R2-irreducible universal covering spaces of P2-irreducible open 3-manifolds

Abstract

An irreducible open 3-manifold W is R 2 -irreducible if it contains no non-trivial planes, i.e. given any proper embedded planein W some component of W −� must have closure an embedded halfspace R 2 ×(0, ∞). In this paper it is shown that if M is a connected, P 2 -irreducible, open 3-manifold such that �1(M) is finitely generated and the universal covering space f M of M is R 2 -irreducible, then either f M is homeomorphic to R 3 or �1(M) is a free product of infinite cyclic groups and fundamental groups of closed, connected surfaces other than S 2 or P 2 . Given any finitely generated group G of this form, uncountably many P 2 -irreducible, open 3-manifolds M are constructed with �1(M) ∼ G such that the universal covering space f M is R 2 -irreducible and not homeomorphic to R 3 ; the f M are pairwise non- homeomorphic. Relations are established between these results and the conjecture that the universal covering space of any irreducible, orientable, closed 3-manifold with infinite fundamental group must be homeomorphic to R 3 .