Triangulations with few vertices of manifolds with non-free fundamental group

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We study lower bounds for the number of vertices in a PL-triangulation of a given manifold M. While most of the previous estimates are based on the dimension and the connectivity of M, we show that further information can be extracted by studying the structure of the fundamental group of M and applying techniques from the Lusternik-Schnirelmann category theory. In particular, we prove that every PL-triangulation of a d-dimensional manifold (d≥ 3) whose fundamental group is not free has at least 3d+1 vertices. As a corollary, every d-dimensional (Zₚ-)homology sphere that admits a PL-triangulation with less than 3d vertices is homeomorphic to Sᵈ. Another important consequence is that every triangulation with small links of M is combinatorial.