An elementary approach to the homological properties of constant-rank operators

TL;DR

This paper presents a simple, constructive method to analyze the homological properties of constant-rank operators, improving existing constructions and establishing a generalized Poincaré lemma for these operators.

Contribution

It introduces a self-contained, improved construction of potentials and annihilators for constant-rank operators, extending previous results and providing a generalized Poincaré lemma.

Findings

Constructed explicit potentials and annihilators with improved order.

Extended Van Schaftingen's optimal annihilator construction.

Established a generalized Poincaré lemma for constant-rank operators.

Abstract

We give a simple and constructive extension of Rai\c{t}\u{a}'s result that every constant-rank operator possesses an exact potential and an exact annihilator. Our construction is completely self-contained and provides an improvement on the order of the operators constructed by Rai\c{t}\u{a}, as well as the order of the explicit annihilators for elliptic operators due to Van Schaftingen. We also give an abstract construction of an optimal annihilator for constant-rank operators, which extends the optimal construction of Van Schaftingen for elliptic operators. Lastly, we establish a generalized Poincar\'e lemma for constant-rank operators and homogeneous spaces on $\mathbb{R}^d$, and we prove that the existence of potentials on spaces of periodic maps requires a strictly weaker condition than the constant-rank property.