Topology of two-dimensional collapsed spaces with lower Ricci bounds

In this talk, I will discuss ​the topology of collapsed Ricci limit spaces. It has been known since the work of Cheeger–Colding that any noncollapsed Ricci limit space is homeomorphic to a manifold away from a set of codimension two. By contrast, the topology in the collapsed case can be very poorly behaved. Indeed, Hupp–Naber–Wang and Zhou recently constructed examples of collapsed Ricci limit spaces, for any rectifiable dimension at least three, that are nowhere locally Euclidean. In joint work with Elia Bruè, we show that the situation is much better when the rectifiable dimension is two. In fact, we show that such spaces are topological manifolds, possibly with boundary. As a corollary, we obtain a partly topological, partly metric classification of such spaces.
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