${\mathbb R}^2$-irreducible universal covering spaces of ${\mathbf P}^2$-irreducible open 3-manifolds
Robert Myers
An irreducible open 3-manifold $W$ is ${\mathbb R}^2$-irreducible if every proper plane in $W$ splits off a halfspace. In this paper it is shown that if such a $W$ is the universal cover of a connected, ${\mathbf P}^2$-irreducible open 3-manifold $M$ with finitely generated fundamental group, then either $W$ is homeomorphic to ${\mathbb R}^3$ or the group is a free product of infinite cyclic groups and infinite closed surface groups. Given any such finitely generated group uncountably many $M$ are constructed with that fundamental group such that their universal covers are ${\mathbb R}^2$-irreducible, are not homeomorphic to ${\mathbb R}^3$, and are pairwise non-homeomorphic. These results are related to the conjecture that closed, orientable, irreducible, aspherical 3-manifolds are covered by ${\mathbb R}^3$.