\(\text{CAT}(0)\) 4-manifolds possessing a single tame point are Euclidean
type: Item
instance of: scholarly article
title: \(\text{CAT}(0)\) 4-manifolds possessing a single tame point are Euclidean
zbMATH Open document ID: 0911.51019
DOI: 10.1007/BF02921662
author: Paul Thurston
published in: The Journal of Geometric Analysis
publication date: 1999-04-28T00:00:00Z
review text: A metric space \((X,d)\) is an inner metric space if \(d(x,y)\) is the infimum of lengths of all arcs in \(X\) joining \(x\) and \(y\) (thus, in the language of K. Borsuk, the intrinsic metric induced by \(d\) coincides with \(d\)). Length spaces are defined to be complete, locally compact inner metric spaces. For any length space \((X,d)\) and a real \(K\), one defines a counterpart of the sectional curvature (of a Riemannian manifold) by comparing ``small'' geodesic triangles in \(X\) with an isometric triangle in the unique simply connected Riemannian surface \(M^2(K)\) of constant sectional curvature \(K\). This generalized notion, \(\text{curv} X \leq K\), is defined locally, but, in view of the Cartan-Hadamard-Aleksandrov Theorem, for \(X\) simply connected the local condition \(\text{curv} X \leq 0\) implies a similar global condition (obtained by comparing all triangles): \(X\) satisfies \(\text{CAT}(0)\). The classical theorem of Cartan-Hadamard characterizes \(\mathbb{R}^n\) (up to diffeomorphism) as the unique simply connected complete Riemannian space of nonpositive sectional curvature. In 1981 Gromov asked whether the synthetic counterpart of that theorem is true (for homeomorphisms). In terms of \(\text{CAT}(0)\) this question can be formulated as follows: Is it true that every \(\text{CAT}(0)\) \(n\)-manifold is homeomorphic to \(\mathbb{R}^n\)? The answer to this question is positive for \(n<4\) and negative for \(n>4\). Paul Thurston gives a partial solution for \(n=4\). As the main result, he proves that if \((M^4,d)\) is a \(\text{CAT}(0)\) 4-manifold with at least one topologically or geometrically tame point, then \(M^4\) is homeomorphic to \(\mathbb{R}^4\) (Theorem 1.6). The proof is based on several very interesting new results. For instance, let us mention the following theorem characterizing \(3\)-manifolds: Theorem 3.3. Let \(Y\) be a locally compact, finite-dimensional Hausdorff space. Then \(Y\) is a 3-manifold if and only if \(Y\) is a h…
reviewed by: Maria Moszyńska
Mathematics Subject Classification ID: 51K10
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 1160131
zbMATH Keywords: intrinsic metric
zbMATH Keywords: inner metric space
zbMATH Keywords: length space
zbMATH Keywords: nonpositively curved space
zbMATH Keywords: CAT(0)
zbMATH Keywords: 4-manifold
zbMATH Keywords: topologically tame point
zbMATH Keywords: geometrically tame point
zbMATH Keywords: homology manifold
MaRDI profile type: Publication
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Wikidata QID: Q126089136
description: scientific article; zbMATH DE number 1160131