Goodbye Thurston

Thurston has left :( so I figure a geometerization of 3 manifolds post is in order.

This is all in his book Three Dimensional Geometry and Topology.

For surfaces, uniformization is prehistoric. We have the easy invariant of the genus to tell us the covering space and it gets a constant curvature metric.

For 3d manifolds, the situation is trickier. We again want to find model manifolds along with their Lie groups of isometries. We also want these actions to be transitive. None of the points should get any special treatment.

The connected component of the point stabilizers are going to be Lie subgroups of SO(3).

If they are the full SO(3), then all directions you stare into look the same so we have a symmetric space. It must be one of euclidean, spherical or hyperbolic depending on the curvature.

If the component of the stabilizer is SO(2), then look in the leftover direction at each point to construct a vector field. The flow lines of the vector field give a foliation of the manifold where every leaf is 1D. The leaves are either circles or lines.
The easiest example of such would be surface cross R. This gives two new geometeries and euclidean 3-space over again.
There are two more of these, the cover of SL2 and the Heisenberg group.

This leaves when the point stabilizer is 1D. The only possibility is Sol geometry.

Virtually Haken

Apparently every compact, orientible, irreducible 3 manifold with infinite fundamental group has a finite sheeted cover that is Haken. Will keep an eye out to see Agol's proof.

What is Haken?
It means it should have an orientable incompressible surface.

So what to do with this orientable incompressible surface? Thicken it up a little bit and then remove that from the 3-manifold. Now you have 3-manifold(s) that have a boundary that is that surface. But if that surface isn't a sphere, then the 3-manifold we are talking about has infinite first homology. This gives a properly embedded 2-sided nonseparating incompressible surface in each of the pieces. So the result after the cut is still Haken. Repeat until the only surface you can find is a sphere. But they are irreducible, so these spheres bound 3-balls. This gives a decomposition of the manifold into 3-balls.

3-balls are easy, and the gluing is doable so we have some control. A nice theorem is that homotopy equivalence is homotopic to an honest homeomorphism. This means we only need to be concerned with fundamental groups. It is also true that these fundamental groups have a solvable word problem. We can actually tell when a complicated curve given as a big product of nasty group generators is actually trivial. This is also true for virtually Haken.

What are some examples of Haken manifolds? There is everybody's favorite type of 3-manifold, link complements. There are also surface bundles over the circle.

Now that we have the definitions written, we play the waiting game for when the preprint goes up to see how this is proved.

http://en.wikipedia.org/wiki/Virtually_Haken_conjecture

http://en.wikipedia.org/wiki/Haken_manifold