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Filtered complexes and cohomologically equivalent subcomplexes

Erlend Grong, Francesca Tripaldi

TL;DR

For any finite-depth filtered cochain complex $(K,d)$, the paper constructs a cohomologically equivalent subcomplex $(\mathcal{P},d|_{\mathcal{P}})$ that computes the same cohomology as $(K,d)$, generalizing the Rumin complex. The construction uses a base differential $d_0$ together with a base transversal $\mathcal{T}$ and cotransversal $\mathcal{C}$ to obtain a splitting $K = \mathcal{T} \oplus d\mathcal{T} \oplus \mathcal{P}$ and a projection $P^\infty$ that identifies $(\mathcal{P},d|_{\mathcal{P}})$ with $(K,d)$ at the cohomology level; this remains invariant under changes of $\mathcal{T},\mathcal{C}$ and under different choices of $d_0$. The authors connect this subcomplex to spectral sequences by treating $(K,d)$ as a multicomplex, deriving explicit formulas for the higher differentials on each page and showing that the $dP^\infty$ operator recovers the same information as the multicomplex differentials. Through detailed examples—including manifolds with negative tangent filtrations, graded Lie algebras, and three-dimensional Lie groups with negative grading—the paper demonstrates how to compute the resulting pages and showcases the precise correspondence between the subcomplex and spectral-sequence data. Overall, the work unifies cohomological techniques in filtered settings and provides practical tools for analyzing filtered complexes in differential geometry and Lie theory.

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.

Filtered complexes and cohomologically equivalent subcomplexes

TL;DR

For any finite-depth filtered cochain complex , the paper constructs a cohomologically equivalent subcomplex that computes the same cohomology as , generalizing the Rumin complex. The construction uses a base differential together with a base transversal and cotransversal to obtain a splitting and a projection that identifies with at the cohomology level; this remains invariant under changes of and under different choices of . The authors connect this subcomplex to spectral sequences by treating as a multicomplex, deriving explicit formulas for the higher differentials on each page and showing that the operator recovers the same information as the multicomplex differentials. Through detailed examples—including manifolds with negative tangent filtrations, graded Lie algebras, and three-dimensional Lie groups with negative grading—the paper demonstrates how to compute the resulting pages and showcases the precise correspondence between the subcomplex and spectral-sequence data. Overall, the work unifies cohomological techniques in filtered settings and provides practical tools for analyzing filtered complexes in differential geometry and Lie theory.

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.
Paper Structure (19 sections, 13 theorems, 103 equations)

This paper contains 19 sections, 13 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Subcomplexes of general filtered complexes
  3. Base differential of filtered complexes
  4. Analogues of Rumin forms and Hodge decomposition
  5. Invariance under choice of transversal and cotransversal
  6. Invariance under choice of base differential
  7. Proof of Theorem \ref{['th:main']}
  8. Examples
  9. Manifolds with a negative filtration of the tangent bundle
  10. Application: Differentiating partial forms
  11. Graded Lie algebras
  12. Spectral methods for the filtered subcomplex
  13. Spectral sequence associated to a multicomplex
  14. Filtered complexes $(K,d)$ of finite depth as multicomplexes
  15. Three dimensional Lie groups with negative grading
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $(K,d)$ be a filtered cochain complex of finite depth with filtration $K = K_0 \supset K_1 \supset K_2 \supset \cdots \supset K_S \supset K_{S+1} =0$ and $dK_j\subset K_j$ for any $j=0,\ldots,S$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7
  • ...and 24 more