Every differential 2-form is exact

Question

Prove that every differential 2-form on $\mathbb R\mathbb P^2$ is exact.

I'm not sure, but does it mean that H$^2_{DR}(\mathbb R\mathbb P^2)$ vanishes? If so, then it's easier statement.

Answer 1

Since $\dim \mathbb{RP}^2 = 2$, every 2-form on $\mathbb{RP}^2$ is closed. So the statement that all 2-forms are exact is the same as the statement that all closed 2-forms are exact, which is the statement that the second de Rham cohomology group vanishes, as you say. By de Rham's theorem, this is the same as the statement that the second homology group vanishes, which you can verify by direct computation using a particular triangulation.