20. COHOMOLOGICA L INVARIANT S O F Quad
n5
( n EVEN ) 47
It follow s fro m a theore m o f Merkurje v (se e e.g . [A r 84, Prop . 2] ) tha t q is uniquel y
determined b y it s invariant s w
2
{q) G H2(k) an d b\(q) G H3(k). I n particular , q i s
a 2-Pfiste r for m i f an d onl y i f bi(q) = 0 .
(For a mor e direc t descriptio n o f b\ i n th e 5 1 case, se e th e lette r o f M . Ros t
reproduced i n Appendi x A. )
20.4. Le t Is denot e th e idea l i n H(ko) consistin g o f those A such tha t (5) -X = 0 .
For n eve n an d 4 , ther e i s a ma p
(20.5) I
6
- Inv
f c o
(Quad
n
^, Z/2Z )
of iJ(/co)-module s give n b y A \— b\, cf . Prop . 20.1.
Let X £ Is li e i n th e kerne l o f (20.5) . Fo r q\ G Quadn _
l 5
w e hav e
0 = b
x
(q + (d(
qi
)6)) = A-ix;
n
_i(9i).
By 17.3, this implie s A = 0 . Tha t is , th e ma p i n (20.5 ) i s injective .
T H E O R E M 20.6 . For n even and 4 , Invfc
0
(Quad
5
, Z/2Z) is a direct sum of
the free H(ko)-module with basis {wo
1
W21 ..., ^ n - 2 } and the image of the ideal Is-
P R O O F . Th e proo f o f thi s theore m i s the sam e a s th e proo f o f 19.1 up t o (19.9)
and (19.10) . Fo r i = n 1 (whic h i s odd) , thes e formula s ar e no t immediatel y
implied b y (19.8) sinc e w
n
-i(qo) = 0 . Instead , examinin g th e coefficien t o f w
n
-i(qo)
in (19.8), w e hav e
(20.7) A
n
_i-((J) = 0 .
Equation (19.11hold ) s wit h n o change , an d instea d o f (19.12) , w e obtai n
n - 2
(20.8) cb(q) = ^2 ^i'
wM)
+
X
n-i -w n-i(qi).
i = 0
i eve n
The secon d ter m i n th e su m ma y b e writte n a s b\
nl
(q) b y (20.7) , whic h prove s
that th e specifie d element s spa n Inv(Quad
n s
, Z/2Z) .
Suppose no w tha t th e invarian t a give n b y (20.8 ) i s 0 . The n th e induce d
invariant o n Quad
n
_
x
i s 0 and th e linea r independenc e (17.3) o f the Stiefel-Whitne y
classes i n Inv(Quad
n
_
l 5
Z/2Z ) implie s tha t al l o f th e coefficient s A ^ ar e 0 . Thus ,
the se t {1, W2,W4,... ,u
n
-2} i s linearl y independent , an d it s spa n ha s intersectio n
{0} wit h th e imag e o f Is.
EXERCISE 20. 9 (th e cas e n = 2) . Le t J$ b e th e idea l o f H(k
0
) generate d b y (-6). I t
is contained i n th e idea l 1$ of 20.4 . Writ e q in Quad
2 5
a s ( a i , ^ ) - Fo r A £ J5 :
(1) Sho w tha t (ai ) - A = (a
2
) -A.
(2) Sho w tha t (ai ) - A onl y depend s o n q.
This give s a n invarian t c\ fo r eac h A £ J5.
(3) Prov e tha t Invfc
0
(Quad2j(5, Z/2Z ) i s th e direc t su m o f th e constan t invariant s
and th e idea l J$.
(Note tha t Is i s th e imag e o f Co r : H(k(VS)) H(k) an d tha t Js i s th e kerne l o f
H(k) - H(k(V^S)), cf . [Ar75 , Cor . 4.6] )
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