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Bounding homotopy and homology groups by curvature and diameter. (English) Zbl 0844.57024

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.
Reviewer: V.Cruceanu (Iaşi)

MSC:

57R19 Algebraic topology on manifolds and differential topology
53C99 Global differential geometry
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