17. COHOMOLOGICA L INVARIANT S O F Qua d
41
Make the following assumptio n o n A:
,y.. Ther e exist s a famil y (a? ) of elements o f Inv/c
0
(A) suc h that , fo r ever y
k/ko, th e images of the a,j in liWk{A) mak e up an ff(/c)-basis o f liWk(A).
Prove tha t CA,A' is then a n isomorphism .
[Hint: Surjectivity . Le t a b e a n elemen t o f luvk
0
(A x A'). I f T' i s a n elemen t o f
i71(/c,A/), th e map T i— a(T,T') i s an elemen t o f Invk(A), henc e ca n be writte n i n a
unique wa y as ^ aj ® bj(Tf), wit h bj(T') G H(k). Th e map T' H ^ bj(Tr) i s an elemen t
6j o f Inv/e
0
(A/), an d it i s clear tha t CA,A'(%2 a3 ® bj) i s equal t o a. Injectivit y i s prove d
similarly.]
Show that , i f A' als o ha s property (*) , so does A x A'. Not e tha t th e case p = 2
gives a n alternate proo f o f Th. 16.4, b y induction o n n. Not e als o tha t assumptio n (* ) is
satisfied (whe n p = 2) by several o f the groups considere d i n later sections , suc h a s On ,
SO
n
( n odd), G2 , i*4, ^6 (split) , an d the symmetric grou p SW.
17. Cohomologica l invariant s o f Quad
n
In thi s sectio n w e determine Inv(Quad
n
, Z/2Z) .
17.1. S T I E F E L - W H I T N E Y CLASSES . I f q is a quadrati c for m o f ran k n ove r fc,
we may write i t as q = (ai , a 2 , . . . , an ) fo r a:; G fc*. Let Wi(q) be the z-th elementary
symmetric polynomia l i n the (a^)' s compute d i n the commutative rin g H(k):
w0 = 1,
^1 = 5Z(«i ) = («1^2 •"ftn) = (%))
i
w2 = ^ ( a
i
) - ( a
J
- ) ,
wn = (ai)-(a
2
) (a
n
) ,
and u ^ = 0 if i 0 or z n.
It i s known (cf . [Delz62 ] o r [Mi70] ) tha t thes e ar e indeed invariant s o f q, i.e.,
they d o not depend o n the particular wa y of writing q as ( a i , . . ., a
n
). The y provid e
n + 1 elements i n Inv(Quad
n
, Z/2Z) .
17.2. T H E TOTA L S T I E F E L - W H I T N E Y CLASS . W e writ e w(q) fo r th e total
Stiefel-Whitney class
n
w(q) = ^2m{q) i n H(k).
i=0
It i s characterized b y the propertie s
(1) w((a)) = l + (a)
(2) wiq^^^w^'wia').
17.3. INVARIANT S O F Quad
n
.
T H E O R E M . The group Inv/e
0
(Quad
n
,Z/2Z) is a free H(ko)-module with basis
{w0,w1,...1wn}.
P R O O F . Le t G = Z/2 Z x x Z/2Z as in Sectio n 16. An arbitrary elemen t o f
Hl(k, G) i s given b y an n-tupl e
( a i , . . . , a
n
) G k*/k*2 x •• xk*/k* 2.
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