Questions tagged [3-manifolds]
A three-manifold is a space that locally looks like Euclidean three-dimensional space
608
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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 $\partial N=M$, ...
6
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1
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282
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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 $Y\times\{1\}$, $K$...
4
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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 ...
2
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1
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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.
...
7
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1
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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 ...
4
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Non compact Seifert manifolds
A Seifert manifold $M$ is a $3$-dimensional orientable smooth manifold with an effective circle action with no fixed points.
Closed connected Seifert manifolds are classified up to an equivariant ...
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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 ...
4
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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 $\varphi_j\circ\varphi_i^{-1}$ ...
2
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1
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312
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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 ...
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Example of three dimensional atoroidal Poincaré duality group with some pathology
I am looking for a 3-manifold which is closed, aspherical, orientable, and atoroidal. And additionally I want to see an example that does not admit a fixed-point-free action on a simplicial tree. As a ...
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2
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"canonical" framing of 3-manifolds
In Witten's 1989 QFT and Jones polynomial paper, he said
Although the tangent bundle of a three manifold can be trivialized, there is no canonical way to do this.
So if I understand correctly, ...
2
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Examples of counting holomorphic curves in cylindrical reformulation of Heegaard Floer
In 2005, Robert Lipshitz reformulated Heegaard Floer in a "cylindrical setting" by counting holomorphic curves in $\Sigma \times [0,1] \times \mathbb{R}$ where $\Sigma$ is a Heegaard surface ...
4
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1
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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 ...
3
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When are homologous embedded surfaces in 3-manifolds related by embedded cobordisms?
Let $M$ be an orientable closed 3-manifold and suppose $A$ and $B$ are embedded incompressible closed orientable surfaces in $M$ with $[A] = [B]$ in $H_2(M,\mathbb{Z})$.
In general, there are a ...
2
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114
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Homotopy type of a 3-manifold produced via Dehn surgery?
My apologizes if this is a fairly elementary question, I am still a novice when it comes to 3-manifold topology.
I am wondering the following: by Kirby calculus, we know that two links (say in $S^{3}$ ...
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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 \...
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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....
5
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515
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Betti numbers of non-orientable $3$-manifolds
Let $M^3$ be a compact $3$-manifold with boundary $\partial M$.
If $M$ is orientable, then it is known (see Lemma 3.5 here) that $2\dim(\ker(H_1(\partial M,\mathbb{Q})\rightarrow H_1(M,\mathbb{Q})))=\...
5
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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 ...
1
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1
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215
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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 ...
8
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Three-dimensional triangulations with fixed number of vertices
My question is the following:
Are there triangulations of $S^3$ which (a) are non-degenerate, (b)
have four vertices, and (c) have no edges of degree two?
A side question:
If one represents this ...
0
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1
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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 ...
5
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252
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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 ...
8
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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) ...
2
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228
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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 $\alpha$ ...
0
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0
answers
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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 ...
4
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1
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138
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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 $(\...
4
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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 ...
3
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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 ...
7
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2
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524
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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 ...
5
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1
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224
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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?
3
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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 ...
12
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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 ...
3
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1
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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 ...
7
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252
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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 $b_2(W)=1$? ($\cdots$...
8
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1
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271
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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 ...
5
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1
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269
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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 ...
5
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3
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224
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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 ...
4
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1
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196
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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}
...
6
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1
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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 ...
5
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1
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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 ...
8
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1
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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 ...
3
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1
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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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1
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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^...
2
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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 ...
3
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158
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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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2
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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-...
1
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0
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106
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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 ...
2
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2
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367
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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 ...
10
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0
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245
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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 ...