Insolubility of the Problem of Homeomorphy ∗
Insolubility of the Problem of Homeomorphy ∗
复制标题
同胚问题的不解性 *
DOI：
10.1007/s00454-004-1105-7
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发表时间：
2001
影响因子：
0.8
通讯作者：
Rolf Herken
中科院分区：
文献类型：
--
作者：
A. Markov;Champaign German;Rolf Herken
1.从一般的同胚问题出发，我们考虑了寻找判定两个给定多面体是否同胚的算法的问题。在这种情况下，多面体是通过它们的三角剖分组合给出的，我们必须准确地理解术语“算法”的含义，即它提供了什么，例如，作为“分类算法”。除了一般的同胚问题外，当然还有不同的子问题，它们本身涉及多面体或那些由此产生的类。例如，可以建立次数不高于n的多面体的同胚问题，n是固定的自然数。如果一个人能够清楚地确定什么是“流形”，就可以用完全相同的方式建立n-流形的同胚问题。可以对要匹配的多面体进行的另一个自然限制是固定其中一个。在这种情况下，给定多面体A的同胚问题包括找到一个算法，该算法对任意多面体确定它是否与多面体A同胚。其中一个问题已经解决了很长时间，即2-流形的同胚问题或给定2-流形的同胚问题。然而，我们发现了以下结果：
1. We consider, from the general problem of homeomorphy, the problem of finding an algorithm that determines whether two given polyhedra are homeomorphic. In this case, polyhedra are combinatorially given through their triangulation and we must understand the term “algorithm” in the precise sense what the it offers i.e., e.g., as a “classifying algorithm”. In addition to the general problem of the Homeomorphy, there are, of course, different subproblems which themselves refer to polyhedra or the those resulting classes. One may, for example, set up the problem of homeomorphy for polyhedra of degree no higher than n, a fixed natural number. One may, in exactly the same way, set up the problem of homeomorphy for the n-manifolds, if one could clearly decide what a “manifold” is. Another natural restriction that can be made to the polyhedra to be matched is fixing one of them. In this case, the problem of the homeomorphy of a given polyhedron A consists of finding an algorithm which, for any polyhedron, determines whether it is homeomorphic to the polyhedron A. One of these problems has been solved for a long time, i.e. the problem of homeomorphy for 2-manifolds or the problem of the homeomorphy of a given 2-manifold. However, we have found the following results: