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25
votes
0answers
895 views

What is the three-dimensional hyperbolic volume of a four-manifold?

Every smooth closed orientable 4-manifold may be constructed via a handle decomposition. Before asking a couple of questions, I recall some well-known facts about handle-decompositions of 4-manifolds. ...
22
votes
0answers
1k views

Are there lots of integer homology three-spheres?

The problem of counting combinatorial three-spheres with $N$ simplices has implications for some partition functions in physics (see a paper by Benedetti and Ziegler for more background and ...
19
votes
0answers
411 views

The oriented homeomorphism problem for Haken 3-manifolds

Haken famously described an algorithm to solve the homeomorphism problem for the 3-manifolds that bear his name (fleshed out by many others, including Hemion and Matveev who fixed some gaps). But ...
10
votes
0answers
155 views

Fox re-imbedding theorem in dimension four

Fox re-imbedding theorem states the following: A compact 3-manifold $M$ with boundary that embeds in the three-sphere $S^3$, can be re-imbedded in $S^3$ so that its complement is a union of ...
9
votes
0answers
89 views

Wanted: a nontrivial weakly inadmissible Heegaard diagram

This is a question asked by a student in my lecture. After drawing pictures for awhile, I thought it was a good one. I seek a nontrivial example of a pointed Heegaard diagram ...
8
votes
0answers
222 views

Krull rings and determinantal invariants

During another attempt to come to grips with Hillman's excellent book Algebraic Invariants of Links, I am having difficulty figuring out why Krull rings are the setting for Chapter 3- the natural ...
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 ...
5
votes
0answers
493 views

Expository accounts of the Thurston norm

Other than Thurston's original paper (which I find quite hard to read), are there any expositions of the basic properties of the Thurston norm? In particular, I'm interested in a proof of his ...
4
votes
0answers
115 views

Construction of 3D Topological Quantum Field Theory from a 2D Modular Functor

I have been reading Bakalov and Kirillov's Lectures on Tensor Categories and Modular Functors in which the authors state that a direct construction of a $C$-extended 3D TQFT from a $C$-extended 2D ...
4
votes
0answers
129 views

Haken manifolds and characterising sutured manifold hierarchies

In Gabai's paper (Knot Theory and Manifolds Lecture Notes in Mathematics Volume 1144, 1985, pp 14-17 An internal hierarchy for 3-manifolds) he considers sutured manifold decompositions of Haken ...
3
votes
0answers
108 views

Euclidean realisation of a polyhedral complex

Let us say that an Euclidean polyhedral manifold is a manifold that is glued from a finite number of Euclidean polyhedrons by identifying isometrically their co-dimension $1$ faces. Let us assume that ...
3
votes
0answers
291 views

What is the behaviour of a smooth 3-manifold acting by a circle?

As Mumford pointed out in his paper 'Topology of Normal Singularities and a Criterion for Simplicity'(1961), every point $p$ on a normal complex surface $V$ has an associated 3-manifold $M$ which is ...
2
votes
0answers
98 views

when is an irreducible SL_2(C) representation of a cusped hyperbolic 3-manifold scheme reduced or smooth

Let $M$ be an orientable cusped hyperbolic 3-manifold. Let $\rho \in hom(\pi_1(M),SL_2(\mathbb{C}))$ be an irreducible representation. Is $\rho$ scheme reduced ? What can prevent $\rho$ from being ...
2
votes
0answers
63 views

Looking for software that computes intersection numbers (Heegaard Diagrams)

As a part of my research I am working with intersection matrices of Heegaard diagrams. Is there some software that could help me compute such matrices for some examples? Thanks.
2
votes
0answers
98 views

Homotoping a 2-plane field on a closed orientable 3-manifold to a contact structure

I am a beginner in Contact Geometry. To prove that the inclusion $\text{Cont}(M)\hookrightarrow \text{Dist}(M)$ induces surjection at $\pi_0$ level, the closest I got was based on Ko Honda's notes. ...
1
vote
0answers
167 views

Toral decomposition

I have a couple of questions on the following theorem: Theorem. (Jaco, Shalen) Let $M$ be a compact, irreducible, orientable 3-manifold with incompressible boundary. There exists a collection ...
1
vote
0answers
226 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?
1
vote
0answers
83 views

Categorification of WRT invariants of integral homology spheres

First, I would like to know how many definitions are there for categorification of WRT invariants. In addition, I wonder if the categorified version of WRT invariants have been explicitly computed for ...
0
votes
0answers
42 views

Is there an example where we cannot lift an analytic arc of irreducible $SL_2(\mathbb{C})$-character to an analytic arc of irreducble representation

Is there an example of an irreducible and boundary irreducible 3-manifold $M$ with torus boundary and a non-abelian representation $\rho: \pi_1(M) \to SL_2(\mathbb{C})$, a non-constant analytic arc ...
0
votes
0answers
73 views

Open Book Decompositions of M^3's : Finding the Projection Map (Hope in Coordinates) in an Abstract Open Book

all: I want to know how to find out , hopefully in coordinates, (but I'll take what's available) , the description of the projection map in an abstract open-book decomposition. Open book ...
0
votes
0answers
151 views

Casson's invariant and intersection homology

EDIT: Immediately after I wrote this question, I remembered the elegant paper "An intersection homology invariant for knots in a rational homology 3-sphere" by Frohman and Nicas, which I believe does ...
0
votes
0answers
63 views

algebraicness of trace field of finite volume hyperbolic 3-manifold and dimension of $SL(2,C)$-character variety

Does the following statement "Let $G$ be a finitely generated group and let $X(G)$ be the $SL(2,\mathbb{C})$ character variety of $G$. If $X(G)$ contains an irreducible component $X_0$ ...
0
votes
0answers
68 views

projective triangulation and holonomy map

Suppose we have a 3-manifold which is triangulated .what can we deduce if we know that the holonomy map is the identity map around each vertex of the triangulation? (Here we imagine the covering space ...