Estimates of covering type and the number of vertices of minimal triangulations

TL;DR

This paper establishes bounds on the covering type of spaces using algebraic invariants and relates it to minimal triangulations, providing new and extended estimates applicable across homotopy classes.

Contribution

It introduces a unified approach to estimate covering type via algebraic invariants and relates it to minimal triangulations, extending previous ad hoc results.

Findings

Derived bounds for covering type using homology and cohomology data.

Connected covering type estimates to minimal triangulation vertex counts.

Provided results valid for entire homotopy classes of spaces.

Abstract

The covering type of a space $X$ is defined as the minimal cardinality of a good cover of a space that is homotopy equivalent to $X$. We derive estimates for the covering type of $X$ in terms of other invariants of $X$, namely the ranks of the homology groups, the multiplicative structure of the cohomology ring and the Lusternik-Schnirelmann category of $X$. By relating the covering type to the number of vertices of minimal triangulations of complexes and combinatorial manifolds, we obtain, within a unified framework, several estimates which are either new or extensions of results that have been previously obtained by ad hoc combinatorial arguments. Moreover, our methods give results that are valid for entire homotopy classes of spaces.