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Questions tagged [3-manifolds]

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

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5 votes
2 answers
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Non-orientable real algebraic three-dimensional manifolds

Smooth real algebraic hypersurfaces of even degree in $\mathbb{RP}^4$ that are maximal (i.e. that are homologically as rich as possible in the sense of the Smith-Thom inequality) are all non-...
Yromed's user avatar
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1 vote
1 answer
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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
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$M^3$ 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
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16 votes
2 answers
486 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
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5 votes
1 answer
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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
0 votes
0 answers
105 views

3-manifold is aspherical if fundamental group is not free and torsion-free

I was wondering if the following statement is true. Let $M$ be a closed, 3-manifold such that $\pi_1(M)$ is not a free group and $\pi_1(M)$ is torsion-free. Then $M$ must be aspherical. My ideas: If ...
JE2912's user avatar
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7 votes
2 answers
290 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
190 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
103 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
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6 votes
1 answer
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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
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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 $\partial N=M$, ...
Zhiqiang's user avatar
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6 votes
2 answers
349 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
4 votes
1 answer
117 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
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2 votes
1 answer
166 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
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7 votes
1 answer
179 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
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4 votes
1 answer
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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 ...
Rei Henigman's user avatar
5 votes
1 answer
165 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
4 votes
0 answers
143 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
  • 346
2 votes
1 answer
327 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
11 votes
1 answer
239 views

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 ...
Peter Kropholler's user avatar
6 votes
2 answers
358 views

"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, ...
zeta's user avatar
  • 447
2 votes
0 answers
132 views

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 ...
semper-lux's user avatar
4 votes
1 answer
277 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
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3 votes
1 answer
107 views

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 ...
Edgar A. Bering IV's user avatar
2 votes
0 answers
116 views

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 $S^{3}$ ...
Elliot's user avatar
  • 295
6 votes
1 answer
403 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
10 votes
1 answer
581 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
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5 votes
1 answer
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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})))=\...
Alessio Di Prisa's user avatar
5 votes
1 answer
151 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
  • 379
1 vote
1 answer
236 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
8 votes
2 answers
779 views

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 ...
Kregnach's user avatar
0 votes
1 answer
162 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
  • 1,185
5 votes
1 answer
284 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
  • 1,185
8 votes
1 answer
852 views

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) ...
Agelos's user avatar
  • 1,896
2 votes
0 answers
231 views

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 $\alpha$ ...
Andrey Ryabichev's user avatar
0 votes
0 answers
81 views

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 ...
Matteo's user avatar
  • 1
4 votes
1 answer
146 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
173 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
99 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
  • 1,065
7 votes
2 answers
553 views

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 ...
Andrey Ryabichev's user avatar
5 votes
1 answer
226 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
  • 1,185
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
599 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
  • 155
3 votes
1 answer
220 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
  • 1,087
7 votes
1 answer
270 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
278 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
  • 1,087
5 votes
1 answer
273 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
230 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
210 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
188 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
  • 1,065

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