On the cohomology algebra of free loop spaces

Abstract

Let $X$ be a simply connected space and $\mathbb{K}$ be any field. The normalized singular cochains $N^{\ast }(X;\mathbb{K})$ admit a natural strongly homotopy commutative algebra structure, which induces a natural product on the Hochschild homology $H{H}_{\ast }{N}^{\ast }X$ of the space $X$. We prove that, endowed with this product, $H{H}_{\ast }{N}^{\ast }X$ is isomorphic to the cohomology algebra of the free loop space of $X$ with coefficients in $\mathbb{K}$. We also show how to construct a simpler Hochschild complex which allows direct computation.

21 pages, to appear in Topology