Deformation theory and rational homotopy type
Deformation theory and rational homotopy type
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发表时间：
2012-11
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通讯作者：
M. Schlessinger;J. Stasheff
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作者：
M. Schlessinger;J. Stasheff
我们把有理同伦类型的分类看作是代数变形理论中的一个问题：任何具有给定上同调的空间都是具有该上同调的“形式”空间的扰动或变形。然后，分类空间是一个“模”空间——扰动的代数变化的某个商。我们对这个模空间的描述将它与同伦理论中相应的结构联系起来，特别是把具有固定纤维F的纤维空间的分类，用基B的映射的同伦类联系到一个由F对自身的同伦等价的单阵构造的分类空间。我们采用了后来由Deligne在对Goldman和Millson的回应中提出的哲学，即变形理论中的任何问题都是由微分梯度李代数“控制”的，直到dg李代数的同调等价（拟同构）为止是唯一的。在这里，我们将这一哲学进一步扩展到由sh- lie代数控制。
We regard the classification of rational homotopy types as a problem in algebraic deformation theory: any space with given cohomology is a perturbation, or deformation, of the "formal" space with that cohomology. The classifying space is then a "moduli" space --- a certain quotient of an algebraic variety of perturbations. The description we give of this moduli space links it with corresponding structures in homotopy theory, especially the classification of fibres spaces with fixed fibre F in terms of homotopy classes of maps of the base B into a classifying space constructed from the monoid of homotopy equivalences of F to itself. We adopt the philosophy, later promoted by Deligne in response to Goldman and Millson, that any problem in deformation theory is "controlled" by a differential graded Lie algebra, unique up to homology equivalence (quasi-isomorphism) of dg Lie algebras. Here we extend this philosophy further to control by sh-Lie-algebras.