Topology Seminar (Main Talk): Classification of diffeomorphism groups of 3-manifolds through Ricci flow

Seminar | January 31 | 4-5 p.m. | 3 Evans Hall

 Richard Bamler, UC Berkeley

 Department of Mathematics

I will present recent work of Bruce Kleiner and myself in which we classify the homotopy type of all spherical and hyperbolic 3-manifolds, except for $RP^3$. This partially resolves the Generalized Smale Conjecture in the spherical case and reproves a theorem due to Gabai in the hyperbolic case.

Our proof is based on a uniqueness theorem for singular Ricci flows, which we established in previous work. Singular Ricci flows are similar to Ricci flows with surgery, which were constructed by Perelman and used in his resolution of the Poincaré and Geometrization Conjectures. In contrast to Perelman’s surgery process, which is carried out at a positive scale and depends on a number of auxiliary parameters, a singular Ricci flow is more canonical, as it “flows through surgeries” at an infinitesimal scale. Our uniqueness theorem allows the study of continuous families of singular Ricci flows, providing important information on the diffeomorphism group of the underlying manifold.

In this talk, I will first give an overview of previous work on the study of diffeomorphism groups of 3-manifolds. I will then discuss Ricci flows with surgery, singular Ricci flows and the uniqueness theorem. Lastly, I will sketch our (partial) proof of the Generalized Smale Conjecture.