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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.

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How can I plot an animated Klein Bottle in Graphing Calculator 3D? [on hold]

How can I plot an animated Klein Bottle in Graphing Calculator 3D? I mean, is it possible to change the shape to cut it and see Moebious bands?
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How can I make a Klein's bottle 2x wider?

I'm using this formula to draw a parameterized Klein's bottle, with $a = 6$, and $b = 16$. It looks pretty. Now I want to "wrap" this shape into another, but a $c$-times bigger one. This task is easy ...
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Klein bottle as two Möbius strips with fundamental polygon

My question is the following. If we operate with the fundamental polygon of a Klein bottle in order to obtain two Möbius bands, really you don´t obtain two new Klein bottles? I think that the ...
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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 ...
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106 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 ...
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How Klein Bottle in R⁴ does not have self intersection?

This is the screenshot from Feko's book on Differential Geometry. In 1.5.10 in the given hint how he deduced that in R^4 the Klein Bottle has no self intersection? Also,in 1.5.11, what does square ...
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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)...
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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 ...
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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 ...
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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 $...
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198 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 ...
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Twisted Strip to boundary of surface A\disc\disc is equal A#K\disc

I have struggles to understand the outline of the proof given in "Donaldson-Riemann surfaces". I work with this source: http://wwwf.imperial.ac.uk/~skdona/RSPREF.PDF. On page 21 bottom he mentions the ...
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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....
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Using polygonal representations to show that a Klein bottle can be constructed by connecting two Mobius strips

I am reading introductory material w.r.t. Differential Geometry and I came across a problem that asks to show that a Klein Bottle can be seen as the connected sum of two Mobius strips using polygonal ...
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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 ...
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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 ...
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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 ...
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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 ...
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190 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 $\...
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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 ...
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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 ...
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441 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 $\Delta^3$ ...
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1answer
231 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.
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129 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 ...
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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 ...
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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 ...
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1answer
653 views

The Klein bottle and its Topology

I have read in several places that the Klein bottle is a 2-manifold, but I cannot find an explicit proof anywhere. How would you show it is locally Euclidean, Hausdorff and second countable (I ...
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108 views

Does Klein bottle homeomorphic to a tube whose two end circles each sewed to mobius bands?

enter image description here I've understood how Klein bottle is homeomorphic to the union of two mobius bands whose boundaries are identified. But I could not understand how it could be homeomorphic ...
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Mobius strip to the Klein bottle

From the Youtube, I noticed that the Klein bottle can be sliced into two Mobius strips. Or this experiment is also explained in this following link: http://www.kleinbottle.com/sliced_klein_bottles.htm ...
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Confusion of classification of closed surfaces

I read that we can distinguish closed topological spaces without boundary up to homeomorphism by orientability and euler characteristic - is this correct? But what confuses me is that the Klein ...
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Is this the projective plane or the Klein bottle? (Fundamental polygon)

I am trying to identify the topological type of this fundamental polygon and I think it is the projective plane or the Klein bottle If we treat the top green and red arrows a single arrow then we can ...
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Link between $\mathbb{Z}$ and the fundamental group's of 'common' topological spaces

I have noticed that some of the most common topological space have fundamental groups related to $\mathbb{Z}$, as we can see below: Why is this the case? Is it is because they are all realted to ...
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1answer
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Homotopy 'diagrams' for Klein bottle and projective plane

Background: I recently discovered that the complement to the circle and vertical axis shown below is homotopy equivalent to a torus Also complement to three infinite straight non-intersecting ...
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Klein bottle in $\mathbb{R}^4$ does not have a couple of normal vector fields

For an orientable 2-manifold in $\mathbb{R}^4$, it's somewhat obvious that if the manifold has a normal vector field then it has a pair of normal vector fields. I am trying to understand why it is ...
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355 views

Fundamental group Klein Bottle triangulation

I have been trying to find the FG of the Klein bottle, and I was wondering if someone could verify that this process is correct. After triangulating it, I then found a maximal tree (shown in yellow) ...
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Is this octogon topologically equivalent to the Klein Bottle?

Note: this is an extension of a previous problem (identify the topological type obtained by gluing sides of the hexagon ) where a hexagon was considered. Is the space below also a Klein bottle ($d$ ...
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1answer
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Is the Klein bottle homeomorphic to the union of two Mobius bands attached along boundary circle?

Question: Determine whether the Klein bottle is homeomorphic to the union of two Mobius bands attached along their boundary circles. The Klein bottle is the quotient space $$ K=I^2 /{\sim}, \quad (x,...
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How to tell whether a combinatorial surface is orientable.

I am learning about combinatorial surfaces, and I've encountered this question: What surface is represented by $a_1a_2\cdots a_na_1^{-1}a_2^{-1}\cdots a_n^{-1}$. I know the Euler characteristic is $...
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1answer
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Whether there exists some compact manifold whose continuous map must have a fixed point?

Whether there is some compact manifold $M$ such that $f:M\rightarrow M$ is continuous $\Rightarrow f$ has a fix point. I mean that any continuous map from $M$ to $M$ must have a fixed point. I ...
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1answer
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Klein Bottle as the connected sum of two projective planes

Can someone explain intuitively why the Klein bottle is diffeomorphic to the connected sum of two projective planes? I can do this using origamis/fundamental graphs )w/e they are called. is it ...
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Is torus w. disc removed homotopic to Klein bottle w. disc removed?

I know that homeomorhic spaces are homotopic, but am not sure if this applies, since I think they are not homeomorphic due to orientability. I know $f$ and $g$ are homotopic if they represent: X$\...
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Connected sum of two “same” Klein bottles

If I take sphere and remove two open disks from it and on the boundary of that space I make identification like on the picture, what do I get? Are both of those objects Klein's bottles? This is what ...
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1answer
280 views

Making a Klein bottle from 2 Möbius bands

I thimk this can be done by idemtifying points on the boundary but I am not sure how to show this Any ideas? E.g. By drawing nets..
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Klein Bottles in the Levine traffic model

The Biham–Middleton–Levine traffic model has recently fascinated me. I started learning about it on the Wikipedia Page found here. One way to run this simulation is on a Klein bottle surface. When ...
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Why is the immersed image of the Klein bottle $S^2$ with three closed discs identified?

Someone said that there are three discs glued together on the Klein bottle self-intersecting itself in a circle. Here is a picture: They said that one from the "near" side of the tiny connecting tube,...
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Properties of a closed solid klein bottle?

Okay, so I'll denote the topological space of the klein bottle as $Kb$ (because I don't know what the proper notation is). I'm curious about a closed version of the solid klein bottle, which I believe ...
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Why is $\mathbb{R}^2/G$ homeomorphic to the Klein bottle?

Let $G$ be the group of transformation generated by $a,b:\mathbb{R}^2\to \mathbb{R}^2$ where $a(x,y)=(x+1,y-1)$ and $b(x,y)=(x,y+1)$. We note than $bab=a$ and that $G$ acts properly discontinuously ...
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The Klein bottle can be immersed in $\mathbb{R}^3$ and embedded in $\mathbb{R}^4$

How can one show that the Klein bottle can be immersed in $\mathbb{R}^3$ and embedded in $\mathbb{R}^4$? We define the Klein bottle as the quotient space of $I^2=[0,1]\times [0,1]$ under the relation ...
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1answer
151 views

How can I write Klein bottle as an adjunction space?

I want to find the homology groups of the Klein bottle by Mayer-Vietoris. For this I want to describe the klein bottle as an adjunction space. I think it can written as a pushout $S^1\cup_f D^2$ but I ...
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1answer
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Computing Klein bottle's cohomology ring in $\mathbb{Z}$

Well I've been struggling with this one. This is the picture of the Klein Bottle. It has two triangles (U upper, V lower), three edges (the middle one is "c") and only one vertex repeated 4x. So my ...