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


Author: Luc Menichi
Journal: Trans. Amer. Math. Soc. 363 (2011), 4443-4462
MSC (2010): Primary 55P50, 55P35, 55P62
DOI: https://doi.org/10.1090/S0002-9947-2011-05374-2
Published electronically: February 8, 2011
MathSciNet review: 2792995
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Abstract: Let $ M$ be a compact oriented $ d$-dimensional smooth manifold and $ X$ a topological space. Chas and Sullivan have defined a structure of Batalin-Vilkovisky algebra on $ \mathbb{H}_*(LM):=H_{*+d}(LM)$. Getzler (1994) 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

$\displaystyle H_*(\Omega^2BG)\otimes\mathbb{H}_*(M)\rightarrow\mathbb{H}_*(LM)$

defined by Félix and Thomas (2004), 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})$.


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Additional Information

Luc Menichi
Affiliation: UMR 6093 associée au CNRS, Université d’Angers, Faculté des Sciences, 2 Boulevard Lavoisier, 49045 Angers, France
Email: luc.menichi@univ-angers.fr

DOI: https://doi.org/10.1090/S0002-9947-2011-05374-2
Received by editor(s): November 10, 2009
Received by editor(s) in revised form: April 17, 2010
Published electronically: February 8, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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