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E^ = E^ ku*(BSL2(3){2)). in the Adams spectral sequence for FIGURE t AA c4 666 v 4 2 3 M v5 i i q 666 4 . cb66 q 66 666 v 3 3 , , q 66 , , 6<$$ 66 666 2 . q6 A A 666 v 6 S 3 , , 66 (jxkcj)^ q 66 4 3 666 666 v2q6(t 2 , q 66bq v v q vaq q 6 4 3 2 . 12. The Adams spectral sequence for ku*(BQs). 5. DIHEDRAL GROUPS. 5. Dihedral groups. The dihedral groups have the same representation rings as the quaternion groups of the same order, but have considerably more complicated H and kucohomologies, since they have rank 2.

The Gysin sequence of the sphere bundle S(an) = BCn —+ BS1 B -^ BS\ th associated to the n tensor power of a faithful simple representation a, splits into short exact sequences because the [n]-series is not a zero divisor. This provides presentations of K*(BCn) and ku*(BCn): K*{BCn) = Z[v,v-l][[e]]/([n](e)) = Z[v,V-%e]]/((l - (1 - e)")) and, sitting inside it, ku*{BCn) = Z[v][[y]]/([n](y)) = Z[v][[y]}/((1 - (1 where y = eku(a) G ku2(BCn), and e = ve#-(a) = (1 - a) e vy)n)/v) K°(BCn). 2. C Y C L I C G R O U P S .

At the prime q, BGVA = BP V B2p V . • V Btq-l Proof: The action of Cp on Cq is the k-th power map, where k is a primitive p-th root of 1 mod q. The induced action on H*(BCq;Fq) is multiplication by kl in degrees 2i — 1 and 2i, so the Cp invariants are exactly the elements in degrees 0 and —1 mod 2p. The inclusion of the wedge of the Bip into BCq composed with the natural mapping to BGVA is therefore an equivalence. • This gives the additive structure of ku*BGVAl and the representation ring will give us the multiplicative relations.