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Pansu pullback and spectral complexes

Filippa Lo Biundo, Francesca Tripaldi

Abstract

In this paper, we prove the commutativity between the Pansu pullback of a smooth contact map between Carnot groups and the differentials appearing in the spectral complexes. As a direct application, we also present a way of "lifting" a Pansu derivative (viewed as a Lie algebra homomorphism) from Carnot groups to their central extensions.

Pansu pullback and spectral complexes

Abstract

In this paper, we prove the commutativity between the Pansu pullback of a smooth contact map between Carnot groups and the differentials appearing in the spectral complexes. As a direct application, we also present a way of "lifting" a Pansu derivative (viewed as a Lie algebra homomorphism) from Carnot groups to their central extensions.
Paper Structure (5 sections, 17 theorems, 184 equations)

This paper contains 5 sections, 17 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. The de Rham complex on Carnot groups as a truncated multicomplex
  3. Pansu pullback of forms
  4. The Pansu pullback on spectral complexes
  5. Constructing lifts to stratifiable and non-stratifiable groups

Key Result

Lemma 2.5

Let us consider $\alpha_1,\alpha_2\in\Omega^k(G)$ two arbitrary $k$-forms. If they are both homogeneous with $w(\alpha_1)\neq w(\alpha_2)$, then they are linearly independent.

Theorems & Definitions (58)

  • Definition 2.1: Dilations
  • Definition 2.2: Carnot groups
  • Remark 2.3
  • Definition 2.4: Weights of forms
  • Lemma 2.5: Forms of different weight are linearly independent
  • proof
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Definition 2.8: Definition 2.1 in livernet2020spectral
  • ...and 48 more