  Suppose M is a connected, open, orientable, irreducible 3-manifold which is
not homeomorphic to R^3. 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^3, 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^3.
