# Van Den Bergh isomorphisms in String Topology

Abstract : Let $M$ be a path-connected closed oriented $d$-dimensional smooth manifold and let ${\Bbbk}$ be a principal ideal domain. By Chas and Sullivan, the shifted free loop space homology of $M$, $H_{*+d}(LM)$ 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 $HH^{-p}(S_*(G),S_*(G))\cong HH_{p+d}(S_*(G),S_*(G)).$ Therefore, the Gerstenhaber algebra $HH^{*}(S_*(G),S_*(G))$ is a Batalin-Vilkovisky algebra and we have a linear isomorphism $HH^{*}(S_*(G),S_*(G))\cong H_{*+d}(LM).$ This linear isomorphism is expected to be an isomorphism of Batalin-Vilkovisky algebras. We also give a new characterization of Batalin-Vilkovisky algebra in term of derived bracket.
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https://hal.archives-ouvertes.fr/hal-00403688
Contributor : Luc Menichi <>
Submitted on : Wednesday, February 10, 2010 - 5:07:51 PM
Last modification on : Monday, March 9, 2020 - 6:15:52 PM
Long-term archiving on: : Thursday, September 23, 2010 - 5:51:10 PM

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• HAL Id : hal-00403688, version 3
• ARXIV : 0907.2105

### Citation

Luc Menichi. Van Den Bergh isomorphisms in String Topology. 2010. ⟨hal-00403688v3⟩

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