4-manifold

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

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.

Donaldson showed that there are some simply connected compact 4-manifolds, such as Dolgachev surfaces, with a countably infinite number of different smooth structures.

Here are some examples: According to Frank Quinn, "Two n-dimensional submanifolds of a manifold of dimension 2n will usually intersect themselves and each other in isolated points.

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.