Cohomology to compute number of holes?

Question

Can one use cohomology to compute the number of holes in a space $E$, where $E=R\times R$, $R$ is a Riemann surface of genus $g$, - i.e., is $\dim(H^n(E))$, and by Künneth's formula, $H^{n}(E) \cong \bigoplus_{k+l=n} H^k(R) \otimes H^l(R)$ in some way related to the number of $n$-dimensional holes in the space $E$? If yes, then how?

Answer 1

The manifold $R$ is compact and orientable and so we can say the same for $E$, in which case we may use Poincaré duality to swap $r$th cohomology for $(4-r)$th homology.

So the number of $r$-dimensional holes in $E$ is $$\mbox{rank }\left(\bigoplus_{k+l=4-r} H^{k}(R)\otimes H^{l}(R)\right)=\sum_{k+l=4-r}\mbox{rank }H^k(R)\cdot\mbox{rank }H^l(R).$$

(Hopefully I didn't mess up in the above - horrible indices)