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Pré-Publication, Document De Travail Année : 2010

Van Den Bergh isomorphisms in String Topology

Résumé

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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Dates et versions

hal-00403688 , version 1 (12-07-2009)
hal-00403688 , version 2 (29-07-2009)
hal-00403688 , version 3 (10-02-2010)

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Citer

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