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

Robert Myers

Vol. 185 (1998), No. 2, 327–345

DOI: 10.2140/pjm.1998.185.327

Abstract

An irreducible open 3-manifold W is R²-irreducible if it contains no non-trivial planes, i.e. given any proper embedded plane Π in W some component of W − Π must have closure an embedded halfspace R² × [0,∞). In this paper it is shown that if M is a connected, P²-irreducible, open 3-manifold such that π₁(M) is finitely generated and the universal covering space M of M is R²-irreducible, then either M is homeomorphic to R³ or π₁(M) is a free product of infinite cyclic groups and fundamental groups of closed, connected surfaces other than S² or P². Given any finitely generated group G of this form, uncountably many P²-irreducible, open 3-manifolds M are constructed with π₁(M)≅G such that the universal covering space M is R²-irreducible and not homeomorphic to R³; the M are pairwise non-homeomorphic. Relations are established between these results and the conjecture that the universal covering space of any irredicible, orientable, closed 3-manifold with infinite fundamental group must be homeomorphic to R³.

Milestones

Received: 11 February 1997

Revised: 26 July 1997

Published: 1 October 1998