All Questions
Tagged with 3-manifolds gt.geometric-topology
65 questions with no upvoted or accepted answers
37
votes
0
answers
2k
views
What is the three-dimensional hyperbolic volume of a four-manifold?
Every smooth closed orientable 4-manifold may be constructed via a handle decomposition. Before asking a couple of questions, I recall some well-known facts about handle-decompositions of 4-manifolds.
...
- 10.5k
19
votes
0
answers
575
views
The oriented homeomorphism problem for Haken 3-manifolds
Haken famously described an algorithm to solve the homeomorphism problem for the 3-manifolds that bear his name (fleshed out by many others, including Hemion and Matveev who fixed some gaps). But it'...
- 25k
14
votes
0
answers
336
views
Are there exotic twisted doubles of 4-manifolds?
Take a smooth 4-manifold $X$ whose boundary has a diffeomorphism $\tau: \partial X \to \partial X$ that extends to a homeomorphism but not a diffeomorphism of $X$. (By Matveyev and Curtis-Freedman-...
- 515
13
votes
0
answers
181
views
Is there a Handle Approximation theorem?
The cellular approximation theorem states that given a continuous map between two CW complexes $f : X \to Y$, then $f$ is homotopic to a cellular map - that is some map $f'$ with $f'(X_n) \subset Y_n$ ...
- 5,349
12
votes
0
answers
229
views
3-manifolds with stacked links
Stacked spheres
A triangulation of a 2-dimensional sphere is called a stacked sphere if it is obtained inductively from the boundary of a 3-simplex by deleting a 2-face (triangle) $T$ adding a new ...
- 24.7k
12
votes
0
answers
281
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 ...
- 121
11
votes
0
answers
179
views
Natural knot homology
All knot homology theories I've seen share a flaw: their definitions explicitly use some combinatorial choices (such as a diagram presentation). The coin, however, has two sides and the other one is ...
11
votes
0
answers
360
views
Fox re-imbedding theorem in dimension four
Fox re-imbedding theorem states the following:
A compact 3-manifold $M$ with boundary that embeds in the three-sphere $S^3$, can be re-imbedded in $S^3$ so that its complement is a union of ...
- 10.5k
10
votes
0
answers
139
views
Space of thick ending laminations
Let $\Sigma$ be a compact closed connected oriented surface of genus $g>1$. Klarreich proved that the space of ending laminations $\mathcal{EL}(\Sigma)$ is the ideal boundary of the curve complex $...
- 68.9k
10
votes
0
answers
288
views
Contact structures associated to taut foliations
Eliashberg and Thurston showed that a taut foliation may be deformed to tight (positive and negative) contact structures. Vogel proved that for a taut foliation without torus leaves, the associated ...
- 68.9k
10
votes
0
answers
127
views
Compatibility of spherical and hyperbolic geometry for fibred knots
Hyperbolic knots and links have a lovely peculiarity that you can always find a position for them in $S^3$ making two groups the same, one defined using the spherical geometry of $S^3$, and the other ...
- 44.4k
8
votes
0
answers
432
views
The figure eight knot complement in $S^3$
Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. Exercise 5.4 (on page 101) gives us a presentation of the fundamental group of $S^3 - K$ where $K$ is the figure-...
- 111
8
votes
0
answers
445
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 ...
- 1,314
7
votes
0
answers
145
views
Long non-deformable hyperbolic fillings
The title and question have been edited in light of Ian Agol's comment. The previous question was stated in terms of the wrong notion of length to discuss deformations:
What is the longest slope $\...
- 5,259
7
votes
0
answers
156
views
Two papers on surface diffeomorphisms
The following two papers appeared in the reference of a paper i was reading.It seems that neither is published formally.Is there a website where i could find them?
A. Casson, Cobordism Invariants of ...
- 103
7
votes
0
answers
719
views
Expository accounts of the Thurston norm
Other than Thurston's original paper (which I find quite hard to read), are there any expositions of the basic properties of the Thurston norm? In particular, I'm interested in a proof of his ...
- 71
6
votes
0
answers
254
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.
- 2,535
6
votes
0
answers
389
views
A conjecture of Thurston and possibly Weeks too
What is the status of the following conjecture:
"... [w]hen the shortest simple closed geodesics are repeatedly removed from any complete hyperbolic 3-manifold of finite volume, eventually one ...
- 1,131
5
votes
0
answers
96
views
$M^3$ admits $Sol$ geometry if and only if $\pi_1M$ is virtually solvable but not virtually nilpotent?
Let $M$ be a closed, orientable, irreducible 3-manifold and having an infinite fundamental group. Is it true that $M$ admits $Sol$ geometry if and only if $\pi_1M$ is virtually solvable but not ...
- 605
5
votes
0
answers
188
views
Triangulating piecewise-linear manifolds
Question 1: Is this the mainstream definition of a PL-manifold?
Definition. A PL-manifold is a manifold with an atlas $(\varphi_i)_{i\in I}$ in which all transition maps $\varphi_j\circ\varphi_i^{-1}$ ...
- 396
5
votes
0
answers
122
views
Heegaard diagrams of prime 3-manifolds
Are there some known results which give a classification of closed prime 3-manifolds up to their Heegaard diagrams? (That is, providing a collection of Heegaard diagrams which exhausts all prime ...
- 459
4
votes
0
answers
161
views
Question on the construction of transversely oriented foliation on a sutured 3-manifold
The question is based on the proof of the main theorem of Gabai's paper on Foliations and the topology of 3-manifolds which is the following:
Theorem 5.1. Suppose $M$ is connected, and $(M,\gamma)$ ...
4
votes
0
answers
191
views
Every closed surface divides a closed 3-manifold $X$ into two parts if and only if $H_1(X)$ is finite
My friend is looking for proof of the following statement
Every closed surface divides a closed 3-manifold $X$ into two parts if and only if $H_1(X; \mathbb{Z})$ is finite.
Rumor source: Justin ...
- 6,962
4
votes
0
answers
172
views
Survey or good reference of taut foliations
I am interested in the topology of foliations.
In particular, I want to understand taut foliations, or projectively Anosov flows, and Anosov flows.
I guess that
A. Candel and L. Conlon, Foliations I (...
- 73
4
votes
0
answers
200
views
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(...
- 5,349
4
votes
0
answers
234
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 ...
- 173
4
votes
0
answers
123
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 $...
- 41
4
votes
0
answers
397
views
Contractibility and orientation double cover
Question. Let $M$ be a triangulated non-orientable 3-manifold with non-orientable boundary. (It is possible to assume that the boundary is the Klein bottle.) Let $\ell$ be a non-orientable loop on the ...
- 976
4
votes
0
answers
273
views
Are triangulations with common refinements PL-homeomorphic?
Do there exist simplicial triangulations $K_1$ and $K_2$ of a topological manifold $M$ such that $K_1$ and $K_2$ have a common subdivision but they are not PL-homeomorphic? Ideally, I would like an ...
- 165
4
votes
0
answers
137
views
Is the mapping class group of a high distance general Heegaard splitting finitely generated?
Let $H_{1}$ and $H_{2}$ are two handlebodies. If $\partial H_{1}$ and $\partial H_{2}$ are homeomorphic, then $H_{1}\cup_{f} H_{2}$ is a Heegaard splitting. To a general Heegaard splitting, one of $H_{...
- 841
4
votes
0
answers
269
views
Action of the mapping class group on separating spheres in a connected sum of aspherical 3-manifolds
Let $M$ be a connected sum of $g$ closed aspherical 3-manifolds $M_1, \ldots, M_g$. [Update: I also assume that all the $M_i$-s are diffeomorphic, i.e. $M$ is a connected sum of copies of the same ...
- 3,203
4
votes
0
answers
302
views
Haken manifolds and characterising sutured manifold hierarchies
In Gabai's paper (Knot Theory and Manifolds Lecture Notes in Mathematics Volume 1144, 1985, pp 14-17 An internal hierarchy for 3-manifolds) he considers sutured manifold decompositions of Haken $3$--...
- 367
3
votes
0
answers
93
views
Finiteness of non-orientable 3-manifolds with the same orientable two-fold cover
Given a compact, orientable and boundary incompressible 3-manifold $M$. Suppose that either $M$ is closed, or $\partial M$ consists of tori.
For which non-orientable 3-manifolds $N$, the orientable ...
- 605
3
votes
0
answers
101
views
Explicit parameterizations of complicated unlinks?
I have a somewhat empirical question which I hope is still welcome here. I would like to know how to write down explicit parameterizations of "complicated unlinks", say with 2 or 10 ...
- 1,075
3
votes
0
answers
72
views
Discreteness of volumes of boundary-parabolic representations
Suppose $M$ is a cusped hyperbolic $3$-manifold of finite volume. Let $\mathfrak{R}_0(M)$ be the space of boundary-parabolic representations $\rho : \pi_1(M) \to \operatorname{PSL}_2(\mathbb C)$. Is ...
- 2,533
3
votes
0
answers
223
views
Can Whitehead manifold admit a properly discontinuous cocompact group action?
Can classical contractible manifolds such as Whitehead manifold admit a properly discontinuous cocompact group action?
Here "properly discontinuous" doesn't have to be fixed point free, but ...
- 2,083
3
votes
0
answers
59
views
Nonuniqueness of Heegaard surfaces for submanifolds of $S^3$
Let $M^3$ be some compact submanifold of $S^3$ with connected boundary. I am interested in the failure of the analog of Waldhausen's theorem for $M^3$ - namely, I would like examples of such $M$ ...
- 5,349
3
votes
0
answers
241
views
Standard sutured (?) Heegaard splitting
I am trying to make sense of what is going on in [Cas16] in terms of diagrams. Let me sum up the construction a bit, where $n\leqslant k$ are integers and $b\geqslant 1$ as well.
$C_{k,b,n}$ denotes ...
- 283
3
votes
0
answers
244
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_{...
- 353
3
votes
0
answers
211
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 ...
- 5,349
3
votes
0
answers
75
views
Approximative extension of the autohomeomorphism of the complement of the trivial knot?
Let $S^1\subset \mathbb{R}^3$ be the unit circle and suppose $h\colon \mathbb{R}^3\setminus S^1\to \mathbb{R}^3\setminus S^1$ is a homeomorphism. Clearly it might be that $h$ cannot be extended to $S^...
- 396
3
votes
0
answers
125
views
Does Heegaard splitting relate topological properties of a $3$-manifold to properties of subgroups of $MCG$
In the proof of Lickorish-Wallace theorem, we use Heegaard splitting of a closed, orientable and connected $3$-manifold and obtain a surface diffeomorphism which glues the two handle-body. I wonder ...
- 113
3
votes
0
answers
154
views
3-manifold being boundary of neighborhood of 2-complex in 4-space
In this question I have asked about boundary of regular neighborhood of $\mathbb RP^2$ in $\mathbb R^4$. I am interested in more general way of producing 3-manifolds in $\mathbb R^4$ namely the ...
user21230
3
votes
0
answers
265
views
Step by step construction of 3-manifolds in $R^4$
My question has survived. Therefore I try another one. Consider some elementary operations on closed compact 3-manifold $M \subset R^4$. These elementary operations are e.g. $0$-surgery or $1$-surgery ...
user21230
3
votes
0
answers
238
views
Lutz twist and open book decompositions
Let $M^3$ be a closed oriented 3-manifold, endowed with an open book decomposition. Consider a section of the open book, that is a knot $K \subset M$ disjoint from the binding and meeting every page ...
- 2,283
3
votes
0
answers
181
views
Definition of the dual spider number and the formula for the first chern class of the triangle
In the process of trying to understand various maps in Heegaard Floer homology I got stuck on the definition of the dual spider number, which, it seems to me, has a combinatorial definition directly ...
- 1,000
3
votes
0
answers
167
views
Euclidean realisation of a polyhedral complex
Let us say that an Euclidean polyhedral manifold is a manifold that is glued from a finite number of Euclidean polyhedrons by identifying isometrically their co-dimension $1$ faces. Let us assume that ...
- 14.3k
3
votes
0
answers
334
views
What is the behaviour of a smooth 3-manifold acting by a circle?
As Mumford pointed out in his paper 'Topology of Normal Singularities and a Criterion for Simplicity'(1961), every point $p$ on a normal complex surface $V$ has an associated 3-manifold $M$ which is ...
- 1,083
2
votes
0
answers
137
views
Are oriented-$h$-cobordant lens spaces orientation-preservingly homeomorphic?
Consider two three-dimensional lens spaces $N_1=L(p,q_1)$ and $N_2=L(p,q_2)$, and assume that there is an oriented-$h$-cobordism between them. In other words, we assume that there is an oriented four-...
- 21
2
votes
0
answers
201
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 ...
- 673