The Alexander Trick for homology spheres.

If Δ is a contractible compact d-manifold, then its boundary Σ will be a homology (d-1)-sphere, but the boundary need not be simply connected and Δ need not be homeomorphic to the d-disk.  In joint work with Randal-Williams, we show that the topological group consisting of homeomorphisms of Δ fixing the boundary pointwise, is nevertheless contractible assuming d≥6.  In the special case of the d-disk this was proved in 1923 by Alexander, by writing down an explicit homotopy now known as the Alexander trick.