Virtually fibered conjecture

In the mathematical subfield of <a href="/facts/3-manifold/48yr0Tzx">3-manifolds</a>, the virtually fibered conjecture, formulated by <a href="/facts/United_States/P0sCdLe5">American</a> <a href="/facts/Mathematician/eaFSR8Fx">mathematician</a> <a href="/facts/William_Thurston/K1Qe8801">William Thurston</a>, states that every <a href="/facts/Closed_manifold/jMG6G6Bl">closed</a>, <a href="/facts/Irreducible_manifold/ECpXQPUM">irreducible</a>, <a href="/facts/Atoroidal/srUhgk57">atoroidal</a> 3-manifold with infinite <a href="/facts/Fundamental_group/VyYtLqSP">fundamental group</a> has a finite <a href="/facts/Covering_space/bDRGasl7">cover</a> which is a <a href="/facts/Surface_bundle_over_the_circle/rTkMt4Ml">surface bundle over the circle</a>.
A 3-manifold which has such a finite cover is said to virtually fiber. If M is a <a href="/facts/Seifert_fiber_space/SledNv2F">Seifert fiber space</a>, then M virtually fibers if and only if the rational <a href="/facts/Euler_number/acpKy6uS">Euler number</a> of the Seifert fibration or the (<a href="/facts/Orbifold/KR6dQNzH">orbifold</a>) Euler characteristic of the base space is zero.
The hypotheses of the conjecture are satisfied by <a href="/facts/Hyperbolic_3-manifold/U8AAYhsw">hyperbolic 3-manifolds</a>. In fact, given that the <a href="/facts/Geometrization_conjecture/FDpqIM7y">geometrization conjecture</a> 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 <a href="/facts/Virtually_Haken_conjecture/4qgD4tWN">virtually Haken conjecture</a>) stemmed from the fact that any of these conjectures, combined with Thurston's <a href="/facts/Hyperbolization_theorem/TJqQt7zX">hyperbolization theorem</a>, 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 "[t]his 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 <a href="/facts/ArXiv/H6EtgnBe">ArXiv</a> on 25 Aug 2009, <a href="/facts/Daniel_Wise_(mathematician)/syYzYmEh">Daniel Wise</a> 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 <a href="/facts/Institut_Henri_Poincar%25C3%25A9/4GxBfjJ9">Institut Henri Poincaré</a> in Paris, <a href="/facts/Ian_Agol/ozxjQ99k">Ian Agol</a> announced he could prove the <a href="/facts/Virtually_Haken_conjecture/4qgD4tWN">virtually Haken conjecture</a> for closed hyperbolic 3-manifolds 
. Taken together with Daniel Wise's results, this implies the virtually fibered conjecture for all closed hyperbolic 3-manifolds.

Virtually fibered conjecture open-in-new

Virtually fibered conjecture