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Proceedings of the American Mathematical Society

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

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Monoid of self-equivalences and free loop spaces
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by Yves Félix and Jean-Claude Thomas PDF
Proc. Amer. Math. Soc. 132 (2004), 305-312 Request permission

Abstract:

Let $M$ be a simply-connected closed oriented $N$-dimensional manifold. We prove that for any field of coefficients $l\!k$ there exists a natural homomorphism of commutative graded algebras $\Gamma : H_* (\Omega \mbox {aut}_1 M) \to \mathbb H_{*}(M^{S^1})$ where $\mathbb H_*(M^{S^1})= H_{* +N}(M^{S^1})$ is the loop algebra defined by Chas and Sullivan. As usual $\mbox {aut}_1 X$ denotes the monoid of self-equivalences homotopic to the identity, and $\Omega X$ the space of based loops. When $l\!k$ is of characteristic zero, $\Gamma$ yields isomorphisms $H^{n+N}_{(1)}(M^{S^1}) \stackrel {\cong }{\to } (\pi _n(\Omega \mbox {aut}_1 M) \otimes l\!k)^\vee$ where $\bigoplus _{l=1}^\infty H^n_{(l)}(M^{S^1})$ denotes the Hodge decomposition on $H^* (M ^{S^1})$.
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Additional Information
  • Yves Félix
  • Affiliation: Département de Mathématique, Université Catholique de Louvain, 2, Chemin du Cyclotron, 1348 Louvain-La-Neuve, Belgium
  • Email: felix@math.ucl.ac.be
  • Jean-Claude Thomas
  • Affiliation: Faculté des Sciences, Université d’Angers, 2, Boulevard Lavoisier, 49045 Angers, France
  • Email: thomas@univ-angers.fr
  • Received by editor(s): May 5, 2002
  • Received by editor(s) in revised form: August 30, 2002
  • Published electronically: May 28, 2003
  • Communicated by: Paul Goerss
  • © Copyright 2003 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 132 (2004), 305-312
  • MSC (2000): Primary 55P35, 55P62, 55P10
  • DOI: https://doi.org/10.1090/S0002-9939-03-07018-7
  • MathSciNet review: 2021275