The 3-manifolds tag has no usage guidance.

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### Construction of 3D Topological Quantum Field Theory from a 2D Modular Functor

I have been reading Bakalov and Kirillov's Lectures on Tensor Categories and Modular Functors in which the authors state that a direct construction of a $C$-extended 3D TQFT from a $C$-extended 2D ...

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### Is there a good notion of morphism between orbifolds?

Following Thurston, an orbifold is a topological space which looks locally like a finite quotient of $\mathbb R^n$ by a finite group of $O(n)$: this is expressed using charts as for differentiable ...

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196 views

### What invariants are of great concern in the field of 3-manifolds and why? How much do we know about them? [closed]

I am curious about 3-manifolds though I know little.
Here I am trying to know what invariants people in this field are interested in.
The following are what I have known and what I particularly want ...

**6**

votes

**1**answer

256 views

### Is there “nonorientable Heegaard Floer homology”?

I have a Heegaard diagram which produces a non-orientable 3-manifold. I want to know any 3-manifold invariant which can be calculated from Heegaard diagrams for non-orientable 3-manifold. (As far as I ...

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**2**answers

293 views

### Getting surgery link from Heegaard splitting

From Lickorish-Wallace theorem, every 3-manifold is an integral surgery on a link in $S^3$. From its proof from Saveliev's book, it seems obvious that if I know the Heegaard splitting of a closed 3-...

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**1**answer

164 views

### Looking for software that computes intersection numbers (Heegaard Diagrams)

As a part of my research I am working with intersection matrices of Heegaard diagrams. Is there some software that could help me compute such matrices for some examples?
Thanks.

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**1**answer

307 views

### Heegaard genus of the hyperbolic dodecahedral space (is it 3 or 4?)

I have a question on the hyperbolic dodecahedral space, first described by C.Weber and H. Seifert in 1933 [Die beiden Dodekaederr\"aume, Math Z. 37 (1933), 237-253]. Is it known whether it admits a ...

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**2**answers

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 ...

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**0**answers

177 views

### Toral decomposition

I have a couple of questions on the following theorem:
Theorem. (Jaco, Shalen)
Let $M$ be a compact, irreducible, orientable 3-manifold with incompressible boundary. There exists a collection $\...

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**2**answers

606 views

### Vector field on 3-sphere

Let $V$ be a vector field on $S^3$ such that its singularity points, namely the points at which the vector field vanishes, are only sinks or sources (i.e. the field is converging or diverging). Is ...

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128 views

### Euclidean realisation of a polyhedral complex

Let us say that an Euclidean polyhedral manifold is a manifold that is glued from a finite number of Euclidean polyhedrons by identifying isometrically their co-dimension $1$ faces. Let us assume that ...

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339 views

### Does Dehn filling always decrease Gromov norm?

As an immediate consequence of Proposition 6.5.2 of Thurston's notes, we have that, if $M$ is a compact 3-manifold with toric boundary and $\tilde M$ is obtained from $M$ via Dehn filling, then for ...

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**1**answer

245 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 ...

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vote

**1**answer

208 views

### isotopy classes of embeddings of the torus

Let's consider $S^1$-bundle $E$ over a 2-manifold $M$. How many isotopy classes of embeddings of the torus $\mathbb{T}^2$ in $E$?
For each free homotopy classes $\gamma$ of mappings of the circle ...

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**2**answers

304 views

### Commutativity in the Fundamental Group and Knot Theory

Let $M$ be a connected $3$-manifold and let $\alpha$ and $\beta$ be elements in $\pi_1(M)$. Then $\alpha$ and $\beta$ can be represented by two knots $a$ and $b$ in $M$. We may further require that ...

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**2**answers

278 views

### question on Thurston-Bennequin number

I have three questions actually:
1- is it true that in a sufficiently small neighborhood of Legendrian knot in a 3-manifold we can find another Legendrian knot?
2- If the above is true, suppose we ...

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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 ...

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**2**answers

216 views

### How does Thurston's Orbifold Geometrization imply that knots with meridional rank 2 are 2-bridge?

Problem 1.11 of Kirby's list asks whether every knot that has a knot group
which can be generated by n meridians, but not less than n, is an n-bridge
knot. There is a one-sentence update, saying that ...

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**1**answer

142 views

### Two links with the same signatures but unknown if they are related by Kirby moves

I am wondering if there are links $L_1, L_2$ in the sphere $S^3$ such that:
the signatures of $L_1, L_2$ are known.
we do not know if they are related by Kirby moves.
If so, could you specify the ...

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votes

**1**answer

420 views

### Geometrization & JSJ decomposition with boundary

Is there any paper where I can find a good explanation of the JSJ decomposition, the geometrization theorem and the relations between them when the manifold has nonempty (and non necessarily toroidal) ...

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**1**answer

286 views

### Casson invariant and signature

In W. Neumann, J. Wahl, "Casson invariant of links of singularities",
Comment. Math. Helv.,1990, Vol. 65, Issue 1, pp 58-78 some connection between the Casson invariant and the signature is ...

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**1**answer

207 views

### A question on Cayley graphs and hyperbolic 3-manifolds

There are two hyperbolic closed 3-manifolds, but I don't know whether they are homeomorphic or not. The only thing I know is that the Cayley graphs of their fundamental groups are quasi-isometric.
...

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336 views

### Unstable Foliations

Let $M$ be a closed compact Riemannian manifold, $\mathcal{F}$ be a $C^1$ foliation on $M$. Let $F(x)\in\mathcal{F}$ be the leaf containing $x$.
Definition. $\mathcal{F}$ is said to be a unstable ...

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166 views

### Haken manifolds and characterising sutured manifold hierarchies

In Gabai's paper (Knot Theory and Manifolds Lecture Notes in Mathematics Volume 1144, 1985, pp 14-17 An internal hierarchy for 3-manifolds) he considers sutured manifold decompositions of Haken $3$--...

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**1**answer

263 views

### open book decomposition of graph manifolds

Let's define a graph manifold as 3-manifold which is obtained by plumbing of a circle bundle over $\Sigma _g$ (with Euler number 0) with a circle bundle over $\Sigma_h$ (with Euler number k). Is there ...

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105 views

### Open Book Decompositions of M^3's : Finding the Projection Map (Hope in Coordinates) in an Abstract Open Book

all:
I want to know how to find out , hopefully in coordinates, (but I'll take what's available) , the description of the projection map in an abstract open-book decomposition.
Open book ...

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**1**answer

288 views

### Andreev's Theorem and Thurston's hyperbolization theorem

I am attempting to get to grips with Thurston's hyperbolization theorem for Haken $3$--manifolds. In particular I was looking at the section related to gluing up along hierarchy surfaces in Otal, Jean-...

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397 views

### The fundamental group of a $3$-manifold with a boundary of genus $>0$

Let $M$ be an orientable $3$-manifold with connected boundary $\Sigma_g$, a surface of genus $g>0$.
I would like to find a reference to the following two statements.
1) $\pi_1(M)\ne 0$.
2) $\...

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322 views

### Standard (special) spines and hyperbolic structure on 3-manifolds

My question relates to constructing angled triangulations or hyperbolic triangulations for $3$--manifolds. Briefly, an angle triangulation can be considered as an assignment of a real number (called ...

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### Homotoping a 2-plane field on a closed orientable 3-manifold to a contact structure

I am a beginner in Contact Geometry.
To prove that the inclusion $\text{Cont}(M)\hookrightarrow \text{Dist}(M)$ induces surjection at $\pi_0$ level, the closest I got was based on Ko Honda's notes.
...

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**1**answer

116 views

### A non-separating surface in a 3-Manifold with Semibundle structure

I have a question of the non-separating surfaces contained in a 3-Manifold with semi-bundle structure.
Suppose $M$ is a $3-$manifold with semi-bundle structure, this means that $M$ is a closed ...

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**1**answer

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### 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
...

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**1**answer

233 views

### Heegaard genus of covers of cusped hyperbolic 3-manifold

Is there a cusped (non-compact) orientable hyperbolic 3-manifold $M$ which has Heegaard genus $g$, and has a finite-sheeted cover with Heegaard genus $<g$?
Moreover, if the cover is index $n$, then ...

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**1**answer

364 views

### What are the homotopy classes of two-component links in $\mathbb{RP}^3$?

This question comes from an unanswered question on Math Stack Exchange.
A two-component link in $\mathbb{RP}^3$ is any embedding $S^1\uplus S^1\to \mathbb{RP}^3$. Two such links are homotopic if ...

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281 views

### contact structure on 3 manifolds

every orientable closed 3-manifold admits a contact structure. how we can construct this contact structure? if the manifold is prime what will happen?

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**1**answer

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### Contact structures and adjunction inequality in 3-manifolds

It is a theorem of Eliashberg that in a tight contact 3-manifold $(M, \xi)$ we have the adjunction inequality $|\langle e(s),[\Sigma] \rangle| \leq -\chi(\Sigma) $ where $s=s(\xi)$ is the spin$^c$-...

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185 views

### Algebraicness of trace field of finite volume hyperbolic 3-manifold and dimension of $\mathrm{SL}(2,\mathbb{C})$-character variety

Does the following statement:
"Let $G$ be a finitely generated
group and let $X(G)$ be the
$SL(2,\mathbb{C})$-character variety
of $G$. Suppose $X(G)$ contains an
irreducible component $...

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**1**answer

128 views

### A self-homeomorphism of $L_{p,q}$ is isotopic to one which preserves heegaard splitting

Consider the lens space $L_{p,q}$, which we can describe using its standard heegaard splitting, i.e. define $L_{p,q}$ as a quotient of two solid tori, identifying meridians on the boundary of one with ...

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963 views

### Tight vs. overtwisted contact structure

I know the familiar differences between tight and overtwisted contact structures. For example, each homotopy class of plane-bundles on a three-manifold has an overtwisted representative but tight ...

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**1**answer

483 views

### How fast does Ricci flow converge on the three-sphere?

Suppose I have a metric $g_0$ on the $\mathbb S^3$, and let $g_t$ be the solution to Ricci flow (with surgery) with initial metric $g_0$. What are some general results which give upper bounds on the ...

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### Is there a way to classify incompressible surfaces in $\Sigma \times [0,1]$ ?

Is there a way to classify incompressible surfaces in $\Sigma \times [0,1]$ where $\Sigma$ is any closed surface? I know of the Hatcher-Thurston classification of incompressible surfaces in 2-bridge ...

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**1**answer

169 views

### construct Seifert fibration on mapping torus of surface with monodromy a periodic mapping class

How do we construct Seifert fibration on mapping torus of surface with monodromy a periodic mapping class. I know that the fiber of the Seifert fibration has to be transverse to the surface fiber of ...

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**1**answer

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### 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 ...

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**1**answer

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### Isotopy classes on the disk and mapping tori

Is the following true?
"The conjugacy classes of two homeomorphisms of the n-times punctured disk have isotopic representatives iff the associated mapping tori are homeomorphic."
By conjugacy class ...

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### Categorification of WRT invariants of integral homology spheres

First, I would like to know how many definitions are there for categorification of WRT invariants. In addition, I wonder if the categorified version of WRT invariants have been explicitly computed for ...

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### Do different Dehn fillings produce homeomorphic 3-manifolds ?

Hi, everyone. I am interested in the dehn filling and Hyperbolic 3-manifold.
Suppose M be an orientable compact 3-manifold with one torus boundary and int(M) admit a
hyperbolic structure. ...

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**1**answer

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### When are isometry groups of hyperbolic 3-manifolds finite?

If $M$ is a finite volume hyperbolic 3-manifold, then its isometry group is finite. I believe this is also true for geometrically finite 3-manifolds. What is the most general condition on a hyperbolic ...

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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 ...

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**1**answer

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### 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 ...

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### 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(...