Couplings and primitives on topological spaces

Abstract

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