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Monoid of self-equivalences and free loop spaces


Authors: Yves Félix and Jean-Claude Thomas
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
Published electronically: May 28, 2003
MathSciNet review: 2021275
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Abstract: Let $M$ be a simply-connected closed oriented $N$-dimensional manifold. We prove that for any field of coefficients $lk$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 $lk $ 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 lk)^\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

DOI: https://doi.org/10.1090/S0002-9939-03-07018-7
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
Article copyright: © Copyright 2003 American Mathematical Society

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