quality:corruption
Topological 4-manifolds

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 Michael Freedman (1982) implies that the homeomorphism type of the manifold only depends on this intersection form, and on a Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } 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 {\displaystyle \mathbb {Z} }, there is a classification similar to the one above using Hermitian forms over the group ring of Z {\displaystyle \mathbb {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.

For any finitely presented group it is easy to construct a (smooth) compact 4-manifold with it as its fundamental group. (More specifically, for any finitely presented group, one constructs a manifold with the given fundamental group, such that two manifolds in this family are homeomorphic if and only if the fundamental groups are isomorphic.) As there can be no algorithm to tell whether two finitely presented groups are isomorphic (even if one is known to be trivial), there can be no algorithm to tell if two 4-manifolds have the same fundamental group. This is one reason why much of the work on 4-manifolds just considers the simply connected case: the general case of many problems is already known to be intractable.