# 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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### Proving the Fundamental group of the Klein Bottle

I'm trying to prove that the fundamental group of the Klein Bottle is isomorphic to $\mathrm{Z}\ast \mathrm{Z}_{2}$. I have considered the square surface, with one pair of opposite sides with the same ...
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### Slicing and gluing topological objects : a Klein bottle circle dance

Consider this Klein bottle model Slicing the model with a horizontal plane from top to bottom gives this beautiful circle dance (also reminiscent of this MRI scan). Hurray! As far as I know, this is ...
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### Relative homology of Klein bottle to figure-eight

I'm trying to compute the relative homology of the Klein bottle to $S^1\vee S^1$ as its 1-skeleton. Denoting these by $X$ and $A$, respectively, I have the homology groups $H_2(X)=H_2(A)=0$ \[H_1(X)...
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### Visualize $\mathbb {RP} ^ 2 \# \mathbb {RP} ^ 2 \# \mathbb {RP} ^ 2 \# \mathbb {RP} ^ 2$ as an immersed surface in $\mathbb R ^ 3$

I have a short question about Munkres chapter 74 question 4 part (b). (b) Show how to picture the $4$-fold projective plane as an immersed surface in $\mathbb R ^ 3$. In the previous part of the ...
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1 vote
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### Dehn surgery on pseudomanifold to make bonafide manifold

Consider four intersecting open cylinders arranged in the unit cube where the caps of the cylinders are of arbitrariy small radius and 'look' globally as if they coincide with the vertices of the unit ...
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### Homotopy class of a path in a Klein Bottle with two points removed

Klein Bottle with two points removed and path $\alpha$ Hi guys, I'm trying to calculate the homotopy class of path $\alpha$ in the Klein Bottle with points Q and R removed (picture above). For now I'...
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### Homology groups of Klein bottle's unit tangent bundle.

Let $K$ denote Klein bottle and $T^1K$ its unit tangent bundle. I want to compute homology group of $T^1K$, I've seen this discussion: Homology groups of unit tangent bundle, I don't understand much ...
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1 vote
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### Klein bottle immersion in 3 space with gaps - why

I watched this video https://youtu.be/q8Umr0BLMiU?t=143 which shows a glass Kleinbottle where the self-intersecting part is "cut out". The professor says that this is a valid immersion into ...
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### Connection examples and embedding dimension theory

I've read about the "Utility Problem" (i.e. three utilities and three customers) requiring three dimensions to accomplish/attaching/embedding; and the Klein bottle requiring four (space-like)...
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### Homology of Klein bottle with Mayer-Vietoris sequence

There are similar questions to mine, but none that touch the issue I am having: On Wikipedia (Link), the Mayer-Vietoris sequence is applied to compute the singular homology of the Klein bottle. It ...
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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 ...
1 vote
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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 $\pi_{1}(K)= \langle a,b \mid a b a b^{-1}=e \rangle$ that ...
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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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1 vote
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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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### 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 ...
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### 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 ...
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