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Passcode:Ìý998749

In dimensions $n \neqÌý4$ the difference between groups ofÌýdiffeomorphismsÌýand 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 ofÌýdiffeomorphismsÌýis simple, byÌýlinearisingÌýit 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.

I will explain some of the history of this problem, as well as recent work with A.ÌýKupersÌýin this direction.