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On the Adams isomorphism for equivariant orthogonal spectra

Holger Reich and Marco Varisco

Algebraic & Geometric Topology 16 (2016) 1493–1566

We give a natural construction and a direct proof of the Adams isomorphism for equivariant orthogonal spectra. More precisely, for any finite group G, any normal subgroup N of G, and any orthogonal G–spectrum X, we construct a natural map A of orthogonal GN–spectra from the homotopy N–orbits of X to the derived N–fixed points of X, and we show that A is a stable weak equivalence if X is cofibrant and N–free. This recovers a theorem of Lewis, May and Steinberger in the equivariant stable homotopy category, which in the case of suspension spectra was originally proved by Adams. We emphasize that our Adams map A is natural even before passing to the homotopy category. One of the tools we develop is a replacement-by-Ω–spectra construction with good functorial properties, which we believe is of independent interest.

Adams isomorphism, equivariant stable homotopy theory
Mathematical Subject Classification 2010
Primary: 55P42, 55P91
Received: 15 September 2014
Revised: 22 July 2015
Accepted: 21 September 2015
Published: 1 July 2016
Holger Reich
Institut für Mathematik
Freie Universität Berlin
Arnimallee 7
D-14195 Berlin
Marco Varisco
Department of Mathematics and Statistics
University at Albany, SUNY
1400 Washington Ave
Albany, NY 12222
United States