東京大学大学院数理科学研究科

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数理談話会

日時: 2006年6月23日(金) 17:00〜18:00
会場: 数理科学研究科棟(駒場) 123号室

講演者

Robert Gompf 氏 (University of Texas at Austin)


講演題目

25 years of exotic $\mathbb{R}^4s$



講演概要

A quarter century ago, 4-manifold theory was revolutionized by the Fields-Medal winning breakthroughs of Freedman and Donaldson, with Freedman showing that topological 4-manifolds behave like their higher dimensional counterparts, but Donaldson showing that smooth 4-manifolds behave in a completely different way. The interplay between these theories produces results unique to dimension 4: A fixed topological 4-manifold often admits infinitely many distinct smooth structures, for which no classification scheme is yet available. The quintessential example is that in contrast with other dimensions, Euclidean 4-space admits exotic smooth structures. That is, there are "exotic R^4s" homeomorphic to R4 but not diffeomorphic to it. We will survey what has been learned about these strange creatures in the last quarter century, and exhibit an explicit example.