Classes Caracteristiques

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C H A R A C T E R I S T I C C L A S S E S - O L D A N D N E W 1 2

BY FRANK LIN P . PETERSON

1. Definition of sphere bundles. L e t M n be an w-dimensional ,

C°°-manifold. Define T(M) to be a l l vec tors t angent to M of uni t

l eng th . Def ine p: T(M)—*M by ^(vec tor ) = in i t i a l po in t o f the vec tor .

T h e n p i s a con t in uo us func t ion w i th p~1(m ) homeomorph ic t o 5 n _ 1

if rnCEM. (T(M), p, M) i s an example of an (n — 1)-sphere bundle .

Le t me now abs t rac t some of the p roper t ies o f th i s example and

define an (» — l) -sphe re bun dle . An (n — l ) - sp her e bu nd le £ i s a t r ip le(E , p, X), w h e r e p: E-+X i s a con t inuous func t ion , X has a cover ing

by ne ighbo rhoods { Va] s u c h t h a t ha: p'KVcd-tVaXS"-1, w h e r e hi s a homeomorph i sm, ha(e) = (p(e), Sa(e)). That i s , we can g ive co

o rd ina t e s t o p~x(V a) us ing V a and 5 n _ 1 . Fu r the rmore , t he r e i s a

condi t ion on changing coord ina tes ; namely , i f e&p"~ 1(V ar\Vp)t t h e n

ha(e) = (p(e), Sa(e)) a n d hp(e) = (p(e), Sp(e)) an d we ob ta in a func

t io n S ^ S * - 1 - ^ - 1 given by Sp(Sa(e)) =5 >(e ) , def ined for e ach

p(e)GV ar\Vp. W e d e m a n d t h a t S%C:0(n), the o r thogona l g roup of

homeomorph i sms o f S* - 1 . F ina l ly , S% depends on p(e) and th i s de

pendence mus t be con t inuous .

T w o (n — l ) - sphe re bu nd le s £ and rj ove r X are ca l led equ iva len t

i f there is a homeomorphism 7^: E$—*E n s u c h t h a t

F

Ez->EV

P\/PX

c o m m u t e s a n d s u c h t h a t F\ p~ 1(x)SO(n) for a l l co ord ina tes on p~l{x).

A ve ry impor t an t example o f an (n — 1) -sphe re bu nd le is th e fol

lowing one . Le t BO(n) = the Grassmann space of a l l w-p lanes th rough

the or igin in J?00

. L e t EO(n) be the set of pairs , an e lement of BO(n)

and a un i t vec tor in tha t w-p lane . Le t p: EO(w)~»BO(w) be the first

e lement o f the pa i r . The impor tance of th i s example i s shown by thefol lowing class i f icat ion theorem.

1An address delivered before the New York meeting of the Society by invitation

of the Committee to Select Hour Speakers, April 13, 1968; received by the editorsApril 21 , 1969.

2In order not to obscure the structure of the subject, I have left out a number of

technicalities; in fact some of the statem ents may be incorrect as stated.

90 7



908 F. P. PETERSON [September

C L A S S I C A L C L A S S I FI C A T IO N T H E O R E M . The equivalence classes of

(n — V)-sphere bundles over X are in one-to-one correspondence w ith the

hom otopy classes of m aps of X into BO(w) .

2. Definition of characteristic classes. Roughly speak ing , a char acter is t ic c lass is a cohomology class in H*(X) ass igned to a bundle

£ over X w hich is na tu r a l w i th r e spec t t o bund le m aps . Ra t he r t ha n

give a precise def ini t ion, le t me give a construct ion.

Le t w G f f* (B O (w ) ) . L e t £ be an (n — l ) - sphe re bund le ove r X cor

r e spond ing t o a map ƒ$: X—»BO(w) by the above theorem, u definesa charac te r i s t i c c lass u(g)E:H*(X) b y u(£)=f£(u), w h e r e f* :

H*(BO(n))—>H*(X) i s th e hom om orph ism induced b y ƒ$ ( reca ll t h a tf* depends on ly on the homotopy c lass o f ƒ$).

Thus , t o s t udy cha rac t e r i s t i c c l a s se s , w e mus t s t udy H*(BO(n)).

The answers, with various fields for coefficients, are as follows. 3

H*(BO(n); Z 2) is a polynomial r ing over Z 2 on gene ra to r s W tG

£T«(BO (»); Z 2 ) , i = l , • • • , n. Wi(Q=f?(Wi) is called the ith Stiefel-Whi tney c l a s s o f £ . H *(BO (oo) ; Zp) a n d # * ( B O ( o o ) ; Q) are po ly

nomia l r ings over Zp (p i s an odd pr ime) and Q respec t ive ly on gene r a t o r s P < e # 4 * ' ( B O ( o o ) ; Zp) or # 4* '( B O (o o ); 0 , i = l , • • • , Pfâ)

=ft(P%) i s cal led the ith Pontr jagin c lass of £ .

3. Some examples of applications of characteristic classes. T h e

study of character is t ic c lasses has been very useful in dif ferent ia l

geomet ry , d i f fe ren t ia l topo logy , and a lgebra ic topo logy . I wi l l nowgive a few examples of such appl icat ions .

I . Cobordism. L e t M n be a closed, connected, C°°-manifold of di

mens ion n. T h e n M n = dWn+l

, w h e r e W n+ 1 i s a compac t , connec t ed ,

C°°-manifold w ith bo un da ry , if an d o nly if ƒ* : H n(BO(n); Z 2)—>H

n(M

n; Z 2 ) i s ze ro where r i s the tangent bundle descr ibed a t the

beginning of th is lecture [1 7 ] .

I I . Homotopy spheres. Le t 0 n be the group of dif feomorphism

classes o f homotopy spheres . Pont r jag in c lasses have been used tos tudy t he se g roups . Fo r example , ® 15 « Z s m [6].

I I I . Em beddings and imm ersions. G i v e n M n, the problem is to f ind

the sma l l e s t k s u c h t h a t M n can be d i f fe ren t iab ly embedded or immersed in R

n+k. The in i t i a l resu l t s were p roved us ing S t ie fe l -Whi tney

c lasses. T h e techn ique s now are qu i te com pl ica ted an d we a re nownear to so lv ing th i s p rob lem for rea l and complex pro jec t ive spaces .

IV . K-theory. Let KO(X) be the se t o f equ iva lence c lasses o fbund le s ove r X, w i th d imens ion X<n. This fo rms a g roup and ac t s

* An excellent introduction to the classical theory, including proofs of the follow

ing assertions, is to be found in [10],



i969] CHARACTERISTIC CLASSES—OLD AN D NEW 9 0 9

as a cohomology theory which has tu rned ou t to be very usefu l . For

example , one can p rove t ha t t he max imum number o f l i nea r ly i nde

pe nd en t t ang en t vec tor fie lds on 5 n _ 1 is 2e+Sd — 1, wh ere n = ( 2 g + 1 ) 2 6 ,

b = c+4d, 0 = ^ 3 [ l ] .

4. More genera l bundles . In recen t years i t has become c lea r tha t

one shou ld s tudy (»— l ) - sphere bundles where the changes o f co

ord ina tes can be a l lowed to be in l a rger g roups than 0(n). E x a m p l e s

of such gro ups , in inc reas ing s ize , a r e : PL(w) =pie cew ise l inear

homeomorph i sms o f 5n _ 1

, Top(w) = hom eom orph ism s of 5n _ 1

, andG(n) = homotopy equ iva l ences o f S"*1.

N E W C L A S S IF I CA T I O N T H E O R E M . Let H= PL, Top, or G. There exists

a space BH(n) such that the equivalence classes of (w— 1) -sphere bundles

with group H over X are in o ne-to-one correspondence with the homo

topy classes of maps of X into BH(n) ( [ l l ] and (14 ] ) .

Us ing th i s theorem, we can g ive the same cons t ruc t ion of charac

ter is t ic c lasses as we did in the c lass ical case . In order to use these

charac te r i s t i c c lasses , we need to know H*(BH(n)) w i th va r iouscoeff ic ients . M os t of the res t of th is pa pe r is de vo ted to de scr ibing

w h a t is k n o w n a b o u t i ? * ( B H ) , w h e r e B H = l im n^oo BH (w ) .

5. 7T*(BH). Before s ta t ing the resul ts on cohomology, le t me f i rs t

g ive the known resu l t s on the homotopy groups of the c lass i fy ing

spaces .

I. 7T*(BO) is periodic of period 8 with 7r8jfc+»(BO) = Z , Z 2 , Z 2 , 0 ,

Z , 0, 0 , 0 , wi th i = 0 , 1 , 2 , 3 , 4 , 5 , 6, and 7 respe ct ive ly [2 ] .

I I . 0—>7r»(BO)—»x»(BPL)—»I\_i—»0 is an exact sequence where I\-_i

is a f in ite gro up which is pa r t ia l ly kn ow n. Also, th e s t r uc tu re of th e

exa c t seque nce i s kn ow n ( [5 ] an d [4] ) .

I I I . 7T f(Top /PL) = 0 if * V 3 , and 7 r 3 ( T o p / P L ) = Z 2 [ 7 ] .

IV. 7 r t(BG) =iri-i+k(S k), k l a rge ; hence , known to a ce r ta in ex ten t .

V. 7T t(G/P L) is pe riod ic of pe riod 4 w ith 7r4fc+i(G/PL) = Z , 0, Z 2 , 0 ,

w i th i = 0 , 1 , 2 , an d 3 respec t ive ly [ l S ] .

6 . H*(BH; Q). Using the above resu l t s on x* , i t i s easy to see tha t

i J * (B G ; 0 = 0 if i> 0 a n d t h a t ü * ( B T o p ; < 2 ) - » i J * ( B P L ; Q)-+

i î * ( B O ; Q)—>Q[Pi, ' ' ' ] are a l l i somorphisms .

7 . i î * ( B H ; Z 2 ) . T he re ex is ts a connec ted Hopf a lgebra C(H) ove r

the mod 2 S t een rod a lgeb ra A2 s u c h t h a t H * ( B H ; Z 2) « i I * ( B O ; Z2)

®C(H), as Hopf a lgebras over ^42 [3] . C(0) is t r iv ia l of course . C(G)

i s 2-conn ec ted , and i t s s t r uc tu re has been de te rm ined rec en t ly [9 ] .

C (PL ) and C(Top) a r e s ti ll unk now n .



910 F. P. PETERSON [September

8. 2ï*(BH; Zp), p odd. Th e situation is a lit tle different from the

case p = 2. Analogous to the case p = 2 we have i7*(BG; Zp)

~(Z9[qi]®E(Pqi))®Cp(G), where j £ f f ^ » ( B G i Z9) is the Wu

class, j8g» is its Bockstein, and CP(G) is a Hopf algebra over Ap [13].

Furthermore, CP(G) is (p(2p — 2)— 2)-connected and its complete

structure has been found very recently [7].

H*(BTop; Zp)—»iî*(BPL; Zp) is an isomorphism by §5, III, so we

need only study iï*(BPL; Zp). It is an unpublished theorem that if

p is an odd prime, BPL is of the same mod p homotopy type as

BO XB Coker / , where B Coker J is a space whose homotopy groups

are the cokernel of the homeomorphism J: 7r*(BO)—»7r*(BG) [16].

However, the map BOXpt.—>BOX£ Coker /—•BPL is not the usual

map so this is quite different from the case p = 2. Also, the map

/ P L : BPL—>BG has the property that /PLG&Z») = 0 if i^p and is not

zero if i^p + 1. The best conjecture at present seems to be that

CP(G)**H*(B Coker / ; Zp). To complete the picture, we need to

know JPL(<Z») explicitly [12].

9. Applications. One expects that a good knowledge of these newcharacteristic classes will lead to many applications as in the classical

case. I ment ion only one, namely tha t §3, I generalizes to the PL case.

Tha t is, let Mn

be a closed, connected PL-manifold. Then Mn

=dWn+1

,

where Wn+1

is a compact, connected PL-manifold with boundary, if

and only if/*: Hn(BPL; Z2)->H

n(M

n; Z%) is zero [3]. A similar

theorem is true for oriented C°°-manifolds, bu t for oriented PL-mani

folds, it fails in dimension 27 (though t rue in lower dimensions) [12].

B I B L I O G R A P H Y

1. J. F. Adams, Vector fields on spheres, Ann. of Math. (2) 75 (1962), 603-632.2. R. Bott, The stable hom otopy of the classical groups, Ann of Math. (2) 70 (1959),

313-337.3. W. Browder, A. Liulevicius and F. P. Peterson, Cobordistn theories, Ann. of

Math. (2) 84 (1966), 91-101.4. G. Brumfiel, On the hom otopy groups of BPL and PL/O, Ann. of Math. (2) 88

(1968), 291-311 .5. M. Hirsch and B. Mazur, Smoothings of piecewise linear man ifolds, Mimeo

graphed notes, University of Cambridge, Cambridge, England, 1964.6. M. Kervaire and J. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2)

77 (1963), 50 ^ 53 7.7. R. C. Kirby and L. C. Siebenmann, On the triangulation of manifolds and the

Hauptvermutung, Bull. Amer. Math. Soc. 75 (1969), 742-749.8. J. P. May (to appear).9. R. J. Milgram (to appear).10. J. W. Milnor, Lectures on characteristic classes, Mimeographed notes, Prince

ton University, Princeton, N. J., 1957.



1969] CHARACTERISTIC CLA SSES-OLD AND NEW 9 1 1

11. , Microbundles . I , Topology 3 (1964) , suppl . 1 , 53-80.

12. F. P. Peterson, Som e resul ts on PL~cobordism ( to appear) .

13. F. P. Peterson and H. Toda, On the s tructure of H* (BSF; Z p), J . Math . Kyoto

Univ . 7 (1967) , 113-121.

14 . J . D . Stasheff, A classif ication theorem f or f ibre spaces^ Topology 2 (1963) , 239-

246.

15 . D. Sul l ivan ( to appear) .

16. ( to appear) .

17 . R. Thorn, Quelques propriétés globales des variétés différentiables t C o m m e n t .

Math. Helv . 28 (1964) , 17-86.

M A S S A C H U S E T T S I N S T I T U T E O F T E C H N O L O G Y , C A M B R I D G E , M A S S A C H U S E T T S 02139

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