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Spectral complexes from truncated multicomplexes

Francesca Tripaldi

Abstract

This paper introduces a new construction of subcomplexes associated with a truncated multicomplex. Inspired by the machinery of spectral sequences, this construction yields a collection of interrelated subcomplexes whose differentials coincide with the spectral sequence differentials. These complexes refine the Rumin complex and retain the cohomology of the underlying multicomplex, providing a new tool for the study of subRiemannian geometry, particularly on Carnot groups.

Spectral complexes from truncated multicomplexes

Abstract

This paper introduces a new construction of subcomplexes associated with a truncated multicomplex. Inspired by the machinery of spectral sequences, this construction yields a collection of interrelated subcomplexes whose differentials coincide with the spectral sequence differentials. These complexes refine the Rumin complex and retain the cohomology of the underlying multicomplex, providing a new tool for the study of subRiemannian geometry, particularly on Carnot groups.
Paper Structure (13 sections, 18 theorems, 171 equations)

This paper contains 13 sections, 18 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. The Rumin complex associated with a truncated multicomplex
  3. Introducing a scalar product on $\mathcal{C}$
  4. Constructing the Rumin complex
  5. Introducing a shorthand notation for the operator $d_c$
  6. Characterising spectral sequences in terms of the Rumin differentials
  7. Expressing the spaces $Z_r^{p,\bullet}$ in terms of Rumin forms and the Rumin differential
  8. Expressing $B_r^{p,\bullet}$ in terms of Rumin forms and the Rumin differential
  9. Expressing the differentials $\Delta_r$ of the spectral sequences in terms of Rumin forms and the Rumin differential
  10. Complexes from spectral sequences
  11. Considerations on spectral complexes
  12. Hodge duality
  13. An explicit example: the Engel group

Key Result

Proposition 2.10

The associated total complex $\mathrm{Tot}\,\mathcal{C}$ admits a direct sum (or a Hodge) decomposition in terms of $\Box_0$, $d_0$, and its adjoint $\delta_0$. More explicitly,

Theorems & Definitions (61)

  • Definition 2.1: Definition 2.1 in livernet2020spectral
  • Remark 2.2
  • Definition 2.3: Definition 2.2 in livernet2020spectral
  • Definition 2.4: The adjoint of $d_0$
  • Remark 2.5
  • Definition 2.6: Rumin forms on $(\mathrm{Tot}\,\mathcal{C},d)$
  • Definition 2.7: The partial inverse $d_0^{-1}$
  • Definition 2.8: Orthogonal projection onto $E_0^\bullet$
  • Definition 2.9: The Laplacian $\Box_0$
  • Proposition 2.10: Hodge decomposition for $\Box_0$
  • ...and 51 more