I am trying to compute the homology groups for the real and complex projective spaces but without use the cw-complex structure.
My idea would be to use the transfer sequence, because we already know the homology groups for the sphere, combined with the relation between the homology with coefficients and the normal homology with coefficients in $\mathbb{Z}$. But my problem is to calculate the Torsion group $Tor(H_{k-1}(\mathbb{R}P^n), \mathbb{Z}_2)$ for the projective space.
I don't know if can be done, because I did a literature search and I only found the calculation using the CW-complex.
And for the complex case, I din't think about it yet.
Thank you!
You Might want to try induction over $n$ and use Mayer-Vietoris. The base step is trivial (you should know the homologises), for the inductive step try to make use of this hint:
$\mathbb{R}P^n$ minus a point deformation retracts in $\mathbb{R}P^{n-1}$ , making use of M-V you should be able to compute the homology of the space.
Analogously for the complex projective space, which should be easier.
Another approach very similar to M-V is to use the homology l.e.s of the pair, since you can identify $\mathbb{R}P^{n-1}\subset \mathbb{R}P^n$ and since they are nice spaces (CW-complex) you have $H_k(\mathbb{R}P^n,\mathbb{R}P^{n-1})\cong \widetilde{H}_k(\mathbb{R}P^n/\mathbb{R}P^{n-1})$ and you should realise what the latter space look like