Lower and Upper Curvature Bounds: Topology vs. Geometry

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This proposal is concerned with the effects of various lower and upper curvature bounds on the topology of Riemannian manifolds and the structure of Gromov-Hausdorff limits of manifolds with lower curvature bounds. Broadly speaking this proposal has three main parts. The first part deals with the structure of manifolds with lower sectional and Ricci curvature bounds. Together with A.Petrunin and W. Tuschmann the PI plans to continue investingating the structure of the fundamental groups of nonnegatively and almost nonnegatively curved manifolds and also look for new topological obstructions to nonnegative and almost nonnegative curvature for simply connected manifolds. The PI also plans to continue his work with B. Wilking on the fundamental groups of manifolds with lower Ricci curvature bounds. In particular we would like to show that the fundamental group of a manifold of almost nonnegative Ricci curvature contains a nilpotent subgroup of finite index with the bound on the index depending only on the dimension.The second part of the proposal (which is a joint project with A. Lytchak) deals with the notion of submetries which is a generalization of Riemannian submersion to singular spaces and its relation to collapsing under a lower curvature bound. The last part is the joint project with I. Belegradek on continuing our study of ends of open negatively and nonpositively pinched manifolds. This proposal deals with the question of how the local geometric picture of a space (i.e the way it's "curved" or "bent" locally) influences the global properties of the space (such as the number of holes of various dimensions the space might have). One of the ways to measure how a space is curved is given by its Ricci curvature. Understanding the influence of various Ricci curvature bounds on the global properties of a space is not only interesting in its own right but it's also important because Ricci curvature plays a fundamental role in the Einstein general relativity theory.