Schoen–Yau's $K(\pi,1)$ conjecture for positive scalar curvature

Let $M^n$ be a closed $n$-dimensional $K(\pi,1)$ manifold, meaning an aspherical manifold whose universal cover is contractible. A Riemannian metric has positive scalar curvature if its scalar-curvature function is everywhere positive. Schoen–Yau's $K(\pi,1)$ conjecture. The manifold $M^n$ does not admit a Riemannian metric with positive scalar curvature.

This extends the Geroch conjecture from tori to closed aspherical manifolds. The conjecture was confirmed in dimensions $4$ and $5$ by Chodosh–Li and by Gromov, while the general-dimensional statement remains open.