The homotopy theory of cyclotomic spectra, 10 years later

TL;DR

This paper advances the foundations of equivariant stable homotopy theory by introducing generalized orbit desuspension spectra, analyzing their properties, and establishing new results on geometric fixed points and multiplicative splittings.

Contribution

It introduces generalized orbit desuspension spectra, constructs derived functors of geometric fixed points, and proves a new multiplicative tom Dieck splitting for equivariant commutative ring spectra.

Findings

Construction of derived functor of geometric fixed points

Comparison of derived functors on different spectra

New multiplicative tom Dieck splitting result

Abstract

This paper studies the foundations of the geometric fixed point functor in multiplicative equivariant stable homotopy theory. We introduce a new class of equivariant orthogonal spectra called generalized orbit desuspension spectra and analyze their homotopical behavior with respect to the geometric fixed point functor and especially the interaction with smash products and symmetric powers. This analysis leads to several new foundational results, including the construction of the derived functor of geometric fixed points on equivariant commutative ring orthogonal spectra and its comparison to the derived functor on the underlying equivariant orthogonal spectra. In addition this theory provides foundations for the category of commutative ring pre-cyclotomic spectra and a formula for the derived space of maps in this category (a formula needed in the authors' paper with Yuan on relative TC). Finally, we prove a new multiplicative tom Dieck splitting for equivariant commutative ring spectra obtained as the pushforward of non-equivariant commutative ring spectra.