Rational BV-algebra in String Topology

Let $M$ be a 1-connected closed manifold and $LM$ be the space of free loops on $M$. In \cite{C-S} M. Chas and D. Sullivan defined a structure of BV-algebra on the singular homology of $LM$, H_\ast(LM; \bk). When the field of coefficients is of characteristic zero, we prove that there exists a BV-algebra structure on \hH^\ast(C^\ast (M); C^\ast (M)) which carries the canonical structure of Gerstenhaber algebra. We construct then an isomorphism of BV-algebras between \hH^\ast (C^\ast (M); C^\ast (M)) and the shifted H_{\ast+m} (LM; {\bk}). We also prove that the Chas-Sullivan product and the BV-operator behave well with the Hodge decomposition of $H_\ast (LM)$

On the cohomology algebra of free loop spaces

Let $X$ be a simply connected space and $\Bbb K$ be any field. The normalized singular cochains $N^*(X; {\Bbb K})$ admit a natural strongly homotopy commutative algebra structure, which induces a natural product on the Hochschild homology $HH_* N^*X$ of the space $X$. We prove that, endowed with this product, $HH_*N^*X$ is isomorphic to the cohomology algebra of the free loop space of $X$ with coefficients in $\Bbb K$. We also show how to construct a simpler Hochschild complex which allows direct computation.Comment: 21 pages, to appear in Topolog

String topology on Gorenstein spaces

The purpose of this paper is to describe a general and simple setting for defining $(g,p+q)$-string operations on a Poincar\'e duality space and more generally on a Gorenstein space. Gorenstein spaces include Poincar\'e duality spaces as well as classifying spaces or homotopy quotients of connected Lie groups. Our presentation implies directly the homotopy invariance of each $(g,p+q)$-string operation as well as it leads to explicit computations.Comment: 30 pages and 2 figure