Geometric diffeomorphism finiteness in low dimensions and homotopy group finiteness
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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 \(m<7\) which admit Riemannian metrics with sectional curvature bounded in absolute value by $\vert K \vert\le C$ and diameter bounded from above by D contains at most finitely many diffeomorphism types. In each dimension \(m\ge 7\) 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 \(k\ge 2\) 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 \(\vert K \vert\le C\) and diameter \(\le D\).
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