CERTAIN THEOREMS ABOUT THREE-DIMENSIONAL MANIFOLDS (I)

Abstract

This chapter discusses certain problems about three-dimensional manifolds. In terms of bounding relations, one cannot analyze the topology of three-dimensional spaces closely enough to isolate the 3-sphere, while more general spaces cannot be isolated in terms of their groups. The chapter explains how any unbounded manifold whose group is null has a rectilinear model in the space of inversion. If the manifold is closed, the model covers the whole space. Otherwise it is an infinite complex covering the residual space of a certain zero-dimensional point set. The methods also provide a finite algorithm for determining whether one of two given circuits is deformable into the other. Thus, one can determine whether or no two combinations of the generators represent the same element of the group provided one is given a representative circuit for each generator.