End reductions, fundamental groups, and covering spaces of irreducible open 3–manifolds

Abstract

Suppose $M$ is a connected, open, orientable, irreducible 3–manifold which is not homeomorphic to $ℝ$. Given a compact 3–manifold $J$ in $M$ which satisfies certain conditions, Brin and Thickstun have associated to it an open neighborhood $V$ called an end reduction of $M$ at $J$. It has some useful properties which allow one to extend to $M$ various results known to hold for the more restrictive class of eventually end irreducible open 3–manifolds.

In this paper we explore the relationship of $V$ and $M$ with regard to their fundamental groups and their covering spaces. In particular we give conditions under which the inclusion induced homomorphism on fundamental groups is an isomorphism. We also show that if $M$ has universal covering space homeomorphic to $ℝ$, then so does $V$.

This work was motivated by a conjecture of Freedman (later disproved by Freedman and Gabai) on knots in $M$ which are covered by a standard set of lines in $ℝ$.