Title: Stably dualizable groups
Author: John Rognes
Author's e-mail address: rognes@math.uio.no
Abstract:
We extend the duality theory for topological groups from the classical
theory for compact Lie groups, via the topological study by (Dwyer and)
J.R. Klein and the p-complete study for p-compact groups by T. Bauer,
to a general duality theory for stably dualizable groups in the
E-local stable homotopy category, for any spectrum E. The principal
new examples occur in the K(n)-local category, where the Eilenberg-Mac
Lane spaces G = K(Z/p, q) are stably dualizable for all 0 <= q <= n.
We show how to associate to each E-locally stably dualizable group G
a stably defined representation sphere S^{adG}, called the dualizing
spectrum, which is dualizable and invertible in the E-local category.
Each stably dualizable group is Atiyah-Poincare self-dual in the E-local
category, up to a shift by S^{adG}. There are dimension-shifting norm-
and transfer maps for spectra with G-action, again with a shift given
by S^{adG}. The stably dualizable group G also admits a kind of framed
bordism class [G] in the homotopy of L_E S, in degree dim_E(G) = [S^{adG}]
of the Pic_E-graded homotopy groups of the E-localized sphere spectrum.