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[Lecture] Lower Ricci Bounds and Nonexistence of Manifold Structure
Apr. 10, 2024

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Speaker: Erik Hupp (Northwestern University, USA)

Time: 09:00-10:00 a.m., April 10, 2024, GMT+8

Venue: online(Zoom: 859 0910 7027, Password: 438807)


By Gromov compactness, any sequence of complete Riemannian n-manifolds with uniform lower Ricci bounds has a subsequence (pointed Gromov-Hausdorff) converging to a limit metric space. How close is this limit to being a manifold itself? A cornerstone result of Cheeger-Colding gives an answer if one also assumes that the limit is volume non-collapsed: it is a topological manifold on an open dense set whose complement has dimension at most n - 2. This talk will describe a family of counterexamples to the corresponding statement in the collapsed setting. These limit spaces can be constructed to approximate any given complete (smooth) Riemannian 4-manifold with lower Ricci bounds, but 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.

Source: School of Mathematical Sciences , PKU