## Homology, Homotopy and Applications

### The cohomology ring of free loop spaces

Luc Menichi

#### 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 $\frac {1}{(p-1)!}\in {\Bbbk}$.

#### Article information

Source
Homology Homotopy Appl., Volume 3, Number 1 (2001), 193-224.

Dates
First available in Project Euclid: 19 February 2006

https://projecteuclid.org/euclid.hha/1140370271

Mathematical Reviews number (MathSciNet)
MR1854644

Zentralblatt MATH identifier
0974.55005

Subjects
Primary: 55P35: Loop spaces
Secondary: 16E40: (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.)

#### Citation

Menichi, Luc. The cohomology ring of free loop spaces. Homology Homotopy Appl. 3 (2001), no. 1, 193--224. https://projecteuclid.org/euclid.hha/1140370271