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9
votes
3answers
205 views

Random links and $3$-manifolds

In Jeffrey Weeks book "The Shape of Space" he explaines at the end of Chapter 18 (on page 255) the following about the geometrization conjecture: A non-trivial connected sum $M_1\# M_2$ admits a ...
3
votes
1answer
121 views

contact surgery diagram on Brieskorn manifolds

For the Brieskorn manifold $\Sigma(p,q,r)=\{z_1^p+z_2^q+z_3^r=0\} \cap S^5 \subset C^3$, replacing zero with $\epsilon$ in the above, realizes $\Sigma$ as boundary of a Stein domain which induces a ...
5
votes
1answer
60 views

stabilization of Legendrian knots

There are two ways to stabilize a Legendrian knot $k$ in standard contact sphere $(S^3,\xi_{st})$ i.e. adding right cusps or left cusps, let's call these two stabilized Legendrian knots $k_R$ and ...
8
votes
2answers
190 views

Quasi-isometric rigidity of Nil

Let $Nil$ be the unique simply connected non-abelian three-dimensional nilpotent Lie group, i.e. the group of upper triangular matrices with all the eigenvalues equal to 1 (this group is also known as ...
1
vote
0answers
77 views

contact structure on double branched covers of $S^3$

We can assign a natural contact structure to double branched cover of a transverse knot $k$ in the $3$-sphere with its standard contact structure $(S^3,\xi_{st})$, as described ...
2
votes
3answers
133 views

Surgery of $S^3$

I have been troubled by this seemingly simple question recently. How do we easily visualize the statement: Surgery of $S^3$ over a trivial unknot gives $S^1 \times S^2$? All I can think of for ...
13
votes
3answers
583 views

Classification of knots by geometrization theorem

I read this interview with Ian Agol, where he says: "...I learned that Thurston’s geometrization theorem allowed a complete and practical classification of knots." My question is: How does ...
1
vote
1answer
226 views

On compact, orientable 3-manifolds with non-empty boundary

I recall my Professor having stated something along the lines of the following, but I am not quite certain about the precise statement she gave: Let $M$ be a compact, orientable 3 manifold with ...
3
votes
0answers
97 views

Constructing a “nice” cobordism

Denote by $\Sigma_g$ the closed, orientable surface of genus $g$. I want to construct a cobordism $M_g$ between $\Sigma_g$ and $\Sigma_{g+1}$ with the following two nice properties: 1) $M_g$ is an ...
23
votes
2answers
489 views

Can one deform an immersion of a 3-manifold in $\mathbb R^4$ to an embedding in $\mathbb R^6$?

Let $M^3$ be an oriented 3-manifold, and let $f:M^3\looparrowright \mathbb R^4$ be a codimension one immersion. Is it possible to find a small deformation of the composite map $$ M^3 \to \mathbb R^4 ...
4
votes
2answers
193 views

Is the following 3-manifold always a trivial I-bundle over a surface?

Let $M$ be a compact, orientable and irreducible 3-manifold with with boundary consisting of two incompressible components $N_0,N_1$, with $N_i \stackrel{f_i}{\cong} S_g$ for some diffeomorphism ...
7
votes
1answer
139 views

Heegard genus of hyperbolic Haken 3-manifolds

Is there an example of a closed Haken hyperbolic 3-manifold of Heegaard genus 2?
4
votes
2answers
415 views

Thurston geometries---the geometry of the universal cover of $SL(2, \mathbb{R})$

In one of the eight Thurston geometries there is the geometry of the universal cover of $SL(2, \mathbb{R})$. But from the algebraic point of view $PSL(2,\mathbb{R})$ is sufficient for building ...
18
votes
3answers
775 views

What are the implications of the simple loop conjecture?

Drew Zemke recently posted a preprint on arXiv proving the Simple Loop Conjecture for 3-manifolds modeled on Sol. Simple Loop Conjecture: Consider a 2-sided immersion $F\colon\, \Sigma\rightarrow ...
2
votes
1answer
69 views

Is every closed Sasakian 3-manifold a circle bundle on a Riemann surface?

It suffices to say that all circle bundles on compact Riemann surfaces admit the structure of a closed Sasakian 3-manifold. The question is, the converse of this statement and/or what are the ...
2
votes
3answers
254 views

Classification of open subset of $\mathbb{R}^{3}$ [closed]

There is a theorem which gives a classification of connected open sets of $\mathbb{R}^{2}$. Unfortunately, I don't remember the correct statement, but it looks like this Theorem ? Let ...
4
votes
1answer
140 views

Which 3-manifolds have positive rank gradient?

For which $3$-manifolds $M$ is the fundamental group $\pi_1(M)$ finitely generated and has positive rank gradient? Recall that the rank gradient of a finitely generated group $G$ is defined to ...
1
vote
1answer
117 views

Engulfing Kleinian groups?

Let $G$ be a Kleinian group, and let $H \lneq G$ be a finitely generated subgroup. Must there be a proper finite index subgroup $U$ of $G$ containing $H$ ? I know that this is true for Fuchsian ...
5
votes
2answers
293 views

3-manifolds homotopy equivalent to a surface

I have heard that an open, orientable 3-manifold $X$ (non-compact, without-boundary) that is homotopy equivalent to an orientable surface $S_g$ must itself already be homemorphic to $S_g \times ...
4
votes
1answer
132 views

Homology 3-sphere with a unique Stein-fillable contact structure

Are there any known examples of oriented integer homology 3-spheres $Y$ (besides $S^3$) which have exactly one Stein-fillable contact structure up to isotopy? Failing that, what are the known examples ...
4
votes
2answers
289 views

Gluing two 3 manifolds along their boundary

Let $X,Y$ be two compact, smooth, orientable 3 manifolds, each with an incompressible boundary component diffeomorphic to some genus $g $ surface $S_g$. Under an orientation-reversig diffeomorphism ...
21
votes
7answers
3k 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 ...
2
votes
2answers
238 views

Is the following 3-manifold irreducible?

We start with product manifold $ X := S_g \times (0,1)$, where $S_g$ is a closed, orientable surface of genus $g \geq 1$. Since this has $\mathbb R^3$ as its universal cover, $X$ is irreducible. Now ...
5
votes
0answers
107 views

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 ...
6
votes
2answers
868 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 ...
7
votes
1answer
159 views

Conditions on the hierarchy for Thurston's hyperbolization theorem

From my understanding the proof of Thurston's hyperbolization theorem for Haken $3$--manifolds consists of cutting the manifold along a hierarchy (collection of incompressible, ...
18
votes
4answers
2k views

How to get convinced that there are a lot of 3-manifolds?

My question is rather philosophical : without using advanced tools as Perlman-Thurston's geometrisation, how can we get convinced that the class of closed oriented $3$-manifolds is large and that ...
7
votes
1answer
533 views

A conjecture of Montesinos

Not every orientable 3-manifold is a double cover of $S^3$ branched over a link. For example, the 3-torus isn't. However, in 1975 Montesinos conjectured (Surjery on links and double branched covers of ...
5
votes
0answers
86 views

Heegaard diagrams of prime 3-manifolds

Are there some known results which give a classification of closed prime 3-manifolds up to their Heegaard diagrams? (That is, providing a collection of Heegaard diagrams which exhausts all prime ...
12
votes
0answers
174 views

Is tightness decidable?

Given a contact structure on a three-manifold, is there an algorithm to decide whether or not it tight? For concreteness' sake, let's agree to represent the given contact three-manifold via an ...
3
votes
1answer
165 views

Non-degenerate periodic orbits in the boundary of Lefschetz fibration over a disk

Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk, and let $\Omega$ be a closed 2-form on $E$ such that it is non-degenerate fiberwise. For any $x \in E$, there is a ...
2
votes
1answer
183 views

The first Betti number of a finite covering space of a closed 3-manifold

Let $\tilde{M}\to M$ be a finite covering of closed 3-manifolds. Is it possible that $\beta_1(\tilde{M})-1> [\pi_1(M):\pi_1(\tilde{M})](\beta_1(M)-1)>0$? Here, ...
7
votes
2answers
206 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 ...
3
votes
2answers
180 views

Brieskorn homology spheres

We know that a Brieskorn homology 3-spheres $\Sigma(p,q,r)$ admit a free $S^1$-action, which makes it a Seifert fibered spaces with three singular fibers: $M(b;r_1,r_2,r_3)$. How should one get from ...
3
votes
2answers
277 views

the space of continuous maps between 3-manifolds

Let $X$ be a connected hyperbolic 3-manifold (without boundary), $S^3$ the 3-sphere and $Map(X,S^3)$ the space of continuous maps between $X$ and $S^3$. Question: Is the space $Map(X,S^3)$ connected ...
3
votes
1answer
325 views

Simple proof for property R conjecture

Gabai's property R theorem is: If the 0-surgery manifold of a knot $K$ is homeomorphic to $S^1\times S^2$, then $K$ is the unknot. Recently, 3-manifold topology has been developed rapidly by Agol, ...
0
votes
0answers
127 views

Hyperbolic manifold of dim 3 with finite volume.

The geometrization Theorem for 3-manifolds classifies all oriented compact (without boundary ) 3-manifolds. Is there a theorem which classifies all hyperbolic oriented 3-manifold (without boundary ) ...
10
votes
1answer
342 views

Morse number of the Poincaré homology sphere

What is the Morse number of the Poincaré homology sphere? What about the stable Morse number?
0
votes
1answer
105 views

Does this count as a canonical decomposition for non-elementary hyperbolic 3-orbifolds?

Let $\Gamma$ be a Kleinian group and let $\mathbb{H}^3$ be the upper half-space model for hyperbolic 3-space. Then $\mathbb{H}^3/\Gamma$ is an orientable hyperbolic 3-orbifold (with the group action ...
7
votes
1answer
309 views

3-manifolds with isomorphic fundamental groups

There are many non-homeomorphic 3-manifolds with isomorphic fundamental groups, for example the lens spaces $L(p,q_1)$ and $L(p,q_2)$ with $q_1 \ne \pm q_2^{\pm 1}\mod p$. Also, Seifert fibered spaces ...
4
votes
3answers
448 views

teichmuller geodesics and hyperbolic mapping torus

Given a pseudoanosov map $\phi$ of a surface $S$, there is a geodesic $\sigma$ in Teichmuller space (with the teichmuller metric) that is an axis for $\phi$, In other words, $\phi$ acts as a ...
3
votes
1answer
163 views

Chern-Simons invariants of 2-bridge knots

2-bridge links $L(p/q)$ are described by the continuous fraction expansion $\frac{p}{q}=\left[a_1,a_2,\ldots,a_n\right]$, where the $a_i$ are the numbers of twists in the boxes below: Looking at ...
5
votes
2answers
214 views

2-bridge knots in the Rolfsen's table

2-bridge knots (aka rational knots) $K(p,q)$ are described by a rational number $\frac{p}{q}$ or likewise its continued fraction expansion $\left[a_1,a_2,\ldots,a_k\right]$. Has somebody worked out a ...
3
votes
2answers
263 views

Modifying the Reeb vector field by multplying by a function

Given a contact 3-manifold $(M,\omega)$ and its Reeb vector field $R$ and contact structure $\Delta$, I want to understand in some sense 'how large' is the set of Reeb vector fields supported by ...
8
votes
5answers
426 views

In a contact manifold, is every tranverse 1-foliation given by some Reeb vector field?

Let $M$ be a contact manifold, and let $F$ be an oriented 1-dimensional foliation that is transverse to the contact structure. Is there a contact form $\alpha$ whose associated Reeb vector field ...
12
votes
1answer
382 views

Geometric intersection with incompressible surfaces

Let $M$ be a oriented compact $3$-manifold, closed or with boundary. For any incompressible surface $F$, define a function $i_F$ on the set of homotopy classes of closed curves in $M$ by $$i_F ...
7
votes
2answers
209 views

What are known examples of a 3-manifold $Y$ embedded into $Y'\times I$ where $Y'$ is another 3-manifold?

The question I have is the following: Let $Y,Y'$ be two integer homology 3-spheres. Can we embed $Y'$ into $Y\times I$ such that $Y'$ separates the two boundary components apart? Do we know any ...
2
votes
0answers
82 views

Self-diffeomorphisms of fibered knot complements

A knot $K \subset S^3$ is fibered if the complement $S^3 \setminus K$ of (a small open neighborhood of) $K$ is a fiber bundle over $S^1$. (The fiber will be a surface with one boundary component.) ...
1
vote
1answer
150 views

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 ...
4
votes
2answers
227 views

Geometrisation of inclusion-like epimorphisms to free groups

Let $H_g$ be the standard $3$-dimensional handle-body, whose boundary is denoted $S_g$, the oriented closed surface of genus $g\geq 1$. Call $F_g$ be the free group of rank $g$. Denote by $i:S_g \to ...