Questions tagged [3-manifolds]

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

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3 votes
0 answers
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4-manifold obtained from a ribbon disk exterior by attaching a 2-handle is simply-connected if its boundary is a homology sphere

I am reading Lemma 2.1 of this paper (https://arxiv.org/pdf/2012.12587.pdf) and I can't see why $W$ is simply-connected. Here is the situation: Let $K$ be a ribbon knot in $S^3$; it bounds a ribbon ...
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3 votes
0 answers
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Integral homology $S^1\times S^2$'s smoothly bounding integral homology $S^1\times B^3$'s

Suppose we are given a compact orientable 3-manifold $M$ which is an integral homology $S^1\times S^2$. Then is there a way to determine whether $M$ bounds a smooth compact orientable 4-manifold which ...
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2 votes
2 answers
255 views

Pullback of $w_1$ for 3-manifolds

Given closed $3$-manifolds $M$ and $N$ and an element $\alpha\in H^1(M;\mathbb{Z}_2)$, when does there exist a map $f:M\to N$ such that $\alpha=f^*(w_1(N))$?
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7 votes
0 answers
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The works of González-Acuña and Duchon from 70s and 80s

I would like to access the following two works of González-Acuña from around nineteen-seventies: González-Acuña, F. Dehn’s construction on knots. Bol. Soc. Mat. Mexicana (2) 15 (1970), 58–79. and ...
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6 votes
1 answer
118 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 ...
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3 votes
1 answer
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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 ...
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2 votes
0 answers
162 views

When does a map between 4-manifolds map boundary to boundary upto homotopy?

Let $f:M\to N$ be a smooth map between smooth 4-manifolds with boundary. When does $f$ map boundary of $M$ to boundary of $N$ upto homotopy i.e. when there is a map $F:M\to N$ homotopic to $f$ such ...
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3 votes
1 answer
86 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 \...
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  • 31
7 votes
2 answers
196 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,...
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6 votes
0 answers
221 views

The figure eight knot complement in $S^3$

Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. In this book, there is an exercise in chapter 5, section 5.6, and exercise number 5.4 (Page - 101). This exercise ...
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2 votes
1 answer
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0-framed smoothly slice knot that can be obtained by blowing down successively a link of unknots

A knot in $S^3$ is called a smoothly slice knot if it bounds a smoothly embedded 2-disk in $D^4$. Every ribbon knot is known to be a smoothly slice knot, and there are known some nontrivial smoothly ...
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3 votes
0 answers
150 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 ...
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11 votes
1 answer
303 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 ...
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3 votes
0 answers
98 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 (...
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4 votes
1 answer
100 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 ...
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3 votes
1 answer
54 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 ...
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6 votes
1 answer
168 views

Heegaard Floer homology of a genus two Heegaard splitting of $S^3$

This is a duplicate of a question (https://math.stackexchange.com/questions/4416204/heegaard-floer-homology-of-a-genus-two-diagram-of-s3) on stackexchange, which did not get any answer. Feel free to ...
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8 votes
1 answer
261 views

Universal covers of non-prime 3-manifolds

Let $M$ be a closed, connected, oriented 3-manifold. If $M$ is prime, then we know what the universal cover of $M$ looks like: it is either $S^3, \mathbb{R}^3$ or $S^2 \times \mathbb{R}$ depending on ...
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1 vote
1 answer
59 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 $\...
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11 votes
0 answers
148 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 ...
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3 votes
0 answers
46 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 $M$ ...
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8 votes
2 answers
196 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 ...
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6 votes
1 answer
233 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 $Ric(M)$ ...
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5 votes
1 answer
200 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 ...
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3 votes
0 answers
168 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 ...
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4 votes
1 answer
251 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 ...
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3 votes
1 answer
175 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 (...
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5 votes
1 answer
259 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?
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  • 321
5 votes
1 answer
179 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 ...
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9 votes
1 answer
329 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 ...
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3 votes
1 answer
102 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 ...
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  • 435
14 votes
3 answers
782 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=...
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5 votes
1 answer
202 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 ...
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6 votes
1 answer
342 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 ...
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7 votes
1 answer
225 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 ...
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13 votes
1 answer
505 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. ...
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12 votes
0 answers
202 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 ...
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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 $\...
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5 votes
1 answer
151 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 $\...
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  • 2,462
3 votes
2 answers
239 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) ...
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  • 321
2 votes
0 answers
135 views

Counterexample to mostow rigidity theorem

I am looking for an example of $M$ and $N$ two orientable hyperbolic complete without boundary 3-manifolds ( with infinite hyperbolic volume) such that $\pi_{1}M\cong \pi_{1}N$ but $M$ is not ...
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  • 321
21 votes
2 answers
3k 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
180 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 ...
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2 votes
1 answer
141 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?
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3 votes
1 answer
100 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?
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  • 435
4 votes
0 answers
136 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 $U$...
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  • 17k
3 votes
0 answers
96 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 ...
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3 votes
1 answer
129 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 ...
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  • 125
4 votes
1 answer
94 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 $G$...
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7 votes
1 answer
165 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 ...
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