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The Alexander trick for homology spheres

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Søren Galatius (Copenhagen)

Abstract: Homeomorphisms of the d-disk which act as the identity on the (d-1)-sphere forms a topological group, which was shown to be contractible by Alexander in 1923. The d-disk is of course itself contractible, but for d>3 there are plenty of examples of contractible compact d-manifolds besides the disk, by work of Kervaire from 1969. In joint work with Randal-Williams, we investigate whether the homeomorphism group relative to the boundary is contractible, or at least weakly contractible, for this more general class of manifolds. Our main result is that this is the case for all d>5.