Simply Connected Manifolds of Positive Scalar Curvature

Hitchin proved that if M is a spin manifold with positive scalar curvature, then the A^O-characteristic number a(M) vanishes. Gromov and Lawson conjectured that for a simply connected spin manifold M of dimension > 5, the vanishing of a(M) is sufficient for the existence of a Riemannian metric on M with positive scalar curvature. We prove this conjecture using techniques from stable homotopy theory. Gromov and Lawson proved that any simply connected manifold of dimension > 5 which is nonspin admits a metric of positive scalar curvature [GL, Corollary C]. On the other hand it was shown by Lichnerowicz that not every spin manifold has a metric of positive scalar curvature [Li], The argument is as follows: If M is a A-dimensional spin manifold of positive scalar curvature, then by the Weizenböck formula kernel and cokernel of the Dirac operator are trivial. In particular, for n = 0 mod 4 the characteristic number A(M) which is the index of the Dirac operator vanishes. This was generalized by Hitchin, who constructed a family of Fredholm operators closely related to the Dirac operator whose index is a A^O-characteristic number a(M) e KO~(pt) [Hi, page 39]. Again using the Weizenböck formula he showed that a(M) = 0 if M has a metric of positive scalar curvature. This is a strict generalization of the result of Lichnerowicz since a(M) can be identified with A(M) (up to a factor) if n = 0mod4. Gromov and Lawson proved that if M is a simply connected spin manifold of dimension > 5 which is spin bordant to a manifold N of positive scalar curvature, then M admits a metric of positive scalar curvature [GL, Theorem B]. Received by the editors January 18, 1990 and, in revised form, June 18, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 53C20, 55T15, 55N22, 57R90.