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2003 The virtual Haken conjecture: Experiments and examples
Nathan M Dunfield, William P Thurston
Geom. Topol. 7(1): 399-441 (2003). DOI: 10.2140/gt.2003.7.399


A 3–manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture says that every irreducible 3–manifold with infinite fundamental group has a finite cover which is Haken. Here, we discuss two interrelated topics concerning this conjecture.

First, we describe computer experiments which give strong evidence that the Virtual Haken Conjecture is true for hyperbolic 3–manifolds. We took the complete Hodgson-Weeks census of 10,986 small-volume closed hyperbolic 3–manifolds, and for each of them found finite covers which are Haken. There are interesting and unexplained patterns in the data which may lead to a better understanding of this problem.

Second, we discuss a method for transferring the virtual Haken property under Dehn filling. In particular, we show that if a 3–manifold with torus boundary has a Seifert fibered Dehn filling with hyperbolic base orbifold, then most of the Dehn filled manifolds are virtually Haken. We use this to show that every non-trivial Dehn surgery on the figure-8 knot is virtually Haken.


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Nathan M Dunfield. William P Thurston. "The virtual Haken conjecture: Experiments and examples." Geom. Topol. 7 (1) 399 - 441, 2003.


Received: 30 September 2002; Accepted: 13 April 2003; Published: 2003
First available in Project Euclid: 21 December 2017

zbMATH: 1037.57015
MathSciNet: MR1988291
Digital Object Identifier: 10.2140/gt.2003.7.399

Primary: 57M05, 57M10
Secondary: 20E26, 20F05, 57M27

Rights: Copyright © 2003 Mathematical Sciences Publishers


Vol.7 • No. 1 • 2003
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