EVENT DETAILS AND ABSTRACT

Title: Lower Ricci bounds and nonexistence of manifold structure
Speaker: Erik Hupp
Speaker Info: Northwestern
Brief Description:
Special Note:
Abstract:

By Gromov compactness, any sequence of closed, uniformly bounded-diameter Riemannian manifolds with uniform lower Ricci curvature bounds has a subsequence (Gromov-Hausdorff) converging to a limit metric space. How close is this limit to being a Riemannian manifold itself? A cornerstone result of Cheeger-Colding gives an answer if one also assumes the non-collapsing condition, i.e. volume bounded away from zero: the limit space is a topological manifold on an open dense set whose complement has dimension at most n - 2. In this talk, I will describe a family of counterexamples to the corresponding statement in the collapsed setting, i.e. where the volume of the sequence of manifolds tends to zero. The limit spaces in these examples have the property that no open set is homeomorphic to R^k, for any k. Everything discussed is joint work with Aaron Naber and Kai-Hsiang Wang.