Questions tagged [3-manifolds]

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

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3
votes
1answer
164 views

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 (...
3
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0answers
117 views

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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0answers
79 views

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 ...
2
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0answers
60 views

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 ...
1
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0answers
132 views

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, ...
6
votes
1answer
174 views

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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0answers
154 views

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
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0answers
53 views

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 ...
2
votes
2answers
182 views

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
votes
1answer
254 views

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 ...
2
votes
1answer
64 views

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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1answer
176 views

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?
-2
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1answer
155 views

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.
8
votes
2answers
336 views

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, ...
6
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0answers
99 views

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
votes
2answers
407 views

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 ...
4
votes
1answer
125 views

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
1answer
217 views

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
2answers
579 views

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 $S^1 \times D^2$?
7
votes
1answer
223 views

Virtually large groups of small rank (related to 3-manifolds)

I am looking for a reason why a 3-manifold group $G$ that is virtually $\mathbb{Z}\times F$, $F$ being either non-cyclic free or a surface group, does not admit a presentation on two generators. These ...
5
votes
2answers
268 views

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 ...
8
votes
2answers
368 views

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 ...
5
votes
1answer
120 views

Exchanging the components of a two-component link

Given a 2-component link in $S^3$ whose components are trivial knots, is it always possible to find a homeomorphism of $S^3$ that exchanges the components? I guess the answer is "no" (but I ...
6
votes
1answer
174 views

low dimensional manifolds by gluing the boundary of a ball

Recall that one way of drawing closed 2-manifolds is to take a disk $D^2$, take a cellular decomposition of $\partial D^2$, pair the vertices in this cellular decomposition so that the pairing ...
0
votes
0answers
126 views

Homology of a closed $3$-manifold with balls removed

This question has been posted on MSE with no answers. Let $M^3$ be a closed, connected and orientable smooth $3$-manifold and let $\mathring{M}$ denote the manifold $M$ with $n$ disjoint open balls $...
3
votes
0answers
187 views

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_{...
6
votes
1answer
182 views

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 ...
3
votes
0answers
130 views

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 ...
2
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0answers
68 views

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 ...
2
votes
1answer
107 views

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 ...
4
votes
0answers
65 views

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 ...
0
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0answers
54 views

Bipartedly slice links and their surgeries

A link L in $S^3$ is said to be strongly slice if $L=∂D$,where $D$ is a disjoint union of smoothly and properly embedded disks in $B^4$. A link $L$ in $S^3$ is called bipartedly slice if $L = L_1 \cup ...
9
votes
1answer
309 views

A strong form of Mostow rigidity without geometrization?

Mostow rigidity theorem says that two closed hyperbolic manifolds with isomorphic fundamental groups are isometric. Here is my question: suppose that $M$ and $N$ are two closed 3-manifolds such that $...
11
votes
5answers
388 views

$0$-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 ...
8
votes
2answers
175 views

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 ...
0
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0answers
80 views

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) ...
1
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0answers
53 views

Rigidity case of a geometric theorem for $3$-manifolds with boundary

Let $(M^3,g)$ be a compact Riemannian $3$-manifold with boundary. In a paper by L. Ambrozio, he considers the set $\mathcal{F}_M$ of all immersed disks in $M$ whose boundaries are curves in $\partial ...
8
votes
1answer
120 views

Separating spheres in $3$-manifolds of positive scalar curvature and mean convex boundary

Recently, A. Carlotto and C. Li proved a complete topological classification of those compact, connected and orientable $3$-manifolds with boundary which support Riemannian metrics of positive scalar ...
5
votes
1answer
225 views

unlinking when relaxing the homeomorphism condition

Say that we have two knots $K_1$ and $K_2$ in $S^3$ linked together in $S^3$ and forming the Hopf link. Usually, we can prove that we cannot unlink them by using a link invariant that shows that the &...
0
votes
1answer
193 views

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?
1
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0answers
134 views

Covers of a 4-manifold pull back a cohomology class to any algebraic multiple

Fix an algebraic integer $x\neq 0$. Is there a closed smooth 4-manifold $M$ with a class $\rho\in H^{1}_{\mathrm {dR} }(M)$ and a smooth covering map $\phi:M\to M$ such that $\phi^*\rho=x\rho$? Is ...
4
votes
2answers
283 views

Heegard diagrams for three-manifolds

I have a basic question about the Heegaard diagrams involved in providing a framework for calculation of Floer-Homology of three-manifolds. Typically such diagrams look like Figure 1 and Figure 2 here ...
5
votes
1answer
135 views

Dehn surgery on $S^3$ along a Hopf link with rational surgery coefficients

Is there an exhaustive list of conditions satisfied by rational surgery coefficients assigned to the components of the Hopf link in $S^3$ such that the resulting 3-manifold by Dehn surgery acting on $...
7
votes
0answers
204 views

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 ...
5
votes
1answer
145 views

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 ...
5
votes
2answers
218 views

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 ...
11
votes
0answers
128 views

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 ...
3
votes
2answers
149 views

Looking through a bunch of links for unlinks?

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 ...
12
votes
2answers
262 views

Relating smooth concordance and homology cobordism via integral surgeries

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 [...
4
votes
1answer
181 views

Irreducibility of 3-manifolds with (non)empty boundary

All manifolds considered here are compact and orientable. A 3-manifold (with possible boundary) is irreducible if any smooth sphere bounds a ball. Note that a closed irreducible 3-manifold is prime, ...

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