The Connective K-Theory of Finite Groups by R. R. Bruner, J. P. C. Greenlees

By R. R. Bruner, J. P. C. Greenlees

Publication by means of Bruner, R. R., Greenlees, J. P. C.

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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.

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