Editing Orientability (section)

===The orientation double cover===

Around each point of ''M'' there are two local orientations.  Intuitively, there is a way to move from a local orientation at a point {{math|''p''}} to a local orientation at a nearby point {{math|''p''&prime;}}: when the two points lie in the same coordinate chart {{math|''U'' → '''R'''<sup>''n''</sup>}}, that coordinate chart defines compatible local orientations at {{math|''p''}} and {{math|''p''&prime;}}.  The set of local orientations can therefore be given a topology, and this topology makes it into a manifold.

More precisely, let ''O'' be the set of all local orientations of ''M''.  To topologize ''O'' we will specify a subbase for its topology.  Let ''U'' be an open subset of ''M'' chosen such that <math>H_n(M, M \setminus U; \mathbf{Z})</math> is isomorphic to '''Z'''.  Assume that &alpha; is a generator of this group.  For each ''p'' in ''U'', there is a pushforward function <math>H_n(M, M \setminus U; \mathbf{Z}) \to H_n\left(M, M \setminus \{p\}; \mathbf{Z}\right)</math>.  The codomain of this group has two generators, and &alpha; maps to one of them.  The topology on ''O'' is defined so that
:<math>\{\text{Image of } \alpha \text{ in } H_n\left(M, M \setminus \{p\}; \mathbf{Z}\right) \colon p \in U\}</math>
is open.

There is a canonical map {{math|&pi; : ''O'' → ''M''}} that sends a local orientation at ''p'' to ''p''.  It is clear that every point of ''M'' has precisely two preimages under {{math|&pi;}}.  In fact, {{math|&pi;}} is even a local homeomorphism, because the preimages of the open sets ''U'' mentioned above are homeomorphic to the disjoint union of two copies of ''U''.  If ''M'' is orientable, then ''M'' itself is one of these open sets, so ''O'' is the disjoint union of two copies of ''M''.  If ''M'' is non-orientable, however, then ''O'' is connected and orientable.  The manifold ''O'' is called the '''orientation double cover'''.