Couplings and primitives on topological spaces

Abstract

AbstractFor an open covering $$\hspace{0.55542pt}\mathcal {U}$$ U of a topological space and a mapping "Equation missing", where $$\mathcal {I}:=\bigl \{(U,V)\in \hspace{0.55542pt}\mathcal {U}\times \hspace{0.55542pt}\mathcal {U}{;}\;U\cap V\ne \varnothing \bigr \}$$ I : = { ( U , V ) ∈ U × U ; U ∩ V ≠ ∅ } , we present conditions for the existence of a mapping "Equation missing" satisfying $$C_V-C_U=d_{UV}$$ C V - C U = d UV for all $$(U,V)\in \mathcal {I}$$ ( U , V ) ∈ I . The result is applied to a Poincaré type theorem concerning distributional potentials. We also put the result into the context of algebraic topology.