THE TRIANGULATION OF 3-MANIFOLDS
THE TRIANGULATION OF 3-MANIFOLDS
复制标题
三流形的三角剖分
DOI：
10.1093/qmath/27.1.63
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发表时间：
1976
影响因子：
0.7
通讯作者：
Andrew J. S. Hamilton
中科院分区：
文献类型：
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作者：
Andrew J. S. Hamilton
本文给出了3-流形可三角化的另一个证明，这个证明首先由Moise（9）在1951年解决，然后由Bing（2）在1957年解决。它遵循Kirby和Siebenmann的建议，即他们用于解决高维三角剖分问题的方法（5），（6）可以适用于三维情况。除非另有说明，流形被认为是仿紧的，有或没有边界。三角剖分问题与手柄拉直问题（定理1）密切相关。熟悉Kirby和Siebenmann的工作的人会记得，通过利用某种巧妙的环面展开思想，他们能够将维度> 5的把手拉直问题减少到PL范畴中的某个问题，即，确定PL同伦等价W -B x T的性质，它们是沿着边界的同胚（其中B = [- 1，1]*，T = Sx）。. . x S（n次），W是k + w维流形）。事实上，他们能够与沃尔、项和沙内森一起证明，对于k + n > 5和k + 3，这样一个映射的2重覆盖是PL同胚的相对边界同伦的，因此在^ 5维中的非3-柄是可直的;更令人惊讶的是，西本曼发现在每个> 5维中都存在一个不可直的3-柄。对于k + n = 3，证明了把手拉直问题的类似简化是可能的，并且Waldhausen已经解决了适当的问题（引理3），表明PL同伦等价W -*· B x T是沿着边界的同胚，它本身是PL同胚的相对边界，但前提是W也是不可约的，即，W中的每一个PL 2-球都有一个PL 3-球。为了达到必要的不可约性，定理1的柄从一开始就假定位于一个图的内部，即R的一个副本，R的不可约性是由亚历山大（1）建立的。我非常感谢我的导师G博士。斯科特，他在黑暗威胁的地方传播光明。
Introduction THIS paper gives another proof of the triangulability of 3-manifolds, which was first solved by Moise (9) in 1951 and then by Bing (2) in 1957. I t follows up a suggestion by Kirby and Siebenmann that the methods they used to solve the problem of triangulation in high dimensions (5), (6) could be adapted to the three dimensional case. Unless otherwise stated, manifolds are taken to be paracompact with or without boundary. The triangulation problem is closely related to the handle straightening problem (Theorem 1). Those familiar with Kirby and Siebenmann's work will recall that by exploiting a certain ingenious torus unfurling idea they were able to reduce the handle straightening problem in dimensions > 5 to a certain problem in the PL category, namely, deciding the nature of PL homotopy equivalences W —»B x T which are homeomorphisms along the boundary (where B = [— 1, 1]*, T = Sx . . . x S (n times), and W is a k + w-dimensional manifold). In fact they were able, with Wall and Hsiang and Shaneson, to show that for k + n > 5 and k + 3 the 2-fold cover of such a map is homotopic relative boundary to a PL homeomorphism, and hence that non 3-handles in dimensions ^ 5 were straightenable; more surprisingly Siebenmann found that a non-straightenable 3-handle exists in each dimension > 5. For k + n = 3 it turns out that a similar reduction of the handle straightening problem is possible, and Waldhausen has solved the appropriate problem (Lemma 3) showing that a PL homotopy equivalence W -*• B x T which is a homeomorphism along the boundary is itself homotopic relative boundary to a PL homeomorphism, but only provided that W is also irreducible, that is, every PL 2-sphere in W bounds a PL 3-ball. To achieve the requisite irreducibility the handles of Theorem 1 are supposed from the first to lie inside a chart, i.e. a copy of R, the irreducibility of which was established by Alexander (1). I would like to thank very much my supervisor Dr. G. P. Scott, who spread light where darkness threatened.