Equivariant Witt Complexes and Twisted Topological Hochschild Homology

TL;DR

This paper introduces equivariant Witt complexes to describe the algebraic structure of twisted topological Hochschild homology's equivariant homotopy groups, extending classical Witt vector theory to an equivariant setting.

Contribution

It defines equivariant Witt complexes and proves their role in structuring the equivariant homotopy groups of twisted THH, advancing equivariant algebraic topology.

Findings

Equivariant homotopy groups of twisted THH have an equivariant Witt complex structure.

The paper generalizes classical Witt vector concepts to an equivariant context.

Provides a new algebraic framework for studying equivariant THH.

Abstract

The topological Hochschild homology of a ring (or ring spectrum) $R$ is an $S^1$-spectrum, and the fixed points of THH($R$) for subgroups $C_n\subset S^1$ have been widely studied due to their use in algebraic K-theory computations. Hesselholt and Madsen proved that the fixed points of topological Hochschild homology are closely related to Witt vectors. Further, they defined the notion of a Witt complex, and showed that it captures the algebraic structure of the homotopy groups of the fixed points of THH. Recent work of Angeltveit, Blumberg, Gerhardt, Hill, Lawson and Mandell defines a theory of twisted topological Hochschild homology for equivariant rings (or ring spectra) that builds upon Hill, Hopkins and Ravenel's work on equivariant norms. In this paper, we study the algebraic structure of the equivariant homotopy groups of twisted THH. In particular, we define an equivariant Witt complex and prove that the equivariant homotopy of twisted THH has this structure. Our definition of equivariant Witt complexes contributes to a growing body of research in the subject of equivariant algebra.