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Orientability

In geometry and topology, a surface in R^3 is called orientable, if, roughly speaking, it is possible to consistently distinguish between the two sides of the surface. Take for instance a sphere: we can distinguish between the inside and the outside, and therefore the sphere is orientable. The whole two-dimensional x-y plane, thought of as a subset of three-dimensional space, is also orientable: we can distinguish between "above" and "below". The Möbius strip is not orientable: it really has only one side. Similarly, the Klein bottle is not orientable, because one cannot distinguish between the inside and the outside.

In general, the property of being orientable is not equivalent to being two-sided; however, this holds when the ambient space (such as R^3 above) is orientable.

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 indpendent of triangulation; this fact is not obvious, but a standard exercise.

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

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 Moebius 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.

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.

The definition above can be generalized to an n-manifold that has a triangulation, but there are problems with that approach: some four manifolds do not have a triangulation, and some 5-manifolds (and some n-manifolds in general) have triangulations that are inequivalent.

Another way of thinking about orientability is thinking of it as a choice of "right handedness" vs. "left handedness" at each point in the manifold.

Formally, an n-dimensional differentiable manifold is called orientable if it posesses a differential form d of degree n which is nonzero at every point on the manifold. Conversely, given such a form d, we say that the manifold is oriented by d. The crucial point to observe here is that such a differential form gives a choice of "right handed" basis at each point. A traveler in an orientable manifold will never change his/her handedness by going on a round trip.



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