The 3-manifolds tag has no wiki summary.

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### Do geodesics in SL2R map to geodesics in the hyperbolic plane?

I am looking for a reference/proof/disproof of the following statement.
Equip the Lie group $SL_2(\mathbb{R})$ with the left-invariant Riemannian metric, whichis given on the Lie algebra by $\langle ...

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**1**answer

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### Reeb orbit and open books

Weinstein conjecture is about existence of a closed orbit of the Reeb vector field on every contact manifold. On the other hand, we know every contact 3-manifold admits a compatible open book, which ...

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### Geometry of the space of circles in the Euclidean plane

We know that Mobius transformations, $z\to\frac{az+b}{cz+d}$, permutes circles and lines in the Euclidean plane, $(\mathbb{R}^2, dx^2 + dy^2 )$.
It may even be possible to write an explicit formula ...

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744 views

### Simplifying triangulations of 3-manifolds

Throughout, by finite triangulation I mean a triangulation consisting of a finite number of triangles.
Suppose $T$ and $T'$ are finite triangulations of a 3-manifold $M$. We will say that $T'$ is ...

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**1**answer

176 views

### Does knot Floer homology detect knot genus in rational homology spheres?

My question is the following:
Does knot Floer homology detect the genus of null-homologous knot in rational homology spheres?
If the answer is yes, I would like to have a reference for the ...

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**1**answer

71 views

### non-isotopic but homotopic tight contact structure

By a theorem of Eliashberg, two overtwisted contact structures on a 3-manifold which belong to the same homotopy class (as plane fields), are also isotopic (through contact structures). Is there an ...

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997 views

### Tetrahedra with prescribed face angles

I am looking for an analogue for the following 2 dimensional fact:
Given 3 angles $\alpha,\beta,\gamma\in (0;\pi)$ there is always a triangle with these prescribed angles. It is ...

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76 views

### Mapping class group and fundamental group of a 3 manifold

If the 3 manifold is Haken, then the natural homomorphism from the mapping class group of this 3 manifold to the outer automorphism of its fundamental group is an isomorphism.
Any other kind of 3 ...

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110 views

### On the realization of a compact surface as a leaf of an analytic foliation

Let $S$ be a compact orientable surface of genus $g \geq 2$. Is there any transversely real (or complex) analytic codimension one foliation $\mathcal{F}$ such that $\mathcal{F}$ has $S$ as a leaf with ...

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266 views

### Software for computing Thurston's unit ball

Is there any software which can be used for computing Thurston's unit ball(for second homology of 3-manifolds) of link complements? In particular can I do that with SnapPy?
PS: even a table for ...

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123 views

### Definition of the dual spider number and the formula for the first chern class of the triangle

In the process of trying to understand various maps in Heegaard Floer homology I got stuck on the definition of the dual spider number, which, it seems to me, has a combinatorial definition directly ...

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153 views

### Quasi-isometry and left invariant orderability for groups

Is the property of left invariant orderability for finitely generated groups preserved by quasi-isometrics? More precisely, if $G$ is a left orderable (finitely generated) group and $H$ is a ...

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297 views

### Heegaard genera of arithmetic 3-manifolds

UPDATE: Because I was hoping that state the question as concisely as
possible, the original post did not include a precise definition of
arithmetic 3-manifold only a reference to Maclachlan and ...

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**1**answer

112 views

### In the definition of the Heegard Floer surgery exact triangle, what exactly is the correspondence between Whitney triangles and periodic domains?

I'm reading Osváth-Szabó's notes on Heegard Floer homology, in particular about the surgery exact triangle.
On page 14 (numbered 42 on the document), they describe an isomorphism between the space of ...

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2k 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 ...

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**1**answer

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### Legendrian knots on pages of a compatible open book

Suppose we have a Legendrian knot embedded on a page of an open book compatible with the given contact structure on the 3-manifold. Is it true that the page framing and Thurston-Bennequin framing of ...

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### Do Heegaard Floer homology detect fibred knot in general oriented 3-manifold?

Do Heegaard Floer homology detect fibred knot in general oriented 3-manifolds other than $S^3$? If the answer is yes could you give a reference.

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### Heegaard Floer Homology of double branched cover

The question is the following:
Let $K\subset S^{3}$ be a knot, consider the double branched cover $Y$ of $S^{3}$ over $K$. We know $Y$ has a unique spin structure $\mathfrak{s}$, now the question is: ...

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### 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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591 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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### References about 3-manifolds

I am working on a subject of geometric group theory closely related to 3-manifolds, and in order to understand these links, I am seeking a good reference about 3-manifolds, as self-contained as ...

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### Question about the fundamental group of rational homology 3-spheres

By a rational homology 3-sphere, I mean a compact oriented manifold three-manifold $Y$ with $H_1(Y)$ finite. My question is whether there exists a reasonable classification of such manifolds such that ...

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**1**answer

346 views

### On the fundamental group of closed 3-manifolds

I know that every finitely presented group can be realized as the fundamental group of a compact, connected, smooth manifold of dimension 4 (or higher). In dimension 2 there are strong restriction on ...

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### 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, ...

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294 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 $ ...

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### Wanted: a nontrivial weakly inadmissible Heegaard diagram

This is a question asked by a student in my lecture. After drawing pictures for awhile, I thought it was a good one. I seek a nontrivial example of a pointed Heegaard diagram ...

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### Did Milnor and Thurston write anything else about characteristic numbers for 3-manifolds?

In Characteristic numbers for 3-manifolds Milnor and Thurston define a characteristic number and this is cited in ch. 6 of Thurston's notes when discussing the Gromov approach to Mostow rigidity.
The ...

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### Action of the mapping class group on separating spheres in a connected sum of aspherical 3-manifolds

Let $M$ be a connected sum of $g$ closed aspherical 3-manifolds $M_1, \ldots, M_g$. [Update: I also assume that all the $M_i$-s are diffeomorphic, i.e. $M$ is a connected sum of copies of the same ...

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### Sutured Manifolds and minimal genus

Is there a result relating sutured manifolds and surfaces of minimal genus? perhaps someone has a very clever point of view of these two notions that can share.
In other matters, do we know how to ...

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### Does the cohomology after Dehn surgery depend only on the original 3-manifold or also how the knot is situated?

For $f:S^1\to M$ a knot in a 3-manifold, we can construct a 3-manifold $N$ by a $0/1$-type Dehn surgery along $f$:
First remove from $M$ a solid torus which is a tubular neighbourhood of the knot ...

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### Is there an example where we cannot lift an analytic arc of irreducible $SL_2(\mathbb{C})$-character to an analytic arc of irreducble representation

Is there an example of an irreducible and boundary irreducible 3-manifold $M$ with torus boundary and a non-abelian representation $\rho: \pi_1(M) \to SL_2(\mathbb{C})$, a non-constant analytic arc ...

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### Do we get a instanton $S^{3}$ if we do $1/n$ surgery on a knot in $S^{3}$?

Consider the following question:
If $K\subset S^{3}$ is a nontrivial knot. Let $Y$ be the manifold obtained by doing $1/n$-surgery ($n\geq1$). Is it possible that the instanton Floer homology of $Y$ ...

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### 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, ...

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### when is an irreducible SL_2(C) representation of a cusped hyperbolic 3-manifold scheme reduced or smooth

Let $M$ be an orientable cusped hyperbolic 3-manifold. Let $\rho \in hom(\pi_1(M),SL_2(\mathbb{C}))$ be an irreducible representation.
Is $\rho$ scheme reduced ?
What can prevent $\rho$ from being ...

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### Does a small-area sphere in a 3-manifold bound a small ball?

Let $M$ be a closed riemannian 3-manifold. I think that the following fact should be true and should have a relatively simple proof, but I cannot figure it out.
For every $\varepsilon>0$ there ...

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### from Dehn twists to surgery diagram [closed]

Assume the relation $(ab)^6=1$, for $a$ and $b$ Dehn twists about the meridian and the longitude of a torus. Now if we glue the two ends of $T\times I$ together by either the diffeomorphism $(ab)^6$ ...

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### Fox re-imbedding theorem in dimension four

Fox re-imbedding theorem states the following:
A compact 3-manifold $M$ with boundary that embeds in the three-sphere $S^3$, can be re-imbedded in $S^3$ so that its complement is a union of ...

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### Ray-Singer torsion of compact 3-manifolds with finite abelian fundamental group

Is the Ray-Singer analytic torsion for an arbitrary compact 3-manifold with finite Abelian fundamental group equivalent to the Ray-Singer analytic torsion of S^3 mod some direct product of Z_N's? It ...

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### Seifert fiberable manifolds with several Seifert fiberings

I have a question on Theorem 2.3 on page 34 of Hatcher's notes on 3-manifolds:
Hatcher: Notes on Basic 3-Manifold Topology.
Regarding the class d), it follows from Proposition 2.1 on page 31, that ...

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### Virtual fibering conjecture for cusped hyperbolic manifolds

I am interested in understanding if the Virtual Fibering Theorem holds in the non-compact case.
Agol proved that every closed hyperbolic $3$-manifold has a finite index cover which fibers over the ...

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### Handlebody decomposition of a 3-manifold adapted to a link

Given a compact connected 3-manifold $M$ with non-empty boundary, and a link $L \subset M$, is there a handlebody decomposition of $M = H^0 \cup (\cup_i H^1_i) \cup \{\text{2-handles}\}$ such that:
...

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### 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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### Automorphisms of Surfaces, Open Books and Contact Structures

Let $S$ be a surface with non-empty boundary $\partial S$ and let $f$ be an element of MCG(S), the Mapping Class Group of S, i.e., the group of self-diffeomorphisms of S up to isotopy, but with $f$ ...

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### Construction of the Casson invariant

What is the easiest construction of the Casson invariant? The original construction using representation spaces (as found, for instance, in Akbulut-McCarthy) is very technical since you have to ...

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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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### Branched coverings over orbifolds with reflector lines

It is well known that if $F\to B$ is a $n$-finite branched covering over an orbifold with cone-points then the orbifold Euler's characteristics are related via ...

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### 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 ...

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### What is the order of the isotopy group of the Brieskorn homology 3-sphere?

Let $\Sigma(p,q,r)$ be the Brieskorn homology 3-sphere with $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}<1$ (so not the 3-sphere or the Poincare sphere). The fundamental group is given by $$ ...

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### Equivariant smoothing of PL structures on $S^3$

Suppose $S^3$ is PL sphere on which a finite group $G$ acts by PL homeomorphisms. Is it always possible to find a compatible smooth structure such that $G$ acts by diffeomorphisms?
I am not quite ...

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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 ...