All 4-Dimensional Smooth Schoenflies Balls Are Geometrically Simply-Connected

Valentin Poénaru ()
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Valentin Poénaru: Université de Paris-Sud, Mathématiques

Chapter Chapter 7 in Surveys in Geometry I, 2022, pp 269-307 from Springer

Abstract: Abstract In a famous paper, Barry Mazur showed (among other things) that a smooth n-dimensional Schoenflies ball, with one boundary point removed, is diffeomorphic to the n-ball, with one boundary point removed. Via the work of Smale and Milnor, that missing point was taken care of, except in dimension four, still mysterious to this day. In dimensions other then four, smooth Schoenflies balls are diffeomorphic to the standard ball. And then, the same dimension four is the only one where (in the compact case) simple connectivity does not imply geometric simple connectivity (GSC). We sketch here the proof that four-dimensional Schoenflies balls are GSC. Strangely enough, the proof requires infinite processes. We explain here, with only hints of proofs, the main ideas contained in a much longer paper where complete proofs are provided, available online, at the site arXiv, carrying the title All smooth four-dimensional Schoenflies balls are geometrically simply connected. See [10].

Keywords: 4-Manifold; 4-Dimensional Schoenflies ball; Geometric simple connectivity; 3-Dimensional Poincaré conjecture; 57M05; 57M10 57N35 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/978-3-030-86695-2_7

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