Math 732: Differential Topology

Assignments

Unless otherwise indicated, the assigned problems are from Guillemin & Pollack. Particularly challenging optional problems will earn gold stars and are denoted with stars (*) below. Problems earmarked for the problem session are indicated with a plus (+).

1. Assignment 1, due W., 1/23.
   
   
2. Assignment 2. due W., 1/30. Required: 1.2.4, 1.2.7, 1.3.3.
   Additional problems: 1.2: 1, 2, 5, 11*, 12*
   
   
3. Assignment 3. due W., 2/06. Required: 1.3.4, 1.4.2, 1.4.5
   Optional: 1.3: 1, 8, 10*


4. Assignment 4. due W., 2/20. Required: 1.5.5, 1.5.10, 1.6.6, 1.6.7 (four problems this week)
   Optional: 1.5: 3, 4+, 6, 8, 9+. 1.6: 4, 5, 10*. 1.7: 1, 5, 6, 11*, 12*, 13*


5. Assignment 5. due W., 2/27. Required: 1.7.8, 2.1.9, and:
   Calculate that the Euler characteristic of Sigma_g, the closed surface of genus g, is 2-2g, using a height function and Morse indices. Argue why the height function is Morse.
   Optional: 1.7.14, 1.7.16*, 1.8.7+, 1.8.10*


6. Assignment 6. due W., 3/6. Required: 2.1.5, 2.1.7, 2.2.3, 2.2.5 (four problems this week)
   Optional: 2.1: 3+, 4+, 6; 2.2: 2+, 6*


7. Assignment 7. due F., 3/22. Required: 2.3: 3,10. (You may assume 2.3.2 without proving it to do 2.3.3. But it's not bad to prove!)
   Optional: 2.3 2,5,6


8. Assignment 8. due Tu., 4/9. Pick a partner. Complete as many of the 12 exercises as possible in section 2.5 which prove the Jordan Curve Theorem and its generalization, the Jordan-Brouwer Separation Theorem. We will discuss these in the problem session (with pizza). G&P promise this to be a "self-guided expedition with gun and camera into the wilds of such jungles, and in n dimensions too!"


9. Assignment 9. due W., 4/17. 4.4: 3,8
   Optional problems: 4.2 -- all; 4.3 -- the one exercise is recommended; 4.4 -- all; problems 3, 6 are recommended


10. Assignment 10. due W., 5/1. 4.5.1, 4.5.2, 4.7.7
    Optional problems: 4.7: 1, 8-14.