On the Adams isomorphism for equivariant orthogonal spectra

Reich, Holger¹ ; Varisco, Marco²

¹Institut für Mathematik, Freie Universität Berlin, Arnimallee 7, D-14195 Berlin, Germany
²Department of Mathematics and Statistics, University at Albany, SUNY, 1400 Washington Ave, Albany, NY 12222, United States

Algebraic and Geometric Topology, Tome 16 (2016) no. 3, pp. 1493-1566

Résumé

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.

Détail

DOI : 10.2140/agt.2016.16.1493

Classification : 55P42, 55P91
Keywords: Adams isomorphism, equivariant stable homotopy theory

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