Questions tagged [3-manifolds]
A three-manifold is a space that locally looks like Euclidean three-dimensional space
608
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The moduli space of finite volume hyperbolic 3-manifolds?
By finite volume hyperbolic 3-manifold, I do mean $M=\mathbb{H}^{3}/\Gamma$ where $\Gamma$ is a torsion-free Kleinian group such that the hyperbolic volume $Vol(M)<\infty$.
I will call
$$\mathcal{M}...
9
votes
1
answer
452
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Noncompact three-manifold with fundamental group isomorphic to a surface group
Let $M$ be an open orientable three-manifold such that $\pi_1 (M)$ is isomorphic to the fundamental group of a closed orientable surface $S\ncong \mathbb{S}^2$. Furthermore, suppose that $\tilde{M} \...
6
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0
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218
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Classifying nested 3-manifolds with fundamental group property
Let $M_1\subseteq M_2\subseteq\mathbb R^3$ be closed connected subsets with smooth boundary. Suppose that every closed loop in $M_1$ is freely homotopic inside $M_2$ to a closed loop inside $M_2\...
7
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1
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330
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Are there non-homeomorphic 3-manifolds with the same Turaev-Viro-Barrett-Westbury invariants?
The Turaev-Viro-Barrett-Westbury invariant of a closed oriented topological $3$-manifold $M$ for a spherical fusion category $\mathcal{C}$ is a number denoted $|M|_{\mathcal{C}}$ computed from (but ...
4
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0
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114
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Surgery diagrams of 3-dim abstract open books
Starting with an abstract open book $(\Sigma,\phi)$, I would like to understand some of the manifolds that I could obtain. Given a surface $\Sigma$ and monodromy $\phi$, it is not hard to find a Kirby ...
7
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Balanced presentation of the fundamental group of a Seifert fiber space
Is there any readily available reference for a balanced presentation of the fundamental group in terms of the classifying invariants of an arbitrary Seifert fibered space? I can't find it, and the ...
5
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76
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Transverse open book decompositions supporting the same contact structure
An open book decomposition on an oriented 3-manifold $M$ is a fibered oriented link $B\subset M$, bounding a foliation by Seifert surfaces $\Sigma_t \subset M$, $t\in S^1$, $\partial \Sigma_t = B$.
A ...
3
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Zagier's "From 3-manifold invariants to number theory"?
Zagier lectures on "From 3-manifold invariants to number theory" - do you know about texts of that or on the discussed web of ideas? ([https://www.mpim-bonn.mpg.de/de/node/10791])
4
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0
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3-manifold proof of Grushko's theorem
Grushko's theorem says that given an epimorphism $\phi: F \to G_1 * G_2$ where $F$ is a finitely-generated free group, there exists. subgroups $F_1$ and $F_2$ of $F$ so that $F = F_1 * F_2$ and $\phi(...
1
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0
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Circle-valued Morse function and minimal genus
I think the following two statements are true, and most likely are in the literature. If so, could someone point me to some references? If not, counterexamples?
Let $Y$ be a closed oriented connected ...
5
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333
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Thurston universe gates in knots: which invariant is it?
Today I discovered this nice video of a lecture by Thurston:
https://youtu.be/daplYX6Oshc
in which he explains how a knot can be turned into a "fabric for universes". For example, the unknot ...
3
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0
answers
56
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What's the Milnor's link group for the trivial knot in a lens space?
For a link $L$ in a 3-manifold $Y$, Milnor's paper "Link Groups" https://link.springer.com/content/pdf/10.1007/BF01393902.pdf defined the link group as some quotient of $\pi_1(Y-L)$. If $L$ ...
1
vote
1
answer
74
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Surface in a product domain
Let $M$ be a connected closed surface. Suppose $N$ is a connected closed surface embedded in the interior of $M \times [0,1]$ such that $N$ separates $M \times \{0\}$ and $M \times \{1\}$. Can we ...
7
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341
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3-dimensional h-cobordisms
Let $W$ be a $3$-dimensional $h$-cobordism of closed surfaces $M_0$ and $M_1$. Can we prove that $W$ is trivial? That is, $W$ is homeomorphic to $M_0 \times [0,1]$.
0
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2
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102
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Tightness/Overtwistedness of genus one open book decomposition
Suppose we have an open book decomposition $(P,\phi)$ of a 3-manifold $Y$, where $P$ is a punctured torus and $\phi$ is the monodromy. We know $\phi$ can be represented by a matrix in $SL(2,\mathbb{Z})...
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118
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Open cone homeomorphic to the Euclidean space
Let $X$ be a topological space and the open cone $C(X)$ over $X$ is defined to be $X \times [0,1)$ with $X \times \{0\}$ identified. Suppose $C(X)$ is homeomorphic to $\mathbb R^4$, can we prove that $...
4
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1
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Reference sought for Seifert fiber spaces
I seek a reference for what is surely a well known basic result about Seifert fibered 3-manifolds. Namely they are all obtained by Dehn-surgery along a regular Seifert fiber (and the surgery slope is ...
1
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0
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79
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Do different decompositions of Dehn twists induce the same cobordisms between mapping tori?
Suppose $Y=T^2\times I/f$ is a mapping torus, where $f$ is a diffeomorphism of $T^2$. Suppose $c$ is a curve on $T^2\times \{1\}$ with surface framing and suppose $D_c$ is the positive Dehn twist ...
2
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131
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Möbius cross energy in $S^3$?
Let $\gamma_i$, $i=1,2$ be two loops in $\mathbb R^3$. The Möbius cross energy of the pair is defined by
$$
E(\gamma_1, \gamma_2)=\iint_{S^1\times S^1}\frac{|\gamma'_1(u)|\cdot|\gamma'_2(v)|}{|\...
7
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Virtually large groups of small rank (related to 3-manifolds)
Edited 25.05.21: the assumptions of the question were incorrect, but as the discussion may be helpful for future MOnauts, I'll strike my mistakes and add clearly marked explanations afterwards.
I am ...
10
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2
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438
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Presentations of mapping class groups in dimension $3$
For any closed oriented surface $M$, its mapping class group $MCG(M)$ can be generated by Dehn twists along certain curves on $M$. A presentation for the group $MCG(M)$ was found in [1] and then ...
4
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211
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Mapping class group of a twisted I-bundle over $RP^2$
$\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Homeo{Homeo}$Let $\Mod(M)=\pi_0(\Homeo(M))$ be the mapping class group of a manifold, possibly with boundary (I'm including the orientation reversing ...
4
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119
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Is finding boundary-reducing discs for PL 3-manifolds with boundary pattern computationally efficient?
I am working with manifolds with boundary pattern, as defined by Matveev in his book Algorithmic Topology and Classification of 3-Manifolds. A manifold with boundary pattern is a pair $(M, P)$ where $...
5
votes
1
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233
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Functoriality of Thurston's norm
Let $M$ be a manifold of dimension $3$ and let $N$ be an embedded submanifold of $M$ (also of dimension $3$).
Then, both second homologies $H_2(M)$ and $H_2(N,\delta N)$ are equipped with a norm (...
15
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1
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How to get 3-manifold, Knots from Number Fields
I'm reading a paper On the Torsion Jacquet-Langlands correspondence by Akshay Venkatesh and Frank Calegari.
Truthfully speaking I have no idea what Jacquet-Landlands is. I'm just trying to ...
6
votes
1
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423
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stabilization of Legendrian knots
There are two ways to stabilize a Legendrian knot $k$ in standard contact sphere $(S^3,\xi_{st})$ i.e. adding right cusps or left cusps, let's call these two stabilized Legendrian knots $k_R$ and $k_{...
2
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0
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173
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The Kirby diagram of a manifold glued along the lens space $L(p,1)$
Suppose $K$ is a knot in $S^3$ with any framing and $m=m_0$ is its meridian with $-1$ framing. Suppose $m_1,\dots,m_{p-1}$ are unknots with framings $-2$, such that $m_{i-1}$ and $m_i$ are linked as a ...
7
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4
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Do you know how to construct a compact hyperbolic 3-manifold with three or four totally geodesic boundary components?
Do you know how to construct a compact hyperbolic 3-manifold with three or four totally geodesic boundary components? The only constructions I could find have one boundary component. A reference ...
1
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0
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272
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Boundary map in Mayer-Vietoris sequence of cohomology
Suppose $M$ is a 3-manifold with connected boundary. Let $T$ be a tangle in $M$, i.e., $T$ is a embedded connected 1-submanifold whose boundary is on $\partial M$ ($T$ is not closed). Moreover, ...
7
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1
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337
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Decidability of knot equivalence in general 3-manifolds? Surface equivalence?
Given a closed orientable 3-manifold $M^3$ and two knots $K_1$ and $K_2$ in $M$, is there an algorithm to decide if $K_1$ and $K_2$ are isotopic? Is there an algorithm to decide if there is a ...
0
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0
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361
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References on Hyperbolic Geometry and Teichmuller Theory
I am asking a soft question here.
I am learning hyperbolic geometry on my own. Recently, I have completed the book "Fuchsian Groups" by Svetlana Katok. Also, I have background in Lee's three ...
2
votes
2
answers
295
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References on Riemann surfaces
I have asked the question in MSE, but did not get an answer.
I am asking a soft question here. I am interested in learning about Hyperbolic Geometry. I have read the book named "Fuchsian Groups&...
8
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1
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339
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Outer automorphism group of Brieskorn homology sphere?
In this post, it is discussed how a Brieskorn homology sphere $\Sigma(a_1,a_2,a_3)$ with $\displaystyle \frac{1}{a_1}+ \frac{1}{a_2}+ \frac{1}{a_3} < 1$ is an aspherical manifold with a ...
12
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1
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Wanted: a nontrivial weakly inadmissible Heegaard diagram
This is a question asked by a student in my lecture. After drawing pictures for awhile, I thought it was a good one. I seek a nontrivial example of a pointed Heegaard diagram $(\Sigma,\mathbf{\alpha},\...
15
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3
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Vector field on 3-sphere
Let $V$ be a vector field on $S^3$ such that its singularity points, namely the points at which the vector field vanishes, are only sinks or sources (i.e. the field is converging or diverging). Is ...
2
votes
1
answer
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Weakly relatively hyperbolicity and asymptotic cone
Drutu, Sapir, Osin showed that
a finitely generated group $G$ is strongly hyperbolic relative to a finite collection $\mathcal{H}$ of subgroups if and only if any asymptotic cone is tree-graded with ...
10
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Where was it first shown that every homotopy self-equivalence of $S^1\times S^2$ is homotopic to a homeomorphism?
The claim in the title is proved on pp.19-20 of Topological rigidity for non-aspherical manifolds
by M. Kreck and W. Lueck. Is there an earlier (classical) reference?
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1
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218
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Existence of a geometric structure on a solid torus
I suppose the solid torus in $\mathbb{R}^3$ is not a geometric manifold. Since I am not an expert in this area, I would like to ask whether there is some easy way to see this.
6
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What is known about a 3-manifold $M$ when its fundamental group is linear?
Suppose we have a 3-manifold $M$ and its respective fundamental group $\pi_1(M)$. An important question about its fundamental group is to ask if it is linear, i.e. they are isomorphic to a subgroup of ...
8
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2
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Quantitative word problem for 3-manifold groups
The word problem for 3-manifold groups is solvable: given a based loop $\gamma$ in $M^3$ there is an algorithm to decide whether $\gamma$ bounds a disk.
What kinds of quantitative results are known ...
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2
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Why does not a closed 3-manifold modelled on SL(2,R) admit a metric of nonpositive curvature?
I was reading the paper `actions of discrete groups on nonpositively curved spaces' written by Kapovich and Leeb.
In this paper, they proved that generic mapping class groups are not Hadamard groups, ...
0
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1
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240
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Invariant knot for finite group actions on $S^3$
Inspired by the Smith conjecture, is there a finite group action on $S^3$ (by smooth or analytic diffeomorphisms) which possesses an invariant knotted circle?
6
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0
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224
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Is a compact aspherical 3-manifold irreducible
Let $M^3$ be a compact $3$-manifold (possibly with boundary). Suppose $M$ is aspherical, can we show that $M$ must be irreducible? Here, irreducible means any embedded sphere in $M$ bounds a $3$-ball.
6
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2
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755
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Higher homotopy groups of irreducible 3-manifolds
A 3-manifold $M$ is irreducible if every embedded 2-sphere bounds a 3-ball. Thanks to Papakyriakopoulos's sphere theorem, irreducibility is the same as having $\pi_2(M)=0$. Does irreduciblity imply ...
5
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2
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843
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3-manifold with fundamental group $\mathbb Z$
Let $M$ be a compact $3$-manifold with nonempty boundary. If $\pi_1(M)=\mathbb Z$, can we prove that $M$ is homeomorphic to $S^1 \times D^2$?
4
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1
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193
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Seifert fiber space with homotopically trivial generic fiber
Let $X$ be a Seifert fiber space, that is, a 3-manifold which is a circle bundle over a 2-orbifold. Suppose all generic fiber of $X$ is homotopically trivial, can we prove that the universal cover of $...
3
votes
1
answer
288
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Circle bundle with homotopically trivial fiber in the total space
Consider a smooth circle fiber bundle
$$
S^1 \to E\to B
$$
where $E$ is a smooth 3-manifold and $B$ is a smooth surface. Assuming any $S^1$ fiber in $E$ is homotopically trivial, can we prove that $E$ ...
5
votes
3
answers
497
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Triangulations of 3-manifolds in Regina and SnapPy
I have been doing some statistical studies on small 3-manifolds, and I note that one can produce larg-ish censuses of triangulations in Regina. Now, the Regina documentation tells us how to convert a ...
5
votes
2
answers
423
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Regular or h-regular CW-complex structure for the Poincaré homology sphere
I am looking for a regular (the characteristic maps of the cells are homeomorphisms) or h-regular (the characteristic maps of the cells are homotopy equivalences) CW-complex structure for the Poincaré ...
5
votes
1
answer
272
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Smoothening pseudo-Anosov flows
A topological Anosov flow on a closed 3-manifold can be replaced by a smooth Anosov flow using an argument of Fried: use Markov partitions to find a surface of section, put in other terms, one can ...