Surjectivity of the maximal coarse assembly map for FCE-by-FCE structured spaces

Establish surjectivity of the maximal coarse assembly map for metric spaces with bounded geometry that admit an FCE-by-FCE coarse fibration structure, i.e., a coarse fibration in which both the base space and the family of fiber spaces admit fibred coarse embeddings into Hilbert space with uniform control and uniform coarse equivalence of fiber neighborhoods. Equivalently, prove the maximal coarse Baum-Connes conjecture (both injectivity and surjectivity) for spaces with an FCE-by-FCE structure, beyond the injectivity already known.

Background

The coarse Baum-Connes conjecture relates geometric properties of metric spaces of bounded geometry to the K-theory of their Roe algebras via the coarse assembly map. For many classes of spaces, injectivity and/or surjectivity of the assembly map are central milestones toward establishing the conjecture.

The FCE-by-FCE framework considers metric spaces equipped with a coarse fibration whose base space admits a fibred coarse embedding into Hilbert space and whose fibers also admit fibred coarse embeddings, with uniform control across the system. In prior work (DGWY23), injectivity of the maximal coarse assembly map was proved for such spaces, but surjectivity—and hence the full maximal coarse Baum-Connes conjecture—remains unresolved.

This paper addresses a related but broader A-by-FCE setting (amenable-by-fibred coarse embedding), proving the maximal coarse Baum-Connes conjecture in that case. The open problem is to extend surjectivity to the FCE-by-FCE setting, completing the picture initiated by DGWY23.