# 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.

126 questions
Filter by
Sorted by
Tagged with
1 vote
42 views

22 views

### 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)...
1 vote
54 views

### 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 ...
1 vote
50 views

232 views

### 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 ...
268 views

1 vote
204 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. ...
226 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 ...
187 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,
80 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 ...
49 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 ...
99 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
836 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 ...
145 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
73 views

### 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?
290 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
416 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 ...
69 views

223 views

### 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 ...
2k views

### 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 ...
910 views

### 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 ...
1 vote
287 views

545 views

### 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 ...
298 views

### 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....
180 views

### 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 ...
1 vote
### 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 ... 