Homology of $\mathbb{P}^2(\mathbb{R}) \times \mathbb{P}^2(\mathbb{R}) $

Question

I'd like to compute the homology of $\mathbb{P}^2(\mathbb{R}) \times \mathbb{P}^2(\mathbb{R})$ in the most efficient way. I thought using universal coefficient and Künneth Theorem but those seem to fail, since I don't know how to compute Tor or Ext which seem non trivial in this case.

I think that using that the homology of $\mathbb{P}^2(\mathbb{R}) \times \mathbb{P}^2(\mathbb{R})$ is equal if calculated with any $CW$ decomposition would be a good idea but unfortunately the problem seems to traduce in understanding the differential of the $CW$ decomposition chosen, and in the case of a product, I'm unable to do so.

There's a standard way to compute this types of Homology? Any help reference or hint would be appreciated, especially if I'm looking in the wrong direction in order to solve the problem.

Answer 1

Actually I realized that the only properties of Tor I knew was the only ones I needed, i.e Tor($B,\mathbb{Z}_n$) = $B/nB$ and that Tor($A,B) = 0$ if either $A$ or $B$ is free.

With this in mind, one just have to compute Tor($\mathbb{Z}_2,\mathbb{Z}_2) = \mathbb{Z}_2$ and apply the Künneth Theorem split sequence in Homology knowing the homology of $\mathbb{P R}^2$.