In sub-Riemannian geometry, Carnot groups play a role analogous to that of Euclidean spaces in the Riemannian setting. Their special structure allows one to define an intrinsic notion of differentiability, namely Pansu differentiability, which in turn gives rise to the Pansu pullback on differential forms. A natural question is whether this pullback commutes with the differentials of the main complexes associated with Carnot groups.
In this talk, I will exhibit counterexamples to this commutativity for the de Rham complex and the Rumin complex, the latter being specifically adapted to the geometric structure of Carnot groups. I will then turn to a recently introduced family of complexes, the spectral complexes associated with the de Rham complex, and explain how, in this setting, the Pansu pullback does commute with the corresponding differentials.