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.

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Robert Myers
Department of Mathematics, Oklahoma State University
Stillwater, OK 74078, USA

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