Combinatorial triangulations of homology spheres

Let M be an n-vertex combinatorial triangulation of a $Z_{2}$-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 cannot 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 $Z_{2}$-acyclic sitnplicial 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