The 3-manifolds tag has no wiki summary.

**4**

votes

**1**answer

218 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 ...

**0**

votes

**1**answer

226 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 ...

**2**

votes

**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 ...

**8**

votes

**1**answer

287 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 ...

**4**

votes

**1**answer

235 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 ...

**1**

vote

**2**answers

290 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) ...

**6**

votes

**1**answer

227 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 ...

**7**

votes

**3**answers

528 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**

votes

**1**answer

287 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 ...

**6**

votes

**2**answers

470 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 ...

**5**

votes

**1**answer

236 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 ...

**18**

votes

**0**answers

440 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 ...

**4**

votes

**1**answer

287 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. ...

**11**

votes

**3**answers

597 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 ...

**3**

votes

**1**answer

404 views

### 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 ...

**5**

votes

**2**answers

249 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 ...

**7**

votes

**2**answers

401 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**

votes

**1**answer

265 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 ...

**4**

votes

**3**answers

384 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. ...

**17**

votes

**6**answers

3k 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 ...

**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 ...

**10**

votes

**3**answers

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

**6**

votes

**2**answers

262 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 ...

**3**

votes

**3**answers

274 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 ...

**3**

votes

**1**answer

209 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.

**8**

votes

**3**answers

836 views

### 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 ...

**1**

vote

**1**answer

153 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 ...

**6**

votes

**2**answers

592 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 ...

**5**

votes

**5**answers

999 views

### 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 ...

**10**

votes

**3**answers

603 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 $x$ ...

**4**

votes

**2**answers

220 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 ...

**9**

votes

**1**answer

385 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 ...

**2**

votes

**1**answer

307 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 ...

**4**

votes

**1**answer

178 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 $G$?

**23**

votes

**0**answers

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 ...

**-1**

votes

**1**answer

443 views

### 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 ...

**13**

votes

**2**answers

339 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 $N$ ...

**2**

votes

**1**answer

332 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 ...

**0**

votes

**1**answer

426 views

### 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 ...

**15**

votes

**3**answers

681 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?

**10**

votes

**1**answer

940 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, ...

**6**

votes

**0**answers

570 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

**1**answer

193 views

### Mapping torus relative to an infinite orbit can be hyperbolic with finite volume?

Consider a homeomorphism of the sphere with an infinite orbit converging forwards and backwards to the same point. Remove the orbit and the accumulation point and make the mapping torus. Can the ...

**17**

votes

**1**answer

880 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 ...

**5**

votes

**1**answer

646 views

### Dehn surgery on handlebody

Assume $V$ is a handlebody and $C$ be a simple closed curve contained in the interior of $V$.
As Sam said, there exists some simple closed curve such that every dehn surgery along it produces a ...

**4**

votes

**1**answer

277 views

### what is the meaning of “inseparable” in this case

Let $i:T^{2}\rightarrow S^{1}\times S^{2}$ be an embedding map.
If $i(T^{2})$ is inseparable in $S^{1}\times S^{2}$, then $S^{1}\times S^{2}-i(T^{2})\cong (O\cup K)^{c}$. Here $(O\cup K)^{c}$ is a two ...

**4**

votes

**3**answers

450 views

### Questions on 3-manifolds with a given boundary

I have the following question:
For a given two-dimensional Riemann surface $C$,
Is there a way to classify all topologically distinct
three-dimensional compact manifolds $M$ whose boundary is $C$,
...

**1**

vote

**1**answer

327 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 ...

**2**

votes

**3**answers

584 views

### what is the meaning of a curve $C$ representing Identity in fundamental group?

Suppose $M$ is a closed 3-manifold, $C\subset M$ is a simple closed curve which represent identity in $\pi_{1}(M)$. Then $C$ bounds an immersed disk in $M$.
My question is:
When does it bound an ...

**12**

votes

**2**answers

643 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 ...