In dimension 4, the works of Freedman and Donaldson led us to the striking discovery that the smooth category is drastically different from the topological category, compared to other dimensions. Since then, it has been extraordinarily successful in investigating the difference in various contexts. In contrast, our understanding of when smooth and topological categories would exhibit similarity in dimension 4 remained, at best, minimal. In this talk, we will introduce some recent progress on new “topological = smooth” results in dimension 4, focusing on embedded disks.
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