Let
Δ
\Delta
be a
(
d
−
1
)
(d-1)
-dimensional homology sphere on
n
n
vertices with
m
m
minimal non-faces. We consider the invariant
α
(
Δ
)
=
m
−
(
n
−
d
)
\alpha (\Delta ) = m - (n-d)
and prove that for a given value of
α
\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
α
(
Δ
)
\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
Δ
\Delta
. Also, we give a short classification of all homology spheres with
n
−
d
≤
3
n-d \leq 3
.