Galois extensions of structured ring spectra. Stably dualizable groups
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Publication:5453522
DOI10.1090/MEMO/0898zbMATH Open1166.55001arXivmath/0502184OpenAlexW1999142206MaRDI QIDQ5453522
Author name not available (Why is that?)
Publication date: 1 April 2008
Published in: (Search for Journal in Brave)
Abstract: We extend the duality theory for topological groups from the classical theory for compact Lie groups, via the topological study by J. R. Klein [Kl01] and the p-complete study for p-compact groups by T. Bauer [Ba04], 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 and nontrivial for 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 pi_*(L_E S), in degree dim_E(G) = [S^{adG}] of the Pic_E-graded homotopy groups of the E-localized sphere spectrum.
Full work available at URL: https://arxiv.org/abs/math/0502184
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