Filtered complexes and cohomologically equivalent subcomplexes

TL;DR

This paper generalizes Rumin's construction of cohomologically equivalent subcomplexes in filtered complexes, demonstrating their dependence on filtration up to isomorphism and relating them to spectral sequences.

Contribution

It introduces a broad construction for subcomplexes from filtered complexes that preserve cohomology and clarifies their dependence on filtration and relation to spectral sequences.

Findings

Subcomplexes depend only on the filtration up to isomorphism.

The construction applies to any filtered cochain complex of finite depth.

Connections between these subcomplexes and spectral sequences are established.

Abstract

Inspired by Rumin's work on a subcomplex in sub-Riemannian manifolds which 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 only depends on the given filtration up to isomorphism. Finally, we show how such subcomplexes relate to spectral sequences, a cohomological machinery that arises naturally when considering a filtered complex.