A primer on A-infinity-algebras and their Hochschild homology
A primer on A-infinity-algebras and their Hochschild homology
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发表时间：
2016-01
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通讯作者：
Stephan Mescher
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作者：
Stephan Mescher
给出了$A\infty$-代数，$A\infty$-双模及其Hochschild同调群和上同调群的初等自包含构造.此外，我们还讨论了Hochschild上同调中的杯积和Hochschild链复形的长度滤子的谱序列。A_\infty$-结构在基环空间和流形几何的研究中自然出现，特别是在Lagrangian Floer理论和莫尔斯同调中。在几种几何情形中，Hochschild同调可以用来描述自由loop空间的同调群。本文的目的不是介绍新的材料，而是对来自几个来源的代数结果进行统一和连贯的讨论。它还包括所有提出的结果的详细证明。
We present an elementary and self-contained construction of $A_\infty$-algebras, $A_\infty$-bimodules and their Hochschild homology and cohomology groups. In addition, we discuss the cup product in Hochschild cohomology and the spectral sequence of the length filtration of a Hochschild chain complex. $A_\infty$-structures arise naturally in the study of based loop spaces and the geometry of manifolds, in particular in Lagrangian Floer theory and Morse homology. In several geometric situations, Hochschild homology may be used to describe homology groups of free loop spaces. The objective of this article is not to introduce new material, but to give a unified and coherent discussion of algebraic results from several sources. It further includes detailed proofs of all presented results.