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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A Batalin-Vilkovisky algebra morphism from double loop spaces to free loops
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by Luc Menichi PDF
Trans. Amer. Math. Soc. 363 (2011), 4443-4462 Request permission

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
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2792995