Counting topological manifolds
Counting topological manifolds
复制标题
计算拓扑流形
DOI：
10.1016/0040-9383(70)90036-4
复制
发表时间：
1970
期刊：
影响因子：
--
通讯作者：
J. Kister
中科院分区：
文献类型：
--
作者：
J. Cheeger;J. Kister
我们考虑所有紧拓扑流形的类%，边界允许。已知Y.[l]和[I]中只有可数个同伦类型。分段线性流形的子类 f8pr 仅具有可数个拓扑不同的元素，因为每个元素都可以被视为有限单纯复形，并且一个简单的论证表明只有可数个这些元素，直至同构。然而，最近，Kirby 和 Siebenmann [3] 发现了一些不承认 PL 结构的拓扑流形的例子，因此用 % 来计算同胚类型的途径相当没有希望（是否可以在没有 PL 结构的情况下对流形进行三角剖分仍然是开放的）。我们采用直接方法，并借助 Edwards 和 Kirby [4] 的非常有用的结果检查重叠坐标邻域，以表明 % 只具有可数的同胚元素。我们的论点与[3]中微分设置中的论点类似。
WE CONSIDER the class% of all compact topological manifolds, boundaries permitted. It is known that there are only a countable number of homotopy types in Y.[l] and [I]. The subclass f8pr, of piecewise linear manifolds has only a countable number of topologically distinct elements, since each could be regarded as a finite simplicial complex and a simple argument shows there are only a countable number of those, up to isomorphism. Recently, however, Kirby and Siebenmann [3] have discovered some examples of topological manifolds admitting no PL structure, so that route for counting homeomorphism types in% is rather unpromising (whether manifolds can be triangulated without a PL structure is still open).We take a direct approach and with the aid of a very useful result of Edwards and Kirby [4] examine overlapping coordinate neighborhoods to show that% has only countably many elements up to homeomorphism. Our argument is similar to one in a differential setting in [3].