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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.

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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)...
Ben Carpenter's user avatar
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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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How can I decide whether two groups defined by finite presentations are (or not) isomorphic?

I have the groups $G_1,G_2$ with presentations $$G_1 = \langle x,y : (y^2x)^2 = x^2, (x^2 y )^2 = y^{-2} \rangle = \langle x,y : x^{-1}y^2 x = y^{-2}, yx^2y^3 = x^{-2} \rangle \\ G_2 = \langle x,y : (...
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Filling a Klein bottle with four-dimensional water

Can you fill a Klein bottle $K \subseteq \mathbb R^4$, embedded in four dimensional Euclidean space $\mathbb R^4$, with hyperraindroplets (hyperballs $B_\varepsilon$) from the outside? Will it enclose ...
Markus Klyver's user avatar
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Non-normal coating of the Klein bottle

(a) Provide an example of a non-normal connected covering map $p:(\tilde X, \tilde x_0) \to (K, x_0)$ where $K$ is the Klein bottle. (b) Choose $x_0 \in K$ and $\tilde x_0 \in p^{-1}(x_0)$. State what ...
Andreadel1988's user avatar
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Calculating fundamental group of Klein bottle

To calculate the fundamental group of the Klein bottle, $X$, we can find path connected open sets, $U$, $V$,and $U \cap V$, satisfying $U\cup V= X$. I know we can choose $U$ and $V$ as follows:[![$U$ ...
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Parametrizing a self-intersecting tubular surface centred around a space curve to form a Klein bottle?

As part of my math project in school I am trying to derive a parametrization of a klein bottle. I am starting by creating a tubular/canal surface around the space curve p: $$x(t)=5(1+\sin(t))$$ $$y(t)=...
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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 ...
zeta space's user avatar
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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'...
fish's user avatar
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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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Calculate the fundamental group of a Klein bottle with two points removed

My attempt Using the van Kampen theorem, we get: $$\pi_1 (U_1) \cong \pi_1 (S^1 \vee S^1) = \mathbb{Z} * \mathbb{Z} = \langle \alpha, \beta \ | \ \varnothing \rangle$$ $$\pi_1 (U_2) = \langle a, b \ |...
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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 ...
Mallacan's user avatar
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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 ...
David Melo's user avatar
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Two presentations of the Klein bottle

In Hatcher's Algebraic Topology, Chapter 1, page 51, he describes two presentations of the Klein bottle: The first one is the usual one, a square with opposite sides identified via the word $aba^{−1}...
Dasheng Wang's user avatar
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How can overflow be defined for the Klein bottle in $\mathbb{R}^4$?

I was watching Dr. Tadashi Tokeida's lecture series on Youtube: https://kevinbinz.com/2017/10/25/isotopy/ For submanifolds $L$ and $K$ being placed in ambient manifold $M$, the overflow can be defined ...
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How many different ways can you write $u^5v^2$?

I have been working on a project trying to say that the Submonoid membership problem is decidable for different examples, one of those being the Klein Bottle. I am now left with a problem that I am ...
Max Pateman's user avatar
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Visualizing shaded region in Klein bottle

Regargind the Klein bottle $K$ as its identification space, I want to know what is the subspace of $K$ obtained by identifying the outer edges of the shaded region of the figure. The main reason I ...
Yeipi's user avatar
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Is there a 3D shape that would be *like* a Klein bottle for 2D?

I am not asking what a Klein bottle would look like if put in the 2 dimensions. I am asking about a 3D shape that in 2D would be like how a Klein bottle is to us in 3D. I am mainly asking this to &...
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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}...
Rabi Kumar Chakraborty's user avatar
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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 ...
Phi's user avatar
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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 ...
Bryan Li's user avatar
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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)...
Arkanoid Arkan's user avatar
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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(...
Caleigh Fisher's user avatar
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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 ...
lixa417's user avatar
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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 ...
Finnegan Manthe's user avatar
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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1 answer
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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 ...
Quin Morris's user avatar
1 vote
2 answers
653 views

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 ...
Zest's user avatar
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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 ...
Enes Ince's user avatar
2 votes
1 answer
78 views

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 ...
astro's user avatar
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3 votes
1 answer
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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 ...
Mister Math's user avatar
10 votes
2 answers
1k views

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 $$...
José Luis  Camarillo Nava's user avatar
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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 ...
Dhruv Bansal's user avatar
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463 views

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 $...
Nome Sobrenome's user avatar
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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 ...
Threnody's user avatar
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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 \\ \...
anderson's user avatar
1 vote
2 answers
706 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. ...
Adam Sheppard's user avatar
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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 ...
peter.petrov's user avatar
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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,
Andrés Villa's user avatar
3 votes
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127 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 ...
Jonathan Hebert's user avatar
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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 ...
aorthey's user avatar
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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- ...
Labba's user avatar
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1 vote
1 answer
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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 ...
George Bentley's user avatar
2 votes
1 answer
288 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 ...
kappa's user avatar
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