Sobolev mappings and the Rumin complex

We consider contact manifolds equipped with Carnot-Caratheodory metrics, and show that the Rumin complex is respected by Sobolev mappings: Pansu pullback induces a chain mapping between the smooth Rumin complex and the distributional Rumin complex. As a consequence, the Rumin flat complex---the analog of the Whitney flat complex in the setting of contact manifolds---is bilipschitz invariant. These two results are Heisenberg group analogs of classical results of Reshetnyak (1967, 1989) and Whitney (1957). We also show that for Sobolev mappings between general Carnot groups, Pansu pullback induces a chain mapping when restricted to a certain differential ideal J^*G⊂Ω^*G of the de Rham complex.