The moduli space of two-convex embedded spheres and tori

Reto Buzano, born Mueller (Queen Mary University of London)

Abstract: It is interesting to study the topology of the space of smoothly 
embedded n-spheres in R^{n+1}. By Smale’s theorem, this space is 
contractible for n=1 and by Hatcher’s proof of the Smale conjecture, 
it is also contractible for n=2. These results are of great importance, 
generalising in particular the Schoenflies theorem and Cerf’s theorem. 
In this talk, I will explain how mean curvature flow with surgery can 
be used to study a higher-dimensional variant of these results, proving 
in particular that the space of two-convex embedded spheres is 
path-connected in every dimension n. We then also look at the space 
of two-convex embedded tori where the question is more intriguing 
and the result in particular depends on the dimension n. 

This is all joint work with Robert Haslhofer and Or Hershkovits.