Some results on Betti numbers of Stanley-Reisner rings

We study the Betti numbers which appear in a minimal free resolution of the Stanley-Reisner ring k[Delta] = A/I-Delta of a simplicial complex Delta over a field k. It is known that the second Betti number of k[Delta] is independent of the base field k. We show that, when the ideal I-Delta is generated by square-free monomials of degree two, the third and fourth Betti numbers are also independent of k. On the other hand, we prove that, if the geometric realization of Delta is homeomorphic to either the 3-sphere or the 3-ball, then all the Betti numbers of [Delta] are independent of the base field k.