Frölicher–Nijenhuis cohomology on G2- and Spin(7)-manifolds

In this paper, we show that a parallel differential form Ψ of even degree on a Riemannian manifold allows to define a natural differential both on Ω∗(M) and Ω∗(M,TM), defined via the Frolicher–Nijenhuis bracket. For instance, on a Kahler manifold, these operators are the complex differential and the Dolbeault differential, respectively. We investigate this construction when taking the differential with respect to the canonical parallel 4-form on a G2- and Spin(7)-manifold, respectively. We calculate the cohomology groups of Ω∗(M) and give a partial description of the cohomology of Ω∗(M,TM).