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