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

TL;DR

This paper establishes lower bounds on the number of vertices in PL-triangulations of manifolds, linking the fundamental group's structure to triangulation complexity, and characterizes certain homology spheres.

Contribution

It introduces new lower bounds based on the fundamental group and applies Lusternik-Schnirelmann theory to triangulation problems, providing novel insights into manifold triangulations.

Findings

Manifolds with non-free fundamental group require at least 3d+1 vertices in their triangulation.

Homology spheres with fewer than 3d vertices are homeomorphic to spheres.

Small link triangulations imply the triangulation is combinatorial.

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 ($d\ge 3$) whose fundamental group is not free has at least $3d+1$ vertices. As a corollary, every $d$-dimensional ($\mathbb{Z}_p$-)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.