Mathematical 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).
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 Michael Freedman (1982) implies that the homeomorphism type of the manifold only depends on this intersection form, and on a formula_1 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 formula_2, there is a classification similar to the one above using Hermitian forms over the group ring of formula_2. 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.
Smooth 4-manifolds.
For manifolds of dimension at most 6, any piecewise linear (PL) 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:
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.
In 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 (indexed by arbitrary integral polynomials) 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. (Some early conjectures that all simply connected smooth 4-manifolds might be connected sums of algebraic surfaces, or symplectic manifolds, possibly with orientations reversed, have been disproved.)
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:
Failure of the Whitney trick in dimension 4.
According to Frank Quinn, "Two "n"-dimensional submanifolds of a manifold of dimension 2"n" 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."
References.
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