Loop homology algebra of a closed manifold

Details Loop homology algebra of a closed manifold Loop homology algebra of a closed manifold

2002-03-14

arxiv.org/abs/math/0203137v2

Mathematics

Algebraic Topology

New version 19 pages

Scientific paper

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.

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