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

Abstract

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 S d . Another important consequence is that every triangulation with small links of M is combinatorial.

Vertex numbers of simplicial complexes with free abelian fundamental group

“…The number of faces of a simplicial complex X can be bounded in terms of the Betti numbers of X [7] or in terms of the minimal number of generators of π 1 (X) [22]. The effect of relations of π 1 (X) on vertex numbers has been studied for cyclic torsion groups [15,23] and for triangulations of manifolds with non-free fundamental group [25]. In this paper, we consider the minimal number of vertices of a simplicial complex with fundamental group Z n : Theorem 1.1.…”

Vertex numbers of simplicial complexes with free abelian fundamental group

“…The number of faces of a simplicial complex X can be bounded in terms of the Betti numbers of X [7] or in terms of the minimal number of generators of π 1 (X) [22]. The effect of relations of π 1 (X) on vertex numbers has been studied for cyclic torsion groups [15,23] and for triangulations of manifolds with non-free fundamental group [25]. In this paper, we consider the minimal number of vertices of a simplicial complex with fundamental group Z n : Theorem 1.1.…”

Estimates of covering type and minimal triangulations based on category weight

“…In [14] we introduced several new estimates for the minimal number of vertices that are needed to triangulate a compact triangulable space X based on the Lusternik-Schnirelmann category of X and on the structure of the cohomology ring of X. In the case of manifolds these estimates were improved by using information obtained from the fundamental group in [26] and on the Lower Bound Theorem in [15]. The information on the number, or respectively rate of growth of number of simplices included in the latter has been used in [21] and [29] respectively.…”