The 3-manifolds tag has no usage guidance.

**15**

votes

**1**answer

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

**1**answer

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

**14**answers

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

**2**answers

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

**0**answers

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

**3**answers

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

**6**answers

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

**2**answers

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

**3**answers

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

**3**answers

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

**3**answers

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

**5**answers

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

**1**answer

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

**3**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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 ...