Bounding homotopy and homology groups by curvature and diameter (Q1902019)

Let \(n \geq 2\) be a positive integer, \(D > 0\) a real number and \({\mathcal M}_D^n\) the set of the isometry classes of the compact Riemannian \(n\)-manifolds which satisfy \(|K|\leq 1\), \(\text{diam} \leq D\), where \(K\) is the sectional curvature. The author shows that many interesting topological invariants of the homotopy and homology groups of \(M \in {\mathcal M}_D^n\) can be bounded by a constant \(N(.,.)\) whose value depends on the parameters in parentheses. Let be mentioned, for illustration, the following theorem: ``Given \(n \geq 2\) and \(D > 0\), let \(M \in {\mathcal M}^n_D\). Then \(\pi_1(M)\) has a normal nilpotent subgroup \(N\) such that the minimal number of generators for \(N\) is not greater than \(n\), and \(\pi_1 (M)/N\) has at most \(N_4(n,D) < \infty\) isomorphic classes up to a possible normal \(\mathbb{Z}_2\)-extension.