The 3-manifolds tag has no usage guidance.

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 an irred. $SL_2(\mathbb{C})$-character to an analytic arc of an irred. 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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### 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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### 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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### 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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### 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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### 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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### 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 ...

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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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### 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 ...

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...