Generalized Topological Poincaré Conjecture

When is a homotopy sphere also a topological sphere?

When does πXGrpπSnXTopSn?

  • n=1: True. Trivial
  • n=2: True. Proved by Poincaré, classical
  • n=3: True. Perelman (2006) using Ricci flow + surgery
  • n=4: True. Freedman (1982), Fields medal!
  • n=5: True. Zeeman (1961)
  • n=6: True. Stalling (1962)
  • n7: True. Smale (1961) using h-cobordism theorem, uses handle decomposition + Morse functions

Smooth Poincaré Conjecture

When is a homotopy sphere a smooth sphere?

  • n=1: True. Trivial
  • n=2: True. Proved by Poincaré, classical
  • n=3: True. (Top = PL = Smooth)
  • n=4: Open
  • n=5: Zeeman (1961)
  • n=6: Stalling (1962)
  • n7: False in general (Milnor and Kervaire, 1963), Exotic Sn, 28 smooth structures on S7

Remarks:

  • It is unknown whether or not $

    {\mathbb{B}}
    ^4 $ admits an exotic smooth structure. If not, the smooth 4-dimensional Poincaré conjecture would have an affirmative answer.

  • Current line of attack: Gluck twists on on S4. Yield homeomorphic spheres, suspected not to be diffeomorphic, but no known invariants can distinguish smooth structures on S4.

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