On rational Pontryagin homology ring of the based loop space on a four-manifold

Abstract

In this paper we consider the based loop space $\Omega M$ on a simply connected manifold $M$. We first prove, only by means of the rational homotopy theory, that the rational homotopy type of $\Omega M$ is determined by the second Betti number $b_{2}(M)$. We further consider the problem of computation of the rational Pontryagin homology ring $H_{*}(\Omega M)$ when $b_{2}(M)\leq 3$. We prove that $H_{*}(\Omega M)$ is up to degree $5$ generated by the elements of degree $1$ for $b_{2}(M)=3$.