Algebraic & Geometric Topology

Covering a nontaming knot by the unlink

Michael H Freedman and David Gabai

Full-text: Open access

Abstract

There exists an open 3–manifold M and a simple closed curve γM such that π1(Mγ) is infinitely generated, π1(M) is finitely generated and the preimage of γ in the universal covering of M is equivalent to the standard locally finite set of vertical lines in 3, that is, the trivial link of infinitely many components in 3.

Article information

Source
Algebr. Geom. Topol., Volume 7, Number 3 (2007), 1561-1578.

Dates
Received: 28 May 2005
Accepted: 27 December 2005
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796754

Digital Object Identifier
doi:10.2140/agt.2007.7.1561

Mathematical Reviews number (MathSciNet)
MR2366171

Zentralblatt MATH identifier
1158.57024

Subjects
Primary: 57N10: Topology of general 3-manifolds [See also 57Mxx]
Secondary: 57M10: Covering spaces 57N45: Flatness and tameness

Keywords
Marden conjecture nontaming knot tameness

Citation

Freedman, Michael H; Gabai, David. Covering a nontaming knot by the unlink. Algebr. Geom. Topol. 7 (2007), no. 3, 1561--1578. doi:10.2140/agt.2007.7.1561. https://projecteuclid.org/euclid.agt/1513796754


Export citation

References

  • I Agol, Tameness of hyperbolic 3–manifolds
  • D Calegari, D Gabai, Shrinkwrapping and the taming of hyperbolic 3–manifolds, J. Amer. Math. Soc. 19 (2006) 385–446
  • D Gabai, Foliations and the topology of $3$-manifolds, J. Differential Geom. 18 (1983) 445–503
  • A Marden, The geometry of finitely generated kleinian groups, Ann. of Math. $(2)$ 99 (1974) 383–462
  • R Myers, End reductions, fundamental groups, and covering spaces of irreducible open 3–manifolds, Geom. Topol. 9 (2005) 971–990
  • R Roussarie, Plongements dans les variétés feuilletées et classification de feuilletages sans holonomie, Inst. Hautes Études Sci. Publ. Math. (1974) 101–141
  • J Stallings, On fibering certain $3$-manifolds, from: “Topology of 3–manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961)”, (M Fort, editor), Prentice-Hall, Englewood Cliffs, N.J. (1962) 95–100
  • T W Tucker, Non-compact 3–manifolds and the missing-boundary problem, Topology 13 (1974) 267–273
  • F Waldhausen, On irreducible 3–manifolds which are sufficiently large, Ann. of Math. $(2)$ 87 (1968) 56–88