Hi people~ My weekends have been unfortunately filled up with grading undergrad assignments for the last couple of weeks >.< I'll try to catch up on blogging by finding some other time slot during the week.
As a grand-student of Thurston's I feel obligated to end my ignorance regarding Haken manifolds. I guess it's a good idea to start by writing my usual kids-friendly exposition here.
In the rest of the post, is a compact (so perhaps with boundary), orientable, irreducible (meaning each embedded 2-sphere bounds a ball) 3-manifold.
Definition: A properly embedded oriented surface is incompressible if is not the 2-sphere and any simple closed curve on which bounds an embedded disc in also bounds one in .
In other words, together with Dehn’s lemma this says the map induced by the inclusion map is injective.
Note that the surface could have boundary, for example:
Definition: is Haken if it contains an incompressible surface.
Okay, at this point you should be asking, what’s good about Haken manifolds? The beauty about it is that, roughly speaking, once you find one incompressible surface in the manifold, you can just keep finding them until the manifold is completely chopped up into balls by incompressible surfaces.
Theorem: (Haken) Any Haken 3-manifold contains a hierarchy where
1. is an incompressible surface in
2. where is an incompressible surface for the closure of some connected component of $latex
3. is a union of 3-balls
Sketch of proof: This is much simpler than it might appear to be. The point is (at least in my opinion), except for trivial cases as long as a manifold has boundary it must be Haken.
Lemma: If has a component that’s not then is Haken.
The proof of the lemma is merely that any such will have infinite hence by the sphere theorem it will contain an embedded surface with non-trivial homology, if such surface is compressible then we just cut along the boundary of the compressing disc and glue two copies of it. This does not change the homology. Hence at the end we will arrive at a non-trivial incompressible surface.
Now back to proving of the theorem, so we start by setting to be an incompressible surface given by being Haken.
Now since is irreducible, we cut along , i.e. take the closure of each component (may have either one or two components) of . Those will have a non-spherical boundary component, hence by lemma containing homologically non-trivial incompressible surface.
This process continuous as long as some pieces has non-spherical boundary components. But since is irreducible, any sphere bounds a 3-ball in , hence all components with sphere boundary are 3-balls. (In particular, the case where a component have multiple sphere boundary components cannot occur since the first boundary component bounds a 3-ball hence it can’t have any non-trivial incompressible surfaces on both sides.)
Now the only remaining piece is to show that this process terminates. We apply a standard ‘normal surface argument’ for this. Essentially if we fix a triangulation of ,
A normal surface in is one that intersects each 3-simplex in a disjoint union of following two shapes:
There can’t be infinitely many non-parallel disjoint normal surfaces in (in fact there can be no more than 6 times the number of 3-simplexes since each complementry component need to contain at least one non-I-bundle part from one 3-simplex).
However, if the above process do not terminate, we would obtain a sequence of non-parallel non-spherical boundary components:
They represent different homology classes hence can be represented by disjoint normal which results in a contradiction.
In general, this gives a way to prove theorems about Haken manifolds by using inductionL i.e. one may hope to just show the property trivially holds for 3-balls and is invariant under gluing two pieced along an incompressible surface. Note that the gluing surface being incompressible is in fact quite strong hence making the induction step possible in many cases.
For example, by applying an incredible amount of brilliant techniques, Thurston was able to prove his revolutionary result:
Hyperbolization theorem for Haken manifolds: Any Haken manifold with tori boundary components that does not contain incompressible tori admits a complete hyperbolic structure of finite volume in its interior.
In other words, this is saying that given a Haken manifold, we cut along any incompressible tori, the resulting manifold with tori boundary must have a complete hyperbolic structure with cusps near each boundary component,
This is the best we could hope for since manifolds with incompressible tori would have their fundamental group split over which of course imply they can’t be hyperbolic.
Now the more manifolds being Haken means the better this theorem is. Many evidences show that in fact a lot of manifolds are indeed Haken, in perticular we have:
Virtual Haken Conjecture: is finitely covered by a Haken manifold as long as is infinite.
We can see that together with Thurston’s hyperbolization theorem, this would give full solution to the geometrization conjecture for general 3-manifolds.
However, although now Perelman has proved the geometrization conjecture, the virtual Haken conjecture remains open. But in light of Perelman’s result now we are able to try to ‘back-solve’ the puzzle and only prove the virtual Haken conjecture for hyperbolic manifolds.
(to be continued)