#### Vol. 300, No. 1, 2019

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Homotopy decompositions of the classifying spaces of pointed gauge groups

### Stephen Theriault

Vol. 300 (2019), No. 1, 215–231
##### Abstract

Let $G$ be a topological group and let ${\mathsc{G}}^{\ast }\left(P\right)$ 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 $B{\mathsc{G}}^{\ast }\left(P\right)$ 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
Revised: 6 August 2018
Accepted: 10 August 2018
Published: 20 July 2019
##### Authors
 Stephen Theriault Mathematical Sciences University of Southampton Southampton United Kingdom