48 2. K-THEORY WITH REALITY
or
BU -→BO-→SO.
The map to the next filtration quotient is given by SO -→ ΣHZ/2, and the
restriction of this to BO is w1. This follows by pullback from G = C2, since every
element in H1(BG, Z/2) is w1 of the corresponding representation BG-→BO(1).
For G = C2, we have
KO7(BC2)
=
hatwidest(C2)/hatwidest(C2)
RO RU =
(Z/2)2,
while
ko7(BC2)
=
0 by Theorem 6.3.1. It follows that w1 is a permanent cycle. Note in passing
that w1 :
hatwider
JO (G)/
hatwider
JU (G) -→
H1(BG;
Z/2) is surjective since
H1(BG;
Z/2) =
Hom(G, Z/2). Next, we have F2 =
hatwide
JSO/
hatwider
JU , realized by [BG,Spin]:
ΣHZ/2 ΣHZ/2
BU
d47 d47
BO
w1
d79 d79
d47 d47
SOd79d79
Ωw2
d79 d79
BU
d47 d47
BSO
d79 d79
d47 d47
Spin
The final step in the filtration is the natural map Spin -→
Σ3HZ
killing the
bottom homotopy group. This restricts to w3 : BSO -→
Σ3HZ,
and this equals
βw2, inducing the map of fibre sequences
Σ2HZ
d47 d47
Σ2HZ/2
β
d47 d47
Σ3HZd79d79
BU
d47 d47
c1
d79 d79
BSO
w2
d79 d79
d47 d47
Spin
BSU
d79 d79
d47 d47
BSpin
d79 d79
d47 d47
String
d79 d79
This gives the final layer, koG 7 = [BG,String] =
hatwider
JSpin/
hatwider
JU
2
by Corollary 2.8.6.
The argument in degree -8 the filtration is longer again
KOG
8
dim
d15 d15
F1

d111 d111
w1
d15 d15
F2

d111 d111
w2
d15 d15
F3

d111 d111
p1/2
d15 d15
koG8

d111 d111
Z
H1(BG, Z/2) H2(BG, Z/2) H4(BG, Z)
Proceeding as before, we see that F1 = ker(dim : RO(G) -→ Z) = JO, that
w1 : JO -→ H1(BG; Z/2) is surjective with kernel F2 = JSO. Since KOG 9 = 0,
this shows that koG 9 is built from the cokernel of w2 : JSO -→H2(BG; Z/2) and
the cokernel of p1/2 : JSpin-→H4(BG; Z). square
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