Couplings and primitives on topological spaces

TL;DR

This paper investigates conditions for the existence of functions on topological spaces that satisfy specific difference relations over open covers, with applications to distributional potentials and algebraic topology.

Contribution

It introduces a new framework for the existence of functions satisfying difference conditions on open covers, connecting to Poincaré theorems and algebraic topology.

Findings

Established criteria for the existence of such functions.

Applied results to distributional potentials.

Connected the framework to algebraic topology.

Abstract

For an open covering $\mathcal U$ of a topological space and a mapping $d\colon\mathcal I\to\mathbb{K}$, where $\mathcal I:=\bigl\{(U,V)\in\mathcal U\times\mathcal U;\ U\cap V\ne\varnothing\bigr\}$, we present a context for the existence of a mapping $C\colon\mathcal U\to\mathbb{K}$ satisfying $C_V-C_U=d_{UV}$ for all $(U,V)\in\mathcal I$. The result is applied to a Poincar\'e type theorem concerning distributional potentials. We also put the result into the context of algebraic topology.