Commutation of pushforward with Mayer–Vietoris coboundary in twisted KR-theory

Determine whether pushforward morphisms in twisted KR-theory, as defined via Poincaré duality for equivariant maps between Real manifolds with graded Real gerbes, commute with the coboundary maps in the Mayer–Vietoris long exact sequence for arbitrary equivariant open covers. Specifically, ascertain if for any continuous equivariant map f and any graded Real gerbe, the canonical pushforward f_* is compatible with the Mayer–Vietoris connecting morphism beyond the special class of covers treated in Proposition \ref{prop:mv}.

Background

In Section 3 the paper develops pushforward maps in twisted KR-theory and establishes key properties (composition, projection formula, Thom isomorphism, embeddings, base change). To enable a Mayer–Vietoris reduction in later arguments, it is necessary that pushforward maps interact well with the Mayer–Vietoris long exact sequence.

The authors prove this compatibility for a particular class of decompositions and covers (Proposition \ref{prop:mv}), which they use to reduce the Fourier–Mukai isomorphism to simpler cases. However, they point out that verifying such compatibility in general is difficult and not clear, leaving open whether pushforward maps commute with the Mayer–Vietoris coboundary maps for arbitrary covers.