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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A problem on finding cap product structure on $\mathbb {RP}^{2}$ and Klein's bottle .

$\mathbf {The \ Problem \ is}:$ Compute all the cap products for $\mathbb{RP}^{2}$ with $\mathbb{Z}$ and $\mathbb{Z} / 2$ coefficients. Do the same for the Klein's bottle $K$. $\mathbf {My \ approach}...
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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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First homology group of Klein bottle (without using the Hurewicz Theorem)

The Klein bottle $K$ is the connected sum of two real projective planes $K = \mathbb{RP}^2\#\mathbb{RP}^2$ and has fundamental group $\pi_1(K) = \pi_1(\mathbb{RP}^2\#\mathbb{RP}^2) = \langle a, b\mid ...
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What are the surfaces that contain an interior volume (space separating) called? Are they related to orientability?

I know that a "closed" surface is defined as a compact surface with no boundary. I don't have it clear if they have something to do with having an interior volume (completely enclosed volume)...
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Rank 1 distribution not coming from a vector field on Klein bottle

I'm trying to solve an exercise which asks to prove the following: there is a 1-distribution $\mathcal{D}$ on the Klein bottle $K$ which isn't of the form $X_m \mathbb{R}$ for some non vanishing ...
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Is this a single sided surface? Is this a Klein bottle?

In constructing a manifold for neutron decay cosmology the following surface evolved and unless I am mistaken this is a single sided closed surface but is NOT a Klein bottle. Shirley’s Surface $$\left(...
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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 ...
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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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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 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 $$...
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Möbius Strip + Möbius Strip = Klein Bottle, What about Klein Bottle + Klein Bottle =?

We know that 2 Möbius-Strips can be joined edge-wise to eliminate that edge producing 0 Edge topological structure. ML (Möbius Left) + MR (Möbius Right) = KOJ (Simple "inverted sock" Klein ...
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Is the Klein bottle a quotient of a $\mathbb{Z}\times\mathbb{Z}$-action on the plane?

One of the basic examples of a group action on a topological space is the $\mathbb{Z}\times\mathbb{Z}-$action on $\mathbb{R}^2$ whose quotient is a torus. There is also an example of a $G-$action on $...
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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 ...
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Relative homology of Klein bottle times two circles

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 \\ \...
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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. ...
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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 ...
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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,
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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 ...
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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 ...
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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- ...
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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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1 answer
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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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How to find an embedding of the Klein bottle into $S^{1}$ $\times$ $S^{2}$ [closed]

This is from an intro topology course. How can I go about finding an explicit embedding for the Klein bottle in this situation?
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5 votes
1 answer
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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)...
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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 ...
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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 &...
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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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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 ...
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Is there any injective parametrization of Klein bottle?

"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 ...
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2 votes
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Different constructions of Klein's Bottle

Consider the following constructions for the Klein's bottle: 1) $\mathbb{R}^{2}/G$ : where $G=\langle f_1, f_2\rangle$, such that, $f_1, f_2:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ given by $...
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6 votes
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Proving that a particular map is an embedding from the Klein bottle into $\mathbb{R}^4$

I would appreciate if anyone could help me with the above: Show that the map $g:\mathbb{R}^{2}\rightarrow \mathbb{R}^{4} $ given by $$g(x, y)=((a+r\cos2\pi y)\cos 2 \pi x, (a+r\cos 2 \pi y)\sin 2 \pi ...
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6 votes
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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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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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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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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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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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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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4 votes
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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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