Open three-dimensional manifolds with finitely generated fundamental groups
Robert Messer · Bulletin of the American Mathematical Society · 1976
Recent results of G. P. Scott [5] and T. W. Tucker [6] indicate that a 3-manifold with a finitely generated fundamental group is in various senses close to being compact. The results announced in this paper are further investigations into the relations between these two properties for open 3-manifolds. With a few additional complications these results also hold for noncompact 3-manifolds with boundary. THEOREM 1. Suppose M is an open, connected 3-manifold, irx(M) is finitely generated, M contains no infinite collection of disjoint fake 3-cells, and M contains no 2-sided projective planes. Then M is homotopy equivalent to the interior of a compact 3-manifold with a tame, closed, ^-dimensional subset deleted. The compact 3-manifold is homeomorphic to a submanifold ofM. Theorem 1 is an extension of Theorem 3.2 of [1]. In general, of course, the homotopy equivalence will not be proper: Whitehead's example [7] of a contractible, open 3-manifold shows that even in simple cases, the structure at infinity can be quite complicated.