Demand

2014-02-02T21:29:08Z

108.242.176.37: Undid revision 577969199 by 14.99.180.21 (talk)

{{Unreferenced|date=December 2009}}
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]]
::<math>h\colon F\times [-1,1]\to M</math>
with <math>h(x,0)=x</math> for each <math>x\in F</math> and
::<math>h(F\times [-1,1])\cap \partial M=h(\partial F\times [-1,1])</math>.
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.

==Examples==
=== Surfaces ===
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.
* An embedded circle in the plane is 2-sided.
* 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.

==Properties==
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'''.

{{DEFAULTSORT:2-Sided}}
[[Category:Geometric topology]]

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