Relative coarse Baum-Connes conjecture for (X, Y)

Establish that for every metric space X with bounded geometry and subspace Y ⊆ X, the relative coarse assembly map μ_{Y,∞}: lim_{d→∞} K_*(C^*_{L,Y,∞}(P_d(X))) → K_*(C^*_{Y,∞}(X)) is an isomorphism.

Background

To refine index-theoretic obstructions to positive scalar curvature to specific directions at infinity, the paper introduces the relative Roe algebra at infinity C^*_{Y,∞}(X) by quotienting C^*(X) by the ghostly ideal generated by Y. The corresponding localization algebra at infinity C^*_{L,Y,∞}(P_d(X)) yields a relative K-homology theory.

The relative coarse Baum-Connes conjecture asserts that the relative assembly map is an isomorphism, providing a method to compute K_(C^_{Y,∞}(X)) and hence to detect refined obstructions to positive scalar curvature outside Y. This conjecture reduces to standard cases when Y is empty or coarsely equivalent to X, and coincides with the boundary coarse Baum-Connes conjecture when Y is bounded.