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We provide the rational-homotopic proof that the ranks of the homotopy groupsof a simply connected four-manifold depend only on its second Betti number. Wealso consider the based loop spaces of the gauge groups and the spaces ofconnections of a simply connected four-manifold and, appealing to~\cite{TI} andusing the models from the rational homotopy theory, we obtain the explicitformulas for their rational Pontrjagin homology rings.

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