The stable Gromov–Lawson–Rosenberg conjecture

Let $M$ be a spin manifold of dimension $n$ with fundamental group $\Gamma=\pi_1M$, and let $B$ be a Bott manifold, namely a simply connected spin $8$-manifold with $\Ahat(B)=1$. Say that $M$ stably admits a positive scalar curvature metric if there exists $k\in\mathbb{N}$ such that $M\times B^k$ admits a positive scalar curvature metric. Let $\alpha(M)\in\KO_n(\Cstar\Gamma)$ be the Rosenberg index.

Stable Gromov–Lawson–Rosenberg conjecture. A spin manifold stably admits a positive scalar curvature metric if and only if

$$\alpha(M)\neq0\in\KO_n(\Cstar\Gamma).$$

The unstable Gromov–Lawson–Rosenberg conjecture is false in general, while the source explains that the stable version is known whenever the real Baum–Connes assembly map is injective. Its general validity remains open in the source.