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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.
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