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5
votes
1answer
203 views

Can the analytic arc of an irred. $\mathrm{SL}_2(\mathbb{C})$-character always be lifted 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 $\rho: \pi_1(M) \to \mathrm{SL}_2(\mathbb{C})$, a non-constant ...
4
votes
1answer
197 views

On Johansson's Theorem on homotopy equivalences of 3-manifolds

Johansson's theorem states the following: Given $f:M_1\rightarrow M_2$ (not a pair map) an homotopy equivalence between 3-manifolds with incompressible boundary. Let $V_i$ be the components of the ...
45
votes
14answers
7k views

What are some of the big open problems in 3-manifold theory?

From what I understand, the geometrization theorem and its proof helped to settle a lot of outstanding questions about the geometry and topology of 3-manifolds, but there still seems to be quite a lot ...
6
votes
2answers
232 views

Homeomorphic but Non-Conjugate Mapping Tori

Suppose we fix a genus $g$ closed surface $S$. Let $f, g \in Map(S)$ be conjugate, for $Map(S)$ the mapping class group of $S$. Then I know that $M_f$ (the mapping torus of $M$ with monodromy $f$) is ...
1
vote
1answer
183 views

Algebraicness of trace field of finite volume hyperbolic 3-manifold and dimension of $\mathrm{SL}(2,\mathbb{C})$-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 $G$. Suppose $X(G)$ contains an irreducible component $...
3
votes
1answer
214 views

When is an irreducible $\mathrm{SL}_2(\mathbb{C})$ representation of a cusped hyperbolic 3-manifold scheme reduced or smooth?

Let $M$ be an orientable cusped hyperbolic 3-manifold. Let $\rho \in \mathrm{Hom}(\pi_1(M),\mathrm{SL}_2(\mathbb{C}))$ be an irreducible representation. Is $\rho$ scheme reduced ? What can ...
3
votes
2answers
454 views

SU(2) representations of alternating knot groups

Suppose that $K$ is an $\textit{alternating}$ knot in $S^3$, and let $R_0$ be the space of homomorphisms from $\pi_1(S^3 - K)\to SU(2)$ which send meridians to trace free matrices. Denote the subset ...
3
votes
2answers
128 views

base point in Heegaard Floer homology

It is stated in Mcduff's overview paper "FLOER THEORY AND LOW DIMENSIONAL TOPOLOGY", end of page 9, that the Heegaard Floer homology without considering a base point depends only on the homology of ...
2
votes
2answers
91 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 ...
4
votes
2answers
159 views

Looking for examples of large hyperbolic two-generator knots or 3-manifolds

We say a knot $K$ in $S^3$ is small if its exterior contains no closed properly embedded incompressible surfaces and we say $K$ is large otherwise. Does anyone know of an example of a large ...
7
votes
2answers
212 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
1answer
102 views

Non-Haken hyperbolic 3-manifolds without nonorientable surfaces

It is well known that there exist infinitely many (non homeomorphic) non-Haken closed hyperbolic 3-manifolds. These can be obtained for example doing Dehn surgery on the figure eight knot complement. ...
4
votes
1answer
163 views

How to understand Taubes' moduli space of holomorphic curves?

Let $(X, \omega)$ be a closed symplectic 4-manifold. Let $\mathcal{C}=(C_i, mi_i)$ be a holomorphic current in $X$, where $C_i$ is a somewhere injective $J$-holomorphic curve in $X$ and $m_i$ is ...
3
votes
1answer
128 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 ...
4
votes
1answer
163 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 ...
7
votes
0answers
80 views

Compatibility of spherical and hyperbolic geometry for fibred knots

Hyperbolic knots and links have a lovely peculiarity that you can always find a position for them in $S^3$ making two groups the same, one defined using the spherical geometry of $S^3$, and the other ...
4
votes
1answer
179 views

Fibered knots vs Heegaard genus

We call a knot $K$ in a 3-manifold $M$ fibered if $M\backslash K$ fibers over $S^1$ with fibers $\Sigma$ and such that $K$ is ambient isotopic to the boundary of the compactified fiber $\overline{\...
0
votes
2answers
121 views

obtaining circle bundle over torus by trefoil surgery

Does any integer surgery on a right or left trefoil knot give the $S^1$-bundle over $T^2$ with Euler number $1$?
3
votes
1answer
172 views

How many non-commensurable non-arithmetic manifolds have a quaternion algebra like this?

I am interested in realizing commensurability classes of hyperbolic $3$-manifolds whose quaternion algebra (note: not invariant quaternion algebra) is isomorphic to one of the form $\Big(\frac{a,b}{F(\...
10
votes
1answer
354 views

What are the different ways of defining 3-manifolds?

I wonder what are the different ways of defining the 3-manifold. Obviously for average human being it is difficult to imagine the 3-manifold. Therefore the presentation or visualisation of such object ...
1
vote
1answer
51 views

Existence of a continuous map of a disk with a given boundary image on a surface to its complement in $\mathbb{R}^3$

I am considering the following problem: given an embedded closed surface in $\mathbb{R}^3$ (unknotted) and a non-trivial simple closed curve on it, does there exist a continuous map of a disk to the ...
13
votes
2answers
353 views

Which closed 3-manifolds can be embedded in $R^4$?

I wonder which closed orientable 3-manifolds can be embedded in $\mathbb R^4$ and which in $\mathbb R^5$. Is there a way to determine whether given closed 3-manifold, obtained, say by Dehn surgery on ...
23
votes
6answers
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 ...
10
votes
1answer
242 views

Closed $3$-manifold, $2$-dimensional subbundle of this manifold, is this form exact or not?

Let $M$ be a closed $3$-manifold, and let $\xi$ be a $2$-dimensional subbundle of $TM$. I know the following. There is a nowhere zero $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any vector ...
4
votes
1answer
150 views

mapping class group of a two-holed torus

It is well-known that in the picture below we have $t_d=(t_at_b)^6$ as elements in the mapping class group of a two-holed torus, ($t_\gamma$ represents positive Dehn twist about the curve $\gamma$). ...
10
votes
2answers
296 views

Rational homology sphere that is not Seifert manifold

I wonder if there is an example of rational homology sphere that is not a Seifert manifold. If there is, how can one construct such a rational homology sphere from a surgery of a knot in $S^3$?
10
votes
3answers
303 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 ...
5
votes
1answer
92 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 $k_{...
8
votes
2answers
228 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
102 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 in:http://arxiv.org/pdf/...
2
votes
3answers
148 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
680 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
263 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 non-...
24
votes
2answers
535 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
206 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 $f_i:...
4
votes
2answers
432 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 3-...
21
votes
3answers
846 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 $F\colon\, \Sigma\rightarrow M$...
2
votes
3answers
262 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 $U\subset\...
4
votes
1answer
152 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 be $...
2
votes
1answer
130 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 ...
6
votes
2answers
324 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
154 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
340 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 $...
22
votes
7answers
4k 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
251 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 $...
6
votes
2answers
949 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
170 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, $\partial$-...
7
votes
1answer
538 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
93 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
186 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 open ...