Virtually fibered conjecture

In the mathematical subfield of 3-manifolds, the virtually fibered conjecture of Thurston states: every closed, irreducible, atoroidal 3-manifold with infinite fundamental group has a finite cover which is a surface bundle over the circle.

A 3-manifold which has such a finite cover is said to virtually fiber. If "M" is a Seifert fiber space, then "M" virtually fibers if and only if the rational Euler number of the Seifert fibration or the (orbifold) Euler characteristic of the base space is zero.

The hypotheses of the conjecture are satisfied by hyperbolic 3-manifolds. In fact, assuming the geometrization conjecture, the only case needed to be proven for the virtually fibered conjecture is that of hyperbolic 3-manifolds.

Proving the virtually fibered conjecture (or any of its weaker cousins such as the virtually Haken conjecture) would prove the geometrization conjecture and so much work has gone into proving such "virtual" conjectures.

This was not actually conjectured by Thurston. Rather, he posed it as a question and has stated that it was intended as a challenge (and not meant to indicate he believed it). There is no widespread consensus on the truth or falsehood of the conjecture. Only a few classes of examples have been proven to virtually fiber but not fiber themselves.

References

*W. P. Thurston, "Three dimensional manifolds, Kleinian groups and hyperbolic geometry", Bull. Amer. Math. Soc, 6 (1982) p. 357-381
*D. Gabai, "On 3-manifold finitely covered by surface bundles", Low Dimensional Topology and Kleinian Groups (ed: D.B.A. Epstein), London Mathematical Society Lecture Note Series vol 112 (1986), p. 145-155.

ee also

*Virtually Haken conjecture
*positive virtual Betti number conjecture
*Surface subgroup conjecture