Questions tagged [3-manifolds]

A three-manifold is a space that locally looks like Euclidean three-dimensional space

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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 $\...
7 votes
1 answer
317 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 ...
7 votes
1 answer
684 views

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 votes
1 answer
299 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 ...
3 votes
0 answers
224 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 ...
6 votes
1 answer
522 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 ...
9 votes
1 answer
392 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 ...
5 votes
1 answer
435 views

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 votes
1 answer
339 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 ...
3 votes
1 answer
264 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 (...
5 votes
1 answer
299 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?
5 votes
1 answer
293 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 ...
18 votes
4 answers
3k views

Classification of homology 3-spheres?

Is there some general description of all homology 3-spheres?
5 votes
1 answer
230 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 ...
5 votes
1 answer
308 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 ...
2 votes
2 answers
368 views

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 votes
1 answer
172 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 ...
14 votes
3 answers
952 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=...
13 votes
1 answer
808 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. ...
5 votes
2 answers
369 views

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 votes
0 answers
222 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 ...
23 votes
2 answers
5k 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
181 views

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 votes
0 answers
155 views

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 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) ...
2 votes
1 answer
161 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?
3 votes
1 answer
172 views

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 votes
14 answers
11k 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 ...
4 votes
2 answers
276 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, ...
24 votes
2 answers
3k views

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 votes
1 answer
224 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 ...
8 votes
0 answers
150 views

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
147 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 ...
4 votes
1 answer
161 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 $G$...
7 votes
1 answer
354 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 ...
3 votes
1 answer
412 views

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 votes
3 answers
1k 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 this ...
11 votes
1 answer
576 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 ...
4 votes
1 answer
98 views

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 votes
1 answer
724 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 ...
1 vote
0 answers
67 views

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 votes
1 answer
182 views

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....
8 votes
2 answers
259 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 ...
3 votes
2 answers
189 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 ...
4 votes
1 answer
371 views

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
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 $\pi_1(M)=\mathbb Z \oplus \mathbb Z_2$ ...
3 votes
1 answer
461 views

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 votes
3 answers
718 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)
3 votes
1 answer
173 views

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 votes
1 answer
196 views

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?

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