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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.

The Klein bottle is a two-dimensional non-orientable surface. It was first described in 1882 by the German mathematician Felix Klein.

The Klein bottle canot be embedded in three-dimensional space. That's why images of the Klein bottle always display self-intersections, which do no exist in the Klein bottle itself. However, it can be embedded in four-dimensional space.

Note that if you slice a Klein bottle in half along its plane of symmetry, we get the mirror image of two Möbias strips. One will have a left twist, the other will have a right twist.

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What is the difference between "immersion" and "embedding"?

Could someone please explain what "embedding" means (maybe a more intuitive definition)? I read that the Klein bottle and real projective plane cannot be embedded in ${\mathbb R}^3$, but is embedded in ${\mathbb R}^4$. Aren't those two things 3D…
surface
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Prove that there is a two-sheeted covering of the Klein bottle by the torus

Prove that there is a two-sheeted covering of the Klein bottle by the torus. OK, so we take the the polygonal representation of the torus and draw a line in the middle as follows: Then there are two Klein bottles in there, but how do I write down…
iwriteonbananas
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Calculating fundamental group of the Klein bottle

I want to calculate the Klein bottle. So I did it by Van Kampen Theorem. However, when I'm stuck at this bit. So I remove a point from the Klein bottle to get $\mathbb{Z}\langle a,b\rangle$ where $a$ and $b$ are two loops connected by a point.…
simplicity
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Klein bottle as two Möbius strips.

I read that glueing together two Möbius strips along their edges creates a surface that is equivalent to the so-called Klein bottle. The Möbius strip comes in two versions that are mirrored versions of each other (wrt the chirality of the…
Gerard
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Is there a Möbius torus?

Does the concept of a Möbius torus make sense: taking a cylinder (instead of a rectangle as in the case of the Möbius strip) and twisting it before joining its ends? Or will the resulting twisted torus be indistinguishable from the normal torus in…
Hans-Peter Stricker
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Minimal triangulation of Klein bottle

What is minimal triangulation of Klein bottle? А triangulation is a subdivision of a geometric object into simplices. Minimal in sense of vertex count. So, I know that minimal count of vertices in the shortest triangulation must be greater than…
Gleb
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Why is this map a covering map of the Klein bottle?

I'm trying to find the universal covering space of the Klein bottle. I know that $\mathbb R^2$ covers the Klein bottle , but I don't know how to prove, I found this proof on internet: Someone knows why this quotient map is a covering map or have an…
user42912
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Homology of the Klein Bottle

I know that in general, $H_{n}(X)$ counts the number of $n$-cycles that are not $n$-boundaries of a simplicial complex $X$. So for the sphere, $H_{0}(X) \cong \mathbb{Z}$ since it is connected. Also $H_{n}(X) = 0$ for all $n>0$ (e.g. all $1$-cycles…
NebulousReveal
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Klein Bottle Embedding on $\mathbb{R}^4$.

First of all, I am aware of the question in How to embed Klein Bottle into $R^4$ , which was inconclusive. Anyway, I've made some progress, but I still have a question. I am using Do Carmo's Riemannian Geometry, and struggling to solve a…
Marra
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Klein-bottle and Möbius-strip together with a homeomorphism

Consider the Klein bottle (this can be done by making a quotient space). I want to give a proof of the following statement: The Klein Bottle is homeomorphic to the union of two copies of a Möbius strip joined by a homeomorphism along their…
Crosscap
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Double-cover of a Klein-bottle-esque Space

I'm trying to complete the following exercise I found in a topology book: Construct a space A which is path-connected with fundamental group equal to $\langle r,s | r^2 s^3 r^3 s^2 = 1\rangle$, and find a unique (connected) double-covering B. Find…
Roy M.
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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 $$BS(1,2)=\langle A,B\rangle…
José Luis Camarillo Nava
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Embed the Klein bottle into the 3-manifold $S^{2} \times S^{1}$

Can the Klein bottle $K$ be embedded into $S^{2} \times S^{1}$? If so, how does it work? If not, what is the obstruction? Thanks in advance.
user223515
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Subgroups - Klein bottle

Let $G$ be the fundamental group of the Klein bottle, $G = \langle x,y \ ; \ yxy^{-1}=x^{-1} \rangle = {\mathbb Z} \rtimes {\mathbb Z} \ .$ What are the nilpotent subgroups of $G$? I was only able to find a normal series of abelian subgroups with…
da Fonseca
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How to calculate all the subgroups of the fundamental group of the Klein bottle?

A problem asks me to find all the covering spaces of a Klein bottle. This needs to calculate all the subgroups of the fundamental group of the Klein bottle. But I don't have any idea how to do it. I googled it and an article says The subgroups of…
Roun
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