# The Poincare conjecture

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: