LAREMA Prépublication n° 162

David CHATAUR, Jean-Claude THOMAS

### Operadic Hochschild chain complex and free loop spaces

We construct, for any algebra $A$ over an operad $\oO$, an Hochschild chain complex, $\hC(\oO, A)$ which is also an $\oO$-algebra. This Hochschild chain complex coincides with the usual one, whenever $A$ is a commutative differential graded algebra. Let $X$ is a simply connected space, $N^\ast (-)$ be the singular cochain functor, $X^{S^1}$ be the free loop space, $\oC_\infty$ be a cofibrant replacement of commutative operad and $M_X$ a $\oC_\infty$-cofibrant model of $X$. We prove that The operadic chain complex $\hC(\oC_\infty, M_X)$ is quasi-isomorphic to $N^\ast (X^{S^1})$ as a $\oC_\infty$-algebra. In particular, for any prime field of coefficients this identifies the action of the large Steenrod algebra on the Hochschild homomology $\hH \left (N^\ast (X)\right)$ with the usual Steenrod operations on H ast

Date d'enregistrement : 05 décembre 2002

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