For all questions about Möbius bands, Klein bottles, projective planes or surfaces built from these (via surgeries, gluings...), intersection or boundary problems, embeddings...

# Questions tagged [non-orientable-surfaces]

62 questions

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### Does the Gauss-Bonnet theorem apply to non-orientable surfaces?

I found statements of the Gauss-Bonnet theorem here, here, here, here, here, here, here, and here. None of them require that the surface be orientable. However, Ted Shifrin claims in a comment to this question that the Gauss-Bonnet theorem actually…

tparker

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### Fixed points in mapping from Möbius strip to disk [Explanation or reference needed]

One of the most elegant demonstrations in topology is the proof of the inscribed rectangle problem (a solved variant of the unsolved inscribed square problem) which states that for any plain, closed continuous closed curve $\Gamma$ in…

David G. Stork

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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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### A space that deformation retracts into the cylinder and Möbius band doesn't embed in $\Bbb R^3$.

Consider the Möbius band, and take the middle circle in it (so that it deformation retracts onto it). Glue the upper boundary of a cylinder through it. This gives a space that deformation retracts into the Möbius band and into the cylinder. How can…

Pedro

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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 is called a "position".
Now, we introduce the…

PyRulez

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### When is the orientable double cover of a product of non-orientable surfaces spin?

Let $M_{k,l}$ denote the orientable double cover of the non-orientable four-manifold $k\mathbb{RP}^2\times l\mathbb{RP}^2$; here $k\mathbb{RP}^2$ denotes the connected sum of $k$ copies of $\mathbb{RP}^2$.
For which $k$ and $l$ is $M_{k,l}$ a spin…

Michael Albanese

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### Classification of surfaces

The Classification Theorem for surfaces says that a compact connected surface $M$ is homeomorphic to $$S^2\# (\#_{g}T^2)\# (\#_{b} D^2)\# (\#_{c} \mathbb{R}P^2),$$ so $g$ is the genus of the surface, $b$ the number of boundary components and $c$ the…

user602039

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### Does this manifold have a name?

EDIT: In an attempt to not let the bounty go to waste, I will consider responses that give reasonable guesses of what the involved surfaces are, WITHOUT requiring parametrizations.
While using Mathematica to alter manifolds and numerically verify…

TribalChief

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### Torus/ Thick Möbius Band homeomorphism

Is a fattened Möbius Spiral Band homeomorphic to a Torus?
(Due to the same Euler Characteristic $\chi$ ?)
Are both non-orientable?
Following (3D printable plastic) Torus has a square section that rotates around a circle. Edges can be four closed…

Narasimham

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### Rational cohomology of punctured closed non-orientable manifold.

Let $M-\{p\}$ be a punctured closed non-orientable even-dimensional manifold. If $M=\mathbb{R}{P}^{2}$ then $\mathbb{R}P^{2}-\{p\}$ is an open Mobius strip. This implies that $H^{*}(\mathbb{R}P^{2}-\{p\};\mathbb{Q})\cong H^{*}(S^{1};\mathbb{Q}).$ …

Michael jordan

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### An orientable surface that cannot be embedded into $\Bbb R^3$?

By Whitney's embedding theorem, every 2-dimensional manifold (aka. a surface) can be embedded into $\Bbb R^4$.
Now, the Wikipedia page on that theorem states in this paragraph that we can even embedd into $\Bbb R^3$ if the surface is
compact and…

M. Rumpy

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### Immersion of non-orientable manifold in a small orientable one

I was trying to prove the following fact: given a non orientable manifold $M$ of dimension $m$, $M$ is always contained in an orientable manifold of dimension $m+1$. I have gotten nothing out of it, so I am asking you. I warn you that my background…

FriedPanda

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### Non-orientable cover of a non-orientable surface

I was quite puzzled by the request of classifying all the $4$-covers of the connected sum of $5$ copies of $\Bbb R P^2$. For oriented covering space, the answer is well known: it's enough to consider the $2$-cover of the orientable cover. But I was…

Luigi M

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### Is the fundamental group of non orientable surface Fuchsian?

Let M be the connected sum of 3 projective planes. Then is the fundamental group of M a Fuchsian group?
This group is an extension of the two element group by the surface group for a genus 2 surface (which is a Fuchsian group). So it seems very…

Ian Gershon Teixeira

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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 class $C^k$, $k\geq 1$, $V_0\subseteq…

Fernando Nazario

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