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

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:


 * For $$n$$ odd, i.e., $$n = 1,3,5,\dots$$ (so real dimensions $$2,6,10,\dots$$), $$\mathbb{C}\mathbb{P}^n$$ possesses an orientation-reversing self-homeomorphism. Viewed on homology, this is a homeomorphism that sends the fundamental class to its negative.
 * For $$n$$ even, i.e., $$n = 2,4,6,\dots$$ (so real dimensions $$4,8,12,\dots$$), $$\mathbb{C}\mathbb{P}^n$$ 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

 * 1) uses::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.