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On trivial mapping class group of 3-manifolds

What are some examples of knots $K\subset S^3$ such that the mapping class group of $S^3_{1/n}(K)$ is trivial? I guess for hyperbolic knots with no symmetry in the complements are good candidate as ...
Anubhav Mukherjee's user avatar
3 votes
1 answer
239 views

Does the Weeks manifold have the smallest volume among all finite volume oriented hyperbolic 3-manifolds?

It is a result of Chinburg-Friedman-Jones-Reid that the arithmetic hyperbolic 3-manifold of smallest volume is the Weeks manifold. There is also a result of Milley that says that if $N$ is a closed ...
Colby's user avatar
  • 33
3 votes
1 answer
173 views

Do taut foliations leafwise branch covering S^2 yield foliations by circles?

In this paper, Danny Calegari shows that taut foliations in (let's say closed for simplicity) 3-manifolds are precisely those which admit a map $f: M \to S^2$ which restricts to a branched cover on ...
Audrey Rosevear's user avatar
6 votes
2 answers
395 views

Dual surfaces of a first cohomology class of a 3-manifold

Let $M$ be closed 3-manifold and $\alpha\in H^1(M;\mathbb Z_2)$ an arbitrary element. (In my case we know that $M$ is non-orientable and $\alpha^3=0$.) It is well known that there is a closed 2-...
Andrey Ryabichev's user avatar
10 votes
0 answers
288 views

Contact structures associated to taut foliations

Eliashberg and Thurston showed that a taut foliation may be deformed to tight (positive and negative) contact structures. Vogel proved that for a taut foliation without torus leaves, the associated ...
Ian Agol's user avatar
  • 68.9k
10 votes
3 answers
493 views

Small examples of exceptional hyperbolic Dehn Filling of hyperbolic manifolds

For experimental purposes, I would like to have a small (i.e. triangulated with few tetrahedra) example of a manifold $M$ with the following properties: $M$ is a hyperbolic manifold with finite ...
user126154's user avatar
9 votes
1 answer
296 views

The works of González-Acuña and Duchon from 70s and 80s

I would like to access the following two works of González-Acuña from 1970: González-Acuña, F. Dehn’s construction on knots. Bol. Soc. Mat. Mexicana (2) 15 (1970), 58–79. and González-Acuña, F. On ...
Oğuz Şavk's user avatar
  • 1,292
6 votes
1 answer
184 views

Uniqueness of the set of decomposing spheres in prime decomposition of a 3-manifold

At the end of Section 1.1 of 3-manifold groups it is written that "the decomposing spheres are not unique up to isotopy, but two different sets of decomposing spheres are related by ‘slide ...
LaFede's user avatar
  • 63
3 votes
1 answer
75 views

Conjugacy of topological actions on aspherical three manifolds to isometric actions

Edited: Due to work of Raymond and Scott, there exist diffemorphisms (of certain three-dimensional nil-manifolds) whose $n$th power is diffeotopic to the identity, but which are not themselves ...
Nicolas Boerger's user avatar
3 votes
1 answer
187 views

Morse functions inducing Heegaard diagrams

Let $(\Sigma, \alpha, \beta)$ be a Heegaard diagram for a 3-manifold $M$, corresponding to a Heegaard splitting $M = H_1 \cup_\Sigma H_2$. There may be many self-indexing Morse functions $f: M \to \...
Pepijn's user avatar
  • 31
7 votes
2 answers
366 views

Boundary of a $4$-manifold and the fundamental group

I am trying to learn $4$-manifolds with boundaries and I don't know much about this topic so these questions may be silly. Given a $4$-manifold $M$ with a boundary say $N$, Assume $\pi_1(N)$ is known,...
piper1967's user avatar
  • 1,177
8 votes
0 answers
432 views

The figure eight knot complement in $S^3$

Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. Exercise 5.4 (on page 101) gives us a presentation of the fundamental group of $S^3 - K$ where $K$ is the figure-...
T ghosh's user avatar
  • 111
3 votes
0 answers
223 views

Can Whitehead manifold admit a properly discontinuous cocompact group action?

Can classical contractible manifolds such as Whitehead manifold admit a properly discontinuous cocompact group action? Here "properly discontinuous" doesn't have to be fixed point free, but ...
Shijie Gu's user avatar
  • 2,083
11 votes
1 answer
386 views

How wild can an open topological 3-manifold be if it has a compact quotient?

Let $M$ be an open, simply connected, 3-manifold. Suppose $M$ admits a properly discontinuous, co-compact topological action by a finitely generated group. Question 1: If $M$ is 1-ended, must it be ...
Agelos's user avatar
  • 1,926
4 votes
0 answers
172 views

Survey or good reference of taut foliations

I am interested in the topology of foliations. In particular, I want to understand taut foliations, or projectively Anosov flows, and Anosov flows. I guess that A. Candel and L. Conlon, Foliations I (...
user473085's user avatar
4 votes
1 answer
177 views

Complex length of geodesic added in hyperbolic Dehn surgery

Suppose $M$ is a cusped finite-volume hyperbolic $3$-manifold, say with a single cusp for simplicity. Following [NZ, Section 4] we can parametrize deformations of the hyperbolic structure with a ...
Calvin McPhail-Snyder's user avatar
3 votes
1 answer
76 views

Given a Heegaard splitting $M = V\cup_F W$, then $V\setminus N(D_1)$ is ambient isotopic to $V\cup N(D_2)$ for a meridian pair $\{D_1,D_2\}$

I sincerely apologize if MathOverflow is not the appropriate place to ask this question. I also tried consulting M.SE but it seems that this question gained little to no interest . Consider a ...
Zest's user avatar
  • 173
1 vote
1 answer
134 views

Annulus theorem for pseudomanifolds

Lets say I take an arbitrary closed and smooth $d$-manifolds $\mathcal{M}$. Now, it is a well-known fact that whenever I take two (sufficiently nice embedded) closed $d$-balls $B_{1}$ and $B_{2}$ in $\...
G. Blaickner's user avatar
  • 1,429
11 votes
0 answers
179 views

Natural knot homology

All knot homology theories I've seen share a flaw: their definitions explicitly use some combinatorial choices (such as a diagram presentation). The coin, however, has two sides and the other one is ...
Mikhail Shkolnikov's user avatar
3 votes
0 answers
59 views

Nonuniqueness of Heegaard surfaces for submanifolds of $S^3$

Let $M^3$ be some compact submanifold of $S^3$ with connected boundary. I am interested in the failure of the analog of Waldhausen's theorem for $M^3$ - namely, I would like examples of such $M$ ...
user101010's user avatar
  • 5,349
9 votes
2 answers
351 views

Gordon's approach: slice knots and contractible $4$-manifolds

Let $K \subset S^3$ be a slice knot. Then it bounds a smooth embedded disk $D \subset B^4$. Let $S^3_{p/q}(K)$ denote a $3$-manifold obtained by $p/q$-surgery on $K \subset S^3$. The following theorem ...
Max Schumann's user avatar
6 votes
1 answer
324 views

Computation of $\pi_1$ for a Mazur manifold and its boundary

If we attach a $4$-dimensional $1$-handle $D^1 \times D^3$ to a $4$-dimensional $0$-handle $B^4$, we obtain $S^1 \times D^3$. The null homologous knot in $S^1 \times S^2$ indicated in the picture ...
Max Schumann's user avatar
3 votes
0 answers
241 views

Standard sutured (?) Heegaard splitting

I am trying to make sense of what is going on in [Cas16] in terms of diagrams. Let me sum up the construction a bit, where $n\leqslant k$ are integers and $b\geqslant 1$ as well. $C_{k,b,n}$ denotes ...
Anthony's user avatar
  • 283
4 votes
1 answer
384 views

A (not existing) self-homeomorphism of the figure eight knot complement

I was recently looking at the figure eight knot complement $M$, as a once-punctured torus bundle over the surface with monodromy $ A=\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix} $ and its ...
Tali's user avatar
  • 111
3 votes
1 answer
363 views

Volume of hyperbolic 3-manifolds with toroidal boundary

A hyperbolic 3-manifold has finite volume if and only if it is either closed or has toroidal boundary and it is not homeomorphic to $T^2\times I$. This statement is from 3-Manifold Groups, page 18 (...
Fredy's user avatar
  • 502
5 votes
1 answer
318 views

Irreducible 3-manifold with boundary of genus greater than 1

Let $M$ be an irreducible 3-manifold with incompressible boundary of genus > 1. When is $M$ homotopy equivalent to an Eilenberg-MacLane space? Or it is never true?
GSM's user avatar
  • 223
5 votes
1 answer
375 views

Integral surgeries on $3$-manifolds

Let $K$ be a knot in $S^3$. Let $N(K)$ be a tubular neighborhood of $K$, a solid torus. On $\partial N(K)$, we may specify a preferred longitude $\lambda$, i.e., a simple closed curve whose linking ...
Max Schumann's user avatar
9 votes
1 answer
424 views

Non-isotopic homology spheres in $S^4$ with equal complements?

Are there two diffeomorphic smoothly embedded homology 3-spheres $M_1^3, M_2^3 \subset S^4$ that have diffeomorphic complements but such that $M_1$ and $M_2$ are not isotopic? I would be interested in ...
user101010's user avatar
  • 5,349
3 votes
1 answer
202 views

How to find the JSJ decomposition in the plumbing tree model of a graph manifold?

A graph manifold can be obtained by plumbing circle bundle over surfaces, where the number in the plumbing tree denotes the Euler number of the bundle (see the picture for an example). The boundary of ...
Faniel's user avatar
  • 673
14 votes
3 answers
1k views

Quotient of solid torus by swapping coordinates on boundary

Let $T$ be the solid 2-torus and let $\sim$ be the equivalence relation on $T$ generated by the relation $\{(\alpha,\beta) \sim (\beta,\alpha) \mid \alpha, \beta \in S^1\}$ on the boundary $\partial T=...
Bipolar Minds's user avatar
6 votes
1 answer
370 views

"Classification" of (orientable) 3-manifolds with genus-g-surface as their boundary

This is in some sense a generalization of the question I asked some time ago. I am very sorry if this question is too basic for MathOverflow, but I just started learning about some more detailed ...
G. Blaickner's user avatar
  • 1,429
7 votes
1 answer
636 views

Knots: locally flat, PL and smooth

In the classical dimension (knots in $S^3$), it is considered standard (I think?) that the following sets are in bijective correspondence: locally flat knots up to ambient isotopy; PL-knots up to PL ...
Anthony Conway's user avatar
7 votes
1 answer
354 views

Decomposition of manifolds with toroidal boundary

Let $\mathcal{M}$ be a compact, connected, oriented 3-manifolds with non-empty connected boundary $\partial\mathcal{M}$. Then, following this article, it is stated that $\mathcal{M}$ can be written as ...
G. Blaickner's user avatar
  • 1,429
14 votes
1 answer
937 views

Classification of 3-dimensional manifolds with boundary

It is well-known that every closed, connected and orientable 3-manifold $\mathcal{M}$ can uniquely be decomposed as $$\mathcal{M}=P_{1}\#\dots\# P_{n}$$ where $P_{i}$ are prime manifolds, i.e. ...
G. Blaickner's user avatar
  • 1,429
12 votes
0 answers
229 views

3-manifolds with stacked links

Stacked spheres A triangulation of a 2-dimensional sphere is called a stacked sphere if it is obtained inductively from the boundary of a 3-simplex by deleting a 2-face (triangle) $T$ adding a new ...
Gil Kalai's user avatar
  • 24.7k
17 votes
3 answers
2k views

Examples of the Thurston geometries with transitive Lie group action

Here are some examples of compact homogeneous 3 manifolds for different Thurston geometries: (1) Spherical: $\mathbb{S}^3 \cong \mathrm{SU}_2$ modulo any finite subgroup (2) Euclidean: 3 torus $\...
Ian Gershon Teixeira's user avatar
3 votes
2 answers
344 views

Hyperbolic volume of hyperbolic knots

Let $G$ be a torsionfree kleinian group. Is there necessary an sufficient conditions on $G$ to be a knot group ? It seems that there is some necessary conditions: $H_{1}(BG) = \mathbb{Z}$ $H_{2}(BG) ...
GSM's user avatar
  • 223
23 votes
2 answers
7k views

Shing-Tung Yau's doubts about Perelman's proof

[EDITED to make the question more suitable for MO. See meta.mathoverflow.net for discussion about re-opening.] According to Wikipedia, Shing-Tung Yau expressed some doubts about Perelman's proof of ...
5 votes
1 answer
249 views

Which elements of the fundamental group can be realized as transversals of a taut foliation?

Specifically in a closed, orientable 3-manifold. I'm not necessarily looking for a complete answer, as I don't expect one. Is there prior literature on this question? Also interested in this question ...
Audrey Rosevear's user avatar
2 votes
1 answer
176 views

A graph manifold without an orientation reversing involution?

Is there a graph manifold (https://en.wikipedia.org/wiki/Graph_manifold) that doesn't admit an orientation reversing involution? If so, what would be a simple example?
aglearner's user avatar
  • 14.3k
3 votes
1 answer
157 views

Formula for the Casson invariant in terms of the linking form

The paper 'Trisections, intersection forms and the Torelli group' by Peter Lambert-Cole quotes the following formula for the Casson invariant of a knot $K$ in a homology $3$-sphere in terms of the ...
Filippo Bianchi's user avatar
3 votes
1 answer
152 views

Reference request: Stallings-Epstein-Waldhausen construction

I am looking for a reference for the Stallings-Epstein-Waldhausen construction (constructing an incompressible surface in a 3-manifold from a nontrivial splitting of the fundamental group). I know of ...
wandersam's user avatar
  • 125
4 votes
1 answer
195 views

Ideal triangulation of hyperbolic 3-manifold with generic mapping class group

I am from physics background so I apologize in advance if my question is trivial. Kojima proves for every finite group $G$, there is a hyperbolic 3-manifold such that its mapping class group equals $G$...
Zhengdi Sun's user avatar
7 votes
1 answer
249 views

Expositions of Stallings's fibration theorem

In his famous paper Stallings, John, On fibering certain 3-manifolds. 1962 Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) pp. 95–100 Prentice-Hall, Englewood ...
Laura's user avatar
  • 353
7 votes
1 answer
372 views

Two details from Stallings's proof of the sphere theorem

EDIT: After a little prompting by Mark Grant, I answered the first question in the comments. The second question remains open. Let $M$ be a compact $3$-manifold with $\pi_2(M) \neq 0$. The sphere ...
Laura's user avatar
  • 353
4 votes
2 answers
306 views

Quadratic cusp shape

Which hyperbolic $3$-manifolds are known to have quadratic cusp shape? Explanations: Cusps of hyperbolic $3$-manifolds are products torus x interval. They lift to horoballs in hyperbolic $3$-space, ...
ThiKu's user avatar
  • 10.4k
12 votes
1 answer
679 views

Revisiting Gordon-Luecke theorem

$\DeclareMathOperator\Diff{Diff}\DeclareMathOperator\GL{GL}$Here is an proof-sketch of a strengthened Gordon-Luecke theorem. This is presumably known, but is it written down somewhere? I am also ...
Ryan Budney's user avatar
  • 44.4k
8 votes
2 answers
283 views

Is there a simple formula to compute the Casson invariant of an homology $3$-sphere from its Heegaard diagram?

Let $(S_g,\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2)$ be a Heegaard diagram of a Heegaard splitting $\Sigma=H_g \cup_{\phi_1\phi_2^{-1}}H_g$ of an integral homology sphere $\Sigma$, i.e. $S_g$ is a ...
Filippo Bianchi's user avatar
3 votes
1 answer
115 views

$\pi_1(M^3)$ containing a normal infinite cyclic subgroup

Let $M^3$ be a compact $3$-manifold such that $\pi_1(M)$ contains a normal subgroup isomorphic to $\mathbb Z$. Can we show either $\pi_1(M)$ is torsion-free or $\pi_1(M)=\mathbb Z \oplus \mathbb Z_2$ ...
Zhiqiang's user avatar
  • 891
3 votes
2 answers
216 views

$P^2$-irreducibility of a $3$-manifold

A $3$-manifold $M$ is called $P^2$-irreducible if it is irreducible and there is no $2$-sided $P^2$ contained in $M$. Can we show $M$ is $P^2$-irreducible iff $\pi_2(M)=0$? Notice that one direction ...
Zhiqiang's user avatar
  • 891

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