# Complex projective space has orientation-reversing self-homeomorphism iff it has odd complex dimension

## Statement

Consider a complex projective space , also denoted , for a natural number. This has complex dimension (as a complex manifold) and real dimension (as a real manifold). Moreover, for all values of , it is a compact connected orientable manifold.

The claim is that:

- For odd, i.e., (so real dimensions ), possesses an orientation-reversing self-homeomorphism. Viewed on homology, this is a homeomorphism that sends the fundamental class to its negative.
- For even, i.e., (so real dimensions ), does
*not*possess an orientation-reversing self-homeomorphism. Viewed on homology, there is no homeomorphism that sends the fundamental class to its inegative.

## Facts used

## Proof

The proof relies on the cohomology ring structure of complex projective space. The key idea is that if we take a generator for , the power of this under the cup product gives a fundamental class.

Any self-homeomorphism of must send a generator of either to itself or to its negative, and there are self-homeomorphisms that do both these. When is even, both these types of homeomorphisms fix the fundamental class because the power of is . When is odd, the homeomorphism that acts as negation on also sends the fundamental class to its negative.