0
votes
1answer
70 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 ...
6
votes
1answer
230 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 ...
2
votes
0answers
49 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 ...
2
votes
2answers
139 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 ...
9
votes
2answers
254 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 ...
5
votes
1answer
98 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 ...
4
votes
1answer
314 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 ...
2
votes
3answers
234 views

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: ...
10
votes
1answer
317 views

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 ...
3
votes
0answers
163 views

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 ...
1
vote
1answer
82 views

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 ...
5
votes
1answer
151 views

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 $Y$ ...
2
votes
0answers
110 views

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 ...
11
votes
1answer
177 views

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 ...
15
votes
2answers
413 views

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 ...
0
votes
1answer
138 views

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 $(ab)^6$ ...
10
votes
0answers
175 views

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 ...
2
votes
1answer
103 views

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 ...
8
votes
1answer
199 views

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 ...
3
votes
2answers
169 views

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: ...
3
votes
2answers
281 views

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 $f$ ...
10
votes
1answer
219 views

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 ...
5
votes
1answer
118 views

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 ...
9
votes
6answers
1k views

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 ...
2
votes
0answers
150 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
1answer
211 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 ...
4
votes
2answers
166 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 ...
9
votes
1answer
192 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 ...
11
votes
2answers
272 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 ...
1
vote
0answers
170 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 ...
10
votes
2answers
411 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 ...
3
votes
0answers
114 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 ...
6
votes
4answers
215 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 ...
1
vote
1answer
153 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 ...
1
vote
2answers
285 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 ...
16
votes
3answers
425 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 ...
7
votes
2answers
178 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 ...
1
vote
1answer
134 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 ...
3
votes
1answer
277 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) ...
4
votes
1answer
240 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 ...
0
votes
1answer
175 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. ...
4
votes
0answers
145 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 ...
0
votes
0answers
83 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 ...
5
votes
1answer
230 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, ...
5
votes
1answer
222 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 ...
5
votes
1answer
434 views

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 ...
1
vote
0answers
241 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?
8
votes
1answer
388 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 ...
7
votes
2answers
277 views

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 ...
1
vote
1answer
135 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 ...