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Graduate Student Geometry-Topology Seminar

Friday, February 16, 2024 - 2:00pm

Albert Yang



University of Pennsylvania


Is the smooth structure unique on the n-dimensional sphere S^n? This is a smooth version of the famous generalized Poincaré conjecture, and it turns out to be extremely difficult. Starting from Milnor's exotic sphere, we know that the number of smooth structures (denoted by N) for higher S^n is highly nontrivial. However, a surprising fact shows that this N is relevant to the calculation of stable homotopy groups of spheres, and it is possible to determine for which S^n, N=1. This is the well-known Kervaire-Milnor theory. In this talk, I will briefly summarize the ideas of this theory and provide a partial answer to this question in low dimensions, using the recent calculation results of stable homotopy groups of spheres by Isaksen-Wang-Xu in 2023.