Mathematics – Algebraic Topology
Scientific paper
2009-08-13
Mathematics
Algebraic Topology
25 pages. Introduction rewritten. Example 35 has been added as application of Theorem 34. Final version. To appear in Trans. A
Scientific paper
Let $M$ 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. Suppose that $G$ acts on $M$. 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 $$H_*(\Omega^2BG)\otimes\mathbb{H}_*(M)\to\mathbb{H}_*(LM)$$ 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 compact Lie group, we compute the Batalin-Vilkovisky algebra $\mathbb{H}_*(LG;\mathbb{Q})$.
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