PROBLEM SETS
There will be six bi-weekly problem sets to be handed in and graded. They will be posted here.
Homework Assignment 1. Due February 24th
- Problem 1 - Section 1.2, in the course notes, exercises 5,6,7,8,9 and 10.
- Problem 2 - Section 1.3, exercise 7 and section 1.4, exercise 8.
- Problem 3 - Section 1.5, exercises 1,2,3,4 and 10.
- Problem 4 - Section 1.5, exercises 5,6,7,8 and 9.
Warning: In exercises 5 and 6 there is a small typographical error: The "T" should be an arbitrary k-tensor.
Homework Assignment 2. Due March 10th
- Problem 1 - Section 1.9 exercises 11 and 12 a,b
- Problem 2 - Section 2.1 exercises 5 and 6
- Problem 3 - Section 2.2 exercises 7 and 8
- Problem 4 - Section 2.4 exercises 5 and 6
Homework Assignment 3. Due March 21th
- Problem 1 - Section 2.5, exercise 8
- Problem 2 - Section 3.2, exercises 4,5,6 and 7
- Problem 3 - Section 3.4, exercise 2
- Problem 4 - Section 3.4 exercise 3
Warning: In Problem 2, exercise 5, there is a typo: the form ω should be supported on UxA not on U.
Homework Assignment 4. Due April 14th
Homework Assignment 5. Due April 25th
- Problem 1 - Section 4.3, exercises 4 a,b and 9
- Problem 2 - Section 4.4, exercises 1, 2 and 3
- Problem 3 - Section 4.4, exercises 6 and 7 a,b
- Problem 4 - Section 4.6, exercise 10
Homework Assignment 6. Due Friday May 9th
- Prove that if a manifold X admits a finite good cover, the DeRham cohomology groups of X are isomorphic to the Cech cohomology groups of this cover. (Prove this by doing the "diagram chasing" exercise at the end of section 5.8.)
