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3
votes
2answers
229 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 ...
4
votes
0answers
141 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 ...
1
vote
1answer
151 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 ...
0
votes
0answers
76 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
224 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
2answers
250 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) ...
5
votes
1answer
216 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 ...
2
votes
0answers
107 views

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. ...
2
votes
1answer
72 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 ...
5
votes
1answer
433 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 ...
7
votes
1answer
180 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 ...
9
votes
1answer
336 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 ...
1
vote
0answers
238 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?
1
vote
1answer
156 views

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 ...
0
votes
0answers
66 views

algebraicness of trace field of finite volume hyperbolic 3-manifold and dimension of $SL(2,C)$-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 $G$. If $X(G)$ contains an irreducible component $X_0$ ...
2
votes
1answer
100 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 ...
4
votes
2answers
451 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 ...
8
votes
1answer
373 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
266 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
123 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 ...
11
votes
1answer
346 views

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 ...
8
votes
1answer
182 views

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 ...
1
vote
0answers
89 views

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 ...
6
votes
2answers
339 views

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. ...
6
votes
1answer
256 views

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 ...
6
votes
0answers
291 views

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 ...
2
votes
1answer
142 views

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 ...
15
votes
1answer
475 views

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 $M$: $$Y=\operatorname{Hom}(\pi_1(M^3),\operatorname{SL}(2,\mathbb C))$$ ...
6
votes
1answer
291 views

integer surgeries on knots

I have constructed a list of surgery coefficients which yield spherical space forms. For instance, there are two knots with different Alexander polynomials on which 29-surgery will give a small ...
7
votes
2answers
203 views

Non-tame 3-manifolds covered by the Euclidean space

An open 3-manifold is tame if it is homeomorphic to the interior of a compact manifold. Is there a (known) example of an open 3-manifold that is not tame, has finitely generated fundamental group and ...
19
votes
2answers
244 views

Hyperbolic exceptional fillings of cusped hyperbolic 3-manifolds

Thurston's Hyperbolic Dehn Surgery Theorem says that all but finitely many fillings of a cusp of a hyperbolic 3-manifold result in hyperbolic manifolds that are deformations of the original manifold. ...
6
votes
1answer
325 views

Hyperbolic 3-manifolds with no geometrically finite structure

Does there exist a compact hyperbolic 3-manifold $M$ that is not diffeomorphic to a geometrically finite hyperbolic manifold? If yes, can such $M$ have incompressible boundary? I think the answer ...
2
votes
2answers
246 views

The knot whose complement is the Hantzsche-Wendt manifold

Can someone identify the knot whose complement in $\mathbb{S}^{3}$ will produce the Hantzsche-Wendt manifold? Thanks
6
votes
2answers
271 views

Hyperbolic Brunnian links and rectangular cusp shapes

My question is as follows. Does every hyperbolic Brunnian link have rectangular cusp shapes on all its components? Here is what I mean: The Borromean rings form a famous link $B$ (a smooth ...
0
votes
0answers
70 views

projective triangulation and holonomy map

Suppose we have a 3-manifold which is triangulated .what can we deduce if we know that the holonomy map is the identity map around each vertex of the triangulation? (Here we imagine the covering space ...
4
votes
1answer
192 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
1answer
214 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
1answer
165 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
1answer
264 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
1answer
206 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
2answers
283 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
1answer
213 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 ...
6
votes
3answers
513 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
1answer
273 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
2answers
438 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
1answer
223 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
0answers
428 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
1answer
286 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 Normalfl├Ąchen. Acta Math. ...
10
votes
3answers
548 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
1answer
379 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 ...