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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.

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

1 Answer 1

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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.

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  • $\begingroup$ Could you please add more details in the first hint? $\endgroup$ Feb 7, 2021 at 13:49
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    $\begingroup$ 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 $\mathbb{R}P^{n}$? $\endgroup$
    – ferhenk
    Feb 7, 2021 at 14:10

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