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arXiv · 1404.4034

On the Adams isomorphism for equivariant orthogonal spectra

Holger Reich·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-Omega-spectra construction with good functorial properties, which we believe is of independent interest.

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Holger Reich, Marco Varisco. 2015-09-22. On the Adams isomorphism for equivariant orthogonal spectra. https://doi.org/10.2140/agt.2016.16.1493

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