On the Adams isomorphism for equivariant orthogonal spectra

Reich, Holger; Varisco, Marco

doi:10.2140/agt.2016.16.1493

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On the Adams isomorphism for equivariant orthogonal spectra

Holger Reich and Marco Varisco

Algebraic & Geometric Topology 16 (2016) 1493–1566

DOI: 10.2140/agt.2016.16.1493

Abstract

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 $G ∕ N$ –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.

Keywords

Adams isomorphism, equivariant stable homotopy theory

Mathematical Subject Classification 2010

Primary: 55P42, 55P91

References

Publication

Received: 15 September 2014

Revised: 22 July 2015

Accepted: 21 September 2015

Published: 1 July 2016

Authors

Holger Reich
Institut für Mathematik Freie Universität Berlin Arnimallee 7 D-14195 Berlin Germany
http://mi.fu-berlin.de/math/groups/top/members/Professoren/reich.html
Marco Varisco
Department of Mathematics and Statistics University at Albany, SUNY 1400 Washington Ave Albany, NY 12222 United States
http://albany.edu/~mv312143/

Volume 16, issue 3 (2016)