1
vote
1answer
36 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
23 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
44 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 ...
25
votes
3answers
373 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
63 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
53 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
28 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
46 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
190 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
119 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
45 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
60 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
42 views

T-shaped polygons

Is there any coefficient that can indicate T-shaped polygons ? Examples of T-shaped polygons:
1
vote
1answer
40 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
35 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
37 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
27 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
44 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
13 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
38 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
27 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
28 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
12 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
84 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
61 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 ...
0
votes
1answer
60 views

countable dense subsets

Let $I$ be some uncountable index set. For $\iota\in I$ and $j\in\mathbb N$ let $A_{\iota,j}\subseteq\mathbb R^d.$ Then there exists a countable dense subset in $A:=\overline{\bigcup_{\iota\in ...
3
votes
2answers
80 views

Cardinality of the set in $\mathbb{R^2}$

I am trying to understand the following exercise: Let $v_1$ and $v_2$ be non-collinear vectors of $\mathbb{R}^2$. Estimate, when $r \rightarrow \infty$, the cardinality of the set $(m,n) \in ...
3
votes
0answers
50 views

Crosscap function in $\mathbb{R}^4$ - and how to show it is proper?

I found the Cross-cap function in $\mathbb{R}^3$ as follows: $$f(x,y,z)=(yz,2xy,x^2-y^2),$$ My questions are (I couldn't show any progress for Q1,2.I have thought hard but had no clue): Q1: Is ...
2
votes
1answer
88 views

Volume of an n-simplex (Without Probabilities) [duplicate]

Compute the volume of $$ S_n=\{(x_1,x_2,...,x_n)\in\mathbb{R^n},x_i\geq 0,\displaystyle\sum_{k=0}^{n} x_i<1\} $$ I don't really have an idea how to solve it. My 'work': Perhaps I could use $$ ...
0
votes
1answer
46 views

preimage of a semi-circle under $p(z)=z^2$

Let $e^{i\theta} \in S^1$, consider $f:\Bbb R \to S^1$ given by $f(t)=e^{it}$. Let $U$ be the image of the set $(\theta-\frac{\pi}{2},\theta+\frac{\pi}{2})$ under $f$ it's the open semicircle centered ...
2
votes
3answers
106 views

finding a topological group with specific conditions

I have a question, it sounds difficult. The question is the following: Let $X$ be a topological group such that the binary operation defined on it is $*$. For any two points $a$ and $b$ in $X$ ...
0
votes
1answer
30 views

Support of form and embedded varieties

I need help with some inclusions. Let $i: S \rightarrow M$ be an embedding between two oriented varieties of dimension k and n respectively. Assume that the $i(S)$ is closed and that $\omega\in ...
1
vote
0answers
53 views

Mixed dimension non-Euclidean geometry?

Is the following a "consistent non-Euclidean geometry"? It seems to satisfy the first 4 Euclidean postulates. Any comments? Any agreements or disagreements? Following are the additional conditions on ...
2
votes
1answer
64 views

Is the map from the circular half cone to the $xy$ plane a local isometry?

This is a text book exercise. And I think that this map is not a local isometry. But, I don't know how to show this question. Please help me explaining this question. Thanks a lot. I posted its ...
1
vote
1answer
54 views

The question related to a regular surface.

Prove that an equation of the form $f(x,y,z)=c$ determines a regular surface if $f$, defined on some open subset $S$ of $\mathbb{R}^3$, is smooth and $\nabla f\neq 0$ everywhere in $S$. I know ...
0
votes
1answer
30 views

Intersection of Dense Sets of Parallel Lines

The question is: In $\mathbb{R}^2$, with the usual topology, let $S_i$ be a dense set of points that can be partitioned into parallel lines. Let $m_i$ be the slope of these lines. For any $n\in ...
1
vote
0answers
76 views

prove that $f$ is a diffeomorphism and an isometry

Let $S_1 : [0, 2\pi r]\times [0, h]$ $S_2: x^2+y^2=r^2$ Let $f: S_1 \to S_2$ $(u,v)=(r\cos (\frac{u}{r}), r\sin (\frac{u}{r}), v)$ for $v\in [0,h]$ and $u\in [0, 2\pi r)$ How do I prove that ...
0
votes
1answer
56 views

Verify this is not orientable.

Verify this is not orientable. Möbius transformation: $$U=\{(t,\theta) \mid \frac{-1}{2}\lt t\lt \frac{1}{2}, 0\lt \theta \lt 2\pi \}$$ $\sigma (t, \theta)=<((1-t\sin (\theta/2))\cos (\theta), ...
2
votes
0answers
41 views

Topology of higher dimensional bodies? [closed]

I don't know wether this is relevant or not. But, I've a question about topology and higher dimensional objects. I'm only in high school so don't expect me to know any undergrad mathematics, but I do ...
1
vote
0answers
70 views

Relative Interior and dense subsets

(Due to no answers, I also posted this question here.) Let $A,B\subseteq \mathbb R^d$ be non-empty, such that $B\subseteq \overline A.$ For $S\subseteq\mathbb R^d$ define the relative interior of $S$ ...
0
votes
0answers
24 views

Constructablilty of regular polygons on a sphere

There is a very clear theory of what polygons can be constructed in the plane. One of my professors said that he believed the same ones could be constructed on a sphere through stereographic ...
0
votes
1answer
49 views

How to enclose a ball more than once with a surface homeomorphic to $S^2$? In 3D.

In 3 dimensional space, how to enclose a monopole (either point-like or ball-like, both types exist.) more than once with a surface homeomorphic to $S^2$?
1
vote
1answer
72 views

Parametrization of $S^3$ embedded in $\mathbb R^4$?

I would like to know of any parametrization of the standard 3-sphere: {$(x_1,x_2,x_3,x_4): x_1^2+x_2^2+x_3^2+x_4^2=1$} embedded in $\mathbb R^4$. I know of parametrizations for $S^1$, for $S^2$ , ...
4
votes
1answer
205 views

Non-Euclidean Space in Dungeons and Dragons

In Dungeons and Dragons, the world is mapped out into five-foot squares. Spheres are represented as cubes, and cones look really weird. However, straight lines remain straight, and a rectangular room ...
2
votes
1answer
85 views

Trivialisation of Moebius strip

I've just started studying Advanced Geometry and I'm in trouble with a (stupid) exercise. It's about finding a trivialisation of the Moebius strip (I'll refer to it as $E $) viewed as a fibre bundle ...
5
votes
3answers
151 views

mapping homotopic to the identity map

Please give me a hand with this problem, It was on my exam, and I just couldn't solve it. Suppose $\phi:\mathbb{S}^2\to\mathbb{S}^2$ is a mapping, homotopic to the identity map. Show that there is ...
1
vote
0answers
38 views

Graphs, Tiling and topological equivalence

Let $X \subseteq \mathbb{R}^2.$ I am fully aware that no subset of the plane it topologically equivalent to a torus. However, it seems to require a lot of heavy machinery to prove this. Does it ...
3
votes
1answer
131 views

What figure does one obtain from a Möbius band if one shrinks the boundary circle to a point?

'Im trying to solve the following problem: What figure does one obtain from a Möbius band if one shrinks the boundary circle to a point? I don't really quite understand the problem. What does it ...
1
vote
0answers
36 views

Does there exists a simple imbedding theorem for general topological $n$-manifolds?

I am interested in finding some paper or book where i can find how to build an imbedding $e\colon M^n \hookrightarrow \mathbb{R}^q$ of an arbitrary Hausdorff topological $n$-dimensional manifold $M$ ...