Rational BV-algebra in string topology

Yves Félix; Jean-Claude Thomas

Bulletin de la Société Mathématique de France (2008)

  • Volume: 136, Issue: 2, page 311-327
  • ISSN: 0037-9484

Abstract

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Let M be a 1-connected closed manifold of dimension m and L M be the space of free loops on M . M.Chas and D.Sullivan defined a structure of BV-algebra on the singular homology of L M , H * ( L M ; k ) . When the ring of coefficients is a field of characteristic zero, we prove that there exists a BV-algebra structure on the Hochschild cohomology H H * ( C * ( M ) ; C * ( M ) ) which extends the canonical structure of Gerstenhaber algebra. We construct then an isomorphism of BV-algebras between H H * ( C * ( M ) ; C * ( M ) ) and the shifted homology H * + m ( L M ; k ) . We also prove that the Chas-Sullivan product and the BV-operator behave well with a Hodge decomposition of H * ( L M ) .

How to cite

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Félix, Yves, and Thomas, Jean-Claude. "Rational BV-algebra in string topology." Bulletin de la Société Mathématique de France 136.2 (2008): 311-327. <http://eudml.org/doc/272416>.

@article{Félix2008,
abstract = {Let $M$ be a 1-connected closed manifold of dimension $m$ and $LM$ be the space of free loops on $M$. M.Chas and D.Sullivan defined a structure of BV-algebra on the singular homology of $LM$, $H_\ast (LM; k)$. When the ring of coefficients is a field of characteristic zero, we prove that there exists a BV-algebra structure on the Hochschild cohomology $HH^\ast (C^\ast (M); C^\ast (M))$ which extends the canonical structure of Gerstenhaber algebra. We construct then an isomorphism of BV-algebras between $HH^\ast (C^\ast (M); C^\ast (M)) $ and the shifted homology $ H_\{\ast +m\} (LM; k)$. We also prove that the Chas-Sullivan product and the BV-operator behave well with a Hodge decomposition of $H_\ast (LM)$.},
author = {Félix, Yves, Thomas, Jean-Claude},
journal = {Bulletin de la Société Mathématique de France},
keywords = {string homology; rational homotopy; Hochschild cohomology; free loop space homology; BV-algebra; Gerstenhaber algebra},
language = {eng},
number = {2},
pages = {311-327},
publisher = {Société mathématique de France},
title = {Rational BV-algebra in string topology},
url = {http://eudml.org/doc/272416},
volume = {136},
year = {2008},
}

TY - JOUR
AU - Félix, Yves
AU - Thomas, Jean-Claude
TI - Rational BV-algebra in string topology
JO - Bulletin de la Société Mathématique de France
PY - 2008
PB - Société mathématique de France
VL - 136
IS - 2
SP - 311
EP - 327
AB - Let $M$ be a 1-connected closed manifold of dimension $m$ and $LM$ be the space of free loops on $M$. M.Chas and D.Sullivan defined a structure of BV-algebra on the singular homology of $LM$, $H_\ast (LM; k)$. When the ring of coefficients is a field of characteristic zero, we prove that there exists a BV-algebra structure on the Hochschild cohomology $HH^\ast (C^\ast (M); C^\ast (M))$ which extends the canonical structure of Gerstenhaber algebra. We construct then an isomorphism of BV-algebras between $HH^\ast (C^\ast (M); C^\ast (M)) $ and the shifted homology $ H_{\ast +m} (LM; k)$. We also prove that the Chas-Sullivan product and the BV-operator behave well with a Hodge decomposition of $H_\ast (LM)$.
LA - eng
KW - string homology; rational homotopy; Hochschild cohomology; free loop space homology; BV-algebra; Gerstenhaber algebra
UR - http://eudml.org/doc/272416
ER -

References

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