Combinatorial triangulations of homology spheres Original Research Article

Let M be an n-vertex combinatorial triangulation of a image-homology d-sphere. In this paper we prove that if image then M must be a combinatorial sphere. Further, if image 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 image shows that the first result is sharp in dimension three. In the course of the proof we also show that any image-acyclic simplicial complex on image vertices is necessarily collapsible. This result is best possible since there exist 8-vertex triangulations of the Dunce Hat which are not collapsible.