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45
votes
14answers
7k views

What are some of the big open problems in 3-manifold theory?

From what I understand, the geometrization theorem and its proof helped to settle a lot of outstanding questions about the geometry and topology of 3-manifolds, but there still seems to be quite a lot ...
41
votes
3answers
3k views

Kirby calculus and local moves

Every orientable 3-manifold can be obtained from the 3-sphere by doing surgery along a framed link. Kirby's theorem says that the surgery along two framed links gives homeomorphic manifolds if and ...
32
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, ...
30
votes
0answers
1k 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. ...
24
votes
2answers
3k views

Level sets of Morse functions

Every compact two dimensional manifold admits a Morse function such that any its regular level set is at most two circles. I am interested in a generalization of that phenomenon. Does there exist a ...
24
votes
2answers
535 views

Can one deform an immersion of a 3-manifold in $\mathbb R^4$ to an embedding in $\mathbb R^6$?

Let $M^3$ be an oriented 3-manifold, and let $f:M^3\looparrowright \mathbb R^4$ be a codimension one immersion. Is it possible to find a small deformation of the composite map $$ M^3 \to \mathbb R^4 \...
24
votes
2answers
2k views

Why is the volume conjecture important?

The volume conjecture, a formula relating hyperbolic volume of a knot complement with the semiclassical limit of a family of coloured Jones polynomials, is widely considered the biggest open problem ...
23
votes
6answers
2k views

How to get convinced that there are a lot of 3-manifolds?

My question is rather philosophical : without using advanced tools as Perlman-Thurston's geometrisation, how can we get convinced that the class of closed oriented $3$-manifolds is large and that ...
23
votes
0answers
2k 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 references)...
22
votes
4answers
2k views

Can all n-manifolds be obtained by gluing finitely many blocks?

Fix a dimension $n\geqslant 2$. Let $S= \{M_1,\ldots, M_k\}$ be a finite set of smooth compact $n$-manifold with boundary. Let us say that a smooth closed $n$-manifold is generated by $S$ if it may ...
22
votes
7answers
4k 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 ...
22
votes
3answers
2k views

How expensive is knowledge? Knots, Links, 3 and 4-manifold algorithms. [closed]

With geometrization, Rubinstein's 3-sphere recognition algorithm and the Manning algorithm, 3-manifold theory has reached a certain maturity where many questions are "readily" answerable about 3-...
21
votes
2answers
2k views

Elevator pitch for the Virtual Fibering Theorem

There has been a great deal of excitement among topologists about the proof of the Virtual Haken Theorem, and in fact of the Virtual Fibering Theorem (for closed hyperbolic 3-manifolds, but I'm ...
21
votes
3answers
850 views

What are the implications of the simple loop conjecture?

Drew Zemke recently posted a preprint on arXiv proving the Simple Loop Conjecture for 3-manifolds modeled on Sol. Simple Loop Conjecture: Consider a 2-sided immersion $F\colon\, \Sigma\rightarrow M$...
19
votes
2answers
307 views

Hyperbolic exceptional fillings of cusped hyperbolic 3-manifolds

Thurston's Hyperbolic Dehn Surgery Theorem says that all but finitely many fillings of a cusp of a hyperbolic 3-manifold result in hyperbolic manifolds that are deformations of the original manifold. ...
19
votes
0answers
460 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 it'...
18
votes
1answer
984 views

Diffeomorphisms vs homeomorphisms of 3-manifolds

For a smooth 3-manifold $M$, is the natural map from the space of diffeomorphisms of $M$ to the space of homeomorphisms of $M$, $${\sf Diff}(M) \longrightarrow {\sf Top}(M),$$ a weak homotopy ...
17
votes
2answers
1k views

Is the 4x5 chessboard complex a link complement?

The 2x3 and 3x4 chessboard complexes (form a square grid of vertices and make a simplex for any set of vertices no two of which are in the same row or column) are a 6-cycle and a triangulated torus ...
17
votes
6answers
1k views

Tetrahedra with prescribed face angles

I am looking for an analogue for the following 2 dimensional fact: Given 3 angles $\alpha,\beta,\gamma\in (0;\pi)$ there is always a triangle with these prescribed angles. It is spherical/euclidean/...
16
votes
3answers
614 views

Looking for “large knot” examples

This question is about knots and links in the 3-sphere. I want to find an example of a "large" knot or link with some special properties. I'm looking for some fairly specific examples, but I'm also ...
16
votes
2answers
2k views

Proofs of Kirby's theorem

Each orientable 3-manifold can be obtained by doing surgery along a framed link in the 3-sphere. Kirby's theorem says that two framed links give homeomorphic manifolds if and only if they are obtained ...
16
votes
2answers
462 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 ...
15
votes
6answers
2k views

Diffeomorphism of 3-manifolds

Surgery theory aims to measure the difference between simple homotopy types and diffeomorphism types. In 3 dimensions, geometrization achieves something much more nuanced than that. Still, I wonder ...
15
votes
3answers
766 views

3-manifolds with solvable fundamental group

Is there a nice reference for the classification of closed 3-manifolds with solvable (nilpotent, abelian, etc.) fundamental group, assuming the Geometrization Conjecture?
15
votes
1answer
627 views

Can the SL_2 character variety of a three-manifold be nonreduced?

Let $M^3$ be a three-manifold and consider the representation variety and the character variety of $M$: $$Y=\operatorname{Hom}(\pi_1(M^3),\operatorname{SL}(2,\mathbb C))$$ $$X=\operatorname{Hom}(\pi_1(...
15
votes
2answers
565 views

The kernel of the map from the handlebody group to Outer automorphisms of a free group

Let $K$ be a compact oriented 3-dimensional handlebody of genus $g$. The group $H_g$ of isotopy classes of diffeomorphisms of $K$ is called the handlebody group. (It embeds as a subgroup of the ...
14
votes
3answers
620 views

Maximal euler characteristic of surfaces bounding two fixed curves

Let $\gamma_0$ and $\gamma_1$ be two simple closed curves in a closed surface $S$. What is the maximum Euler characteristic of a compact properly embedded surface $\Sigma \subset S\times [0,1]$ ...
14
votes
1answer
709 views

Where are $+$, $-$ and $\infty$ in bordered Heegaard-Floer theory?

Here goes my first MO-question. I've just read Lipshitz, Ozsváth and Thurston's recently updated "A tour of bordered Floer theory". To set the stage let me give two quotes from this paper. ...
14
votes
1answer
845 views

$SL_2 R$ Casson invariant?

Casson's invariant is an invariant of a homology 3-sphere, obtained by ``counting" representations of the fundamental group into $SU(2)$. I was wondering if there is an analogous invariant counting ...
13
votes
3answers
857 views

Irreducible homology 3-spheres that bound smooth contractible manifolds

Some examples of irreducible homology 3-spheres that bound smooth contractible 4-manifolds are listed in the comment to problem 4.2 in Kirby's problem list, and all of them happen to occur among the ...
13
votes
3answers
686 views

Classification of knots by geometrization theorem

I read this interview with Ian Agol, where he says: "...I learned that Thurston’s geometrization theorem allowed a complete and practical classification of knots." My question is: How does ...
13
votes
2answers
380 views

Heegaard splitting of covering hyperbolic manifold.

I am curious about how the Heegaard genus changes after a finite covering. Is there anyone constructing an closed hyperbolic 3-manifold $N$ such that the Heegaard genus of a finite covering of $N$ ...
13
votes
1answer
1k views

Detecting a cover of the figure-8 knot complement

I have a specific concrete example of a complete, finite volume, cusped hyperbolic 3-manifold $M$, and I am trying to determine whether or not $M$ is a cover of the figure-8 knot complement (call this ...
13
votes
2answers
356 views

Which closed 3-manifolds can be embedded in $R^4$?

I wonder which closed orientable 3-manifolds can be embedded in $\mathbb R^4$ and which in $\mathbb R^5$. Is there a way to determine whether given closed 3-manifold, obtained, say by Dehn surgery on ...
13
votes
1answer
1k views

Question about Thurston's paper “A norm for the homology of 3-manifolds”

I have a question about the proof of Theorem 5 in Thurston's paper "A norm for the homology of 3-manifolds". It's the theorem that asserts that the fibered faces are, well, fibered. More precisely, ...
13
votes
1answer
402 views

Distinguishing 3-manifolds by homologies of covers

In a blog post on ldtopology, a recent arxiv posting of Lins-Lins is discussed. The main argument of that paper is difficult to algorithmically distinguish two 3-manifolds and to that end the authors ...
12
votes
2answers
699 views

How to see isometries of figure 8 knot complement

The figure 8 knot complement $M$ is the orientable double cover of the Gieseking manifold, which implies that $M$ has a fixed-point free involution. If we think of $M$ with its hyperbolic metric, this ...
12
votes
3answers
872 views

Flat SU(2) bundles over hyperbolic 3-manifolds

Can someone give me a non-trivial example of a flat SU(2)-connection over a compact orientable hyperbolic 3-manifold? The literature on such bundles over 3-manifolds is huge and my naive searches don'...
12
votes
1answer
394 views

Geometric intersection with incompressible surfaces

Let $M$ be a oriented compact $3$-manifold, closed or with boundary. For any incompressible surface $F$, define a function $i_F$ on the set of homotopy classes of closed curves in $M$ by $$i_F (\alpha)...
12
votes
1answer
491 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 $(\Sigma,\mathbf{\alpha},\...
12
votes
0answers
188 views

Is tightness decidable?

Given a contact structure on a three-manifold, is there an algorithm to decide whether or not it tight? For concreteness' sake, let's agree to represent the given contact three-manifold via an open ...
11
votes
3answers
310 views

Random links and $3$-manifolds

In Jeffrey Weeks book "The Shape of Space" he explaines at the end of Chapter 18 (on page 255) the following about the geometrization conjecture: A non-trivial connected sum $M_1\# M_2$ admits a ...
11
votes
1answer
280 views

Construction of the Casson invariant

What is the easiest construction of the Casson invariant? The original construction using representation spaces (as found, for instance, in Akbulut-McCarthy) is very technical since you have to ...
11
votes
2answers
366 views

The homeomorphism problem for hyperbolic 3-manifolds and the virtual Haken theorem

If $N$ and $N'$ are two closed hyperbolic 3-manifolds, then one would like to have an algorithm which determines whether or not $N$ and $N'$ are homeomorphic. If $N$ and $N'$ are Haken, then such an ...
11
votes
1answer
436 views

Did Milnor and Thurston write anything else about characteristic numbers for 3-manifolds?

In Characteristic numbers for 3-manifolds Milnor and Thurston define a characteristic number and this is cited in ch. 6 of Thurston's notes when discussing the Gromov approach to Mostow rigidity. The ...
11
votes
1answer
252 views

Closed $3$-manifold, $2$-dimensional subbundle of this manifold, is this form exact or not?

Let $M$ be a closed $3$-manifold, and let $\xi$ be a $2$-dimensional subbundle of $TM$. I know the following. There is a nowhere zero $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any vector ...
11
votes
2answers
865 views

Geometrization for 3-manifolds that contain two-sided projective planes

Often when people write about the geometrization conjecture they assume (for simplicity) that the manifold is orientable. I never seriously thought of non-orientable 3-manifolds, but while reading ...
11
votes
3answers
778 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 ...
11
votes
0answers
230 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 ...
10
votes
3answers
498 views

Torsion in cuspidal cohomology

Following Lemma 2.7 from Vogtmann's Rational Homology of Bianchi Groups, I want to define cuspidal cohomology as $$H_{\mathrm{cusp}}(M)=\frac{H_1(M)}{i_*(H_1(\partial M))}$$ where $i:\partial M\to M$ ...