Frolicher-Nijenhuis cohomology on G2- 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* (M) and Omega*(M,T M), defined via the Rolicher-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 G(2)- and Spin(7)-manifold, respectively. We calculate the cohomology groups of Omega*(M) and give a partial description of the cohomology of Omega*(M,TM).