A 4-manifold is a topological space that locally resembles \(\mathbb{R}^4\) (four-dimensional Euclidean space) and is a type of manifold—a fundamental concept in topology and differential geometry. Formally, a 4-manifold \(M\) is a space that is Hausdorff, second countable, and locally homeomorphic to \(\mathbb{R}^4\).
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A 4-manifold is a topological manifold locally modeled on , with a half-space model if a boundary is allowed. Examples include the four-sphere, the Complex projective plane and the product of two spheres . Its middle-dimensional intersection form pairs degree-two homology classes. Dimension four is special because an embedded two-dimensional surface has the same dimension as its complementary normal directions; intersections and their removal become particularly subtle. A smooth four-manifold carries additional differentiable structure, which is not determined merely by its local topological model.
