# Questions tagged [3-manifolds]

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

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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 (...
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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])
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### Is finding boundary-reducing discs for PL 3-manifolds with boundary pattern computationally efficient?

I am working on a project that involves 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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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### Parametric general position theorem for foliations

The situation is the following: let $M$ be a manifold endowed with a smooth foliation $\mathcal{F}$ of codimension one (suppose orientable, transversely orientable) and let $F_t : S \rightarrow M$ be ...
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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&...
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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 ...
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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 ...
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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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### 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.
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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, ...
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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.
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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 ...
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### Hyperbolic metrics and the general Ahlfors-Bers theorem

Let $M$ be an oriented smooth compact 3-manifold with non-empty boundary and hyperbolizable interior such that all boundary components have genus greater than $1$. Denote $N:={\rm int}(M)$ and HM_{...
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### Difficulty with “On fibering certain 3-manifolds” by Stallings

I am reading the paper "On fibering certain 3-manifolds" by John Stallings and I was hoping someone could help me through a certain detail. In particular, I am confused at the very end of ...
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### Using a 4th dimension to make Seifert surfaces isotopic?

Let $L$ be a link in three manifold $M^3$ and let $F_1$ and $F_2$ be two homeomorphic surfaces in $M$ with $L = \partial F_1 = \partial F_2$. Suppose that $F_1$ and $F_2$ are not isotopic rel ...
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### Is this family of minimal tori compact?

Let $\Sigma$ be a smooth $2$-sphere and let $M = \Sigma \times \mathbb{S}^1$. Fix an integer $n \geq 0$. Is there generic set $\mathcal{S}$ of Riemannian metrics on $\Sigma$ such that the following ...
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### Space of embedded minimal surfaces of fixed genus in a generic $3$-manifold

Let $M^3$ be a closed, connected and oriented smooth $3$-manifold, and fix an integer $g \geq 1$. Is it true that for a generic set of Riemannian metrics on $M$ the set of closed, connected and ...
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### Infinitely many distinct minimal tori

Let $M = \Sigma_g \times \mathbb{S}^1$ be endowed with the product metric, where $\Sigma_g$ is a compact orientable surface of genus $g$ with an arbitrary fixed metric. Is it true that there are ...
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### $0$-surgeries on trefoil and figure-eight

Let $M$ and $N$ be $3$-manifolds obtained by zero-surgery on (left-handed) trefoil and figure-eight knot respectively. What is the easy way to prove that $M$ and $N$ are not homeomorphic? Note: When ...
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### Toroidal Heegaard splittings

Suppose I have a Heegaard splitting of a closed oriented irreducible 3-manifold $M$, defined by the Heegaard diagram $(\Sigma_{g},\{\alpha_{1},\dots,\alpha_{g}\},\{\beta_{1},\dots,\beta_{g}\})$. Are ...
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### JSJ-type decompositions for knots

According to Wikipedia, JSJ decomposition for 3-manifolds is the following statement: "Irreducible orientable closed (i.e., compact and without boundary) 3-manifolds have a unique (up to isotopy) ...
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### Integer surgeries along links yielding lens spaces

Does there exist an integer $N$ such that any lens space $L(p,q)$ can be obtained by integer surgery from $S^3$ along a link $L$ with at most $N$ components? EDIT: I have worked out the comment by ...
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### Manifolds with boundary admitting no closed embedded minimal hypersurface

The following Theorem is proved in the paper entitled "Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex ...
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### Negative surgeries on negative knots

This question is two-fold. The first question is rather specific: what are some small examples of negative surgeries on negative knots that give rise to the same 3-manifold? I know one class of ...
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### 3-manifold foliated by circles is Seifert fibered

Let $M$ be a compact 3-manifold with boundary equipped with a 1-dimensional foliation all of whose leaves are circles. An old theorem of Epstein says that $M$ is a Seifert fibered space. The proof of ...
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I am looking for a bit of orientation with regards to computational topology resources, as I am personally totally ignorant on the subject. I have lots of different links in $S^3$ (hundreds of ...
Let $K_0$ and $K_1$ be knots in $S^3$. They are called smoothly concordant if there is a smoothly properly embedded cylinder $S^1 \times [0,1]$ in $S^3 \times [0,1]$ such that \$\partial (S^1 \times [...