Fr\"olicher-Nijenhuis bracket on manifolds with special holonomy

In this article, we summarize our recent results on the study of manifolds with special holonomy via the Fr\"olicher-Nijenhuis bracket. This bracket enables us to define the Fr\"olicher-Nijenhuis cohomologies which are analogues of the $d^c$ and the Dolbeault cohomologies in K\"ahler geometry, and assigns an $L_\infty$-algebra to each associative submanifold. We provide several concrete computations of the Fr\"olicher-Nijenhuis cohomology.