Fundamental Group of Self-Dual Four-Manifolds with Positive Scalar Curvature

Abstract

Main Theorem (3.3): Let $M$ be a compact four-dimensional manifold either with curvature, positive on complex isotropic two-planes, or self-dual of positive scalar curvature. If $π_1 (M)$ admits a nontrivial unitary representation, and $M$ is orientable, then there exists a surjective homomorphism from $π_1 (M)$ on $\bbz$. Corollary: If $π_1 (M)$ is finite, then either $π_1 (M) = 1$, or $π_1 (M) = \bbz_2$. Observe that finitely presented groups which do not admit a nontrivial unitary representation, are extremely rare (see 3.4).