Smooth four-dimensional Poincaré conjecture

The smooth four-dimensional Poincaré conjecture asks: must a closed, connected, smooth $4$-manifold that is homotopy equivalent to $S^4$ be diffeomorphic to $S^4$? The smooth four-dimensional formulation crystallized around 1960 out of the generalized Poincaré program, so its date is only approximate.

The topological counterpart is a theorem: Freedman proved that every homotopy $4$-sphere is homeomorphic to $S^4$ [Freedman1982TopologyFourManifolds]. The remaining question is therefore purely about smooth structures, and dimension four is genuinely delicate: exotic homotopy spheres in dimensions at least $7$ show that the naive smooth statement in all dimensions is false, while in dimension four the techniques that settle the other dimensions do not apply. The problem has been a fixture of the standard problem lists in low-dimensional topology [Kirby1997ProblemsLowDimensional] [K3ProblemList2026].

The current state is symmetric ignorance: no exotic smooth $4$-sphere has been constructed, and no argument rules one out. A resolution requires either exhibiting a homotopy $4$-sphere with a detectably nonstandard smooth structure, or a proof that all such manifolds are standard. The conjecture remains open.