Estimates of covering type and minimal triangulations based on category weight

TL;DR

This paper extends a method for estimating the minimal size of triangulations of topological spaces by incorporating the fundamental group's properties, leading to stronger lower bounds for various complex spaces.

Contribution

It introduces a new approach that considers the fundamental group, enhancing previous cohomology-based estimates for triangulation sizes.

Findings

Developed weighted estimates using fundamental group properties.

Applied the method to orbit spaces of cyclic group actions.

Provided explicit lower bounds for triangulations of complex spaces.

Abstract

In a recent publication (D. Govc, W. Marzantowicz, P. Pavesic, Estimates of covering type and the number of vertices of minimal triangulations, Discr. Comp. Geom. 63 (2019), 31-48) we have introduced a new method, based on the Lusternik-Schnirelmann category and the cohomology ring of a space X, that yields lower bounds for the size of a triangulation of X. In this paper we present an important extension that takes into account the fundamental group of X. In fact, if it contains elements of finite order, then one can often find cohomology classes of high 'category weight', which in turn allow for much stronger estimates of the size of triangulations of X. We develop several weighted estimates and then apply our method to compute explicit lower bounds for the size of triangulations of orbit spaces of cyclic group actions on a variety of spaces including products of spheres, Stiefel manifolds, Lie groups and highly-connected manifolds.