3-manifold

For the case of a 3-manifold given by a connected sum of prime 3-manifolds, it turns out there is a nice description of the second fundamental group as a

Another distinctive property is the amount of space covered by the 3-ball in hyperbolic 3-space: it increases exponentially with respect to the radius of the ball, rather than polynomially.

In 2003, lack of structure on the largest scales (above 60 degrees) in the cosmic microwave background as observed for one year by the WMAP spacecraft led to the suggestion, by Jean-Pierre Luminet of the Observatoire de Paris and colleagues, that the shape of the universe is a Poincaré sphere.

In mathematics, Seifert–Weber space (introduced by Herbert Seifert and Constantin Weber) is a closed hyperbolic 3-manifold.

It is a quotient space of the order-5 dodecahedral honeycomb, a regular tessellation of hyperbolic 3-space by dodecahedra with this dihedral angle.

The Gieseking manifold can be constructed by removing the vertices from a tetrahedron, then gluing the faces together in pairs using affine-linear maps.

In the hyperbolic structure of the Gieseking manifold, this ideal tetrahedron is the canonical polyhedral decomposition of David B.

Contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle and specified by a one-form, both of which satisfy a 'maximum non-degeneracy' condition called 'complete non-integrability'.

A Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface.

Assuming smoothness the existence of a Heegaard splitting also follows from the work of Smale about handle decompositions from Morse theory.

Equivalently, by a result of Dennis Sullivan, a codimension 1 foliation is taut if there exists a Riemannian metric that makes each leaf a minimal surface.

Taut foliations were brought to prominence by the work of William Thurston and David Gabai.

Kneser-Haken finiteness says that for each compact 3-manifold, there is a constant C such that any collection of disjoint incompressible embedded surfaces of cardinality greater than C must contain parallel elements.

If M admits an essential map of a torus, then M admits an essential embedding of either a torus or an annulus[8] The JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem: The acronym JSJ is for William Jaco, Peter Shalen, and Klaus Johannson.

[13] The Lickorish–Wallace theorem states that any closed, orientable, connected 3-manifold may be obtained by performing Dehn surgery on a framed link in the 3-sphere with

Friedhelm Waldhausen's theorems on topological rigidity say that certain 3-manifolds (such as those with an incompressible surface) are homeomorphic if there is an isomorphism of fundamental groups which respects the boundary.

The Smith conjecture (now proven) states that if f is a diffeomorphism of the 3-sphere of finite order, then the fixed point set of f cannot be a nontrivial knot.

The cyclic surgery theorem states that, for a compact, connected, orientable, irreducible three-manifold M whose boundary is a torus T, if M is not a Seifert-fibered space and r,s are slopes on T such that their Dehn fillings have cyclic fundamental group, then the distance between r and s (the minimal number of times that two simple closed curves in T representing r and s must intersect) is at most 1.

In fact, the theorem states that volume decreases under topological Dehn filling, assuming of course that the Dehn-filled manifold is hyperbolic.

Also, Gabai, Meyerhoff & Milley showed that the Weeks manifold has the smallest volume of any closed orientable hyperbolic 3-manifold.

One form of Thurston's geometrization theorem states: If M is a compact irreducible atoroidal Haken manifold whose boundary has zero Euler characteristic, then the interior of M has a complete hyperbolic structure of finite volume.

The Mostow rigidity theorem implies that if a manifold of dimension at least 3 has a hyperbolic structure of finite volume, then it is essentially unique.

The ending lamination theorem, originally conjectured by William Thurston and later proven by Jeffrey Brock, Richard Canary, and Yair Minsky, states that hyperbolic 3-manifolds with finitely generated fundamental groups are determined by their topology together with certain "end invariants", which are geodesic laminations on some surfaces in the boundary of the manifold.

After nearly a century of effort by mathematicians, Grigori Perelman presented a proof of the conjecture in three papers made available in 2002 and 2003 on arXiv.

Thurston's hyperbolization theorem implies that Haken manifolds satisfy the geometrization conjecture.

Grigori Perelman sketched a proof of the full geometrization conjecture in 2003 using Ricci flow with surgery.

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.

In a posting on the ArXiv on 25 Aug 2009,[14] Daniel Wise implicitly implied (by referring to a then unpublished longer manuscript) that he had proven the Virtually fibered conjecture for the case where the 3-manifold is closed, hyperbolic, and Haken.

[17] In March 2012, during a conference at Institut Henri Poincaré in Paris, Ian Agol announced he could prove the virtually Haken conjecture for closed hyperbolic 3-manifolds.

A proof of this case was announced in the Summer of 2009 by Jeremy Kahn and Vladimir Markovic and outlined in a talk August 4, 2009 at the FRG (Focused Research Group) Conference hosted by the University of Utah.