The Atiyah--Segal completion theorem in twisted K-theory

TL;DR

This paper proves a generalized version of the Atiyah-Segal completion theorem for equivariant twisted K-theory applicable to any compact Lie group and arbitrary twistings, filling a gap in the literature.

Contribution

It extends the Atiyah-Segal completion theorem to all compact Lie groups and general twistings, providing a complete proof where none previously existed.

Findings

The theorem holds for twistings from graded central extensions of G.

The proof uses a two-stage approach: special case and Mayer-Vietoris argument.

Generalization to arbitrary compact Lie groups and twistings is established.

Abstract

In this note we prove the analogue of the Atiyah-Segal completion theorem for equivariant twisted K-theory in the setting of an arbitrary compact Lie group G and an arbitrary twisting of the usually considered type. The theorem generalizes a result by C. Dwyer, who has proven the theorem for finite G and twistings of a more restricted type. While versions of the general result have been known to experts, to our knowledge no proof appears in the current literature. Our goal is to fill in this gap. The proof we give proceeds in two stages. We first prove the theorem in the case of a twisting arising from a graded central extension of G, following the Adams-Haeberly-Jackowski-May proof of the classical Atiyah-Segal completion theorem. After establishing that the theorem holds for this special class of twistings, we then deduce the general theorem by a Mayer-Vietoris argument.