The Gromov-Lawson conjecture: closed aspherical manifolds admit no metric of positive scalar curvature

Theorem 1.1: if a closed connected oriented smooth $M^n$, $n\ge2$, admits positive scalar curvature then $(c_M)_*[M]=0$ in $H_n(B\pi_1(M);\mathbb Q)$. Corollaries: no closed aspherical manifold (orientable or not) admits positive scalar curvature; every nonnegative-scalar-curvature metric on a closed aspherical manifold is flat; a closed oriented manifold mapping with nonzero degree to an aspherical one has no positive scalar curvature; and nonnegative scalar curvature forces zero real simplicial volume. Prior cases: dimensions 4 and 5 (Chodosh-Li, Gromov). It does not give the quantitative scalar-curvature/simplicial-volume inequality (that is the companion entry).

Theorem 1.1: if a closed connected oriented smooth $M^n$, $n\ge2$, admits positive scalar curvature then $(c_M)_*[M]=0$ in $H_n(B\pi_1(M);\mathbb Q)$. Corollaries: no closed aspherical manifold (orientable or not) admits positive scalar curvature; every nonnegative-scalar-curvature metric on a closed aspherical manifold is flat; a closed oriented manifold mapping with nonzero degree to an aspherical one has no positive scalar curvature; and nonnegative scalar curvature forces zero real simplicial volume. Prior cases: dimensions 4 and 5 (Chodosh-Li, Gromov). It does not give the quantitative scalar-curvature/simplicial-volume inequality (that is the companion entry).