An algorithm for the construction of 3-manifolds from 2-complexes

Abstract

In order to study 3-manifolds with some particular group as fundamental group, one likes to have examples. There is an absence in the literature of means of constructing such examples. In this paper, theorems are proved which give necessary and sufficient conditions for the canonical 2-complex which corresponds to a group presentation to be a spine (2-dimensional skeleton in a cell decomposition with one 3-cell) of a connected closed† orientable 3-manifold. By enumerating all presentations of a group given by a particular presentation, two of the theorems provide an effective algorithm which permits one to attempt in a systematic way to construct 3-manifolds with a given group as fundamental group. Since every closed 3-manifold has a spine which corresponds to a group presentation, if the group is the fundamental group of an orientable 3-manifold, then the algorithm will eventually yield a 3-manifold, if not, then one is out of luck. There is no way out of this since Stallings has shown (1) that no algorithm exists which can, for every finitely presented group G , answer the question: Is G the fundamental group of a 3-manifold?