1) Let $S$ be a topological space with an equivalence relation $~,$ and let $S/~$ denote the quotient space. Show that if the quotient space $S/~$ is Hausdorff, the set $E$ is closed. (See Lemma 2.2 of the notes `Topological Prelimnaries' for the definition of $E.$)
2) Exercise 1.1 in Lee, pg. 3.
3) Show that stereographic chart for the n-sphere is smoothly compatible with the chart of Example 1.20 in Lee.
1) Problem 1-7, pg 29 of Lee.
2)Exercise 2.4, pg 32 of Lee.
3) Exercise 2.5, pg 33 of Lee.
1) Exercise 2.12, pg 41 of Lee.
2) Problem 2-10, pg 58 of Lee.
1) Problem 3-2, pg 78 of Lee.
2) Problem 3-3, pg 79 of Lee.
1) Prove that the cotangent bundle $T^*M$ is a smooth manifold and that the natural projection is a submersion.
2) Exercise 4.3, pg 85 of Lee.
3) Exercise 4.6, pg 92 of Lee.
4) Problem 4-1, pg 100 of Lee.
1) Problem 17-1, pg 460 of Lee.
2) Problem 17-2, pg 460 of Lee.
1) Problem 9-1, pg 236 of Lee.
2) Problem 9-2, pg 237 of Lee.
3) Problem 9-4, pg 237 of Lee.
1) Problem 18-2, pg 491 of Lee.
2) Problem 18-5, pg 491 of Lee.
1) Problem 12-3, pg 319 of Lee.
2) Problem 12-4, pg 491 of Lee.
3) Problem 13-1, pg 346 of Lee.
4) Exercise 14.1, pg 356 of Lee.