On the fundamental group and homotopy type of open 3-manifolds

Abstract

It is shown that a finitely-generated subgroup of the fundamental group of a 3- manifold is the fundamental group of a compact 3-manifold if the subgroup can be suitably approximated by a finitely-presented group. This result is then applied to study certain subgroups of 3-manifold fundamental groups: those subgroups which are finitely-generated and have finite abelianizations. In many cases, these subgroups are shown to be isomorphic to fundamental groups of closed 3-manifolds. Corollaries concerning the embedding of spheres and cells in 3- space are given. A necessary and sufficient condition is given in order that an open 3-manifold with no 2-sided projective planes should be homotopy-equivalent to a compact 3-manifold.