The Hochschild cohomology of a closed manifold

Yves Felix; Jean-Claude Thomas; Micheline Vigué-Poirrier

Publications Mathématiques de l'IHÉS (2004)

  • Volume: 99, page 235-252
  • ISSN: 0073-8301

Abstract

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Let M be a closed orientable manifold of dimension dand 𝒞 * ( M ) be the usual cochain algebra on M with coefficients in a fieldk. The Hochschild cohomology of M, H H * ( 𝒞 * ( M ) ; 𝒞 * ( M ) ) is a graded commutative and associative algebra. The augmentation map ε : 𝒞 * ( M ) → 𝑘 induces a morphism of algebras I : H H * ( 𝒞 * ( M ) ; 𝒞 * ( M ) ) → H H * ( 𝒞 * ( M ) ; 𝑘 ) . In this paper we produce a chain model for the morphism I. We show that the kernel of I is a nilpotent ideal and that the image of I is contained in the center of H H * ( 𝒞 * ( M ) ; 𝑘 ) , which is in general quite small. The algebra H H * ( 𝒞 * ( M ) ; 𝒞 * ( M ) ) is expected to be isomorphic to the loop homology constructed by Chas and Sullivan. Thus our results would be translated in terms of string homology.

How to cite

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Felix, Yves, Thomas, Jean-Claude, and Vigué-Poirrier, Micheline. "The Hochschild cohomology of a closed manifold." Publications Mathématiques de l'IHÉS 99 (2004): 235-252. <http://eudml.org/doc/104207>.

@article{Felix2004,
abstract = {Let M be a closed orientable manifold of dimension dand $\mathcal \{C\}^*(M)$ be the usual cochain algebra on M with coefficients in a fieldk. The Hochschild cohomology of M, $H\!H^*(\mathcal \{C\}^*(M);\mathcal \{C\}^*(M))$ is a graded commutative and associative algebra. The augmentation map $\varepsilon : \mathcal \{C\}^*(M) \rightarrow \{\textbf \{\textit \{k\}\}\}$ induces a morphism of algebras $I : H\!H^*(\mathcal \{C\}^*(M);\mathcal \{C\}^*(M)) \rightarrow \{H\!H^*(\mathcal \{C\}^*(M);\{\textbf \{\textit \{k\}\}\})\}$. In this paper we produce a chain model for the morphism I. We show that the kernel of I is a nilpotent ideal and that the image of I is contained in the center of $H\!H^*(\mathcal \{C\}^*(M);\{\textbf \{\textit \{k\}\}\})$, which is in general quite small. The algebra $H\!H^*(\mathcal \{C\}^*(M);\mathcal \{C\}^*(M))$ is expected to be isomorphic to the loop homology constructed by Chas and Sullivan. Thus our results would be translated in terms of string homology.},
author = {Felix, Yves, Thomas, Jean-Claude, Vigué-Poirrier, Micheline},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {Hochschild cohomology; free loop space},
language = {eng},
pages = {235-252},
publisher = {Springer},
title = {The Hochschild cohomology of a closed manifold},
url = {http://eudml.org/doc/104207},
volume = {99},
year = {2004},
}

TY - JOUR
AU - Felix, Yves
AU - Thomas, Jean-Claude
AU - Vigué-Poirrier, Micheline
TI - The Hochschild cohomology of a closed manifold
JO - Publications Mathématiques de l'IHÉS
PY - 2004
PB - Springer
VL - 99
SP - 235
EP - 252
AB - Let M be a closed orientable manifold of dimension dand $\mathcal {C}^*(M)$ be the usual cochain algebra on M with coefficients in a fieldk. The Hochschild cohomology of M, $H\!H^*(\mathcal {C}^*(M);\mathcal {C}^*(M))$ is a graded commutative and associative algebra. The augmentation map $\varepsilon : \mathcal {C}^*(M) \rightarrow {\textbf {\textit {k}}}$ induces a morphism of algebras $I : H\!H^*(\mathcal {C}^*(M);\mathcal {C}^*(M)) \rightarrow {H\!H^*(\mathcal {C}^*(M);{\textbf {\textit {k}}})}$. In this paper we produce a chain model for the morphism I. We show that the kernel of I is a nilpotent ideal and that the image of I is contained in the center of $H\!H^*(\mathcal {C}^*(M);{\textbf {\textit {k}}})$, which is in general quite small. The algebra $H\!H^*(\mathcal {C}^*(M);\mathcal {C}^*(M))$ is expected to be isomorphic to the loop homology constructed by Chas and Sullivan. Thus our results would be translated in terms of string homology.
LA - eng
KW - Hochschild cohomology; free loop space
UR - http://eudml.org/doc/104207
ER -

References

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  10. 10. Y. Félix, S. Halperin and J.-C. Thomas, Rational Homotopy Theory, Grad. Texts Math. 205, Springer-Verlag, New York (2000). Zbl0961.55002MR1802847
  11. 11. M. Gerstenhaber, The cohomology structure of an associative ring, Ann. Math., 78 (1963), 267–288. Zbl0131.27302MR161898
  12. 12. J. D. S. Jones, Cyclic homology and equivariant homology, Invent. Math., 87 (1987), 403–423. Zbl0644.55005MR870737
  13. 13. M. Vigué-Poirrier, Homologie de Hochschild et homologie cyclique des algèbres différentielles graduées, in: Astérisque: International Conference on Homotopy Theory (Marseille-Luminy-1988), 191 (1990), 255–267. Zbl0728.19003MR1098974

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