Homology, Homotopy and Applications

Volume 25 (2023)

Number 1

On the topological $K$-theory of twisted equivariant perfect complexes

Pages: 173 – 187

DOI: https://dx.doi.org/10.4310/HHA.2023.v25.n1.a9

Authors

Michael K. Brown (Department of Mathematics and Statistics, Auburn University, Auburn, Alabama, U.S.A.)

Tasos Moulinos (Institut de Mathématiques de Toulouse, Université Paul Sabatier, Toulouse, France)

Abstract

We construct a comparison map from the topological $K$-theory of the dg-category of twisted perfect complexes on certain global quotient stacks to twisted equivariant $K$-theory, generalizing constructions of Halpern-Leistner–Pomerleano [HLP15] and Moulinos [Mou19]. We prove that this map is an equivalence if a version of the projective bundle theorem holds for twisted equivariant $K$-theory. Along the way, we give a new proof of a theorem of Moulinos that the comparison map is an equivalence in the non-equivariant case.

Keywords

dg-category, topological $K$-theory, twisted equivariant $K$-theory

2010 Mathematics Subject Classification

18E30, 19L47, 19L50

Received 29 July 2021

Received revised 20 March 2022

Accepted 23 March 2022

Published 22 March 2023