4-manifolds

A smooth 4-manifold homeomorphic yet not diffeomorphic to euclidean space

In mathematics, a 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).

invariant of smooth four-manifolds defined by gauge theory

Study in mathematical gauge theory

A simply connected 4-manifold with intersection form the E8 lattice