Questions tagged [klein-bottle]

The Klein bottle is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. It was first described in 1882 by the German mathematician Felix Klein.

Filter by
Sorted by
Tagged with
4
votes
1answer
51 views

What do you get when you cut a disk out of a Klein Bottle?

I heard that you can obtain a real projective plane by gluing a disk to a Mobius band. But then I thought: if you cut a disk out of a Klein bottle (1 face, 0 edges) you'd get a shape with 1 face and 1 ...
0
votes
0answers
30 views

How I prove that the Klein´s bottle is a Hausdorff space?

I have tried to do it separating in cases depending on the different types of points, but I didn´t convince myself.
1
vote
1answer
72 views

Nonnormal covering space of Klein bottle by Torus

I've been trying to construct a non-normal covering space for a Klein bottle $K$ by some torus T. I've found some non-normal subgroups of $\pi_{1}(K)=<a,b|abab^{-1}=e>$ that should correspond ...
2
votes
1answer
59 views

A quotient space of a closed annulus is homeomorphic to a Klein bottle

I was told that a quotient space of a closed annulus centered at the origin obtained with a relation $x \sim - x$ for $x$ in the boundary is homeomorphic to a Klein bottle, which is a connected sum of ...
1
vote
2answers
115 views

Computing the cohomology groups of the Klein bottle as a $\Delta$-complex

I am currently working on how to compute the cohomology and ring structure of certain surfaces who are given as $\Delta$-complexes such as the Kein bottle pictured below. For this i encountered this ...
0
votes
0answers
31 views

Triangulation of Klein Bottle [duplicate]

Is this triangulation of klein bottle right? I need to make a 10 vertices triangulation of klein bottle and i don't know almost anything about triangulation but tried to do one anyway and not sure if ...
2
votes
1answer
65 views

maps on quotients.

I'm trying to define a map over a Klein bottle $\mathbb{K}^2$ but I'm not totally sure on how to do it the right way. My approach is to define over a fundamental domain (a square) and try extending it ...
3
votes
1answer
89 views

How to obtain the Klein bottle as a product of manifolds?

I know the Klein bottle $K$ is a fiber bundle over $S^1$, but my question is: is it possible to find a manifold $M$ such that $K = S^1 \times M$ without the need to take an equivalence relation ...
10
votes
2answers
418 views

Fundamental group of Klein Bottle

It is well know that the fundamental group of the Klein Bottle $G$ is defined by $$G=BS(1,-1)=\langle a,b: bab^{-1}=a^{-1}\rangle.$$ I know, for example that $BS(1,2)$ can be defined as the group $$...
0
votes
0answers
148 views

Möbius Strip + Möbius Strip = Klein Bottle, What about Klein Bottle + Klein Bottle =?

We know that 2 Möbius-Strips can be joined edge-wise to eliminate that edge producing 0 Edge topological structure. ML (Möbius Left) + MR (Möbius Right) = KOJ (Simple "inverted sock" Klein ...
6
votes
0answers
181 views

Is the Klein bottle a quotient of a $\mathbb{Z}\times\mathbb{Z}$-action on the plane?

One of the basic examples of a group action on a topological space is the $\mathbb{Z}\times\mathbb{Z}-$action on $\mathbb{R}^2$ whose quotient is a torus. There is also an example of a $G-$action on $...
0
votes
0answers
24 views

How exactly are fundamental planes derived for topological surfaces?

I've seen a couple examples of these Fundamental Polygons/Planes, but I do not understand how they're derived or what they even mean. I'm fairly new to the subject ...
2
votes
0answers
40 views

Relative homology of Klein bottle times two circles

Using Künneth formula I calculated $H_n(K\times S^1) = \begin{array}{cc} \{ & \begin{array}{cc} \mathbb{Z} & n= 0 \\ \mathbb{Z}^2\bigoplus \mathbb{Z}_2& n=1 \\ \...
1
vote
2answers
95 views

Möbius strips with 3 twists to make a Klein bottle

I've been looking into Klein bottles and Möbius strips. What would happen if you took two "Möbius" strips with three twists in them, each oriented opposite eachother, and then connected the edges. ...
0
votes
0answers
134 views

Sphere with 1 disk removed and replaced with a Möbius strip

1) I read that a Klein bottle is in fact a sphere with 2 disks removed and replaced by Möbius strips. I find it hard to imagine how this constructs a Klein bottle. Any ideas how I can convince myself ...
0
votes
1answer
133 views

Show that the circle represented by arc b is not a rectract of the Klein Bottle.

You tried to solve the following exercise, but the truth is I do not have many ideas, can I help you solve it please? The figures of the exercise $ 12.12 $ are: As for the suggestion,
2
votes
0answers
45 views

Is the 3d model of a Klein bottle a cross-section of its 4d embedding?

(or homeomorphic to it?) Here's what prompts this question. The way we draw a 3d cube on 2d paper is essentially the image of the cube's skeleton under a retract to a plane cross section of the 3d ...
0
votes
1answer
31 views

Inverse of Parameterization of Klein Bottle

On wikipedia i found a parameterization of the immersion of the Klein Bottle into 3-d space (see Image). Does anyone know how to compute its inverse? Given coordinates $(x,y,z),$ I would like to ...
0
votes
0answers
80 views

Is $\langle a, b \mid baba^{-1}\rangle$ truly isomorphic to $\langle a, c \mid a^2c^2\rangle$? [duplicate]

Kosniowski's A First Course in Algebraic Topology sure claims that $$\langle a, b \mid baba^{-1}\rangle$$ and $$\langle a, c \mid a^2c^2\rangle$$ are isomorphic to each other (see ch. 23 "The Seifert- ...
1
vote
1answer
532 views

Fundamental groups of the Klein bottle and torus

I'm confused. I've seen some materials saying that the torus and Klein bottle do not have the same fundamental group. However, although I understand the standard presentations of both groups (the ...
2
votes
1answer
54 views

Is a crosshandle homeomorphic to a Klein bottle?

I am aware that a Klein bottle is homeomorphic to two Möbius bands, and by Conway's zip proof a crosshandle is homeomorphic to two crosscaps. Now, since you can think of a crosscap as a Möbius band ...
1
vote
0answers
71 views

How to find an embedding of the Klein bottle into $S^{1}$ $\times$ $S^{2}$ [closed]

This is from an intro topology course. How can I go about finding an explicit embedding for the Klein bottle in this situation?
3
votes
1answer
181 views

Unit (co)tangent bundle of Klein bottle

The unit (co)tangent bundle of the 2-torus is trivial, whose total space is the 3-torus. Since the torus is a double cover of the Klein bottle, I would imagine this 3-torus double covers the unit (co)...
1
vote
1answer
223 views

Number of vertices in the Klein bottle triangulation given in “Topology and geometry for physicists” by Nash and Sen

I stumbled upon the following triangulation of the Klein Bottle in page 76 of the book Topology and geometry for physicists by Charles Nash and Siddhartha Sen. Below the figure, it says "the ...
2
votes
0answers
61 views

Why the following space is homeomorphic to the Klein Bottle?

Why the following space is homeomorphic to the Klein Bottle? $$\Bbb R^2/ \Gamma,\qquad \Gamma:=<\alpha,\beta>$$ where $$\alpha=\begin{pmatrix} 1 & 0& 1/2\\ 0 & -1 & 1/2\\ 0 &...
6
votes
1answer
405 views

Proving Quotient Space of Torus Homeomorphic to Klein Bottle

Problem. Let $T=S^1\times S^1$, where $S^1=\{z\in\mathbb{C}:|z|=1\}$. Prove the quotient space of $T$ by the equivalence relation $(z,w)\sim(\bar{z},-w)$ is homeomorphic to the Klein bottle. Theorem ...
2
votes
1answer
108 views

Show that a function induces an embedding of the Klein bottle in $\mathbb{R^{5}}$

Show that the function $f :[0,2\pi]$×$[0,\pi]$ $\rightarrow$$\mathbb{R^{5}}$ defined by \begin{equation} f(x,y)=(\cos x,\cos2y,\sin2y,\sin x\cos y,\sin x\sin y) \end{equation} induces an embedding ...
2
votes
1answer
166 views

Is there any injective parametrization of Klein bottle?

"Let $K$ be (the topological space that is known to topologists as) the Klein bottle. There's a standard immersion $f:K \to \mathbb{R}^3$, whose image is known, in popular culture, as "the Klein ...
2
votes
1answer
101 views

Different constructions of Klein's Bottle

Consider the following constructions for the Klein's bottle: 1) $\mathbb{R}^{2}/G$ : where $G=\langle f_1, f_2\rangle$, such that, $f_1, f_2:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ given by $...
6
votes
1answer
201 views

Proving that a particular map is an embedding from the Klein bottle into $\mathbb{R}^4$

I would appreciate if anyone could help me with the above: Show that the map $g:\mathbb{R}^{2}\rightarrow \mathbb{R}^{4} $ given by $$g(x, y)=((a+r\cos2\pi y)\cos 2 \pi x, (a+r\cos 2 \pi y)\sin 2 \pi ...
6
votes
3answers
2k views

Is there a non-trivial covering of the Klein bottle by the Klein bottle?

Let K be the Klein bottle obtained by the quotient of $[0, 1] × [0; 1]$ by the equivalence relation $(x, 0) ∼ (1 − x, 1)$ and $(0, y) ∼ (1, y)$. Is there a non trivial covering of $K$ by $K$? The ...
2
votes
1answer
615 views

Homology of Klein bottle with Mayer-Vietoris

I'm practicing with using the Mayer-Vietoris sequence, and found this computation. I thought it would be a good exercise to try cutting the Klein bottle into two cylinders, instead of into two mobius ...
1
vote
0answers
244 views

Klein bottle is homeomorphic to unit square quotient

let $I=[0,1]^2$. Let's provide $\mathbb{Z}^2$ with the group operation: $(k, l) ∗ (k' , l' ) = (k + (−1)^l k' , l + l' )$ Let's then define $\mathbb{Z}^2$ action on $\mathbb{R}^2$ given by: $((k, l)...
2
votes
0answers
499 views

What's the surface area of a Klein bottle?

I am creating a 3D model of a Klein Bottle based on the Robert Israel formula: Then I need to apply algorithms on the model and I need to know the surface area of this 3D model, then what's the ...
1
vote
0answers
57 views

How to determine boundary of a $3D$ Klein bottle?

A question struggled me for a long time: If a $3D$ Klein Bottle represented by Robert Israel function. How to determine the exact boundary. I mean if you put it in a cuboid, what will the exact values ...
2
votes
0answers
86 views

Fundamental group of $K\# K/\{3\text{ points}\}$

The fundamental group of Klein bottle has been well discussed in many materials, I'm think if it is requested to calculate the fundamental group of $K\# K/\{3\text{ points}\}$ or even more generally $...
3
votes
1answer
459 views

The fundamental group of Klein bottle/{3pt}

The fundamental group of Klein bottle has been well discussed in many materials, I'm think if it is requested to calculate the fundamental group of Klein bottle/{3pt} or even more generally Klein ...
2
votes
0answers
242 views

Topological embedding of Klein bottle into $\mathbb{R}^4$ that projects to usual “beer-bottle” surface in $\mathbb{R}^3$?

What is an explicit topological embedding of the Klein bottle into $\mathbb{R}^4$ whose projection, of some sort, down to $\mathbb{R}^3$ gives the usual "beer-bottle" immersed surface (https://upload....
6
votes
1answer
149 views

Does Day and Night on a Klein bottle have a steady state?

Place a $m \times n$ ($m,n \ge 3$) square grid on a Klein bottle. On each square, we select an arbitrary non-mirror symmetric marker, and arrange them on the Klein bottle in some way. This arrangement ...
1
vote
2answers
219 views

From $[0,1]\times [0,1]$ construct the Klein bottle

The argument by Wikipedia is enough or complete for the solution of the exercise? Exercise: From $[0,1]\times [0,1]$ construct the topological space known as Klein bottle. Wikipedia says ...More ...
4
votes
1answer
389 views

Why an ordered pair gives a torus?

I saw a video (Who cares about topology) that explained the inscribed square problem. The problem say's that you have a simple closed loop in the plane, prove that there's at least one square such ...
1
vote
0answers
351 views

Simplicial complex underlying space homeomorphic to torus/klein bottle

I have read a lot of books and a lot of posts in the Internet and I still can't understand few problems. Professor, who presents us a lecture "Algebraic topology", is doing his job like he must to do ...
0
votes
1answer
587 views

Klein bottle and torus in mod $p$ homology

I read that for mod $2$ homology, the Klein bottle and the torus are indistinguishable. Q1) How does the second homology work for the Klein bottle? It is $0$ in integral homology. I suppose it is $\...
2
votes
1answer
134 views

Circle such that the one point compactification of the Klein bottle minus the circle is homeomorphic to $\mathbb{P}^2$

The first part of this exercise is showing that for any circle $C$ embedded in the Klein bottle $K$, $K-C$ is locally compact. This is not really hard, since $K$ is embedded in $\mathbb{R}^4$ and ...
-2
votes
2answers
501 views

Klein Bottle self intersecting in $\mathbb{R^3}$ and not in $\mathbb{R^4}$ [closed]

The Klein bottle is a surface that has an oval of self-intersection when it is shown in 3-space. It can live in 4-space with no self-intersection. How? I'm having a hard time approaching how to solve ...
1
vote
1answer
1k views

Hatcher exercise 2.1.2 deformation retract of $\Delta$-complex to Klein bottle by edge identifications

I saw the same question posted here. However, by the answer of Ka Ho, I get a torus instead of a Klein bottle. Since they are not homotopy equivalent, this would mean that the quotient of $\Delta^3$ ...
2
votes
1answer
287 views

Is this really a Klein Bottle?

In an exercise it is asked to triangulate the Klein Bottle, and it is presented by this octagon. I really can't see a Klein Bottle here.
0
votes
1answer
167 views

Klein bottle being homeomorphic to the surface with 2 crosscaps

I know this statement is true and I can see the reasoning behind it by the Classification Theorem, but I am still having trouble seeing why it holds. I know the form of the Klein bottle using a square ...
1
vote
0answers
319 views

Hatcher exercise 1.2.13

In this exercise we consider a disk with two holes and we identify the three boundary circles. There are only two essentially different ways of identifying the three boundary circles: one gives a ...
0
votes
0answers
235 views

A peculiar connected sum of two projective “planes”

I start from the knowledge that the connected sum of two projective planes is homeomorphic to the Klein bottle. For the first projective plane I want to use a normal projective plane, e.g. an ...