# 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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### Intuition for the homology of the klein bottle, specially the $\mathbb{Z}_2$ part

I understand the method of getting the homology but I dont know how to interpret it: Ive always interpreted 1-homology as the number of independent 1-cycles you can do on a surface. For example for ...
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### "Exact" continuous form on the Klein Bottle

Let $S,T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ be defined through $S(x,y)=(x+1,-y)$ and $T(x,y)=(x,y+1)$. Show that if $\omega$ is a continuous and invariant ($S^*(\omega)=T^*(\omega)=\omega$) 2-form ...
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### Kneser theorem about the Klein bottle

I know that in $1923$ H. Kneser showed that a continuous flow in a Klein bottle without singular points has a periodic trajectory. The original article is this, but does anyone know another old or new ...
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### Why the fiber of the normal bundle of the Klein bottle is $\mathbb{R}^2$ instead of $\mathbb{R}$?

What I know: If M is an m-dimensional manifold embedded in $\mathbb{R}^{{m+k}}$, the normal bundle $NM$ is with the typical fiber $\mathbb{R}^k$. My Question: When I think about a torus ($T$) or a ...
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### 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 ...
Using Künneth formula I calculated $H_n(K\times S^1) = \begin{array}{cc} \{ & \begin{array}{cc} \mathbb{Z} & n= 0 \\ \mathbb{Z}^2\bigoplus \mathbb{Z}_2& n=1 \\ \... 1 vote 2 answers 567 views ### Möbius strips with 3 twists to make a Klein bottle I've been looking into Klein bottles and Möbius strips. What would happen if you took two "Möbius" strips with three twists in them, each oriented opposite eachother, and then connected the edges. ... 0 votes 0 answers 418 views ### Sphere with 1 disk removed and replaced with a Möbius strip 1) I read that a Klein bottle is in fact a sphere with 2 disks removed and replaced by Möbius strips. I find it hard to imagine how this constructs a Klein bottle. Any ideas how I can convince myself ... 0 votes 1 answer 281 views ### Show that the circle represented by arc b is not a rectract of the Klein Bottle. You tried to solve the following exercise, but the truth is I do not have many ideas, can I help you solve it please? The figures of the exercise$ 12.12 $are: As for the suggestion, 3 votes 0 answers 117 views ### Is the 3d model of a Klein bottle a cross-section of its 4d embedding? (or homeomorphic to it?) Here's what prompts this question. The way we draw a 3d cube on 2d paper is essentially the image of the cube's skeleton under a retract to a plane cross section of the 3d ... 0 votes 1 answer 99 views ### Inverse of Parameterization of Klein Bottle On wikipedia i found a parameterization of the immersion of the Klein Bottle into 3-d space (see Image). Does anyone know how to compute its inverse? Given coordinates$(x,y,z),$I would like to ... 0 votes 0 answers 135 views ### Is$\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 1 answer 2k 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 1 answer 272 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 0 answers 77 views ### How to find an embedding of the Klein bottle into$S^{1}\timesS^{2}$[closed] This is from an intro topology course. How can I go about finding an explicit embedding for the Klein bottle in this situation? 5 votes 1 answer 445 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 1 answer 732 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 0 answers 76 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 $$\alpha=\begin{pmatrix} 1 & 0& 1/2\\ 0 & -1 & 1/2\\ 0 &... 6 votes 1 answer 1k views ### Proving Quotient Space of Torus Homeomorphic to Klein Bottle Problem. Let T=S^1\times S^1, where S^1=\{z\in\mathbb{C}:|z|=1\}. Prove the quotient space of T by the equivalence relation (z,w)\sim(\bar{z},-w) is homeomorphic to the Klein bottle. Theorem ... 2 votes 1 answer 216 views ### Show that a function induces an embedding of the Klein bottle in \mathbb{R^{5}} Show that the function f :[0,2\pi]×[0,\pi] \rightarrow$$\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 ...
"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 ...