Filtered complexes and cohomologically equivalent subcomplexes

Abstract

Inspired by Rumin’s work on a subcomplex in sub-Riemannian manifolds that is cohomologically equivalent to the de Rham complex, we present a more general construction that produces subcomplexes from any filtered cochain complex of finite depth and still computes the cohomology of the original filtered complex. A priori, these subcomplexes depend not only on the filtration itself, but also on the choice of additional structures. However, we show that the construction depends only on the given filtration up to isomorphism. We give examples with application to partial forms and partial connections. Finally, we show how such subcomplexes relate to spectral sequences, a cohomological machinery that arises naturally when considering a filtered complex.