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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Pré-publication, Document de travail
Final version. To appear in J. Noncommut. Geom. A few typos corrected including a sign in the mai.. 2010
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Contributeur : Luc Menichi <>
Soumis le : mercredi 10 février 2010 - 17:07:51
Dernière modification le : lundi 5 février 2018 - 15:00:03
Document(s) archivé(s) le : jeudi 23 septembre 2010 - 17:51:10

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

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Luc Menichi. Van Den Bergh isomorphisms in String Topology. Final version. To appear in J. Noncommut. Geom. A few typos corrected including a sign in the mai.. 2010. 〈hal-00403688v3〉

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