The Hochschild cohomology of a closed manifold
The Hochschild cohomology of a closed manifold
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闭流形的 Hochschild 上同调
DOI：
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
Micheline Vigué
中科院分区：
文献类型：
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作者：
Y. Félix;Jean;Micheline Vigué
Let M be a closed orientable manifold of dimension d and $mathcal{C}^*(M)$ be the usual cochain algebra on M with coefficients in a field k. The Hochschild cohomology of M, $H!H^*(mathcal{C}^*(M);mathcal{C}^*(M))$ is a graded commutative and associative algebra. The augmentation map $varepsilon: mathcal{C}^*(M) o{ extbf{ extit{k}}}$ induces a morphism of algebras $I : H!H^*(mathcal{C}^*(M);mathcal{C}^*(M)) o{H!H^*(mathcal{C}^*(M);{ extbf{ extit{k}}})}$. 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 $H!H^*(mathcal{C}^*(M);{ extbf{ extit{k}}})$, 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.
Let M be a closed orientable manifold of dimension d and $mathcal{C}^*(M)$ be the usual cochain algebra on M with coefficients in a field k. The Hochschild cohomology of M, $H!H^*(mathcal{C}^*(M);mathcal{C}^*(M))$ is a graded commutative and associative algebra. The augmentation map $varepsilon: mathcal{C}^*(M) o{ extbf{ extit{k}}}$ induces a morphism of algebras $I : H!H^*(mathcal{C}^*(M);mathcal{C}^*(M)) o{H!H^*(mathcal{C}^*(M);{ extbf{ extit{k}}})}$. 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 $H!H^*(mathcal{C}^*(M);{ extbf{ extit{k}}})$, 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.