The 3-manifolds tag has no wiki summary.

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### 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 ...

**2**

votes

**1**answer

162 views

### Once punctured torus bundles in snappy/twister

I have been trying to learn about snappy's method for encoding once-punctured torus bundles (http://www.math.uic.edu/t3m/SnapPy/manifold.html#snappy.Manifold). As you can see from the link, they are ...

**15**

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**1**answer

547 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 $M$:
$$Y=\operatorname{Hom}(\pi_1(M^3),\operatorname{SL}(2,\mathbb C))$$
...

**6**

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**1**answer

361 views

### integer surgeries on knots

I have constructed a list of surgery coefficients which yield spherical space forms. For instance, there are two knots with different Alexander polynomials on which 29-surgery will give a small ...

**7**

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**2**answers

211 views

### Non-tame 3-manifolds covered by the Euclidean space

An open 3-manifold is tame if it is homeomorphic to the interior of a compact manifold. Is there a (known) example of an open 3-manifold that is not tame, has finitely generated fundamental group and ...

**19**

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**2**answers

272 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. ...

**6**

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**1**answer

361 views

### Hyperbolic 3-manifolds with no geometrically finite structure

Does there exist a compact hyperbolic 3-manifold $M$ that is not diffeomorphic to a geometrically finite hyperbolic manifold? If yes, can such $M$ have incompressible boundary?
I think the answer ...

**3**

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**2**answers

285 views

### The knot whose complement is the Hantzsche-Wendt manifold

Can someone identify the knot whose complement in $\mathbb{S}^{3}$ will produce the Hantzsche-Wendt manifold?
Thanks

**6**

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**2**answers

312 views

### Hyperbolic Brunnian links and rectangular cusp shapes

My question is as follows.
Does every hyperbolic Brunnian link have rectangular cusp shapes on all its components?
Here is what I mean:
The Borromean rings form a famous link $B$ (a smooth ...

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**0**answers

76 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 ...

**5**

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**1**answer

233 views

### Seifert surfaces via Alexander duality

If we take a knot $K$ in $S^3$, there are several ways to construct the associated Seifert surface. One way, which I am not familiar with, I just came across in a paper I am reading. It goes like ...

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**1**answer

230 views

### slam-dunk operation

I study surgery theory on 3-manifolds using the text book written by Gompf and Stipsicz.I can't understand \bf{slam-dunk} operation.
Let $K_{1}$ be the meridian of a knot $K_{2}$ in $S^3$, and $T$ is ...

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**1**answer

174 views

### the carrier graph and Heegaard surface

Let $M$ be orientable 3-manifold admitting a Heegaard splitting $V\cup_{S}W$.
Let $X$ be a carrier graph of $M$ such that rank($X$)=rank($\pi_{1} M$).
Note: A connected graph is called a carrier ...

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**1**answer

307 views

### Geodesic cuffs of pairs of pants in a hyperbolic manifold- why are they disjoint?

I'm trying to understand Kahn-Markovic's celebrated Immersing almost geodesic surfaces in a closed hyperbolic three manifold. There is something probably quite basic which I can't figure out.
We have ...

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**1**answer

239 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 ...

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**2**answers

293 views

### faraway curves in surface

Let $S$ be a compact orientable 2-surface with $\chi(X)\leq -3$. Y.Minsky and H.Masur proved that the curve complex of $X$ is $\delta$-hyperbolic and infinite.
E.Klarreich (see also U.Hamenstadt) ...

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**1**answer

236 views

### Can bilipschitz models of hyperbolic 3-manifolds be made effective?

In their proof of the Ending Lamination Conjecture, Brock, Canary, and Minsky prove existence of bilipschitz models of hyperbolic 3-manifolds (homeomorphic to a surface times $\mathbb{R}$) depending ...

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**3**answers

535 views

### Judging whether a finitely presented group is a 3-manifold group?

Given a finitely presented group $G$, how many necessary conditions do people know for $G$ to be isomorphic to the fundamental group of some closed connected 3-manifold? (e.g. residually finite)

**10**

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**1**answer

290 views

### Examples of 3-manifolds with RFRS fundamental group

I'm wondering if anyone knows how to construct hyperbolic 3-manifolds whose fundamental group is RFRS. Clearly the recent work of Agol, Wise, etc. says that such manifolds are abundant, and in ...

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**2**answers

482 views

### heegard diagram

It seems like there is an algorithm to find the Heegard diagram of a 3 manifold obtained by surgery on a link. Also someone told me I can find it in the Gompf and Stipciz's book. But I could not find ...

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**1**answer

246 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 ...

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445 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 ...

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**1**answer

291 views

### When are incompressible surfaces isotopic into a two-skeleton?

Haken proved that an incompressible surface in a triangulated irreducible 3-manifold is isotopic to a surface which is normal with respect to the triangulation (Theorie der Normalflächen. Acta Math. ...

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640 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 ...

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**1**answer

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### slam-dunk or Kirby move

I have difficulty understanding the slam-dunk. In slam-dunk you have two knots K1 and K2 with framings r (rational) and n (integer) such that K1 is the meridian of K2. Now we perform surgery on K2 ...

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254 views

### What is a higher genus analogue of the Pontryagin product?

Given a compact oriented aspherical $3$--manifold $M$ with torus boundary $\partial M\simeq T^2$ (e.g. a knot complement), the condition that the images in $\pi_1 M$ of basis $x,y\in \pi_1 T^2$ under ...

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414 views

### Isotopy in 3-manifolds

Assume $\Sigma_1$ and $\Sigma_2$ are two embedded compact surfaces
(say orientable) in an orientable 3-manifold $M$. Assume $\Sigma_1$ and $\Sigma_2$
are homotopic in $M$. Then are they isotopic?

**6**

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**1**answer

268 views

### What is known about the distribution of average edge-degrees for 3-manifold triangulations (with the number of 3-simplices less than a fixed constant)?

This is my first question on mathoverflow! It relates to a project I'm undertaking with a student.
Work by Tamura (extending results by Luo and Stong) shows that for any closed 3-manifold $M$ and any ...

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**3**answers

397 views

### Second Homotopy Group of Graph Manifolds

A graph manifold is a closed 3-manifold $M$ that admits a finite collection of disjoint embedded tori $\mathcal{T}$ so that $M \setminus \mathcal{T}$ is a disjoint union of Seifert fibred spaces (i.e. ...

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### 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 ...

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### 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 ...

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453 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 $i:\partial M\to M$ ...

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265 views

### Hyperbolic structures on $S\times\mathbb{R}$

Let $S$ be a closed incompressible surface in a finite volume hyperbolic 3-manifold $M$ without cusps. Let $N$ be the cover of $M$ associated to $\pi_1(S) \subset \pi_1(M)$. The cover $N$ is ...

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281 views

### When is a three-manifold deck transformation group solvable?

Suppose that $\pi:Y \to Y'$ is a regular covering of closed, connected, orientable three-manifolds and let $G$ be the deck transformation group. Furthermore, suppose that $Y$ is a rational homology ...

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**1**answer

223 views

### Is there a torsion element in the homology cylinder group?

The homology cylinder group is the monoid of homology cylinders modulo homology cobordism. I wonder whether there is any known finite order element in this group.

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### Why S^3-K and SL(2,R)/SL(2,Z) are diffeomorphic? Here K is a trefoil in S^3.

I've heard this result from my differential manifold class, and I don't know how to prove it.
Does anyone know how to construct such diffeomorphism? Please tell me, thanks a lot.
Any comments are ...

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**1**answer

159 views

### Does the coefficient of the meridian determine the coefficient of the longitude?(on Dehn surgery)

I'm studying Dehn surgery, and it says that the coefficient $(p,q)$ which says how the meridian curve on solid torus is attached will determine the entire resulting manifold. I'm wondering whether the ...

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**2**answers

633 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 ...

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### What is the possible usefulness of étale topology and cohomology apart from the resolution of the Weil conjecture ?

My question is as stated in the title:
What is the possible usefulness of étale topology and cohomology apart from the resolution of the Weil conjecture ?
I am particularly interested to know if ...

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**3**answers

619 views

### Best known Margulis constants?

A Margulis number for a hyperbolic $n$-manifold $M=\mathbb{H}^n/\Gamma$ is a number $\epsilon>0$ such that for each $x\in\mathbb{H}^n$ the group generated by the elements in $\Gamma$ which move $x$ ...

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**2**answers

226 views

### Pseudoanosov mapping torus and length of curves.

Let $M_{\phi}$ be a hyperbolic mapping torus coming from a pseudo-Anosov map $\phi$ in a surface $S$. Is there any way to estimate the length of the geodesic representing a given curve in the surface ...

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**1**answer

394 views

### Complexity of surfaces bounding knots in 4-ball and 3-sphere respectively

I'm interested in a complexity question related to problems like the slice-ribbon problem.
To be specific, if $K \subset S^3$ is a knot, it might be non-trivial yet still bound a smoothly-embedded ...

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411 views

### teichmuller geodesics and hyperbolic mapping torus

Given a pseudoanosov map $\phi$ of a surface $S$, there is a geodesic $\sigma$ in Teichmuller space (with the teichmuller metric) that is an axis for $\phi$, In other words, $\phi$ acts as a ...

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**1**answer

183 views

### Rank of a group generated by side-pairing isometries of a polyhedron

Let $P$ be a compact convex polyhedron in $\mathbb{H}^3$. Let $G$ be a group generated by side-pairing isometries of $P$. Is there an algorithm to find the rank of $G$?

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### 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 ...

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**1**answer

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### Is this manifold orientable? [closed]

Let $C$ be the set of points $(a,b,c,d) \in \mathbb{C}^4$ which satisfy
1) $ \left|a\right|^2+\left|c\right|^2=\left|b\right|^2+\left|d\right|^2 =1 $.
2) $ a\bar{b}+c\bar{d}=0 $
There is a ...

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**2**answers

349 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 $N$ ...

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**1**answer

339 views

### Explicit tetrahedralizations of closed 3-manifolds and connections between convex polytopes and hyperbolic knot complements

There are three consistent ways to glue opposite faces of a dodecahedron to get a closed 3-manifold. With a $\frac{1}{10}$ twist we receive the Poincaré dodecahedral space, with a $\frac{3}{10}$ twist ...

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**1**answer

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### 3-manifold theorem reference request or proof

The following is a theorem of which I have great interest in but cannot find anything about on the internet,
Every 3-manifold of finite volume comes from identifying sides of some polyhedron
I'm ...

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695 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?