The Smooth Poincaré Conjecture in Dimension Four

Claim: every smooth 4-manifold homotopy-equivalent to S⁴ is diffeomorphic to S⁴ (equivalently, S⁴ admits no exotic smooth structure).

Freedman (1982) proved the topological version, earning a Fields Medal; Donaldson's gauge-theoretic constraints (also Fields work) revealed that dimension four admits exotic smooth phenomena invisible to topology — indeed exotic ℝ⁴s exist (the only dimension with them). Every other dimension's smooth Poincaré is settled: exotic spheres abound in high dimensions (Kervaire–Milnor), dimensions ≤ 6 are classical, dimension 5 follows from surgery. Only n = 4 resists, defended by gauge theory's rigidity and the absence of handle-moving freedom.