In the mathematical subfield of

3-manifold In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (geometry), plane (a tangent ...

s, the virtually fibered conjecture, formulated by American

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

William Thurston William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurst ...

, states that every closed, irreducible,

atoroidal In mathematics, an atoroidal 3-manifold is one that does not contain an essential torus. There are two major variations in this terminology: an essential torus may be defined geometrically, as an embedded, non- boundary parallel, incompressible t ...

3-manifold with infinite

fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...

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 Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...

of the Seifert fibration or the (

orbifold In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space that is locally a finite group quotient of a Euclidean space. D ...

) Euler characteristic of the base space is zero. The hypotheses of the conjecture are satisfied by hyperbolic 3-manifolds. In fact, given that the

geometrization conjecture In mathematics, Thurston's geometrization conjecture (now a theorem) states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theor ...

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 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''. That is, it has a finite cover (a covering s ...

) 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, writing only that " is dubious-sounding question seems to have a definite chance for a positive answer". The conjecture was finally settled in the affirmative in a series of papers from 2009 to 2012. In a posting on the

ArXiv arXiv (pronounced as "archive"—the X represents the Chi (letter), Greek letter chi ⟨χ⟩) is an open-access repository of electronic preprints and postprints (known as e-prints) approved for posting after moderation, but not Scholarly pee ...

on 25 Aug 2009, Daniel Wise implicitly implied (by referring to a then-unpublished longer manuscript) that he had proven the conjecture for the case where the 3-manifold is closed, hyperbolic, and Haken. This was followed by a survey article in Electronic Research Announcements in Mathematical Sciences. Several other articles have followed, including the aforementioned longer manuscript by Wise. In March 2012, during a conference at

Institut Henri Poincaré The Henri Poincaré Institute (or IHP for ''Institut Henri Poincaré'') is a mathematics research institute part of Sorbonne University, in association with the Centre national de la recherche scientifique (CNRS). It is located in the 5th arrondi ...

in Paris, Ian Agol announced he could prove the

for closed hyperbolic 3-manifolds . Taken together with Daniel Wise's results, this implies the virtually fibered conjecture for all closed hyperbolic 3-manifolds.

Notes

References

* *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. *

External links

* {{Cite web, last=Klarreich, first=Erica, author-link=Erica Klarreich, date=2012-10-02, title=Getting Into Shapes: From Hyperbolic Geometry to Cube Complexes and Back, url=https://www.quantamagazine.org/from-hyperbolic-geometry-to-cube-complexes-and-back-20121002/, website=

Quanta Magazine ''Quanta Magazine'' is an editorially independent online publication of the Simons Foundation covering developments in physics, mathematics, biology and computer science. History ''Quanta Magazine'' was initially launched as ''Simons Science ...

, language=en 3-manifolds Conjectures

See also

Notes

References

External links