\(R^2\)-irreducible universal covering spaces of \(P^2\)-irreducible open 3-manifolds (Q1366646)

An irreducible open 3-manifold \(W\) is \(\mathbb{R}^2\)-irreducible if it contains no non-trivial planes, i.e. given any proper embedded plane \(\Pi\) in \(W\) some component of \(W-\Pi\) must have closure an embedded halfspace \(\mathbb{R}^2\times [0,\infty)\). In this paper it is shown that if \(M\) is a connected, \(\mathbb{P}^2\)-irreducible, open 3-manifold such that \(\pi_1(M)\) is finitely generated and the universal covering space \(\widetilde{M}\) of \(M\) is \(\mathbb{R}^2\)-irreducible, then either \(\widetilde{M}\) is homeomorphic to \(\mathbb{R}^3\) or \(\pi_1(M)\) is a free product of infinite cyclic groups and fundamental groups of closed, connected surfaces other than \(S^2\) or \(\mathbb{P}^2\). Given any finitely generated group \(G\) of this form, uncountably many \(\mathbb{P}^2\)-irreducible, open 3-manifolds \(M\) are constructed with \(\pi_1(M)\cong G\) such that the universal covering space \(\widetilde{M}\) is \(\mathbb{R}^2\)-irreducible and not homeomorphic to \(\mathbb{R}^3\); the \(\widetilde{M}\) 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 \(\mathbb{R}^3\).