4-manifold

Summary

A $4$-manifold is a topological manifold of dimension $4$. Dimension $4$ is special: the methods that classify manifolds of dimension at least $5$ fail, and topological and smooth $4$-manifolds behave very differently. Freedman classified simply connected closed topological $4$-manifolds through their intersection form, together with the Kirby–Siebenmann invariant when the form is odd. Donaldson's theorem and Seiberg–Witten theory show that many of them have no smooth structure, or many smooth structures. Even $\R^4$ has exotic smooth structures, which happens in no other dimension.