# The virtual Haken conjecture: Experiments and examples

@article{Dunfield2003TheVH, title={The virtual Haken conjecture: Experiments and examples}, author={Nathan M. Dunfield and William P. Thurston}, journal={Geometry \& Topology}, year={2003}, volume={7}, pages={399-441} }

A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture says that every irreducible 3-manifold with innite fundamental group has a nite 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 HodgsonWeeks census of 10,986 small-volume closed hyperbolic 3…

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## References

SHOWING 1-10 OF 93 REFERENCES

Virtual Haken 3-manifolds and Dehn " lling

- Mathematics
- 1999

A stronger conjecture is that any closed, connected, orientable, irreducible 3-manifold= with in"nite fundamental group has a virtually positive "rst Betti number, i.e.= has a "nite cover which has…

Virtually Haken Dehn-lling

- Mathematics
- 1997

We show that \most" Dehn-llings of a non-bered, atoroidal, Haken three-manifold with torus boundary are virtually Haken. 1 Results Suppose that X is a compact, oriented, three-manifold with boundary…

Heegaard splittings of branched coverings of

- Mathematics
- 1975

This paper concerns itself with the relationship between two seemingly different methods for representing a closed, orientable 3-manifold: on the one hand as a Heegaard splitting, and on the other…

Alexander and Thurston norms of fibered 3-manifolds

- Mathematics
- 1999

For a 3-manifold M, McMullen derived from the Alexander polynomial of M a norm on H^1(M, R) called the Alexander norm. He showed that the Thurston norm on H^1(M, R), which measures the complexity of…

Bounded, separating, incompressible surfaces in knot manifolds

- Mathematics
- 1984

This generalizes and strengthens the main theorem of [13]. Note that the hypothesis of Theorem 1 is satisfied whenever M is a knot manifold, i.e. the complement of an open tubular neighborhood of a…

Kneser-Haken finiteness for bounded 3-manifolds, locally free groups, and cyclic covers

- Mathematics
- 1998

Free quotients and the first betti number of some hyperbolic manifolds

- Mathematics
- 1996

In this note we present a very simple method of proving that some hyperbolic manifoldsM have finite sheeted covers with positive first Betti number. The method applies to the standard arithmetic…

Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature

- Mathematics
- 1982

Let N be a three dimensional Riemannian manifold. Let E be a closed embedded surface in N. Then it is a question of basic interest to see whether one can deform : in its isotopy class to some…

Problems in low-dimensional topology

- Mathematics
- 1997

Four-dimensional topology is in an unsettled state: a great deal is known, but the largest-scale patterns and basic unifying themes are not yet clear. Kirby has recently completed a massive review of…