58
VII. COHOMOLOGICA L INVARIANT S O F ETAL E ALGEBRA S
by pullin g bac k th e cohomolog y o f S
n
a s i n 4.2 : Ever y E G Etn(fc) define s a ma p
¥E
:
T/c—• Sn a s i n 3.2 , an d w e set a u{E) = (f* E(u).
We hav e obviou s embedding s S
n
-^ On c - GL
n
wher e O
n
an d GL
n
denot e
the group s ove r R endowe d wit h th e usua l topology . Le t Bo
n
an d Bs
n
b e th e
"classifying spaces " o f O
n
an d S n, cf . [Bor55 , §8] . Th e ma p Bs
n
Bo
n
define s a
map o f singular cohomolog y group s
H(B0n, Z/2Z ) - H(B
Sn
,Z/2Z).
Since S ^ i s discrete , th e cohomolog y o f S
n
(i n th e algebrai c sens e o f th e word )
is th e sam e a s th e (topological ) cohomolog y o f th e classifyin g spac e Bs n\ tha t is ,
we hav e th e (sometime s puzzling ) identit y H(Bs n,Z/2Z) = H{S niZ/2Z). Thu s
any element o f H{Bo
7l
-) Z/2Z ) define s a corresponding elemen t o f H(Bsn, Z/2Z ) =
H(Sn, Z/2Z) . I n particular, th e Stiefel-Whitney classe s w\, ... , w
n
o£H(Bo
nl
Z/2Z )
(see e.g. [Bo r 55, §10]) define element s o f H(S
n
, Z/2Z ) whic h we still denote by w\,
..., w
n
. Th e associate d element s o f Inv(Et
n
, Z/2Z) ar e denote d b y wf a .
25.2.
TOTA L GALOI S STIEFEL-WHITNE Y CLASS .
W e adop t th e conventio n
that WQ & = 1 and tha t wf a = 0 when i 0 or i n, an d w e set
As in 17.2, we hav e
w^\k[x}/(x2-a)) = l^(a)
ws&l(E x E') = w^\E)w^\E')
EXAMPLE
25.3 . w
x
e iJ
1(5
n
,Z/2Z) = Hom(5 n,Z/2Z) i s th e sig n homomor -
phism, an d wf
a
i s the discriminant .
W2 G
H2(Sn,
Z/2Z ) determine s a central extension S
n
o f Sn b y Z/2Z describe d
by Schu r [Schu r 11]: Ther e i s an exac t sequenc e
1 -+ {±1} ^!&-S n-*l
characterized b y th e fac t tha t a transpositio n i n S
n
lift s t o a n elemen t o f orde r 2
of S n, bu t a produc t o f 2 disjoint transposition s lift s t o a n elemen t o f orde r 4 . S o
u2 is not trivia l i f n 4.
EXAMPLE
25.4 . Le t E = E\ x x Em wher e Ej
Xj G /c*//c*2. Th e propertie s i n 25. 2 giv e
3
25.5. DESCRIPTIO N O F Inv(Et
n
, Z/2Z) I N TERMS O F THE wf a[. Le t m = [n/2] ,
the integra l par t o f n/2 , an d le t H = Z/2 Z x x Z/2 Z ( m copies) . W e identif y
i7 wit h th e subgrou p o f S
n
generate d b y th e m transposition s (12), (34) , .. . B y
Th. 24.11the restrictio n ma p
Invfco(Etn, Z/2Z ) = Inv
fco
(5n, Z/2Z) - Inv
ko
(H, Z/2Z )
is injective . Moreover , b y 13.2, it s imag e lie s i n th e subgrou p Inv/
Co
(if, Z/2Z)5rT1
of Invfc 0(il, Z/2Z) fixe d b y the actio n o f the symmetri c grou p 5
m
(actin g o n H b y
permuting th e factors) .
= k[x]/(x
2
Xj) fo r som e
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