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# Homology, Homotopy and Applications

## Volume 3 (2001)

### Number 1

### The cohomology ring of free loop spaces

Pages: 193 – 224

DOI: http://dx.doi.org/10.4310/HHA.2001.v3.n1.a9

#### Author

#### Abstract

Let $X$ be a simply connected space and $\Bbbk$ a commutative ring. Goodwillie, Burghelea and Fiedorowicz proved that the Hochschild cohomology of the singular chains on the space of pointed loops $HH^{*}S_*(\Omega X)$ is isomorphic to the free loop space cohomology $H^{*}(X^{S^{1}})$. We prove that this isomorphism is compatible with the usual cup product on $H^{*}(X^{S^{1}})$ and the cup product of Cartan and Eilenberg on $HH^{*}S_*(\Omega X)$. In particular, we make explicit the algebra $H^{*}(X^{S^{1}})$ when $X$ is a suspended space, a complex projective space or a finite CW-complex of dimension $p$ such that $\frac {1}{(p-1)!}\in {\Bbbk}$.

#### 2010 Mathematics Subject Classification

Primary 55P35. Secondary 16E40.