Төрт өлшемді кеңістіктер және математикалық ерекшеліктері

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.

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(M,Z/2Z) 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 (it is usually false in dimensions at least 7; see exotic sphere). The Poincaré conjecture for PL manifolds has been proved for all dimensions other than 4, but it is not known whether it is true in 4 dimensions (it is equivalent to the smooth Poincaré conjecture in 4 dimensions). 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 (as shown by Donaldson). 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. In dimension at least 5 the existence of topological manifolds not homeomorphic to a simplicial complex was an open problem. Ciprian Manolescu showed that there are manifolds in each dimension greater than or equal to 5, that are not homeomorphic to a simplicial complex.

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."