4-dimensional Busemann \(G\)-spaces are 4-manifolds
The notion of \(G\)-space was introduced by \textit{H. Busemann} [Metric methods in Finsler spaces and in the foundations of geometry, Ann. Math. Studies, No. 8, Princeton University Press, Princeton, N. J., (1942); On spaces in which points determine a geodesic, Trans. Am. Math. Soc. 54, 171-184 (1943)]. Definition 1.1. Let \((X,d)\) be a metric space. \(X\) is said to be a \(G\)-space provided it satisfies the following axioms: (1) Menger convexity: Given distinct points \(x,y\in X\), there is a point \(z\in X-\{x,y\}\) so that \(d(x,z)+ d(z,y) = d(x,y)\). (2) Finite compactness: A given \(d\)-bounded infinite set has an accumulation point. (3) Local extensibility: To every \(w\in X\), there is a positive radius \(\rho_w\), such that for any pair of distinct \(x,y\in\text{int} B(w,\rho_w)\), there is \(z\in\text{int} B(w,\rho_w)-\{x,y\}\) such that \(d(x,y)+d(y,z)=d(x,z)\). (4) Uniqueness of the extension: Given distinct \(x,y\in X\), if there are points \(z_1,z_2\in X\) for which both \(d(x,y)+d(y,z_i)=d(x,z_i)\) for \(i=1,2\) and \(d(y,z_1)= d(y, z_2)\) hold, then \(z_1=z_2\). In a classic text [The geometry of geodesics (1955; Zbl 0112.37002), pp. 3 and 49] \textit{H. Busemann} conjectured that \(G\)-space theory is the study of geometry on manifolds. Conjecture 1.2 (The Busemann Conjecture): Every finite dimensional \(G\)-space is a topological manifold. The modern theory of 4-manifolds developed by \textit{M. H. Freedman} and \textit{F. S. Quinn} [J. Differ. Geom. 17, 357-453 (1982; Zbl 0528.57011); 17, 503-521 (1982; Zbl 0533.57009); Topology of 4-manifolds, Princeton Math. Ser. 39 (1990; Zbl 0705.57001)] has rendered the 4-dimensional version of the problem accessible. This article is devoted to establishing the following result: Theorem 1.3. Every 4-dimensional \(G\)-space is a 4-manifold. The argument involves elements from algebraic topology, controlled surgery and modern decomposition theory and is organized as follows. Step 1. shows that every finite dimensional \(G\)-space is an ANR homology manifold. This involves use of elementary properties of \(G\)-spaces together with sheaf-theoretic results of \textit{J. Dydak} and \textit{J. Walsh} [Lect. Notes Math. 1283, 65-87 (1987; Zbl 0631.57014)]. Step 2 specializes to 4D \(G\)-spaces and shows that the geometry of these objects allows to embed arbitrarily small 2-spheres, 3-cells and 3-spheres in such a \(G\)-space. A recent result of \textit{W. J. R. Mitchell} [Proc. Am. Math. Soc. 110, 509-513 (1990; Zbl 0835.57011)] is used in combination with classical 2-manifold recognition theory. This step verifies that \(X\) has a useful local structure, that of a semi-nicely spherical space. Step 3. uses the geometric objects identified above) to show Quinn's resolution obstruction on every 4D \(G\)-space. That is, every 4D \(G\)-space \(X\) is the image of a genuine 4-manifold \[ M^4 @>f>> X\tag{1} \] where \(f\) is a continuous surjection such that for all \(x\in X\), \(f^{-1}(x)\) is a cell like subset of \(M^4\). Step 4 establishes the 4D semi-nicely spherical shrinking theorem. This can then be applied to show that all maps of the form (1) are approximable by homeomorphisms and thus establish the desired result.
scientific article; zbMATH DE number 937119
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