Real connective K-theory and the quaternion group

Source abstract

Let k o ko be the real connective K-theory spectrum. We compute k o ∗ B G ko_*BG and k o ∗ B G ko^*BG for groups G G whose Sylow 2-subgroup is quaternion of order 8. Using this we compute the coefficients t ( k o ) ∗ G t(ko)^G_* of the G G fixed points of the Tate spectrum t ( k o ) t(ko) for G = S l 2 ( 3 ) G = Sl_2(3) and G = Q 8 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 ( k o ) G t(ko)^G is not a wedge of Eilenberg-Mac Lane spectra, as occurs for groups of prime order.