On the Geometry and Topology of Positively Curved Eschenburg Orbifolds

TL;DR

This paper investigates how positive sectional curvature influences the geometry and topology of Eschenburg orbifolds, revealing restrictions on singular sets and distinctive cohomology properties compared to non-negatively curved cases.

Contribution

It proves restrictions on singular sets due to positive curvature and computes orbifold cohomology rings for all Eschenburg orbifolds, confirming a conjecture and highlighting topological differences.

Findings

Restrictions on singular sets in positively curved orbifolds

Computed orbifold cohomology rings for all Eschenburg orbifolds

Identified distinctive cohomology behavior compared to non-negatively curved cases

Abstract

The present article explores the relationship between positive sectional curvature and the geometric and topological properties of Eschenburg $6$-orbifolds. First, we prove that positive sectional curvature imposes restrictions on the their singular sets, thereby confirming a conjecture posed by Florit and Ziller. Then we compute the orbifold cohomology rings for those with a specific singular locus. This reveals a distinctive behavior in the cohomology groups of positively curved Eschenburg orbifolds compared to their non-negatively curved counterparts. Furthermore, we compute the orbifold cohomology rings of all Eschenburg orbifolds.