# Questions tagged [non-orientable-surfaces]

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

30 questions
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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, ...
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### Existence af a Frame on the Klein Bottle

I was trying to disprove the fact that there exist a global tangent frame on the Klein bottle, i.e. two global vector fields everywhere indipendent. Since my background involves no charachteristic ...
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### When you divide the real projective plane into two subsets, does it always have exactly one non-orientable component?

Let's say you divide the real projective plane into two subsets, are exactly one these subsets non-orientable? In particular, we will require that each subset $S$ is "nice" in the sense their common ...
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### Criterion for being in a non-orientable 3 manifold?

I'm trying to wrap my head around the concept of orientability as an intrinsic property of a manifold. Assume I'm in some (3-dim) manifold for which I'd like to decide its orientability; what could I ...
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### What is the 'center circle' of a Mobius Band?

What is the 'center circle' of a Mobius band? The question I am working on asks me to cut (literally) a Mobius band in half 'along its center circle.' What exactly does this mean? I know the plane ...
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### Is there a way to prove algebraically that a Möbius strip is non-orientable?

I am doing my HL Maths coursework on non-orientability of surfaces and am trying to prove whether a möbius strip is orientable or not (of course it isn't) Is there a way to prove algebraically that a ...
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### Example of a non-orientable 3-manifold

I was reading a paper and it was affirmed in there that $\mathbb{R}\mathbb{P}^2\times\mathbb{S}^1$ was a non-orientable 3-manifold. Does anyone knows how to prove it? if not, is there another (simple)...
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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 ...
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### Why is Klein bottle non-orientable?

I am doing the homework of differential geometry and encounter this problem: The Klein bottle $K^2$ is defined to be the identification space [0, 1] \times [0, 1]/{\sim}, \text{ where the ...
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### Cell decomposition of non-orientable surfaces

I saw that a cell-decomposition of a genus g non-orientable surface is $D^0\cup D^1\cup ...\cup D^g$. Can anyone explain why this is true?
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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 ...
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### Orientable Surface Covers Non-Orientable Surface

I need to describe how a 4-genus orientable surface double covers a genus 5-non-orientable surface. I know that in general every non-orientable compact surface of genus $n\geq 1$ has a two sheeted ...
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### Klein bottle contains Möbius band

I read the following: "The Klein bottle contains a copy of the Möbius band". I assume this means that there is a subspace of the Klein bottle that is homeomorphic to the Möbius band. How do we obtain ...
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### Torus/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 ...