4-manifold

In mathematics, 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure and even if there exists a smooth structure it need not be unique (i.e. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic).

4-manifolds are of importance in physics because, in General Relativity, spacetime is modeled as a pseudo-Riemannian 4-manifold.