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

Abstract

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

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