Virtually fibered conjecture
Encyclopedia
In the mathematical subfield of 3-manifold
s, the virtually fibered conjecture, formulated by American
mathematician
William Thurston
, states that 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-manifold
s. In fact, given that the geometrization conjecture
is now settled, the only case needed to be proven for the virtually fibered conjecture is that of hyperbolic 3-manifolds.
The original interest in the virtually fibered conjecture (as well as its weaker cousins, such as the virtually Haken conjecture
) stemmed from the fact that any of these conjectures, combined with Thurston's hyperbolization theorem
, would imply the geometrization conjecture. However, in practice all known attacks on the "virtual" conjecture take geometrization as a hypothesis, and rely on the geometric and group-theoretic properties of hyperbolic 3-manifolds.
The virtually fibered conjecture 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). For many years, there has been no widespread consensus on the truth or falsehood of the conjecture. However, starting in 2008-2009, breakthrough results by Agol and Wise in the direction of the conjecture have convinced many experts that the conjecture should be true.
3-manifold
In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...
s, the virtually fibered conjecture, formulated by American
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
William Thurston
William Thurston
William Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...
, states that every closed
Closed manifold
In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....
, irreducible, atoroidal
Atoroidal
In mathematics, there are three definitions for atoroidal as applied to 3-manifolds:*A 3-manifold is atoroidal if it does not contain an embedded, non-boundary parallel, incompressible torus....
3-manifold with infinite fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
has a finite cover which is a surface bundle over the circle
Surface bundle over the circle
In mathematics, a surface bundle over the circle is a fiber bundle with base space a circle, and with fiber space a surface. Therefore the total space has dimension 2 + 1 = 3. In general, fiber bundles over the circle are a special case of mapping tori....
.
A 3-manifold which has such a finite cover is said to virtually fiber. If M is a Seifert fiber space
Seifert fiber space
A Seifert fiber space is a 3-manifold together with a "nice" decomposition as a disjoint union of circles. In other words it is a S^1-bundle over a 2-dimensional orbifold...
, 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-manifold
Hyperbolic 3-manifold
A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant sectional curvature -1. In other words, it is the quotient of three-dimensional hyperbolic space by a subgroup of hyperbolic isometries acting freely and properly discontinuously...
s. In fact, given that the geometrization conjecture
Geometrization conjecture
Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed canonically into submanifolds that have geometric structures. The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces...
is now settled, the only case needed to be proven for the virtually fibered conjecture is that of hyperbolic 3-manifolds.
The original interest in the virtually fibered conjecture (as well as its weaker cousins, such as the virtually Haken conjecture
Virtually Haken conjecture
In topology, an area of mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is virtually Haken...
) stemmed from the fact that any of these conjectures, combined with Thurston's hyperbolization theorem
Hyperbolization theorem
In geometry, Thurston's geometrization theorem or hyperbolization theorem implies that closed atoroidal Haken manifolds are hyperbolic, and in particular satisfy the Thurston conjecture.-Statement:...
, would imply the geometrization conjecture. However, in practice all known attacks on the "virtual" conjecture take geometrization as a hypothesis, and rely on the geometric and group-theoretic properties of hyperbolic 3-manifolds.
The virtually fibered conjecture 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). For many years, there has been no widespread consensus on the truth or falsehood of the conjecture. However, starting in 2008-2009, breakthrough results by Agol and Wise in the direction of the conjecture have convinced many experts that the conjecture should be true.
See also
- Virtually Haken conjectureVirtually Haken conjectureIn topology, an area of mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is virtually Haken...
- positive virtual Betti number conjecture
- Surface subgroup conjectureSurface subgroup conjectureIn mathematics, the surface subgroup conjecture of Friedhelm Waldhausen states that the fundamental group of every closed, irreducible 3-manifold with infinite fundamental group has a surface subgroup. By "surface subgroup" we mean the fundamental group of a closed surface not the 2-sphere...