# Can I have more than $2$ orientations on a connected component of a variety?

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?

## migrated from mathoverflow.netAug 13 '17 at 14:06

This question came from our site for professional mathematicians.

• This is immediate from the homological definition of orientation. You find it nicely explained in chapter 22 of Greenberg-Harper: "Algebraic Topology". – user39082 Aug 13 '17 at 10:34
• 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). – NAC Aug 13 '17 at 14:18
• Understand the answer to your question just for a vector space! (After that, do a standard connectedness argument.) – Ted Shifrin Aug 13 '17 at 16:37