Bruce KLEINER - Ricci flow, diffeomorphism groups, and the Generalized Smale Conjecture
Channel: Institut des Hautes Études Scientifiques (IHÉS)
Date: 2017-12-27

The Smale Conjecture (1961) may be stated in any of the following equivalent forms:
• The space of embedded 2-spheres in R3 is contractible. 
• The   inclusion   of   the   orthogonal   group   O(4)   into   the   group of diffeomorphisms of the 3-sphere is a homotopy equivalence. 
• The space of all Riemannian metrics on S3 with constant sectional curvature is contractible. 
While the analogous statement one dimension lower can be proven in many ways -for instance using the Riemann mapping theorem -Smale's conjecture turned out to be surprisingly difficult, and remained open until 1983, when it was proven by Hatcher 
using a deep combinatorial argument. Smale's Conjecture has a natural generalization to other spherical space forms: if M
is a spherical space form with a Riemannian metric of  constant  sectional  curvature,  then  the  inclusion  of  the  isometry
group  into  the diffeomorphism group is a homotopy equivalence. The lecture will explain how Ricci flow  through  singularites,  as  developed  in  the  last  few  years  by  John  Lott,  Richard 
Bamler, and myself, can be used to address this conjecture.
This is joint work with Richard Bamler.