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A phenomenon that is unique to dimension 4 is the existence of infinite families of manifolds that are homeomorphic but not diffeomorphic. This is shown via a combination of gauge theory (Seiberg-Witten theory or Yang-Mills theory) with Freedman’s topological classification results. In a joint project with Dave Auckly, we find similar `exotic’ behavior comparing the topology of the groups of diffeomorphisms and homeomorphisms of a smooth 4-manifold. Our main theorem is that the kernel of the map on homotopy groups induced by the inclusion Diff(X) -> Homeo(X) can be infinitely generated. The same techniques yield similar results about spaces of embeddings of surfaces and 3-manifolds in 4-manifolds.