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.

The homotopy type of a simply connected compact 4-manifold only depends on the intersection form on the middle dimensional homology. A famous theorem of M. Freedman (1982) implies that the homeomorphism type of the manifold only depends on this intersection form, and on a Z/2Z invariant called the Kirby–Siebenmann invariant, and moreover that every combination of unimodular form and Kirby–Siebenmann invariant can arise, except that if the form is even then the Kirby–Siebenmann invariant must be the signature/8 (mod 2).

Examples:

Freedman's classification can be extended to some cases when the fundamental group is not too complicated; for example, when it is Z there is a classification similar to the one above using Hermitian forms over the group ring of Z. If the fundamental group is too large (for example, a free group on 2 generators) then Freedman's techniques seem to fail and very little is known about such manifolds.

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