Problem Sets

Homework Assignment 1. Due February 25th

• 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 11th

• Problem 1: Section 1.9, exercises ii and iii.
• Problem 2: Section 2.1, exercises v and vi.
• Problem 3: Section 2.2, exercises vii and viii.
• Problem 4: Section 2.4, exercises v and vi.

Homework Assignment 3. Due March 22nd

• Problem 1 Section 2.5, exercise viii.
• Problem 2 Section 3.2, exercises iv, v, vi, and vii.
• Problem 3 Section 3.4, exercise ii.
• Problem 4 Section 3.4, exercise iii.

Homework Assignment 4. Due April 17th

• PDF

Homework Assignment 5. Due May 1st

• Problem 1, section 4.3, exercise 1v and v.
• Problem 2. section 4.6, exercise ix.
• Problem 3, section 4.6, exercise x.
• Problem 4, section 4.7, exercises iii, iv and v.

Homework Assignment 6. Due May 15th

This homework assignment will consist of the following two "diagram chasing" problems.

• Problem 1 - Prove Mayer-Vietoris. (Hint: Exercises 1,2 and 3 in section 5.2.)
• Problem 2 - 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 the cover. (Hint:the diagram chasing argument at the end of section 5.8.)