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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.

The Möbius band or Möbius strip is a surface with only one side and only one boundary (a simple closed curve which means it is homeomorphic to a circle). The Möbius strip has the mathematical property of being non-orientable. It is named after the German mathematician August Ferdinand Möbius. The Euler characteristic of the Möbias strip is zero.

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Why is the Möbius strip not orientable?

I am trying to understand the notion of an orientable manifold. Let M be a smooth n-manifold. We say that M is orientable if and only if there exists an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ such that $\textrm{det}(J(\phi_{\alpha} \circ…
Richard G
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Is it possible to determine if you were on a Möbius strip?

I understand that if you were to walk on the surface of a Möbius strip you would have the same perspective as if you walked on the outer surface of a cylinder. However, would it be possible for someone to determine whether they were on a Möbius…
Steven Wallace
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Cutting a Möbius strip down the middle

Why does the result of cutting a Möbius strip down the middle lengthwise have two full twists in it? I can account for one full twist--the identification of the top left corner with the bottom right is a half twist; similarly, the top right corner…
AndJM
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Klein bottle as two Möbius strips.

I read that glueing together two Möbius strips along their edges creates a surface that is equivalent to the so-called Klein bottle. The Möbius strip comes in two versions that are mirrored versions of each other (wrt the chirality of the…
Gerard
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Is there a Möbius torus?

Does the concept of a Möbius torus make sense: taking a cylinder (instead of a rectangle as in the case of the Möbius strip) and twisting it before joining its ends? Or will the resulting twisted torus be indistinguishable from the normal torus in…
Hans-Peter Stricker
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Retraction of the Möbius strip to its boundary

Prove that there is no retraction (i.e. continuous function constant on the codomain) $r: M \rightarrow S^1 = \partial M$ where $M$ is the Möbius strip. I've tried to find a contradiction using $r_*$ homomorphism between the fundamental groups, but…
user32847
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In how many dimensions is the full-twisted "Mobius" band isotopic to the cylinder?

There is a question on this site about the distinctions between the full-twisted Mobius band and the cylinder, but I would like to ask something different, so I start a new question. Let us call $C$ the standard cylinder embedded in $\mathbb{R}^3$,…
Kaius
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Klein-bottle and Möbius-strip together with a homeomorphism

Consider the Klein bottle (this can be done by making a quotient space). I want to give a proof of the following statement: The Klein Bottle is homeomorphic to the union of two copies of a Möbius strip joined by a homeomorphism along their…
Crosscap
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Topologically distinguishing Mobius Strips based on the number of half-twists

We can distinguish between a (closed) Mobius strip and 'regular' (untwisted) strip by examining the set of points which have no neighborhood homeomorphic to a disk (intuitively, the 'boundary' of the strip). For a Mobius strip, this set is…
user88319
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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 the same for the second horizontal side we get a…
Antimony
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How do Flatlanders represent Möbius strips?

There are 3D representations of Klein bottles that give people in our 3D universe a pretty good idea of how one is constructed: We can sort of see how this thing needs to be 'twisted' in the fourth dimension. But how do Flatlanders create 2D…
Paul Reiners
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Homework: The Klein bottle retracts onto a loop.

From the UCLA topology qualifying exam, Fall 2011: Let $K$ be the Klein bottle. Show that there exist homotopically nontrivial simple closed curves $\gamma_1,\gamma_2$ such that $K$ retracts to $\gamma_1$, but does not retract to $\gamma_2$. A…
Ted Dokos
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Drawing a thickened Möbius strip in Mathematica

I would like to have Mathematica plot a "thickened Möbius strip", i.e. a torus with square cross section that is given a one-half twist. Ideally, I would like this thickened Möbius strip to be transparent with a (non-thickened) solid Möbius strip…
Zev Chonoles
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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 useless here because the Möbius band is homotopic…
Akiva Weinberger
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Klein bottle homeomorphic to union of Möbius strip

I'm having trouble showing that the Klein bottle defined as a quotient space of $I^2$ with relation $(x,-1)R(x,1)$ and $(-1,y)R(1,-y)$ is Hausdorff and that it can be expressed as $X\cup Y$ where $X,Y$ are homeomorphic to the Möbius strip and…
user62931
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