Homotopy decompositions of the classifying spaces of pointed gauge groups

Abstract

Let G be a topological group and let G∗(P ) be the pointed gauge group of a principal G-bundle P −→M . We prove that if G is homotopy commutative then the homotopy type of the classifying space BG∗(P ) can be completely determined for certain M . This also works p-locally, and valid choices of M include closed simply-connected four-manifolds when localized 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.