The topology of 4-dimensional G-spaces and a study of 4-manifolds of non-positive curvature

The 40-year-old Busemann G-space conjecture states that all n-dimensional G-spaces (with n < ∞) are n-manifolds. I have shown that the conjecture is true when n = 4, and the bulk of this paper is dedicated to assembling the elements of metric geometry, algebraic topology, and modern decomposition theory required in the proof. In his classic text The Geometry of Geodesies , Herbert Busemann set out to create an analogue of differential geometry in metric spaces having apriori no analytic structure. He invented the notion of a geodesic space, or G-space, and his theory has a significant impact in the study Riemannian geometry. Examples of non-Riemannian G-spaces are found by considering certain Finsler spaces and also the Teichmüller moduli space of complex structures on any closed Riemann surface of genus ≥ 2. In 1955, Busemann established that all 2-dimensional G-spaces are surfaces using purely geometric techniques. In 1968, the Polish mathematician Krakus applied a topological 2-sphere recognition theorem of Borsuk to show that the conjecture is true in dimension 3. The next part of this paper is a study of non-positively curved 4-manifolds. The classic theorem of Cartan-Hadamard states that every n-manifold with non-positive sectional curvature is covered by euclidean space. If the notion of curvature is relaxed to that of Aleksandrov's CAT(ε)-inequalities, Gromov asked in 1981 whether every simply connected, non-positively curved n-manifold is necessarily euclidean. For n≤3, work of Rolfsen and Brown yield an affirmative solution. For n≥5, counter-examples 111 have been constructed by Davis and Januszkiewicz in 1991. When n = 4, I am able to establish a partial result in the affirmative, provided the 4-manifold has a single metrically tame point. Finally I establish a 3-manifold recognition criterion. A compact, finite dimensional metric space is a 3-manifold if and only if it is a homology 3-manifold and it admits an inner metric having curvature bounded above by some positive constant. As there exist non-manifold homology 3-manifolds, the curvature condition can not be discarded.