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In [[topology]], a [[compact set|compact]] [[codimension]] one [[submanifold]] <math>F</math> of a [[manifold]] <math>M</math> is said to be '''2-sided''' in <math>M</math> when there is an [[embedding]] |
In [[topology]], a [[compact set|compact]] [[codimension]] one [[submanifold]] <math>F</math> of a [[manifold]] <math>M</math> is said to be '''2-sided''' in <math>M</math> when there is an [[embedding]] |
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::<math>h\colon F\times [-1,1]\to M</math> |
::<math>h\colon F\times [-1,1]\to M</math> |
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A submanifold which is not 2-sided is called 1-sided. |
A submanifold which is not 2-sided is called 1-sided. |
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==Examples== |
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=== Surfaces === |
=== Surfaces === |
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For curves on surfaces, a curve is 2-sided if and only if it preserves orientation, and 1-sided if and only if it reverses orientation: a tubular neighborhood is then a [[Möbius strip]]. This can be determined from the class of the curve in the [[fundamental group]] of the surface and the [[orientation character]] on the fundamental group, which identifies which curves reverse orientation. |
For curves on surfaces, a curve is 2-sided if and only if it preserves orientation, and 1-sided if and only if it reverses orientation: a tubular neighborhood is then a [[Möbius strip]]. This can be determined from the class of the curve in the [[fundamental group]] of the surface and the [[orientation character]] on the fundamental group, which identifies which curves reverse orientation. |
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* An embedded circle generating the [[fundamental group]] of the [[real projective plane]] (such as an "equator" of the projective plane – the image of an equator for the sphere) is 1-sided, as it is orientation-reversing. |
* An embedded circle generating the [[fundamental group]] of the [[real projective plane]] (such as an "equator" of the projective plane – the image of an equator for the sphere) is 1-sided, as it is orientation-reversing. |
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==Properties== |
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Cutting along a 2-sided manifold can separate a manifold into two pieces – such as cutting along the equator of a sphere or around the sphere on which a [[connected sum]] has been done – but need not, such as cutting along a curve on the [[torus]]. |
Cutting along a 2-sided manifold can separate a manifold into two pieces – such as cutting along the equator of a sphere or around the sphere on which a [[connected sum]] has been done – but need not, such as cutting along a curve on the [[torus]]. |
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maybe the simplest example of this is seen when one cut a [[mobius band]] along its '''core curve'''. |
maybe the simplest example of this is seen when one cut a [[mobius band]] along its '''core curve'''. |
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{{DEFAULTSORT:2-Sided}} |
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[[Category:Geometric topology]] |
[[Category:Geometric topology]] |
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[[Category:Articles lacking sources (Erik9bot)]] |
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This article does not cite any sources. Please help improve this articlebyadding citations to reliable sources. Unsourced material may be challenged and removed.
Find sources: "2-sided" – news · newspapers · books · scholar · JSTOR (December 2009) (Learn how and when to remove this message) |
Intopology, a compact codimension one submanifold 
![{\displaystyle h\colon F\times [-1,1]\to M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e733af655eb9618d5f6b2625f7c58d2eace6ba67)
with 
![{\displaystyle h(F\times [-1,1])\cap \partial M=h(\partial F\times [-1,1])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b14478b5beef0d22f67c8da0d0d04b15be1e570e)
.In other words, if its normal bundle is trivial.
This means, for example that a curve in a surface is 2-sided if it has a tubular neighborhood which is a cartesian product of the curve times an interval.
A submanifold which is not 2-sided is called 1-sided.
For curves on surfaces, a curve is 2-sided if and only if it preserves orientation, and 1-sided if and only if it reverses orientation: a tubular neighborhood is then a Möbius strip. This can be determined from the class of the curve in the fundamental group of the surface and the orientation character on the fundamental group, which identifies which curves reverse orientation.
Cutting along a 2-sided manifold can separate a manifold into two pieces – such as cutting along the equator of a sphere or around the sphere on which a connected sum has been done – but need not, such as cutting along a curve on the torus.
Cutting along a (connected) 1-sided manifold does not separate a manifold, as a point that is locally on one side of the manifold can be connected to a point that is locally on the other side (i.e., just across the submanifold) by passing along an orientation-reversing path.
Cutting along a 1-sided manifold may make a non-orientable manifold orientable – such as cutting along an equator of the real projective plane – but may not, such as cutting along a 1-sided curve in a higher genus non-orientable surface, maybe the simplest example of this is seen when one cut a mobius band along its core curve.
