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 $b_{2}(M)=3$.

Keywords: Rational Pontryagin homology; based loop space; four-manifolds.

MSC: 55P35, 55P62, 57N13

Pages:  90$-$103     

Volume  71 ,  Issue  1-2 ,  2019