Homework Assignments for
Differential Topology

This page is always under construction, so you should check it regularly. Assignments with the word homework in bold face are set in stone. Other assignments are still tentative.

Homework # 1: (due January 29) Available here. Solutions available here.

Homework # 2: (due February 5) Available here. Solutions available here.

Homework # 3: (due February 12) Just Guillemin and Pollack this week:
Page 18, #2, 3, 4, and 9.
Page 25, #1, 2, 6, 7, 12 and 13.
Page 32, #1, 2, 4. (We'll do some more from this section next week).
Solutions available here.

Homework # 4: (due February 19) All but one problem from Guillemin and Pollack this week:
Page 32, #5, 7, 10
Page 38, #7, 8, 9, 11. On problem (8), either prove part (e) of the theorem or make sure you understand the proof in the book, since we did not do this part in class.
Page 45, #5, 16, 17, 18.
Extra problem a) Let X be a k-manifold and Y be the n-sphere (i.e. the unit sphere in Rn+1), and suppose that n>k. Show that any smooth map from X to Y is homotopic to a constant map.
b) Give an example of a k-manifold X and an n-manifold Y (not a sphere!) with n>k, and a map X to Y that is not homotopic to a constant map. (You may need facts from other courses to show that the map isn't null-homotopic. We'll develop our own tools later in the semester.)
Solutions available here.

Homework # 5: (due February 26)
Page 54, problems 6, 7, 8, 9.
Page 62, problems 1, 2, 6, 7, 8.
Page 66, problems 6 and 7.
Let X be an arbitrary manifold, Y=S2, and Z a closed submanifold of Y. Let f: X ⇒ Y be a smooth map. Show that, for almost every rotation R in SO(3), the map fR = R ° f is transversal to Z.
Solutions available here.

Homework # 6: (due March 4) Available here.
Solutions available here.

Homework # 7: (due March 21) Note that this set, which wraps up Chapter 2, is due the Monday AFTER Spring Break.
Page 74, problems 1, 16, 17, 18
Page 82, problems 3, 4, 5, 6, 9, 10. On Problem 5, assume that dim(X)>0. Contrary to the parenthetical comment, there IS a zero dimensional anomaly!
Write a 1-paragraph description of what you intend to write your term paper about.
If you're feeling REALLY energetic, or bored over Spring Break, work through the full proof of the Jordan-Brouwer separation theorem on pages 87-89. That's the best way to really get a feel for mod-2 intersection theory.
Solutions available here.

Homework #8 (due WEDNESDAY, March 30).
Page 103, problems 10, 11, 14, 17, 22. Note that there are some typos on problem 11. The formulas for v1, v2 and n should involve partial derivatives of f(x,y), not of F(x,y) (which doesn't even make sense, since F is a function of (x,y,z)).
Page 116, problems 3, 4, 6, 7, 9, 10, 11.
Solutions available here.

Homework # 9: (due FRIDAY April 8) Available here.
Solutions available here.

Homework #10 (due Friday, April 15) is a series of exercises embedded in part 1 of the lecture notes on differential forms. Do all of the exercises.
Solutions available here.

Homework #11 (due Friday, April 22). Do all the exercises from part 2 and the first 6 exercises from part 3 of the lecture notes on differential forms.
Solutions to the problems on part 2 of the notes available here. Solutions to the questions from part 3 are available here.

Homework #12 (due Friday, April 29). Do the remaining exercises from part 3 of the lecture notes on differential forms, and all of the exercises from part 4. Here are solutions to the questions from part 3 and part 4.

Homework #13 (due Friday, May 6). Do all of the exercises from sections 1 through 6 of part 5. Solutions to the questions from part 5 are available here.

Extra homework for your long-term benefit (not to be turned in). Do exercises 12 and 13 from part 5 of the notes, and all the exercises from the final part 6. These go beyond what you will be expected to know for the prelim exam, but are Very Good Things To Know. Solutions to the questions from part 6 are available here.