All Questions
Tagged with 3-manifolds gt.geometric-topology
423 questions
8
votes
1
answer
223
views
Preserving non-conjugacy of loxodromic isometries in a Dehn filling
Suppose that $g$ and $h$ are non-conjugate loxodromic isometries in a cusped hyperbolic $3$-manifold $M$ of finite volume. Fix a cusp $T$ of $M$. Can I choose a hyperbolic Dehn filling of $M$ along $...
5
votes
1
answer
378
views
Why is this Brieskorn manifold a rational homology sphere?
In Némethi's book "Normal surface singularities", Example 5.1.17, there is a formula to find the Seifert invariants of a Brieskorn complete intersection $\Sigma(a_1,...,a_n)$. I am ...
- 197
1
vote
1
answer
91
views
When is a 2-bridge knot hyperbolic?
It is known that 2-bridge knots in $S^3$ can be classified by the Schubert form. My question is: which 2-bridge knots are hyperbolic? (Do we have a complete classification for hyperbolicity in 2-...
- 605
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
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
8
votes
1
answer
352
views
Can I endow the following 3-manifold with a hyperbolic metric?
Consider the following three-dimensional topology. Start with $S^3$ and drill out four unlinked tori as shown in the picture. Then, fill in the gaps with the same tori but with their longitudes and ...
- 355
4
votes
1
answer
88
views
$\partial$-incompressibility of a surface obtained when attaching a 2-handle to an irreducible 3-manifold produces a reducible 3-manifold
This question arises in my previous question.
Let $M$ be a compact, orientable, irreducible 3-manifold with incompressible boundary. Let $\alpha\subseteq \partial M$ be a simple closed curve, which is ...
- 605
5
votes
1
answer
215
views
Is it possible to fill a boundary component of an irreducible 3-manifold using a handlebody so that the resulting manifold is still irreducible?
Let $M$ be a compact, orientable, irreducible 3-manifold with boundary (possibly more than one component). Let $S\subseteq\partial M$ be one of its boundary components, which is an orientable surface ...
- 605
3
votes
1
answer
160
views
Does the fundamental group of a compact 3-manifold induce full profinite topology on the fundamental group of its boundary?
Let $M^3$ be a compact, orientable, irreducible 3-manifold with incompressible boundary. Let $S\subseteq\partial M$ be one of its boundary components. Does $\pi_1(M)$ induce the full profinite ...
- 605
7
votes
2
answers
534
views
Does there exist a Dehn filling of an irreducible 3-manifold with toroidal boundaries which is still irreducible?
Let $M$ be a compact, orientable, irreducible 3-manifold with incompressible toroidal boundary (there might be more than one boundary component). Is it always possible to choose appropriate slopes on ...
- 605
12
votes
2
answers
308
views
Property P and R for general 3-manifolds
Let $Y$ be a closed oriented $3$-manifold and $K$ be a knot in $Y$. We say $K$ is the unknot if $K$ is contained in a local $3$-ball in $Y$ and is unknotted therein.
Generalized Property R:
If a Dehn ...
- 557
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 $(M,\gamma)$ ...
3
votes
1
answer
268
views
If the complement of a knot $K$ fibers over the circle is $K$ necessarily fibered?
Let $K \subseteq S^3$ be a knot in the $3$-sphere and assume there exists a smooth map $p \colon S^3\setminus K \to S^1$ which is a fiber bundle.
For every point $\require{enclose} \enclose{...
- 195
1
vote
1
answer
114
views
The diameter of the projection of a convex core
Let $M$ be a closed hyperbolic 3-manifold and $H_{g}$ a genus g handlebody. Assume that $\pi: int(H_{g})\rightarrow M$ is a cover. Denote $N\subset H_{g}$ the convex core.
My question is: If the ...
- 841
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
17
votes
2
answers
554
views
$3$-manifold that is a surgery on a knot
By the Lickorish-Wallace theorem, every oriented closed $3$-manifold can be obtained by a surgery on a link in $S^3$. In the statement of this result, links are required: not every such manifold can ...
- 331
5
votes
1
answer
430
views
Linking number and intersection number
Consider a disjoint union of two circles $A$ and $B$ smoothly embedded in $\mathbb{R}^3$ with linking number more than $1$. Suppose we know that there exists a disc $D$ in $\mathbb{R}^3$ such that $\...
- 125
7
votes
2
answers
320
views
Existence of a surface group ensures the existence of a $\pi_1$-injective immersed surface
The question is simple:
For a $3$-manifold $M$, if $\pi_1(M)$ contains a surface group $\Gamma$ (i.e. the fundamental group of some surface) then $M$ contains a $\pi_1$-injective immersed surface $S$....
4
votes
1
answer
200
views
Residual finiteness and a gluing problem
The below flowchart is from Thurston's paper Hyperbolic structures on 3-manifolds I. I don't know if I interpreted it correctly but at the bottom it says that
Residual finiteness "implies" ...
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
6
votes
1
answer
263
views
Euler number of a Seifert bundle as a generalization of an Euler number of a circle bundle over a surface
In classic, Euler numbers associated to circle bundles over a fixed surface classify all possible such bundles. But the construction of Euler class in general requires the fact that any fiber bundle ...
1
vote
0
answers
107
views
Extend a circle action on $3$-manifolds
Let $M$ be an oriented closed $3$-manifold equipped with an effective smooth circle action.
Can we have a classification of all such $M$ such that there exists a $4$-manifold $N$ with $\partial N=M$, ...
- 891
6
votes
2
answers
395
views
Slice knots in 3-manifolds
Is there a nonslice knot $K\subset S^3$ that is slice in some closed oriented $3$-manifold $Y$? Here, when we say $K$ is slice in $Y$, it means that when regarded as a local knot in $Y\times\{1\}$, $K$...
- 557
4
votes
1
answer
128
views
Rigidity/flexibility of Sol-structures on closed 3-manifolds
This is a follow-up to the question
Rigidity/flexibility of Nil-, Sol-, $\widetilde{\rm SL}_2$- structures on closed 3-manifolds
From the answers/comments there and from an excellent survey by Bonahon ...
- 353
3
votes
1
answer
202
views
Guts of 3-manifolds for sutured manifolds and pared manifolds
I found the notion "guts of three-manifolds" unclear to me. There exists "sutured guts" and "pared guts" in the literature, the well definedness of both are vague to me.
...
- 502
7
votes
1
answer
190
views
Rigidity/flexibility of Nil-, Sol-, $\widetilde{\rm SL}_2$- structures on closed 3-manifolds
It is known that closed spherical and hyperbolic 3-manifolds are rigid. I.e., if two such manifolds are diffeomorphic, then they are isometric (moreover, I think, that every diffeomorphism is isotopic ...
- 353
5
votes
1
answer
182
views
Volume of the Weeks manifold and of the 5.2 knot complement
Some computations show that the Weeks manifold and the 5.2 knot complement have the same trace field (which is $\mathbb{Q}[x]/(x^3-x+1)$) and the (hyperbolic) volume of the second is 3 times the ...
- 1,181
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
2
votes
1
answer
350
views
Realizable geometrically finite hyperbolic 3-manifolds with prescribed conformal boundaries
By Bers' simultaneous uniformization theorem, if $\Gamma$ is a Fuchsian group, then $\operatorname{QC}(\Gamma)\cong \mathcal{T}(S)\times\mathcal{T}(\overline{S})$ where $S = \Bbb H^2/\Gamma$. In ...
4
votes
1
answer
308
views
Casson's knot invariant
$\DeclareMathOperator\SU{SU}$Informally speaking, the Casson invariant counts half the number of conjugacy classes of representations of the fundamental group of a homology $3$-sphere $M$ into the ...
- 954
6
votes
1
answer
444
views
Branched coverings of non-orientable 3-manifolds
A continuous map of 3-dimensional manifolds $f \colon M^3 \to N^3$ is called a branched covering if there is a link $L \subset N^3$, such that the restriction $f \colon M \setminus f^{-1}(L) \to N \...
11
votes
1
answer
605
views
If the universal cover has three boundary components, does it have infinitely many?
Suppose that $M$ is a compact, connected three-manifold with boundary. Suppose that $\pi_1(M)$ is infinite. Suppose that $\tilde{M}$, the universal cover of $M$, has at least three boundary components....
- 28.2k
5
votes
1
answer
188
views
Properly embedded surfaces in handlebodies are compressible or boundary compressible?
I've read in a couple of different places (a paper and a blog) the following fact:
if $F$ is a surface, properly embedded in a three-dimensional handlebody of genus at least two, then $F$ is either ...
- 421
1
vote
1
answer
256
views
Reference request and prerequisites for understanding the Sphere Theorem and the Loop Theorem in 3-manifold theory
As part of my directed studies project, my advisor has suggested that I completely understand the proof of the Sphere Theorem and the Loop Theorem in 3-manifold theory and explain it to him. I have ...
- 131
0
votes
1
answer
168
views
Mappings of reducible 3 manifolds with boundary
In section 3 of his paper "Mappings of reducible 3 manifolds" McCullough, proves that every self-homeomorphism of a reducible 3 manifold can up to isotopy be written as a composition of ...
- 1,174
5
votes
1
answer
340
views
0-surgery on a fibered hyperbolic ribbon knot
Does there exist a fibering hyperbolic ribbon knot such that the 0 surgery is exceptional? If so does there exist such an example where the result of 0-surgery is Seifert fibered?
I tried looking at ...
- 1,174
4
votes
1
answer
152
views
Topological type of complement of Heegaard curves in Heegaard surface $(\Sigma - \alpha - \beta)$
Suppose $(\Sigma, \alpha, \beta)$ is a genus-$g$ Heegaard diagram for a closed, oriented $3$-manifold $Y$, i.e. $\Sigma$ is an orientable genus-$g$ surface, and $(\alpha_1, \dots, \alpha_g)$ and $(\...
- 281
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
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
5
votes
1
answer
232
views
Amenable link groups
The unknot and the Hopf link are (as far as I know) the only links whose complements have abelian fundamental groups. Are there more examples whose complement have amenable fundamental group?
- 1,174
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
12
votes
1
answer
623
views
Definition of Thurston's skinning map
A key construction in Thurston's proof of the existence of hyperbolic structures on Haken manifolds is the so-called "skinning map" associated to a 3-manifold $M$ with boundary whose ...
- 197
3
votes
1
answer
235
views
Example of a non $\pi_1$-injective, degree one, self-map of a three-manifold
All manifolds will be assumed to be closed, oriented, and connected.
Let $f\colon M\to M$ be a map of degree $\pm 1$. It is not hard to show that $\pi_1(f)$ is surjective.
What is an example of a non ...
- 1,097
7
votes
1
answer
319
views
Lens space bounding a topological, simply-connected 4-manifold with $b_2=1$
The following is written in section 1.6 (p.7) of this paper: https://arxiv.org/pdf/1010.6257.pdf.
($\cdots$) Which lens spaces bound a smooth, simply-connected 4-manifold $W$ with $b_2(W)=1$? ($\cdots$...
- 335
8
votes
1
answer
281
views
Non-compact three-manifolds with the same proper homotopy type are homeomorphic?
I am looking for some literature with some (counter) examples of the following fact (though I don't know if the fact is true or not):
Let $M, M'$ be two non-compact connected $3$-manifolds with the ...
- 1,097
5
votes
1
answer
283
views
Homology of spherical $3$-manifold group
I have been studying $3$-manifolds recently and I got stuck in the following situation. For lens spaces the below fact is true.
Let $G$ be a finite group acting freely and orthogonally on $S^3$ so ...
- 179
5
votes
3
answers
245
views
Ideal triangulations of $3$-manifolds with "cusps" of genus $\ge 2$
Typically when one thinks about ideal triangulations of a $3$-manifold the link of each ideal vertex is a circle, so the ideal points correspond to toroidal cusps; alternatively, one can truncate the ...
- 2,533
4
votes
1
answer
223
views
Stallings' fibration theorem - Explicit description
Stallings' celebrated Fibration Theorem states that if a closed irreducible $3$-manifold $M$ admits a short exact sequence
\begin{equation}
1 \to N \to \pi_1(M) \to \mathbb{Z} \to 1,
\end{equation}
...
- 143
6
votes
1
answer
196
views
Bigon criterion in dimension 3?
The bigon criterion for surfaces says that if two simple closed curves $\alpha$ and $\beta$ embedded on a surface $\Sigma$ intersect in points $\{p_1,\dotsc,p_n\}$ and $\alpha$ and $\beta$ can be ...
- 1,075
5
votes
1
answer
246
views
Is there a way to calculate the Froyshov $h$-invariant for Seifert homology spheres?
In 2002, by using Floer theory, Froyshov defined the $h$-invariant for intergal homology 3-spheres, which is a surjective group homomorphism $\Theta^3_{\Bbb Z}\to \Bbb Z$, where $\Theta^3_{\Bbb Z}$ is ...
- 335