Comparing cyclotomic structures on different models for topological Hochschild homology

TL;DR

This paper constructs stable equivalences between two models of topological Hochschild homology (THH) for orthogonal ring spectra, showing their resulting topological cyclic homologies are equivalent.

Contribution

It establishes a chain of stable equivalences between different models of THH, bridging the cyclic bar construction and B"okstedt's definition.

Findings

The two models of THH are stably equivalent.

Resulting topological cyclic homologies are equivalent.

Provides a unified framework for different THH constructions.

Abstract

The topological Hochschild homology $THH(A)$ of an orthogonal ring spectrum $A$ can be defined by evaluating the cyclic bar construction on $A$ or by applying B\"okstedt's original definition of $THH$ to $A$. In this paper, we construct a chain of stable equivalences of cyclotomic spectra comparing these two models for $THH(A)$. This implies that the two versions of topological cyclic homology resulting from these variants of $THH(A)$ are equivalent.