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 R^{n+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=S^{2}, 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 f_{R} = 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 v_{1}, v_{2}
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.