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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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How to find an embedded submanifold in $\mathbb{R}^{n}$ which pass through some given points in $\mathbb{R}^{n}$ and has the lowest dimension?

How to find an embedded submanifold in $\mathbb{R}^{n}$ which pass through some given points in $\mathbb{R}^{n}$ and has the lowest dimension? I have no idea with this question. I know, at least, the ...
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Stiefel Whitney class and intersection form

Why is the second Stiefel-Whitney Class of a closed oriented 4-manfifold, $M^{4}$, a characteristic element for its intersection form? Precisely, why must the following identity hold for closed ...
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Why a special ball in $S^3$ is unique?

I'm studying : An introduction to knot theory(by: W.B. Raymond Lickorish). To prove composition of two oriented knots is unique, Lickorish has written:"regarded $K_1$ and $K_2$ as being in distinct ...
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Can a compact surface-with-boundary admit arbitrarily high genus subsurfaces-with-boundary?

More precisely, does there exist a compact surface-with-boundary $\Sigma$ with the following property? For every $g\geq 0$, there exists a subsurface-with-boundary $\Sigma'\subseteq\Sigma$, where the ...
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Covering space of a compact connected surface without boundary is a compact surface without boundary

Let $p: X' \rightarrow X$ be an n-sheeted cover of $X$. I proved that $X$ being compact implies that $X'$ is compact in the standard way. I started with an open cover of $X'$, projected it under $p$, ...
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The fundamental groups of 3-dimensional spherical space forms

Let $S^3/\Gamma_i\,(i=1,2)$ be a $3$-dimensional spherical space form, where $\Gamma_i \subset SO(4)$ is a finite subgroup acting freely on $S^3$. If $S^3/\Gamma_1$ is homotopy equivalent to $S^3/\...
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Hyperbolic 3-manifolds of finite volume as link complements

This is an improved version of this question (sorry if I wasn't so clear there and sorry if this is well-known and I didn't find the reference). Let $N$ be a hyperbolic 3-manifold of finite volume ...
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Understanding a comment by Thurston

In page 359 (right after Theorem 2.3) of the following paper Thurston, William P., Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Am. Math. Soc., New Ser. 6, 357-379 (...
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Is every hyperbolic 3-manifold of finite volume a link complement in some closed 3-manifold?

The question says all I need to know, but I will try to clarify it a little more. Let $M$ be a compact 3-manifold with nonempty torus boundary such that ${\rm int}(M)$ admits a complete hyperbolic ...
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Is every 3-dimensional Poincaré complex a 3-dimensional topological variety?

I have this question, if every 3-dimensional Poincaré complex is a 3-dimensional topological manifold? Definition (Poincaré complex) $X$ is a n-dimensional Poincaré complex if $X$ have the same ...
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Covering space of a $3$-manifold with infinite fundamental group

For a $3$-manifold $X$, s.t. $\pi_1(X)$ is infinite, how to see the universal covering space of $X$, $\tilde{X}$ is a non-compact $3$-manifold and $H_3(\tilde{X})=0$?
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Homotopically vs geometrically atoroidal

In the book Aschenbrenner, Matthias; Friedl, Stefan; Wilton, Henry, 3-manifold groups, EMS Series of Lectures in Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-154-5/pbk). ...
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Hyperbolization Theorem

I know that for experts this question is trivial, but it's been a while I'm having trouble understanding this... The version of the hyperbolization theorem I found on Aschenbrenner, Friedl and Wilton'...
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Reference request for gauge theory in low dimensional topology

I've been studying 3 and 4 manifold topology and it seems to me that lots of very powerful invariants come from a mysterious place called "gauge theory". When I peer into this place, I am confronted ...
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Obtaining the three torus via Dehn surgery

It is a well known theorem from the '60 (Lickorish-Wallace) that any closed orientable three dimensional smooth manifold can be obtained performing a sequence of integral Dehn surgeries along knots in ...
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Geometry as a Group Action

At 38:45 in this lecture by Thurston he defines a geometry as a an action by a group $G$ on a simply connected topological space $X$ such that the action is transitive and the stabilizer of a point $x\...
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Word length vs hyperbolic length of curves on a hyperbolic surface

Suppose S is a surface which admits a hyperbolic metric - by this I mean a complete Riemannian metric of constant negative curvature, with totally geodesic (possibly empty) boundary. Fix some ...
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Heegaard Diagrams Understanding

I have a question /understanding problem with some Heegaard diagrams (see below). As far as I understand the concept correctly then the Heegaard diagram explains uniquelly how two handlebodies $H_1, ...
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Is there any method to embed $K_p$ into a orientable surface?

It is known that $K_p$ can be embedded into genus $g=\lceil{\frac{(p-3)(p-4)}{12}}\rceil$ orientable surface. Do we know how to embed $K_p$ into the genus $g$ orientable surface?
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Is there a natural family of finite volume hyperbolic $3$-manifolds parametrized by $n$ distinct hyperbolic points?

Let $x_1,\ldots,x_n$ denote $n$ distinct points in the open $3$-ball, thought of as the Poincaré model of hyperbolic $3$-space. Let $X$ denote the complement of these $n$ points $x_i$, $1 \leq i \leq ...
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Khovanov Homology

I'm reading this and trying to understand how he computed the Khovanov homology of the Hopf link. The construction of the chain complexes and the maps look fine to me but the only problem is, I do ...
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Low dimensional structures in a high dimensional space

Let $n \ge 3$ and $x(t) \in \mathbb{R}^n$ be a vector-valued function of $t \in I = [0,M]$. Suppose $x_1(t) = [t,\cdots,t]^T = \textbf{1}t$ which is a line, a 1-dimensional manifold. Then $$x_1(I) = \...
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Geometric presentation of fundamental group of a surface

Let $S = S_g$ be a closed surface. An author of a paper writes: We say $\langle a_1, b_1, \cdots, a_{2g}, b_{2g} \ | \ R \rangle$ is a geometric presentation of the fundamental group $\pi_1(S)$ ...
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Explicit determination of the open book in $S^3$

I'm familiar with the fundamental concepts of algebraic and differential topology. How can I determine explicitly the topology of a page of the open book in $S^3$ given by for example $$f: \mathbb{C}^...
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Three Octagons on a Surface with $\chi=-2$

Given a cubic bipartite graph $G$, living happily on an orientable surface with Euler characteristic $\chi$. Euler's formula then reads: $$ F+V=E+1+\chi , $$ where $F=\sum f_k$, the number of $k$-...
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When does the link of an algebraic singularity determines it algebraic type?

Let $X \subset \Bbb C^n$ be an algebraic hypersurface with an isolated singularity $x$ which is locally irreducible, i.e the local ring $\mathcal O_{X,x}$ is an integral domain (this is a necessary ...
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What are the Jones polynomials for the torus links and the closure of the other braid word below?

I am working on a project to determine the Jones polynomial for the torus links and a class of links which I call tst links. Their braid words are respectively given by $$(\sigma_1 \sigma_2 \cdots \...
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1answer
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Fundamental Group of Connected Sum of two $3$-manifolds

I've been asked to find the fundamental group of $(S^1\times S^2)\#(S^1\times S^2)$. However, connected sum, as briefly defined in my class (in a vague way) was as follows: You remove a $2$-disk from ...
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1answer
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The radial plane is not locally compact

I'm trying to prove that the radial plane is not locally compact. I assume on the contrary that is locally compact and take any arbitrary point in it (say 0). Now 0 has a compact neighborhood $K$. $\...
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Definition of connected sum and orientation problem

I am reading Kosinski's book. To define the connected sum of $M_1^n$and $M_2^n$ (oriented and closed manifolds) we choose two embeddings of the disk $h_i:\mathbb{D}^n\to M_i$ such that $h_1$ preserves ...
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How to show $\operatorname{Int}(z)$ is not empty?

For any closed smooth curve $z:[a,b]\to \Bbb C$, we define the interior $\operatorname{Int}(z)$ of $z$ as follows: Because $\operatorname{Im}(z)$ is bounded, we can find a circle $C$ such that $\...
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Bounded 4-manifold signature

My understanding is that the same 3-manifold may have more than one bounded 4-manifold depending on what particular Heegaard decomposition you use to construct your 3-manifold. I would like to be ...
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Field Theory Phase Factor vs Anomaly

In this paper on topological quantum field theories the authors discuss something called the anomaly in section 5. In Witten's paper on field theory and the Jone's polynomial he discusses something ...
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Brieskorn 4-manifold

As continuation of this question consider subset $N^4=\{Re(x^p+y^q+z^r)=0:x,y,z\in\mathbb C, |x|^2+|y|^2+|z|^2=1\}$ in sphere $S^5$. It is 4-manifold in sphere which contains Brieskorn $\Sigma(p,q,...
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Does the monoid of knots form a ring?

Consider the collection of all oriented knots modulo ambient isotopy. There is a composition operation which turns this collection into a monoid. I would like to know if there is a second operation ---...
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Set of stable laminations dense in $PL(S)$?

I'm working through a paper and I had a question. For a given surface of genus $g>1$, is the set of stable laminations dense in $PL(S)$? I know (or at least, I think) that not every element in $PL(...
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How to think about exotic differentiable structures in manifolds?

I apologize in advance for the vagueness of this question. It is known that there exist differential manifolds that are homeomorphic but not diffeomorphic to spheres (Milnor), and likewise there are ...
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Modularity vs Antipode In the category $Bord^{\operatorname{or}}_{1,2,3}$

Supposedly the category $Bord^{\operatorname{or}}_{1,2,3}$ carries with it the structure of a Hopf algebra. If that's the case I would like to understand what the antipode is. To that end, I've ...
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1answer
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To what extent are homeomorphisms just deformations?

Background. It is often said that two spaces are homeomorphic if, roughly speaking, one space can be continuously deformed into the other without any tearing and gluing. It is then emphasized that ...
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Grid diagram and Seifert surfaces

I have been studying the book "Grid Homology for Knots and Links". In section 3.4 it gives a method to construct a Seifert surface by a grid diagram of a knot. I tried to follow the steps for the ...
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1answer
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Cyclic covers and “cut-open” curves

For any surface $\sum_g$ and any cyclic group $C_n$ we can build a surjection $\phi:\pi_1(\sum_g,x_0)\to C_n$ by building a corresponding surface $\sum_{n(g-1)+1}$ and build $p:\sum_{n(g-1)+1}\to\...
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Dehn Surgery Presentation of the Figure Eight Knot Complement

If $K$ is a figure eight knot how can I realize $S^3-K$ as a Dehn filling on a genus $g$ handle-body? I had the simplistic thought that a genus 5 handle-body and a (5,1) Dehn filling would do the ...
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Euler characteristic in $\mathbb{R}^2/\mathbb{Z}^2$

Define $\hat{\mathbb{R}}^2=\mathbb{R}^2/\mathbb{Z}^2$ and define $p=P(\hat{\mathbb{R}}^2)$, the powerset of $\hat{\mathbb{R}}^2$. I think it follows that $(\hat{\mathbb{R}}^2, p)$ is a topological ...
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1answer
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General topology - challenging problem.

While revising my knowledge in general topology, I tackle various difficult problems. I came across one particular problem at which I got completely stuck. Below is the problem: Assume: on $\mathbb{R}...
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snapPy Simple Example

I'm trying to learn how to use snapPy by doing some examples that I'm familiar with. So, how would I create a genus 1 handleboy and then do a double Dhen filling to create $\mathbb{R}P^3$? Sorry, I ...
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Mapping Class Group of Knot Complements and Their Heegaard Splitting

Using this algorithm and the ideas from this paper (page 3) I gather that I can present $S^3-K$ where $K$ is the figure eight knot as the gluing of two genus 5 handle bodies, $H_5$, i.e. $$ S^3-K=H_5\...
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Finite-sheeted covers of integer homology spheres

For an integer homology sphere $M$, one can show that non-trivial connected finite-sheeted covers of $M$ must have at least $5$ sheets. This is because, for a given $n$-sheeted cover $M'$, one can ...
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1answer
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Is there a solvable subgroup with finite index and finite type in the mapping class group of a surface?

I want to find a subgroup $H$ of the (orientation preserving) mapping class group $G=MCG(g,n)$ of a surface with genus $g$ and $n$ boundary components that satisfies the following properties: $[G:H]&...
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Reference for Dehn Surgery

I'm looking for an introductory reference for the basics on Dehn surgery on links. Does anybody have any recommendations?
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Heegaard Diagrams Via Polyhedra Identification

Thurston described the figure eight knot complement by identifying faces and edges of two tetrahedra here. Just prior to that he briefly introduced Heegaard decomposition. Is there a method to get ...