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

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Statement

Consider a complex projective space \mathbb{P}^n(\mathbb{C}), also denoted \mathbb{C}\mathbb{P}^n, for n a natural number. This has complex dimension n (as a complex manifold) and real dimension 2n (as a real manifold). Moreover, for all values of n, it is a compact connected orientable manifold.

The claim is that:

Facts used

  1. Cohomology of complex projective space

Proof

The proof relies on the cohomology ring structure of complex projective space. The key idea is that if we take a generator for H^2(\mathbb{P}^n(\mathbb{C});\mathbb{Z}), the n^{th} power of this under the cup product gives a fundamental class.

Any self-homeomorphism of \mathbb{P}^n(\mathbb{C}) must send a generator of H^2(\mathbb{P}^n(\mathbb{C});\mathbb{Z}) either to itself or to its negative, and there are self-homeomorphisms that do both these. When n is even, both these types of homeomorphisms fix the fundamental class because the n^{th} power of -1 is 1. When n is odd, the homeomorphism that acts as negation on H^2(\mathbb{P}^n(\mathbb{C});\mathbb{Z}) also sends the fundamental class to its negative.