%0 Journal Article
%T Rational BV-algebra in String Topology
%A Yves Felix
%A Jean-Claude Thomas
%J Mathematics
%D 2007
%I arXiv
%X 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) $.
%U http://arxiv.org/abs/0705.4194v1