Loop homology algebra of a closed manifold

Abstract

The loop homology of a closed orientable manifold $M$ of dimension $d$ is the ordinary homology of the free loop space $M^{S^1}$ with degrees shifted by $d$, i.e. ${\mathbb{H}}_{\ast }(M^{S^1})=H_{\ast +d}(M^{S^1})$. Chas and Sullivan have defined a loop product on ${\mathbb{H}}_{\ast }(M^{S^1})$ and an intersection morphism $I:{\mathbb{H}}_{\ast }(M^{S^1})\to H_{\ast }(\Omega M)$. The algebra ${\mathbb{H}}_{\ast }(M^{S^1})$ is commutative and $I$ is a morphism of algebras. In this paper we produce a model that computes the algebra ${\mathbb{H}}_{\ast }(M^{S^1})$ and the morphism $I$. We show that the kernel of $I$ is nilpotent and that the image is contained in the center of $H_{\ast }(\Omega M)$, which is in general quite small.

New version 19 pages