Combinatorial triangulations of homology spheres

Let $M$ be an $n$-vertex combinatorial triangulation of a ${\text{ZZ}}_2$-homology $d$-sphere. In this paper we prove that if $n\le d+8$ 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 ${\text{ZZ}}_2$-acyclic simplicial complex on $\le 7$ vertices is necessarily collapsible. This result is best possible since there exist 8-vertex triangulations of the Dunce Hat which are not collapsible.