# Tagged Questions

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

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### p-simplex spanned by elements in the boundary of the unit ball of Thurston norm is in the the boundary of this unit ball.

in completing my thesis I have reached a momentary impass. I am trying to solve an exercise given in the book "Foliations II" by Candel and Conlon. In particular, Exercise 10.4.1, and I can't seem to ...
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### How can a Kirby diagram fail to determine a handle decomposition?

I've read that a handle decomposition for 4-manifold determines a unique smooth structure, and I've also read that every 4-manifold admits a Kirby diagram. So when does a Kirby diagram fail to ...
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### Gluing 3 dimensional tetrahedra with orientation reversing edge

I am not sure how to proceed on exercise 3.2.3 in Thurston's book "Three Dimensional Geometry and Topology". The wording is as follows: "In a gluing of three dimensional simplices, each edge enters ...
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### The first Kirby move and $\mathbb{C}P^2$

A surgery on a link in $S^3$ can be regared as 2-hadnle attachements to $D^4$ (resulting 4-manifold $W$ whose boundary is the result of the surgery). I would like to know how the first Kirby move (...
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### Compute the fundamental and homology groups of $S^3 \setminus K$, where $K$ is two linked copies of $S^1$ in $\mathbb R^3$

Compute the homology groups of $S^3 \setminus K$, where $K$ is two linked copies of circles in $\mathbb R^3$. How about the homology group of $S^3 \setminus K'$ where $K'$ is just one copies of ...
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### Alexander polynomial of sums

A standard argument of knot theory reveals that the Alexander polynomial of the sum of two knots $K= K_1 \# K_2$ is equal to the product of the Alexander polynomial of the two summands $K_1$ and $K_2$....
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### Does pushing point along a loop on a surface induce a homotopy from identity to a homeomorphism of the surface?

Suppose $S$ is a closed surface (assume $\chi(S)\leq 0$ if necessary). Let $p$ be a fixed point on the surface $S$, and $\beta$ be an oriented loop based at $p$. Intuitively, I can put finger on $p$ ...
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### Looking for a “Guide for the Perplexed by Low-dimensional Topology”

The following excerpt is from pp. 56-57 of Loring Tu's (so far very enjoyable) textbook An Introduction to Manifolds (2nd ed.): One of the most surprising achievements in topology was John Milnor’...
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### Presentation of Fundamental Group for a Seifert Fibred Space

I am trying to understand how to make sense of the presentation for a seifert fibred space geometrically. I understand that for each exceptional fibre (with index a/b) you get a generator with torsion ...
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### Can a trefoil knot be stretched to look like a triangle with three knots at the vertices?

Can a trefoil knot be stretched to look like a triangle with three knots at the vertices, like in the right side of the image below, or is that transformation impossible to happen? If possible, what ...
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### 1-surgery on the figure-eight knot: reference request

As far as I know, 1-surgery on the figure-eight knot gives ($\pm$) the Brieskorn sphere $\Sigma(2,3,7)$. However, is there a citeable source for this? Sometimes Thurston's notes are mentioned, but I ...
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### What is the underlying geometry of the space in this advertisement

There is this really intriguing new ad by a large Japanase car manufacturer: Here. The universe in the ad looks to be an orbifold with a cone-axis of order 6, but this is not the math I am comfortable ...
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### How equivalent are the homeomorphism and diffeomorphism groups of 3-manifolds?

Every topological 3-manifold carries a smooth structure, unique up to diffeomorphism. In addition, "up to homotopy", the maps between them are the same: every continuous map is homotopic to a smooth ...
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### Visualization of the fact that the integers defining lens spaces must be coprime

This is related to this question I asked: Visualization of Lens Spaces and is also related to this question by @Earthliŋ: Why are the integers appearing in lens spaces coprime? I understand the ...
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### Visualization of Lens Spaces

I am trying to visualize lens spaces geometrically. While I am aware of the fact that most manifolds which cannot be embedded in $\mathbb{R}^3$ are hard to visualize because of the obvious ...
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### Incompressible tori in 3-manifolds obtained by Dehn surgery on knots

Let $K \subset S^3$ be a knot. Given $r \in \mathbb{Q} \cup \{ \infty\}$, denote by $S^3_{r}(K)$ the 3-manifold obtained by Dehn surgery on $K$ with coefficient $r$. Is it true that: $S^3_r(K)$ ...
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### A(nother ignorant) question on phantom maps

My last question (Is such a map always null-homotopic?) is quite similar. If you do not care about my motivation for these questions, you can skip to the last line. I asked if some assumptions were ...
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### Using Alexander's Theorem to show that the sphere $S^3$ is a prime manifold

I'm completely aware of the triviality of this question, but for some reason, I can't visualize the argument. In Hatcher's 3-manifold notes, the form of Alexander's theorem stating that Every ...
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### A question on Lebesgue's covering dimension

Roughly, a compact, Hausdorff space $X$ has covering dimension $\leqslant n$ if each finite cover $\mathcal{U}$ of $X$ can be refined by a cover $\mathcal{V}$ such that each point $x\in K$ belongs to ...
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### Convention of a continued fraction presentation of a lens space

I want to clarify the following two conventions on a surgery description of a lens space. Let $p$ and $q$ are relatively prime integers. Express  p/q=x_1-\cfrac{1}{x_2-{\cfrac{1}{x_3-\dots\cfrac{1}{...
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### Introduction to Euler structures

I am looking for a basic text on Euler structures, in particular smooth Euler structures, and the relation to combinatorial Euler structures; It is known that given a combinatorial Euler structure on ...
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### Reeb orbit and open books

Weinstein conjecture is about existence of a closed orbit of the Reeb vector field on every contact manifold. On the other hand, we know every contact 3-manifold admits a compatible open book, which ...
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### Lens Space Orientation Reversing Homeomorphism

I am thinking of an example where the connected sum of two three Manifolds depends on the chosen orientation. Hempel gives in his book "3-Manifolds" an example, namely lens spaces. He shows that two ...
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### Heegaard splitting and mapping class group

I would like to ask questions about the definition of the Heegaard splitting. The following are the facts I know. A Heegaard splitting says that any 3-manifold is built up from two ...
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### Plane models from the “word”

I have a "word" for a plane model $abacdc^{-1}db^{-1}$. From what I reckon, it's a torus. But I am not too sure of it. I sketched it up and did some "adjustments". Could it be a projective plane ...
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### Classification of compact 3-delta-complexes made of a single simplex

With a single 3-simplex (by identifying its faces in couples) it is possible to make 39 compact delta-complexes that can be grouped in 8 classes of complexes having the same homology groups (see ...
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### Gluing oriented manifold along boundaries

Let $M_1$ and $M_2$ be oriented manifolds with boundaries. Suppose they have homeomorphic boundaries. I want to glue $M_1$ and $M_2$ along the boundaries via some homeomorphism. To ensure that the ...
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### Complement of a solid genus-2-handlebody in $S^3$

I'm not sure if this is a stupid question or not but is the complement of a solid genus-2-handlebody in $S^3$ also a solid genus-2-handlebody? Thanks!
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### Gluing the ends of a cylinder. Can we get other than a torus?

Let $X=S^1 \times I$ be a cylinder, where $S^1$ is the 1-dimensional circle. If we glue the "bottom" boundary $S^1 \times 0$ and the "top" boundary $S^1\times 1$ by a homeomorphism sending $x\times 0$ ...
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### Is a Möbius Strip in > 4 dimensions impossible?

I seem to remember reading, on a plaque in the math building at Penn State, that Möbius Strips are only possible in 3 and 4 dimensions. In higher dimensional spaces, a Möbius strip will use the extra ...
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### What is the point of triangulating topological spaces?

In a general sense, what is the purpose to triangulating, for example, a 3-dimensional topological space? What advantages does it give if we can triangulate a Seifert-Weber space into 23 tetrahedra? ...
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### Examples of finding norm-minimizing surfaces and Thurston polytope

Let $Y$ be a (compact, oriented, connected) $3$-manifold. Thurston introduced a norm on $H_2(Y, \partial Y)$, which is defined as follow: any class $x \in H_2(Y, \partial Y; \mathbb{Z})$ is ...
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### How to distinguish between knots and links based on knot diagrams/projections

I'm interested in the distinction between knots and links in $\mathbb{R}^3$/$S^3$. In particular, is there an algorithmic way (as in not by sight/intuition) that we can examine the arcs and crossings ...
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### High-Dimensional Topology vs. Low-Dimensional Topology: What are the hard questions in the former?

This is a somewhat vague/non-technical question. I've heard a lot about how the topology of manifolds (smooth or otherwise) is simpler in dimension at least 5, due to the applicability of surgery ...
### fibered knots in $S^3$
Given a fibered knot $k$ in $S^3$, we have the decomposition of $S^3$ as union of $M$ and $S^1\times D^2$, where M is a fiber bundle over $S^1$, with fiber $F$ such that its boundary is the knot $k$. ...