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 .