On the Adams isomorphism for equivariant orthogonal spectra

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 $\mathbf{X}$, we construct a natural map $A$ of orthogonal $G∕N$–spectra from the homotopy $N$–orbits of $\mathbf{X}$ to the derived $N$–fixed points of $\mathbf{X}$, and we show that $A$ is a stable weak equivalence if $\mathbf{X}$ 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.