As I stated in the title I'm struggling with the problem of computing $H_{\bullet}(\mathbb{P}^n\mathbb{R}, \mathbb{Z})$ for $n\geq0$. I am trying to use the cellular homology. I know that $\mathbb{P}^n\mathbb{R}$ is given a cellular structure inductively as:
$$\mathbb{P}^n\mathbb{R} = \mathbb{P}^{n-1}\mathbb{R} \cup_{\pi} D^{2n}$$
where the attaching map $\pi : S^{n-1} \rightarrow \mathbb{P}^{n-1}\mathbb{R}$ is the quotient map. So the cellular complex is:
$$0 \rightarrow \mathbb{Z} \rightarrow \dots \rightarrow \mathbb{Z} \rightarrow 0$$
I now need to understand how the boundary maps work. This is the point where I am stucked. I am trying to use the definition of boundary map given here. My problem is how to compute the Brouwer degree of the map $\chi^{\alpha\beta}_n$ (I am keeping the notation used in the link above). In the case of $\mathbb{P}^n\mathbb{R}$ we have only one map. I tried to write it down and it seems to me that the map is:
$$S^{n-1} \rightarrow \mathbb{P}^n \rightarrow \mathbb{P^n}/(\mathbb{P}^n - Int(D^n)) \rightarrow S^{n-1}$$
Where the first and second maps are the quotient and the third is the homeomorphism between $D^n/S^{n-1}$ and $S^{n-1}$. It seems to me the quotient $\mathbb{P^n}/(\mathbb{P}^n - Int(D^n))$ can be seen as: all the point in $\mathbb{P}^n$ scaled to have norm equal to 1, then if they have the first coordinate equal to zero they are identified all to a single point, otherwise they are not identified to any other point. If this is true then the last map can be sen as the one that sends all the point with first coordinate equal to zero in the North pole and the others in: $$[x_0: \dots : x_n] \rightarrow \left( \frac{2 \tilde{y}}{1+||\tilde{y}||^2}, \frac{||\tilde{y}||^2-1}{1+||\tilde{y}||^2}\right)$$
where $y = (x_1, \dots, x_n)$ and $$\tilde{y} = \frac{y}{1-||y||^2}$$
So the whole map would be:
$$(x_0, \dots, x_n) \rightarrow \left( \frac{2 \tilde{y}}{1+||\tilde{y}||^2}, \frac{||\tilde{y}||^2-1}{1+||\tilde{y}||^2}\right)$$
I am not sure all the things I wrote are true, but even if they are I can't succeed in calculating the Brouwer degree of this map. If I wrote something wrong please tell me. If what I wrote is true how do I proceed in calculating the Brouwer degree of that map? Is there a general way to approach the problem of calculating the Brouwer degreee of a map? Thank you in advance!