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 $\alpha (\Delta ) = m - (n-d)$ 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 $\alpha (\Delta )$ up to four and, as a corollary, all homology spheres with up to eight minimal non-faces. To prove these results we consider the lcm-lattice and the nerve of the minimal non-faces of $\Delta$. Also, we give a short classification of all homology spheres with $n-d \leq 3$.