I'm quite at a loss with this...I want to use Mayer-Vietoris with open covers $A=\Sigma_{2}\times (S^{1}\setminus \{p\})$ and $B=\Sigma_{2}\times (S^{1}\setminus \{q\})$ so that $A$ and $B$ both deformation retract to $\Sigma_{2}$ and $A\cap B$ deformation retracts to $\Sigma_{2}\times\{0,1\}$, but I don't understand how to think about the inclusion maps $H_{n}(A\cap B) \hookrightarrow H_{n}(A)\bigoplus H_{n}(B)$.
$\Sigma_2$ denotes the orientable surface of genus two.