4-manifold - Special Phenomena in 4-dimensions

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(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 countable infinite number of non-diffeomorphic smooth structures.

• 4 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 is an open problem (as of 2007).