Real connective $K$-theory and the quaternion group
Dilip Bayen and Robert R. Bruner
Mathematics Department
Wayne State University
Detroit, Michigan, 48202
dbayen@math.wayne.edu
rrb@math.wayne.edu
April, 1995
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 [Generalized Tate Cohomology, Conj 13.4], by showing, in
particular, that t(ko)^G is not a wedge of Eilenberg-Maclane spectra,
as occurs for groups of prime order.