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- Tags: - #open/conjectures - Refs: - #todo/add-references - Links: - smooth structures - homology sphere - h-cobordism

Generalized Poincare Conjecture

History

Poincaré, Analysis Situs papers in 1895. Coined “homeomorphism”, defined homology, gave rigorous definition of homotopy, established “method of invariants” and essentially kicked off algebraic topology.

Generalized Topological Poincaré Conjecture

When is a homotopy sphere also a topological sphere?

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

• \(n=1\): 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 \(S^4\).

Proofs