Combinatorial triangulations of homology spheres

Let Mu be an n-vertex combinatorial triangulation of a Ζ2-homology d-sphere. In this paper we prove that if n ≤ d+8 then Mu must be a combinatorial sphere. Further, if n=d+9 and M is not a combinatorial sphere then Mu 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 anyΖ2-acyclic simplicial complex on ≤7 vertices is necessarily collapsible. This result is best possible since there exist 8-vertex triangulations of the Dunce Hat which are not collapsible