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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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

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.