Triangulated 3-manifolds: from Haken to Thurston

Feng LuoRutgers University

May, 19, 2010Barrett Memorial Lecture University of Tennessee

Part of this is a joint work with Stephan Tillmann.

Recall of surfaces

Uniformization Thm. Given a Riemann surface S, there exists a conformal constant curvature metric g on S.

Question. How to compute g?

Input: triangulated surface (S, T) with some structure.

Output: a constant curvature metric g on (S, T).

The work: Thurston on circle packing, Colin de Verdiere, Stephenson, Bobenko-Springborn, Gu-Yau, Leibon, Rivin, Braegger, Chow-Luo, and many others

The work of Colin de Verdiere, 1990

He constructed a variational principle for circle packings.

1. wrote down the energy function

2. Introduced Euclidean angle structures (EAS) on

(S, T): assign each corner a positive real number so that sum in each triangle is π, sum at each vertex is 2π.

.

THM(Colin de Verdiere). For any (S, T), then

(a) either there are no EAS on (S, T), or

(b) the energy function has a max point on the space of all radii.

max point of energy → a metric g on (S,T), the topological obstruction →modify triangulation T.

3-manifolds: the geometrization thm of Perelman-Thurston

M closed, oriented.

1. Cut M open along S2’s to obtain

M = N1 # N2 # … # Nk

where Ni cannot be decomposed further (irreducible)

2. M irreducible, cut open along (essential) tori T2’s to obtain W1, W2, …, Wn. Wi cannot be decomposed further (atoriodal)

THM(Perelman-Thurston) W atoriodal, then either W is hyperbolic or W admits an S1-action.

By Mostow, the hypebolic metric is unique.

Similarity between 2-d and 3-d

There are 1701935 prime knots with < 17 crossing, all but 32 of them are hyperbolic.

Problem

Given triangulated (M3, T), compute the spheres, the tori and the hyperbolic metrics.

There are softwares available: Snapea etc, mainly for compact manifolds M with torus boundary.

Work of Haken, Jaco-Rubinstein, Li, Weeks and others

We propose a variational principle to approach it.

1. collection of oriented tetra A1, …, Ak

2. Identify faces in pairs by affine orientation reversing homeomorphisms.

The result is an oriented closed pseudo 3-manifold with a triangulation (M, T).

Triangulation of 3-manifolds

Thurston’s example of

Normal quads and triangles in tetra

properties of quads

1. Each quad = pair of opposite edges

2. q1, q2, q3 are 3 quads in an oriented tetra, then cyclic order q1 ->q2 ->q3 -> q1 depends only on the orientation.

Haken’s theory of Normal surfaces

A surface S in (M, T) is normal if for each tetra K, S∩K is a collection of triangles and quads:

Haken’s question

Given a collection of t’s and q’s in T, when can you produce a normal surface out of these?

Haken’s equation

S normal, its coordinate x is in R∆ X R□

x(t)= # copies of t in S

x(q) = # copies of q in S

Q. Is x in R∆ X R□ a coordinate?

Ans. It satisfies Haken’s equation

x(t)+ x(q) = x(t’) + x(q’) (*)

Def. a 2-quad-type solution x of (*): exist two quads q1 ≠q2 so that x(q)=0 for all other q’s and x(q1)≠0.

Thurston’s way to produce geometry from T• Make each tetra ideal hyperbolic tetra

• Glue by isometries• Match them nicely at edges

Thurston’s parametrization

Shape of an ideal tetra in H3 is given by z є C-{0,1}.

Opposite edges have the same parameter, i.e. defined on quads.

Thurston’s equation on (M, T)

Shape parameter z in (C-{0,1})□ so that,

1. For q->q’, z(q’) = 1/(1-z(q)).

2. For each edge e,

∏q ~e z(q) = 1.

If the right-hand-side is ±1, it is the generalized Thurston’s

equation.

Remark Thurston used a solution of his equation to produce the

hyperbolic metric on the figure-8 knot complement in 1978.

Theorem 1.(M, T) closed oriented pseudo 3-manifold,

then either

(1) there exists a solution to generalized Thurston’s equation,

or

(2) there exists a 2-quad-type solution to Haken’s equation.

A variational principle in 3-D

Def. S1-angle structure (SAS) on (M, T) assigns each quad q, x(q) є S1 so that

(1) If x1, x2, x3 assigned to 3 quads in a tetra, then x1x2x3=-1

(2) For each edge e,

∏q ~e x(q) = 1.

SAS(T)= the set of all SAS’s on T. It is non-empty and is a smooth closed manifold.

RM. This generalizes Casson, Rivin’s earlier definition.

Volume of SAS

Lobachevsky formula for volume of tetra

Def. The volume of an SAS x is

Vol(x) =sum of volume of tetra

The Lobachevsky function is periodic of period π and is continuous, non-smooth at Zπ.

So vol: SAS(T) -> R is continuous, non-smooth. Vol has a maximum point p.

Thm 1’. If p is a smooth point, then (1) holds.

If p is non-smooth, then (2) holds.

Conj. For a minimal triangulated closed 3-manifolds, if all max points are smooth, then there is a solution to Thurston’s equation. (Thanks to Burton and Segerman).

Sketch of proof

Sketch of proof

A triangulation of M is minimal if it has the smallest number of tetra among all triangulations of M.

Thm 2(L-Tillmann). If (M,T) minimal triangulated closed 3-manifold and vol: SAS(T) -> R has a non-smooth max point, then M is either

(1) reducible, or

(2) toriodal, or

(3) admitting an S1-action, or

(4) contains the connected sum of the Kleinbottle and the projective plane.

A very recent development

I was informed by Tillmann 3 weeks ago that he and Segerman can prove.

Thm(Segerman-Tillmann) If (M, T) a closed 3-manifold supports a solution to Thurston’s equation, then each edge in T is homotopically essential.

This thm and the conjecture (all smooth max -> sol to Th.

eq) give a new proof of the Poincare conjecture (without using Ricci flow).

the argument

Suppose M is a closed 3-manifold homotopic to S3.

By Kneser (1929), may assume M irreducible.

Take T to be a minimal triangulation of M.

By Jaco-Rubinstein (2001), T has only one vertex.

Since each edge is null homotopic, by Segerman Tillmann, T has no solution to Thurston’s eq.

By the conjecture (all smooth max-> sol. to Th), then T has a non-smooth max point.

By L-Tillmann and irreducibility of M, thus M is S3.

ConclusionVolume maximization on circle-valued angle structures

on (M, T).

1. non-smooth max point links to Haken’s normal surfaces,

2. smooth max points links to Thurston’s equation which produces hyperbolic structures.

Ref. F. Luo: http://arxiv.org/abs/0903.1138

F. Luo: http://arxiv.org/abs/1003.4413

Thank you