Questions tagged [low-dimensional-topology]

Low-dimensional topology generally refers to the study of 3 or 4 dimensional topological manifolds and knot theory.

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Is johnson-lindenstrauss lemma valid for a set of high dimensional complex vectors? [closed]

I apologize in advance if this is a simple question, but is it valid in the complex domain?
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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. In this book, there is an exercise in chapter 5, section 5.6, and exercise number 5.4 (Page - 101). This exercise ...
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Compressible torus in irreducible 3-manifold bounds a solid torus

In his 3-manifolds notes (page 11, item $(4)$), Hatcher shows that a 2-sided compressible torus $T$ in an irreducible 3-manifold $M$ either bounds a solid torus $S^1 \times D^2$ or is contained in a ...
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Obstruction Theory in Seifert Surfaces

Background: This is from Livingston unplished notes in knot concordance: Proposition 1.7.1 claims: Let $K$ be a knot in $S^3$. Then every Seifert surface $F$ for $K$ has a function $f:S^3-\nu(K)\to S^...
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Is the pair of pants (3 punctured sphere) surface an incompressible surface?

Is the pair of pants (3 punctured sphere) surface an incompressible surface? My intuition says no since a simple closed curve around a boundary component will not bound a disk on the surface.
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Can a rational Seifert surface have n boundary components?

I was wondering if it is possible for a knot to have a rational Seifert surface with 3 boundary components (pair of pants) and n many in general.What would such a knot look like?
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Intersection of Satellite, Torus, and Hyperbolic Knots

Thurston famously proved every knot that is neither a satellite nor a torus knot is hyperbolic. I was thinking about the definition of satellite knots, in particular which cases we should eliminate to ...
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handle decomposition of three sphere $S^3$

Prove by attaching a 0- handle(the three diemsnion disc $D^3$) to the empty set,then attaching a 3-handle to the boundary of $D^3$ result in the $S^3$. I don't quite understand how the process really ...
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Examples of Seifert fibered homology spheres

The following is taken from chapter 1 of Saveliev's book Invariants for Homology 3-Spheres, there is the following definition: Let $a_1,\dots,a_n$ be positive integers, $n\geq 3$. Let $B=(b_{ij})$ be ...
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Reidemeister Torsion Example with Lens Spaces

In Andrew Ranicki's notes on Reidemeister torsion (https://www.maths.ed.ac.uk/~v1ranick/papers/torsion.pdf), he gives the following example with lens spaces: where I copied the beginning of the ...
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Seifert surface with components

I was wondering what the knot of a Seifert surface with 3 or more boundary components would be. Doesn’t every Seifert surface have one boundary component by definition ?
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'Flask' four manifold

I've seen in the text the construction of a 'flask' 4-manifold described by adding a punctured 4-sphere to a tubular end $S^3\times \mathbb{R}^+$. Also it is said in the context that this manifold ...
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Why is the number of components in the torus link $T_{p,q}$ equal to $\gcd(p,q)$?

I was not able to prove this by myself and also have not found any proof online. Since it is often stated as a fact, I assume it should not be a difficult statement to prove. The definition of $T_{p,q}...
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Heegaard Floer homology of a genus two diagram of $S^3$

I am reading the introductory paper "Heegaard diagrams and holomorphic disks" by Ozsváth and Szabó (https://arxiv.org/abs/math/0403029v1, Section 2.2), and I do not understand one of the ...
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What is the geometry of the twisted I-bundle over the Klein bottle?

Let $K^2$ be the Klein bottle and $M = K^2\widetilde{\times}I$ be the twisted, orientable $I = [0,1]$-bundle over $K^2$. So, $M$ is geometrically atoroidal (there is only one $\mathbb{Z}\times\mathbb{...
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Finitely many Speiser graphs for a given entire holomorphic map of finite type?

Recently, I read definition of Speiser graph or, also called, line complex (see, for example there ). There is a certain ambiguity in its definition and I am going to formulate my question about it. ...
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Atoroidal closed 3-manifold

A 3-manifold $N$ is called atoroidal if any incompressible torus is boundary parallel, i.e. can be isotoped into the boundary. To me, this definition assumes that $N$ has boundary, but I have read a ...
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Formula for action of $\operatorname{SL}(2,\mathbb{C})$ on hyperbolic 3-space [duplicate]

It's pretty standard in 3-manifold topology and hyperbolic geometry that $\operatorname{PSL}(2,\mathbb{C})$ is the orientation-preserving isometry group of hyperbolic 3-space $\mathbb H^3$. I haven't ...
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Klein bottle immersion in 3 space with gaps - why

I watched this video https://youtu.be/q8Umr0BLMiU?t=143 which shows a glass Kleinbottle where the self-intersecting part is "cut out". The professor says that this is a valid immersion into ...
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Borsuk-Ulam theorem for real projective plane

For every continuous map $f$ : $S^2$$\rightarrow$$\Bbb{R}P^2$ (from sphere to real projective plane) does there exist a pair of antipodal points that landed together? A.k.a. there exist $$x,-x \in S^...
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Manifolds with Compressible Boundary

I recently stumbled over the following terminology, but since I am not really familiar with geometric topology I having a hard time to understand it correctly. So, lets start with the following ...
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Explicit form of element in a link group

I have a link which is union of knots $K_1\cup\ldots\cup K_n.$ I do know how to find link group $\pi_1(\mathbb{R^3}-K_1\cup\ldots\cup K_{n-1})$, for example, using Wirtinger presentation. What I want ...
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Homeomorphism type of Dehn filling depends only on the isotopy class of meridian

I am reading Dehn filling which is defined as follows in this lecture note(page 20): Let $M$ be a $3$-manifold and $T\subseteq \partial M$ be an embedded torus. For a homeomorphism $\varphi\colon \...
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Complement of a neighborhood of a smooth conic in $\Bbb CP^2$

In p.4 of this pdf: https://www.intlpress.com/site/pub/files/_fulltext/journals/pamq/2008/0004/0002/PAMQ-2008-0004-0002-a001.pdf, there is the following paragraph and I have some questions about it. ...
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Connection examples and embedding dimension theory

I've read about the "Utility Problem" (i.e. three utilities and three customers) requiring three dimensions to accomplish/attaching/embedding; and the Klein bottle requiring four (space-like)...
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Gluing up points of Torus to get a new 2-manifold.

Lets define Torus as set all pair (z1, z2) where z1,z2 $\in$ $\mathbb{C}$ and |z1| = 1,|z2| = 1. Now if we glue together (z1, z2) $\sim$ (-z1, -z2) What the new 2-manifold look like? My guess is its a ...
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How to prove that braid index of a specific knot is at least 3?

I have a specific knot $K$ (it seems that it is $6_3$) and want to find its braid index. I managed to construct a braid with 3 strings whose closure is $K$, however I do not know whether 2 strings ...
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signature of 4-manifold

I am currently reading Quantum Invariants of Knots and 3-manifolds by G.Turaev. In Section 2.2, the book gives a way to define invariant of 3-manifolds by Dehn surgery and signature. I have known what ...
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Proving the connectedness of a connected surface deleting a proper closed disc

Let $S$ be a connected surface. Pick $x\in S$, then by definition there exists an open neighborhood $U_x\ni x$ of $S$ homeomorphic to an open disc $D(0,1)\subset \mathbb{R}^2$. To exclude the case $S=\...
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Pure mapping class group of the sphere with marked points

Consider Pure Mapping Class Group of the sphere $\mathbb{S}^2$ with finitely many marked points $P$, i.e. $$ \mathrm{PMCG}(\mathbb{S}^2, P) = \{\varphi \in \mathrm{Homeo}^+(\mathbb{S}^2) : \varphi|_{P}...
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Definitions of irreducible three manifold

I am trying to understand the definition of irreducible $3$-manifold. Let $M$ be a connected $3$-manifold. Definition 1: We say $M$ is irreducible if given any smoothly embedded submanifold $S$ of $M$...
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Taking the boundary of a tubular neighborhood

By $\mathrm{Tub}({\bf RP}^2)$, I mean a small tubular neighborhood of the standardly-embedded ${\bf RP}^2$ in ${\bf CP}^2$ (embedded as the fix-point set of conjugation $\mathrm{conj}:{\bf CP}^2\to{\...
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$s$-isotopy of braids is an equivalence relaiton

So, I have been reading Chapter 6 of "A Study of Braids" by Murasugi and Kurpita. The main goal of the chapter is to prove that many notions of equivalence of braids actually coincide. I ...
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3 votes
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Equivalence of triangulations and piecewise-linear triangulations in dimension $d\leq 4$

There are two different notions of triangulations for a manifold $\mathcal{M}$: A (simplicial) triangulation is an abstract simplicial complex $\Delta$ such that $\vert\Delta\vert\cong \mathcal{M}$. ...
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3 votes
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criterion to show manifold only admits trivial principal bundle

Suppose $M$ a $n-$ dimension manifold and $N$ a $n-$ dimension manifold with boundary. If there exist an embedding $N\to M$ and $M$ only admits trivial principal $G$ bundle. Can we show that $N$ only ...
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Quick proof clarification: Show 1 dimensional manifold is homeomorphic to $\mathbb{R}$ or a circle

I am trying to see it in the case where we just have to open sets $U,V$ that cover the connected manifold $M$, each of which is homeomorphic to the real line. I know that the answer depends on the ...
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2 votes
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3-manifolds with one spherical boundary component

My question is the following: Can all compact, connected and orientable 3-dimensional topological manifolds $\mathcal{M}$ with connected boundary $\partial\mathcal{M}\cong S^{2}$ be obtained by ...
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Is the Alexander polynomial of the 2-cable of a knot zero?

I am trying to find a proof (or counterexample) of the following claim: the Alexander polynomial of the 2-cable of a knot (where the blackboard framing is considered) is zero. That is, the Alexander ...
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Intersection form of a homology manifold

Recall that for a closed orientable 4-manifold $M$ with fundamental class $M$, there is a intersection form $H^2(M)\times H^2(M)\to \Bbb Z$ defined by $(a,b)\mapsto \langle a\cup b, [M]\rangle$, where ...
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Prove that $\frac{H_1(\Sigma)}{[\alpha_1], \ldots, [\alpha_g], [\beta_1], \ldots, [\beta_g]} \cong H_1(Y)$

I am looking at Definition 2.11 in this paper: https://arxiv.org/pdf/math/0101206.pdf. In particular, given a Heegaard diagram $(\Sigma, \alpha, \beta)$ for a 3-manifold $Y$, how to prove the ...
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Homology of 4-manifold minus a small ball

Let $X$ be a simply connected compact 4-manifold without boundary. We know that $H_4(X)=\mathbb Z$. Intuitively, I can see why removing a small (open) 4-ball $D^4$ would kill the top homology, but I'm ...
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Explicit calculus of Hilbert-Smith conjecture

According to the answer of this related question, isometry actions on any manifold form a locally compact group and are closely related to Hilbert-Smith conjecture. As the $3$-dimensional case is ...
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3 votes
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How to recognize torus knots

I have a list of 185 knots with less than 15 crossings and I want to check which of them are torus knots. I know there are many invariants being able to exclude some of them being torus, but I was ...
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1 vote
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Essential surfaces in 3-manifold

I am reading about essential surfaces and I know that $F\times \{pt\}$ is essential within the mapping torus of $F$. I also know there are essential tori in $F\times S^1$. Are there other broad ...
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2 votes
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Example of a 3-dimensional manifold with toroidal boundary, which is not a solid torus

I am looking for an example of a $3$-dimensional compact and orientable topological manifold $\mathcal{M}$ with boundary, such that $\partial\mathcal{M}$ is homeomorphic to a $2$-torus $T^{2}=S^{1}\...
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4 votes
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How to tell if a straight line will pass through a closed loop in 3-space?

The loops I am talking about are unknots (rubber bands in 3-space). What is the mathematical difference between a straight line that passes through the "inside" a closed loop in 3-space ...
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what is the universal covering of surface with genus 2 [closed]

I'm looking for the universal covering for any closed surface, and I gusse that we need only to find the universal covering of $S^2$, $M_1$, and $M_2$, where $M_j$ is the closed surface of genus $j$. ...
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4 votes
2 answers
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If an embedding of $S^1\times I$ has nice edges, are the edges ambiently isotopic?

Let $f_t(x)$, $~t\in I$, $~x\in S^1$ be an embedding of $f:S^1\times I\to\Bbb R^3$. (Thus for example the images of $f_{t_0}$ and $f_{t_1}$ are disjoint if $t_0\ne t_1$.) It's known that the images of ...
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Essential Surface in $F\setminus\!\{\text{point}\} \times S^1$

So I am reading about essential surfaces, and I know that there is an essential torus in a punctured surface times $S^1$. I just don't see it? The only torus I can think of would be one $\partial$-...
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1 vote
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Intersection form of a 4-manifold with boundary

For a closed oriented 4-manifold $X$, the bilinear intersection form $H_2(X)\times H_2(X)\to \Bbb Z$, $(a,b)\mapsto \langle PD(a)\cup PD(b), [X]\rangle$ is unimodular, which can be shown by Poincare ...
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