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Orientability

 For a surface to be non- orientable there must be a closed path around which a figure can be moved only to look like it's mirror image when it gets back to the start. The möbius strip is perhaps the most important example of a non-orientable surface. This is because a strip can be cut from all other non-orientable surfaces, such as the one bellow, which is homeomorphic to ("looks like") the möbius strip. For most other shapes, such as spheres, tori, cubes etc… this can not be done so they are orientable.

 

Closed paths

 

Another equivalent definition of orientability which turns out to be of importance latter is seen as follows. Start by considering a closed path on a surface. Then construct another path by starting infinitesimally close to the original path and running along side parallel to it until the new path is parallel to the starting point (see möbius strip bellow). At which point there are two things that can happen. Firstly the new path could join up with it's start point to form a closed path, in which case we say the original loop is orientable. Alternatively the end of the new path could find it's self on the opposite side of the original closed path to which it started, in which case we say the loop is non-orientable. The surface is then orientable if there exists non-orientable loops.

It is the existence of non-orientable loops and the necessity of having to cross that loop at some point in order to create a new loop next to it that is ultimately responsible for exempt points.

 

 

 

 

paths_on_mobius.gif

 

Extension to higher dimensional manifolds

So far we have only considered surfaces, but this can easily be extended to any n-manifold. For an n-manifold the definitions of orientability are directly analogous to those for surfaces. For example non-orientability could be shown by moving a n-dimensional ball around a loop on the manifold where the ball is found to be reflected upon returning to it's start position.

Another way would be by finding a subset of the manifold homeomorphic to the direct product of a n-1 dimensional ball, Sn, and the interval [0, 1], where Sn × {0} is identified with Sn ×{1} with a reflection in between. Such a method is an extension of trying to find a strip of the surface which is homeomorphic to the möbius strip as mentioned above.