Geometric Tate-Swan cohomology of equivariant spectra

TL;DR

This paper introduces a geometric approach to Tate-Swan cohomology of equivariant spectra, with potential applications to Atiyah-Segal K-theory and topological cyclic homology.

Contribution

It proposes a new geometric perspective on Tate-Swan cohomology and suggests conjectural applications to important areas in algebraic topology.

Findings

A geometric model for Tate-Swan cohomology is sketched.

Potential applications to Atiyah-Segal K-theory of circle actions.

A possible geometric model for topological cyclic homology of the sphere spectrum.

Abstract

We sketch a quick and dirty geometric approach to the Tate-Swan cohomology of equivariant spectra, illustrating it with conjectural applications to Atiyah-Segal $K$-theory of circle actions, and a possible geometric model for the topological cyclic homology of the sphere spectrum.