Relative Novikov and Baum–Connes conjectures

Let $Y\subseteq X$ be aspherical compact spaces, and let

$$\mu_{\max}\colon KO_*(X,Y)\to KO_*(C^\ast_{\max}(\pi_1(X),\pi_1(Y)))$$

be the maximal relative Baum–Connes map. If $j\colon \pi_1(Y)\to \pi_1(X)$ is injective, let

$$\mu_{\mathrm{red}}\colon KO_*(X,Y)\to KO_*(C^\ast_{\mathrm{red}}(\pi_1(X),\pi_1(Y)))$$

be the reduced relative Baum–Connes map. Relative Novikov and Baum–Connes conjectures. The map $\mu_{\max}$ is injective; moreover, when $j$ is injective, the map $\mu_{\mathrm{red}}$ is an isomorphism. The paper introduces these relative maps to encode information from both a compact space and its subspace; the supplied text gives no resolution status for either assertion.