A Batalin-Vilkovisky algebra morphism from double loop spaces to free loops

13 pagesLet $M$ be a compact oriented $d$-dimensional smooth manifold and $X$ a topological space. Chas and Sullivan~\cite{Chas-Sullivan:stringtop} have defined a structure of Batalin-Vilkovisky algebra on $\mathbb{H}_*(LM):=H_{*+d}(LM)$. Getzler~\cite{Getzler:BVAlg} has defined a structure of Batalin-Vilkovisky algebra on the homology of the pointed double loop space of $X$, $H_*(\Omega^2 X)$. Let $G$ be a topological monoid with a homotopy inverse. We define a structure of Batalin-Vilkovisky algebra on $H_*(\Omega^2BG)\otimes\mathbb{H}_*(M)$ extending the Batalin-Vilkovisky algebra of Getzler on $H_*(\Omega^2BG)$. We prove that the morphism of graded algebras $$H_*(\Omega^2BG)\otimes\mathbb{H}_*(M)\rightarrow\mathbb{H}_*(LM)$$ defined by Felix and Thomas~\cite{Felix-Thomas:monsefls}, is in fact a morphism of Batalin-Vilkovisky algebras. In particular, if $G=M$ is a connected Lie group, $H_*(\Omega^2 BG)$ is a trivial sub Batalin-Vilkovisky algebra of $\mathbb{H}_*(LG)$