# 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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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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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### 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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### 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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### 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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### 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 ...