A century ago, J. W. Alexander showed that the space of homeomorphisms of a disc (which are the identity near the boundary) is contractible, using an explicit radial deformation now known as the "Alexander trick". I will explain recent joint work with Soren Galatius which shows that the same conclusion holds for all compact contractible manifolds of dimension at least 6. In this generality there can be no such explicit deformation, and the question must be approached more obliquely. Along the way, I will explain a new result on spaces of smooth embeddings of one-sided h-cobordisms into other manifolds.
