We give a natural construction and a direct proof of the Adams isomorphism for equivariant orthogonal spectra. More precisely, for any finite group , any normal subgroup of , and any orthogonal –spectrum , we construct a natural map of orthogonal –spectra from the homotopy –orbits of to the derived –fixed points of , and we show that is a stable weak equivalence if is cofibrant and –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 is natural even before passing to the homotopy category. One of the tools we develop is a replacement-by-–spectra construction with good functorial properties, which we believe is of independent interest.
"On the Adams isomorphism for equivariant orthogonal spectra." Algebr. Geom. Topol. 16 (3) 1493 - 1566, 2016. https://doi.org/10.2140/agt.2016.16.1493