Generalized Topological Poincaré Conjecture

When is a homotopy sphere also a topological sphere?

When does \(\pi_* X \cong_{Grp} \pi_* S^n \implies X \cong_{Top} S^n\)?

  • \(n=1\): 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)
  • \(n\geq 7\): 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)
  • \(n\geq 7\): False in general (Milnor and Kervaire, 1963), Exotic \(S^n\), 28 smooth structures on \(S^7\)

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 \(S^4\). Yield homeomorphic spheres, suspected not to be diffeomorphic, but no known invariants can distinguish smooth structures on \(S^4\).

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