A homotopy classification of $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(7)$-structures with applications to exceptional Riemannian holonomy

We use classical obstruction theory à la Eilenberg-Steenrod to obtain a homotopy classification of $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(7)$-structures on compact $8$-manifolds with abelian fundamental group. As an application, we show that a compact, connected Riemannian $8$-manifold with holonomy contained inside the group $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(7)$ has exactly two $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(7)$-structures extending the induced $G_2$-structure on the boundary.