Consider a surface $S$ whose homology is $H_*(S)$. At least for sufficiently nice surfaces, whatever this vague adjective means, one can calculate the Betti number (or Hodge numbers) and find the dimensions of the homology groups. Then it is easy to construct the Poincare polynomial. For example, for the torus we have $b_1=1, b_2=2, b_3=1$ and thus $P(T^1)=x+2x^2+x^3$.
My questions is wether this polynomial is in some sense fundamental. Are there examples where one cannot directly calculate Betti numbers but can find the Poincare polynomial and as such find out the dimensions of the homology groups? And if so what is the (or a) method to calculate the Poincare polynomial?