Volume 16, issue 1 (2016)

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Classifying spaces of twisted loop groups

Thomas J Baird

Algebraic & Geometric Topology 16 (2016) 211–229
Abstract

We study the classifying space of a twisted loop group ${L}_{\sigma }G$, where $G$ is a compact Lie group and $\sigma$ is an automorphism of $G$ of finite order modulo inner automorphisms. Equivalently, we study the $\sigma$–twisted adjoint action of $G$ on itself. We derive a formula for the cohomology ring ${H}^{\ast }\left(B{L}_{\sigma }G\right)$ and explicitly carry out the calculation for all automorphisms of simple Lie groups. More generally, we derive a formula for the equivariant cohomology of compact Lie group actions with constant rank stabilizers.

Keywords
loop groups, twisted conjugacy, twisted adjoint action, equivariant cohomology, classifying spaces, gauge groups
Primary: 22E67
Secondary: 57S15
Publication
Received: 8 May 2014
Revised: 25 May 2015
Accepted: 16 June 2015
Published: 23 February 2016
Authors
 Thomas J Baird Department of Mathematics & Statistics Memorial University of Newfoundland St. John’s NF A1C 5S7 Canada http://www.thomasjohnbaird.com