4-manifold - Topological 4-manifolds

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 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:

• In the special case when the form is 0, this implies the 4-dimensional topological Poincaré conjecture.
• If the form is E₈, this gives a manifold called the E8 manifold, a manifold not homeomorphic to any simplicial complex.
• If the form is Z, there are two manifolds depending on the Kirby–Siebenmann invariant: one is 2 dimensional complex projective space, and the other is a fake projective space, with the same homotopy type but not homeomorphic (and with no smooth structure).
• When the rank of the form is greater than about 28, the number of positive definite unimodular forms starts to increase extremely rapidly with the rank, so there are huge numbers of corresponding simply connected topological 4-manifolds (most of which seem to be of almost no interest).

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.

For any finitely presented group it is easy to construct a (smooth) compact 4-manifold with it as its fundamental group. As there is no algorithm to tell whether two finitely presented groups are isomorphic (even if one is known to be trivial) there is 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.