Frölicher–Nijenhuis cohomology on $G_2$ - and Spin(7) -manifolds

In this paper we show that a parallel differential form $\Psi$ of even degree on a Riemannian manifold allows to define a natural differential both on $\Omega^\ast(M)$ and $\Omega^\ast(M, TM)$, defined via the Fr\"olicher-Nijenhuis bracket. For instance, on a K\"ahler manifold, these operators are the complex differential and the Dolbeault differential, respectively. We investigate this construction when taking the differential w.r.t. the canonical parallel $4$-form on a $G_2$- and ${\rm Spin}(7)$-manifold, respectively. We calculate the cohomology groups of $\Omega^\ast(M)$ and give a partial description of the cohomology of $\Omega^\ast(M, TM)$.