# Van den Bergh isomorphisms in string topology

### Luc Menichi

Université d’Angers, France

## Abstract

Let $M$ be a path-connected closed oriented $d$-dimensional smooth manifold and let $\mathbb{k}$ be a principal ideal domain. By Chas and Sullivan, the shifted free loop space homology of $M$, $H_{*+d}(\mathrm{L}\mkern-1.5mu M )$ is a Batalin–Vilkovisky algebra. Let $G$ be a topological group such that $M$ is a classifying space of $G$. Denote by $S_*(G)$ the (normalized) singular chains on $G$. Suppose that $G$ is discrete or path-connected. We show that there is a Van Den Bergh type isomorphism

Therefore, the Gerstenhaber algebra $\mathrm{HH}^{*}(S_*(G),S_*(G))$ is a Batalin–Vilkovisky algebra and we have a linear isomorphism

This linear isomorphism is expected to be an isomorphism of Batalin–Vilkovisky algebras. We also give a new characterization of Batalin–Vilkovisky algebra in terms of the derived bracket.

## Cite this article

Luc Menichi, Van den Bergh isomorphisms in string topology. J. Noncommut. Geom. 5 (2011), no. 1, pp. 69–105

DOI 10.4171/JNCG/70