Covering Homology

TL;DR

This paper introduces covering homology, a new invariant of commutative ring spectra inspired by topological cyclic homology, and explores its properties and potential applications to algebraic K-theory and the red shift conjecture.

Contribution

It defines covering homology for commutative ring spectra, connecting it to topological cyclic homology and iterated topological Hochschild homology, providing new tools for algebraic K-theory.

Findings

Covering homology with respect to circle isogenies equals topological cyclic homology.

Covering homology with respect to torus isogenies is constructed from iterated topological Hochschild homology.

The structure of covering homology may aid in exploring the red shift conjecture.

Abstract

We introduce the notion of "covering homology" of a commutative ring spectrum with respect to certain families of coverings of topological spaces. The construction of covering homology is extracted from Bokstedt, Hsiang and Madsen's topological cyclic homology. In fact covering homology with respect to the family of orientation preserving isogenies of the circle is equal to topological cyclic homology. Our basic tool for the analysis of covering homology is a cofibration sequence involving homotopy orbits and a restriction map similar to the restriction map used in Bokstedt, Hsiang and Madsen's construction of topological cyclic homology. Covering homology with respect to families of isogenies of a torus is constructed from iterated topological Hochschild homology. It receives a trace map from iterated algebraic K-theory and the hope is that the rich structure, and the calculability of covering homology will make covering homology useful in the exploration of J. Rognes' ``red shift conjecture''.