M ar 1 99 5 Fundamental Group of Self - Dual Four - Manifolds with Positive Scalar Curvature

1. Background. It is a fundamental goal in Riemannian geometry to understand the topology of manifolds of positive curvature. The only general facts so far known are: finiteness of fundamental group (Myers’ theorem), vanishing of Â-genus (Lichnerowicz’ theorem and the modification of Hitchin) and a universal bound on Betti numbers (Gromov’s theorem). In a well-known paper [10] Micallef and Moore introduced a new notion of positivity for the curvature tensor, that is, positivity on complex isotropic two-planes. For x ∈ M , a Riemannian manifold, let R : ΛTxM → Λ TxM be the curvature tensor. After complexification we get a Hermitian operator in ΛT x M . We say that z ∈ Λ T x M comes from a complex isotropic two-plane if z = ξ ∧ η with (ξ, ξ)C = (ξ, η)C = (η, η)C = 0. Here (·, ·)C is the canonical symmetric (not Hermitian!) complexification of the Euclidean scalar product in TxM . The condition above says that (Rz, z) > 0 for such z. The theorem of Micallef and Moore reads: