On homology spheres with few minimal non faces

Let \Delta be a (d-1)-dimensional homology sphere on n vertices with m minimal non-faces. We consider the invariant \alpha := m - (n-d) and prove that for a given value of \alpha, there are only finitely many homology spheres that cannot be obtained through one-point suspension and suspension from another. Moreover, we describe all homology spheres with \alpha up to four and, as a corollary, all homology spheres with up to eight minimal non-faces. To prove these results we consider the nerve of the minimal non-faces of \Delta.