  We provide the 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, appealing to~\cite{TI} and
using the models from the rational homotopy theory, we obtain the explicit
formulas for their rational Pontrjagin homology rings.
