The coarse geometric Novikov conjecture

Let $\Gamma$ be a discrete metric space with bounded geometry. For each $d\geq 0$, let $P_d(\Gamma)$ be its Rips complex, and let $C^*(P_d(\Gamma))$ denote the Roe algebra. Write

$$K_*(P_d(\Gamma))=KK_*(C_0(P_d(\Gamma)),\mathbb{C})$$

for the $K$-homology group of the locally compact space $P_d(\Gamma)$, and let $Ind$ be the index map

$$Ind:\lim_{d\rightarrow\infty}K_*(P_d(\Gamma))\longrightarrow\lim_{d\rightarrow\infty}K_*(C^*(P_d(\Gamma))).$$

The coarse geometric Novikov conjecture. The map $Ind$ is injective.

This conjecture is a coarse-geometric analogue of the Novikov conjecture and implies Gromov's conjecture on uniformly positive scalar curvature for uniformly contractible manifolds of bounded geometry, as well as the zero-in-the-spectrum conjecture. It is false when the bounded geometry hypothesis is removed; the status under the stated hypothesis is not resolved in the supplied source.