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I have to calculate the cohomology of complex projective spaces $\mathbb{C}P^{n}$ using cellular cohomology (I know that we have a CW decomposition of $\mathbb{C}P^{n}$ in $n+1$ cells of even dimention). How can I do it explicitly? For example I can think $\mathbb{C}P^{2}$ as $ \mathbb{C}^{2} \cup \mathbb{C}P^{1}$. Then how can conclude my computation?

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As you have the decomposition $\mathbb C P^n = \mathbb C^n \cup \mathbb C^{n-1} \cup ... \cup \mathbb C^0$, you deduce a chain complex $$ \mathbb Z \to 0 \to \mathbb Z \to 0 \to \mathbb Z \to ... $$ It shows that $H^k(\mathbb CP^n) = \mathbb Z$ if $k$ is even and it equals zero when $k$ is odd.

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