I do not remember somewhere in specific the Betti numbers but rather their Dolbeault cohomology counterparts, the Hodge numbers . The idea is that if we are given some topological space (satisfying the Calabi-Yai conditions etc) we need a way to count the number of free parameters in the representative of its $SU(N)$ holonomy metric. This is not a new idea, we do the same when we choose the compactification radius when we perform compactifications on $S^1$. The number of parameters (for the topological space I started talking about) is given in terms of the Hodge numbers which are a topological invariant of the corresponding complex manifold.