15
votes
2answers
396 views

Does a small-area sphere in a 3-manifold bound a small ball?

Let $M$ be a closed riemannian 3-manifold. I think that the following fact should be true and should have a relatively simple proof, but I cannot figure it out. For every $\varepsilon>0$ there ...
2
votes
1answer
90 views

Ray-Singer torsion of compact 3-manifolds with finite abelian fundamental group

Is the Ray-Singer analytic torsion for an arbitrary compact 3-manifold with finite Abelian fundamental group equivalent to the Ray-Singer analytic torsion of S^3 mod some direct product of Z_N's? It ...
3
votes
2answers
244 views

Automorphisms of Surfaces, Open Books and Contact Structures

Let $S$ be a surface with non-empty boundary $\partial S$ and let $f$ be an element of MCG(S), the Mapping Class Group of S, i.e., the group of self-diffeomorphisms of S up to isotopy, but with $f$ ...
1
vote
0answers
228 views

contact structure on 3 manifolds

every orientable closed 3-manifold admits a contact structure. how we can construct this contact structure? if the manifold is prime what will happen?
7
votes
1answer
359 views

How fast does Ricci flow converge on the three-sphere?

Suppose I have a metric $g_0$ on the $\mathbb S^3$, and let $g_t$ be the solution to Ricci flow (with surgery) with initial metric $g_0$. What are some general results which give upper bounds on the ...
6
votes
0answers
282 views

A conjecture of Thurston and possibly Weeks too

What is the status of the following conjecture: "... [w]hen the shortest simple closed geodesics are repeatedly removed from any complete hyperbolic 3-manifold of finite volume, eventually one ...
13
votes
5answers
2k views

The Chern-Simons/Wess-Zumino-Witten correspondence

I have often seen a relationship being alluded to between these two theories but I am unable to find any literature which proves/derives/explains this relationship. I guess in the condensed matter ...
6
votes
2answers
486 views

Morse Function on 3-Torus from Heegaard Splitting

It is easy to visualize some Morse functions on surfaces (such as the torus) via the height function, but this seemingly doesn't work for 3-manifolds. I am looking for an explicit one on the 3-torus ...
6
votes
2answers
279 views

$S^1$-action in three dimensions

Let $M$ be a closed, orientable 3-manifold with a non-trivial differentiable $S^1$-action. What does this imply for $M$? What are examples except for (products of) spheres?
6
votes
1answer
487 views

Geometry of Whitehead manifolds.

I'm currently studying some problems about the Whitehead manifold $W$ (the open 3-manifold which is contractible but not homeomorphic to $\mathbb{R}^3$). Does there exists some survey paper on its ...
6
votes
4answers
329 views

In a contact manifold, is every tranverse 1-foliation given by some Reeb vector field?

Let $M$ be a contact manifold, and let $F$ be an oriented 1-dimensional foliation that is transverse to the contact structure. Is there a contact form $\alpha$ whose associated Reeb vector field ...
9
votes
3answers
612 views

Is the cut locus of a generic point in a hyperbolic manifold a generic polyhedron?

Let $p\in M$ be a point in a closed riemannian manifold $M$. Recall that the cut locus of $p$ is the subset of $M$ consisting of all points that are connected to $p$ by at least 2 distance-minimizing ...
26
votes
2answers
2k views

Drawing of the eight Thurston geometries?

Do you know of a picture, drawing, or other concise visual representation of the eight three-dimensional Thurston geometries? I am imagining something akin to the standard picture (of a sphere, ...