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

Abstract

In this paper we consider the based loop space $\mathrm{\Omega }M$ on a simply connected manifold $M$. We first prove, only by means of the rational homotopy theory, that the rational homotopy type of $\mathrm{\Omega }M$ is determined by the second Betti number $b_2(M)$. We further consider the problem of computation of the rational Pontryagin homology ring $H_{\ast }(\mathrm{\Omega }M)$ when $b_2(M)\le 3$. We prove that $H_{\ast }(\mathrm{\Omega }M)$ is up to degree $5$ generated by the elements of degree $1$ for $b_2(M)=3$.