# Computing the Euler characteristic of real projective space $\mathbb{R}P^{n}$

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.

• What definition of Euler characteristic are you using? Feb 7, 2021 at 10:40
• @Kajelad I define $\chi (M)$ to be the sum $#(0-simplex)-#(1-simplex)+...$ for a partition of $M$ with simplexes. Feb 7, 2021 at 10:48
• In that case, are you able to describe $\mathbb{RP}(n)$ as a simplicial complex? Feb 7, 2021 at 10:52
• @Kajelad No I am not, that's my problem. Feb 7, 2021 at 10:54
• Alternatively, would you be able to use the double covering map $c:S^n\to\mathbb{RP}(n)$ to relate $\chi(S^n)$ and $\chi(\mathbb{RP}(n))$? Feb 7, 2021 at 11:16

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.
• Do you need any help finding $\chi(\mathbb{S}^{n})$? Once you have the Euler characteristic of the $n$-sphere, you may use the following fact: If $X$ is a finite CW complex and if $Y \to X$ is a $n$-sheeted covering, then $Y$ is a finite CW complex and $\chi(Y) = n \chi(X)$. Can you establish $\mathbb{S}^{n}$ as a double (i.e. 2-sheeted) cover of $\mathbb{R}P^{n}$? Feb 7, 2021 at 14:10