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

In: Surveys in Geometry I

Author

Listed:
  • Valentin Poénaru

    (Université de Paris-Sud, Mathématiques)

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].

Suggested Citation

  • Valentin Poénaru, 2022. "All 4-Dimensional Smooth Schoenflies Balls Are Geometrically Simply-Connected," Springer Books, in: Athanase Papadopoulos (ed.), Surveys in Geometry I, chapter 0, pages 269-307, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-86695-2_7
    DOI: 10.1007/978-3-030-86695-2_7
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