Topology of non-positively curved manifolds

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A fundamental question of Geometry and Topology is how the global topology of a space is determined by local geometric data, in particular, the curvature of the space. Nonpositively curved manifolds (spaces) has been a subject where there are a lot of rich theorems, and it has been a central object of study not only in geometry, but also in related fields of mathematics, such as geometric group theory and dynamics. The field, however, seems to have been abandoned in the last two decades although there are a lot of unanswered questions, some of which are within reach. Constructions of smooth, nonpositively curved manifolds are poorly understood in the sense that most examples known are of a very restrictive type, such as hyperbolic manifolds and pinched negatively curved manifolds, which do not illustrate more generic properties of nonpositively curved spaces. One of the aims of the proposed research is to construct more nonpositively curved manifolds, especially with interesting properties that are not seen in hyperbolic or pinched negatively curved manifolds. If a space is known to have a nonpositively curved metric, one can deduce a lot about it. The question whether a space admits a nonpositively curved metric has become of great interest in different areas in topology. The proposed research also aims to find topological properties shared by large classes of nonpositively curved manifolds and topological obstructions to having a nonpositively curved metric. Specifically, the PI will study the topology of ends as obstructions to having nonpositively curved metrics. The project aims to study noncompact, complete, finite volume, bounded, nonpositively curved manifolds M. Objects to be studied are the topology of ends, invariants of the fundamental groups of these manifolds (such as the cohomological dimension and the action dimension), and the relation between the topology of the end of such a manifold of the set of non-horospherical limit points. The problem of how to distinguish which locally CAT(0) manifolds have a smooth nonpositively curved Riemannian metric will also be studied. In particular, in the case when M has tame ends, the question which manifold C can occur as the cross section of each end of M will be studied extensively. In low dimension, such as when M has dimension 4, the goal is to show that each cross section of an end of M is aspherical by computing the second homotopy group of the ends of M. Once this has been proven, and with the result of the Geometrization Theorem in three-manifold theory, a classification of all manifolds C that can occur as cross sections may be within reach, or at least progress can be made using recent techniques of Ontaneda's Riemannian hyperbolization procedure. The relation between the topology of the end of such a manifold and the set of non-horospherical limit points will be studied. For the case when M is a locally symmetric space of noncompact type, it will be shown that the set of non-horospherical limit points is the same as the rational Tits building using Saper's tilings. The question whether a similar phenomenon happens in more general nonpositively curved settings will be investigated. This project also aims to construct new examples of locally CAT(0) manifolds that do not admit a smooth nonpositively curved metric. Obstructions that will be used to distinguish these from smooth nonpositively curved manifolds are invariants at infinity that come from embedded flat tori in the manifolds that are locally linked with linking number greater than one.