42 VI. COHOMOLOGICA L INVARIANT S O F O
n
, S O
The ma p ( a i , . . . , a
n
) i— ( a i , . . ., a
n
) define s a morphism fro m th e functo r H 1 (* , G)
to th e functo r Quad
n
.
Thus an y invarian t a G Inv(Quadn , Z/2Z ) give s an invarian t a
G
i n Inv(G, Z/2Z) ;
one ma y vie w a
G
a s the restrictio n o f a relative t o th e diagona l embeddin g G On ,
cf. §13. B y Theore m 16.4, a
G
i s o f th e for m
aG(ai,...,an)= ^ A/-(a)/fo r A/ G #(&) ,
/C[l,n]
where (a) / is , a s above , th e cu p produc t o f th e (c^ ) fo r i E I. Bu t sinc e a i s a n
invariant o f Quad
n
, i t i s unchange d whe n w e permut e th e c^'s . Consequently , th e
value o f A / depend s onl y o n th e cardinalit y \I\ o f / . Sinc e J2\i\=i( a)i = w i(q)i a 1S
of th e desire d form . Thu s th e Stiefel-Whitne y classe s spa n Inv(Quad
n
, Z/2Z) .
If a i s 0 , the n b y th e linea r independenc e o f th e (a)j' s i n Inv(G , Z/2Z ) (16.4)
all o f th e A/' s mus t b e 0 . Consequentl y th e Stiefel-Whitne y classe s ar e linearl y
independent. (Alternately , on e ma y als o extrac t th e A/' s fro m a a s i n th e proo f o f
16.4.)
The proo f show s tha t th e natura l restrictio n ma p
Inv(O
n
, Z/2Z ) - Inv(G , Z/2Z )
is injectiv e an d tha t it s imag e i s th e subgrou p o f Inv(G , Z/2Z ) mad e u p o f th e
elements fixed unde r th e natura l actio n o f S
n
o n G (i.e. , fixed unde r th e actio n o f
the normalize r o f G i n O
n
, cf . 13.2).
R E M A R K 17.4. A consequenc e o f Theore m 17.3 is :
(1) Th e cu p produc t o f tw o Stiefel-Whitne y classe s ca n b e writte n a s a linea r
combination o f th e Stiefel-Whitne y classes , sinc e suc h a produc t i s a n
invariant. A n explici t formul a ca n b e foun d i n [M i 70, p . 331, lin e 2] : T o
compute w
r
-w
s
, writ e r i n dyadic for m a s r = J2ieR 2 * for i ? C {0,1, 2,...}
and similarl y writ e s = J2ies ^• The n
wr(q) "u
3
(q) = £ m "w
r
+3-m(q), wher e m = ^ 2 l
ieRns
and e (—1) G i/ 1(fc), th e power s £ m o f e bein g relativ e t o th e cu p
product. Equatio n (19.3) i s th e specia l cas e r 1, wher e m = 0 i f s i s
even an d m = 1 if s i s odd .
This show s i n particula r (cf . [M i 70]) tha t ever y w
r
ca n b e writte n a s
a monomia l i n w\ , W2 , w^, wg , . More precisely , i f r = J2ieR ^ a s a ^ove,
then w
r
(q) = H
ieR
w2i(q).
(2) Ther e ar e exterio r power s X pq associate d wit h ever y quadrati c for m q.
(See 27. 1 fo r a definitio n an d a n explici t formula. ) Thei r Stiefel-Whitne y
classes ca n b e writte n (b y a universa l formula ) a s a linea r combinatio n o f
the Stiefel-Whitne y classe s o f q. A n exampl e o f thi s i s
,,
9 x
I 0 i f rankfg ) i s od d
Wl(-X q) = \ i \ -t w (
I wi(q) i t rank(7 ) i s even .
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