2025/05/16 by David Chan, Chan, David, Teena Gerhardt +3
Mathematics · #16E40 #19D55 #55P91 #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT)
paper · pdf · doi:10.48550/arxiv.2505.11327
openalex publication_date 2025/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the past decades, one of the most fruitful approaches to the study of algebraic K-theory has been trace methods, which construct and study trace maps from algebraic K-theory to topological Hochschild homology and related invariants. In recent years, theories of equivariant algebraic K-theory have emerged, but thus far few tools are available for the study and computation of these theories. In this paper, we lay the foundations for a trace methods approach to equivariant algebraic K-theory. For G a finite group, we construct a Dennis trace map from equivariant algebraic K-theory to a G-equivariant version of topological Hochschild homology; for G the trivial group this recovers the ordinary Dennis trace map. We show that upon taking fixed points, this recovers the trace map of Adamyk--Gerhardt--Hess--Klang--Kong, and gives a trace map from the fixed points of coarse equivariant A-theory to the free loop space. We also establish important properties of equivariant topological Hochschild homology, such as Morita invariance, and explain why it can be considered as a multiplicative norm.