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## Main Question or Discussion Point

Hi, I'd like to compute the de Rham cohomology of the 3 following objects :

-A connected sum of ##g \in \mathbb{N}## reals projectives plans ##P_{2}(\mathbb{R})##.

-A connected sum of ##g \in \mathbb{N}## torus without ##n## points ##\mathbb{T}^{2} - \{x_{1}, x_{2}, ..., x_{n}\}##.

- A connected sum of ##g \in \mathbb{N}## reals projectves plans without ##n## points ##P_{2}(\mathbb{R}) - \{x_{1}, x_{2}, ..., x_{n}\}##.

I'll wrote ##[]## to talk about the class of an element in ##M = P_{2}(\mathbb{R})##.

By connexity I get : ##H^{0}(M) = \mathbb{R}## and by dimension I get ##H^{p}(M) = \{0\}## for ## p > 2##.

Here is what I tried to : let's focus on the first sum :

If ##g = 1## I consider the 2 opens ##U = P_{2}(\mathbb{R}) - [((1, 0, 0)]## : it has the same homotopy as ##P_{1}(\mathbb{R}) \simeq \mathbb{S}_{1}## so the same de Rham cohomology. I consider ##V = \{ [(x, y, z)] \in P^{2}(\mathbb{R}) | x \neq 0\}## which as the same homotopy as ##\mathbb{R}^{2}##. We have that ##U \cup V = M## and ##U \cap V## has the same cohomology than a plan miness a point so the cohomology

than a circle.

Thanks to Mayer-Vietoris, I got the following exact sequences :

##0 \mapsto \mathbb{R} \mapsto \mathbb{R} \times \mathbb{R} \mapsto \mathbb{R} \mapsto H^{1}(M) \mapsto \mathbb{R} \mapsto \mathbb{R} \mapsto H^{2}(M) \mapsto

0 ~ (1)##

The 3th ##\mapsto## is the application ##(x, y) \mapsto x - y## which is surjective. By exactitude we get that ##H^{1}(M) \hookrightarrow \mathbb{R}## so it's ##\mathbb{R}## or ##\{0\}##.

I also know that if ##0 \mapsto E_{1} \mapsto ... \mapsto E_{n} \mapsto 0## is an exacte seqeunces between finite dimension spaces I got ##\sum_{i = 0}^{n} (-1)^{i}dim(E_{i})## but I have to unkown dimension in by sequence ##(1)##.

So how solve this? Whatever the open ##U, V## I take, I always got twoo unkown.

Feel free to move the thread in homework if you think I'll get more answer.

The problem comes from here : http://www.math.u-psud.fr/~paulin/notescours/cours_geodiff.pdf : page 246 exercice 141.

Notice that the method the author use above to look for the de Rham cohomology of a connected sum of thorus is such that he has twoo unknown dimension. But he claims he compute ##H^{2}## with another method or by knowing the morphism (but if I look Mayer-Vietoris theorem proof, we only know the morphism between ##H(M), H(U) \times H(V)##).

Could you help me please?

I wish you a good day.

-A connected sum of ##g \in \mathbb{N}## reals projectives plans ##P_{2}(\mathbb{R})##.

-A connected sum of ##g \in \mathbb{N}## torus without ##n## points ##\mathbb{T}^{2} - \{x_{1}, x_{2}, ..., x_{n}\}##.

- A connected sum of ##g \in \mathbb{N}## reals projectves plans without ##n## points ##P_{2}(\mathbb{R}) - \{x_{1}, x_{2}, ..., x_{n}\}##.

I'll wrote ##[]## to talk about the class of an element in ##M = P_{2}(\mathbb{R})##.

By connexity I get : ##H^{0}(M) = \mathbb{R}## and by dimension I get ##H^{p}(M) = \{0\}## for ## p > 2##.

Here is what I tried to : let's focus on the first sum :

If ##g = 1## I consider the 2 opens ##U = P_{2}(\mathbb{R}) - [((1, 0, 0)]## : it has the same homotopy as ##P_{1}(\mathbb{R}) \simeq \mathbb{S}_{1}## so the same de Rham cohomology. I consider ##V = \{ [(x, y, z)] \in P^{2}(\mathbb{R}) | x \neq 0\}## which as the same homotopy as ##\mathbb{R}^{2}##. We have that ##U \cup V = M## and ##U \cap V## has the same cohomology than a plan miness a point so the cohomology

than a circle.

Thanks to Mayer-Vietoris, I got the following exact sequences :

##0 \mapsto \mathbb{R} \mapsto \mathbb{R} \times \mathbb{R} \mapsto \mathbb{R} \mapsto H^{1}(M) \mapsto \mathbb{R} \mapsto \mathbb{R} \mapsto H^{2}(M) \mapsto

0 ~ (1)##

The 3th ##\mapsto## is the application ##(x, y) \mapsto x - y## which is surjective. By exactitude we get that ##H^{1}(M) \hookrightarrow \mathbb{R}## so it's ##\mathbb{R}## or ##\{0\}##.

I also know that if ##0 \mapsto E_{1} \mapsto ... \mapsto E_{n} \mapsto 0## is an exacte seqeunces between finite dimension spaces I got ##\sum_{i = 0}^{n} (-1)^{i}dim(E_{i})## but I have to unkown dimension in by sequence ##(1)##.

So how solve this? Whatever the open ##U, V## I take, I always got twoo unkown.

Feel free to move the thread in homework if you think I'll get more answer.

The problem comes from here : http://www.math.u-psud.fr/~paulin/notescours/cours_geodiff.pdf : page 246 exercice 141.

Notice that the method the author use above to look for the de Rham cohomology of a connected sum of thorus is such that he has twoo unknown dimension. But he claims he compute ##H^{2}## with another method or by knowing the morphism (but if I look Mayer-Vietoris theorem proof, we only know the morphism between ##H(M), H(U) \times H(V)##).

Could you help me please?

I wish you a good day.