I would like to compute the Euler characteristic of $\mathbb{P}^n(\mathbb{R)}$. I do not know if cohomology could help but I should avoid it because I did not studied it yet. I would like to use only basic tools of differential topology.
I though to consider $\mathbb{P}^n(\mathbb{R)}$ as the quotient of $\mathbb{D}^n$ where its edge is identified: $x \sim y$ is and only if $x=y$ or $\|x\|=\|y\|=1$ and $x=-y$. But I am not sure on how to continue.
Hint 1. $\mathbb{R}P^{n} \cong \mathbb{S}^{n}/x \sim -x$. What is the Euler characteristic of $\mathbb{S}^{n}$? Can you deduce $\chi(\mathbb{R}P^{n})$ from there?
Hint 2. $\mathbb{R}P^{n} = \mathbb{R}P^{n-1} \cup_{f} D^{n}$, where $f: S^{n-1} \to \mathbb{R}P^{n-1}$ is the 2-to-1 covering map and $D^{n}$ is the interior of an $n$-disk.