The Gromov–Lawson conjecture on higher Â-genera

Let $M$ be a connected closed spin Riemannian manifold of positive scalar curvature. Let $\pi$ be a discrete group and let $f\colon M\to B\pi$ be a continuous map. Define the higher $\widehat A$-genus by

$$f_*(\widehat{\mathcal A}(M)\cap[M])\in H_\bullet(B\pi,\mathbb Q).$$

Gromov–Lawson conjecture. The higher $\widehat A$-genus vanishes.

This generalizes the Lichnerowicz obstruction to positive scalar curvature. The source says it remains open in general and records that rational injectivity of the $K$-theory assembly map implies it.