\(\text{CAT}(0)\) 4-manifolds possessing a single tame point are Euclidean
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 homology 3-manifold and each component \(Y'\) of \(Y\) admits a compatible, complete inner metric for which \(\text{curv }Y' \leq K\) for some \(K \in R\).
scientific article; zbMATH DE number 1160131
scientific article