The Insolvability of the Problem of Homeomorphy

摘要

The author considers the problem of homeomorphy to be the problem of searching for an algorithm which, for any two given polyhedrons, discerns whether or not they are homeomorphic. The polyhedrons are set combinatorially by their triangulation, making it possible to understand the term algorithm in its exact sense (as a normalized algorithm). The author has developed two theorems in the discussion of his work (1) For each natural number n greater than three, some n-dimensional manifold M to the nth power can be shown so that the problem of finding manifolds homeomorphic to manifold M to the nth power is insolvable; and (2) For any natural number n greater than 3, the problem of the homotopic equivalence of manifolds to manifold M to the nth power is insolvable.