Subcomplexes on filtered Riemannian manifolds

Abstract

In this paper, we present a general construction to extract subcomplexes from two distinct complexes on filtered Riemannian manifolds. The first subcomplex computes the de Rham cohomology of the underlying manifold. On regular subRiemannian manifold equipped with a compatible Riemannian metric, it aligns locally with the so-called Rumin complex. The second complex instead generalises the Chevalley-Eilenberg complex computing Lie algebra cohomology of a nilpotent Lie group. Our approach offers key insights on the role of the Riemannian metric when extracting subcomplexes, opening up potential new applications in more general geometric settings, such as singular subRiemannian manifolds.