2-sided
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In mathematics, specifically in topology of manifolds, a compact codimension-one submanifold F of a manifold M is said to be 2-sided in M when there is an embedding :: h\colon F\times [-1,1]\to M with h(x,0)=x for each x\in F and :: h(F\times [-1,1])\cap \partial M=h(\partial F\times [-1,1]) . 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.
In mathematics, specifically in topology of manifolds, a compact codimension-one submanifold F of a manifold M is said to be 2-sided in M when there is an embedding h\colon F\times [-1,1]\to M with h(x,0)=x for each x\in F and h(F\times [-1,1])\cap \partial M=h(\partial F\times [-1,1]) . 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
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.
References
Text derived from https://en.wikipedia.org/w/index.php?oldid=1122780865 under CC BY-SA 4.0; modifications have been made.