In 1988 Friedman and Morgan used Donaldson polynomial invariants to show that several simply connected algebraic surfaces possess self-homeomorphisms which are non-smoothable i.e. are not C^0-isotopic to any self-diffeomorphism. Since then many other pairs (X,phi) with X a simply-connected 4-manifold and phi:X->X a non-smoothable homeomorphism have been found. In all known examples phi acts non-trivially in homology; when X is closed, this is a necessary condition for non-smoothability for otherwise phi would be isotopic to the identity (Perron-Quinn). I will show that this is not necessary anymore when X has non-empty boundary. More precisely, I will show how to construct simply connected 4-manifolds with non-empty boundary possessing non-smoothable self-homeomorphisms which fix the boundary pointwise and act trivially in homology. This is a joint work with Daniel Galvin.

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