Difficulty of proving the Generalized Poincaré Conjecture as a function of dimension:

n=1: Trivial

n=2: Easy (19th century)

n=3: Perelman (2003), declined Fields Medal

n=4: Freedman (1982), won Fields Medal

n≥5: Smale (1961), won Fields Medal

The difficulty of the generalized Poincaré conjecture is non-monotonic in dimension!

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