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

TL;DR

This paper investigates the fundamental groups of certain four-dimensional manifolds with positive curvature properties, establishing conditions under which their fundamental groups are trivial, infinite cyclic, or finite, with implications for their topological structure.

Contribution

It proves that for specific self-dual four-manifolds with positive scalar curvature, the fundamental group admits a surjective homomorphism onto the integers, revealing new topological constraints.

Findings

Fundamental group surjects onto  for certain manifolds.

Finite fundamental groups are limited to trivial or ,  cases.

Finitely presented groups without nontrivial unitary representations are rare.

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 $\pi_1 (M)$ admits a nontrivial unitary representation, and $M$ is orientable, then there exists a surjective homomorphism from $\pi_1 (M)$ on $\bbz$. Corollary: If $\pi_1 (M)$ is finite, then either $\pi_1 (M) = 1$, or $\pi_1 (M) = \bbz_2$. Observe that finitely presented groups which do not admit a nontrivial unitary representation, are extremely rare (see 3.4).