Real connective K-theory and the quaternion group

Bayen, Dilip; Bruner, Robert

doi:10.1090/S0002-9947-96-01516-4

Real connective K-theory and the quaternion group
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by Dilip Bayen and Robert R. Bruner PDF

Trans. Amer. Math. Soc. 348 (1996), 2201-2216 Request permission

Abstract:

Let $ko$ be the real connective K-theory spectrum. We compute $ko_*BG$ and $ko^*BG$ for groups $G$ whose Sylow 2-subgroup is quaternion of order 8. Using this we compute the coefficients $t(ko)^G_*$ of the $G$ fixed points of the Tate spectrum $t(ko)$ for $G = Sl_2(3)$ and $G = Q_8$. The results provide a counterexample to the optimistic conjecture of Greenlees and May [ J. P. C. Greenlees and J. P. May, Generalized Tate cohomology, Memoirs AMS 543 (1995)], Conj. 13.4, by showing, in particular, that $t(ko)^G$ is not a wedge of Eilenberg-Mac Lane spectra, as occurs for groups of prime order.

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