Homotopy decompositions of the classifying spaces of pointed gauge groups Seminar

We show that if G is a topological group with a homotopy commutative

multiplication (integrally or p-locally) then there is an (integral or p-local) homotopy decomposition of the classifying space of the pointed gauge group of principal G-bundles over simply-connected four-manifolds. This is used to exhibit a large subalgebra of the mod-p cohomology of the classifying space of the full gauge group.