Orientability

:This article discusses orientability and orientation on surfaces and manifolds. For orientation of vector spaces see orientation (mathematics). For alternate uses, see orientation.

Orientation by a triangulation

Orientability, for surfaces, is easily defined, regardless of whether the surface is embedded in an ambient space or not. Any surface has a triangulation: a decomposition into triangles such that each edge on a triangle is glued to at most one other edge. We can orient each triangle, by choosing a direction for each edge (think of this as drawing an arrow on each edge) so that the arrows go from head to tail as we go around the boundary of the triangle. If we can do this so that in addition triangles sharing an edge have arrows on that edge going in opposite directions, then we call what we've done an orientation for the surface. Note that whether the surface is orientable is independent of triangulation; this fact is not obvious, but a standard exercise.

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This rather precise definition is based on intuition gathered from observing the following phenomenon:

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Imagine a capital "R" written on the surface, that can freely slide along the surface but cannot be lifted off the surface (that letter is chosen because of its asymmetry). If the surface is a Möbius band, and the letter slides all the way around the band and returns to its starting point, then it will look like a mirror-image of an "R" rather than the "R" it looked like originally. If the surface is a sphere, on the other hand, that cannot happen.

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The relation to the definition above is that sliding the "R" around from triangle to triangle in a triangulation gives an orientation for each triangle; the "R" in a triangle induces an obvious choice of arrow for each edge. The only obstruction to consistently orienting all the triangles is that when the "R" returns to its original starting triangle, it may induce choices of arrows going opposite to the original choice. Clearly, if this never happens, then we want the surface to be orientable, whereas if this does happen, then we want to call the surface non-orientable.

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The definition above can be generalized to an n-manifold that has a triangulation, but there are problems with that approach: some 4-manifolds do not have a triangulation, and in general for n > 4 some n-manifolds have triangulations that are inequivalent.

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