R 2 -irreducible Universal Covering Spaces of P 2 -irreducible Open 3-manifolds

An irreducible open 3-manifold W is R 2-irreducible if it contains no non-trivial planes, i.e. given any proper embedded plane in W some component of W ? must have closure an embedded halfspace R 2 0; 1). In this paper it is shown that if M is a connected, P 2-irreducible, open 3-manifold such that 1 (M) is nitely generated and the universal covering space f M of M is R 2-irreducible, then either f M is homeomor-phic to R 3 or 1 (M) is a free product of innnite cyclic groups and fundamental groups of closed, connected surfaces other than S 2 or P 2 : Given any nitely 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 innnite fundamental group must be homeo-morphic to R 3 .