Orientable manifold

Consider an example. The surface of an ordinary, flat sheet of paper is an orientable, two-dimensional manifold. If you draw the characters p and q on this surface, you will note that there is no way you could rotate either of the letters, or otherwise move it around the manifold (the sheet of paper) so that it could be placed directly over the other letter. You could do this only if, so to speak, you could lift the letter off of the paper and turn it over. The p and q have the same shape in an obvious sense, but they are oriented differently, so that they are reflections of one another. A simply connected two-dimensional space which obeys Euclidean geometry is orientable for two-dimensional objects: it is possible to describe two objects that are reflections of one another but cannot be transformed into one another.

However, such a two-dimensional manifold is non-orientable for any one-dimensional objects: that is, it is impossible to describe two one-dimensional objects that are reflections of one another but could not be rotated into one another. An example might be a straight line colored red at one and and blue at the other. All such lines could be rotated into one another in two-dimensional space, but in a one-dimensional space (a string-shaped universe), it would be possible to have two different ones (with the red at opposite ends).

Correspondingly three-dimensional space is non-orientable for two-dimensional objects. But it is orientable for three-dimensional objects. If your reflection stepped out of the mirror she could not be rotated in such a way as to appear identical to you. For example, if you had a tattoo that said "pop" on your right arm, your reflection's tattoo would say "qoq" and be on her left arm.

The best-known non-orientable two-dimensional manifold is the Möbius strip. If you drew a p and q next to one another, then slid one of them along the length of the strip until it returned to its starting point, it would have the same shape as the other one. There is no global difference between ps and qs in such a space. Another example is a Klein bottle, which is, roughly speaking, the product of two Moebius strips glued together along each of their lone edges. (A proper Klein bottle can only exist in four-dimensions; it can be only imperfectly represented in three.)

The space-time manifold of the actual universe is believed to be orientable. However, it is also believed to be closed, meaning if you travel in a straight line far enough you will arrive back where you started. If space-time were non-orientable in the same manner as a Moebius strip or Klein bottle, then when you arrived back you (or the rest of the universe, from your perspective) would have become left-right reversed, like a mirror image of itself.

See also: chirality, handedness

(This is written by a topological non-professional; probably rife with error, please correct)

Orientation cover

To make topological sense of the notion of orientation, one can use the idea of covering space (see local homeomorphism). For a connected manifold M one defines a covering space M*, also a manifold, and a 2-to-1 local homeomorphism from M* to M. The two points of M* mapping to a given p in M are the two orientations of a small open ball near p. Then M* is either in two connected components - in which case M is orientable as we see by selecting one - or M* is connected, meaning that by following an orientation round a path we can get to the other one. In the latter case M isn't orientable.