Steenrod operations for $4$-dimensional toric orbifolds

TL;DR

This paper characterizes when non-trivial Steenrod operations exist on the mod-2 cohomology of 4-dimensional toric orbifolds and explores implications for their homotopy type, gauge groups, and spin structures.

Contribution

It provides necessary and sufficient conditions for Steenrod actions on these orbifolds and links cohomological properties to their geometric and topological features.

Findings

Criteria for non-trivial Steenrod actions established

Determination of stable homotopy types and gauge groups

A combinatorial criterion for spin structures in smooth cases

Abstract

We prove necessary and sufficient conditions for the existence of non-trivial Steenrod actions on the mod-$2$ cohomology of 4-dimensional toric orbifolds. As applications, the stable homotopy type and the gauge groups of a $4$-dimensional toric orbifold are determined, a partial solution to the cohomological rigidity problem for $4$-dimensional toric orbifolds is provided, and, in the smooth case, a combinatorial criterion is established for when the toric orbifold is spin.