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

Abstract

We use classical obstruction theory \`{a} la Eilenberg-Steenrod to obtain a homotopy classification of $\mathrm{Spin}(7)$-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 $\mathrm{Spin}(7)$ has exactly two $\mathrm{Spin}(7)$-structures extending the induced $G_{2}$-structure on the boundary.