  Let $M$ be an $n$-vertex combinatorial triangulation of a $\ZZ_2$-homology
$d$-sphere. In this paper we prove that if $n \leq d + 8$ then $M$ must be a
combinatorial sphere. Further, if $n = d + 9$ and $M$ is not a combinatorial
sphere then $M$ can not admit any proper bistellar move. Existence of a
12-vertex triangulation of the lens space $L(3, 1)$ shows that the first result
is sharp in dimension three.
  In the course of the proof we also show that any $\ZZ_2$-acyclic simplicial
complex on $\leq 7$ vertices is necessarily collapsible. This result is best
possible since there exist 8-vertex triangulations of the Dunce Hat which are
not collapsible.
