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Multicomplexes on Carnot Groups and Their Associated Spectral Sequence.

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  • 1SISSA, Mathematics Area, via Bonomea, 265, 34146 Trieste, Italy.

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Summary
This summary is machine-generated.

This study explores the Rumin complex on Carnot groups and its connection to spectral sequences. It details how weight filtrations compute the de Rham cohomology for these groups.

Keywords:
Carnot groupsMulticomplexesRumin complexSpectral sequences

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Area of Science:

  • Differential Geometry
  • Algebraic Topology

Background:

  • Carnot groups are a class of Lie groups with a graded structure.
  • The de Rham cohomology is a fundamental invariant in topology.
  • The Rumin complex is a differential complex on Carnot groups.

Purpose of the Study:

  • To investigate the relationship between the Rumin complex and spectral sequences on Carnot groups.
  • To demonstrate how a spectral sequence, derived from a filtration on differential forms, computes the de Rham cohomology of Carnot groups.

Main Methods:

  • Utilizing the filtration of differential forms by homogeneous weights.
  • Applying spectral sequence techniques to analyze the Rumin complex.
  • Computing the de Rham cohomology of Carnot groups.

Main Results:

  • Established a connection between the Rumin complex and a specific spectral sequence.
  • Showcased the utility of this spectral sequence in computing de Rham cohomology.
  • Provided insights into the structure of Carnot groups through their cohomology.

Conclusions:

  • The spectral sequence effectively computes the de Rham cohomology of Carnot groups.
  • The Rumin complex plays a crucial role in this computational framework.
  • This work offers a new perspective on the interplay between analysis and topology on Carnot groups.