1
vote
0answers
48 views

A Homeomorphism that is not unique even upto Isotopy

I'm currently reading the following paper by Richard Skora, entitled Cantor sets in $S^3$ with simply connected complements found here, and on page 2, just before Theorem 1, it says "the homeomorphism ...
-1
votes
0answers
64 views

why some solid can have no surface? [closed]

For solid construction, I can understand the closed surface has no edges. But i cannot understand why some solid can have no surface (except just lines?), any other solid which can have no surface?
9
votes
1answer
80 views

Packing infinitely many ellipses into a circle

Given a circle $C$, and an infinite set $S$ of mutually disjoint ellipses which are inside and tangent to $C$, prove that there must exist a disk $D$ which lies inside $C$ but outside every ellipse. ...
-1
votes
0answers
25 views

What does it mean to say “Resolving intersections”

Consider a surface (with boundary) $S$ with marked points on the boundary such that we may may triangulate the surface. Call a line joining two marked points in a triangulation an arc. Consider a ...
4
votes
1answer
62 views

Why doesn't a metric give an isomorphism $TX \cong T^*X$?

Any smooth manifold $X$ admits a Riemannian metric $g$, and we have a map $$ TX \to T^*X, \qquad (x, v) \mapsto (x, g(v,-)) $$ which is smooth if $g$ is. Why isn't this an isomorphism of vector ...
-2
votes
1answer
75 views

Cw complex $\Sigma_g$

Consider the oriented connected compact surface $\Sigma_g$ of genus $g$ with its standard CW structure. How do I write down the attaching map for the single $2$-cell and how can it be proven that it ...
74
votes
18answers
7k views

How to distinguish walking on a sphere or on a torus?

Imagine that you're a flatlander walking in your world. How could you distinguish if the world is a sphere or a torus ? I can't see the difference from this point of view. If you are interested, this ...
4
votes
0answers
76 views

Rolfsen exercise, chord theorem

Here's a problem from Rolfsen's Knots and Links that has me scratching my head: Show that there is always a counterexample to the "chord theorem" if $n$ is not an integer. [Hint: In attempting to ...
0
votes
0answers
27 views

Find close points by grouping points in n-dimensional space

I know I already asked a similar question over here: Finding the closest point in a set to another point in n-dimensional space: efficiently But now my question is different. I have many points ...
1
vote
1answer
49 views

Finding the closest point in a set to another point in n-dimensional space: efficiently

I'm a programmer and am working on writing an efficient algorithm that, given a point P in n-dimensional space, can find the closest point from a set of points. For ...
0
votes
1answer
42 views

Definition singular manifold

I'm looking for the definition of a singular manifold. I haven't found it yet. For instance, in $\mathbb{R}^4$, with $f(x,y,z,t)=xy-zt$, $f^{-1}(0)$ is a singular submanifold. I only found a few ...
1
vote
0answers
37 views

Covering of Riemann sphere

The question consists of several parts: What is the simply connected ramified covering of the Riemann sphere with ramification indexes {2. 3. 5} over three points of RS in every preimage of these ...
1
vote
1answer
41 views

Finite family of subtori in the torus $(S^{1})^{n}$

Working on a problem on matroids, I've already ask a question about some subtori. Here's the link to a previous problem: Topological subspace in $(S^{1})^{n}$ Anyway, here's another problem related ...
0
votes
1answer
47 views

Luzin's theorem, finding a continuous function under a certain condition

Let $X:\mathbb R^2\rightarrow\mathbb R,$ be the map defined by $(x,y)\mapsto y-x.$ Let $h:\mathbb R\rightarrow\mathbb R$ be Borel measurable. Let $\mu$ be a Borel probability measure on $\mathbb ...
2
votes
1answer
92 views

Independence of $H(f)=\int_M \alpha \wedge f^* \beta$ on choice of $d\alpha=f^*\beta$?

I came across the following UCLA qual question while studying for my upcoming qual: Let $f: M^{4n-1} \to N^{2n}$ be a smooth map between closed connected oriented manifolds of the indicated ...
4
votes
1answer
73 views

A norm on $\mathbb{R}^2$ such that $\partial C$ is the unit sphere?

Suppose we are on $\mathbb{R}^2$. Assume that $C \subset \mathbb{R}^2$ is a convex bounded neighborhood of the origin invariant by central symmetry. Let $\partial C$ denote the boundary of $C$. My ...
6
votes
1answer
141 views

Fractals - when the number of seed shapes that can fit into the scaled-up copy is non-integer.

I've heard people say (for eg. here) that the dimension of fractal patterns (particularly, in this question, Lindenmayer fractals) can be formulated as follows: $$D=\frac{\ln N}{\ln S}$$ Where $N$ ...
0
votes
1answer
34 views

Transforming a curve on an arc to a line

I have a function, actually a point cloud, (similar to a sine wave) on an arc with a known radius of curvature. I need to remove the curvature to regenerate the original function (or point cloud). ...
0
votes
1answer
19 views

Relation between dense subsets in the product map and dense subsets in each component

Let $\Omega$ bea Polish space and $X_1,\dots,X_n:\Omega\rightarrow\mathbb R^d$ be Borel measurable maps. Consider now the map $X:\Omega\rightarrow(\mathbb R^d)^n$ defined by ...
3
votes
1answer
44 views

Constructing The Cayley Graph and quasi-isometry to $\mathbb{Z}$

If we have a group $G$ defined by: $G=\langle a,b\mid b^2=1\rangle$ then I first need to construct the cayley graph of this, now I think that this is going to look like the "telephone pole" metric ...
1
vote
1answer
40 views

$T_3$ is quasi isometric to $T_4$

I have a question which asks me to show that $T_3$ is quasi isometric to $T_4$, that is the three and 4 valence trees. I know that this means that I have to define a map $f:T_3\rightarrow T_4$ such ...
0
votes
0answers
17 views

What is the nth barycentric simplicial subdivision?

I read it in a paper, but, unfortunately, am not familiar with it. Here's my guess: For example, in $\mathbb{R}^2$, a simplex is a triangle, and if we divide it by three lines connecting each vertex ...
1
vote
3answers
55 views

What is a “control point”?

I'm trying to figure out a good definition of control point for use in wikipedia (see https://en.wikipedia.org/wiki/Control_point_(mathematics) ) There seems to be a bias towards ascribing a ...
5
votes
1answer
121 views

How can a simple closed curve not look locally like the rotated graph of a continuous function?

A simple closed curve is a continuous closed curve without self-intersections. The question of whether you can inscribe a square in every simple closed curve is currently an open problem, but this ...
1
vote
1answer
52 views

Is it true there exists $f:S^{2n}\longrightarrow S^{2n}$ making the diagram commutative?

Let $g:\mathbb R\mathbb P^{2n}\longrightarrow \mathbb R\mathbb P^{2n}$ be a continuous map where $\mathbb R\mathbb P^{2n}=\mathbb S^{2n}/\{\pm x\}$. Is it true there exists $f:\mathbb ...
0
votes
1answer
30 views

Is there a smooth map from the square to the deltoid?

Is there a $C^\infty$ map between a unit square in $\mathbb R^2$ and a deltoid like this one The deltoid is obtained by varying the angles $\theta_1$, $\theta_2$ in the equations \begin{align} x_2 ...
1
vote
1answer
65 views

How to use Markov-Kakutani fixed point theorem to show that abelian groups are amenable?

Recall that a group $\Gamma$ is amenable if there exists a state $\mu$ on $l^{\infty}(\Gamma)$ which is invariant under the left translation action: for all $s\in \Gamma$ and $f\in ...
29
votes
3answers
457 views

Are there surfaces with more than two sides?

I'm watching a naive introduction to the Möbius band, the lecturer asks if it's possible to construct a one sided surface and then she says that there is one of these surfaces, namely the Möbius band. ...
1
vote
3answers
78 views

Is convex hull of a finite set of points in $\mathbb R^2$ closed?

Is the convex hull of a finite set of points in $\mathbb R^2$ closed? Intuitively, yes. But not sure how to show that. Thanks!
1
vote
2answers
69 views

What is this quotient space of the torus?

Suppose we have a $\mathbb{Z}/2\mathbb{Z}$ action on torus $\mathbb{T} \times \mathbb{T}$ by $(\xi,\theta)$ goes to $(-\xi,\bar{\theta})$. Then what is the quotient space?
0
votes
0answers
31 views

Antipodal map and parallel transport on $S^3$

I'm reading Thurston and Levy's "Three-dimensional geometry and topology" where they have an informal explanation of what life in $S^3$ would look like. For the following detail that I am concerned ...
1
vote
1answer
51 views

Computing the intersections of arbitrary number of planes

Imagine three sets of planes (A, B, and C). The planes in each set are parallel and evenly spaced and so can be defined by a plane equation and a scalar representing the distance between planes. If ...
3
votes
3answers
258 views

Visualizing mathematics and geometry

Im writing a paper on the role of visualization in mathematics and specifically geometry. I was wondering if it is possible to represent any arbitrary system of relations and manipulable objects ...
3
votes
1answer
156 views

Shortcut in proof of continuity/differentiability in inverse function theorem

The messiest, least interesting part of the various proofs of the inverse function theorem comes after you have constructed the inverse function and must now establish continuity and ...
2
votes
1answer
54 views

What are the formulas for topological transformations? How to obtain them?

I'm reading Flegg's From Geometry to Topology, the author says that in Euclidean geometry, translation and rotation are: $$T:(x,y)\to(x+a,y+b)$$ $$R:(x,y)\to(x \cos \phi - y \sin \phi, x \sin \phi +y ...
2
votes
2answers
80 views

Are a finite cylinder and the corresponding planes iso/homeomorphic?

Let me give some context first. In the scope of physics, I often have to compute the area of the side of a right circular cylinder with height $h$ and radius $r$, namely $2\pi rh$. I think this can ...
15
votes
3answers
2k views

Is an infinite line the same thing as an infinite circle?

Imagine that you are sitting next to a line that extends infinitely in both directions. Is it possible to distinguish it from an infinite circle? From my poor understanding of topology, I would ...
1
vote
1answer
50 views

T-shaped polygons

Is there any coefficient that can indicate T-shaped polygons ? Examples of T-shaped polygons:
1
vote
1answer
45 views

Sets defined by distance to a convex set

Let $Y \subset \mathbb R^n$ be a bounded convex set, let $R>0$, and let $$Z := \left\{z \in \mathbb R^n : d(z,Y) > \dfrac12R \right\}$$ where $$ d(z,Y) = \inf_{y\in Y}|y-z|. $$ If you like, ...
1
vote
2answers
36 views

Suggestions on how to view an (anti) self folded triangle.

Consider a triangle. Now chose two edges of this triangle and glue them together by the following orientation. .----->-----.------>-----. |____________| (Where the line underneath is meant to join ...
1
vote
2answers
40 views

Point as an element of an affine space vs point as an element of a topological space?

I am searching for the "most natural" definition of a (geometrical/space) point as an element of "something" in mathematics (I am trying to design a small computational geometry library on strong ...
0
votes
1answer
31 views

Relations between an affine space and a topological space

What is the relation between an affine space and a topological space? Is one a specialization of the other? Moreover, what do we call a point in geometry: an element of a topological space or an ...
4
votes
1answer
51 views

Does an infinite collection of circles accumulates at a circle?

There is an infinite collection of closed circles in the plane, all within a finite bounding square. Does it contain an infinite sequence of circles that converge to a circle? Assume that a point is ...
0
votes
0answers
19 views

Extendability of Contact Structures, Foliations of S^2

I am currently reading Eliashberg's paper on the classification of overtwisted contact structures (http://bogomolov-lab.ru/G-sem/eliashberg-tight-overtwisted.pdf). In it, there is the following ...
1
vote
0answers
42 views

Prove that A has a geometric realization in $\mathbb{R}^d.$

A flag in a simplical complex K in $\mathbb{R}^d$ is a nested sequence of proper faces, $\sigma_0 < \sigma_1 < ... < \sigma_k$. The collection of flags forms an abstract simplical complex A ...
0
votes
1answer
49 views

Connectivity relation

Tiling of a plane fills the plane with a set of polygons. Connectivity of a vertex is # edges coming into it which equals # of faces it touches. Suppose the average # of edges on a polygon is x. ...
0
votes
1answer
32 views

Genus 2 drawing

I am asked to draw a picture of a genus two using only polygons and then calculate the euler characteristic. I take it I'm aiming to use the least amount of polygons for the sake of the diagram but ...
0
votes
0answers
13 views

Countable product of metric spaces

Let $X=\prod X_i$ of countably many metric spaces $(X_i,d_i)$. Prove that the function which associates to $x=(x_i)$,$y=(y_i) \in \prod X_i$ the number $d(x,y)\in [0,\infty]$ defined by ...
3
votes
0answers
96 views

Why can't I tie a infinite rope in hard knots?

I think this is a genuine math problem. And it's somehow related to knot energy but not directly solved by the latter. Why can't I tie a hard knot on a rope of infinite length? By infinity I mean ...
5
votes
1answer
67 views

Continuous curve, traps itself outside the unit circle.

Lets say i have an injective continuous curve $\sigma$ in $\mathbb{C}$, indexed on $[0,\infty)$ and converging to $\infty$. If $\vert \sigma(0)\vert>0$ , is it possible that it can trap itself ...