Algebraic & Geometric Topology

On the Adams isomorphism for equivariant orthogonal spectra

Holger Reich and Marco Varisco

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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.

Article information

Source
Algebr. Geom. Topol., Volume 16, Number 3 (2016), 1493-1566.

Dates
Received: 15 September 2014
Revised: 22 July 2015
Accepted: 21 September 2015
First available in Project Euclid: 28 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1511895855

Digital Object Identifier
doi:10.2140/agt.2016.16.1493

Mathematical Reviews number (MathSciNet)
MR3523048

Zentralblatt MATH identifier
1350.55015

Subjects
Primary: 55P42: Stable homotopy theory, spectra 55P91: Equivariant homotopy theory [See also 19L47]

Keywords
Adams isomorphism equivariant stable homotopy theory

Citation

Reich, Holger; Varisco, Marco. On the Adams isomorphism for equivariant orthogonal spectra. Algebr. Geom. Topol. 16 (2016), no. 3, 1493--1566. doi:10.2140/agt.2016.16.1493. https://projecteuclid.org/euclid.agt/1511895855


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