Editing Orientability (section)

==Orientable double cover==
[[File:Orientation cover of Mobius strip.webm|thumb|450px|right|Animation of the orientable double cover of the [[Möbius strip]].]]
A closely related notion uses the idea of [[covering space]]. For a connected manifold <math>M</math> take <math>M^*</math>, the set of pairs <math>(x,o)</math>  where <math>x</math> is a point of <math>M</math> and <math>o</math> is an orientation at <math>x</math>; here we assume <math>M</math> is either smooth so we can choose an orientation on the tangent space at a point or we use [[singular homology]] to define orientation.  Then for every open, oriented subset of <math>M</math> we consider the corresponding set of pairs and define that to be an open set of <math>M^*</math>.  This gives <math>M^*</math> a topology and the projection sending <math>(x,o)</math> to <math>x</math> is then a 2-to-1 covering map.  This covering space is called the '''orientable double cover''', as it is orientable.  <math>M^*</math> is connected if and only if <math>M</math> is not orientable.

Another way to construct this cover is to divide the loops based at a basepoint into either orientation-preserving or orientation-reversing loops.  The orientation preserving loops generate a subgroup of the fundamental group which is either the whole group or of [[index of a subgroup|index]] two.  In the latter case (which means there is an orientation-reversing path), the subgroup corresponds to a connected double covering; this cover is orientable by construction.  In the former case, one can simply take two copies of <math>M</math>, each of which corresponds to a different orientation.