The homotopy classification of four-dimensional toric orbifolds

TL;DR

This paper classifies 4-dimensional toric orbifolds up to homotopy, showing how their cohomological torsion properties determine their homotopy types, with implications for their classification.

Contribution

It provides a homotopy classification criterion for 4D toric orbifolds based on their cohomology torsion and ring isomorphisms, extending understanding of their topological structure.

Findings

Orbifolds with non-trivial odd primary torsion in H^3 are homotopy equivalent to a wedge of a Moore space and a CW-complex.

Two 4D toric orbifolds without 2-torsion are homotopy equivalent iff their cohomology rings are isomorphic.

The classification depends on the torsion properties of the third cohomology group.

Abstract

Let $X$ be a $4$-dimensional toric orbifold. If $H^3(X)$ has a non-trivial odd primary torsion, then we show that $X$ is homotopy equivalent to the wedge of a Moore space and a CW-complex. As a corollary, given two 4-dimensional toric orbifolds having no 2-torsion in the cohomology, we prove that they have the same homotopy type if and only their integral cohomology rings are isomorphic.