The second half of the XX-th century has witnessed an explosion of results on the existence of smooth structures on topological manifolds. Following various sources in Wikipedia, a rough timeline goes like this (I am forgetting many other important results, but these are the ones I am most interested in).
- In 1952, Moise proved that topological 3-manifolds admit exactly one smooth structure
- In 1956, Milnor proved the existence of exotic 7-spheres, believing at first that they could be counterexamples to the generalized Poincaré conjecture
- He eventually would classify the exotic 7-spheres together with Kervaire in 1963
- In the meantime, Smale had proved the the generalized Poincaré conjecture in dimension $\geq 5$ in 1961, and the h-cobordism theorem, which would simplify Milnor's construction, in 1962
- In 1962, Stallings proved that $\mathbb{R}^n$ has a single smooth structure for $n \neq 4$
- The generalized Poincaré conjecture in dimension 4 was then proved by Freedman in 1982. Together with Kirby, he also proved the existence of an exotic $\mathbb{R}^4$
- The results of Donaldson on the intersection form of 4-manifolds of 1983, together with work by Freedman, could be used to construct examples of topological 4-manifolds without any smooth structure
As an algebraic geometer, I only know about these important developments from folklore, but I have never taken the time to study them in detail. I am trying to find a reference which would recap the whole story on exotic structures. So far I am aware of
- Topology of 4-manifolds, Freedman and Quinn, 1990
- The wild world of 4-manifolds, Scorpan, 2005
- Differential Algebraic Topology: From Stratifolds to Exotic Spheres, Kreck, 2010
The first two seem to be very specific on dimension 4, while the latter does not talk about dimension 4 at all, and also takes a rather non-standard approach through stratifolds.
Is there any book going through all (or most) of the above results?
