Mayer-Vietoris for $\mathbb C\mathbb P^n$?

Question

Does anyone have any idea of how calculating the De Rham cohomology $H^k(\mathbb C\mathbb P^n)$ of the complex projective space using Mayer-Vietoris?

Answer 1

Hint: Take $U=\mathbb{C}P^n\setminus\mathbb{C}P^{n-1}$ where $\mathbb{C}P^{n-1}$ is the obvious inclusion in to $\mathbb{C}P^n$, i.e the set of points in homogeneous coordinates with non-zero last coordinate. Take $V=\mathbb{C}P^n\setminus\{[0:0:\ldots:0:1]\}$ in homogenous coordinates.

Can you find homotopy equivalences between $U, V, U\cap V$ and some spaces which you know the cohomology of (you will need to use an inductive argument here for one of these spaces)?

Answer 2

Hint: Consider $\mathbb{C}P^n=e^0\cup e^2 \cup \dots\cup e^{2n}$ (cell complex, $e^i$ are balls) then the intersection is just the boundary of balls. Also, for $k=0,1$ we can just use Hurewicz that states $\pi_1(\mathbb{C}P^n)^{ab}=H^1(\mathbb{C}P^n)$ and that $\mathbb{C}P^n$ is path-connected the get that $H^0(\mathbb{C}P^n)=\mathbb{Z}$. Hope this helps.