We study the geometry induced on the local orbit spaces of Killing vector fields on (Riemannian) $G$-manifolds, with an emphasis on the cases $G=\textrm {Spin}(7)$ and $G=G_2$. Along the way, we classify the harmonic morphisms with one-dimensional fibres from $G_2$-manifolds to Einstein manifolds.

We construct a cycle in higher Hochschild homology associated to the two-dimensional torus which represents 2-holonomy of a nonabelian gerbe in the same way as the ordinary holonomy of a principal G-bundle gives rise to a cycle in ordinary Hochschild homology. This is done using the connection 1-form of Baez–Schreiber. A crucial ingredient in our work is the possibility to arrange that in the structure crossed module $\unicode[STIX]{x1D707}:\mathfrak{h}\rightarrow \mathfrak{g}$ of the principal 2-bundle, the Lie algebra $\mathfrak{h}$ is abelian, up to equivalence of crossed modules.

We prove that among all compact homogeneous spaces for an effective transitive action of a Lie group whose Levi subgroup has no compact simple factors, the seven-dimensional flat torus is the only one that admits an invariant torsion-free $\text{G}_{2(2)}$-structure.

We consider some deformations of ${{G}_{2}}$ -structures on 7-manifolds. We discover a canonical way to deform a ${{G}_{2}}$ -structure by a vector field in which the associated metric gets “twisted” in some way by the vector cross product. We present a system of partial differential equations for an unknown vector field $w$ whose solution would yield a manifold with holonomy ${{G}_{2}}$ . Similarly we consider analogous constructions for Spin(7)-structures on 8-manifolds. Some of the results carry over directly, while others do not because of the increased complexity of the Spin(7) case.