Diffeomorphisms of discs

-
Oscar Randal-Williams, University of Cambridge

In dimensions $n \neq 4$ the difference between groups of diffeomorphisms and of homeomorphisms of an $n$-manifold $M$ is governed by an $h$-principle, meaning that it reduces to understanding these groups for $M=\mathbb{R}^n$. The group of diffeomorphisms is simple, by linearising it is equivalent to $O(n)$, but the group $Top(n)$ of homeomorphisms of $\mathbb{R}^n$ has little structure and is difficult to grasp. It is profitable to instead consider the $n$-disc $M=D^n$, because the group of homeomorphisms of a disc (fixing the boundary) is contractible by Alexander's trick: this removes homeomorphisms from the picture entirely, and makes the problem one purely within differential topology.