The $Z_2$-homology of a surface viewed as a simplicial complex allows us to extract interesting invariants from the resulting homology groups. $\beta_0$ is the number of connected components, $\beta_1$ is the number of "tunnels" and $\beta_2$ is the number of "cavities" ($\beta_i$, the $i^{th}$ Betti number, being the rank of the $i^{th}$ homology group)

Are there other "interesting" invariants of surfaces (or even graphs) that can be obtained by going from $Z_2$ to some other field (like the rationals), or are all of these equivalent in some sense ?