In
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, an area of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the virtually Haken conjecture states that every
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
,
orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". A space is o ...
,
irreducible three-dimensional 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 ...
is ''virtually Haken''. That is, it has a finite cover (a
covering space
In topology, a covering or covering projection is a continuous function, map between topological spaces that, intuitively, Local property, locally acts like a Projection (mathematics), projection of multiple copies of a space onto itself. In par ...
with a finite-to-one covering map) that is a
Haken manifold.
After the proof of 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 ...
by
Perelman, the conjecture was only open for
hyperbolic 3-manifolds.
The conjecture is usually attributed to
Friedhelm Waldhausen in a paper from 1968, although he did not formally state it. This problem is formally stated as Problem 3.2 in
Kirby's problem list.
A proof of the conjecture was announced on March 12, 2012 by
Ian Agol in a seminar lecture he gave at the
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 ...
. The proof appeared shortly thereafter in a preprint which was eventually published in
Documenta Mathematica. The proof was obtained via a strategy by previous work of
Daniel Wise and collaborators, relying on actions of the fundamental group on certain auxiliary spaces (CAT(0) cube complexes, also known as
median graphs)
It used as an essential ingredient the freshly-obtained solution to the
surface subgroup conjecture
In mathematics, the surface subgroup conjecture of Friedhelm Waldhausen states that the fundamental group of every closed manifold, closed, irreducible manifold, irreducible 3-manifold with infinite fundamental group has a surface subgroup. By "su ...
by
Jeremy Kahn and
Vladimir Markovic.
Other results which are directly used in Agol's proof include the Malnormal Special Quotient Theorem of Wise and a criterion of
Nicolas Bergeron and Wise for the cubulation of groups.
In 2018 related results were obtained by Piotr Przytycki and Daniel Wise proving that mixed 3-manifolds are also virtually special, that is they can be cubulated into a cube complex with a finite cover where all the hyperplanes are embedded which by the previous mentioned work can be made virtually Haken.
See also
*
Virtually fibered conjecture
*
Surface subgroup conjecture
In mathematics, the surface subgroup conjecture of Friedhelm Waldhausen states that the fundamental group of every closed manifold, closed, irreducible manifold, irreducible 3-manifold with infinite fundamental group has a surface subgroup. By "su ...
*
Ehrenpreis conjecture
Notes
References
*.
*.
External links
*
3-manifolds
Theorems in topology
Conjectures that have been proved
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