This survey article provides an excellent introduction to the explosive discoveries of the 1980’s concerning the structure of topological and smooth four dimensional manifolds. The author provides a historical background which led to the distinguished work of Michael Freedman on the classification of simply-connected topological 4-manifolds, outlines the key ideas in this body of work, and then points out that one can at this stage show, utilizing the example of Cappel and Shaneson of a smooth 4- manifold that is h-cobordant to \({\mathbb{R}}P^ 4\) but not diffeomorphic to \({\mathbb{R}}P^ 4\), that \({\mathbb{R}}P^ 4\) connected sum with the Kummer surface is homeomorphic, but not diffeomorphic, to \({\mathbb{R}}P^ 4\#11(S^ 2\times S^ 2).\)
This brings the author to 1982 and the incredulous work of Simon Donaldson concerning the severe restrictions placed on the intersection form of a smooth 4-manifold. An outline of how the self-dual Yang-Mills equations are brought into the scheme of things is presented, and then it is shown how the disparate results of Donaldson and Freedman collide to yield an exotic smooth structure on \({\mathbb{R}}^ 4\) (in fact a continuum’s worth of exotic structures due to the end periodic analysis of C. Taubes). The author concludes by pointing out that Freedman and Taylor have constructed a universal exotic \({\mathbb{R}}^ 4\) and that Donaldson can show that the smooth h-cobordism theorem fails in dimension four.
This is not the end of the story. Donaldson’s introduction of the self- dual Yang-Mills equations into the study of smooth 4-manifolds is being further pursued by Donaldson and others, improving upon the above results, and tantalizing relationships between the study of two and three dimensional manifolds with that of four manifolds are being made. A book on this subject is sorely needed to introduce new students to this exciting and evolving area of mathematics.