Exercise 1. Let , . Show that is a 2-dimensional smooth manifold homeomorphic to the torus .
Exercise 2. Let be the stereographic projection from the north pole , and be the stereographic projection from the south pole .
- Show that for , .
- Show that if is a non constant polynomial, the map , is smooth.
- More generally, if is smooth, find a condition on so that , is smooth.
- By following Milnor’s argument in the proof of the fundamental theorem of algebra, find sufficient conditions so that a smooth map is onto.
Exercise 3. By using Sard’s theorem, prove that the set of regular values of a smooth map is dense in .
John Milnor is a renowned mathematician who made fundamental contributions to differential topology and was awarded the Fields medal in 1962. One of his most striking result is the existence of several distinct differentiable structures on the 7 dimensional sphere (!).
The following video is an introductory lecture to differential topology given in 1965. The first 26 minutes will give you a good overview of what differential topology is about.