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Let $M$ be a smooth oriented variety with $c$ connected components. It's easy to build up a new orientation from the original. So, if there are no other orientations, we have $2^c$ possible orientations for the variety.

How can I show that there exist only two orientations for each connected component?

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migrated from mathoverflow.net Aug 13 '17 at 14:06

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    $\begingroup$ This is immediate from the homological definition of orientation. You find it nicely explained in chapter 22 of Greenberg-Harper: "Algebraic Topology". $\endgroup$ – user39082 Aug 13 '17 at 10:34
  • $\begingroup$ The top exterior power of the tangent (or cotangent) bundle has fibers of real dimension 1. Remove 0 from it and you get two connected components (hence two orientations). $\endgroup$ – NAC Aug 13 '17 at 14:18
  • $\begingroup$ Understand the answer to your question just for a vector space! (After that, do a standard connectedness argument.) $\endgroup$ – Ted Shifrin Aug 13 '17 at 16:37

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