- Irreducible Universal Covering Spaces of P 2 - Irreducible Open 3 - Manifolds

An irreducible open 3-manifold W is R-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 R × [0,∞). In this paper it is shown that if M is a connected, P-irreducible, open 3-manifold such that π1(M) is finitely generated and the universal covering space M̃ of M is R-irreducible, then either M̃ is homeomorphic to R or π1(M) is a free product of infinite cyclic groups and fundamental groups of closed, connected surfaces other than S or P. Given any finitely generated group G of this form, uncountably many P-irreducible, open 3-manifolds M are constructed with π1(M) ∼= G such that the universal covering space M̃ is R-irreducible and not homeomorphic to R; the M̃ are pairwise nonhomeomorphic. 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.