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

TL;DR

This paper constructs a BV algebra structure on the tensor product of homologies related to double loop spaces and free loop spaces, proving a morphism between them preserves BV algebra structure, with explicit computations for Lie groups.

Contribution

It introduces a BV algebra structure on $H_*( ext{double loop space of } BG) imes ext{homology of } M$ extending Getzler's structure, and proves a BV algebra morphism to the free loop space homology.

Findings

Established a BV algebra structure on the tensor product involving double loop space homology.

Proved the algebraic morphism between double loop and free loop space homologies is a BV algebra morphism.

Computed the BV algebra for the free loop space of a connected compact Lie group.

Abstract

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 $\mathbb{H}_*(LG;\mathbb{Q})$.