Geometric diffeomorphism finiteness in low dimensions and homotopy group finiteness

The main results of this note consist in the following two geometric finiteness theorems for diffeomorphism types and homotopy groups of closed simply connected manifolds: 1. For any given numbers C and D the class of closed smooth simply connected manifolds of dimension which admit Riemannian metrics with sectional curvature bounded in absolute value by $vert K vertle C$ and diameter bounded from above by D contains at most finitely many diffeomorphism types. In each dimension there exist counterexamples to the preceding statement. 2. For any given numbers C and D and any dimension m there exist for each natural number up to isomorphism always at most finitely many groups which can occur as the k-th homotopy group of a closed smooth simply connected m-manifold which admits a metric with sectional curvature and diameter . Received: 21 August 1999 / Accepted: 20 April 2001 / Published online: 19 October 2001