# Questions tagged [3-manifolds]

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

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### Problem 3.14 from Kirby's list

In his famous list of Problems in Low-Dimensional Topology, Kirby states the following as Problem 3.14 (B), which is attributed to Thurston:
Conjecture: Suppose $G$ (an arbitrary group I suppose) ...

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### Is the square of a primitive cohomology class always primitive?

Let $M$ be a closed manifold (in my case $\dim M=3$).
Take $\alpha\in H^1(M;\mathcal{Or})$, where $\mathcal{Or}$ is the orientation local system for $M$ with coefficients $\mathbb Z$.
Suppose $\alpha$ ...

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### Determinant of SU(N) elements, and radius of associated manifold

I'm wondering if the fact $SU(2)$ group elements have $det = 1$ is connected with the radius of the unitary $S^{3}$ manifold associated.
The context is demonstration of dU being an Haar invariant ...

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### 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 $(\...

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### 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 ...

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### 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 ...

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### Covering of a knot complement

Let $B=S^3\setminus K$ for some (tame) knot $K$. Suppose we have a covering $E\to B$ with a finite fiber.
Question: is $E$ homeomorphic to a knot/link complement?
On this question I found only the ...

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### 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?

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $b_2(W)=1$? ($\cdots$...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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}
...

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### 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 ...

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### 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 ...

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### On trivial mapping class group of 3-manifolds

What are some examples of knots $K\subset S^3$ such that the mapping class group of $S^3_{1/n}(K)$ is trivial? I guess for hyperbolic knots with no symmetry in the complements are good candidate as ...

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### Does the Weeks manifold have the smallest volume among all finite volume oriented hyperbolic 3-manifolds?

It is a result of Chinburg-Friedman-Jones-Reid that the arithmetic hyperbolic 3-manifold of smallest volume is the Weeks manifold.
There is also a result of Milley that says that if $N$ is a closed ...

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### Homotopy type of Diff(ℝP³)

$\DeclareMathOperator\Diff{Diff}$I am looking for the paper computing the homotopy type of the group $\Diff(\mathbb{R}P^3)$ of diffeomorphisms of the $3$-dimensional real projective space $\mathbb{R}P^...

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### Every surface of sufficiently large genus separates

Let $M^3$ be a smooth closed orientable manifold.
Does there exist a non negative integer $g_0$ such that every closed orientable embedded surface $\Sigma \subset M$ of genus $g \geq g_0$ represents ...

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### Do taut foliations leafwise branch covering S^2 yield foliations by circles?

In this paper, Danny Calegari shows that taut foliations in (let's say closed for simplicity) 3-manifolds are precisely those which admit a map $f: M \to S^2$ which restricts to a branched cover on ...

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### Dual surfaces of a first cohomology class of a 3-manifold

Let $M$ be closed 3-manifold and $\alpha\in H^1(M;\mathbb Z_2)$ an arbitrary element. (In my case we know that $M$ is non-orientable and $\alpha^3=0$.) It is well known that there is a closed 2-...

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### About the classification of simply connected homogeneous 3-manifolds

I've read somewhere (but cannot locate the source) that the following classification holds: simply connected homogeneous 3-manifolds are either isometric to $S^2 \times \mathbb{R}$ or to a metric Lie ...

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### Under what conditions can an orientable Riemannian 3-manifold be defined implicitly?

Under what conditions can an orientable Riemannian 3-manifold $\Sigma$ be defined implicitly?
What I mean by implicitly is that there exists a smooth function $f:\mathbb{R}^n\to \mathbb{R}^m$, such ...

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### 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 ...

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### Small examples of exceptional hyperbolic Dehn Filling of hyperbolic manifolds

For experimental purposes, I would like to have a small (i.e. triangulated with few tetrahedra) example of a manifold $M$ with the following properties:
$M$ is a hyperbolic manifold with finite ...

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### 4-manifold obtained from a ribbon disk exterior by attaching a 2-handle is simply-connected if its boundary is a homology sphere

I am reading Lemma 2.1 of this paper (https://arxiv.org/pdf/2012.12587.pdf) and I can't see why $W$ is simply-connected. Here is the situation:
Let $K$ be a ribbon knot in $S^3$; it bounds a ribbon ...

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### Integral homology $S^1\times S^2$'s smoothly bounding integral homology $S^1\times B^3$'s

Suppose we are given a compact orientable 3-manifold $M$ which is an integral homology $S^1\times S^2$. Then is there a way to determine whether $M$ bounds a smooth compact orientable 4-manifold which ...

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### Pullback of $w_1$ for 3-manifolds

Given closed $3$-manifolds $M$ and $N$
and an element $\alpha\in H^1(M;\mathbb{Z}_2)$,
when does there exist a map $f:M\to N$
such that $\alpha=f^*(w_1(N))$?

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### The works of González-Acuña and Duchon from 70s and 80s

I would like to access the following two works of González-Acuña from around nineteen-seventies:
González-Acuña, F. Dehn’s construction on knots. Bol. Soc. Mat. Mexicana (2) 15 (1970), 58–79.
and
...

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### Uniqueness of the set of decomposing spheres in prime decomposition of a 3-manifold

At the end of Section 1.1 of 3-manifold groups it is written that "the decomposing spheres are not unique up to isotopy, but two different sets of decomposing spheres are related by ‘slide ...

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### Conjugacy of topological actions on aspherical three manifolds to isometric actions

Edited: Due to work of Raymond and Scott, there exist diffemorphisms (of certain three-dimensional nil-manifolds) whose $n$th power is diffeotopic to the identity, but which are not themselves ...

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### When does a map between 4-manifolds map boundary to boundary upto homotopy?

Let $f:M\to N$ be a smooth map between smooth 4-manifolds with boundary. When does $f$ map boundary of $M$ to boundary of $N$ upto homotopy i.e. when there is a map $F:M\to N$ homotopic to $f$ such ...

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### Morse functions inducing Heegaard diagrams

Let $(\Sigma, \alpha, \beta)$ be a Heegaard diagram for a 3-manifold $M$, corresponding to a Heegaard splitting $M = H_1 \cup_\Sigma H_2$. There may be many self-indexing Morse functions $f: M \to \...

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### Boundary of a $4$-manifold and the fundamental group

I am trying to learn $4$-manifolds with boundaries and I don't know much about this topic so these questions may be silly. Given a $4$-manifold $M$ with a boundary say $N$,
Assume $\pi_1(N)$ is known,...

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### 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-...

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### 0-framed smoothly slice knot that can be obtained by blowing down successively a link of unknots

A knot in $S^3$ is called a smoothly slice knot if it bounds a smoothly embedded 2-disk in $D^4$. Every ribbon knot is known to be a smoothly slice knot, and there are known some nontrivial smoothly ...

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### 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 ...

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### How wild can an open topological 3-manifold be if it has a compact quotient?

Let $M$ be an open, simply connected, 3-manifold. Suppose $M$ admits a properly discontinuous, co-compact topological action by a finitely generated group.
Question 1: If $M$ is 1-ended, must it be ...

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### 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 (...

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### Complex length of geodesic added in hyperbolic Dehn surgery

Suppose $M$ is a cusped finite-volume hyperbolic $3$-manifold, say with a single cusp for simplicity. Following [NZ, Section 4] we can parametrize deformations of the hyperbolic structure with a ...

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### Given a Heegaard splitting $M = V\cup_F W$, then $V\setminus N(D_1)$ is ambient isotopic to $V\cup N(D_2)$ for a meridian pair $\{D_1,D_2\}$

I sincerely apologize if MathOverflow is not the appropriate place to ask this question. I also tried consulting M.SE but it seems that this question gained little to no interest .
Consider a ...

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### Heegaard Floer homology of a genus two Heegaard splitting of $S^3$

This is a duplicate of a question (https://math.stackexchange.com/questions/4416204/heegaard-floer-homology-of-a-genus-two-diagram-of-s3) on stackexchange, which did not get any answer. Feel free to ...

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### Universal covers of non-prime 3-manifolds

Let $M$ be a closed, connected, oriented 3-manifold. If $M$ is prime, then we know what the universal cover of $M$ looks like: it is either $S^3, \mathbb{R}^3$ or $S^2 \times \mathbb{R}$ depending on ...