A note on the topology of irreducible ${\rm SO}(3)$-manifolds

TL;DR

This paper establishes topological criteria for the existence of irreducible SO(3)-structures on 5-manifolds and provides new examples of such manifolds.

Contribution

It offers necessary and sufficient conditions for irreducible SO(3)-structures on 5-manifolds and constructs new examples based on these criteria.

Findings

Derived topological conditions for SO(3)-structures

Identified new 5-manifolds with irreducible SO(3)-structures

Enhanced understanding of 5-manifold topology

Abstract

We give necessary and sufficient topological conditions for the existence of an irreducible ${\rm SO}(3)$-structure on a $5$-manifold. Using these conditions we provide some new examples of $5$-manifolds with an irreducible ${\rm SO}(3)$-structure.