Homotopy decompositions of the classifying spaces of pointed gauge groups

Stephen Theriault

Vol. 300 (2019), No. 1, 215–231

DOI: 10.2140/pjm.2019.300.215

Abstract

Let $G$ be a topological group and let ${\mathcal{G}}^{\ast }\left(P\right)$ be the pointed gauge group of a principal $G$-bundle $P\to M$. We prove that if $G$ is homotopy commutative then the homotopy type of the classifying space $B{\mathcal{G}}^{\ast }\left(P\right)$ can be completely determined for certain $M$. This also works $p$-locally, and valid choices of $M$ include closed simply connected four-manifolds when localised at an odd prime $p$. In this case, an application is to calculate part of the mod-$p$ homology of the classifying space of the full gauge group.

Keywords

gauge group, mapping space, homotopy type, homology

Mathematical Subject Classification 2010

Primary: 55P15, 55R35

Secondary: 54C35, 81T13

Milestones

Received: 2 March 2017

Revised: 6 August 2018

Accepted: 10 August 2018

Published: 20 July 2019