4manifold

A four-manifold is a topological manifold of dimension four; when equipped with a compatible atlas of smooth charts it is a smooth four-manifold. In the smooth category, the transition maps between charts are smooth. In four dimensions, the relationship between topology and smooth structure is unusually intricate, with phenomena that are not present in other dimensions.

Examples include the Euclidean space R^4 and the four-sphere S^4, as well as products and connected sums such as S^2 × S^2, the complex projective plane CP^2, and CP^2 with reversed orientation (CP^2bar). Other well-known smooth examples are the K3 surface, the four-torus T^4, and various connected sums of these. These manifolds can carry a range of geometric structures, such as complex or symplectic structures in appropriate cases.

Two foundational strands describe the limits and possibilities of smooth structures in four dimensions. Freedman’s classification of simply connected closed topological 4-manifolds shows that such manifolds are determined up to homeomorphism by their integral intersection forms. Donaldson’s gauge-theoretic invariants, by contrast, demonstrate that many forms cannot be realized by any smooth 4-manifold, illustrating a fundamental separation between the topological and smooth categories. The E8 manifold is a classical example: a topological 4-manifold with a definite unimodular form that does not admit a smooth structure.

A striking consequence is the existence of exotic smooth structures on R^4: there are uncountably many smooth manifolds homeomorphic to R^4 but not diffeomorphic to the standard R^4. It remains open whether the sphere S^4 admits any exotic smooth structures. Four-manifolds thus occupy a central role in low-dimensional topology, linking algebraic invariants, gauge theory, and geometric constructions in ways unique to dimension four.