19. COHOMOLOGICA L INVARIANT S O F Quad
n5
( n ODD ) 4 5
The valu e o f a(q) mus t remai n unchange d i f w e writ e q differently , i n particula r
if w e swa p th e las t tw o entrie s (tha t is , i f w e interchang e th e term s a
n
_ i an d
8a\ .. . a
n
_ i ) . Thu s
/
i g 7
x a(q) = a^qo £ (Sai - an-!))
= 127=o K-Wi(q
0
e (Sai---a
n
-i)).
The differenc e o f (19.6) an d (19.7) give s b y (19.2):
n - l
] P Xi -(5ai... a
n
-2) -^i-i(^o ) = 0 .
2=0
We hav e (5a\ ... a
n
-2) {8) + Wi(qo), an d applyin g (19.3) t o expan d th e produc t
wi(qo)wi-i(q0), w e obtai n
n—l n —1
(19.8) Yl V(-*)-«i-i(?o ) + J2 Xi ' P ) ^ i - i ( 9 o ) +m(qo)} = 0 .
i=0 i=0
2 even z od d
Since {wo, w±, ..., u
n
-2} i s a basi s fo r Inv(Quad
n
_
2
, Z/2Z ) b y 17.3, the coefficient s
of thes e wi mus t al l b e 0 . Examinin g th e coefficient s o f th e Stiefel-Whitne y classe s
of eve n degree , w e hav e
(19.9) Xi (£) = 0 fo r i odd , 1 i n.
Collecting th e coefficient s o f th e classe s o f od d degree , w e obtai n
(19.10) A , = A
i +
i (-S) fo r i odd , 1 i n.
For i eve n (includin g th e cas e wher e i = 0 , wher e w e se t A_ i = 0 ) w e have :
Ai_i -Wi-i^i ) 4 - Xi "Wi{q{) =
= X
l
-[(-l)'Wi-1) (q1 + (S)'W^
1
(q1 )-i-wt(q1 )} b y (19.10)
(19.11) = X
i
'[w1)wi-1) (q1 (q1 + {S)'W
i
-1 {q1 )-hwi(q1 )] b y (19.3)
= Xi [(Sai an _ i ) - ^ - i ( ^ i ) + Wi{qi)] b y (19.4)
= Xi-Wi(q) b y (19.2)
Combining thi s wit h (19.5) an d usin g th e fac t tha t n i s odd , w e obtain :
n - l
(19.12) a(q)= J2 V^ifa) .
i = 0
i eve n
This prove s tha t th e eve n Stiefel-Whitne y classe s spa n Inv(Quad
n 5
, Z/2Z) .
Suppose no w tha t th e invarian t a given b y (19.12) i s 0. The n s o i s a\. Sinc e th e
Stiefel-Whitney classe s are linearl y independen t i n Inv(Quad
n
_
1 ?
Z/2Z ) b y 17.3, th e
A; must b e 0 fo r al l i , henc e i n particula r fo r i even . Thu s th e eve n Stiefel-Whitne y
classes {uo , W2,..., w
n
-i} ar e linearl y independen t i n Inv(Quad
n 5
, Z/2Z).
EXERCISE 19.13 . Fo r n odd , defin e a n isomorphis m o f functor s
Quadn
5
x Quad
1
Quadn
by (q, qi) •— q ® #i, an d us e Exercis e 16.5 to recove r Theore m 19.1.
Previous Page Next Page