Hausdorff convergence and the fundamental group
When investigating general limits of Hausdorff convergent sequences of Riemannian manifolds, only little is known about relations between topological properties of the terms of the sequence and the topology of the limits. In the present paper the following theorem is proved: Let \(X\) be a limit of a Hausdorff convergent sequence \((M_ k)_{k \in \mathbb{N}}\) of \(n\)-dimensional compact connected Riemannian manifolds with uniformly bounded curvatures and diameters. Then for sufficiently large \(k\) there exist surjective group homomorphisms \(\pi_ 1 (M_ k) \to \pi_ 1 (X)\) from the fundamental groups of the \(M_ k\) to the fundamental group of \(X\). Furthermore it is shown that any such \(X\) is locally simply connected and possesses a universal covering. Applications of the above theorem, in particular to classes of aspherical manifolds, and its extension to sequences of Riemannian manifolds subject only to a lower curvature bound are discussed.
scientific article; zbMATH DE number 737181