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7 votes
1 answer
218 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 $...
Emily Hamilton's user avatar
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 ...
user13121312's user avatar
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-...
YC Su's user avatar
  • 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 ...
YC Su's user avatar
  • 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 $...
Ian Agol's user avatar
  • 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 ...
Holomaniac's user avatar
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 ...
YC Su's user avatar
  • 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 ...
YC Su's user avatar
  • 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 ...
YC Su's user avatar
  • 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 ...
YC Su's user avatar
  • 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 ...
Qiuyu Ren's user avatar
  • 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 $(M,\gamma)$ ...
one potato two potato's user avatar
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{...
Patrick Perras's user avatar
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 ...
yanqing 's user avatar
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 ...
YC Su's user avatar
  • 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 ...
mathmo's user avatar
  • 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 $\...
user429294's user avatar
7 votes
2 answers
319 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 $S$....
one potato two potato's user avatar
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" ...
one potato two potato's user avatar
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-...
Nathan's user avatar
  • 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 ...
one potato two potato's user avatar
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 $\partial N=M$, ...
Zhiqiang's user avatar
  • 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 $Y\times\{1\}$, $K$...
Qiuyu Ren's user avatar
  • 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 ...
Roman's user avatar
  • 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. ...
Fredy's user avatar
  • 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 ...
Roman's user avatar
  • 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 ...
Julien Marché's user avatar
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 $\varphi_j\circ\varphi_i^{-1}$ ...
Vadim's user avatar
  • 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 ...
one potato two potato's user avatar
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 ...
Partha's user avatar
  • 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 \...
vladimir smurygin's user avatar
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....
Sam Nead's user avatar
  • 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 ...
luthien's user avatar
  • 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 ...
ZSMJ's user avatar
  • 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 ...
ThorbenK's user avatar
  • 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 ...
ThorbenK's user avatar
  • 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 $(\...
Matija Sreckovic's user avatar
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 ...
Arshak Aivazian's user avatar
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 ...
Sprotte's user avatar
  • 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?
ThorbenK's user avatar
  • 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 ...
Calvin McPhail-Snyder's user avatar
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 ...
mrburch's user avatar
  • 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 ...
Random's user avatar
  • 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 $b_2(W)=1$? ($\cdots$...
user302934's user avatar
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 ...
Random's user avatar
  • 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 ...
gola vat's user avatar
  • 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 ...
Calvin McPhail-Snyder's user avatar
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} ...
Frieder Jäckel's user avatar
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 ...
Sprotte's user avatar
  • 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 ...
user302934's user avatar

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