On the Topology of Scalar-Flat Manifolds

Abstract

Let M be a simply connected closed manifold of dimension greater than 4 which does not admit a metric with positive scalar curvature. We give necessary conditions for M to admit a scalar-flat metric. These conditions involve the first Pontrjagin class and the cohomology ring of M. As a consequence, any simply connected scalar-flat manifold of dimension greater than 4 with vanishing first Pontrjagin class admits a metric with positive scalar curvature. We also describe some relations between scalar-flat metrics, almost complex structures and the free loop space.