Relative coarse Novikov conjecture for (X, Y)

Show 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 injective.

Background

The relative Novikov conjecture is the injectivity part of the relative coarse Baum-Connes conjecture. It ensures that nontrivial classes in relative K-homology at infinity map nontrivially into K-theory of the relative Roe algebra at infinity, thereby detecting obstructions to positive scalar curvature at infinity outside Y.

The paper demonstrates consequences of this conjecture, including implications for scalar curvature at infinity and connections to groupoid-based formulations.

References

Let $X$ be a metric space with bounded geometry, $Y$ a subspace of $X$. \begin{itemize} \item The relative coarse Baum-Connes conjecture for $(X,Y)$: the relative coarse assembly map $\mu_{Y,\infty}$ is an isomorphism; \item The relative coarse Novikov conjecture for $(X,Y)$: the relative coarse assembly map $\mu_{Y,\infty}$ is injective. \end{itemize}

— Relative higher index theory on quotients of Roe algebras and positive scalar curvature at infinity  (2509.23380 - Guo et al., 27 Sep 2025) in Section 3.1 (Relative coarse assembly maps)