I'm trying to compute $H_i(\mathbb{RP}^n \times \mathbb{RP}^m; G)$ for $G = \mathbb{Z}, \mathbb{Z_2}$ respectively by using the cellular chain complexes.
I'm not really sure how to get started, though I do know each real projective space has 1 cell in each dimension up to its dimension. I also know that the chain complex for $\mathbb{RP}^n$ should look like:
$\rightarrow 0 \rightarrow G \rightarrow G \rightarrow \ldots \rightarrow G \rightarrow 0$ where $G$ is repeated $n$ times as a result of the projective plane having 1 cell in each dimension up to its dimension.
So I've computed the case for when $G = \mathbb{Z}_2$. We simply get if $ 0 \leq i \leq n$ $H_i(\mathbb{RP}_m \times \mathbb{RP}_n) = \mathbb{Z}^{i+1}$, if $ n \leq i \leq m$ $H_i(\mathbb{RP}_m \times \mathbb{RP}_n) = \mathbb{Z}^{n+1}$, if $ m \leq i \leq n + m$ $H_i(\mathbb{RP}_m \times \mathbb{RP}_n) = \mathbb{Z}^{n+m+1-i}$ and $0$ otherwise.
The case where $G = \mathbb{Z}$ on the other hand has me stumped. I'm reduced to computing homologies with Macaulay2 and trying to find a pattern. I can find the homologies myself by hand from the chain complexes I've computed but this is very slow and tedious. Even so the patterns are not immediately obvious to me.
Thoughts?