Insolubility of the Problem of Homeomorphy∗

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: