関連

下記がよくまとまっている
http://en.wikipedia.org/wiki/4-manifold#cite_note-2
（抜粋）
4-manifold
From Wikipedia, the free encyclopedia
Contents
1 Topological 4-manifolds
2 Smooth 4-manifolds
3 Special phenomena in 4-dimensions
4 Failure of the Whitney trick in dimension 4

Smooth 4-manifolds
For manifolds of dimension at most 6, any piecewise linear (PL) structure can be smoothed in an essentially unique way,[1]
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:

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

There is an almost complete answer to the first problem of which simply connected compact 4-manifolds have smooth structures.