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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5
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1answer
42 views

Help with moduli spaces of four-marked spheres

I'm thinking about the moduli space of the four-punctured sphere where some of the removed points are distinguishable and some are indistinguishable. I believe there should be some covering maps ...
2
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1answer
34 views

Finiteness in Prime Decomposition Theorem for 3 manifolds

In Allen Hatcher's text on 3 manifolds, he proves the Prime Decomposition Theorem by showing that a collection S of 2 spheres embedded in a smooth and compact 3-manifold M satisfying the condition ...
2
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0answers
56 views

Notes on Low-Dimensional Topology

I am studying algebraic topology at the moment and I'm halfway done with Hatcher's book. I am extremely interested in low-dimensional topology, so I was wondering if anybody knows a good set of notes ...
3
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0answers
58 views

Understand the topology of this 4-dimensional data.

Now I have a set of 4 dimensional data points. If I project them onto the first and second dimension, they look like figure 1. If I project them onto the third and forth dimension, they look like ...
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1answer
63 views

Neighbourhood retract of a loop

Suppos $f:S^1\to M$ is a smooth map and $M$ is a smooth manifold. Does there exist a neighborhood $\mathcal N$ of the image $f(S^1)$ which retract (or deformation retract) to $f(S^1)$? If so, is there ...
3
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1answer
69 views

References on hyperbolic geometry and Teichmuller Theory

I am asking a soft question here. I am learning hyperbolic geometry on my own. Recently, I have completed the book "Fuchsian Groups" by Svetlana Katok. Also, I have background in Lee's three ...
1
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1answer
52 views

The 3-component hopf link covering the trefoil

Let $H$ be the complement of the 3-component Hopf link, which is homeomorphic to the complement of the 3-chain link if you prefer, and let $T$ be the trefoil knot complement. I recently encountered ...
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3answers
70 views

Recommendation for Low-dimensional topology textbook [closed]

Can anyone here recommend a low-dimensional topology textbook that contains knot theory and 3,4-manifolds?Or should I look for these subjects in separate textbooks?
2
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1answer
65 views

Reference request for 3-manifold

I am asking a soft question. I am planning to learn $3$-manifold using the book "Geometry and topology of three-manifolds" by William Thurston. I want to know how much of Riemannian geometry,...
3
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3answers
199 views

What are the dimensions of a 4D cube represented in 3D?

I'm hoping to construct a physical model of a 4D cube. However, I'm struggling to work out the proper size ratio between the inner and outer cubes. From the graphics I've seen, the inner cube seems ...
0
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1answer
29 views

Triangulation that includes given subcomplex

I was reading Prasolov and Sossinsky's book Knots, Links, Braids and Three-Manifolds and came across the following statement in the proof of Theorem 9.2: We can assume $L^3$ has a triangulation $K$ ...
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1answer
32 views

Connected sum of handlebodies has the homotopy type of a $1$-dimensional CW-complex

Let $M^3$ and $N^3$ be two compact connected $3$-dimensional handlebodies. It is easy to see that they have the same homotopy type of $1$-dimensional CW complexes. Is it true that their connected sum $...
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1answer
37 views

Connected sum of a link and a knot

Let $L=K_1 \cup K_2$ be a two-components link in a copy of $S^3$ and let $K$ be a knot, thought in a different copy of $S^3$. In other words, we have two couples $(S^3, L)$ and $(S^3, K)$. Let us ...
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0answers
53 views

Product of Mobius Band with Itself

I have a few questions about the Mobius band $M$. The first two questions are pretty direct, whereas the third is a bit vague. Let $F_2(M)$ denote the collection of non-empty subsets of $M$ with at ...
3
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2answers
68 views

Is every fundamental group of a mapping torus of a surface homeomorphism a CAT(0) group?

TLDR; Is every fundamental group of a mapping torus of a surface homeomorphism a CAT(0) group? Let $S$ be a compact surface of negative Euler characteristic and let $f :S\to S$ be a homeomorphism. ...
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0answers
25 views

Some version of the Annulus theorem

Let A,B,C,D be tame (not wild) n-1-spheres inside the n-sphere S such that $A\cap B = C \cap D = \emptyset$. Let also $f: A \rightarrow C$ and $g: B \rightarrow D$ be two homeomorphisms. Is there ...
3
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1answer
95 views

Homology of Alexander Horned Sphere

I am taking a course in homology this semester, and so far we have only examined spaces/surfaces that the simplicial structures are rather easy to find. I was curious about the Alexander Horned Sphere,...
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1answer
75 views

How to find generators in simplicial cohomology? For example, for $S^1$ and $S^2\dots$

Is there a way to find generators in cohomology groups? Any algorithm or even philosophical remark would be highly appreciated. For example, I have $S^1\cong K$ represented by $1$-dimensional ...
1
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1answer
56 views

Alexander polynomial of some 2-component links

I would like to understand whether the multivariate Alexander polynomial of a link $L$ does not vanish everywhere for a certain class of links; I don't know the links' diagrams in general but I have ...
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1answer
77 views

Proving that the complementary of a surface is connected

I must prove that, being $S$ a differentiable surface of dimension $d$, which is a closed subset of $\mathbb{R}^k$, where $k\geq d+2$, one have $\mathbb{R}^k-S$ connected. This is trivial for $d=1$, ...
3
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1answer
80 views

Can Isotopic Arcs be Isotoped So That They Don't Intersect Until $t = 1$?

Suppose $A, B$ are two disjoint arcs in $X = \mathbb{R}^3$. An 'arc' is a homeomorphic image of $[0,1]$. Let $F$ be an ambient isotopy on $X$ carrying $A$ to $B$ - that means there is a ...
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0answers
45 views

Question about $3$-Sphere Characterization

Suppose that $X$ is a $3$-dimensional ENR continuum (a compact, connected, metric space that is a Euclidean neighborhood retract). Suppose that $X$ also satisfies the following: If $a, b \in X$ and $...
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0answers
34 views

Topological name of nearly-reversed torus

I have a mountain floating in space. Topologically, it's a sphere. I put a cave in the mountain. Topologically, it's still a sphere. I add a stalactite (a cone of rock hanging from the roof of the ...
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0answers
41 views

Double of a drilled out hyperbolic 3-manifold

If one starts with a closed hyperbolic 3-manifold and removes the interior of a some number of solid tori. The resulting manifold with boundary can be doubled along its boundary. Does this doubling ...
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0answers
16 views

Equivalent bridge representations of a link

I'm looking for something like an analogue of Markov's theorem (which states necessary and sufficient conditions for two braids to be closure equivalent) in the context of bridge representations of ...
1
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1answer
47 views

Compact 3-manifold with compressible boundary tori?

There is much to be said about 3-manifolds with boundary consisting of a possibly empty collection of incompressible tori. I don't seem to know where to look to find much about 3-manifolds with ...
1
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1answer
54 views

Surface bundle with flat connection and monodromy representation

I saw the following statement: An $S_g$-bundle $p: E\to B$ admits a flat connection if and only if the monodromy representation $\rho:\pi_1(B)\to\text{Mod}(S_g)$ lifts to $\rho':\pi_1(B)\to\text{Diff}...
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2answers
66 views

Existence of solution for a linear system mod 2

Let $A$ be a (skew-) symmetric matrix over $\mathbb{Z}/2$. (In fact, I would take $A$ as the linking matrix of an oriented framed link in $S^3$ or the matrix representing the intersection form on a ...
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1answer
45 views

Boundary of a connected sum: is it connected?

Let $M^3$ and $N^3$ be compact orientable manifolds with connected boundary. When we perform the connected sum operation, will the resulting manifold $X = M \#N$ have a connected or disconnected ...
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2answers
63 views

proof two general dimension inequalities

I have just showed that $A+B$ is indeed a subspace by showing it is non empty and closed under multiplication and addition, however I am not sure how to do these types of problems with $\mathbf{dim}$. ...
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0answers
41 views

Homology of a closed $3$-manifold with balls removed

Let $M^3$ be a closed, connected and orientable smooth $3$-manifold and let $\mathring{M}$ denote the manifold $M$ with $n$ disjoint open balls $B_1, \dots, B_n$ removed. I am trying to compute $H_2(\...
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1answer
36 views

Relationship between ends of a hyperbolic 3-manifold and ideal boundary components

I've been reading Sullivan's paper "Quasiconformal homeomorphisms and dynamics II: Structural stability implies hyperbolicity for Kleinian groups", and I was hoping I could get some ...
3
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1answer
72 views

Realization of subgroups of $\pi_1(M)$ by submanifolds

Let $M$ be a smooth oriented connected 3-manifold, let $G$ be the fundamental group $\pi_1(M)$. If $M$ is compact, by Kneser-Milnor Decomposition Theorem $M$ can be written as a connected sum of prime ...
1
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1answer
95 views

Haken's Algorithm for Unknot Recognition

I'm an undergrad studying knot theory and I'm exploring the problem of unknot recognition. I believe I understand Haken's algorithm at a high level but I'm having trouble understanding why it is ...
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0answers
72 views

What does the commutator subgroup of a knot group “look like”?

For $X = S^3 - K$ a knot complement, the abelianization $$1 \to [\pi_1(X),\pi_1(X)] \to \pi_1(X) \to \mathbb{Z} \to 1$$ has a right split, where $\mathbb{Z} \cong \pi_1(X)^{ab}$ is realized by a ...
0
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1answer
39 views

Can we see both sides of faces of 4D tesseract from 3D?

To see the whole surface of a cube it is sufficient to have two viewpoints above two opposite vertices of a cube. We can see three faces of a cube from the first viewpoint and other three faces from ...
1
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1answer
55 views

Are topological 1-manifolds embedded in 2-manifolds always locally flat?

I would like to know whether the following is true: Let $M$ be a topological 2-manifold (without boundary), and let $i: [0,1] \to M$ be a continuous embedding. Then $\mathrm{Im}(i)$ is locally flat in ...
2
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0answers
47 views

Assigning a “canonical geometry” to a Seifert surface

Edit: I have since crossposted this on mathoverflow: https://mathoverflow.net/questions/375024/soft-question-assigning-a-canonical-geometry-to-a-seifert-surface Suppose I have a knot $K: S^1 \...
3
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1answer
172 views

Covering space of the 2-torus with finitely many sheets?

I am trying to find all possible covering spaces of the 2-torus $T = S^1 \times S^1$ that is finitely-sheeted of degree $n$, that is, each point $x \in T$ has a neighborhood $U_x$ whose preimage under ...
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0answers
42 views

Why do we thicken $\Sigma$ up when we construct a 3-manifold from a Heegaard diagram?

From a Heegaard diagram $(\Sigma, \alpha,\beta)$, the usual way to construct a 3-manifold is to look at $\Sigma \times [0,1]$ and attach 3D 2-handles to each of the beta curves in $\beta \times \{1\}$ ...
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0answers
24 views

How to represent the intersection of two 3 dimensional spaces

In a 2 D space (a plane) the intersection of 2 lines is a point. In a 3 dimensional space, the intersection of two planes is a line. So, in a 4 dimensional space, the intersection of two 3 dimensional ...
2
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0answers
35 views

Reference Request: Software for Viewing Seifert Surfaces (Linux-compatible Seifertview alternatives)

As the title suggests, I'm aware of and have used Seifertview before. It's an excellent tool for folks running Windows machines. However, I recently switched to Ubuntu and won't be able to use this ...
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0answers
49 views

Repelling curves on low genus surface

I'd guess this is well studied and I just don't have the keywords. I have $N$ non-intersecting lines crossing from the left side of a rectangle to the right side. Inside the rectangle, there are ...
9
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2answers
238 views

$4$-manifold with fundamental group $\Bbb Z/4\Bbb Z$

Before I write my question, I want to write some thoughts. Let $M$ be a connected topological manifold such that $\pi_1(M)=\Bbb Z/3\Bbb Z$. Then, considering its orientation $2$-fold cover, which is ...
2
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1answer
49 views

Fibration of the $n$-holed torus

Let $X$ be the $n$-holed torus, a $2$-dimensional manifold. For $n = 0$, there is a fibration $S^3 \rightarrow X \cong S^2$ with fibers $S^1$. For $n = 1$, there is a fibration $\mathbb{R}^2 \...
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0answers
25 views

What is the point of the differing appearances of the HOMFLY skein?

What is the point of the differing appearances of the HOMFLY skein? For example, can somebody explain to me why one may prefer the relation $tP_P(q,t)-t^{-1}P_-(q,t)=(q-q^{-1})P_0(q,t)$ to the ...
4
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0answers
78 views

Are these two links equivalent?

Are the following two links equivalent (orientation preserving isotopies)? The two links have the same linking number. The only difference is the crossing that in one case is positive while in the ...
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0answers
49 views

The Seifert form of a fibered knot.

Is the Seifert form of a fibered knot always unimodular? It is known that $H_1 (F)$ $\cong$ $H_1 (S^3- F)$ for any knot. Probably we have to use the Alexander duality, but I don't see how.
1
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1answer
64 views

Orientation of torus links and fiberedness

I know that there is this result of Milnor that all algebraic links are fibered. And the $(p,q)$-torus link is an algebraic link. But then I'm reading this paper of Baader and Graf (http://dx.doi.org/...
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0answers
42 views

Difficulty in understanding Towards principles methods for training GANs.

I was reading the research paper TOWARDS PRINCIPLED METHODS FOR TRAINING GENERATIVE ADVERSARIAL NETWORKS. I got stuck at Section 2 Lemma 1 and it's proof in Appendix A. I have following questions, ...

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