When is a homotopy sphere also a topological sphere?
When does ?
- : True. Trivial
- : True. Proved by Poincaré, classical
- : True. Perelman (2006) using Ricci flow + surgery
- : True. Freedman (1982), Fields medal!
- : True. Zeeman (1961)
- : True. Stalling (1962)
- : True. Smale (1961) using h-cobordism theorem, uses handle decomposition + Morse functions
Smooth Poincaré Conjecture
When is a homotopy sphere a smooth sphere?
- : True. Trivial
- : True. Proved by Poincaré, classical
- : True. (Top = PL = Smooth)
- : Open
- : Zeeman (1961)
- : Stalling (1962)
- : False in general (Milnor and Kervaire, 1963), Exotic , 28 smooth structures on
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 . Yield homeomorphic spheres, suspected not to be diffeomorphic, but no known invariants can distinguish smooth structures on .