# 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

## Access Full Article

top## Abstract

top## How to cite

topFelix, 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

top- 1. J. F. Adams, On the cobar construction, Proc. Nat. Acad. Sci., 42 (1956), 409–412. Zbl0071.16404MR79266
- 2. D. Anick, Hopf algebras up to homotopy, J. Am. Math. Soc., 2 (1989), 417–453. Zbl0681.55006MR991015
- 3. M. Chas and D. Sullivan, String topology, Ann. Math. (to appear) GT/9911159. Zbl1185.55013
- 4. R. Cohen and J. Jones, A homotopy theoretic realization of string topology, Math. Ann., 324 (2002), 773–798. Zbl1025.55005MR1942249
- 5. R. Cohen, J. D. S. Jones and J. Yan, The loop homology algebra of spheres and projective spaces, in: Categorical Decomposition Techniques in Algebraic Topology, Prog. Math. 215, Birkhäuser Verlag, Basel-Boston-Berlin (2004), 77–92. Zbl1054.55006MR2039760
- 6. R. Cohen, Multiplicative properties of Atiyah duality, in preparation (2003). Zbl1072.55004MR2076004
- 7. P. Deligne, P. Griffiths, J. Morgan and D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math., 29 (1975), 245–274. Zbl0312.55011MR382702
- 8. Y. Félix, S. Halperin and J.-C. Thomas, Adams’s cobar construction, Trans. Am. Math. Soc., 329 (1992), 531–549. Zbl0765.55005
- 9. Y. Félix, S. Halperin and J.-C. Thomas, Differential graded algebras in topology, in: Handbook of Algebraic Topology, Chapter 16, Elsevier, North-Holland-Amsterdam (1995), 829–865. Zbl0868.55016MR1361901
- 10. Y. Félix, S. Halperin and J.-C. Thomas, Rational Homotopy Theory, Grad. Texts Math. 205, Springer-Verlag, New York (2000). Zbl0961.55002MR1802847
- 11. M. Gerstenhaber, The cohomology structure of an associative ring, Ann. Math., 78 (1963), 267–288. Zbl0131.27302MR161898
- 12. J. D. S. Jones, Cyclic homology and equivariant homology, Invent. Math., 87 (1987), 403–423. Zbl0644.55005MR870737
- 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

## Citations in EuDML Documents

top- Luc Menichi, Batalin-Vilkovisky algebra structures on Hochschild cohomology
- Yves Félix, Jean-Claude Thomas, Rational BV-algebra in string topology
- David Chataur, Jean-François Le Borgne, On the loop homology of complex projective spaces
- Katsuhiko Kuribayashi, The Hochschild cohomology ring of the singular cochain algebra of a space
- Grégory Ginot, Thomas Tradler, Mahmoud Zeinalian, A Chen model for mapping spaces and the surface product

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.