Orientability
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 p to a local orientation at a nearby point p ′: when the two points lie in the same coordinate chart U → R, that coordinate chart defines compatible local orientations at p and p ′. 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 H n (M, M ∖ U; Z) {\displaystyle H_{n}(M,M\setminus U;\mathbf {Z})} is isomorphic to Z. Assume that α is a generator of this group. For each p in U, there is a pushforward function H n (M, M ∖ U; Z) → H n (M, M ∖ { p }; Z) {\displaystyle H_{n}(M,M\setminus U;\mathbf {Z})\to H_{n}\left(M,M\setminus \{p\};\mathbf {Z} \right)}. The codomain of this group has two generators, and α maps to one of them. The topology on O is defined so that

is open.

There is a canonical map π : O → M that sends a local orientation at p to p. It is clear that every point of M has precisely two preimages under π. In fact, π 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.
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