The rational homology ring of the based loop space of the gauge groups and the spaces of connections on a four-manifold

We provide a rational-homotopic proof that the ranks of the homotopy groups of a simply connected four-manifold depend only on its second Betti number. We also consider the based loop spaces of the gauge groups and the spaces of connections of a simply connected four-manifold and, using the models from rational homotopy theory, we obtain explicit formulas for their rational Pontryagin homology rings.