#### Geometry & Topology, Vol. 9 (2005)
Paper no. 22, pages 971--990.

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

### Robert Myers

**Abstract**. 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.
**Keywords**.
3-manifold, end reduction, covering space

**AMS subject classification**.
Primary: 57M10.
Secondary: 57N10, 57M27.

**E-print:** `arXiv:math.GT/0407172`

**DOI:** 10.2140/gt.2005.9.971

Submitted to GT on 14 July 2004.
(Revised 18 May 2005.)
Paper accepted 18 May 2005.
Paper published 29 May 2005.

Notes on file formats
Robert Myers

Department of Mathematics, Oklahoma State University

Stillwater, OK 74078, USA

Email: myersr@math.okstate.edu

URL: http://www.math.okstate.edu/~myersr/

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