Loop homology algebra of a closed manifold

TL;DR

This paper develops a model to compute the loop homology algebra of a closed manifold and analyzes the properties of the intersection morphism, revealing its kernel and image characteristics.

Contribution

It introduces a new model for calculating the loop homology algebra and studies the algebraic properties of the intersection morphism in this context.

Findings

Kernel of the intersection morphism is nilpotent

Image of the morphism is contained in the center of the loop space homology

Provides explicit computation method for loop homology algebra

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. $\mathbb H_*(M^{S^1}) = H_{*+d}(M^{S^1})$. Chas and Sullivan have defined a loop product on $\mathbb H_*(M^{S^1})$ and an intersection morphism $I : \mathbb H_*(M^{S^1}) \to H_*(\Omega M)$. The algebra $\mathbb H_*(M^{S^1})$ is commutative and $I$ is a morphism of algebras. In this paper we produce a model that computes the algebra $\mathbb H_*(M^{S^1})$ and the morphism $I$. We show that the kernel of $I$ is nilpotent and that the image is contained in the center of $H_*(\Omega M)$, which is in general quite small.