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15
votes
1answer
625 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(...
8
votes
1answer
343 views

3-manifolds with isomorphic fundamental groups

There are many non-homeomorphic 3-manifolds with isomorphic fundamental groups, for example the lens spaces $L(p,q_1)$ and $L(p,q_2)$ with $q_1 \ne \pm q_2^{\pm 1}\mod p$. Also, Seifert fibered spaces ...
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 ...
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, ...
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
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-...
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 ...
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 ...
13
votes
3answers
851 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 ...
11
votes
3answers
768 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 ...
10
votes
3answers
302 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 ...
8
votes
5answers
437 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 ...
8
votes
1answer
483 views

Topological rigidity of compact manifolds in dimension three

The Borel Conjecture asserts that homotopy equivalent aspherical closed manifolds are homeomorphic, which is still open in general. But, for three-dimensional manifolds, this conjecture holds (I ...
10
votes
3answers
496 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 $i:\partial M\to M$ ...
5
votes
1answer
251 views

What is known about a 3-manifold $M$ when its fundamental group is linear?

Suppose we have a 3-manifold $M$ and its respective fundamental group $\pi_1(M)$. An important question about its fundamental group is to ask if it is linear, i.e. they are isomorphic to a subgroup of ...
4
votes
1answer
266 views

Equivalent to Oriented knot complement conjecture

I would like to see why the following two statements in Kirby's list of problem are equivalent: Statement 1: If $K_1$ and $K_2$ are knots in a closed oriented 3-manifold $M$ whose complements are ...
2
votes
2answers
250 views

Is the following 3-manifold irreducible?

We start with product manifold $ X := S_g \times (0,1)$, where $S_g$ is a closed, orientable surface of genus $g \geq 1$. Since this has $\mathbb R^3$ as its universal cover, $X$ is irreducible. Now $...
1
vote
1answer
367 views

Smooth structures on closed $3$-manifolds are unique up to diffeomorphism?

Hi! I'm using the theorem stated in the question, but so far, I haven't found a source that does explicitely state and prove it or at least give a proper citation. It must be me though, since the ...
13
votes
2answers
379 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 $N$ ...
5
votes
1answer
474 views

sufficient conditions on Non-Haken manifolds

Is there an algorithm to detect the Non-Haken Manifold? Or, is there a sufficient condition for a manifold to be a non-Haken manifold? (off course, I hope that condition is not the ones in its ...
4
votes
1answer
151 views

Which 3-manifolds have positive rank gradient?

For which $3$-manifolds $M$ is the fundamental group $\pi_1(M)$ finitely generated and has positive rank gradient? Recall that the rank gradient of a finitely generated group $G$ is defined to be $...
3
votes
1answer
172 views

How many non-commensurable non-arithmetic manifolds have a quaternion algebra like this?

I am interested in realizing commensurability classes of hyperbolic $3$-manifolds whose quaternion algebra (note: not invariant quaternion algebra) is isomorphic to one of the form $\Big(\frac{a,b}{F(\...
2
votes
1answer
130 views

Engulfing Kleinian groups?

Let $G$ be a Kleinian group, and let $H \lneq G$ be a finitely generated subgroup. Must there be a proper finite index subgroup $U$ of $G$ containing $H$ ? I know that this is true for Fuchsian ...
1
vote
1answer
262 views

On compact, orientable 3-manifolds with non-empty boundary

I recall my Professor having stated something along the lines of the following, but I am not quite certain about the precise statement she gave: Let $M$ be a compact, orientable 3 manifold with non-...
1
vote
1answer
350 views

Kropholler's Conjecture and 3-manifolds

Suppose that $G$ is a group and that $H$ is a subgroup, both finitely generated, and assume that there is a non-trivial H-almost invariant set $X$ with $HXH=X$. Kropholler's Conjecture asserts that $ ...
0
votes
1answer
244 views

mapping class group of a surface

I want to know what techniques are known to present a diffeomorphism on a surface with boundary (the diffeomorphism is not necessarily the identity restricted to the boundary) as product of Dehn ...