4-manifold

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.
4-manifolds are important in physics because in general relativity, spacetime is modeled as a pseudo-Riemannian 4-manifold.

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 implies that the homeomorphism type of the manifold only depends on this intersection form, and on a 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 equal to 1/8 of the signature.
Examples:

In the special case when the form is 0, this implies the 4-dimensional topological Poincaré conjecture.
If the form is the E8 lattice, this gives a manifold called the E8 manifold, a manifold not homeomorphic to any simplicial complex.
If the form is, 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.
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.

Freedman's classification can be extended to some cases when the fundamental group is not too complicated; for example, when it is, there is a classification similar to the one above using Hermitian forms over the group ring of. If the fundamental group is too large, 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 compact 4-manifold with it as its fundamental group. As there can be no algorithm to tell whether two finitely presented groups are isomorphic, 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.

Smooth 4-manifolds

For manifolds of dimension at most 6, any piecewise linear structure can be smoothed in an essentially unique way, so in particular the theory of 4 dimensional PL manifolds is much the same as the theory of 4 dimensional smooth manifolds.
A major open problem in the theory of smooth 4-manifolds is to classify the simply connected compact ones.
As the topological ones are known, this breaks up into two parts:

Which topological manifolds are smoothable?
Classify the different smooth structures on a smoothable manifold.

There is an almost complete answer to the first problem asking which simply connected compact 4-manifolds have smooth structures.
First, the Kirby–Siebenmann class must vanish.

If the intersection form is definite Donaldson's theorem gives a complete answer: there is a smooth structure if and only if the form is diagonalizable.
If the form is indefinite and odd there is a smooth structure.
If the form is indefinite, we may make its signature ≤ 0 by reversing orientation if necessary: then it is homeomorphic to a sum of m copies of II_1,1 and 2n copies of E₈ for some m and n. For m ≥ 3n, there is a smooth structure: the manifold is homeomorphic to a connected sum of n K3 surfaces and m − 3n copies of S²×S². For m ≤ 2n, Furuta proved no smooth structure exists. This leaves a small gap between 10/8 and 11/8 where the answer is mostly unknown, but the "11/8 conjecture" states that smooth structures do not exist if the dimension is less than 11/8 times |signature|; i.e. there are no smooth structures in the gap. The smallest case not covered above has n = 2 and m = 5, but this has also been ruled out, so the smallest lattice for which the answer is not currently known is the lattice II_7,55 of rank 62 with n = 3 and m = 7.

By contrast, very little is known about the second question of classifying the smooth structures on a smoothable 4-manifold; in fact, there is not a single smoothable 4-manifold where the answer is fully known. Donaldson showed that there are some simply connected compact 4-manifolds, such as Dolgachev surfaces, with a countably infinite number of different smooth structures. There are an uncountable number of different smooth structures on R⁴; see exotic R⁴.
Fintushel and Stern showed how to use surgery to construct large numbers of different smooth structures on many different manifolds, using Seiberg–Witten invariants to show that the smooth structures are different. Their results suggest that any classification of simply connected smooth 4-manifolds will be very complicated. There are currently no plausible conjectures about what this classification might look like.

Special phenomena in 4 dimensions

There are several fundamental theorems about manifolds that can be proved by low-dimensional methods in dimensions at most 3, and by completely different high-dimensional methods in dimension at least 5, but which are false in dimension 4. Here are some examples:

In dimensions other than 4, the Kirby–Siebenmann invariant provides the obstruction to the existence of a PL structure; in other words a compact topological manifold has a PL structure if and only if its Kirby–Siebenmann invariant in H⁴ vanishes. In dimension 3 and lower, every topological manifold admits an essentially unique PL structure. In dimension 4 there are many examples with vanishing Kirby–Siebenmann invariant but no PL structure.
In any dimension other than 4, a compact topological manifold has only a finite number of essentially distinct PL or smooth structures. In dimension 4, compact manifolds can have a countably-infinite number of non-diffeomorphic smooth structures.
Four is the only dimension n for which Rⁿ can have an exotic smooth structure. R⁴ has an uncountable number of exotic smooth structures; see exotic R⁴.
The solution to the smooth Poincaré conjecture is known in all dimensions other than 4. The Poincaré conjecture for PL manifolds has been proved for all dimensions other than 4. In 4 dimensions, the PL Poincaré conjecture is equivalent to the smooth Poincaré conjecture, and its truth is unknown.
The smooth h-cobordism theorem holds for cobordisms provided that neither the cobordism nor its boundary has dimension 4. It can fail if the boundary of the cobordism has dimension 4. If the cobordism has dimension 4, then it is unknown whether the h-cobordism theorem holds.
A topological manifold of dimension not equal to 4 has a handlebody decomposition. Manifolds of dimension 4 have a handlebody decomposition if and only if they are smoothable.
There are compact 4-dimensional topological manifolds that are not homeomorphic to any simplicial complex. Ciprian Manolescu showed that there are topological manifolds in each dimension greater than or equal to 5, that are not homeomorphic to a simplicial complex.
Failure of the Whitney trick in dimension 4

According to Frank Quinn, "Two n-dimensional submanifolds of a manifold of dimension 2n will usually intersect themselves and each other in isolated points. The "Whitney trick" uses an isotopy across an embedded 2-disk to simplify these intersections. Roughly speaking this reduces the study of n-dimensional embeddings to embeddings of 2-disks. But this is not a reduction when the dimension is 4: the 2-disks themselves are middle-dimensional, so trying to embed them encounters exactly the same problems they are supposed to solve. This is the phenomenon that separates dimension 4 from others."

Geometrization in dimension four

The uniformization theorem for two-dimensional surfaces states that every simply connected Riemann surface can be given one of three geometries. In dimension 3, it is not always possible to assign a geometry to a closed 3-manifold but the resolution of the geometrization conjecture, proposed by, implies that closed 3-manifolds can be decomposed into geometric pieces.
Each of these pieces can have one of 8 possible geometries: spherical, Euclidean, hyperbolic, Nil geometry, Sol geometry,, and the products, and.
In dimension four the situation is more complicated. Not every closed 4-manifold can be uniformized by a Lie group or even decomposed into geometrizable pieces. This follows from unsolvability of the homeomorphism problem for 4-manifolds. But, there is still a classification of 4-dimensional geometries due to Richard Filipkiewicz. These fall into 18 distinct geometries and one infinite family. An in depth discussion of the geometries and the manifolds that afford them is given in Hillman's book. The study of complex structures on geometrizable 4-manifolds was initiated by Wall.

The Four Dimensional Geometries

The distinction in to the following classes is somewhat arbitrary, the emphasis has been placed on properties of the fundamental group and the uniformizing Lie group. The classification of the geometries is taken from. The descriptions of the fundamental groups as well as further information on the 4-manifolds that afford them can be found in Hillman's book