Questions tagged [3-manifolds]
A three-manifold is a space that locally looks like Euclidean three-dimensional space
480
questions
1
vote
0answers
102 views
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 ...
3
votes
1answer
235 views
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
votes
0answers
41 views
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
0answers
90 views
Smooth Schoenflies Theorem for compact $3$-manifolds
Let $M^3$ be a compact $3$-manifold with $\partial M=N$ a connected surface. Suppose one has a smooth embedding of $N$ into the interior of $M$ and $N$ bounds a domain $D$ in $M$. Can we show that $D$ ...
1
vote
1answer
64 views
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
votes
1answer
175 views
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
votes
2answers
48 views
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})...
1
vote
0answers
81 views
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
votes
1answer
96 views
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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0answers
60 views
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 ...
1
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0answers
123 views
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)|}{|\...
9
votes
2answers
245 views
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 ...
3
votes
0answers
153 views
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
votes
1answer
196 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
0answers
156 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
0answers
102 views
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 $...
2
votes
0answers
81 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
vote
0answers
141 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
180 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
0answers
164 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
0answers
58 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
188 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
259 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
votes
1answer
184 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
1answer
156 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
352 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
0answers
103 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
421 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
128 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
218 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
583 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
299 views
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 ...
5
votes
2answers
270 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
122 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
175 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
195 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
131 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
votes
0answers
69 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
111 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
66 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
votes
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
311 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
395 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 ...
9
votes
2answers
179 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
votes
0answers
81 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
vote
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 ...
