It is interesting to study the topology of the space of smoothly embedded $n$-spheres in $\mathbb{R}^{n+1}$. 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 result in particular depends on the dimension $n$. This is all joint work with Robert Haslhofer and Or Hershkovits.