$\mathbf {The \ Problem \ is}:$ Compute all the cap products for $\mathbb{RP}^{2}$ with $\mathbb{Z}$ and $\mathbb{Z} / 2$ coefficients. Do the same for the Klein's bottle $K$.
$\mathbf {My \ approach}:$ As $H_{0}(\mathbb {RP}^{2};\mathbb Z)=\mathbb Z$ and $H_{1}(\mathbb {RP}^{2};\mathbb Z)=\mathbb Z_{2}$ and rest homology groups are $0.$
Then for $\mathbb {RP}^{2}$ , we only need to find :
$H_{0}(M) × H^{0}(M) \to H_{0}(M)$ and $H_{1}(M) × H^{1}(M) \to H_{0}(M)$
where $M=\mathbb {RP}^{2}$
The simplicial complex is : 
But, I can't figure out how to start calculating it and also for $\mathbb Z_{2}$ co-efficients .
And, I also couldn't do anything with $K.$
$H_{0}({K};\mathbb Z)=\mathbb Z$ and $H_{1}({K};\mathbb Z)= \mathbb Z \oplus \mathbb Z_{2}.$ and rest are $0.$
Hence we only have to find :
$H_{0}(M) × H^{0}(M) \to H_{0}(M)$ and $H_{1}(M) × H^{1}(M) \to H_{0}(M)$
where $M=K.$
The simplicial complex is :
A help is much needed, thanks in advance .