Questions tagged [3-manifolds]
A three-manifold is a space that locally looks like Euclidean three-dimensional space
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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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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
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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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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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1
answer
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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
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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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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
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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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2
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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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0
answers
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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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answers
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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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1
answer
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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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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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1
answer
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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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1
answer
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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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answer
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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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answer
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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
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1
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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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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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answers
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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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2
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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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1
answer
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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 $Ric(M)$ ...
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answer
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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 ...
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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 ...
4
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1
answer
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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 ...
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3
votes
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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votes
1
answer
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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?
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1
answer
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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 ...
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votes
1
answer
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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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1
answer
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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 ...
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votes
3
answers
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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=...
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1
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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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votes
1
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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 ...
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votes
1
answer
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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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votes
1
answer
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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. ...
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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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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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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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votes
2
answers
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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) ...
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2
votes
0
answers
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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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votes
2
answers
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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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votes
1
answer
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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 ...
2
votes
1
answer
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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?
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answer
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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?
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0
answers
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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 $U$...
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answers
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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
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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 ...
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4
votes
1
answer
94
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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 $G$...
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7
votes
1
answer
165
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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 ...
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