Maximal relative coarse Novikov conjecture for (X, Y)

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

Background

This is the injectivity part of the maximal relative coarse Baum-Connes conjecture. It asserts that nontrivial classes in maximal relative K-homology at infinity survive under the maximal assembly map, reflecting obstructions to positive scalar curvature at infinity in the maximal Roe setting.

The paper develops tools (relative fibred coarse embeddings, coarsely proper algebras, Bott generators) to verify these conjectures in significant cases and to deduce geometric applications.

References

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