#### Volume 16, issue 3 (2016)

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On the Adams isomorphism for equivariant orthogonal spectra

### Holger Reich and Marco Varisco

Algebraic & Geometric Topology 16 (2016) 1493–1566
##### 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-$\Omega$–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
##### Publication
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/