Questions tagged [3-manifolds]
A three-manifold is a space that locally looks like Euclidean three-dimensional space
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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 $\...
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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
...
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Relation between Ricci curvature and sectional curvature for 3-manifolds
Let $(M^n,g)$ be a smooth Riemannian manifold. It is well known that if $sec(M)\geq \kappa$ then $Ric(M)\geq (n-1)\kappa$.
If I understand correctly in dimensions $n\geq 4$ a lower bound on $Ric(M)$ ...
6
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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 ...
3
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0
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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 ...
6
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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 ...
9
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1
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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 ...
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Rotation part of short geodesics in hyperbolic mapping tori
Otal [Sur le nouage des géodésiques dans les variétés hyperboliques. C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 7, 847--852.] showed that "short" simple closed geodesics in 3-dimensional ...
4
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1
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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 ...
3
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1
answer
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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 (...
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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?
5
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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 ...
18
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4
answers
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Classification of homology 3-spheres?
Is there some general description of all homology 3-spheres?
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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 ...
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"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 ...
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2
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Does the coefficient of the meridian determine the coefficient of the longitude?(on Dehn surgery)
I'm studying Dehn surgery, and it says that the coefficient $(p,q)$ which says how the meridian curve on solid torus is attached will determine the entire resulting manifold. I'm wondering whether the ...
3
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1
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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 ...
14
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3
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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=...
13
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1
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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. ...
5
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2
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Does WRT invariant detect hyperelliptic involution on the genus 2 surface?
The Witten-Reshetikhin-Turaev invariant cannot detect the hyperelliptic involution on the genus 1 surface, and that if $M_U$ is the mapping torus for a mapping class group element $U\in \mathrm{Mod}(\...
12
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0
answers
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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 ...
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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 ...
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1
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3-manifolds with all minimal surfaces closed
Question. Let the manifold $(M^3,g)$ be compact without boundary. Suppose that every complete, embedded minimal surface $\Sigma \subset M^3$ is closed. Must $M$ be diffeomorphic to $\mathbf{S}^3$ or $\...
4
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0
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Extension of smooth structure on three dimensional topological manifolds
Let $M$ be a three dimensional compact topological manifold and $U$ an open set in $M$ homeomorphic to some smooth $C^k$ manifold $U'$, $1 \leq k \leq \omega$. Can we extend the smooth structure on $U$...
3
votes
2
answers
324
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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) ...
2
votes
1
answer
161
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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?
3
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1
answer
172
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Stein fillable tight contact structures on the 3-torus
Kanda classified tight contact structures on the 3-torus. Which of them is Stein fillable? Is there any good reference?
61
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14
answers
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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 ...
4
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2
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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, ...
24
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2
answers
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Proofs of Kirby's theorem
Each orientable 3-manifold can be obtained by doing surgery along a framed link in the 3-sphere. Kirby's theorem says that two framed links give homeomorphic manifolds if and only if they are obtained ...
7
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1
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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 ...
8
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0
answers
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Is the number of prime factors of 3-manifolds obtained by Dehn surgery along a link with $N$ components in $S^3$ bounded from above?
For a given $N$, is the number of prime factors of 3-manifolds obtained by Dehn surgery along a link with $N$ components in $S^3$ bounded from above?
The Two Summands Conjecture states that surgery ...
3
votes
1
answer
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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 ...
4
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1
answer
161
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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 $G$...
7
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1
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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 ...
3
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1
answer
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Euler characteristic of pseudomanifolds with boundary
It is a well-known fact that for every compact oriented odd-dimensional manifold $\mathcal{M}$ with boundary it holds that
$$\chi(\mathcal{M})=\frac{1}{2}\chi(\partial\mathcal{M}).$$
In particular, if ...
18
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3
answers
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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 this ...
11
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1
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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 ...
4
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1
answer
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Normal form of framed links under Kirby moves
It's well known that any oriented closed 3-manifold (topological or smooth) can be obtained by surgerizing along a (framed oriented) link $L$ in the 3-sphere $S^{3}$. Even better, Kirby found a ...
5
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1
answer
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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
0
answers
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Isotopy of open book supporting same contact structure
In dimension 3, the Giroux correspondence gives us a bijection between contact structures (up to isotopy) and open book decompositions (up to positive stabilisation). Moreover, Giroux shows that two ...
2
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1
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Do once-punctured torus bundles have integral traces?
Once-punctured torus bundles are a well-studied class of hyperbolic 3-manifolds, but unfortunately I have been unable to find out whether they always have integral traces (in the sense of [1], Def. 5....
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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 ...
3
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2
answers
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$P^2$-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 ...
4
votes
1
answer
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3-manifold with boundary containing a projective plane
Let $M$ be a compact $3$-manifold such that no component of $\partial M$ is $S^2$ and one component $F$ of $\partial M$ is the projective plane.
If $i_*:\pi_1(F) \to \pi_1(M)$ is an isomorphism, can ...
3
votes
1
answer
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$\pi_1(M^3)$ 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 $\pi_1(M)=\mathbb Z \oplus \mathbb Z_2$ ...
3
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1
answer
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Ambiguity in the unoriented knot connected sum
It is well known that there might be ambiguity in the unoriented knot connected sum if the knots concerned are not invertible.
E.g., consider 8_17, the only knot with crossing number 8 which is non-...
9
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3
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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)
3
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1
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Special kind of 3-manifolds
Is there an open connected orientable 3-manifold $M$ with the following properties:
$M$ admits a complete hyperbolic metric with finite hyperbolic volume.
$H_{i}(M,\mathbb{Z})=0$ for any $i>0$.
3
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1
answer
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One-sided incompressible surface in 3-manifolds
Let $M^3$ be a closed orientable $3$-manifold. If $H_2(M,\mathbb Z)=0$ and $H_2(M, \mathbb Z_2)\ne 0$, can we show that $M$ contains a 1-sided incompressible surface?