Bounding homotopy and homology groups by curvature and diameter
type: Item
instance of: scholarly article
title: Bounding homotopy and homology groups by curvature and diameter
zbMATH Open document ID: 0844.57024
DOI: 10.1215/S0012-7094-95-07820-X
published in: Duke Mathematical Journal
publication date: 1996-08-29T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 57R19
Mathematics Subject Classification ID: 53C99
zbMATH DE Number: 815770
zbMATH Keywords: Riemannian manifolds
zbMATH Keywords: sectional curvature
zbMATH Keywords: homotopy
zbMATH Keywords: homology
author: Xiaochun Rong
reviewed by: Vasile Cruceanu
MaRDI profile type: Publication
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full work available at URL: s0012-7094-95-07820-x
OpenAlex ID: W1592633708
description: scientific article; zbMATH DE number 815770