The covering type of closed surfaces and minimal triangulations

TL;DR

This paper computes the covering type for all closed surfaces, linking topological complexity with minimal triangulations, and resolves a problem posed by Karoubi and Weibel.

Contribution

It provides a complete calculation of the covering type for closed surfaces, clarifying the relationship between surface topology and minimal triangulations.

Findings

Covering type of all closed surfaces is explicitly determined.

Results resolve a previously posed open problem.

Insights into the link between topology and minimal triangulations.

Abstract

The notion of covering type was recently introduced by Karoubi and Weibel to measure the complexity of a topological space by means of good coverings. When X has the homotopy type of a finite CW-complex, its covering type coincides with the minimum possible number of vertices of a simplicial complex homotopy equivalent to X. In this article we compute the covering type of all closed surfaces. Our results completely settle a problem posed by Karoubi and Weibel, and shed more light on the relationship between the topology of surfaces and the number of vertices of minimal triangulations from a homotopy point of view.