# Questions tagged [mobius-band]

The Möbius band or Möbius strip is a surface with only one side and only one boundary. The Möbius strip has the mathematical property of being non-orientable. It is named after the German mathematician August Ferdinand Möbius.

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### Which space do we obtain if we take a Möbius strip and identify its boundary circle to a point?

I know that the boundary circle of a Möbius strip is actually formed by the horizontal sides of $[0 ,1] \times [0,1]$.If we identify all the points of the 1st horizontal side to a single point and do ...
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### How can the Möbius band be an image of a parametrization if it is not orientable?

I'm using a book of analysis on $\mathbb{R}^n$ with the following definitions: m-dimensional parametrization of class $C^k$ of $V\subseteq \mathbb{R}^n$: a homeomorphism $\phi:V_0\rightarrow V$ of ...
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### Mobius strip and homeomorphism of a circle [closed]

I'm assigned a homework problem as follows: Let $X$ be the Möbius strip, obtained as a quotient space of $[0,1] \times [0,1]$ (with subspace topology of $\mathbb{R}^2$) by identifying the pairs of ...
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### Intuition behind equivalence of two identifications obtained from Möbius band.

Consider the real projective plane $\Bbb RP^2$ which can be realized as a quotient space obtained from a square by identifying points on each pair of it's opposite edges in reverse order. It has been ...
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### Proving a result concerning quotient spaces.

Let $M$ be the Möbius strip and $C$ it's boundary circle. Prove that $M/C$ is homeomorphic to $\Bbb RP^2.$ I know that $\Bbb RP^2 \approx D^2/x \sim -x, x \in \partial D^2.$ So if we can able to get ...
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### What is the tangent bundle of Mobius band?(It is trivial or not?)

I know that Mobius band is a quotient space of unit square with the equivalence relation $(0,t)\sim (1,1-t)$. I want to find the tangent bundle of Mobius band upto diffeomorphism. I am not getting any ...
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### Is there a fixed-point free involution of the Möbius strip?

Is there a fixed-point free continuous involution of the Möbius strip? (Meaning a function $f:M\to M$ such that $f\circ f={\rm id}$ and $f(x)\ne x$ for all $x$.) The Lefschetz fixed-point theorem is ...
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### Is a thick Möbius strip genus 2?

A cylinder has genus 0, but a thick cylinder has genus 1. This gets me into wondering, since a Möbius strip has genus 1, would a thick Möbius strip have genus 2? I can't quite comprehend if the ...
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### The boundary of a closed Mobius band

The Mobius band is defined by the quotient: $r:I\times I\rightarrow M$, with the equivalence relation $(0,x)\sim (1,1-x)$ for all $x\in I$. The boundary of Mobius band $M$ is defined as the set of ...
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### Attaching disks and contracting is equivalent to taking quotients.

Take a Mobius band $M$ and attached a disk $D^2$ along its boundary, which is a copy of $S^1$ embedded in $M$. It is known that what we obtain is something homeomorphic to the projective plane. Now ...
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### how to chart a moebius strip

This is a question from a tutorial for topology manifold. It is stated as: How many charts do you need to cover the Moebius strip that has a river printed on it? The chart was defined as"Chart: -...
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### Understanding Mobius bundle.

I'm trying to solve the following question: Consider the vector bundle $$E:= [0,1] \times \mathbb{R}/\sim \quad \to S^1$$ where $\sim$ is the equivalence rleation $(0,t) \sim (1,-t)$ for ...
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### Matrices for symmetries of Möbius band

Taken from Groups and Symmetry by M.A. Armstrong, question 18.8: Setup: A 5x1 rectangular strip of paper is marked off on both sides into 5 unit squares. The two ends of the paper are then put ...
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### Which is the function that the zero-set of which is Möbius strip?

Many surfaces can be expressed by implicit functions. As described later, the spherical surface, the side surface of the cylinder, and the torus can be expressed by implicit functions. However, I have ...
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### Why is the Möbius strip not a product manifold despite being parameterizable as $(\theta, h)$?

I was reading this† lecture note and came across to the section on fibre bundles. On page 2 they mention that there is no unambiguous and continuous way to write a point $m$ on a Möbius strip $M_0$ ...
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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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### Why does the boundary of mobius strip wrap twice around core circle but not any other line?

A space $X$ deformation retracts onto a subspace $A$ if there exists continuous map $F:X\times [0,1]\rightarrow X$ such that $F(x,0)=x,F(x,1)\in A,F(a,t)=a$ $\forall a\in A$. The mobius strip ...
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### Possibility of representing the Möbius strip by ways other than specifying a parameterization?

I have looked up some texts, they all represent the Möbius strip by ways of specifying a parameterization. This is the only way how the Möbius strip is represented founded in the Wikipedia currently. ...
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### Knots on $\mathbb{R}P^2$ and on n-half-twist Möbius strips

Can you embed a non-trivial knot on the surface of $\mathbb{R}P^2$? I know we can on a torus, and we can't on a 2-sphere. My hand-wavy argument/intuition based on the little I know from self study of ...
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### When is an open subset of an orientable manifold an orientable submanifold?

I am trying to complete problem $4$ from the introductory chapter of Riemannian Geometry by do Carmo. The question asks us to show that the projective plane $P^2(\mathbb{R})$ is non-orientable, first ...
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### Which surface is homeomorphism to mobius strip?

I'm a bit confused since I read many versions of this. First of all, I couldn't understand how to define homeomorphism. Intuitively, I think that if I can deform an object, without tearing it, to ...