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摘要翻译：
高等同伦如今在数学以及理论物理的某些分支中发挥着突出的作用。我们回顾过去和现在事态发展之间的一些联系。至少早在20世纪40年代，高等同伦就被孤立在代数拓扑中。由于Alexander-Whitney乘法的同义词不能交换，斯汀罗德发展了某些运算，以一种连贯的方式来度量这种失败。Dold和Lashof将Milnor的分类空间构造扩展到结合H-空间，对这种扩展的仔细研究使Stasheff发现了a-空间和Ainfty-空间，它们是以一种连贯的方式控制结合性失败的概念，以便分类空间构造仍然可以被推进。我们都知道，高等同伦的代数版本使Kontsevich最终证明了形式猜想。同调微扰理论(HPT)在50年代早期由Eilenberg和Mac Lane首先提出，以一种简单的形式出现，现在已成为处理高等同伦代数化身的标准工具。一个基本的观察是，较高的同伦结构相对于同伦比严格的结构表现得好得多，而HPT使人们能够在各种具体情况下利用这一观察，特别是导致各种不变量的有效计算，而这些不变量在其他情况下是难以解决的。高等同伦比比皆是，但它们很少被明确认识，其意义也很难被理解；有时，它们的出现乍一看甚至会令人惊讶，例如在Kodaira-Spencer关于复杂流形变形的方法或叶状理论中。
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英文标题：
《Origins and breadth of the theory of higher homotopies》
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作者：
Johannes Huebschmann (Universite de Lille 1)
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最新提交年份：
2007
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分类信息：
一级分类：Mathematics 数学
二级分类：Algebraic Topology 代数拓扑
分类描述：Homotopy theory, homological algebra, algebraic treatments of manifolds
同伦理论，同调代数，流形的代数处理
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一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
Higher homotopies are nowadays playing a prominent role in mathematics as well as in certain branches of theoretical physics. We recall some of the connections between the past and the present developments. Higher homotopies were isolated within algebraic topology at least as far back as the 1940's. Prompted by the failure of the Alexander-Whitney multiplication of cocycles to be commutative, Steenrod developed certain operations which measure this failure in a coherent manner. Dold and Lashof extended Milnor's classifying space construction to associative H-spaces, and a careful examination of this extension led Stasheff to the discovery of An-spaces and Ainfty-spaces as notions which control the failure of associativity in a coherent way so that the classifying space construction can still be pushed through. Algebraic versions of higher homotopies have, as we all know, led Kontsevich eventually to the proof of the formality conjecture. Homological perturbation theory (HPT), in a simple form first isolated by Eilenberg and Mac Lane in the early 1950's, has nowadays become a standard tool to handle algebraic incarnations of higher homotopies. A basic observation is that higher homotopy structures behave much better relative to homotopy than strict structures, and HPT enables one to exploit this observation in various concrete situations which, in particular, leads to the effective calculation of various invariants which are otherwise intractable. Higher homotopies abound but they are rarely recognized explicitly and their significance is hardly understood; at times, their appearance might at first glance even come as a surprise, for example in the Kodaira-Spencer approach to deformations of complex manifolds or in the theory of foliations.
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PDF链接：
https://arxiv.org/pdf/0710.2645
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