Thom's theory of differential forms on simplical sets
Thom's theory of differential forms on simplical sets
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汤姆的简单集合微分形式理论
DOI：
10.1016/0040-9383(75)90008-7
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发表时间：
1975
期刊：
影响因子：
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通讯作者：
R. G. Swan
中科院分区：
文献类型：
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作者：
R. G. Swan
在一个过程中给予芝加哥大学在1957年夏天，索恩表明，有一个交换cochain复杂的真实的数字与每一个拓扑空间，产生正确的奇异上同调环。这种结构的细节从未公布，尽管在[6]中给出了一些应用。在[3]中，Quillen证明了在有理数上甚至存在这样一个复形，从而得到了正确的有理上同调环。Quillen的结构相当复杂。由于索恩的构造是如此简单和优雅，自然会产生这样的问题：它是否可以被修改以处理有理数。我将在这里证明情况确实如此。Sullivan [4]已经宣布了一个类似的结构，有许多有吸引力的应用，并指出基本思想包含在Whitney的书中[7]。我要感谢沙利文和索恩提供的关于建筑历史的信息，特别是前者提出的参考文献[I，2和51]。索恩的基本思想是将单纯复形上的微分形式定义为各种单纯形上的C”形式的集合，这些形式适当地组合在一起。在本文中，我将给出这个理论的一个更函子的版本，然而，所得到的上链与Thorn构造所给出的上链相同。设S是其对象是集合是集合（0，1，...，r}上的一个单变函子，因此，根据定义，一个单纯集是C上的一个逆变函子。设a是定义在域1上的代数簇的范畴（我们实际上只需要仿射（空间和仿射映射）的子范畴）。定义一个函子：H：9+ a如下。令H（{0，1，.，r}）= H，be
IN A COURSE given at the University of Chicago in the summer of 1957, Thorn showed that there is a commutative cochain complex over the real numbers associated with every topological space which yields the correct singular cohomology ring. The details of this construction were never published although some applications were given in [6]. In [3], Quillen showed that there is even such a complex over the rational numbers, yielding the correct rational cohomology ring. Quillen’s construction is rather complicated. Since Thorn’s construction is so simple and elegant, the question naturally arises as to whether it can be modified to work over the rational numbers. I will show here that this is indeed the case. Sullivan [4] has announced a similar construction, with many attractive applications, and pointed out that the essential idea is contained in Whitney’s book [7]. I would like to thank Sullivan and Thorn for information on the history of the construction, and particularly the former for suggesting references [I, 2 and 51. Thorn’s basic idea was to define a differential form on a simplicial complex to be a collection of C” forms on the various simplexes which fit together properly. In the present exposition, I will give a more functorial version of this theory, the resulting cochains, however, being the same as those given by Thorn’s construction.Let S be the category whose objects are the sets are the sets (0, 1,..., r} with monotone maps, so that, by definition, a simplicial set is a contravariant functor on C. Let a be the category of algebraic varieties defined over a field 1 (we actually only need the subcategory of affine (spaces and affine maps). Define a functor: H: 9+ a as follows. Let H ({O, 1,..., r})= H, be