# Questions tagged [3-manifolds]

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

467
questions

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votes

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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 ...

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vote

**0**answers

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

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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

**2**answers

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**

votes

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**2**answers

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

**2**answers

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

**1**answer

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

**1**answer

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 ...

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votes

**0**answers

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 $...

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votes

**0**answers

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

**1**answer

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

**0**answers

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 ...

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votes

**0**answers

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

**1**answer

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 ...

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votes

**0**answers

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 ...

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votes

**0**answers

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

**1**answer

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

**5**answers

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

**2**answers

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 ...

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votes

**0**answers

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) ...

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vote

**0**answers

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 ...

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votes

**1**answer

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

**1**answer

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 &...

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votes

**1**answer

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?

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vote

**0**answers

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 ...

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votes

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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 ...

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votes

**2**answers

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

**0**answers

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

**2**answers

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

**2**answers

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

**1**answer

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, ...