Cochain multiplications
Cochain multiplications
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上链乘法
DOI：
10.1353/ajm.2002.0017
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发表时间：
2002
影响因子：
1.7
通讯作者：
Michael A. Mandell
中科院分区：
文献类型：
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作者：
Michael A. Mandell
我们描述了Eilenberg-Steenrod公理的一个改进，它提供了从空间到微分分次代数或E∞微分分次代数的函子与奇异上链函子自然拟同构的充分必要条件.导论. 20世纪40年代Eilenberg-Steenrod公理的引入彻底改变了对上同调的理解。公理化框架提供了一个强有力的定理来识别普通的上同调，并通过隔离通常用于其基本计算的关键元素来重组上同调理论的研究，最大限度地提高灵活性。人们很快就发现，上同调的其他结构，如乘法和Steenrod运算，也承认类似的基本公理化。空间上的奇异上链函子，或者更一般地说，单纯集上的规范化上链函子，给出了一个特殊的普通上同调模型，其中所有已知的附加结构都是可见的。例如，乘法和Steenrod运算来自“E∞代数”结构[12，6]。我们现在知道，对于有限型幂零空间（或单纯集），空间的所有p-adic同伦信息都编码在这个E∞代数结构的拟同构类型中[10]。事实上，我们可以从其上链的E∞代数（直到拟同构）恢复任何连通空间的p-亲有限完备化（直到弱等价）[10，App B]。这几乎是理论上的最大同伦信息量，可以保留任何模型的拟同构类普通Z/pZ上同调。虽然对携带如此多同伦信息的普通上同调理论进行改进是有用的，但为了使计算可行，最好在模型中具有更大的灵活性。虽然[10]的论点可能会被修改和扩展以适用于任何合适的模型，但在个案基础上这样做是乏味的。一个更好的选择是在E∞代数范畴中有公理来识别直到拟同构的奇异上链函子。在本文中，我们提供这样的公理。它们是对Eilenberg-Steenrod公理的上链级细化。我们将它们编码在上链理论的定义中。2000年12月11日收到手稿; 2001年9月24日修订。研究部分由NSF博士后研究奖学金DMS-9804421支持。American Journal of Mathematics 124（2002），547-566.
We describe a refinement of the Eilenberg-Steenrod axioms that provides a necessary and sufficient condition for functors from spaces to differential graded algebras or E∞ differential graded algebras to be naturally quasi-isomorphic to the singular cochain functor. Introduction. The introduction of the Eilenberg-Steenrod axioms in the 1940’s revolutionized the understanding of cohomology. The axiomatic framework provided a powerful theorem for identifying ordinary cohomology and reorganized the study of cohomology theory by isolating key elements typically used for its basic calculations, maximizing flexibility. It was quickly seen that additional structures on cohomology such as multiplication and Steenrod operations also admit similarly elementary axiomatizations. The singular cochain functor on spaces or, more generally, the normalized cochain functor on simplicial sets gives a particular model for ordinary cohomology where all known additional structure is visible. For example, the multiplication and Steenrod operations come from an “E∞ algebra” structure [12, 6]. We now understand that for a finite type nilpotent space (or simplicial set), all p-adic homotopy information about the space is encoded in the quasi-isomorphism type of this E∞ algebra structure [10]. In fact, we can recover the p-pro-finite completion (up to weak equivalence) of any connected space from the E∞ algebra of its cochains (up to quasi-isomorphism) [10, App B]. This is nearly the theoretical maximum amount of homotopy information that can be preserved by the quasi-isomorphism class of any model for ordinary Z/pZ cohomology. While it is useful to have a refinement of ordinary cohomology theory carrying so much homotopy information, for calculations to be feasible, it would be preferable to have more flexibility in the model. While it is probable that the argument of [10] could be modified and extended to apply to any suitable model, this would be tedious to do on a case by case basis. A better alternative is to have axioms to identify the singular cochain functor up to quasi-isomorphism in the category of E∞ algebras. In this paper, we provide such axioms. They turn out to be a cochain-level refinement of the Eilenberg-Steenrod axioms. We encode them in the definition of a cochain theory. Manuscript received December 11, 2000; revised September 24, 2001. Research supported in part by NSF Postdoctoral Research Fellowship DMS-9804421. American Journal of Mathematics 124 (2002), 547–566.