The next few sections of Chapter 1 are intended to introduce some key examples of constructions with manifolds:
- The high-dimensional Poincaré conjecture and the h-cobordism theorem
- Milnor’s exotic spheres
- Variation of Pontrjagin classes within a homotopy type
In the version as written I started with Milnor’s examples. That’s because I wanted to get the reader to a point where Milnor has recorded that he arrived in the middle 1950s; he had a smooth, 7-dimensional homotopy sphere, but he didn’t know whether his example was an exotic smooth structure on \(S^7\), or a counterexample to the Poincaré conjecture in dimension 7. But looking at this again, I’ve come to feel that that makes the exposition a bit hard to follow. So I’d like to move the Poincaré discussion earlier (perhaps even before characteristic classes) and then pick up Milnor’s examples, even though this reverses the historical order of Milnor and Smale.
Now the Poincaré discussion has to begin with the original Analysis Situs papers and the 3-dimensional conjecture, even if dimension 3 is too low for surgery-type methods to be applicable. I’m working on the exposition here but I’d just like to link to this very beautiful talk by John Morgan at the Clay Institute Conference in 2010:
Here are some additional references:
Kirby, R. C., and M. G. Scharlemann. 1979. “Eight faces of the Poincaré homology 3-sphere.” Pp. 113–146 in Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977). New York: Academic Press Retrieved August 29, 2013 (http://www.ams.org/mathscinet-getitem?mr=537730).
Milnor, John. 1956. “On Manifolds Homeomorphic to the 7-Sphere.” Annals of Mathematics 64(2):399–405. Retrieved March 5, 2013.
Poincaré, Henri. 2010. Papers on Topology: Analysis Situs and Its Five Supplements. American Mathematical Soc. (translated by John Stillwell)
Smale, S. 1990. “The story of the higher dimensional Poincaré conjecture: what actually happened on the beaches of Rio.” Mathematical Intelligencer 12:44–51.
Stillwell, John. 2012. “Poincaré and the early history of 3-manifolds.” American Mathematical Society. Bulletin. New Series 49(4):555–576. Retrieved August 29, 2013.