Forms

shall imitate the work of J.F.Adams to deal with G - (co)reducibility (5 4 ) . In all this, we adopt and follow Becker's point of view. The reader who is interested in the various problems related to vector fields on manifolds is referred to the initial articles of H.Hopf 11201 and E.Stiefe1 12331 (written, respectively, in 1927 and 1936), then to the papers of E.Thomas [240] and M.F.Atiyah [181 which state general results; next, he or she should go to 1531 for n - manifolds, to [239] for low-dimensional manifolds and finally, to 12221 for quotient manifolds. The notion 'Spherical Space Form' came up in connection with the Clifford-Klein Problem, which consisted in giving a description of all connected, complete riemannian manifolds of constant curvature. Indeed, the work of Killing 11501 and Hopf 11191 show that a riemannian manifold M of dimension n 2 2 is a connected, complete manifold of constant curvature k > 0 if, and only if, M is isometric to Sn/G, where G is a finite subgroup of o(n+l) which acts freely on the sphere Sn. It is such a manifold M that we call a S p h e r i c a l Form; by abuse of language, any representative of the isometry class of Sn/G will also be called a Spherical Form. Furthermore S', which is homeomorphic to S 1 / G I will also be considered as a Spherical Form. The classical properties of group actions give the following result. 1.1. E v e r y S p h e r i c a l Form Mn = Sn/G i s a c o n n e c t e d , compact, o r i e n t a b l e m a n i f o l d w i t h o u t boundary, w i t h a n CW- s t r u c t i r r e and s u c h t h a t n l (M) = G and ni(M) Z ni(S ) i f i * l . The classification of Spherical Forms can be reduced to the determination of all finite groups having fixed point free real orthogonal representations and therefore, of all those groups admitting fixed point free unitary representations. (Our reference for all the facts concerning representation theory of finite groups is 1671). This work, done first in dimension 3 by Seifert and Threlfall [211], was continued by J.A,Wolf [2591 after the decisive contribution of G.Vincer [2471. The results obtained may be stated as follows.

Proposition

Introduction

Proposition

1.2.

Let

l i n e a r l y and f r e e l y o n

If N

1.

=

2n, G

G

be a n o n - t r i v i a l

SN

.

i s isomorphic t o

141

f i n i t e group a c t i n g

Z2 and t h e o n l y e v e n - d i -

.

m e n s i o n a l s p h e r i c a l f o r m s a r e t h e r e a l p r o j e c t i v e s p a c e s IRP 2n

N = 2n + 1 t h e f o l l o w i n g h o l d : (i) i f G i s abelian, then G i s cyclic; (ii) i f G i s n o n - a b e l i a n , i t h a s t h e f o l l o w i n g e q u i v a -

2.

I f

lent properties:

(a) a l l t h e a b e l i a n s u b g r o u p s of G a r e - c y c l i c ; (b) t h e p - S y l o w s u b g r o u p s of G a r e o f one o f t h e f o l l o w in g two t y p e s :

(I) a l l o f t h e m a r e c y c l i c ; (11) t h e y a r e c y c l i c f o r p 2 and g e n e r a l i z e d q u a t e r n i o n i c g r o u p s f o r p = 2. The spherical forms corresponding to 2(i) are called L e n s S p a c e s ; if G is a generalized quaternionic the associated spherical form is called Q u a t e r n i o n i c group Q, spherical form. S p h e r i c a Z Form or, more simply, Q,-

*

Let us mention that the complete classification of the groups satisfying 2(ii)(a) results from the papers of Zassenhaus 12701 and Suzuki [2351 : there are six classes of groups among which two are constituted by non-solvable groups (see [260; pages 179 and 1951 1 . Let M be a spherical form viewed as S"/G where G acts freely on Sn by way of an orthogonal fixed point free be a p - Sylow subgroup of G : representation p . Let G P the canonical inclusion i : G G induces a ring homomorP P phism i* : R O ( G ) + R O ( G ) and defines a fixed-point free P P orthogonal representation = i*(p) of G by way of which PI OP P G acts freely on Sn. Let M = S /GP be the quotient maniP P fold thus obtained. If G I is another p-Sylow subgroup of P G , the spherical form MI associated to G I is isometric to P P M indeed, since G and G I are conjugated in G , the P ' P P and p ' are equivalent. This justifies representations P pP us to call M the S p h e r i c a l p - f o r m a s s o c i a t e d t o M. +

-

P

142

Span of Spherical Forms

Notice that because of Proposition 1 . 2 , the spherical p - forms associated to a given spherical form are real projective spaces, lens spaces of order pm or Qm - spherical forms. We finally n : S ” / G ~= M + S”/G = M is a covering finote that P P bration; this is one of the interesting facts about associated spherical p - forms. Lemma 1.3. L e t M b e a n n - d i m e n s i o n a l s p h e r i c a l f o r m w i t h nl(M) = G T h e n t h e i n t e g r a Z fordinaryl cohomoZogy g r o u p s of M are: ( Z if i = O or i = n ;

.

I

Hi(M;e)

=

J Hi(G;e) , if

0
0

,

isomorphism

p - Sylow subgroup G of P t h e i n c l u s i o n G -D G i n d u c e s a n P >

H1(G;Z)

i H (GpiZ)

r

i s t h e p - p r i m a r y c o m p o n e n t of Hi(G,Z). (PI Proof. Let r (resp. s ) be the canonical inclusion of N (resp. GP) into G . Let us consider the Hochschild-Serre spectral sequence

where

Hi(G;Z)

=

fIi(c/~; ~ I J ( N , z ) )

"i+j H (G;z)

which relates.the Artin-Tate cohomology of a group to those of one of its normal subgroups. Since (IN1 , p ) = 1 are led to a split exact sequence [127; page 1271 0 ->

^i H ( G ;Z) P

ti

->

Ai r H (G;Z) > hi(N;E)

, we

-> o

for any i E Z . In the preceeding sequence, ti and ri are respectively, the natural transfer and restriction homomorphisms (the latter induced by r) , fIi(N;E)G is the subgroup of defined by the elements which are invariant under the (N;Z) action of G Let ui (resp. si) be the section of ti (resp. the homomorphism induced by s ) ; the definition of the transfer homomorphism ti (cf. [62;page 2551) shows that p i = t.s is multiplication by [G : G ] = IN1 in hi(G;Z). i i P It follows that p i is onto; moreover, by killing the IN1 primary component of hi(G;Z) we obtain an automorphism of "i H (G;Z) Since

ii

.

.

si

=

it is clear that isomorphism. Theorem with G group

N

(U.t.)Si = u.p. 1 1 1 1 si

is onto and that

[169] L e t

1.6.

a semi-direct

M = S 2n+ 1 /G

p r o d u c t of a

"i silH (G;Z) (PI

b e a s p h e r i c a l form

p-group

.

S 2n+l /GP

of

M

and of u

G P

whose o r d e r s a r e r e l a t i v e l y p r i m e . L e t

c o v e r i n g space

is an

M

P

be t h e

i n d u c e d b y t h e i n c l u s i o n of

Span of Spherical Forms

144

N

i n G . T h e n , KF( Mp) is a d i r e c t summand of P f w h e r e 7F = IR o r C ) More p r e c i s e Z y ,

G

KF (M)

.

N

N

is i s o m o r p h i c t o

(i) KU(M)

KU(Mp)

:

.

Proof. Let TI be the covering map M + M The P are onhomomorphisms Ti : Ei(M;Z) -P Fi(Mp;Z) induced by IT to for i 2n + 1 this follows from 1.3 and 1 . 5 , and for i = 2n + 1 , because n is orientation preserving. Since Z the functor (resp. Torl) is right exact (resp. additive), it follows that for every integer i the homomorphisms induced by

*

in the .Atiyah-Hirzebruch spectral sequence are onto. More precisely,except for the case in which IF =IR, p t 2 , i = 2n + '1 and n 0 (mod.4), the restriction of y i to the p - primary of Ei'-i(M) is an isomorphism onto component (E; ,-i (PI H from the abelian Ei'-i(M ) The additivity of the functor ' P category of complexes of abelian groups into the category of abelian groups, shows that the same is still true for "i : E:r-i (MI -* Er i,-i (Mp) for any r > 2 ; hence the result. *r 2 and n = 0 (mod.4) , the term When IF =IR, p 2n+lI-2n-1(Mp) is isomorphic to Z2 and s o , the preceeding E2 argument is still valid if M is replaced by its 2n- skele-

.

ton

(MpIo

.

P

Remark. The proof of the previous Theorem does not require any particular assumption on the Atiyah-Hirzebruch spectral sequences. Besides, it does not depend on the (free) action of G on S2n+' ; indeed, it still holds for any manifold having

Introduction

145

the same homotopy type as a spherical form. Applying the Theorem to the lens spaces (a > I ) , we obtain immediately (compare with 1166]), Corollary

Let

1.7.

a

s r fl pi i i=l

=

be t h e p r i m e d e c o m p o s i t i o n o f

2. (i)

n

if

+

0

a

,

with

.

Then,

2

5

p,

S

...
0 )

e= 1

e= 2

e=3

8c+2 8c+2 8~

8c+4 8c+4 8c+4

8c+8 8c+6 8c+5

v2(m)

=

141

Vector Fields on Spherical Forms

The previous table will enable us to obtain some interesting upper and lower bounds of the span of some spherical forms. We begin by observing that for any N-dimensional spherical form M , we have N (2.1) span(M) 5 SpanlR ( S ) = p (N+lrR) -1

.

Next, notice that the methods used to determine spanlR(Sn ) prove that a system of p (N+1r I R 1 - 1 vector fields on SN remains invariant under the antipodal action. It follows that

for any

N

.

Let us now suppose that : and let us consider the rotation y S2n+1 = {(z o,...,zn) E c n+l I 1 lZk k=o

where 5 that a

is odd, that is on the sphere = 11

1

,

N

=

2n + 1,

defined by

exp(2ni/a), with a,aoral,... ,an E N* and such 1 and for every k E {O,lr...,n) r (a,,a) = 1. i Under these conditions, the group r = { y }l 2 ) and, writing its

.

with elements as xky' classes of Q, are

0< k < 2m

-

1,

II = 0,1,

the conjugacy

2 Since the commutator group [QmIQm] is isomorphic to (1,x 1 , it follows that Q, has 4 irreducible unitary representaand c 3 defined by tions of degree 1 : So(x) = 1

;

5,

(XI

= 1

: [,(x)

= -1

-

,

S3(X)

= -1

:

The number of conjugacy classes of Q, shows that Q, has 2m- 1 - 1 other irreducible unitary representations (all of degree 2 ) , given by

Span of Spherical Forms

150

where 5 = e~p(in/2~-’) and 1 < r < 2m-1. The construction of their character table shows that the representations quoted before are not equivalent; furthermore, it enables us to obtain the multiplicative structure of R U ( Q m ) by means of the tensor product of characters. Proposition

2.5.

As a group,

(cf 11961)

abelian group generated by

(5,)

is t h e f r e e I t s muZtipZicative

RU(Q,)

o 0), the action being given by way of a the spheres S virtual representation p of the type

Q,

S

P =

1

i=1

nipi

l< i -< s . Definition

2.6.

Let

ro,...,r

be

is t h e G e n e r a l i z e d

Q,-

n + I

* ,m- 1 - 1 1 .

essariZy d i s t i n c t ) o f {l,3151...,

S p h e r i c a l Form

i n t e g e r s (not necn T h e n , N (m;ro,rl...,rn)

S4n+3/Qm

n a c t s b y way of

=

1 5ri+3. .When

i=o

aZZ t h e

where

Qm

ri’s a r e e q u a l

Vector Fields on Spherical Forms

151

t o a same number r , we s h a l l w r i t e N"(m;r) f o r N n (m;r,r,...,r) ; f i n a l l y , i f r = 1, t h e f o r m Nn(m;l) w i l l b e d e n o t e d b y Nn(m) and c a l l e d O r d i n a r y Q m - S p h e r i c a l E'orm. If we consider the unit sphere S3 as the (multiplicative) group of quaternions with norm 1 that is to say, 3 S 3 = Sp( 1 ) , one can identify Qm with the subgroup of S generated by 5 = exp(in /2m-1) and j , by the correspondence x + 5 , y + j . With these assumptions, the ordinary Q m - spherical form Nn(m) is no other than the form which corresponds to the restriction to Qm of the standard representation of s3 in 7R4n+4 .-*n+1 . When N 3 (mod.4), the quaternionic vector fields on SN (referred to in 2.0) are invariant under such an action; we conclude that I

(2.7)

span(Nn(m)) >

P

(4n+4,H) - 1

.

The reader will notice that the span of Nn(m) coincides with span7R (S4n+3) when v2 (n+l) 0 (mod.4) (use table (2.0)). For convenience, we summarize (2.2) and the inequalities (2.11, (2.4) and (2.7) in the following Proposition. Proposition 2.8. (i) For a n y N - d i m e n s i o n a l s p h e r i c a l f o r m M , span(M) < - spanR(SN ). In p a r t i c u l a r , span(RPN ) = spanR(SN 1 . (ii) M o r e o v e r , span(Ln(a)) > p(2n+2,~) - 1 span(Nn(m)

2

p

(4n+4,H) - 1

r

.

The above proposition and the arguments which preceeded it show how the determination of the span of an arbitrary spherical form depends on the kind of its associated spherical 2 - forms. For this reason, we shall classify the spherical forms according to the following terminology. Definition

2.9.

A S p h e r i c a l Form i s s a i d

t o b e of

cyclic type

f r e s p . of q u a t e r n i o n i c t y p e ) i f i t s a s s o c i a t e d s p h e r i c a l form h a s a c y c l i c g r o u p q u a t e r n i o n i c group

Remark 2.10. (cf. [26] or

Qm

2 -

Z ( m >0 ) f r e s p . a g e n e r a l i z e d 2" (m > 2))a s i t s d e f i n i n g g r o u p .

The reader who is familiar with Clifford Algebras [134]) will easily deduce the preceeding results

Span of Spherical Forms

152

from the fact that if IRN+l is a

Ck-l 0 IF -module then

SN

admits k - 1 linearly independent vector fields which are S d ' l - equivariant under the standard action of Sd- 1 on FN+l'd mN+l (see beginning of 9 2 ) . In this context,denoting by ak QF) the smallest integer n such that lRdn is a Ck-, @ IF-module, the integer p(mJ!?) appears as the greatest integer k such that m E 0

(mod. d.akIE'))

.

The reader can find more observations on Clifford Algebras in the previous chapter.

3.

G - Fibre Homotopy

J-Equivalence.

Two vector bundles 5 = (E,p,X) and 5 ' = (E',p',X) over a finite connected C W - complex X are said to be J - equivalent, if the associated sphere bundles S ( S ) = ( S ( E ) ,ps,X) and S ( 5 ' ) = (S(E'),p$,X) are of the same stable fibre homotopy type (see chapter 3 , § 3 ) . Let us recall the following result due to Dold [74]. Suppose that we can find a fibrewise map f : S ( E ) + S(E') such that, for each x E X the map fx : S(Ex) + S(Ei) induced on the fibres is a homotopy equivalence; then, f is a fibre homotopy equivalence. If the sphere bundles S ( 5 ) and S ( 5 ' ) have the same fibre homotopy type, then the Thom spaces T ( 5 ) and T(5') are homotopy equivalent; in the stable range this assertion admits a partial converse (see [121, pages 2 9 8 - 2 9 9 ) . Let

G be a finite group (with discrete topology) and let 5 = (E,p,X) be a G - vector bundle such that X is a finite, path-connected C W - complex and G acts trivially on X. We also suppose that has a riemannian metric such that the G - action restricted to every fibre is orthogonal: with these conditions G will act (always by restriction) orthogonally on each fibre of the associated sphere bundle S ( 5 ) = ( S ( E ) ,p,X). In this context, if dimIF 5 = n and G acts freely on its fibres IFn , we can associate to 5 the spheriSdn- 1 / G , where d = 1 , 2 , or 4 , according to cal form

G-Fibre Homotopy J - Equivalence

153

F =IR,C or M , respectively. These considerations motivate the following terminology.

Definition 3.1. T w o G - v e c t o r b u n d l e s 5 = (E,p,X) and 5 ' = (E',p',X) a r e G - f i b r e homotopy J - e q u i v a Z e n t i f , and o n l y i f , t h e r e i s a G - f i b r e homotopy e q u i v a l e n c e b e t w e e n t h e a s s o c i a t e d s p h e r e b u n d l e s S ( 5 ) and S ( [ ' ) . By this we mean the following: (i)

there exist maps

S

E)

KG(D(E) r S(E) 1

Span of Spherical Forms

154

Using the Thom isomorphisms KG(X) KG(D(E) , S ( E ) ) and KG(D(E') , S ( E ' ) ) , we see that there exists an element KG(X) u E KG(X) such that

-

Q*(P(S')) = u

(3.3)

-

P(51

I

where ~ ( 5 ) is the Thom class associated with tion of the Bott classes now shows that

okw -

J,

k

(u) = u

-

5

.

The defini-

ekw) .

Let JI' be an inverse G - fibre homotopy equivalence of then, as before we have that

-

Q ' * ( P ( 5 ) ) = u'

(3.4)

-

J, :

ll(5')

-

-

T'*i: T* and Q* e for some element u' E KG(X). Since are the identities, (3.3) and ( 3 . 4 ) show that uu' = 1 = U'U in KG(X). After this technical result we are going to consider relations between the G - fibre homotopy and G -fibre the P homotopy J - equivalences, where G is a p - Sylow subgroup P of G Since the problems we shall examine turn around the possibibity of extending these homotopy J - equivalences, the machinery used to measure the possible obstructions to such extensions will be obviously based on the manipulation of the homotopy groups of appropriated spaces. We shall begin by defining some objects which will prove to be very useful. Let X and Y be two G - spaces and let MG(X,Y) The be the space of all G - equivariant maps from X to Y inclusion G c G defines clearly a map from MG(X,Y) into P MG (X,Y) which, for a fixed f E MG(X,Y), induces the homoP morphisms $ I *

.

.

l r

:

rr(MG(X,Y),f)

->

Tr(MG (X,Y) , f )

.

P Let ( I ~ be ) ~the restriction of lr to the p - primary component nr(MG(,Y) ,f) of nr(MG(X,Y),f). Moreover, let A be the subspace of X which remains invariant under the action of G : denote by MG(X,A;Y;f) the subspace ~hEMG(X,Y)lh]A=fl

G-Fibre Homotopy

Lemma

Let

3.5.

X

and

J - Equivalence

be

Y

155

G - s p a c e s such t h a t :

(i) X is a f i n i t e C W - complecc o n u h i c h G a c t s f r e e Z y and c e 1 lu l a r Zy; (ii) Y is s i m p l y c o n n e c t e d ; (iii) t h e a c t i o n of G o n X ( r e s p . Y) i s s u c h t h a t f o r X f r e s p . Y) d e f i n e d b y e v e r y q E G , t h e a u t o m o r p h i s m of x gx ( r e s p . q -+ gyl is h o m o t o p i c t o l X I r e s p . ly) ; -+

(iv)

*;

the inclusion onto

(G;Z)

G

i n d u c e s an i s o m o r p h i s m f r o m

c G

,.P H*(G ; e ) P

.

(PI T h e n , for a f i x e d G - e q u i v a r i a n t map r > 0 , t h e homomorphism ( 1 r1 P :

f

:

X

and f o r e v e r y

Y

+

nr(MG(X,Y) ,f) (P) -> n r (MG (X,Y)r f )

P

is a n i s o m o r p h i s m . Proof. We shall only sketch it. Let fs-l be the restriction of f to the ( s - 1) - skeleton Xs-l of X ; the homotopy exact sequence of the fibration

gives rise to an exact couple

E~~~ .I

=

[171]

in which

~ , ( M ~ ( X ~ ,; X~ ~; -f ~~ - ~. ) )

By the exponential law, there is a bijection between Errs and a certain subset of the set of the homotopy classes of maps in MG(Xs,M(Sr,Y)). Then, by (ii) and (iii), we can exhibit an isomorphism between E r r s and HS(Xs/G,Xs_l/G:~r+s(Y)) that the E 2 term of the spectral sequence is (3.6)

2 HS(X/G;nr+s(Y))

E:,’

Considering also the isomorphism ‘rr

will see that :

I*

,

so

.

(3.6)

for

G

P I

the reader

may be viewed as the homomorphism

H’(X/G;~~+~(Y)I ->

HS ( x / G ~ ; T ~ + ~ ( Y ) )

r induced by the canonical projection (iii) for

X

implies that

G

X/Gp

-+

X/G

. As

acts trivially on

condition

71r+s(Y)

,

Span of Spherical Forms

156

the Universal Coefficient Theorem and (iv) show that the reis an isomorphism; to 'H ( x / G ; ~(Y) ~+~ striction of :I (PI then the result holds for ir. We shall suppose, f r o m now t o t h e end of t h i s s e c t i o n , that the groups G and G satisfy condition (iv) of lemma

P

3.5.

Let us consider two n-dimensional G-vector bundles t and t' with base X , such as described at the beginning of the section. We also suppose that G acts freely on their fibres (which are isomorphic to IFn) by equivalent actions. Hence, we can consider the fibres as n-dimensional G-modules, which will be denoted simply by L Finally, if s is the cellular dimension of x , let es be an s - cell of X and set Xs = X - int es

.

.

Lemma

Let

3.7.

$

be a

G

-

f i b r e homotopy

J - equivalence

CIXs and < ' ] X s ( s -> 1 ) . If 4 i s G - f i b r e homoP t o p i c t o J , l X s where J, i s a G - f i b r e homotopy J - e q u i v a P a Zence b e t w e e n 5 and 5 ' , t h e n we c a n f i n d an i n t e g e r

between

-

p and a G - f i b r e homotopy J - e q u i v a l e n c e JI b e at and a t ' , s u c h t h a t J, is an e x t e n s i o n o f 9 (a)

prime t o

.

tween

Furthermore, Lemma

3.8.

between to J , l X s tween g

p and

and a

aS'

5

Let 4 and 5 '

.

be a I f

G - f i b r e hornotopy J - e q u i v a l e n c e $ /PS X ( s > 1) i s G - f i b r e homotopic -

P

where J, i s a G - f i b r e homotopy J - e q u i v a l e n c e b e and t ' , t h e n we c a n f i n d an i n t e g e r a p r i m e t o G - f i b r e homotopy such t h a t

J,

is

-

J - e q u i v a l e n c e $ b e t w e e n ag G - f i b r e h o m o t o p i c t o 4 (a)

P

These two Lemmas, as well as Lemma 3.5 from which 1 4 0 ; pages 1 0 1 8 - 1 0 2 4 1 . they derive, are due to J.C.Becker Since the proofs of ( 3 . 7 ) and (3.8) are obtained following the same kind of considerations, we shall only establish the proof for the first of these two results. of ( 3 . 7 ) . Clearly there are equivalences of G-bundles f : (aes) x S(L) + S(ElaeS) and f' : S(E'laeS) -, (aes) x S(L) ; the G - fibre homotopy equivalence Proof

G - Fibre Homotopy

S

+ / a e : S(ElaeS)

J - Equivalence

->

157

S ( E * jaes)

N

and let

I$ be the composition

where the last map is the canonical projection. N O W , since aes ss-1 , we associate to + an element h(+) in TI (MG(S(L) ,S(L)) (with base point l S (L)) , via the expos- 1 nential law. Obviously, + can be extended to a G - fibre homotopy J-equivalence between 5 and 5 ' if, and only if, h($) = 0 But, by assumption, @I being G - fibre homotopic P to $lXs, the homomorphism N

.

~ ~ - ~ ( h ( +=) ) 0 . Thus, for s > 1, the injectivity of (is-,)p (Lemma 3.5) shows that there is an integer a prime to p such that ah(+) = 0, that is to say, +(a) has an extension $ from a< to ag'. If s = 1 , the result is trivial, for l o is itself injective.

yelds

Proposition

Let

3.9.

valence betueen

6

+

and

be a G - f i b r e homotopy J - e q u i P 5 ' . We a s s u m e t h a t + , u p t o G -

homotopy e q u i v a l e n c e , i n d u c e s maps of d e g r e e o n e o n Then, t h e r e a r e an i n t e g e r homotopy

-

J-equivalence

-

a

$

prime t o

between

ac

p , and a and aE,'

S(L)

.

G-fibre such t h a t

+(a). In this case, the proof is obtained by induction on the cellular dimension s of X using Lemmas 3.7 and 3 . 8 . We need only to see that the assertion is true when s = 0 : this is guarantedbythe assumption made on the degree of t$ Indeed, it means that we can find G-homotopy equivalences $

is

G

P

-fibre

homotopic t o

.

S(L)

->

'f


such t h a t

rSlS';a;f-4'IG

F u r t h e r m o r e , w e s h a l l assume t h a t

,

a c t s c e l l u l a r l y and ( i . e . , Hi ( A ; Z ) = 0 for G

CW- complex

on which

i > 0)

If

CW- complex on which

i s an a c y c l i c

A

.

: [5rS';a;4'lG

is a f i n i t e

every

5 , and

is a n o t h e r G - b u n d l e morphism, l e t

b e t h e map i n d u c e d by X

G-bundles with base

G - s u b - b u n d l e of

G - b u n d l e morphism

IS,S';a;$'lG

f ,

a

be

G - b u n d l e morphism; w e s h a l l d e n o t e t h e s e t of

a

homotopy c l a s s e s of

$15'

a

and

acts freely.

G

is ca2Zed a G - p a i r i f t h e r e i s a n a c t i o n o f G on Y Zeaving 2 i n v a r i a n t . I f G a c t s f r e e l y on Y ( r e s p . Y-Z) t h e n

Definition

(YrZ)

A pair of

3.11.

topological spaces

i s s a i d t o be a f r e e

Definition

3.12.

nected f o r

G

i s s a i d t o be

(Y,Z)

n

i

,

Z XA/G;II)

and e v e r y

let

NOW,

Sn

E,

* 5

t h e j o i n of e a c h f i b r e of

Sn. Note t h a t

with

o*n :

z E Sn

* sn

is a

+

tgz

t E [0,1]

and

. S

be t h e map g i v e n by Proposition

be a

3.14.

G - sub-bundle

a:(+) = $

Let of

5,a

6

I

[ ~ * s ~ ~ s ' * s ~ ; ~ * s ~ ; ~ *Gs ~ ; ~ '

-->

[ ~ r c '

5

i s g i v e n by

G

q ( ( l - t ) e + t z ) = (1-t)ge g E GI e E E,

.

Il

n - s p h e r e on which G i s supSn t h e b u n d l e o b t a i n e d by t a k i n g

G - b u n d l e f o r which t h e a c t i o n of

Let (3.13)

n-cocon-

= 0

G-group

be an

p o s e d t o a c t ; d e n o t e by

for every

G-pair.

i f , a n d onZy i f ,

H1 (Y X A / G

f o r every

(resp. reZativeZy f r e e )

G-pair

A

(Y,Z)

*

1 Sn

f o r every

be g i v e n

+ E tSIS';a;$'lG-

G - b u n d l e s and l e t

; we assume t h a t

(E,E')

5'

i s a reZa-

t i v e z y f r e e G - p a i r , t h a t t h e G - a c t i o n s a r e ceZZuZar and (r-1) - c o n n e c t e d . T h e n , t h u t the f i b r e type of a i s

G - F i b r e Homotopy

i f

(i)

159

J-Equivalence

- coconnected

(E,E')

is

(2r-1)

(E,E')

is

2r-coconnected

for

n

G,

i n in-

U*

jective; i f

(ii)

n G I o*

for

i s surjec-

tive. Proof.

a

of

x

over

b : Dn+'

+

s i o n of

Sn

Ax

BX

and l e t

*

Sn

*

Ax

q ( b ) = x , is a G - bundle.

the G

- action

every

z E Sn

G

a c t s on

B

Sn

and by

5

(g,b)

f : a

-+

5

p

+

Sn; t h e n , @ = ( B =

s p a c e of E

x

B,,q,X)

I n f a c t , w e r a d i a l l y extend

1 , we set

g

-

-

pz = p

g z ) ; then

L e t u s consider t h e

gbg-I.

g E GI

G - bun-

d e f i n e d by

@

f(a)(pz) = (l-p)a + f o r every

xE

( i n o t h e r words, f o r e v er y

aDn+'

0

is the natural inclu-

bl aDn+'

with

on

be t h e f i b r e

Ax

b e t h e s p a c e of t h e maps

such t h a t

into the join

d l e morphism

x E X, let

F o r any g i v e n

PZ

I

a E E a t z E Sn and 0 5 p 2 1 (Ea i s t h e t o t a l a). Then w e c o n s t r u c t t h e isomorphism

: [ S , S ' ; @ ; f . + ' l G ->

e E E l z E Sn

by s e t t i n g , f o r e v e r y E(+)

B e c a u s e of

and

( ( 1 - t ) e + t z ) = + ( e )( t z ) and

(3.10)

S

I

t E [0,1]

,

[S*S~,5'*S~;arS";+'*1

(3.13)

.

w e have t h a t

E

D

f, =

0:.

The reader s h o u l d n o t i c e t h a t t h e c o n c l u s i o n of t h e P r o p o s i t i o n c a n be now o b t a i n e d f r o m t h e f o l l o w i n g Lemma.

Lemma

3.15.

G - sub-bundle

assume t h a t



... ,vk) =vk.Next,

Vk(Ln (+> Lm)

.

m (Here Vk(Ln) * Vk(Lm) is the j o i n of Vk(Ln) and Vk(L ) already defined in Chapter 2, 9 2.) Given (u,v,t) E Vk(Ln)+ Vk(Lm), t E [ 0 , 1 ] , we define hk (u,v,~)= w = (~,r.-.rW~)r

162

Span of Spherical Forms

1 where w i = (ui cos a , v 1 . sin a ) , with a = nt (the reader 2 is referred to [138; pages 513 and seq.], for a study of the properties of the intrinsic map). One can verify that the map

ek given by

P ~ ( L ~ ( ~ ; ---> R ~ ) V~( L" @R~ )

:

.

= (v,,v2,. .,v,l

€!,[XI

1 < i < k , defines a morphism of the same way)

with

x

E

S(Ln@

i -< 2(n+m-k-l) Definition

Dk

such t h a t

The p a i r

4.4.

+

are isomorphisms for

(

(L" 6.nm ,pk mm))+ ( a;

and

=

is G - P B - morphism

Pk(Ln $ Wm ) ,PkWm))

G - coreduciblel i f there i s a

: ( pk

ak

6.IRm ) )

(see [213]).

ducible (resp.

resp.

ek

induced by

))

and

space pairs (denoted in

G -

It can be shown that the homomorphisms ni( Pk(Ln ni(Vk (Ln @ lRm

Rm)

aklSWm)

G

s (L" 9 IR m-k+l

(Fesp.

8,

and

81;=DklPk(~m:

are reductions (resp. coreductionsl.

Under certain conditions on the G - action, and on n and m I the reducibility of a; is sufficient to induce the reducibility of ak More precisely,

.

Lemma with

4.5.

Let

n

t

0

'

(mod. akW))(*), m

E

0

(mod. a k W ) )

m > k, and l e t u s s u p p o s e t h a t G a c t s simply on t h e pk(Ln @ IRm ) ) ( + IT.h e n , if t h e r e i s a G-map

hornotopy g r o u p s

1

(*)

See Remark

(+)

This means that ple

.

2.10

for the definition of n i (Pk(Ln9 IR" )

)

as a

a,@?). G

- module

is sim-

G-

s u c h that

is

G

a i

(Co)

Reducibility

163

(P~(L" c+, I R )~ ,pkmm))

is a reduction, t h e pair

- reducible.

(S(Ln @ I R m ) ,S(IR")) defined by f = (S(Ln IRm) ,S(Rm)) = (V,(L" 6 IRm) ,V1(Rm) O1 Xk ak. By (3.101, this composition leads to the map Proof.

-

(1

e

Let

f

be the map

-

S(L")

*

f')!

:

[s(L"C+.IRm),SWm); S(L"@IRrn)

[ s L" (

which, according to (3.13)), the map

(+; IRm

,s Wrn) :s ( L" E

f$

-

1 S aRm, IG

:

lRm )

+

']G

(3.15), is an isomorphism. Besides (see

being also bijective, it follows that is an isomorphism. Since, for every [I$] E [S(Ln);S(Ln)IG

(lS(Ln)*

;

o:-"@l

,

(1 s (L") * f')!

=

we have that

1

[I$*ls(Rm)

= [$

*

f'],

we see that it is possible to find a G - map I$ of S(Ln) into itself (unique up to G - homotopy) such that f is G - homotopic to $ * f'. NOW, since a i is a reduction, we obtain that deg(@+f') = deg($) deg(f') = I deg($), and consequently, deg(f) = f deg(@) = a [GI f 1 , with a E Z . If a = 0 , it follows easily that a k is a reduction. Otherwise, by modifying a k slightly, we can find a G-morphism

-

-a k

:

(S

(L"

+, IR"

-

, s OR^) )

( p k (L" 3 I R )~,pk

--> N

.

which is a reduction (take a; = a i ) Since n and m 3 0 (mod. a,@?)), there is an element u E nn+m-l

(Pk(Ln 9lRm))

We need only to choose

N

ak

such that

so that

(0,

*

=

mrn)1 0

(mod. akaR)

X k ) , (u) =

Span of Spherical Forms

164

ak

(i) the restrictions of skeleton are equal, N

(ii)

by setting

€ = 0

1

-

Ak

and

-

N

ak

N

ak

,

to the

(n+m-2)

-

the difference cohomo-

logy class [?I - t f ~ in Hn+m-1 ( S (L") /G n n+m- 1 (pk (L" 0 nm 1 ) is precisely -au.

* s mm),s mm);

Indeed, these conditions guarantee that deg ('f) = deg (f) aIGI = f 1 ; the argument is standard. A s far as the relation existing between the span of the spherical form S(Ln)/G and the G - reducibility of a pair (Pk(Ln,,+. Rm ,Pk(Rm 1 ) is concerned, an answer is given by the following lemma, Lemma 4 . 6 . Let m > k a n d m = 0 (mod.ak(R)) If span(S(Ln)/G) k - 1, then (Pk(LnG Rm),Pk(Rm)) is G - r e d u c i b t e . If n > - 2k, the converse is true. Proof. A s m = 0 (mod.ak(R)), the results obtained for the spheres imply immediately the existence of a reduction ai : SQRm) + PkWm). To establish the G - reducibility of

.

N

(

pk(Ln @ IRm ,PkWm)) , we must extend

-

z/

S(Ln IRm) + S D " ) so that a Let us notice that the map

a

k -

induced by 8 i need to extend

(4.3) 0 i

w

to a

a i

G - morphism

is also a reduction.

,

is an isomorphism ( 3 . 1 5 ) . S o , we only a i into a G - morphism between S (Ln 3 nrn)

N

c

and Vk(LnF I R m ) . Since span(S(Ln)/G) 2 k - 1, it is obvious that the projection pk : Vk(L n) + S(Ln) has a G - equivariant S(Ln) + Vk(L n 1 . NOW, the extension wanted cross-section s k *

is no other than the composition given by the following diagram S(L"

TI

IRm

z. S(L") * s W m )

Vk(L")

Vk(L"li

>

*

v,m

m

1

.

nm)

G - ( C o ) Reducibility

165

To prove the converse, that is to say, that the G - reducibility of (Pk(Ln G IRm ) ,PkWm)) implies the existence of a G equivariant cross-section sk , the reader needs only to exhibit a G - map sk : S(Ln) + Vk(Ln) such that pk G sk is - 2k beG-homotopic to the identity. The assumption that n > N

N

comes then necessary to obtain Proposition

N

sk

(use (3.14) (ii) and (3,15)).

n - dimensional G - m o d u l e s on which G a c t s f r e e l y . Then, i f n 2k, > k - 1. span(S(Ly)/G) 2 k - 1 i s e q u i v a l e n t to span(S(L:)/G) 4.7.

Proof.

Let

and

Suppose that

b e two

span(S(Ly)/G) 2 k

-

1; to get

the same lower bound for the span of the other spherical form S(L;)/G , the two Lemmas just given show that we need only to find a G - m a p

such that a i is a reduction. By assumption, we know that there is a G - reduction

We are now going to build G-maps y and 6 so that 6' a; y' is a reduction; then the following diagram will define a 2 :

-

-

a2

I

I

3

3 k'(

a'

Ly 17 IRm

,PkWm))

with Y defined as the join of a G - equivariant map S(L;) + S(L7) with 1 Likewise, to obtain 6 , we choose a G s mml n equivariant map S ( L 7 ) + S ( L 2 ) , which commutes with the antipodal maps (such a map always exists), and take its join with 1 before passing to the quotient; we shall have S Wm)

.

166

Span of Spherical Forms

6' = 1

PkaR")

*

Remark 4.8. Invokingthe table given at the beginning of section 2 , the reader will notice that if n if n F = C

(i)

(ii)

1,2,4,8,16, then n 2 2 - ~ ( n IR) , , + 1,2,4,8, then n -> 2. p f n , IF) , with or M .

$

Let us examine now the relations between G - coreducibility, G - reducibility and G - fibre homotopy type. Let us recall that we have already indicated that the G - fibre homotopy J - equivalence between two G - vector bundles 5 and F,' induces a G - homotopy equivalence between their Thorn spaces T(c) and T(S'). In this context, if we denote by the canonical line bundle over the (k-1) -dimensional projective space, we have:

qk

Lemma 4.9. S u p p o s e t h a t n > k and m > k, w i t h m I k (mod.ak ( R ) ) . T h e n q k @ Ln and Ln a r e G - f i b r e h o m o t o p i caZZy J - e q u i v a Z e n t if, and o n l y i f , t h e p a i r (Pk(Ln 6 IRm) , Pk ORm) ) i s G - c o r e d u c i b Z e . Proof. The asumption that there is a coreduction

a;,

N

:

p,aRml

m

-,

I

k

(mod.akW))

SWm-k+l

means

I *

Then, we must characterize Bk so that [ B,] E Pk(Ln ? IRm , m -k+l ) , Z1;lG Because of the well-known identiP p m ) ;S(L"@ lR fication

[

.

(Pk(Ln@

lRm) ,PkaRm))

=

(T(qk

@

(Ln @ lRm-k) ) , T ( n k '31Rm-k))

page 3041 the assertion will follow quickly by using on one hand, the construction of the Thorn space, and on the other hand, (3.13) and (3.16). The duality between reducibility and coreducibility can be expressed in the following way [12;

Lemma k, m

4.10. R

I

0

Let

k,II and m frnod.akW)) and

be i n t e g e r s such t h a t

m = k

(mod.ak@)).

II > 2k,

Then,

the

G - ( C O ) Reducibility

167

(Pk(Ln 0 IR '1 , P k WII ) 1 i s G - r e d u c i b l e i f , and o n l y i f , (Pk(Ln@ lRm ,Pk ORm)) i s G - c o r e d u c i b l e

pair

As a consequence we have the following. Proposition 4.11. If span(S(Ln)/G) = k - 1 t h e n qk C 3 Ln a r e G - f i b r e homotopicaZ1y J - e q u i v a l e n t ; on t h e o t h e r hand, if tlk C 3 Ln and Ln a r e G - f i b r e h o m o t o p i c a l l y J - e q u i v a l e n t a n d n 1. 2k , t h e n span(S(Ln)/G 2 k - 1. This proposition, given by Becker as a generalization of Atiyah's work [12; 9 61, is an immediate consequence of Lemmas 4.6, 4.9 and 4.10. Finally, we wish to notice that, while Proposition 4.1 shows that the spans of two spherical forms with the same dimension and same defining group have the same lower bound (which is independent of the free representation selected), the next Proposition proves that a similar result holds for some spherical forms and their associated spherical 2 - forms. More precisely: Proposition let

n

1. 2k

4.12.

Let

G2

be a

2-Sy20w

s u b g r o u p of

G

and

; suppose t h a t t h e canonical i n c l u s i o n

G2 + G i n i H (G;Z)(2) and T h e n , span(S(Ln)/G) 2 k - 1 i f , k - 1.

duces an isomorphism between t h e groups

Hi(G2;E) , for e v e r y i 2 0 . and o n l y i f , span(S(Ln)/G2) 2

Proof. The only implication that we have to consider is, of course, the one for which we assume that span(S(Ln)/G2), k - 1 . Since n 2k, according to Proposition 4.1 1 , 'lk % Ln is G2 - fibre homotopy J - equivalent to Ln ; then, because of (3.9) , there is an odd integer a such that arlk 8 Ln is G - fibre homotopy J - equivalent to aLn. Let i 0 , (IRPk-') be the Grothendieck group associated to the class of all G-vector bundles over IRPk-' (the G actions being free); then, by analogy to the standard construcj G W Pk-1 the tion of the J-theory, let us denote by quotient of iOG(IRPk-') by the subgroup generated by the elements of type [ 5 ] - [ E l ] where 6 and 5 ' are G - fibre homotopically J - equivalent, and let us denote by

Span of Spherical Forms

168

iG

:

i O G W Pk- 1 ) ->

3,aRpk-5

the corresponding "J - homomorphism". With this terminology, the G - fibre homotopy J - equivalence between aqk 8 Ln and aLn is written

Since kOG@Pk-')

2 - torsion, it follows that

has only

In other words, there is a G-module M (the G - action be€ 3 + M and Ln + M ing fixed point free) such that ( ~ ~ Ln) are G - fibre homotopically J - equivalent. Indeed, by (3.16), this J - equivalence still holds for Q Ln and L" : Pro' k n position 4.11 now shows that span(S(L )/G) L k - 1. Span of Spherical Forms of Cyclic Type. In this section, as well as in the following one, we shall eventually assume that the defining group G of the spherical form Sn/G is not trivial. Notice that span(S 1 /G) = spanR(S1 ) = 1 and that, because any 3-dimensional orientable manifold is parallelizable [233], span(S3/G) =spanR(S 3)=3. We must examine the spherical forms whose defining group has a 2-Sylow subgroup G2 isomorphic to a cyclic group Z ( m 2 0 ) (see (2.9)). When m = O or 1, we always have that zrn span(S 2n+1 /G2) = spanW (S2n+1). From Remark 4 . 8 we conclude 5.

that if

n

+

0,1,3 n

then, Propositions

or 7 , + 1 > p(2n+2,R 4.7

and

4.12

span(S2"+'/G) Furthermore, using

)

;

show that > spanR(S 2n+l ) -

.

(2.8) we conclude the following.

3 or Lemma 5.1. If n 2n+ 1 span(S / G ) = p (2n+2,IR )

7

-

and 1

.

v2(lGl) =

0

or

1

,

Let us now suppose that m 2 2 and n + 0,l or in this case we also have that n + 1 2 p(2n+2,C). Because

3;

Spherical Forms of Cyclic Type

169

span(L"(2")) > p(2n+2,C) - 1 (see Proposition 2.81, if (4.7) and (4.12) prove that

n

+

3,

2n+ 1 span(S /G) 2 p(2n+2,C) - 1 If we consider the canonical inclusion

z4 z -+

2m L"(2m)r (here m 2 2) which induces the surjective map Ln(4) we see that ~pan(L"(2~) ) 5 span(L"(4) ) Thus, by using (4.12) (and eventually ( 4 . 7 ) ) , we a l s o have (for n * 3) that -+

.

2n+l /GI 5 span(~"(4)) span(s

.

This shows that we must study the span of Ln(4) for n * 0,l and 3 . Let 5 , be the unitary representation of degree 1 of Z4 defined by C , ( i ) = 5 l a , with 5 = exp(ir/2) ; we know that Ln(4) is obtained from the standard complex representation o = (n+l)C1 (cf. (2.3)). We would like to establish the fact that span(~"(4)) = p(2n+2,~) - I

.

Since we already know that > p(2n+2,~)- 1 span(~"(4)) -

,

let us suppose that span(L"(4)) L k - 1 , with k > p(2n+2,C). Then (4.1) shows that n k C3 u and o are Z4- fibre homotopy J - equivalent. But then, this means that we can find a unit u of KU (IRPk-') such that z4 ej(qk dD a)

*

$ j (u) =

u

*

ej(u)

(see Proposition 3.2). In order to go on, we must use the $ structure of KU,4(RPk-1) ; the following lemma gives the necessary information. Lemma

5.2.

generutor

xk

(i) =

'I?U(IRPk-l) i s i s o m o r p h i c to

[c(nk)3

-

1 =

[nk

@

@I

-

'ak(C)

1 ; moreover

with

Span of Spherical Forms

170

The first assertion was already given in the previous chapter. The other assertion, which means that (Js is but the identity on KU (RPk-') , results from chapter 4, Q 3 and 24 from the following. 5.3.

Proposition

[ 3 6 ; page 2741

t h e n t h e Adams o p e r a t i o n s i n u l a r , +lGl+j = $1 .

-

Let

G

be a f i n i t e group;

a r e p e r i o d i c ; in p a r t i c -

RU(G)

All this implies that

e5[nk@

.

(n+l)S11 = o5[(n+1)C11

On the other hand, RU(Z4) is additively generated by 5, = 1, E l , 5, = 5 ; and 5 , = 5: : hence, the properties of the Bott operations (see Chapter 0 ) show that

e 5 q ) 8 5 ( nk

@

=

l + c , + c l +2c l + c3l 51)

4

=

1+co+c1+c2+c3

1+[nk@ c 1 '61+52+[qk@

=

5, + 5 , + 5, + 5, = a

and

5, + 6,

=

6

c1

' p(851+8,H) is not true. Thus, span (N2R+1(2)) = p(851+8,H) - 1 and consequently, ~ p a n ( S ~ ~ + ~=/ p(811+8,IH) G)

(6.8) (with II

*

0).

-

1

Span of Spherical Forms

176

We now state the main result of this section by summarizing 16.1) and (6.8). n Theorem 6.9. L e t S / G be a s p h e r i c a l f o r m of q u a t e r n i o n i c type with n + 7 We assume t h a t t h e n a t u r a l i n c l u s i o n of i - i G2 = Q , i n t o G i n d u c e s an i s o m o r p h i s m H ( G ; Z ) ( 2 ) = H ( G 2 ; E ) .

.

N

Then,

span(Sn/G) =

p

(n+l,M) - 1

.

The case n = I has to be excluded in our theorem; in fact, we shall see in the next chapter that no Q m - spheric a l form

7 S /am

is parallelizable.

CHAPTER 6 IMMERSIONS AND EMBEDDINGS OF MANIFOLDS

Throughout this chapter, except when otherwise speThe dicified, all manifolds and morphisms are of class Cm mension of a manifold M will often appear a s a superscript of M.

.

Background. Let f : M + N be a morphism between two manifolds and let x be a point of M By definition, the rank of f at x is the rank of the tangent linear transformation Txf : TxM + Tf(x)N. Clearly, 1.

.

rankX f

=

dim(1m Txf) 5 inf (dimxM, dimf(x)N)

.

It seems then natural to study the morphisms f whose rank coincides with the local dimension of one of the manifolds considered. We say that f is a l o c a l i m m e r s i o n at x (resp. a t o e a 2 s u b m e r s i o n at x) whenever dimxM 5 dimf (x)N and rankxf = dimxM (resp. dimf(x)N < - dimxM

and

rankX f

=

dimf(x)N);

in other words, f is a local immersion at x (resp. a local submersion at x) when Txf is injective (resp. surjective). If f is a local immersion (resp. a local submersion) at every point x E M , we say that f is an i m m e r s i o n (resp. a s u b m e r s i o n ) of M into N . It follows that an immersion is 31ways locally injective (and thus, a topological immersion); however, a morphism can be globally injective without being an immersion (for example, this is the case of the function 3 f : IR --t IR given by f (x) = x ) From now on we shall be dealing only withimmersions; some of the facts that we shall recall can be transposed "mutatis mutandis" to submersions. Fur-

.

177

178

Immersions and Embeddings of Manifolds

thermore, we shall consider only c o n n e c t e d manifolds. The reader should observe that the composition of two immersions is an immersion; moreover, if f : M + N and f' :M' +N' are immersions, f x f' : M x M ' + N x N ' is also an immersion. Finally, given a morphism f : M + N , its graph r f : M + M x N defined by rf(x) = (x,f(x)), for every x E M, is an immersion. From this last property it follows that the diagonal morphism,

is an immersion. The notion of immersion can be characterized geometrically as follows: a morphism f : M + N is an immersion if, and only if, there exists an open covering "i'iE I of M such that, for every i E I, flui is an isomorphism of In particular, it follows that Ui onto a sub-manifold of N if M is a sub-manifold of N, the canonical injection i : M -+ N is an immersion. If f : M + N induces a homeomorphism of M onto f(M), then f(M) is a sub-manifold of N. In this situation, f is an isomorphism (of manifolds) from M onto f(M); we say that f is an ernbedding of M into N. In order to emphasize the topological difference between the notions of immersion and embedding, we recall the example of the curve C on the torus T = S1 x S1 c I R 3 obtained from a line of irrational slope: C is a 1 - dimensional manifold immersed in T; however, this immersion is not an embedding because the open sets of C are not necessarily the intersections of open sets However, applying the well-known "inverse of T with C function theorem" one can easily see that, for every x E M, the restriction of an immersion of M to a certain neighborhood of x is an embedding. One of the most important results on immersions and embeddings is the famous theorem of H.Whitney: "every n - d i mensional manifold can be embedded in R Z n [ 2 5 5 ] and, if R2n- 1 [2561." We n 2 , such a manifold can be immersed in

.

.

shall call this theorem the Embedding (resp. Immersion) Theorem. At this point we want to observe that in the preceeding remarks

Background

179

we have not made any statements in terms of local coordinates; however, their proofs are local indeed and based on the classical "Rank Theorem" (see, for instance, [ 7 2 1 ) . We now take the point of view of fibre bundles. We begin by observing that if f : M + N is an immersion, then there is a monomorphism of vector bundles over M , namely f*r(N) , where T ( M ) (resp. T(N)) is the tangent T(M) >-> bundle of M (resp. N). Because this monomorphism has constant rank, we can form the quotient bundle W(f) of f * T ( N ) by the image of T (M) ; the bundle w (f) is called the norma2 b u n d l e a s s o c i a t e d t o t h e i m m e r s i o n f . Hence, attached to each immersion f : M + N , there is an exact sequence of bundles (1.1)

0 ->

T(M)

->

v(f) ->

f*T(N) ->

0

.

It is well-known that such a sequence is split whenever paracompact: (1.2)

T

(M) @ W(f)

IY

f*T (N)

M

is

.

In particular, if M is an n-dimensional paracompact manifold immersed into the euclidean space IRn+k via a morphism f , then n+k (1.3)

T(M)

F v(f)

3

E

Now let f : M +lR n+k be an embedding; the total space of its associated normal bundle can be viewed as the set of pairs (arb)E IR n+k nn+k such that: (i) there is x E M such that f(x) = a; (ii) for every z E Im Txf , the scalar product ( b l z ) = 0. Hence, M can be identified with the image of the trivial cross-section so of w(f). From this standpoint, whenever M is compact we can show that there exists an embedding f : M +lR n+k (k sufficiently large) and a homeomorphism 4 : D(w(f)) + V - where D(w(f)) is the total space of the disc bundle associated to w(f) and V is a neighborhood of f(M) in IRn+k - such that @ s o = f; V is a t u b u l a r n e i g h b o r h o o d of f(M) in IRn+k (see, for instance, [1611).

The relevance of ( 1 . 3 ) in K-theory is clear: it just means that the stable classes of T ( M ) and w(f) are inverse to each other in KO(M). This remark shows that the stable

180

Immersions and Embeddings of Manifolds

class of v(f) is independent of the immersion f selected; it depends only on the manifold M . Indeed, this result is even more interesting since it has an inve se: we are referring to the well-known theorem of M.W.Hirsch [ 11; 5 51

.

Theorem 1 . 4 . An n-dimensional manifold IRn+k in if, and only if, there is a real 5 such that T(M) @ 5 is trivial (k, 1 )

M can be immersed k - vector bundle

This characterization enables us to obtain several non-immersion and non-embedding criteria of a manifold into an euclidean space. They consist in taking the appropriate characteristic classes relative to the (co)-homology theory being utilized and then, observing the nullity of these classes for the tangent bundle. In this chapter we shall state the criteria of the type mentioned before obtained by M.F.Atiyah, via K-theory and Grothendieck operations [ 1 3 ] . Hence, we shall apply them to certain spherical forms introduced in the previous chapter. We should mention that the method we just referred to presupposes our ability to determine the tangent bundle of a given manifold (or, at least, its stable class). This determination is not always easy: besides the case of homogeneous spaces, whose tangent bundles have been studied by A.Bore1 and F.Hirzebruch [ 4 4 ] , we shall work based in two general situations. First, if M is a manifold with boundary, there is a relation between the tangent bundles of M and aM, namely, T

(MI I aM

=T (3M)

6; E 1

;

this relation can be established by observing that there is an embedding f : aM x I + M such that f(x,O) = x. The second general situation is that of the quotient manifolds. Let 5 = (E,p,M) be a (differentiable) G - bundle over a manifold M , with fibre F and E compact. Let T , ( ( ) = (TF( k ; i (ii) i f M i s embeddabze . i n IRn+kl y ( - T ~ ) = O r f o r e v e r y i > k . 3.

The proof of this theorem consists essentially in studying the geometric dimension of -T = n - [-r(M)] ; we shall see it in 0 a little while. More generally, let X be a (finite) connected C W complex and let 4 : Vectn (X) + KO(X) be the canonical morphism which associates to the isomorphism class of a (real) vector bundle over X its class in KO(X); then, we say that an element x E KO(X) is p o s i t i v e (x 2 0 ) provided x E Im g. Now, let x E 8O(X) be given: recalling that KO(X) =KO(X) z, we define the g e o m e t r i c d i m e n s i o n of x (notation: gdim x) to be the smallest integer a such that x + a 1 0. (Recall, that each element of KO(X) is of the form [ [ I - n (see ( 0 . 1 ) ) Hence, for any x E 20 (X) there are integers n such that x+n L 0; this implies that the geometric dimension exists for any xEFO(X Moreover, for all x E KO(X): gdim x 5 dim X; see (0.4).) Proposition 3.2. If x E FO(X), t h e n y i (x) = 0 f o r e v e r y

e.

.

N

i > gdim x. Proof.

Suppose that

gdim x

= a;

then

Atiyah' s Criterion

Yt(X)

=

At/(l-t)

=

189

lt/ ( 1-t) (-a)At/(l-t)(x+a)

and hence , (3.3)

Yt(X)

=

1 xi (x+a)ti (1-t)a-i .

ilo

such that Since x + a > 0 there is a real vector bu.ndle 5 i [El = x + a : but rank[E] = a so A (5) = 0 for every i > a. Relation (3.3) shows that yt(x) is a polynomial of degree < a , which is precisely our statement. I

3.4. L e t 5 = (E,p,X) b e a r e a l v e c t o r b u n d l e of k a n d l e t S ( 6 ) = (S(E),.rr,X) b e i t s a s s o c i a t e d i t s p h e r e bundZe. If x = [5] - k E %(X), y (.rrTT'x) = 0 for every i l k . Proof. It is easy to see that the vector bundle

Corollary dimension

.rr*( gdim n!(x).

=

gdim ([ql + I-k) 5 k

-

1

I (3.2) y i (~'(x)) = 0 for every

(i) If f is an immersion Proof of Theorem 3.1. n+k of Mn into IRn+k, (1.3) shows that T ( M ) (+ u(f) = E Hence, [v(f)1 = k - T in K O ( M ) and therefore, gdim(-.ro)< k. O i Now ( 3 . 2 ) shows that y ( - T ~ ) = 0 for every i > k.

.

(ii) Let us assume now that f is an embedding of Mn into Rn+k.Let S(u(f)) = (S(NfM),r,M) be the sphere bundle associated to the normal bundle u(f); because of ( 3 . 4 ) we obtain, for every i 2 k, f (3.5) Y i (n*(-To)) = 0 But S(NfM) can be viewed as the boundary of a tubular neighborhood N of M and so, from considerations developed in 9 1 we conclude that the trivial cross-section s is actually a homotopy equivalence between M and N I so that the diagram below is homotopy-commutative: I

Immersions and Embeddings of Manifolds

190

M

.

This shows that the exact sequence of the pair KO-theory can be written as

(N,aN) in

N

(3.6)

where

N

1

1

a'

=

1

1

a' KO(N,aN) ->

... -> 1

...

N

KO(M) ->

I

, with j.

s.j.

71'

N

KO(aN) ->

N

:

KO(N,aN)

-+

ZO(N). Now let us map

(N,aN) into (Dn+k,Dn+k-N) where D " + ~ is a sufficiently large ball of IRn+k it is then clear that

KO

(D"+~,D~+~-N)

>

i?O(Dn+k)

I

commutes, where P' is the excision isomorphism. It follows that a! = 0 because 80(Dn+k) = 0 and hence, the morphism IT! of ( 3 . 6 ) is injective. The naturality of the yi operai tions and ( 3 . 5 ) now show that y ( - T ~ ) = 0 , €or every i,k. Of course, it is very tempting to compare the criterion just obtained to the classical ones. For example, if we consider rational Pontrjagin classes, we know that if Mn is immersed (resp. embedded) in IRn+k , then (3.7)

P~(-T~ = )0

if

2i > k

(resp.

2i - k).

This relation shows that the Grothendieck operations have the advantage of giving more informations for k > [n/2]

.

More generally, let x be an element of '?ijO(X) and let let : KO(X) H*(X;Q) be the composition of the complexlfication homomorphism with the Chern character. We claim that -+

(3.81

ch r2J(x) = ( - 1 ) J pj(x) + higher terms.

Atiyah's Criterion

191

This relation shows that ( 3 . 7 ) is a consequence of Atiyah's theorem. To prove (3.8) let r\ be an m - dimensional vector bundle such that x = ~ - m , and let us suppose €or the presentr that the complexification 0 is a sum of complex line bundles, C

qC

n1

- - - anm

0

dim

r

We denote the first Chern class of Chern class of we then have

r(

i

= 1

by

qi

.

ai : for the total

nc

m

and the

j - t h Pontrjagin class of

x

is therefore given by

(3.9)

where

s

23

denotes the

2j - t h

For the complexification Yt(x)

c

elementary symmetric function. of

yt(x)

we compute

Hence m

ch y , ( x )

n {l+(exp(ai)-l)t)

=

i=1 (3.10

Relations

(3.9)

and

(3.10)

imply our claim

(3.8)r

in case rl decomposes into a sum of line bundles. The geneC ral case follows invoking the splitting principle. Finally, whenever we restrict ourselves to i n t e g r a l Pontrjagin classes, the previous result can be stated as follows. Proposition 3 . 1 1 . H*(X;E)

Suppose t h a t

is f r e e ) . L e t

5,

X

is t o r s i o n - f r e e

(i.e.,

b e t h e s t a b l e c Z a s s of a r e a l v e c -

192

Immersions and Embeddings of Manifolds

t o r bttndZe

5

ouer y

X. T h e n , t h e a s s e r t i o n

2i (€, 1 S

= 0

for ezrery

i

f o r euery

i > k

k

r

implies that

pi(cs)

= 0

.

We shall see, later on, based on the examples mentioned in [155], that for spaces with torsion (3.11) is not always valid; other comments about 3.11 will also be made. We give next a first application of Theorem 3 . 1 . Let us recall that a DoZd m a n i f o Z d D(m,n) of d i m e n s i o n m + 2 n is obtained from Sm x cPn by identification of every (XrZ) E Sm x CPn with (-x,z). Clearly D(m,O) (resp. D(0,n)) is just the real (resp. complex) projective space lRpm (resp. CP") The cellular structure and the mod.2 cohomology ring of D(m,n) were described by Albrecht Dold in [75] : let (ai,bJ)r 0 5 i 5 rn, 0 5 j 5 n , be the (i+2j)th- integral cohomology class of D(m,n) which corresponds to the (i+2j) cell (e;,e{) of D(m,n); then, using the same notation for such a class or for its reduction mod.2, we obtain:

.

Proposition

3.12.

The cohomoZogy r i n g

c i d e s w i t h t h e t r u n c a t e d poZynomia2 r i n g 1 1 w h e r e a, = (a ,bo) and bl = (ao,b ) .

H* (D(m,n); Z 2 )

coin-

m+ 1 ,bn+ 1 ), Z2[a,b]/ (al

Let n : D(m,n) +lRPm be the canonical projection. The reader can check that (D(m,n),naprn) is a fibre bundle with fibre CPn-' and group Z 2 ; we shall denote by i the inclusion of the fibre CP"" into D(m,n). Now let (resp 5 ) be the canonical line bundle over IRP" (resp. CP"-' ) M.Fujii [ 9 2 ; Theorem 2.21 and J.Ucci [244; Proposition 1.41 proved independently that there is a 2-dimensional real vector bundle u over D(m,n) such that i*p = rg (if n = 0, p O! E & n * n ) . From the two papers quoted before it fOllOWSr firstly , that the tangent bundle T = -t(D(m,n)) can be expressed by the relation

.

T

&

(4 n*V

(n*Tmprn)) c+. (n+l)p

or, in other words (using 1.5),

r

Atiyah's Criterion

(3.13)

T mEL

and secondly

,

qn*n

((m+l)n*n) 3 (n+l)p

Y

and

IT*Q

:

m,n > 2 , the total Stiefelare given by

assuming that

Whitney classes of

193

.

w(n*~)= 1 + al , w p ) = I + a l + b l

(3.14)

This last result implies that c ( ( ~ * n ) ~ =) 1 + a2 where (3.15)

a2

=

(a2 ,b0

C(T

and

b2

=

(ao,b2)

.

Hence,

n+ 1

m

= (l+a2) (l+a2-b2)

a: )

Let us now compute y i (-To), where T o = [ T I - (m+2n1~K0(D(m,n)). We observe first that the KU-theory of the Dold manifolds was studied by Ucci [244] and Fujii [92], [93]; as for their KO-theoryl very complete results were obtained by M.Fujii and T.Yasui in [941. For our purposes it will be sufficient to know that 'ijO(D(m,n)) contains a direct factor which is isomorphic to

generated by

with the relations (3.16)

U f(m)+l = 0 ,

u2

=

-201 ux

= 0,

x[n/21+l+h

= 0

,

where h = 0 or 1 according to n 4 1 or n I 1 (mod.4) and also where f(m) is the Radon-Hurwitz number defined by f(m) = cardinality { O < s <mls

P

0,1,2

or

4

(mod.8))

(see Chapter 4, Proposition 3.12). From (3.13) we conclude that (3.17)

Because

Y ~ ( - T ~= )

gdim

CJ

=

1

and

yt(-(m+n+l)a

gdirn x

= 2,

-

(n+l)x)

it follows that

194

Immersions and Embeddings of Manifolds

+ xt + y 2 (x)t2

yt(d = 1 + ot, yt(x) = 1

.

,

We now compute y 2 (x). To this end, we take (0.12) to obtain that 2 2 1 1 2 y (XI = y ([).I] - 2) + y ([pl - 21y (-(I) + y ( - 0 ) But

.

yt(-a)

-1

=

(l+ot)

=

1

-

at

= 1

-

-

at

+

02t2

- ...

...

2ot2 +

(utilize 3.16) and s o , y

2

(x)

2

= y ([).I1

-

- 2)

o

1 y ( [ ! A 1 - 2)

*

-

.

2o

On the other hand Y t h l - 2 ) = Y (t/l-t)([p]-2)

= (l+(t/l-t))-2(I+[p] (t/l-t)

2 2 2 2 2 + [ A pJ(t/l-tI 1 = (1-t) + [pJt(l-t)+ [ A p]t

-t

,

which shows that y 1 ([pI-21

= [pl

-

2

and

2 y ([pl-2) = 1 -

[PI+

2

[ A p1

.

It follows that y

2 (x)

= 1

-

= 1

- [PI

[PI +

[A

+ [A

2 ~1 - 0([~1-2) - 20 2

PI

since, as one can easily prove, y

o[pl

= 0 ; thus,

2 ( x ) = [2 A ~ ] - u - l - x .

.

Actually, the line bundle A L p is isomorphic to a*n To see this, we compare their first Stiefel-Whitney classes; since wl(r*rt) = al = wl(p) (see 3.14), it suffices to show that (3.18)

w,(p)

=

wl(A

2

).I)

.

But (3.18) holds for every real vector bundle of dimension 2 ; we leave the proof of this result to the reader. Putting Going back together these facts, we conclude that y 2 (x) = -x to (3.17) we now see that

.

Atiyah’s Criterion

195

Yt(-T0) = (l+ut)- (m+n+l ( 1 +xt-xt2) -(n+l) finally, because O X = 0 and u i+’ = (-1)i2iu i > 0 , it is not difficult to conclude that

,

I

for every

where

.

Ai k = (-1)i-k (i?k)(n+;+l) Consider the integer

A(m,n)

{

= sup i12i-l(m+;+ij

+

0

(mod.2f(m))\

2f(m)

and the term

J

and define

then, because the order of

u

is

(this term is part of the coefficient of t2[n/21 in the development of (l+xt-xt ) -(n+l)) , it follows that yA(mrn) ( - T ~ ) $ 0 . This result together with (3.1) shows that Proposition 3.21. -12441 - (i) D(m,n) c a n n o t b e i m m e r s e d in IRm+2n+A(m,n)-1 ,. (ii) D(m,n) c a n n o t b e e m b e d d e d i n IRm+2n+A(mrn) Setting A(m)

=

A(m,O) = A(m,O), we obtain

Corollary 3.22. - [13] - (i) n p m c a n n o t b e i m m e r s e d i n IRm+A(m) - 1 (ii) I R P ~ c a n n o t be e m b e d d e d i n IRm+A (m) I

Remarks. For the benefit of the reader interested in comparing the results obtained above for IRPm and those obtained by different methods (the literature is full of examples) we trans-

Immersions and Embeddings of Manifolds

196

cribe next some of the best possible known values: in the following tables, m + k is the dimension of the smallest euclidean space in which lRPm can be immersed.

m

1 2 3 4 5 6 7

8

9

10 1 1 1 2 1 3 1 4 1 5 16 1 7 1 8 1 9 20

lm+k12 3 4 7 7 7 8 1 5 1 5 1 6 1 6 1 8 22 2 2 22 3 1 3 1 32 3 2 3 4

I

I

I

Im

S

= 2 s> -4

-1

m+k

1

s

0 (mod.4);s

2m

-

2s

= 2m

I

1 (mod.4) s

-

2s+ 1

p

2 (mod.4)

2m- 2s + 1

S = 3

(mod.4)

2m- 2 s - 1

;; We want to observe that Atiyah had already indicate1 that in certain cases 3 . 2 2 could give better informations than the Stiefel-Whitney classes (particularly for m = 2'-1) while in other cases (for example, m = )'2 the results are less interesting. We leave to the reader the task of obtaining from 3 . 2 1 , analogous criteria of non-immersion and non-embedding for the complex projective space CPm ; at any rate, we shall go back to this problem at the end of the chapter. We now revert back to the discussion of the result given in 3 . 1 1 . Suppose that X = lRPm and let cx be a real vector bundle over it. Since KOWPm) is generated by u = - 1 the stable class a S of a in KOWPm) is an integral multiple of a , say a S = uo Computing the total Chern class of ua c one obtains that the ith integral Pontrjagin class of a s is given by N

N

.

-

Atiyah's Criterion

197

z2 (if 4i > m where y is the generator of H2@pm;Z) have that pi(as) = 0).Moreover, (3.16) shows that

1 +

=

1

we

(-1) i-12i-l (iu) ati

i> -1 and hence, Y

2i

(as) =-2

Because KO @Pm) Z z 2f ( m ) (see Chapter 41, to say that y 2i ( a s ) = 0 is equivalent to saying that 22i-1(:i) = 0 N

(mod.

.

2f(m)) But a scrutiny of the values assumed by f ( m ) reveals that if 4i 5 m, 21 - 1 < f(m): hence, = 0 implies that = 0 (mod.2) and consequently, p. (a ) = 0 With 1 s this we observe that the conclusion of 3.11 holds for X = RPm , although such a space is not torsion-free. On the other hand, if X = D(m,n) with m = , ' 2 n = 2V ( v > s > 1) and s-2 2v-l k = 2 r (3.15) and (3.17) show that y 2i (--r0) = 0 > k , while P ~ ( - T ~ )0 . Finally, using lRPm we can for i give a counter-example to show that the converse of 3.11 is false, in general. It suffices, for instance, to take m = ' 2 1, S s-3 n = 2 and i = 2 ( s > 3); in fact, since the dyadic valuation of (2s-2) 2s is

.

(Ji)

+

it follows that

p2S-3(as) = 0

2s-2v 2(2 2s-2 i.e.,

.(*)

y

(as)

see Lemma

*

,

while

y2))

0

4.5.

=

.

2s-2+1

KU

Y

->

L (a;ao.a.,,...,an)) KU

”(L

RU(za)

A>RU(za)

(a;ao,a,,...,an)) ->

(chapter 0) gives the following description of

. .

a,,, . ,a,) 1 Proposition

-->

o

n XU(L (a;ao,

. 4 .1 .

-

[166]

-

n

is i s o m o r p h i c to

(i) T h e r i n g

z [ ~ I / (n ( 1 - [ 5 1 r=o

ar

),[51a-

1)

n KU(L (a;ao,a,,...,an)) n idhere

(

n

r=o

(1

a

[ 2 - Actually, Proposition 4.8 (ii) and Atiyah's criterion imply lowing detailed results. Let L(n,m)

=

I

sup 1 5 i < [n/211

("') 4

0

the fol-

(mod.2m+n+c-zi

1

)J

where E = 1 or 0 according to n being even or odd respectively. Then, similarly to (4.12) we have: Proposition immersed

4.16,

- [157]

-

(resp. imbedded) i n

If

m

2 , Ln(2m)

R 2n+2L(nIm)

cannot be

( r e s p a 2n+2L (n,m)+I 1.

For example, if we compute the dyadic valuation of

Immersions and Embeddings of Manifolds

206

( n+kn’21)

using the well-known arithmetic relation

v (N!) = 1 “/pi] (p prime) ( * ) P i> 1 we obtain the following. S Corollary 4.18. G i v e n m a n d s > 2, f o r every n = 2 fresp. 2’+11 Ln(2m) c a n n o t b e i m m e r s e d i n IR3n ( r e s p . I R ~ I ~ - ~ 3n+l (resp. I R ~ ~ ) . a n d c a n n o t b e i m b e d d e d i n IR Finally, the reader who wishes to state the analogous results for the g e n e r a l i z e d lens s p a c e s , can do so by using (4.7)(iii) and (4.10) Furthermore, the expression for r([nIk-l) given in 4.7 (iii) can also be employed to show that, for every real vector bundle over Ln(a), with a odd, the implication of 3.11 holds. (4.17)

~

.

5.

The Case of the

Q, - Spherical Forms

The results concerning the KIF-theory of the Qmspherical forms which we are about to present follow essentially from the work of one of the authors (see [169]) and from D.Pitt‘s paper [196]. Another useful reference is 1901. We shall keep the notation introduced in 2.5 and 2.6 of Chapter 5, ren lative to the generalized Q m - spherical forms N (m;ro,rl, rn) n defined by the action p = @ 3

6r-1 I

if

, 2< r -

zm-’ -

2.

-

Q,

Spherical Forms

207

Now recall that n+ 1

1

(-1)A'p

II (2-Eri+3) i=o

=

j=o

and that the cohomology groups of

z,H

HO(Q,;Z)

o~

~ (Q,;z)~

+=

if

k > 0

H4m(Qm:z)+~2m+l , H4k+2(Q,;Z)

are

Q,

,

Z2@Z2 , if k

s

[62; page 2531

> 0

These results, together with Atiyah's exact sequence (0.9) and the Atiyah-Hirzebruch Spectral 'Sequence (0.10) lead US immediately to the following statements.

.

n (i) The r i n g s KU(N (m:ro,rl ,. . ,rn)) Proposition 5 . 2 . n RU(Q,) /( ll (2-5r.+3)) a r e i s o m o r p h i c . More p r e c i s e l y , i=o

and

1

n KU(N (m;ro,rl,...,rn)), a s a n a d d i t i v e g r o u p , i s g e n e r u t e d b y {6r)l 2 , t h e g r o u p FU(L 3 ( 2m ) ) i s i s o m o r p h i c Lemma 6.3. I f m to

where e a c h f a c t o r i s g e n e r a t e d r e s p e c t i v e l y b y x1 = w + 40 and x3 = uu, w i t h w = u 2 f 2 u . If m = 1,

XU(L3 (2m)

=

z8 ,

i s isomorphic t o

KuQRP’)

For the real case we obtain: 3 m The g r o u p aO(L (2 ) Proposition 6 . 4 .

ut

x2 -

g e n e r a t e d by

u.

i s d e s c r i b e d by t h e

)

following table:

m = 1

->

‘8

Z2m+l @

z2

T

generator:

Y1

generators:

y, = ru, y2 =

= ru

K

.

Proof. The case m = 1 is well-known; let us then > 2 Using 4.9 and the notation and results assume that m of Lemma 6 . 3 , we conclude that the images of y , ‘ ~Ei?O(L3 (2m) by the homomorphism c are given by

.

c(Y,)

+

= - 1 8 ~ ~3x2

-

x3

C(K) = -(22m+ 22m-2)xl

-

)

I

2m-2x2

.

It follows that the orders of c(yl) and c(y2) in FU(L 3 (2m)) are 2m+1 and 2 respectively. Indeed, these elements gene, Since, on the one hand, rate the subgroup z ~ ~@+ z2

.

21 4

Immersions and Embeddings of Manifolds

and on the other hand (see Proposition IXO(L3 ( 2m1 ) I = 2m+2

4.6)

,

we have t h a t

?O(L

3

m (2 ))

namely, to the direct sum A

(6.5)

is isomorphic to the image of Z 2m+l

OZ,

c ,

*

consequence of this result is that

c : “ O ( L 3 (2m1 )

->

3

m

K U ( L (2 1 )

is an injection. Thus, relation 4 . 7 (iii) holds for p = 2. From (4.3) we obtain that u 4 = 0 in X U ( L 3 ( 2m ;ao,al,a2,a3)); thus , 2 = (cruI2 = u 4 / ( ~ + u ) = ~o c((ru) (see 4.7 (i)). The injectivity of c shows also that (ra)2= O in E O ( L 3 (2m ;ao,al,a2,a3)). In other words, (6.2) every prime number p

.

Theorem

6.6.

t h e Zens space

holds for

(i) F o r every m -> 1 a n d every odd prime p L 3 (pm.,ao,al,a2,a3) is paraZZeZizabZe ifJ a n d

onZy i f ,

2 +a2 a0 l + a 22 + a23 = 0

( m o d . pm!

;

(ii) the o n l y paraZZeZizabZe space a m o n g s t the Zens spaces L 3 (2m ;ao,al,a2,a3) is t h e reaZ projective space I R P ~= L3 (2). Proof. The necessary conditions are evident (compare to [178]) because the order of ru f i?O(L3~pm;a0,al, if a2,a3)) is either pm if p + 2 and m 2 1, or 2“+l p = 2 and m 2 2. Statement (ii) follows trivially, since the congruence 2 + a3 2 = 0 :a + a21 + a2 (mod. 8) has no solution

(aola1,a2,a3) with

(ai,8)

=

1, 0 < - i 0, ((r-l)m,n) = 1, rn = 1 (mod.m), d with

d

=

1,2

or

4,

equal to the order of r in the multiplicative sub-

group of the invertible elements of em . One can check that necessarily m is odd and that n is a multiple of d (besides , d corresponds to the order of the free representations of G). Since the order of G is mn , only the groups for which d = 1 can give rise to parallelizable spherical forms; G is therefore cyclic. Hence, the only parallelizable 7 spherical forms S /G are the lens spaces (G non-trivial)! These are completely determined by Theorem 6.6. Remark. Working with Grothendieck operations on lens spaces and Q m - spherical forms and applying Propositions 6.2.8 and 3.2, we can show that span(S 7/C ) = 5 (m? 2) and span(S 7/Qm)=5. 2m

Immersions of Complex Projective Spaces In the previous sections we have shown how we can take advantage of Atiyah's Theorem ( 3 . 1 ) to obtain information about immersions and embeddings of Dold Manifolds and Spherical Forms into euclidean spaces. Furthermore, as we have already indicated in 9 2, K-theory gives other techniques leading to interesting results on the subject; for example, the reader is directed to the work of Atiyah and Hirzebruch [ 2 7 1 and to the methods developped by Mayer 11731, which employ the famous Index Theorem. Actually, without recurring to primary operations in K O - theory, excellent results can be 7.

21 8

Immersions and Embeddings of Manifolds

obtained with the use of just complex K-theory and the Adams Jlk - Operations. To illustrate this last remark we shall present in this section some criteria for non-immersion of complex projective spaces (the reader should compare them with those deduced from (3.21) for m = 0).We investigate immersions of CP" in IR 4n-2a(n) , where a(n) is the number of 1 ' s in the dyadic decomposition of the integer n (see Conjecture 2.3) From this point and to the end of the chapter, we shall denote the Complex Projective space CPn simply by Pn

.

and the truncated projective space more, we shall write (7.1)

s(n'k)

=

(2n-a(n)+l I (2n-a(n)+l+k) 1

Pn/Pn-k

*

by

PE

. Further-

S (2n-a(n)+l+k,2n-a (n)+I )

where the integers S(m+q,m) are'the Stirling numbers of the first kind defined as it is well known, by (7.2)

(log(l+t)/t)m

1

= P

(m!/(m+q)

!)

.

S(m+q,m)tq

O

Regarding the parity of these numbers, notice that S(m+q,m) =

(7.3)

(ml:)

(mod.2).

The result we have in mind is the following. Theorem 7.4. - [2201 - If t h e r e i s a n i m m e r s i o n of Pn IR4n-2a(n) t h e n , t h e r e e x i s t s a n i n t e g e r eo s u c h t h a t : e v e n i n t e g e r , if

eos(n,k)

i s an

{ odd

if

integer,

0 5 k 5 a(n)

k = a(n)

-1

or

into

2 k = a(n)

.

The proof of this Theorem is essentially based on the study of the multiplicative structure of KU(T(V)), where T(V) is the Thorn space of the normal bundle v associated to the considered immersion. This structure can be determined working with a convenient suspension of T(v), which is homeomorphic to a truncated projective space. We shall go over the details of the proof for (7.4) later on; presently, we prove the following. N

Immersions of Complex Projective Spaces

21 9

7.5. If Pn c a n be immersed in m2n+k, t h e r e i s an integer b s u c h that t h e (2b-k-2n-2)- s u s p e n s i o n o f T(v) is b- 1 Pn+l homeomorphic t o Lemma

.

Proof. Let n be the Hopf vector bundle over Pn. d The order of J(n) in the groupe J(Pn) is equal to the James' number bn (*I;taking b as a multiple of bn , the bundle bn will have the same fibre homotopy type as the trivial real 2b-vector bundle. Then the usual relation N

N

in KO (P") lence,

becomes, in terms of stable fibre-homotopy equivaJ

u @ (2b-k-2n-2)~ r (b-n-l)n

.

In the language of Atiyah's Theory of Thom Complexes b- 1 last relation means that C 2b-k-2n-2 ( v ) 'n+l *

[12], the

(y

Proposition 7.6. If there < s an immersion o f Pn in R 4n-2cr(n), the muZtipZicative structure of KU(T(u)) is n o t N

trivial.

Proof. Since I$ 2 ( x ) s x2 (mod.2) (see 0.12) it suffices to show that on %U(T(u)) the Adams operation +2 is non-trivial modulo 2. We set q = b - 2n + a(n) 1. In view of the suspension isomorphism

-

(C2q)!

:

?U(T(u)) 'Z &J(C2qT(v))

and the relation

N

it follows from Lemma 7.5 that the $-operations on KU(T(u)) can be determined from those of the truncated projective spaces. Let us recall that &(Pb-') is generated by 1 subject to the relation ab = 0 , where 6 is the canonical complex line bundle over Pb-', moreover, the

(J

=

[5]

-

(*I See Theorem

4.5.8.

Immersions and Embeddings of Manifolds

2 20

Adams operations are such that ter 4 , § 3). The cofibration pb-n-2

pb- 1

---->

J,

R

(a)

= (a+l)'

-

1. (see Chap-

b- 1 'n+l

-->

gives rise to an exact sequence ) ->

0

N

can be identified with the ideal of %J(Pb-') 1 Let gi ( O < i(n) b-na be the element of -

Hence, KU!P:;i)

.

generated by b-1 KU (Pn+l) that corresponds to cation. From the relations

-

5b-n-l+i

under this identifi-

(i) 2i-k oi+k

Y2(ai) = (a2+2aIi =

k=o we obtain

in i;v(pb-l) (7.7)

2 (g,)

'y

We claim that reduce to

=

2

+i (b-n-1 k ) 9i+k

'

can be chosen so that the latter relations

b

I (go)

n-i 2b-n-l +i-k 1 k=o

3

b- 2n+a (n)- 1 2 ( gn- +gn 1

b-2n+a (n) (mod.2

1

1

(7.8) Y

if

2

(gi) :0

b-2n+a(n) (mod.2 1 ,

l< i c n .

To this end (assuming that b is already a multiple of it is sufficient to adjust the power of 2 dividing b such a way that a(b-a)

+

a(a-1) = a(b-l), for every

a E {1,2,...,n)

bnl, in

.

In fact, utilizing ( 4 . 1 7 1 , such hypothesis allows us to guarantee that for every i ,

221

Immersions of Complex Projective Spaces

.

= v2() "':-n(

Since f o r every integer N and consequently,

we have that

-

N = v2((2N)!)

v2(N!)

it follows that the coefficient of same dyadic valuation as

.

2b-2n+a(n)-l

n!

v2(N!)

-

a(N)

,

gi+k

.

= N

(2i)! i!

in

.

(7.7)

k! (2k)!

has the

-

The latter number is equal to

.

2b-2n+ a(n) -1

(2i)! (n-i+k)! (2k)!

.( 2)

is odd only for (21) (n-i+k)! (2k)! or n : this establishes ( 7 . 8 ) .

It is clear that and

k = n - 1

Let p o l v l Un corresponds to g,,gl,...,gn I . .

h(T(u)) 2 &J(C relation (7.9)

2qT(u))

2

Y

and Proposition Remark.

be the basis of h((T(u)) which under the isomorphisms

ikJ(P:yi)

I

q = b

-

2n + a(n)

-

I . The

implies

(7.8)

vo =

z

i = O

2

(vo) = P (7.6)

Using Lemma

0 5 i 5 n, given in

~ ++u n ~ (mod.2)

is proved.

7.5 (7.81,

and the description of Y 2 (gi)

,

the same kind of argument shows the cannot be immersed in IR4n-2u(n) -1

well-known result that Pn [173], [203] (resp. cannot be embedded in JR 4n-2a(n)

[271).

222

Immersions and Embeddings of Manifolds

Proof of Theorem 7 . 4 . Let us first note that the J - equivalence v @ 2 (b-2n+a(n)- 1 ) J (b-n-1) rl and the behavior of the Thom class with respect to Whitney sums (see [134; 16.8.11 give rise to the following commutative diagram of isomorphisms N

H*(P";R)

B > -

H*(T(u) :R)

N

a'

q = b - 2n + a(n) - 1, R = E or Q: (here, 0 and denote the Thom isomorphism of the appropriate vector bundle). In the following we will identify the two groups in the bottom line. Let a = c,(C) be the canonical generator of H* (Pn;Z); then the elements z

n-a (n)+i

=

@(ai)

,

N*

i

=

o ,...,n ,

form a basis of H ( T ( v ) ; Z ) . The cup-product structure of is completely controlled by the Euler class

T(v)

e0 E H 2n-2a(n)(CPn;Cl

t , which we interpret as an integer. We

have

u zr

zs

=

e z s+r 0

By ( 7 . 6 ) the integer eo is n o n - z e r o . (The Chern character G*(T(v);Q) is injective, since T ( v ) is torch : KU(T(V)) sion free. ) As in the proof of (7.6) let pi be the K U theory generator of T(v) obtained desuspending the generator gi of KU(Pn+l). b-I We intend to determine the integers ak in N

+

.

a (n)

(7.10)

v2 o -

1

k=o

This will be achieved using the Chern character, which is multiplicative and compatible with suspension. Let

Immersions of Complex Projective Spaces

-

223

.

b-1 ;Z) Desuspending the equabe the canonical basis of H* (Pn+l tion ch gi = (eY-l)b-n-l+i we get

We then compute

and b-2n+a (n)-1 ch

uo

Substituting eZ ch u: we obtain

=

1

+

t

-

C

k

ak(e z -1) 2n-2a(n)+k

in the equation

b-2n+a (n)-1

ch po

U

ch p o

=

k

(7.11) =

i akt

From this latter relation and in view of (7.9) and (7.10) we conclude: If Pn immerses into IR 4n-2a (n) there exists an integer eo such that in the power series of (7.11) the coefficients ao,...,aa(n)-2 are even integers, whereas aa(n)-1 and a are odd integers. Q (n) Since we may chose b such that the relevant coefficients of the power series

(logil+t)b )

are even integers

(except for the constant term, which is of course plication of

(7.11) by

( logt(l+t))b

l), multi-

does not change the in-

tegrality nor the parity of the first a(n) + 1 coefficients of the power series involved and theorem 7.4 is proved. Finally, congruence (7.3) allows us to establish the following non-immersion criterion Corollary IR4n-2a(n)

7.12.

.

If

),A:(

is odd

Pn

c a n n o t be immersed i n

This Page Intentionally Left Blank

CHAPTER 7 GROUP HOMOMORPHISMS AND MAPS BETWEEN CLASSIFYING SPACES; VECTOR BUNDLES OVER SUSPENSIONS

It is well known that, for any two topological groups G and H , there exists a function a from the set GrH Hom(G,H) of all classes of conjugate continuous homomorphisms from G to H into the set of free homotopy classes from BG into BH In the first part of this chapter we give an example is injective but not surjective; for this exin which a GrH ample, we take G to be compact, connected Lie group and H to be the infinite unitary group. This work is developed in sections 1 to 4 ; it follows the argument of [163]. The second part of the chapter presents some results about embeddings of vector bundles into trivial bundles; it is bases on work of Chan and Hoffman [641, [ 6 5 1 .

.

Generalities We begin with a quick review of the Milnor construction of classifying spaces: details can be read in [134]. Let G be a topological group and let SG be the set of all formal .) where each xi E G and sequences (toxor...,t x i i'" ti E [ 0 , 1 ] , with Cti = 1 and ti = 0 except possibly for a finite number of indices. We define an equivalence relation in SG by saying that (tOxo,...,tixi,...)i (t'x' , t;x;,. *.) if ti = ti and xi = x! whenever ti = ti > 0 Give to 1 EG = SG/= the initial topology with respect to the functions -1 ti : EG + [0,1] and xi : ti (0 , 1 ] c EG G which take the into ti and xi , respectively. classes {(toxo,..., tixi,...)) It is easy to see that there is a continuous action of G on the right of EG ; hence define BG to be the orbit space EG/G with the final topology relative to the quotient function EG + EG/G. 1.

-

225

.

.

226 Group Homomorphisms and Maps between Classifying Spaces

A continuous homomorphism

h : G H between two topological groups induces a continuous function Eh : EG -+ EH : Eh (tOxor. ,tixi,. . ) 1 = { ( toh(xo) ,.. ,tih(xi),.. . ) 1 for every {(taxor...,tixi,...)] E EG. By passing to the quotient we define a map Bh : BG -t BH. One can easily show that if h,h' : G -B H are conjugate, i.e., if there is an element -1 z E H such that h' = zhz , then Bh and Bh' are homotopic. In this way we obtain a function a * Hom(G,H) + [BG,BH] GrH * in some particular cases. Let us study a GrH Example 1.1 G = finite abelian group, H = S 1 : in this case,

..

+

.

.

.

a

GrH

is an isomorphism. In fact, the resolution

2

+

R

-t+

S1 shows that

is exact [165; IV, 5.51. Since H 1 (G;IR) = H 2 ( G n ) = 0 1165; I V r 5-41 H 1 (GrS1 3 H 2 (G;Z). NOW H 2 (G;Z)N- [BG,K(Z,2)I + [BG,K (Zr 2) 1 (the last isomorphism follows because K (Z,2) is BS 1 a path-connected H - space 178; 4.111) and since K(Z,2) 2 1 it follows that H (G;Z) CY [BG,BS 1. On the other hand regarding 5' as a trivial Z(G) -module, H 1 (G;S1 ) N Hom(G,S 1 [165; 1 1 page 1061. Hence Hom(G,S = [BG,BS 3. Example 1.2 G = z , H = IR; then a is surjective but not GiH injective. In fact, Bh = S1 and BIR = * . Example 1.3 G compact, connected Lie group, H = U, the infinite unitary group; in this case, a is injective but not GrH surjective. The proof of this result requires some preparation; this will be done in sections 2 and 3. Cartan-Serre-Whitehead Towers and H - Spaces Let Y be a based path-connected CW complex, Y 1 be its universal cover and p1 : Y 1 + Y be the covering map; it is well known that Y 1 is a C W - complex [253; 9 5 ( N )I r is 1 - connected and that the nap p1 induces isomorphisms P1# : 1 2. Suppose that we have contructed a senj(Y ) 3 n (Y), j j quence of based spaces and fibrations 2.

-

Cartan- Serre- Whitehead Towers

227

such that: (i) Y ' is the universal cover of Y; (ii) Y 2 ,...,Yi have the homotopy type of CW-complexes; (iii) for every k, 1 5 k 5 i, Yk is k - connected and (pl .pk) : k = n . (Y), j 1 k + 1 . We are going to construct a (i+l)n . (Y 7 7 connected space Yi+l of the homotopy type of a C W - complex, together with a map pi+l : Yi+l + Yi such that (pl -pi+l)++ : n . (Yi+') IY n (Y), j 2 i + 2 In this way we obtain, by induc7 j tion, an infinite sequence of based spaces progressively more connected starting at Y ; this sequence is called a C a r t a n S e r r e - W h i t e h e a d t o w e r of Y

-.

--

.

.

The construction of n

Yi+l

goes as follows. Take

i+l (Yl) IY n i+l (Y) and recall that i i+l i [Y ,K(n,i+l)]* = H (Y ; n ) IY Hom Hi+l (Yi;Z),n)

= n

(

let [ g ] E [Yi,K(n,i+l)], morphisms of the inverse of Hi+l(Yi) Notice that [g] [K(n;i+l) ,K(n,i+l)] * under

;

be the image under the above isoi the Hurewicz isomorphism T T ~ + ~ (Y )=+ is also the image of [ I K ( n,i+l)I € the homomorphism

[K(n,i+l) ,K(n,i+l) I * -> [Yi,K(n,i+l)I *

.

induced by g On the other hand, [lK(n,i+,) ] corresponds to -b the inverse of the Hurewicz isomorphism (K(n, i+l)1 Hi+l (K(n,i+l) ; Z ) and thus, by naturality, Hom(g,, 1 ) takes an isomorphism into an isomorphism. It follows that g, : i Hi+, (Y ;z) + Hi+? (K(n,i+l);Z) is an isomorphism and therefore,

(2.1)

is an isomorphism. Now consider the pull-back diagram

Pi+l V

Ip

V

228 Group Homomorphisms and Maps between Classifying Spaces

where PK(n,i+l) is the space of paths starting at the base point of K(n,i+l) and p(A) = A(1), for every path h E PK(n,i+l) The Theorem of [205] shows that Yi+' is of the homotopy type of a CW- complex; furthermore, Yi+l is (i+l)i+l ) PL TI (Y), j 2 i + 2, as connected and (p,---pi+, ) : n.(Y j one car? see comparing the exact sequences of homotopy groups of the appropriate fibrations and applying (2.1). The tower over Y we have described is precisely the Moore-Postnikov factorization of the inclusion of the base point in Y given by Spanier in [225; 8.3.81; this means that, for every n 1 0 there is a map f h : {yo} + Yn such m that pnfA = fh-l. Furthermore, there is a map f' : {yo] -. Y = lim Yn which is a weak homotopy equivalence [225; 8.3.21 c We shall see next that if Y is an H - space, then so are the spaces at every stage of the tower. We begin with the following.

.

,

.

Lemma 2.2. L e t Y b e a p a t h - c o n n e c t e d CW-comptex and l e t Y' b e i t s u n i v e r s a Z c o v e r . I f Y is an H-space, t h e n s o is 1 Y . Proof. Let e be the unit of Y , which is a l s o viewed as its base point, and let PY be the space of paths over Y, starting at e We say that A , A ' E PY are equivalent if where A, is the constant x(1) = A'(1) and A * A'-'=A e r path at e and A * A' -1 is the path defined by

.

A * A'-l(t)

=

-2I < t L l

.

1 Then PY/- = Y : now, for every pair of classes { A ) , { p } of Yi , define the multiplication {XI { P I = CALI) , where Ap(t) = A (t)l~ (t) This multiplication is well-defined: say L : A * A'-1 c* and M : LI * p'-' ry A e are homotopies and my : Y x Y + Y Ae is the multiplication in Y The reader can easily verify that

.

.

L M : I x I -

is a homotopy

on Y'

> Y x Y (LrM)

>

~

Y

mY

.

A ~ J (A'l~'1-lY A Finally, the multiplication e is continuous and has a two-sided unit, namely { A e )

.

KU-Theory of certain Classifying Spaces

map

229

and ( Z ,mZ) are H - spaces, a if mZo(g x g)ci g o m y .

Recall that if (Y,%) g : Y + Z is an H - m a p

Lemma 2.3. L e t Y be a n (n-I) - c o n n e c t e d H - s p a c e , w i t h n - 1 > 0. Then a n y map g : Y + K(n,n) i s an H - m a p . Proof. The obstructions to g being an H - m a p lie in H i (Y~Y;n~(K(rn,n))); now Y A Y is (2n-1)-connected and ni(K(n,n)) = 0 for all i n

.

Theorem and l e t

2.4.

-

(Stasheff, [226])

QZ -P E -B Y 9 H-map g : Y

Let

Y,Z

be

H-spaces

be t h e f i b r a t i o n i n d u c e d f r o m

.

QZ+PZ-rZ

E i s an H - space. 9 Proof. Recall that E = ((y,A) E Y x PZlg(y) = A(1)). g An obvious candidate for a multiplication on E is (y,A) 9 (y', A ' ) = (yy', A h ' ) , where Ah' (t) = A (t)A ' (t) However, g(yy') in general is not equal to g(y)g(y') and thus (yy',XX') is not necessarily in E We correct this incong venience using the fact that g is an H-map. Let by an

+

Z

Then

.

.

>

h : Y x Y x I -

z

be a homotopy with h(y,y',O) =g(yy'), h(y,y',l) = g(y)g(y'l now define a multiplication on E by g (y,h)(y',X') = (yy',XX' Lemmas

2.2, 2.3

*

;

h-l (y,y', 1 ) .

and Theorem

2.4

show the following.

Theorem 2 . 5 . L e t Y be a p a t h - c o n n e c t e d CW- comp2ex w i t h an H - s p a c e s t r u c t u r e . Then a22 t h e s p a c e s of a C a r t a n - S e r r e Whitehead tower o v e r Y a r e H - s p a c e s . Remarks about the KU - Theory of certain Classifying Spaces Let X be a based C W - complex filtered by its skeleta Xnl n 2 0 For every H - space Y there is an obvious group homomorphism 3.

.

Q : [X,YI* ____ >

l$m[Xn,Yl,

.

It is easy to show that 0 is an epimorphism: if ~[fol,[fll, is an arbitrary element of the inverse limit, fn : [f2], . . . I X n + Y d n+,lXn for every n 2 0. Because of the Homotopy I

Group Homomorphisms a n d Maps b e t w e e n C a s s i f y i n g S p a c e s

230

,

Extension Property for t h e p a i r Hn+l : 'n+l x I + Y , I ) IXn = f n ; w r i t i n g

topy Hn+l(

.

- fn

fA+llXn

(Xn+,,Xn) s u c h t h a t Hn+l

=

(

,1 )

f

: X + Y

n 2 0, f l X ,

phism; f o r example, i n

Adams a n d W a l k e r c o n s t r u c t t w o

[7]

f : X

-*

nected L i e group, and

Y"

->

... ->

->

yn-1

y 1

> Y o = Y

Pn

PI

be a C a r t a n - S e r r e - W h i t e h e a d

tower over

Y

.

f n : X + Yn

Let

n 2 1, fn a r e i n e s s e n t i a l .

n I 0

be a s e q u e n c e o f maps s u c h t h a t , f o r e v e r y

Pnfn

fn-1

*

Then, a l l m a p p i n g s

B e c a u s e o f t h e C o v e r i n g Homotopy P r o p e r t y w e

Proof.

c a n d e f o r m t h e maps

i n t o maps

fn

n 2 1 , pnfA = f & .

Yw = l i m Y"

.- Y"

+

Y

n

fn

: X

+

such t h a t

f h = qnf'

. Let

3.2.

(i)

f o r every

and

X

so t h a t , f o r

Y"

f o r every

i s t h e c a n o n i c a l map. H e n c e , b y

Lemma

n

f' 0

:

, where

[225; 7.6.231

are i n e s s e n t i a l , and so are

be based

Y

n 2 O,Xn

fh

CW-complexes such t h a t :

i s finite;

~~

i s a path-connected (iii) f o r e v e r y n 1 , nn(Y (ii)

f:

Hence there i s a u n i q u e map

qn w e c o n c l u d e t h a t a l l t h e maps t h e maps

whose re-

Xn

T,et

3.1.

... ->

-B

Y

are i n e s s e n t i a l . I n t h i s secX = BG , w h e r e G i s a c o m p a c t , conY = BU , t h e n @ i s a n isomorphism.

t i o n we prove t h a t i f

X

such

i s n o t n e c e s s a r i l y a monomor-

a n d a n e s s e n t i a l map

X,Y

w e see t h a t

fn.

p1

@

strictions to a l l skeleta

every

fn+,l

I n t h i s way w e o b t a i n a map

for

fA+,

The homomorphism

Lemma

10)

Hn

t h a t , for every

CW-complexes

t h e r e i s a homo-

Y

H

- space; i s a f i n i t e l y generated abelian

group;

(iv)

for every

n > 0 , H"(X

->

0 : tX,Y],

i s an isomorphism. Proof.

Q)

Q

l;mtxn,Y1*

(Compare w i t h Let

€

n n +,f Y) = 0 . Then,

: X -+ Y

[ 6 0 ; Theorem 1 . 1 1 ) . be s u c h t h a t

fIXn=constant

KU-Theory of certain Classifying Spaces

231

map, for every n 0 : consider a tower over Y (notation of 3 . 1 ) . We are going to construct by induction - a sequence of maps {fn : X + YnIn,O] such that: (a) fo = f; (b) for every 0 , fnlXk CY constant map; c) for every k 2 0 and every n n 2 1 I P,fn = fn-l Then, by 3.1, all maps fn (in particular, fo = f) will be inessential, thereby proving the Lemma. Suppose that fn : X + Y" has been constructed and is such that for every k 0, fnlXk = constant m P and Pnfn = fn-l * We have seen that the space yn+l has been defined

-

.

by an appropriate pull-back diagram yn+ 1

>

(Y) ,n+l

, clearly

fn

if, and only if, n+l gfn = constant map. But [gfA €Hn+l (X;R,+~ (Y)) = H (Xn+2; nn+ 1 (Y)) , where the isomorphism is induced by the inclusion in+2 : 'n+2 -> X ; thus, (in+2)*[gfnl = [ gfnlXn+21 = 0 because fnlXn+2 = constant map, and therefore, [gf,] = 0. Let :

X

Y"

-r

can be lifted to

Yn+l

n+l be such that ~ , + ~ =h fn' Recall now that the h : X-rY base-homotopy classes of liftings of fn to Yn+l are in a n one-to-one correspondence with the elements of H (X;IT~+~(Y)) [225; 8.21. A l s o

n H ( X ; R ~(Y)) +~ @ Q

-

n H (X;Q) 63

R

~

(Y) + 0 ~ Tor(H"+'

)

(X),Rn+l (Y) @ @ = O

n is a finite because of the hypothesis and thus, H (X;TT~+~(Y)) group. Consider the following diagram of groups (use 2.5):

n+1 I ,

> ltm [Xk,y

1&m[Xk,Y

n

It

*;

H"

it shows that

->

K =

232 Group Homomorphisms and Maps between Classifying Spaces

.

Indeed, the homotopy class nite group and that @n+l[h] E K of any lift of fn to Yn+’ is taken by into K Since K is finite and Qn+l is onto, there is a lift fn+l : X + Y n+ 1 such that On+l[fn+l] = 0 , that is to say, fn+llXk Y constant map for all k 0 Notice that pn+lfn+l~fn.

.

.

The reader should recall that if G is a compact Lie group then BG can be filtered by compact spaces BG1 c Indeed, it is possible to construct a classifying BG2 c space BG of G which is a C W - complex and whose skeleta (BG), are finite C W - complexes; furthermore, if G is connected, BG and all (BGIn’s are connected.

... .

Theorem

3.3.

For e v e r y compact c o n n e c t e d L i e g r o u p

tBG,BUl, = lJm[ (BGIn,BU1,

G,

.

Proof. The hypothesis of 3 . 2 hold: BU is a pathconnected H - space (see, for example, [236; page 4051) ; (iii) is true because of the Bott Periodicity Theorem and (iv) holds because H2n+1 (BG;Q) = 0 [ 43; 7.2 and 19.11 and n 2n+l (BU) = 0. Remark. If we also require the Lie group G to have torsion free integral cohomology, Theorem 3.3 follows from an obstruction theory argument 11951

.

4.

A Theorem of Non-Surjectivity for

aG I

In this section we prove the following result. Theorem 4.1. If G i s a c o m p a c t , c o n n e c t e d L i e g r o u p and i s t h e i n f i n i t e u n i t a r y g r o u p u, aGIU : Hom(G,U) + [BG,BUl an i n j e c t i o n b u t n o t a s u r j e c t i o n . .-Proof.

nectivity of

Theorem 3.3, Satz (BG), show that

[BG,BU]

C.L

4.11

[BG,BU], = l&m[ (BG)n,BU],

H is

of [ 9 1 and the con-

C.L

1$m

k ((BG),)

.

According to Atiyah and Hirzebruch, lim KU( (BG),) is expressible in terms of the ring R(G) of unitary representations In fact, let E : R(G) -P 2 be the homomorphism obof G tained by sending each representation of G into its dimension.

.

Theorem of Non- surjectivity for

A

a

233

GtH

Define I (GI = ker E and give to R(G) the I (G) - adic topology, that is to say, the topology which has the set {I (G)" n L 1) as a sub-basis for the system of neighborhoods of 0 E R(G) Then

.

lfim KU((BG)~)= lfim R(G)/I(G)" =

ii(~)

.

128; 9 41

Since G is compact and connected, it has a maximal torus T which can be written as the group of k-tuples of reals mod 1 ; every irreducible representation of T is 1dimensional and given by a homomorphism 0 : T + U ( 1 ) , $(x,,

... ,xn) = exp[2ni(alxl + ... +

akxk I

,

.. .,exp(2nixk),

ai E 2. Hence R(T) = Z[exp(Znix,) ,exp(-2nixl), for exp(2nix.) - 1 exp(-2nixk)l : writing z 7

j

,

we see that

..

I (TI is the maximal ideal generated by zl,. ,zk, (1+zl)-' - 1, -1 I(T)" = 0 because an element of ...,(l+zk) - 1 ) . Hence, this intersection is a power series whose lowest term has arI(T)" , it follows bitrarily high degree. Since I(G)" c that I(G)" = 0 and so, R(G) is Hausdorff; now 1 5 1 ; chap. 111, 9 5, n.41 shows that R(G) is isomorphic to a totally We are going to show that dense proper sub-ring of f i ( G ) there is an injection Hom(G,U) + R(G) ; with this, the proof of 4.1 is complete: in fact, it follows from the commutative diagram

.

Hom(G,U) >

>

R(G) >not a surjection >

I=

V

To obtain the injection Hom(G,U) >->

R ( G ) we first notice that because G is compact, Horn(G,U) l&m Hom(G,U(n)) On the other hand, if p , p ' : G + U(n) are t w o conjugate homomorphisms (actually, unitary representations of G ) they have the same character and so, are identified in R ( G ) . This defines an injection of Hom(G,U(n)) into R ( G ) and ultimately, due t o QL

.

234 Group Homomorphisms and Maps between Classifying Spaces

the compactness of

G

,

an injection of

Hom(G,U)

into

R(G).

Vector Bundles over Suspensions. In this section we shall consider finite dimensional complex vector bundles over a compact path-connected Hausdorff space X It is well known that if E is a bundle of this type, there is a bundle r) over X such that 5 @ r) is trivial. 5.

.

We shall1 prove that if X is the suspension of a space Y there exists such a bundle n which has the same dimension as 5 Furthermore, K-theory will be used to show that the result obtained is the best possible.

.

L e t 5 = (E,p,SY) b e a c o m p l e x n - b u n d l e , Theorem 5.1. w i t h Y a b a s e d , compact p a t h - c o n n e c t e d H a u s d o r f f s p a c e . Then t h e r e is a c o m p l e x n - b u n d l e r) o v e r SY s u c h t h a t 6 @ is t r i v i a l .

Proof. First we construct a vector bundle which is isomorphic to 6 but which allows a certain degree of manipulation. Let us regard SY as the union of two cones C-Y and , where Y'C c-Y = Y x [ - 1 , 0 1 / ( - 1 x Y u [ - 1 , 0 1 x {yo}) and + C Y = Y x IOrll/(l x Y u IOrlI x {yo}) v

Since both cones are contractible the restrictions of to C-Y + and C Y are isomorphic to trivial vector bundles; let a- : + EIC-Y -B C-Y x C n 'and a + , : EIC'Y -B C Y x C n be bundle iso+ morphisms. Notice that Y = C-Y n C Y and consider the isomorphism of trivial vector bundles p : Y x Cn + Y x C n defined by p = ( a - l (Sly))( a y ' l (Sly)) Now form the vector bundle p ( 5 ) over SY having for total space ( C ' Y x Cn) U (C-Y x Cn) the adjunction space of Y'C x Cn and C-Y x C n by the function p Next, define the bundle map

.

.

a : c

as follows: for every

->

P(6)

e E E ,

, a(e) =

{ a+(e) (el a-

if

e E EIC+Y

, if e

E

EIC-Y

.

Vector Bundles over Suspensions

235

Since a+ and a- are isomorphisms on the fibres, so is a and hence, 5 and @ ( < ) are isomorphic. (5) over SY Now consider the vector bundle rl = whose total space is the adjunction space (C’Y x En) UB-,

@-’

(C-Y x C“)

,

and whose bundle map is the projection on the first component. We are going to show that B ( E ) @ 8 - l ( 5 ) is trivial, so proving the theorem. The isomorphism B and its inverse -1 B correspond respectively to maps 0 and 0-’ from Y into GL(n,C); one should notice that for every y E Y,O(y)-’= -1 0 (y). Furthermore, one observes that

I

with B @ p-’ by the matrix

corresponding to a map

Y

:.

Y

+

GL (2n,C) given

for every y E Y . Following [ 1 7 ; Theorem 2.4.61 we define a homotopy 0 : Y = 12n , the 2n x 2n identity matrix: In cos t n/2 -1

n sin t n/2

In sin t n/2

In cos t n/2

In cos t n/2 -In sin t n/2 In sin t n/2

In cos t n/2

This, of course, completes the proof. ‘The previous theorem is saying, in particular, that because SY is the union of two contractible spaces, any n-dimensional vector bqndle over SY can be embedded in a 2n-dimensional trivial vector bundle. This result can be extended. Theorem

5.2.

If

X

is a l s o n o r m a l a n d h a s an o p e n c o v e r i n g

Group Homomorphisms and Maps between Classifying Spaces

236

IU,,...,Uk) tor bundle

F;

of c o n t r a c t i b l e s p a c e s , e v e r y n - d i m e n s i o n a l v e c o v e r X c a n be embedded i n a k n - d i m e n s i o n a l

t r i v i a l vector bundle.

Proof. If 5 = (E,p,X) , for every integer j between 1 and k , there is a homeomorphism $j : EIUj + U x c n j whose restrictions to the fibres are isomorphisms of vector spaces. Let {qjll< j < k ) be a partition of unity subordinated to {Uj 1 1 < j < k } ; define, for each j I Jlj

:

>

E

XXC"

by

I

,

otherweise.

i : X x c" + X x (Cn)k takes j + ( x , v ) into (x, (O,...,v ,O)) with v sitting on the jthcomponent of the k - tuple. The morphism J, : E + X x gives the embedding. We now comment on the previous two theorems and for this, we need K-theory. First, we show that in Theorem 5.1, the condition that 5 is a vector bundle o v e r a s u s p e n s i o n is necessary; our forthcoming Theorem will also show that the result of (5.2) is the best possible €or n = 1 , k = 3 .

Then set

J, = C i $

j j

where +

-D

,...

X = CPk, k 2 2, and 5 b e t h e c a n o n i c a l l i n e b u n d l e o v e r CPk Then e v e r y (r-1) - d i m e n s i o n a l u e c t o r b u n d l e rl o v e r X w i t h r 1 < k i s such t h a t 5 r\ i s not t r i v i a l .

Theorem

5.3.

Let

.

e)

-

Note: This result shows that there is no line bundle CP2 such that 5 @ rl is trivial. Proof. A - operations to

and thus ,

Suppose that [ [ I + 1111 :

5 @n

n

over

is trivial and apply

Vector Bundles over Suspensions

k Since K U E P 1

237

Z[v]/(vk+') (see Chapter 4, 9 3) , where 0 This contradicts the fact that v = [ 5 ] - 1, Ar[O] Ar[r,] = 0 (the triviality of Ar[n] comes from dim T-, = r-1) Next, we show that in ( 5 . 1 ) , n is the minimum dimension for T-, so that < @ rl is trivial; our argument will also show that (5.2) is the best result if k = 2. Theorem 5 over 0 over

5.4.

X X

=

.

I f X = S2n , there exists an n-vector bundZe such that, for every r-dimensiona2 vector bundle with r < n, 5 is not trivial.

or,

Proof. Let a be a generator of HZn(Szn,Z). As a consequence of the Integrality Theorem (0.11) There is an element z E ker KU(S2") + K U ((S2") 2n-, ) ] 5 %J(S2") such that

.

I

But ffU(S2") ZY [S2nlGn(C2n)]* (where Gn(C2n) is ch(z) = a the Grassman manifold of n spaces in C 2n) and so there is an n - dimensional vector bundle 5 over S2n whose image in k(S2") is z Therefore ch[c] = c1 0 showing that the nth - Chern class cn(c) is non-trivial. Suppose that n is a vector bundle over S2n such that < @ n is trivial. Then,

.

hence

cn(q) + 0 and then,

dim q 2 n

.

We complete this section with an example which answers in the negative the following question: if X is a normal space which is covered by k contractible open sets and q is an n-dimensional vector bundle over X embedded in a trivial vector bundle of dimension less than kn , can we decompose n as a Whitney sum of non-zero bundles? Example 5.5. Let X = G P ' , 5 be the canonical line bundle over X and n be a 2-dimensional vector bundle over X such 3 that 5 @ n GS (x x C ,prl,X). Then n cannot be decomposed into the Whitney sum of two line bundles. In fact, suppose that n is the sum of two line and n2 bundles

'I,

.

238

Group Homomorphisms and Maps between Classifying Spaces

Then,

[n,] [q21

-

and because

1 = a([EI 1 =

[nl

-

- 1) + b([El - 1 1 2 a"51- 1)

lo1 + [51 =

3

-

for some

b([El- 112

-

2

a,b

E Z

,

I

,

Recall that for any line bundle ( s e e Chapter 0) and hence,

T

, yt([?1

- 1)

= 1

+ ([TI - 1) t

on the other hand,

(in both cases one must remember that K U ( C P 2 ) = Z " E I Thus, -a(a+l) = 1 , absurd.

- ll/([Cl-l)3;

CHAPTER 8 ON THE INDEX THEOREM OF ELLIPTIC OPERATORS

Introduction. It would be inconceivable to present a certain number of geometric and topological problems in which K-theory appears, either as their adequate theoretical framework or as a powerful tool capable of simplifying greatly their solution, without mentioning the deep connections between the Theory of Differential Operators and K-theory. One should recall moreover that this relationship has played a large role in the development of K - theory itself and that it has produced some of the most important results obtained in the mathematics of the last fifteen years. This motivation and some other gcod reasons, easily convinced us to present in this book - even if sketchily - some aspects of the Atiyah-Singer Theorem and of its several formulations. The purpose of this chapter is two-fold: we first wish to direct the attention of the reader to this important area of mathematics we mentioned above and second, to present enough bibliographical material as to permit a more profound independent study of the subject. It should therefore be understood more as a presentation of the Index Theorem and some of its classical consequences, rather than a detailed and complete exposition of the subject. At any rate, one must recognize that the large number of works relative to that delicate and complex subject would not leave us with any other alternative. It is our belief that we shall have reached part of our objectives if the study of the following pages will help in vulgarizing this beautiful mathematical Theory. It is neither new nor surprising that there are tight relations between Analysis and Algebraic Topology. For example, 1.

239

On the Index Theorem of Elliptic Operators

240

let y be a closed curve of lR2 which does not contain the origin 0 E lR' and consider the number of times y turns around 0 ; this numerical "invariant" is the index of y with respect to 0 , or the degree d(f) of a continuous function 3

f : S

1

-> c*

.

such that f(S 1 ) = y The number d(f) can be computed analytically, for example, via Cauchy's Integral Formula

Y

.

where g is an holomorphic function approximating f We recall that d(f) does not change when f is deformed with continuity (homotopically): furthermore, given an integer d , we can always find a function f such that d(f) = d. This is not a fortuitous example. In fact, let C (Sn-' ,GL (m,C1 ) be the set of all maps from Sn" into GL(m,C) ; then, if 2m 2 n I 2, it was proved by Bott in 1 4 6 1 that (i) if

n = even, there exists a surjection d

such that motopic ;

:

C(Sn-',GL(m,C))

d(f) = d(g)

->

z

if, and only if, f

and

g

are ho-

(ii) if n = odd, all maps f E C(Sn-',GL(m,C)) are homotopically trivial. Incidentally, we wish to observe that the periodicity of the stable homotopy groups of the general linear groups the most remarkable property of K-theory was established for the first time in that same paper. The proof of the periodicity given subsequently by Atiyah and Bott in [22] is another example of the connections between K-theory and Analysis - in this case, speciffically with the theory of Complex Functions. Let u be an open subset of IRn and let C"(U) (resp. Cz(u)) be the space of all COD differentiable maps on U (resp. with compact support). Then, a pseudo-differen-

-

-

-

241

Introduction

tial operator D

where

c(y) =

I

of order

r

on

u(x)e-i(x'Y)dx

U

is of the form

i s the Fourier transform of

u

U

(here (xly) is the scalar product of x and y in IR") and p is a C w - differentiable function on U x 7Rn with convenient properties over each compact subspace of U (these are necessary conditions for Du E Cw(U) and which allow differentiation under the integral - see [126]); furthermore, we require that for every ar(x,y)

=

x E U

and

every

y E XIn

-

COI

I

lim P (x,ay) a-rm ar

.

exists. This limit is the symbol of order r of D Whenever D is a differentiable operator, or is a polynomial in y of degree r and with coefficients in Cm(U) : moreover, if ar(x,y) E GL(m,C) for every x E U and every y E IRn - 10) , then D is said to be an elliptic differential operator. In this case, for a fixed x and by restriction, we obtain a map n- 1 > GL(m;C) ; +XI-) : s n is even, this map has a degree, called the degree of the elliptic operator D . As expected, in view of the local chart structure of a differentiable manifold, all these notions can be extended globally to a compact manifold. With this, we now make a crucial observation relative to an elliptic operator D defined on a compact manifold: the solution space of the equation Du = 0 if

is finite dimensional: the same happens to D* , the adjoint of D We then define the index of the elliptic operator D as the integer

.

i(D) = dim(ker D)

-

dim(ker D*)

.

Let us finish this paragraph with the observation that Atiyah,

On the Index Theorem of Elliptic Operators

242

at the end of his interesting paper [ 1 6 1 , after commenting upon the (real) Bott periodicity and the idea of defining the index of an elliptic operator as a function of its degree was lead to say that "analysis and topology are now inextricably mixed" One can probably say that the first study about the index of an elliptic operator appeared in F.Noether's paper on tidal movements, published in 1 9 2 1 [ 1 9 0 ] . Then, except for a few sparse results (e.g., those obtained by S.G.Mihlin in 1 9 4 8 1 , very little was done in this context and it was only during the sixties that the question attracted considerable interest. In fact, in 1 9 6 0 Gelfand clearly formulated the problem of computing explicitly the index of an elliptic operator in pure topological terms: this problem was taken up by the,analysts among whom we quote M.S.Agranovich, A.S.Dymin, R.T.Seeley and A.I.Vol'pert - who established some particular results. It was only in 1 9 6 3 that Atiyah and Singer obtained the first general answer (see Section 4 ) . The strength of the Index Theorem lies partly in the fact that it unifies in one same formulation, results apparently as different as the Chern-Gauss-Bonnet formula (cf. [ 9 7 ] ) , the Hirzebruch-Riemann-Roch Theorem and the Hirzebruch Signature Theorem (cf. [ 1 1 6 ] ) , which we presently recall. For a compact riemannian surface M with gaussian curvature k , the classical Gauss-Bonnet formula tells us that

.

1 X(M) = 2n

I

k dv

,

M where X(M) is the Euler-PoincarG Characteristic of M and dv is the density defined by the riemannian metric. The original Riemann-Roch problem consisted in computing the number of holomorphic cross-sections of a 1 - dimensional holomorphic bundle 0 over a riemannian compact surface M , that is to say, find H o ( M : f i ( ~ ) ) where n ( q ) is the sheaf defined by the germs of the holomorphic cross-sections of q : we conclude that 1 dim Ho(M:fi(~))- dim H ( M ; R ( r l ) ) = X(n) + 1 - g

Introduction

243

where x ( n ) is the Euler characteristic of rl and g is the Finally, Hirzebruch's Signature Theorem shows genus of M that if M is a 4!L-dimensionalI compact, oriented manifold, the signature of the quadratic form

.

H~'(M~)

x

H~'(M~)

->

H4'(M%)

= IR

defined by the cup-product is the L - genus of M (see Section 3 ) . It is important to observe that the Atiyah-Singer Theorem is much more than a useful unifying proposition: indeed, several interesting generalizations of the results we quoted were derived from it. For example, the Theorem of Hirzebruch-RiemannRoch was extended to every compact complex manifold following this route. In this context, we quote Bott in saying that the Index Theorem "may also be thought of as a beautiful and far reaching generalization of Hirzebruch's Riemann-Roch Theorem both in statement and in the spirit of the proof". The reader is invited to get acquainted with the different proofs and multiple applications of the Index Theorem by reading the following sections and browsing through the standard bibliography (a recent introductory study of the Atiyah-Singer Index Theorem written by Shanahan [216] contains a good bibliography). In the meantime, we observe that Atiyah and Bott proposed in [23] a proof of the Index Theorem using the 5 - function < ( s ) = CA-S

,

usually introduced for the determination of the proper values of an operator; the index formula obtained in this way is actually too complicated. Nevertheless, the idea of approaching the Index problem to the study of the Froper values of a differential operator proved itself to be an efficient and rewarding technique. We bring to our testimony the papers of MacKean and Singer [164], Patodi 11941 and Gilkey [961; these papers actually suggested a new proof of the Index Theorem to Atiyah, Bott and Patodi 1251 , using the Heat Equation's method. Recently the Atiyah-Singer Theorem crossed its pure mathematical boundaries to make its entrance in the physics of elementary particles, specially in problems related to Gauge

On the Index Theorem of Elliptic Operators

24 4

Theories. Such theories have excited the interest of theoretical physicists and owe their present success to the experimental proof of some of their predictions: the discovery of neutrius's diffusion reactions on matter (1973) and the detection of charmed particles (1974). We observe also that the right mathematical framework for Gauge Theories is precisely the geometry of principal bundles; for example, the result obtained by Atiyah, Hitchin and Ward about certain solutions (instantons) of the Yang-Mills equations (these are, roughly speaking, generalizations of the Maxwell equations of ElectroMagnetism) concerns SU(n) - bundles [ 311 !*I Finally, Atiyah and Ward were lead towards Algebraic Geometry problems, by interpreting the equations defining the instantons as equations in the 3-dimensional complex projective space [38]. 2.

The Index of an Elliptic Differential Operator

Given define

x = (x,x2,...,xn) EIR"

and

p

=

(p,~p2,...~pn)E N n ,

.

with i2 = - 1 On the other hand, for any open set U 5 IRn and any finite dimensional complex vector space V , we denote by C"(U,V) the space of all Ca - differentiable functions from U into V Then, if A and B are finite dimensional vector spaces over I: , a linear function

.

D

:

Cm(U,A)

-> Ca(U,B)

is a d i f f e r e n t i a 2 o p e r a t o r of o r d e r such that

r

if there exist functions

gp E Cm(U,Hom(A,B))

Df

(2.1)

1

=

gp(x)DPf

.

Ipl L r

To such an operator we associate a function ar(D)

:

U

x

IRn >

Hom(A,B)

defined by (*)

It is worth noticing that this result was obtained using the Index formula.

Index of Elliptique Operators

where

245

and

.

ar(D) is called the symbol of D (it depends on rf If for every x E U and every y E IRn {O), ur(D) (x,y) is an isomorphism (thus, we are admitting that dim A = dim B ) , we say that D is an elliptic differential operator of order r As an example we quote the Laplacian n a2 A = 1 * ' i=l ax 2

-

.

indeed,

a2(A) (x,y) = -(y, 2

+

i 2 y2 +

-.-

+ yi)

*

0

if

y

$

0. We

finally observe that any operator D of order r is also of and u,+~(D) = 0 for every i 2 1 ; order r + 1,r + 2,..., furthermore, ur is additive for operators of order 5 r

.

We now transpose all the previous notions to differentiable manifolds: we shall assume, from this point to the end of the chapter that ali! manifolds c o n s i d e r e d a r e finite dimensionai!, C m - differentiable (smooth), compact a n d p r o v i d e d w i t h a riemannian metric. Let

6

and rl be two C"-differentiable vector bundles over an n-dimensional manifold M We denote by r ( 5 ) and r ( q ) the spaces of cross-sections of 5 and 11 , respectively. Then, a linear function

.

D :

r(o

-'

r(0)

is said to be a linear differential operator of o r d e r r if it is locally of type (2.1) : in other words, we shall assume that: of M so that each (i) there is an open covering { " j ' j C J U is a local chart of M (and s o f it is Cm diffeomorphic j to an open subset of IR") and moreover, both 5 and q are trivial over U

-

-

j '

(ii) for every j E J , the restriction D of D to U is j j a differential operator of order r (in the sense of ( 2 . 1 ) ) , D

j

: C m (LJj, A)

->

C"'(U.,B) 3

On the Index Theorem of Elliptic Operators

246

where A and B are typical fibres of 5 and r\ , respectively. We also want to define the symbol or(D) so to retrieve locally the description given by ( 2 . 2 ) . We assume that ur(D.) is given (over the local chart U j ) by ur(Dj) (x,y) = 3

1

gp(x)yp , where y is a cotangent vector of M with IPl’r This approach permits us to give an actual inorigin x E U j’ trinsic definition of the symbol (*I : let ?*(M) be the cotangent bundle of M : then, the s y m b o l of D is a morphism of vector bundles N

(2.3)

or(D)

:

?*(MI

->

Hom(S,n)

defined by

for every x E M, y E T:(M) , z E 6, and where f is a Cm map such that f(x) = 0, df(x) = y and s is a cross-section of E ( U (U is a local chart containing x) such that s(x) = z Of course, we must show that ( 2 . 4 ) is independent of the choice of f and s : to this end, it suffices to express the righthand side of ( 2 . 4 ) in local coordinates. In fact,

Here again, we say that D is an e l l i p t i c o p e r a t o r of o r d e r r if, and only if, ur(D)(x,y) is an isomorphism for every ; in x E M and every non-zero cotangent vector y E T:(M) other words, the symbol must be an isomorphism on the complement of the trivial cross-section of ?*(M) for D to be elliptic. This suggests an alternative definition of ellipticity. Let (D(M),n,M) be the disc-bundle associated to T*(M); then, the existence of the symbol ( 2 . 3 ) is equivalent to the existN

( * ) For more details over the motivation conducting to the definition of the symbol, please consult the excellent lecture notes of L.Schwartz [ 2 0 6 1 ; for other historical notes, see

12091.

247

Index of Elliptique Operators ence of a fibre-preserving homomorphism or(D)

(2.5)

:

n*S

>

n*r)

N

given by ar(D) (o,z) = (w,ar(x,w)z) for every w E D(M) and z E 5, , where x = n ( w ) (after conveniently identifying the elements of D(M), with the vectors of T:(M) of length < 1). Then, if S(M) is the total space of the sphere bundle associated to T*(M) we have the following. Definition 2.6. A linear d i f f e r e n t i a l o p e r a t o r D of o r d e r r is e l z i p t i c if, a n d only if, t h e r e s t r i c t i o n of or(D) to n*ClS(M) is a n i s o m o r p h i s m . The latter definition of ellipticity has the advantage of setting that notion in a convenient Algebraic Topology framework: indeed, one can see that the correct procedure is to interpret it with the aid of the group KU(D(M) , S ( M ) ) . We recall that if 5 is a Cm-differentiable complex vector bundle over a manifold M (compact, riemannian, according to our convention), then 5 is endowed with an herHence, the space r ( < ) of cross-secmitian metric ( , ) tions has automatically the structure of a real pre-Hilbert space: the scalar product is defined by ~

.

(SltlC =

I

(Srt)C vg

M

for every s,t E r ( c ) , where v is the canonical measure g Thus, for every associated to the riemannian metric g of M differential operator D : r ( < ) + r ( r ) ) , there exists a f o r m a l a d j o i n t D* : r ( u ) + r ( < ) defined by

.

(Dslt) = (slD*tIC u for every s E r ( c ) and t E r ( u ) . Since the hermitian structures of < and Q can be transmitted to n*C and n*q respectively, it is possible to define the adjoint a:(D) : n*r) of the symbol ur(D). The crucial results concerning us + n*< at this moment are the following (for their proof, consult [ 1931, pages 70-73) : (2.7)

1)

the differential operator adjoint D* :

D

has one and only one

248

On the Index Theorem of Elliptic Operators

From relations ( 2 . 7 ) and ( 2 . 8 ) we obtain the following two conclusions (I) ker D and im D* are orthogonal: more precisely, if s is orthogonal to im D* , from (slD*Ds) = 0 we deduce that 5 (DSIDS)~ = 0 and hence, Ds = 0 , showing thereby that ker D is the complement of im D* in r ( c ) ;

D is elliptic, its adjoint D* is also such. On the other hand, as shown by Gelfand 1 9 5 1 , ker D i s a f i n i t e d i m e n s i o n a l c o m p l e x v e c t o r s p a c e w h e n e v e r D i s e l l i p t i c . This fundamental result plus (I) and (11) justify the notion of index. Definition 2.9. The a n a l y t i c i n d e x o f a n e l l i p t i c d i f f e r e n t i a l operator D i s the integer (11) if

i a (D) = dim(ker D) = dim(ker D)

-

dim(ker D*) dim(coker D)

.

At that point, mathematicians started to contend with the verification of Gelfand's conjecture (see 1 9 5 1 ) namely, that ia(D) could be expressed uniquely in terms of topological invariants. In the remainder of this section we shall express the analytic index of an elliptic operator in different ways: thus, we hope to expose some facets of the deepness and richness of the integer ia(D). To begin with, since the ellipticity of D implies that of D* , it is clear that the operators A[ = D*D and A,, = DD* are elliptic: moreover, they are positive-definite and self-adjoint. Then, because ker A = 5 ker D and ker A = ker D* , 17

.

ia (D) = dim(ker A ( ) - dim(ker A n ) This is a convenient formula to have since it allows the use of some fundamental results concerning the eigenvalues of operators like A E For every real number X , consider the set (2.10)

.

r,(c)

= IS E

then, Hodge's Theory Theorem

2.11.

r([) [118]

~ A ~ ( s= )h s l

;

implies the following.

- (i) The s p a c e

r,(C)

i s f i n i t e dimensional,

Index of Elliptique Operators

for e v e r y

h E lI? ; f u r t h e r m o r e ,

able number o f

r,(C)

249

f o r aZZ but a count-

= 0

A's ;

T X ( 5 ) is an o r t h o g o n a l d i r e c t s u m ; m o r e o v e r , t h e c o m p l e t i o n of r , ( S ) is t h e H i l b e r t s p a c e (ii) t h e s u m of t h e s p a c e s

L2(r(S))

(here

L2(r(C))

w i t h r e s p e c t to t h e m e t r i c

X

,

4.

r(5)

is j u s t t h e c o m p l e t i o n o f (

1

;

D t o FA( 0). If the operator considered is (or A q ) there is an assymptotic development for t + 0

A E, given

by (2.14)

where n = dim M and the coefficients vk(Ag) can be conThis important structed locally from the coefficients of A C result, first proved for the Laplacian by Minakshisundaram and Pleijel [183], was shown in the general case by Seeley 12081. With its aid, the index can be expressed by the formula

.

i,(D)

(2.15)

= u0(Ac)

-

u0(A,.,)

.

More precisely, let {cp,) be a complete orthogonal set of eigenfunctions associated to the eigenvalues {Am} of A (here For every t > 0 , let Ht be A means either A < or All) the differential operator

.

Ht = e-At notice that, on the one hand, (-&+A)

,. Ht

Ht = 0

satisfies the Heat Equation

,

and on the other hand, the eigenvalues of Ht are given by - Xmt Then, writing an arbitrary cp E L2(r(c)) in the form e cp = Ca cp m m (Theorem 2.11) we have:

.

(Htcp)( X I = because

it follows that

(*I

The book of Berger et al. [42] this subject.

is a good reference about

Index of Elliptique Operators

251

or, by setting h (t;xryl

(Htv)(XI .erIn other wordsr h ing the vector bundle 5 % < * r Hom(Er[) over M x M (the hermitian product on 5 induces an isomorphism between its conjugate 5 and its dual t * l r the series h(t;x,y) - which converge for t > 0 - can be viewed as elements of the fibre ( 5 @ ( * ) (x,y) Hence, the trace of h(t,x) = h(t;x,xI is gi-

.

ven by -Xmt

tr(h(t,x))

1e

=

(cPmrcPm)

m

-Amt = C e m

=

h(t)

.

1 ( * I The reader can easily verified that if M = S , 5 = rl = d2 trivial line bundle and A = - the kernel of Ht is the 2 dO - function

ern

h(t;O,O')

=

1 e-m

2

t

.

im(0-0') e ,

m

2

this function converges and

h(t,O,O)

=

1 e-mt,"-

m t - 0 .

1

2v n t

when

On the Index Theorem of Elliptic Operators

252

Since ia(D) = h (t) - hn(t) (this equality is a consequence 5 of (2.12) and (2.13)) we obtain the following integral expression for the index:

When

t

+

0, h(t,x)

has an assymptotic development

(2.18)

then, the format of formula (2.17) can be approximated to that of (2.15). Utilizing (2.14) and (2.16), it follows that tr(lJk(XiA))Vg(X) !lk(h) = M and hence, (2.17) can also be written in the form

I

)

.

(tr(uo(xrAc)1 - tr(po(xrAq)) vg(x) M The trouble with this approach to the analytic index via the Heat Equation (as initially tried by Atiyah and Bott) was the difficulty in obtaining a good description of the coefficients uk(x,A) of (2.18). Meanwhile, Patodi's decisive contribution [I941 gave an inkling of the simplified formula (2.19). For a characterization of the uk(x,A)Is , the reader may consult Gilkey's papers [96] and 1971 , and of course, the work of Atiyah, Bott and Patodi [25] Next, we describe certain arithmetic considerations which yield information about the coefficients p ( A 1 . The k 5 central idea is to replace hS (t) = 1 e ltdim r ( 5 ) by a kind (2.19)

ia(D)

=

.

of Riemann's (2.20)

5

-

x

function, namely

cS(s) =

1

dim

A-'

h

r,cE,) ,

with s complex. On the other hand, it is known that the gamma function is given by +oo s-I -u e du r(s) = u

I

0

Index of Elliptique Operators

with

Re(s) > 0

.

Writing

I

u = At

253

we obtain

+m

T(s) =

pts-le-Xtdt

0

and therefore, +m

0

Because of

(2.20)

,

1

+m

5 5 ( s ) T ( s )=

ts-’hg(t)dt

.

0

Now we can show that the points sk = -k/2r, k > -n are the simple poles of C g ( s ) r ( s ) ; indeed, at these points the residues are precisely ( A ) . In particular, k 5 ‘ ; g ( 0 ) = ~ . I ~ ( A -~ 1)

and we obtain the equation (2.21)

ia(D) = 5 ( 0 )

5

-

~

~

(

.0

)

The previous considerations are intrinsecally connected to the idea of assymptotic development: remarks of this kind were already made in the work of Minakshisundaram and Pleijel. The reader who desires to find out how extensive are the connections between Number Theory and the Index Theorem should consult the excellent book of Hirzebruch and Zagier [ 1 1 7 1 . To conclude this section we shall indicate how the definition of the index of an elliptic differential operator can be generalized to a family of differential operators. A differentiable complex ( r ( 5 ) ,d) over a manifold M is a finite sequence {

n*cn

->

o

is exact.

Given an elliptic complex ( r ( S ) , d ) over M , it is natural to study its cohomology, namely, the spaces H i ( r ( 5 ) ) = ker di/im di-, These spaces happen to be finite dimensional; this suggests us to view the index of such a complex as its Euler-Poincar& characteristic.

.

Definition 2.23. The analytic index of an elliptic c o m p z e x ( r ( 5 ),d) over a manifoZd M is the integer ia(S,d)

=

C(-l)i dim Hi(r(

r (5,)

such that

where Si is a C"- operator on r(Si) : next, we use the finiteness properties of the Fredholrn operators 1 si

r (5,)

-

.

Four Standard Complexes (I) Let M be a compact, oriented, riemannian manifold of dimension n and let - r * ( M ) be its cotangent bundle. Consider the complex vector bundle n n E = @ Sp = 0 AP(,*(M)) 63 E p=o p=o 3.

.

where E is the trivial complex line bundle over M T h e elements of the set r ( 5 ) of all cross-sections of 5 will be called (complex) exterior differential forms over M ; in particular, a cross-section of the fibration Sp will be an ex-

255

Four Standard complexes

terior differential p-form or just a p-form, for short. We denote the set of all p - forms over M by nP(M). with the aid of the differential operators

(R (M)rd)

we form the De Rham Complex

R 0( M ) -> d

>

0 -

Q

1

:

...

(M) -> d

->

d

n

$2 (M)

-> o ;

because of De Rham’s Theorems, its cohomology groups are naturally isomorphic to the ordinary cohomology groups of M with complex coefficients. A s we have seen in the previous section r ( c ) is automatically endowed with an inner product ( 1 )5 ; furthermore, let us recall that there is a well-known automorphism * of Q(M) - the Hodge star operator - characterized by the following properties: (3.0)

for every integer

*

:

RP(M)

p

->

,

0< p < n

Rn-‘(M)

,

;

(3.1)

for every

a,B E QP(M), a

A

*B = ( a / B ) S * l

(3.2)

for every

a E QP(M), **a

=

(-l)p(n-p)a

Notice that

(3.1)

implies

(alB)< =

a

A

*B.

,

1 E Ro(M);

. We use Hodge’s

M star operator to define the exterior codifferentiation 6 : nP(M)

-> QP”(M)

6 = (-1)

np+n+l

d

:

.

One should observe that given any a E nP-l (M) and since d ( a A * B ) = da A * B + (-lIp-’a A d*B

I

f3 E

RP(M) ,

.

it follows that (dalB)S = (a16B1c d(a A * B ) = 0 M An element s E r ( < ) Hence, 6 is just the adjoint of d cZosed (resp. c o c l n s e d ) if ds = 0 (resp. 6 s = 0 ) . and

.

is

256

On the Index Theorem of Elliptic Operators

We now define the H o d g e - L a p l a c e o p e r a t o r A = (d+6)2 = d6

+ 6d

and claim it to be elliptic. In order to prove this assertion it suffices to study the 1 - symbols of d and 6 To this end, let x E M I y E T P and s like in ( 2 . 4 ) ; if f is a smooth function such that f(x) = 0 and df(x) = y ,

.

N

u1 (d)(x,y)s(x) = i(df = i(y

A

A

s

+ f

ds) (x)

A

s(x)).

Thus, the symbol of d is just left exterior multiplication by iy and therefore, because of (2.8), the symbol of 6 is over FY , up to a factor equal to the internal product w Y i (Recall that w y : 5 , 5, is defined from

.

-

where {e,,...,e,I moreover, f o r every conclude that

= II y

ItL

is an orthonormal basis for M:T a,b E tX, (y A a,bI5 = (a,o (b))5) Y

.

and We

. 2

N

In other words, a 2 ( A ) (x,y) is scalar multiplication by llyll in Ex and s o , A is elliptic. The self-adjointness of A implies that its analytic index is 0 ; this and other considerations severely reduce its usefulness. However, from the ellipticity of A , the decomposition A = (d+6)(d+6) and ( 2 . 8 ) , we deduce that D = d + 6 is elliptic. Let us now assume that, Moreover, let besides our usual requirements on M , aM = (d

.

b

P

=

dim H P ( M ; d : )

,

O < p c n

be the Betti numbers of the manifold

M

I

and set

257

Four Standard Complexes

proposition

3.3.

D = d

The differentia2 operator

+

->

6 : I'(Ceven)

rtCodd)

is elliptic and

n ia(D) = X ( M ) =

1

(-1)'b

p=o

As =

0

P *

Proof. A differential form s E r ( c ) such that is said to be h a r m o n i c ; in that case, because

(Asls)

5

=

(ddsls)S + (6dsls)5 = ( 6 ~ 1 6 s ) + (dslds)S

5

.

,

ds = 0 and 6 s = 0 It is clear from the definition of A that if s E r ( 5 ) is such that ds = 6s = 0 then s is harmonic; thus, in other words, a differential form over M is harmonic if, and only if, it is closed and coclosed. Notice that every harmonic form over M defines a cohomology class in H* (M;C) ; but, as one learns in Hodge Theory, each cohomology class of M contains a unique harmonic representative (see [98]). Hence, = ker(Alr(S even) )

r e

H~~(M;c)

P>O and ker D*

=

@ H 2p+! ( M ; C )

ker(Ajr(Sodd))

.

P>O Definition

then shows

(2.9)

From the previous Proposition and the Poincar6 Duality Theorem we conclude that if M (oriented) is odd-dimensional, ia(D) = 0 . Remark.

If

n

I

1

(mod 4 )

one can use Hodge Theory to prove

258

On the Index Theorem of Elliptic operators

that the differential operator K , defined in the algebra of even differential forms from the formula Ka

=

(-1IPd*a

+

*da

for every a E i22P(M) , 0 5 p 5 [n/21 , is elliptic and furthermore, ia(K) (mod 2) is the (real) Kervaire semi-characteristic [421

k(M)

dim H2p(MB)

=

.

(mod 2)

p=o

Finally, Proposition 3.3 and Gauss-Bonnet Theorem show that the total curvature of a closed riemannian manifold is ia(D). In fact, the Chern-Gauss-Bonnet Formula can be obtained through a direct treatment (for more details, please consult [97, Chapter 41 1 . M will be a c l o s e d , oriented, riemannian manifoZd of even dimension n = 2m We use the same notation as in the previous example. n Let T be the endomorphism of 5 = @ cp , defined p=o on g p by the formula (11) For our second example,

.

2

(i = - 1 )

(3.4)

Because of

Since T so that

(3.2)

and the definitions of

d

is an involution, we can decompose

. and

5

6

as '6

,

F s 5-

Thus, with the usual notational abuses, r ( 5 ) is decomposed in two proper subspaces r + and r , associated respectively to the eigenvalues +1 and - 1 of T , so that r' = r ( c ' ) . Moreover, the second equality of ( 3 . 5 ) shows that D = d + 6 f + + sends r+ (resp. r-) onto r (resp. r 1 . Writing D = D I r and D = D l r - , the operators

-

D+

:

D

:

-

r+ r-

> ->

-

r r+

Four Standard complexes

259

are adjoint to e a c h other (because D

is self-adjoint) and elliptic (this follows from the ellipticity of D and the functoriality of the symbol). Let us study the analytic index of D+ , namely, ia(D+ )

dim (ker D+ )

=

-

dim (ker D-)

.

We begin by observing that A T = T A ; hence, identifying ker A (space of the harmonic forms of r ( S ) ) with H*(M;C) , the involutivity of T implies the decomposition H*(M;c) where Hf(M;C) values 5 1 of ia(D

(3.6)

=

H:(M;c)

3 H*(M;c)

are the eigenspaces associated to the eigenT , respectively. Hence,

+)

=

-

dim H:(M;C)

dim H*(M;C)

,

because the solutions of D+s = 0 (resp. D-s = 0) are precisely the harmonic forms of r + (resp. r - ) . The subspaces = HP(M;C) mH2m-p(M;C) 0 < p < m and Hm(M;C) of H*(M;C) P are stable under the action of T on H*(M;C) ; actually, Definition 3 . 4 shows that T exchanges the two factors of E P; hence, the subspace of E associated to the eigenvalue + 1 P of T equals the one associated to the eigenvalue - 1 Consequently, (3.6) becomes

E

.

m ia(D+ ) = dim H+(M;C) If m

** -

is o d d , -1,

T(s)

dim Hm(M;C)

.

shows that T = i* As * is real and = -To for every s E Hm(M;C) ; hence T is

.

(3.4)

an Isomorphism of

HT(M;C) ia(D

If m

-

is e v e n ,

T

=

*

+)

onto = 0

Hf(M;C)

and therefore

.

is real and therefore, we can write

m ia (D+ ) = dim H+(M;IR) But, for every non-trivial s E

-

dim Hm(M;IR)

.

Hy(M;IR) (resp. t E Hm(M;IR))

On the Index Theorem of Elliptic Operators

260

I I

S A

=

S

M

t

A

t

M

S

*S

A

=

(SlS)

=

-1

t

5

> 0

*t = -(tlt)5 < 0

A

;

M

this means that +>

I

M

ia(D+)

1

i s the signature of the quadratic

.

S A s d e f i n e d b y c u p p r o d u c t on M This topoM logical invariant Sign(M) is called the H i r z e b r u c h S i g n a t u r e of M If we make the convention that Sign(M) = 0 whenever dim M = 2m with m odd, we can state the following.

form

s

.

Proposition 3 . 7 .

The a n a l y t i c i n d e x o f t h e o p e r a t o r

.

t h e Hirzebruch Signature o f

D+

is

M This result explains why the elliptic operator D+ is usually called the S i g n a t u r e O p e r a t o r of M , We suggest that the reader searches in the "ad hoc" literature to evaluate the fundamental r61e this operator plays in the general theory of the Index. (111) Our third example is supplied by Complex Analytic Geometry. In this case, M will be a (compact) complex manifold of complex dimension m We begin by noticing that an exterior differential form over M can be written as

.

adzk1

A

...

A

k dz

A

...

dzgl A 1 m z ,...,z

dzeq and -1 z , . . . , z-m A

(form of type (p,q)) where are holomorphic local coordinates of M Because of this, the complex vector bundle 5 = A ( T * (M)) 8 E decomposes into the direct s um m 5 = 9 c r , = 9p q r=o p+q=r

.

cr

and r(gPrq) is the set of all forms of type (p,q). Hence, r ( cPrq) the exterior differentiation d defined on r ( E ) =

-

Pr9

decomposes into two operators a and a ; more precisely, a and a are defined by linearity from the formulas

Four Standard Complexes

so

that

aPrq

(resp. zprq) takes

26 1

r(gptq) onto

r(< P+l l q )

(resp. r(gprq+l)). In this situation, the complex analogue to De Rham's is the D o l b e a u l t ' s Complex

--

(it is easy to see that a a = 0 ) . We now call the riemannian metric of M into the play. More precisely, the underlying 2m-dimensional real differentiable structure of M has a riemannian metric such that the canonical involution J : T ( M ) + T (M) defined by

...

,xm ,ym such that zk = xk + (for local coordinates x 1 ,y 1 , k 2 iy I 1 = -1, 1 < k < m) is an isometry. Then, we can talk about the adjoint of a (we have the equality a * = and one easily verifies that

a*

-*a*

=

a a* + a*a

-

is just the complex Laplacian over M . ( * I It follows that a i s e Z l i p t i c a n d so i s D o l b e n u Z t ' s C o m p l e x . More generally, let r- be an holomorphic vector bundle over M and let n ( q ) be the sheaf of germs of all Suppose that Q is endowed holomorphic cross-sections of 0 with an hermitian metric; then one can extend a to r ( < 8 q) so that

.

-

o

--->

r ( c O i OB

q ) --->

a

-

r(co"

8

a

->.

-

. . --> a

r ( co,m@

q)

-ro

becomes elliptic (for the details, the reader is referred to 11161 , p. 187 and on). This complex is the so-called T w i s t e d D o l b e a u l t Complex. For two given integers p and q , O~p,q(m, we denote by GPpq the space of all complex harmonic forms of type (p,q) and with coefficient in Q , that is to say (*)

In fact, if a =1/2A.

M

is a Kahler manifold

(cf. Chapter 3 ,

5

11,

On the Index Theorem of Elliptic Operators

262

The results of Dolbeault [ 731 , Kodaira [ 1581 and Serre [ 2 1 5 1 form the equivalent of Hodge's Theory in Complex Analytic Geometry. We should point out, in particular, that the complex vector spaces G p r q are finite dimensionai and that GOrq Hq(M;Q(n)). Thus, if we write ,leven=

@;

p q g l

q

,

q odd =

4Lo

gor2q+l

r l r

C O

we obtain the following. Proposition

-a +

3.8.

3*

The differential operator :

r(qeven1 -

>

r (TI odd

is elliptic a n d

m ia(T+T*) = x(M;n)

=

1

dim Hq(M;Q(v))

g=0

.

m The characteristic x ( M ; q ) =

1

(-l)q

dim G O r q is known usu-

q=o

ally as the Riemann-Roch Characteristic of M (if rl = E I x(M;E) is the arithmetic genus of M ) . Slight generalizations of the preceeding considerations are the departure point for extremely fruitful developments in the theory of kshlerian manifolds, notably in questions connected to Algebraic Manifolds ( * ) and the Riemann-Roch Theorem, one of the most profound results of Algebraic Geometry.

In our last example we shall be concerned with the Dirac Operator. Since this operator can be defined only on Spin-manifolds, we shall give a brief account of those results on Clifford Algebras and Spin groups which are relevant to our work (for the details one can consult Chapters 1 1 and 1 3 of 11341 (IV)

(*IWe recall that a compact, complex manifold M is algebraic if ,and only if, M admits a kzhlerian metric whose fundamental class belongs to the image of the natural homomorphism H2(M;Z)

+

2

H (M;W) (cf. 1 1 5 9 1 ) .

Four Standard Complexes

263

or the basic paper of Atiyah, Bott and Shapiro t 2 6 1 ) . Please notice that we have already made our acquaintance with Chifford Algebras, namely, during the proof of the Hurwitz-Radon-Eckmann Theorem (see Chapter 4 ) .

cal base fined by

Let A =IRn {el, ,en}

...

be a real vector space with the canoniand let q be the quadratic form de-

.

for every a = (al,...,an) E A The CZifford A Z g e b r a Cn is the quotient of the tensor algebra @A by the ideal generated by all elements of the form a 3 a - q(a) 1 Hence, Cn is an associative algebra with a unit element 1 , generated by the elements el,...,en with the relations

-

' e i

=

-1, e.e = -e e 1 1 j i

if

i

j, i,j = Ir...,n

(cf. Chapter 4 ) . The multiplication on

Cn

r n u Z t i p Z i c a t i o n . Notice that we can regard the products

ekl

e k2

...

.

t

is called Clifford Cn as generated by

e , k , < k 2 < . . . < k r , 1 ~ r 5 n kr

and A(A) are isomorphic and thus, dim Cn = 2"; moreover, n' as left O(n) -modules. Using this isomorphism we define a Z 2 graduation on Cn , namely,

cn

=

+ cn

Q

c;

which corresponds to A (A) = A even (A) G) Aodd(A) . Clearly A - One of the readily visible facts is naturally embedded in Cn about Clifford Algebras is that for n = 2m,thecomplexification is a matrix algebra over the complex numbers; it Cn = C n N n C follows that Cn is a matrix algebra over IR or M On the other hand, if n is odd, Cn is a direct sum of matrix algebras. Next, consider the automorphism a : Cn -. Cn defined from e a (eklek2...ekr) = ( - 1 1 ekr ~ ek2 k l

.

N

.

N

264

On the Index Theorem of Elliptic Operators

and let Spin(n) such that

be the set of all invertible elements g E ~ + n,a(g) .

(3.9)

{

(i) (ii)

gag-1 E A

g

=

for every

g E Cn

~:

a E A

.

This set, together with the multiplication induced from Cn, is a group. It is easy to verify that, for every element g of the group Spin(n), the function n : A -+ A defined by 9 n (a) = gag-' is an orientation preserving isometry; hence, 4 we can define a surjection

n : Spin(n) >

SO(n)

by taking every g E Spin(n) into and moreover, ker n = g {-1,l). This shows that we can regard Spin(n) as a double covering of SO(n). Indeed, if n 3 , the Lie group Spin(n) is the universal covering of SO(n) , because n 1 (SO(n)) = z 2 . Spin(n) has a complex representation of dimension 2" , denoted usually by S and called the S p i n R e p r e s e n t a t i o n ; its representation space (denoted also by S ) is called the S p a c e o f S p i n o r s . We wish to observe that Spin(n) acts on S and this action is induced by the Clifford multiplication (3.10)

> s ;

p : A @ S -

furthermore, since for every

g E

Spin(n) , a E A

and

x E S,

the function p is actually a Spin(n) -module homomorphism. Now let M be a compact, connected, oriented, riemannian manifold of dimension n We say that M is a S p i n m a n i f o l d if the structural group SO(n) of T ( M ) can be lifted to Spin(n). More generally,

.

3.11. A S p i n - s t r u c t u r e o n M is a f i b r e - p r e s e r v i n g i s o m o r p h i s m b e t w e e n t h e o r i e n t e d b u n d l e s T ( M ) and P x Spin (n)7Rn , where P i s a p r i n c i p a l Spin(n) - b u n d l e o v e r

Definition

M .

Each lifting of SO(n) to Spin(n) defines a Spinstructure on M and so, the Spin-structures of M are clas-

Four Standard Complexes

265

sified (up to isomorphism) by H 1 (M;Z2). If M is simply connected, M has essentially a unique Spin-structure. It is possible to show that M has a Spin structure if, and only if, its second Stiefel-Whitney class w 2(M) is trivial [ 4 4 ] . Thus, CP", n = odd, is an example of Spin-manifold. From now to the end of the section we shall assume that M has a Spin-structure'defined by a principal Spin-bundle P over M Let 5 be the bundle with fibre S associated to P ; its total space is given by E = P x Spin (n) s . This bundle 5 is called the S p i n o r B u n d l e over M; the D i r a c O p e r a t o r & over M is a differential operator of order 1 over r ( 5 ) . More precisely, & is defined as the composition

.

where v is the covariant derivative induced by the riemannian connection of M (Levi-Civita Connection) and p * is the morphism induced by the Clifford multiplication (3.10) (recall that we can identify T ( M ) and T * ( M ) because M is riemannian). Thus, if {v,, vn) is an orthonormal basis of T(MI, for every s E r ( 5 ) ,

...,

k= 1

where

Vv s is the covariant derivative of s in the direck Since &2 = -A = Laplacian of M , it follows that tion vk & is elliptic. A particularly interesting situation arises whenever the dimension of M is even, say n = 2m. In this case, the Spin Representation S becomes the direct sum of two irreduc- of dimenible (non-equivalent) representations S+ and S sion 2n- 1 More precisely, the Space of Spinors, identified to a minimal left-ideal of Cn (for example, to the left ideal generated by (el+ iem+l)( e 2 + iem+2) (em + ieZm)) decomposes

.

.

N

. ..

26 6

On the Index Theorem of Elliptic Operators

as

, with

S = S + q S-

S+

=

5;

n

and

S

S

-

.

N-

= Cn n S

Because the Clifford multiplication exchanges the factors S+ and S among themselves, one obtains two Spin(n) -module homomorphisms:

+

: A @ S+-

> S - ,

-

l~

: A @ S - - > S

+

.

Setting E+ = P x S* , 5 becomes the direct sum 5 = Spin (n) + and p al'5 @ 5- and the properties of the morphisms p low us to define the operators

-

a+ : r ( c+

r(5-1,

-+

A+-

:

r(c-)

->

r ( c+)

H be the space of the Harmonic S p i n o r s over M , that is to say, H = ( s E r ( c ) I& = 0 ) Since & is self-adjoint (consider the usual inner product on r ( 5 ) ) it follows that 3- is adjoint to Thus, H = H+ Hwith w h i c h a r e still e l l i p t i c . Let

.

.

H+ = ker

a+, H -

= ker &-

coker &+

and consequently ia(&+)

=

dim H

+ -

dim H-

,

3.12.

Spin-index o f

The a n a l y t i c i n d e x o f

+

is c a l l e d t h e M and i s d e n o t e d b y Spin(M). Whenever n 1~ 0 (mod 4 ) Spin(M) coincides with the 2 - g e n u s o f M (see next Section): this is one of the interesting topological facts about Spin(M1. To conclude this section, we wish to observe that there are interesting connections between the Dirac Operator and the other elliptic operators we described before (the reader is referred in particular to Atiyah's lectures in [20]). Finally, we notice that the study of elliptic operators over Spin-manifolds allows one to retrieve certain classical results concerning compact Riemann surfaces [ 1 9 ] .

Definition

The Index Theorem

267

The Index Theorem A s we have said at the beginning of this chapter, we do not intend to give here a proof of the Index Theorem, but rather, to present the various concepts which make up its statement, comment on the main ideas involved and finally, apply the Theorem to some of the situations discussed in the previous section. Let D : r ( < ) + T(n) be an e l l i p t i c d i f f e r e n t i a l o p e r a t o r of order r on a manifold M (not necessarily oriented) but which we suppose to be riemannian, closed and of dimension n In view of Definition 2.6,. the ellipticity of C is equivalent to saying that the symbol ar(D) : n*S + n*n is a vector bundle isomorphism when the bundles involved are restricted to S(M). The presence of the bundles n * < , n*n and of the bundle isomorphism ar(D)lS(M) indicates that we are in a classical situation of K-theory. We associate the element d = d(n*S,n*n,ar(D) IS(M)) E KU(D(M) ,S(M) to the triple (n*

-

KU(D(+I),S(M)1

in K U - Theory (see (0.8)) is related to the cohomology Thom isomorphism 'p* by the formula ch

(4.6)

'p,

(x) = ( - 1 )

n 'p*(ch x

-

(Td a ) - ' )

for every x E KU(M) , where n = dimension of 0 (see [ 4 9 ] 1 . Moreover, the Chern character ch D of the difference bundle d(n*S ,n*q ,ur (D)I S (MI) is such that e(~* (M)

(4.7)

-

-1

'p*

(ch D) = ch 5 - c h q

-

where e(T* (MI is the Euler class of the real SO(2n) bundle T * (M) We are now ready to concentrate all our attention on the celebrated A t i y a h - S i n g e r I n d e x Theorem 1341.

.

Theorem 4 . 8 . L e t 5 and q b e d i f f e r e n t i a b l e v e c t o r b u n d l e s o v e r a c l o s e d d i f f e r e n t i a b l e m a n i f o l d M and l e t D : r(5) + r(q) be a n e l l i p t i c d i f f e r e n t i a l o p e r a t o r . The n,

Note that from this Theorem it follows immediately that it(D) is actually an integer. Furthermore, it is easy to see directly that it(D) = 0 w h e n e v e r dim M = n is odd (it suffices to use the antipodal map of D(M)); this generalizes the analogous result given by Proposition 3 . 3 relative to the particular elliptic operator D = d + 6 : r(ceven 1 + r (Eodd) and which yields the Euler-Poincar6 characteristic of M The proof of the equality ia(D) = it(D) rests on the following key idea : extend the definitions of the analytic and topological indices so that they become homomorphisms from KU(D(M) ,S(M)) P KU(T*M) (*Iinto 0 and then, show that these homomorphisms coincide. The proof of that equality is obtained after a careful analysis of the properties which characterize it ; indeed, from that analysis one constructs a system of

.

(*IIf X is locally compact, we agree to define KU(X) =&(X where Xf is the one-point compactification of X. Hence KU (T*M) K U (D(M),s (MI

=

.

+ ),

The Index Theorem

27 1

axioms which completely determine an "index function". At this point, one has to show that i satisfies the axioms obtained. a Observe that the expression

is well-defined for every x E KU(D(M) ,S (MI) Z KU(T*M) (cf. (4.5)); furthermore, if M' is another manifold, one can show that it I s multiplicative with respect to tensor product, that is to say, (4.9)

.

for every x E KU(T*M) and y E KU(T*M') Hence, it can be into Q viewed as a homomorphism of KU(D(M) , S ( M ) ) The following construction is essential for a more systematic investigation of the properties satisfied by it. Let f be an embedding of M into X = IRn+q (q sufficiently large) and let uf(M) be the normal bundle associated to f We have seen in Chapter 6 that the total space D(uf(M)) of the disc-bundle associated to uf (M) is homeomorphic to a tubular neighborhood V of f(M) in IRn" ; an analogous observation can be made with respect to the tubular neighborhood T*V obtained in the embedding T*f of T*M into T*X (induced by f). We should also recall from Chapter 3, Section 2, that the 2n- dimensional manifold T*M has an almost complex structure induced by a = n * ( ~ (M)) ; since

.

.

C

(cf. ( 1 . 3 1 , Chapter 6 1 , it follows that v ~ * ~ ( T * M ) n*(vf(M) 9 vf(M)) is endowed with an almost complex structure, namely that defined by v = n*(uf(M) @ C ) . These facts let us define a homomorphism JI,

:

KU(T*M)

> KU(T*V)

via the Thom isomorphism KU(T*M) KU(D(V~, (T*M)),S(V~*~ ~ (T*M))) ~ 1 ) = KU(T*V) . On the and the identification K U ( D ( U ~ ,(T*M) other hand, if U is an opep set of a locally compact space

272

On the Index Theorem of Elliptic Operators

X and, as befores X+ is the one-point compactification of X , the map X+ 4 X+/(X+-U) Z U+ gives rise to a natural homomorphism KU(U) = h ( U + )

Hence, to each embedding phism

f,

:

-> f

:

KU(T*M)

N

KU(X+) = KU(X)

M -. X

.

we associate a homomor-

->

KU(T*X)

defined by f ! = i, * $! , where i, is the natural homomor(*) phism induced by the open inclusion i : T*V c T*X Let us now observe that for every x E KU(T*M),

.

(ch f ! ( X I ) [T*Xlv = (ch i*$!

( X I ) [T*XIw = (ch $ , (x)1 [T*Vlv

;

moreover, considering $ * : H*(T*M;Q) + H*(T*V;Q) ( $ * is defined analogously to I$,) and relation (4.6) , we obtain (ch $ , (x))IT*Vlv = (-1Iq$*(ch x

*

(Td W)-~)[T*VI~

and therefore,

In particular, consider the case in which the manifold M is just the origin xo of X =?Rn+q and the embeddBecause j, is an ing f is the inclusion j : {xo) c X isomorphism, relation (4.10) is written simply as

.

(ch j,(y))[T*XI,, = -

-1 )"+'ch

y[T*(x,)

]U

,

-1)"+q y

for every y E KU(T*IX,I) z z (since, in this case, the normal bundle v is trivial, Td v = 1 ) . In other words, for every is defined by z E KU(T*X), j ;' jy'(z)

= (-1)"+"

ch(z)[T*X],,

.

( * I This construction is valid €or any manifold M is embedded.

X

on which

The Index Theorem This and

show that

(4.10)

-1 j! o f ,( x ) = j;'

j;'

0

f ,(XI

273

( f , (x) = (-l)"+'(ch

= (ch x

-

f ,(x) [T*XIy

(Td v ) - ' ) [T*Mla

,

.

Finally, because a @ v is the trivial (n+q) - dimensional complex vector bundle, we obtain from ( 4 . 3 ) that (Td v ) - ' Tda and hence, -1 1 , o f ,(X) = (ch x

(4.11)

for every

*

(Td a ) [T*M],

x E KU(T*M) = KU(D(M) ,S(M)

ind :KU(T*M)

->

KU(T*{xo))

. We Z

=

, now define

Z

to be the homomorphism which makes the following diagram commutative : ( * )

KU (T*X) Using ( 4 . 1 ) and ( 4 . 1 1 ) we conclude that the function ind c o i n c i d e s w i t h t h e t o p o Z o g i c a Z i n d e x it :KU(T*M) + (9 This coincidence shows immediately that it is a f u n c t i o n u i t h int e g r a l v a l u e s . We wish to note that the homomorphism ind is f u n c t o r i a z : if h : M + M' is a diffeomorphism, the following diagram commutes:

.

KU(T*M')

h*

Z moreover, the index functions indM properties:

(*)

M

is a point,

I

satisfy the following

ind = identity; M The homomorphism ind is independent of the embedding f : M + X selected (cf. 1 3 5 , Part I, page 4 9 8 1 ) .

(A 1 )

if

KU(T*M)

274

On the Index Theorem of Elliptic Operators

(A 2) if g is an embedding of nifold M ' , the diagram

M

into'another compact ma-

commutes. According to 1 3 5 , Part I, Proposition 4.11, (A 1) and ( A 2) characterize completely the index functions indM : in other words, if for every compact manifold M there is a function yM : KU(T*M) -+ Z satisfying ( A 1) and ( A 21, then, y M = indM. The uniqueness property of the index functions we alluded to suggests that a path one could follow in order to prove that ia(D) = it(D) would be to redefine ia(D) as a function on KU(T*M), with values on Z , and satisfying conditions ( A 1) and ( A 2). However, when we attempt to give a meaning to ia(x) for every x E KU(D(M) , S ( M ) ) , we immediately run into two difficulties: (i) one must make sure that if

D1 and

D2

are

two elliptic operators whose symbols Or1 (D1) and

ur2(D2)

define the same difference bundle in

,

ia(D,1

=

KU(D(M) , S ( M ) )

then

ia(D2);

(ii) every x E KU(D(M) ,S(M)) must be the class defined by the difference bundle of some elliptic operator. But, two symbols which determine the same difference bundle are homotopic and two homotopic operators have the same analytic index; hence (i) is actually a consequence of (i') a homotopy between

u

r 1 (D1) and

plies the existence of a homotopy between

D1

u

(D2) imr2 and D2.

Unfortunately, (i') and (ii') are in general not true for elliptic differential operators! To circumvent these problems, Atiyah and Singer [35, 55 5 and 6 1 considered a large class of pseudo-differential

The Index Theorem

275

operators constructed from the singular integral Calder6n-Zygmund operators extended by Seeley [207] to vector bundles over compact manifolds without boundary. ( * ISince the notion of symbol still exists in such class one can consider the elliptic operators within the class; interestingly enough, the notions of analytic and topological index still have a meaning for these elliptic operators and moreover, conditions (i') and (ii) hold. Because ia is an integral valued function, (A 1 ) follows easily. However, the verification of (A 2) appears to be a delicate process; the rub is that for a given operator D over M it is necessary to construct an operator D' over M' with symbol u ( D ' ) = g ,( D ( D ) ) and such that ia (D') = ia(D). Having observed that ( A 2) can be proved whenever the manifolds involved are spheres, the idea is then to make a reduction to such case. With this in mind, Atiyah and Singer introduced three new axioms which imply (A 2) for any index function: these are the excision, normalization and multiplicative axioms. The excision axiom is the following. Y' Let V be a non-compact manifold and i : V -D Y, i' : V be two o p e n embeddings into the compact manifolds Y and Y ' ; then, the diagram +

KU(T*Y)

commutes. The multiplicative axiom - "the most significant one" according to Atiyah and Singer - makes it possible to compute the index function of a product manifold, while the normalization axiom is concerned with certain operators over spheres. Let us describe, very briefly, how these axioms lead towards (A 2 ) . Let V be a tubular neighborhood of M g(M) in M' and let k : M -D 2 be the inclusion of M into the double of (*)

We refer the reader to 119-31 (particularly to Chapters XI and XVI of that paper) for the detailed construction and properties of the "Seeley Algebra" thus obtained.

27 6

V Y

, =

On the Index Theorem of Elliptic operators

.

indicated here by Z Then, the excision axiom applied to M' and Y' = Z shows that

o g! = indMl o i, o q ! = indZ o ii o $ ! = indZ o k , ind M'

Since Z can be viewed as a sphere bundle over M , the multiplicative axiom and then, the normalization obtain that indZ o k , = indM and so, indM, o g! = we wanted. The proof of the Index Theorem whose main just retraced here is not the one given initially by and Singer in [ 3 4 ] , but rather, that contained in

.

applying axiom, we indM as steps we Atiyah [35]

.( * )

Indeed, Cobordism Theory was used in [ 3 4 1 to show that the analytic index satisfies the characteristic properties of the topological index. Furthermore, the earlier proof did not allow one to envisage certain generalizations, like the study of the index in the equivariant situation. We should point out that in [35] the Index Theorem is stated and proved within the framework of equivariant KU-theory; there the index functions are given as homomorphisms KUG(T*M) + RU(G) of RU(G) - modul e s . In our previous description we supposed G to be trivial for the sake of simplicity: however, all the preceeding constructions can be transposed "mutatis mutandis" to the equivariant case (in the next section we shall describe an important application of the Index Theorem in the equivariant case). We also wish to observe that the theorem was extended to manifolds with boundary in [Zl], and in 135, IV], to families of elliptic operators parametrized by the points of a compact space (here the index of an operator is viewed as an element of the Grothendieck group of that given space); finally, the case for the real operators was discussed in 135, V ] (the appropriate K-theoretical framework in this situation is given by the KRtheory studied in [ 1 5 ] ) . In 1973 , Atiyah, Bott and Patodi published a new proof of the Index Theorem (see [25]), based on the assymptotic developments of the Heat Equation (cf. Section 2). In its main lines, this proof follows that of 1 3 4 1 ,

( * IThe remarks about cohomology we utilized are taken from [ 3 5 , Part 1111

.

The Index Theorem

277

with local Riemannian Geometry replacing Cobordism. As a technical remark, we observe that in 1251, the authors use systematically Invariant Theory (see Gilkey's paper [96]) for the orthogonal group. ( * I Hence, the Riemann-Roch Theorem (for kahlerian manifolds) and the generalized Hirzebruch Signature Theorem are totally proved within the framework of Riemannian Geometry. Furthermore, although the proof given in [251 is less general than that of [35,I] it emphasizes, for example, in a very clear fashion, the interesting connections between the eigenvalues of an operator and the geometric properties of manifolds (see [32], in particular). NOW, just as an example (sic ! ) we apply the Index Theorem to the Signature O p e r a t o r D+ = d + 6 : r (5') + ' l ( c - ) over a closed oriented manifold M of dimension n = 2m (see Section 3). Because of Proposition 3.7 , we know that ia(D

+)

= Sign(M) ;

we also know that if m is odd, both sides of this relation are equal to zero. Using (4.5) we obtain it(D

+

NOW, because of

and

e(r*(M)) =

=

(4.4)

+ (ch D

)

=

=

(*I

ch D+

(

-

Td a ) [MI

.

we have that

m ll yk , we conclude from k= 1

-1 'P*

-1

'p*

m fl k= 1

e k'-

- eyk yk

m (-l)m2m II k=1

Please read Bott's lectures

( 4 . 7 ) that

sinh 1/2.yk.cosh 1/2*yk

1501

1/2.yk

.

On the Index Theorem of Elliptic operators

278

(note that

1/2(eYk

- emYk) =

-

sinh yk = 2 sinh 1/2-yk cosh 1/2-y k)

It follows that

Thus, the topological index of D+ is nothing else but the L - genus L(M) of M (observe that if m is odd, the righthand side of (4.12) is trivial, because the m functions yk

+’

1/2 Yk tan h 112 yk

are even)

. This shows that the Atiyah-

Singer Index Theorem (4.8) contains, as a particular case, the Hirzebruch Signature Theorem(stated in 1953, [1131, Theorem 3.1): the latter theorem represents Sign(M) as a polynomial in the Pontrjagin classes of the cotangent bundle of M Putting together the previous remarks, we have:

.

Theorem 4.13. dimension,

If M

is a compact, oriented manifozd of even

Sign(M) = L ( M )

.

We have already seen that the analytic index of the operator

a

+ T * : r ( neven

>

r ( nodd)

(where Q is a holomorphic vector bundle over a complex compact manifold M of dimension m) is equal to the Riemann Roch characteristic of M , namely to m x(M;rl)

=

1

(-1)’

dim Hq(M;n(~))

q=o

,

where Q ( Q ) I s the sheaf defined by the germs of the holomorphic cross-sections of n (see Proposition 3.8). Let 0 = B ( M ) be the complex tangent bundle of M Using the notation introduced before, a is just the complexification of the real bundle T ( M I subjacent to 0 (M): hence,

.

where

B(M)

is conjugated to

B(M).

It follows that

.

The Index T.heorem

Td a = Td(0 +

B)

Td 0

=

-

279

.

Td 7j

Since M is endowed with an hermitian metric, B 8* and so, we can identify t o r s to AqS (in this context B(M) is viewed as the set of all differential forms of type (1,O)). Then, the relation

where c,(B) shows that

is the -1

‘p*

Because of

(ch(3 +

mth

Chern class of

a*))=

(-1)

m

ch rl

-

(cf. (4.7)),

f3

(Td

B*)-’

.

(4.5) we have it(T +

T*) =

(ch n

-

(Td 6*)-l

-

Td B -Td 8)[Ml

that is to say, it(T +

z*)=(ch rl -

Td B)[MI

.

Hence, the Index Theorem implies the following G e n e r a l i z e d

-

Riemann

Theorem and

TI

Roch Theorem: ( * I

4.14.

1341

Let

M

be a holomorphic v e c t o r bundle ov e r

x(M;TI)=(ch n Td 6

#here over

M

be a c o m p l e x , compact m a n i f o l d

-

Td 6)[M1

M

then

,

i s t h e Todd c l a s s o f t h e c o m p l e x t a n g e n t b u n d l e

.

We wish to observe that Hirzebruch had established the general format (4.14) of Riemann - Roch‘s Theorem whenever M was an algebraic manifold 11141; it was only with the aid of the Index Theorem that the result could be extended to complex, compact manifolds. Observe also that Theorem 4.14 says, in particular, that the arithmetic genus of a complex manifold coincides with its Todd genus (just take 17 to be the trivial vector m - bundle). ( * I The Riemann - Roch problem is stated in the introduction to this Chapter.

280

On the Index Theorem of Elliptic Operators

The reader should have noticed that the application of the Index Theorem to the operators D+ and a + consisted ultimately, in the evaluation of ch(D+) and ch(a + Indeed, the computations one must make are taken out of a general result concerning manifolds with a G - structure 134; Theorem 21, which utilizes a characterization of the cohomology of the classifying space BG [44]. In the previous section we described the notion of Spin-structure on a manifold; now we generalize it. Let G be a Lie group and V an oriented real G-module; a G - s t r u c t u r e on M is a fibre (and orientation) preserving isomorphism between the oriented bundles r ( M ) and PXGV where P is a principal G-bundle over M Thus, for the Signature Operator D+ (resp. the Dolbeault Operator 71 + , G = ~0(2m) (resp. G = U(m) x U(&) , &=dim q ) 2m and V =IR Therefore, if M is a 2m-dimensional Spin-manifold (Def. 3.11) the same kind of computations show that the topological index of the Dirac Operator

z*

.

.

a*).

a*)

A++

:

r ( r + ) -> r ( C )

is given by

or, using

(4.4) I

This means that the topological index of a+ is precisely the - g e n u s of M (notation: i(M)) defined by Hirzebruch (see [1151) in case m = even (if m is odd, (4.15) implies that it(a+) = 0 because of the parity of the function 1/2'Yk yk +> sinh 1/2 yk obtain the following. Theorem

4.16.

If M

)

. Using Definition

3.12, one can thus

is a S p i n - m a n i f o z d of d i m e n s i o n

411

I

Generalized Lefschetz Fixed-Point Formula

5.

281

The Generalized Lefschetz Fixed-Point Formula. A s we pointed out in the previous section, the Atiyah-

Singer Index Theorem was stated for an equivariant situation. For every compact Lie group G , the topological index i t is viewed as an index function from KUG(T*M) into RU(G) , where M is a compact G-manifold (in what follows we shall identify T*M with TM because we assume that M is endowed with a riemannian G - invariant metric). We construct it precisely as in Section 4 : we begin by associating a homomorphism f, : KUG(TM) + KUG (TX) to each G - embedding f : M + X (here X' is a real representation space of G) (*Ithen we take the isomorphism j , : KUG(T{xo}) -D KUG(TX) which corresponds to the G - embedding of the origin xo E X into X and finally, we define it *- KUG(TM)

___ >

RU(G) 2 XUG(Tlxo))

.

as the composition ( j , )- 1 o f , It goes without saying that it satisfies condition ( A 1) and also, the excision, normalization and multiplicativity axioms of the index functions; hence, it satisfies (A 1) and ( A 2). On the other hand, it is still possible to define the analytic index of an elliptic operator D : T ( C ) + T(0) over the G-manifold M I or more generally, of an elliptic complex (I'(c),d) over M (see Definition 2.22)

where the s i t s ( 0 5 i - n) are complex vector G - bundles whose G - actions commute with the differentials di (in other words, ( r ( 6 ) ,d) is a G - module) ; in the latter case, the analytic index is defined by

Thus, ia(€,,d) is itself viewed as an element of

(*)

RU(G). How-

Such an embedding is allways possible, according to R. Palais [1921.

28 2

On the Index Theorem of Elliptic Operators

ever, one should note that in order to talk about the adjoints of the differentials di , it is necessary to assume that df each bundle ti is endowed with a G-invariant metric. The first objective of this section is to study the topological index in relation to the fixed point set of the action of G on M : the result we are aiming at is Theorem 5.3 , due to Atiyah and Segal [ 3 3 ] . Let us notice "en passant" that such result, when conveniently interpreted in Cohomology, allows us to obtain specific formulas representing the topological index in terms of the characteristic classes (compare with the remarks about G - structures made at the end of Section 4 ) . The second objective is to describe briefly how the topological index leads to a generalized Lefschetz formula (relation ( 5 . 4 ) ) . For a fixed g E G ,. let Mg be the set of points in M which are invariant by the action of g on M or, in short, Mg

= {X

E Mlg

-

X

= X)

.

The canonical G - embedding f : Mg + M gives rise to f : KUG(TMg) + KUG(TM) and induces f* : KUG(TM) + KUG(TMg)

,

;

thus, composing these homomorphisms and using the Thom isomorphism in K U G - theory, we obtain a homomorphism f*

0

f , : XU,(M~)

> KU,(M~I

which is just multiplication by the element

is the complexification of the normal bundle vf (Mg) of Mg c M (see [35,I, ( 3 . 1 ) 1 ) . Next, we describe an essential result established by Segal. Let G be a compact Lie group, y be a conjugacy class in G and My = U M g The set gEY 7 = 7, of all characters vanishing on y is a prime ideal of RU(G) ; for every RU(G) -module A , let be the module obtained from A by localizing at 7 We can now state the

.

.

9

Generalized Lefschetz Fixed-Point Formula

283

following localization theorem [210] - Let i! : KUG(M) -* KUG(My) b e t h e RU(G) - m o d u l e s d e f i n e d by t h e c a n o n i c a l ini : My + M. T h e n , t h e i n d u c e d homomorphism

Proposition

5.1.

homomorphism of clusion

(ill3

:

3 ->

KU~(M)

K U ~ ( M ~ ) ~

is an i s o m o r p h i s m . From now on we s h a l l s u p p o s e t h e e x i s t e n c e of a n e l e m e n t g E G whose p o w e r s f o r m a d e n s e s u b g r o u p of G (then G is the product of a cyclic group and a torus). In this case G acts trivially on Mg : in fact, Mg = MG = {x E MI (vh E G) hx = x) ; moreover, i f y is the conjugacy class of this d i s -

tinguished element g , My = M g - . Then, using the homeomorphism (TMIg TMg , Proposition 5 . 1 and the Thorn Isomorphism Theorem, we obtain that

is an isomorphism. Since X - , ( v g ) i s invertible in KU,(M~)? 1 3 3 ; Lemma 2 . 7 1 , it follows that the RU(G) -module homomorphism f! induces an isomorphism

its inverse is given by the isomorphism

Consider the following commutative diagram f

!

>

K U (TM)

L!

V

284

On the Index Theorem of Elliptic Operators

its top is given by Axiom (A 2 ) of 9 4 , its bottom is the localization of the top and the vertical arrows are the localization homomorphisms. Next, consider the evaluation map €

:

RU(G)

-> c

.

€(x) = x ( g ) for every x E RU(G) Defining so that E7 5 r = E , we have RU(G)? + 6:

given by

€7

:

€

0

€7

indM G =

o r I using the inverse

0

(5.2)

In other words, for every

Since

-

r

of

=

€7

c

(ind;)?

':

II

,

(f!)?

x E KUG(TM)

acts trivially on

G

indM G

,

T M ~,

and by localization, K U ~ ( T M ~ + , ZKU(TM~)60 RU(G)

7 .

Because of these isomorphisms we make the following identifications:

so that

RU(G)7 (2)

Where

Rg ( f * (x))

can be viewed as an element of

KU(TMg) C3

:

(indzg)rs

ind M9

ind Mg

:

C 3

KU(TM~)

(RU(GI7

is the topological index

RU(G)

+

7 it : KU(TMg)

E 8 RU(G)

7l

+

z

cor-

Generalized Lefschetz Fixed-Point Formula

responding to the case in which (3)

€3

@

€7 :

z

€3

is trivial (just as in 9 4 ) ;

G

RU(G)

285

7

-> Z € 3 6 : ? ' @ . (v,) (g) -

The reader should also notice that the element

obtained evaluating A (v ) E KU(TMg) at g - is, in this -1 g P context, viewed as an invertible element of the ring KU(TMg) €3 c Finally, if

.

is the complexification of the topological index, the following diagram commutes:

Hence, from the preceeding considerations, ( 5 . 3 ) relation

gives the

or

With all these results one can show the following.

On the Index Theorem of Elliptic Operators

28 6

-

M b e a compact G - m a n i f o l d . Supp o s e t h a t t h e r e i s an e l e m e n t g € G w hose p o w e r s f o r m a d e n s e subgroup o f G L e t Mq b e t h e s e t o f p o i n t s of M w h i c h a r e g ; let v be the compZexificai n v a r i a n t b y t h e a c t i o n of 9 t i o n of t h e n o rmal b u n d l e d e f i n e d b y t h e e m b e d d i n g M9 t M . T h e n , t h e e l e m e n t X-, ( v g ) ( g ) , o b t a i n e d b y e v a l u a t i n g X ( v ) = 1 ( - 1 ) k A k (ug) a t g , i s i n v e r t i b l e i n t h e r i n g klo KU(TM~)o c M o r e o v e r , for e u e r y x E KU~(TM)

Theorem

5.4.

[33]

Let

.

-'

whe re

.

f*

:

I

KUG(TM)

onical inczusion

+

f

:

KUG(TMg) ~g

+

M

i s t h e map i n d u c e d b y t h e c a n -

.

Let f be a finite rank endomorphism of a (finite) complex of R-modules; its L e f s c h e t z number is the alternate sum of the traces of the homomorphisms induced by € in the cohomology of the complex. For example, the Lefschetz number of the identity homomorphism is the Euler-PoincarG characteristic of the complex. For a G-invariant elliptic complex (r([) defined over a compact G - manifold M , the Lefschetz number of an element g E G is given by L(g;s) =

1

i,o

i (-l)itr(g[H ( r ( c ) ) ) ;

notice that L(g;E) is just the evaluation of the analytic index of ( r ( [ ) ,d) at g (see Definition 2.23). Let x = [ a ( 5 ) ] E KUG (TM) be the class of the symbol sequence u ( 5 ) for the complex ( I ' ( [ ) , d ) (see [ 3 5 ; I l ( s 2 1 ) ; then, restricting the action of G to that defined by the closure of the cyclic group generated by an element g E G I applying Theorem 5.4 and the Index Theorem, we obtain (5.5)

This is the g e n e r a l i z e d L e f s c h e t z f i x e d p o i n t fo rm u Za obtained by Atiyah and Segal [33; Theorem 2.121. I t s discovery opened

General zed Lefschetz Fixed-Point Formula

281

the way to several diversified results; unfortunately, due to their large number we can only suggest that the reader consult the extensive "ad hoc" literature. At any rate, we wish to observe that if we apply the Lefschetz formula to the elliptic complexes described in § 2 , we obtain the equivariant version of the results described in 5 3 ; thus, besides the case of the De Rham complex, for which ( 5 . 5 ) gives back the classical Lefschetz formula, we find an equivariant form of the Riemann-Roch Theorem and also, some G - signature theorems (see [33] and [ 3 5 ; 1111). If G is a finite group, L ( g , c ) is an algebraic integer; thus (5.5) becomes the source of several integrality theorems. If the action of G is such that Mg is finite, say Mq = {yl,...,yr) , the Atiyah-Segal formula ( 5 . 5 ) becomes particularly simple; this is due to the fact that in such case, the topological index is easily obtained. Let us explain o u r statement. Because

the map it : KU(TMg) + Z is the sum of the natural isomorphisms KU({yk}) z Z . In other words, if fk is the natural inclusion {yk} c M and v ~ is , the ~ complexification of the normal bundle induced by such an inclusion, then

Notice that

f;

)i[5i,k1

[U

I

where

(ilk

is

and hence,

the restriction of f p o 5)

on the other hand, because the total space of the normal bundle over {yk} coincides with T (M) , if we write T~ = fE[T (MI 1 I 'k

we obtain

On the Index Theorem of Elliptic Operators

= detn ( l - g ] T k )

(here det (resp. detlR) indicates the complex (resp. real) 6 determinant). Finally, we obtain the formula

(5.6)

Formula ( 5 . 6 ) was announced by Atiyah and Bott in [ 2 3 1 and proved in E24; I (Theorem A ) I If ( r ( 6 ),a) is the De Rham's Complex, ( 5 . 6 ) reduces to the Lefschetz formula : L(g;S) = L ( g ) = number of points kept fixed by g Clearly, ( 5 . 6 ) is particularly adapted to the study of the fixed point set €or the isometries of a group G acting on M If (I'(E),d) is the Dolbeault Complex, formula ( 5 . 6 ) gives rise to the following.

.

.

.

-

M be a complex compact m a n i f o l d , ri a hoZ omorphi c v e c t o r b u n d l e o v e r M and g : M + M , a h o l o m o r p h i c f u n c t i o n w i t h s i m p l e f i x e d p o i n t s . T a k e L(g,f) t o b e t h e L e f s c h e t z number g i v e n by t h e a c t i o n of (g,f) on H* (M;R(n)1, w h ere f : g*n + is a m o r p h i s m o f h o l o m o r p h i c b u n d l e s . Then,

Proposition

[ 2 4 ; 111

Let

i s the differentiaz of E -endomorphism i n d u c e s by f

where the

5.7.

dgk

g

at

y E Mg

on t h e f i b r e

a nd

i s

fk of

0

.

Yk Proposition (5.7) has several applications to Representation Theory; for example, the Atiyah-Bott Formula is connected to the Hermann Weyl Formula, which evaluates the trace of certain representations over a maximal torus of a compact Lie group. Furthermore, we point out that whenever M is

Generalized Lefschetz Fixed-Point Formula

289

a curve and g is a transversal morphism of M into M (every fixed point yk of g is simple, i.e., det (l-dgk) 0) we 6, can use ( 5 . 7 ) to show that the degree d of g is related to the number r of points kept fixed by g ; the relation is given by the formula

.

real part of the complex number z ) Now we apply the Atiyah-Bott Formula Signature Operator (Re(z)

=

D+ = d

+

6 :

r(c+)

(5.6)

to the

r(C)

___ >

defined over a closed, oriented, riemannian manifold M of dimension n = 2m (see 5 3). We suppose that the G - a c t i o n p r e s e r v e s t h e o r i e n t a t i o n of M (we shall continue to identify T * (MI to T (M) via the G - invariant riemannian metric of M) and furthermore, we assume that there is a g E G whose action on M is an i s o m e t r y h a v i n g o n l y i s o l a t e d f i x e d p o i n t s (since M is compact, Mg is finite). The action induced by g on T(c+) (resp. 'i'(

r(Y3

9+

14-

V

V

The S i g n a t u r e of g (notation: Sign(g;M)) is defined to be the Lefschetz number of g To compute Sign(g;M) we shall determine its contribution Sign(g;M)k at each point yk € Mg ( 1 2 k 5 r). Since g is an isometry its differential dgk : T (M) + T (M) is also an isometry. Hence, Yk k'

.

On the Index Theorem of Elliptic operators

290

NOW, it is possible to decompose sum of orthogonal 2 - planes m TkM = @ E

T (M) = TkM k'

jrk

.

so that (dgk)(Ejrk) Let us select a basis €or each E j,kf 1 5 j 5 m , so that e l,k

A

1 rk

...

A

into a direct

A

em,k

CejrkrejfkI

exk,k

A

in the 2m- differential form defined by the orientation of M in yk ; as a consequence, the restriction of dgk to each is just a rotation by a certain angle 0jfk , that is to E j ,k say r

The set of angles {Ojlkll 5 j ,L m} is called a coherent system €or dgk. Using the properties of the operator T defined by ( 3 . 4 ) one can show that

2

(with i

= -1)

and s o f

(5.8)

Sign(g;M) =

r

1

Sign(g;MIk

k= 1 r

-

1

k=l

m (-i)m II

cot(+

Eljrk)

.

j=l

There are many applications of the G - Signature Theorem expressed by relation (5.8) (for a more general form of the Theorem, please consult [ 3 5 ; 1111). For example, Atiyah

Generalized Lefschetz Fixed-Point Formula

291

and Bott proved the following result in [ 2 4 ; 111. For a prime 2 ) , consider a smooth action of the cyclic number p (p

*

group zp on a homology sphere and suppose that such action has only two fixed points; then, the induced representations of Z on the tangent spaces to these two points are isomorphic P (Theorem 7 . 1 5 ) . In particular, this result allows one to show that two h - cobordant lens spaces are isometric (cf. Milnor [ 1 7 8 ] ) . Other interesting applications of the G - Signature theorems can be found in [ 1 1 7 ] ; these will consent the reader to have a better grasp of the relations between Number Theory and the G - Signature theorems. We conclude the Chapter with an example about e x o t i c i n o o Z u t i o n s , also taken from 1 2 4 ; Part 111. This time we shall utilize the Fixed Point Formula concerning the Lefschetz number of an isometry of a Spin-manifold (see 3 . 1 1 and 3 . 1 2 ) . Let M be a Spin-manifold of dimension n = 2m , 5 the Spinor -+ r ( r , - ) , the Dirac Operator. bundle over M and &+ : r )'5( In addition, let g : M + M be a Spin- structure preserving isometry whose fixed points are all isolated. Then, it is pos: g * < + 5 which sible to lift g to a bundle isomorphism + induces an automorphism g (resp. g - ) of r ( c + ) (resp. r ( < - ) ) , such that the following diagram commutes:

9+

According to the Atiyah-Bott Formula, the Lefschetz number Spin(G;M) corresponding to the action of @ over the Dirac complex is given by r (5.9)

Spin($;M) =

1

k= 1

Ek(G)($)m

m Il j=l

(i

cosec - 0 Ilk)

29 2

On the Index Theorem of Elliptic Operators A

les for dgk and E ( g ) k A g ‘k and the lifting and an odd number a 2 3 manifold defined by

. ,

,

depending on the fixed point NOW, for a given complex number t let Xt a) 5 Cn+’ be the Brieskorn

= +1

Consider its intersection with the unit sphere s2n+l (respec2n+2 tively, the unit disc D ) of Cn+’ , that is to say,

We agree to write X(a), S ( a ) and D(a) for Xo(a) , So(a) and Do(a) , respectively. Then, if t is sufficiently Small, the following statements are true (see [ 5 4 1 ) . (1) 12)

Dt(a)

St(a)

is the boundary of the

2n - manifold

Dt(a)

;

if n 2 3 , Dt(a) is (n-1)-connected (indeed, is homotopically equivalent to a bouquet of n - spheres);

(3)

St(a)

and

S(a)

(4)

if

n 2 3, S(a)

(5)

if

n

are diffeomorphic

;

(n-2)- connected:

is

n = 2m+l, t h e n S(a) i s a t o p o (actually S(a) is diffeomorphic to 4m+ 1 , except for m 2 2 and a *3

i s o d d , say

4m+1

l o g i c a l sphere

the standard sphere (mod.8)1 . Let y be the involution of C y

-

if

Zk = -Zk

y - z n = zn

=

C2m+2

given by

0 5 k 5 n-1

.

Clearly y preserves Dt(a) and its boundary St(a) ; furthermore, for t small, the action of y on St(a) is isomorphic to its action on S(a). The action of y on a topological sphere S(a) will be called an e x o t i c i n v o l u t i o n . We are interested in knowing when two exotic involutions, say on S ( a )

Generalized Lefschetz Fixed-Point Formula

293

and S(b) , are isomorphic. Before studying this problem, we observe that the fixed point set of Dt(a) under the action of y is given by

furthermore, the rotation Pa :

c

n+ 1

>

n+ 1

defined by

I

Pa(Zn) = 5a

-

zn

,

where

5, = exp

(%)

commutes with the action of y on Dt(a) and interchanges the points of (Dt(a))' . Suppose now that the actions induced by y on S(a) and S(b) are isomorphic. Then, for t sufficiently small, there is a diffeomorphism f : St(a) 4 St(b) which commutes with the actions induced by y on St(a) and St(b) Let M be the space obtained by the adjunction of Dt(a) and Dt(b) via the map f ; observe that if m 1, M is at least 2-connected. Thus (cf. 3 . 1 1 and the following lines) M is a Spin-manifold of dimension 4m + 2 Moreover, y acts on M and respects its Spin- structure; hence, let be a lifting of y l M and Spin(?;M), its Lefschetz number. Consider the Spin- structures of Dt(a) and Dt(b) : let Spin(?;Dt(a)) (resp. Spin(?;Dt(B))) be the Lefschetz number relative to the restriction of Y to Dt(a) (resp. Dt(b)). Because

.

.

relation

(5.9)

implies that

€A(?)).

(or To this end, let y We want to compute Ej (?) j Dt(a) be a fixed point: since the order of Pa is odd and

On the Index Theorem of Elliptic Operators

29 4

further, pa follows that

commutes with the action of

y

on

Dt(a) , it

But pa is a circular permutation on the fixed point set of Dt (a) Hence, the E . I s , 1 5 j 5 a , are equal among them3 selves; the same happens to the Ei(?) I s , 1 5 k 5 b I and consequently,

(c)

.

(5.10)

Furthermore, y is an involution on C n+' and in view of the construction of M it follows that has a period equal to at most 4 From this we conclude that the complex number Spin(;;M) is a Gaussian integer. Thus, relation (5.10) shows that if t h e e x o t i c i n v o l u t i o n s on S(a) and S ( b ) a r e i s o I

.

morp h i c

a

3

*b

(mod

2m+')

.

In 1521, Bredon proved that if m = 1 and a = 3 , the action of Y on the topological 5 - sphere S ( 3 ) is not isomorphic to the standard antipodal map; the previous remark shows that 1 and every topological (4m+l) -sphere to be the case for m S(3).

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-

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,

Vector fields on spheres. Ann. of Math. 7 5

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,

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ADAMS, J.F.; ATIYAH, M.F. - K-theory and the Hopf invariant. Quart. J.Math. Oxford (2) 1 7 ( 1 9 6 6 ) , 31-38. ADAMS, J.F.; WALKER, G. - An example in homothopy theory. Proc. Cambridge Philos. SOC. 6 0 ( 1 9 6 4 1 , 6 9 9 - 7 0 0 . ADAMS, J.F.; WALKER, G. - On complex Stiefel manifolds. Proc. Cambridge Philos. SOC. 61 ( 1 9 6 5 ) , 8 1 - 1 0 3 . ADEM, J. - The iteration of the Steenrod squares in algebraic topology. Proc. Nat. Acad. Sci. U.S.A. 3 8 ( 1 9 5 2 1 , 720-726.

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-

Topologie. Springer, Berlin:

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Adams operations in

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CW

INDEX

Adams conjecture

10

Adams operations

8

Closed dual n-cell

31

Coherent system

290

A-genus 266, 280 Almostcomplexstructure 41, 83 a-orientation 85, 267

Completely regular map Complex James number Continuous vector product

183 130 42

Analytic index

Coreducible space

160

Coreduction

160

248, 254

Arithmetic genus Atiyah exact sequence Atiyah -Hlrzebruch spectral sequence Atiyah's criterion

262 5

6 188

31

Block dissection

32

Bott homomorphisms

4

Bott operations

9

Bott periodicity theorems

3

Brieskorn manifold Cartan-Serre-Whitehead towers Cauchy integral' formula Cayley projective plane

17 255

Difference bundle Differential operator

Atiyah-Singer index theorem 270 Barycenter

Degree of a map De Rham complex

267 244, 245

Dihedral spherical forms

145

Dlrac operator

265

Division algebra

40

Dolbeault complex

261

Dold manifold

192

Dual cell subdivision

32

Eilenberg-Mac-Lane space

67

292

Elliptic complex

226 240 86

Chern character Chern classes

253

Elliptic differential operator

245

Embedding

178

Exotic involution

292

Exterior differential p-form 254

Clifford algebra 121, 263 Clifford multiplication 263 Closed (co-closed) 255 differential form 315

Finite H-space

43

Formal adjoint

247

316

Index

Free G-pair F-vector field

158

Lefschetz number

286

94

Lens spaces Like-lens space

198

Gauss-Bonnet formula

242

G- (co)reducibility Generalized Lefschetz fixed-point formula 281, Generalized quaternionic group Generalized Riemann-Roch theorem G-fibre homotopy J-equivalence

160

Geometric dimension Grothendieck operations Harmonic differential form

286 145

152 188 7 257

143 255

Hopf construction 28, 48 Hopf invariant 13, 18, 25 Hopf invariant one problem 13 Hurwitz-Radon-Eckmann theorem

183 6

Intrinsic map Immersion

161 177

Kahlerian manifold Kervaire semicharacteristic

67

Multiplication on a sphere 37 Milnor construction of classifying space 225 n-coconnected G-pair Normal bundle associated to an immersion

158 179

Operations in KIF Original cycle

7 26

p-adic valuation

126

Parallelizability

212

Parametrix Positive element

249

Projective plane of an H-space

188 48

Quaternionic James number 133 Quaternionic spherical form, Qm-spherical form 141, 150

121

Intersection number Integrality theorem

James number

199 19

279

Harmonic spinor 266 Hilton-Roitberg H-space 43 Hirzebruch signature of manifold 260 Hirzebruch signature theorem278 Hochschild-Serre spectral sequence Hodge star operator

Linking number Localization theory

95 82, 262

Rank of anH-space 44 Rank of a manifoldmorphism 177 Rationalization map 67 Rationalization of a CW complex 67 Reduced projective plane 54 of an H-space 160 Reducible space 160 Reduction 182 Regular map Regular self intersection

258

183

Index

264

Thom isomorphism theorem Thom space Todd classes Topological index Torsion free rank one H-space Torsion free rank two H-space Tubular neighborhood Twisted Dolbeault complex Type ofanH-space

265

Type of a map

28

Vector field Vector field problem

94

Relatively free G-pair 158 Riemann-Roch characteristic 262 Riemann-Roch formula 242 Semi-regular map Separation element (of two maps 1 Signature operator Space of spinors Spin-index Spin-manifold Spinor bundle Spin-representation Spin-structure Spherical (space) form 139, Spherical form of cyclic (resp. quaternionic) type Splitting principle Stably complex structure Stiefel manifold Stiefel-Whitney classes Stirling numbers Submersion Symbol

317

184 38 260 264 266

4 4 268 267 44 44 179 261 44

264 264 140 151 4 85 98 4 218

177 241, 245, 246

94

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