Surjectivity of the right map in the coarse cohomology Mayer–Vietoris sequence

Determine whether the rightmost map CX^•(X) ⊕ CX^•(Y) → CX^•(X ∩ Y) in the cochain sequence 0 → CX^•(X ∪ Y) → CX^•(X) ⊕ CX^•(Y) → CX^•(X ∩ Y) is surjective for arbitrary big families X and Y in a proper metric space M. Establish necessary and sufficient conditions for surjectivity or construct explicit counterexamples, thereby clarifying the exactness of the sequence in the coarse cohomology setting.

Background

The paper develops a measure-theoretic framework for coarse homology and cohomology on proper metric spaces and formulates Mayer–Vietoris sequences for big families. For homology, a short exact sequence of chain complexes is established, yielding a standard long exact sequence. For coarse cohomology, the authors construct an exact sequence of cochain complexes dual to the homological one and prove injectivity of the left map and exactness in the middle.

However, unlike the homological case, the authors explicitly state that they do not know whether the right map in the cohomological sequence is surjective in general. They provide auxiliary results (e.g., a localization lemma and a short exact sequence under additional structural assumptions) that yield surjectivity in specific settings, but the general case remains unresolved.

Resolving this question would clarify the exactness properties of the Mayer–Vietoris sequence in coarse cohomology for arbitrary big families and strengthen the duality between the homological and cohomological frameworks presented in the paper.