**12/9** - The exam has been corrected as stated in the email.

**12/7** - There was a slight gap in the argument before. This has been
corrected and an updated version is now posted.

**12/7** - A small typo has been fixed in the notes posted earlier
today. ("p to q in M" in the footnote should have been "p to q in
C".)

**12/7** - Here is a proof of de Rham's Theorem
for those of you with exposure to singular cohomology.

**12/2** - The final exam is now up. I
decided to add an easy problem, so there are now 7 problems.

**11/30** - The final exam will be posted Friday, December 2 by 6pm. It
will be due Thursday, December 15 at 5pm. Collaboration and use of
outside resources are prohibited, except for the textbook and your
personal course notes.

**11/28** - Notes on The Mayer-Vietoris sequence in
de Rham cohomology for those interested.

**11/16** - Assignment #6

Complete the following exercises in Spivak:

8-3, 8-4, 8-7, 8-8, 8-10, 8-14, 8-23, 8-31.

Due December 2 at 5pm.

**11/1** - Tomorrow's office hour (Wednesday, 11/2) is postponed to
Monday, 11/7 at 3-4pm.

**10/31** - Assignment #5

Complete the following exercises in Spivak:

6-6, 6-8, 7-2, 7-4, 7-5, 7-6, 7-16, 7-27.

Due November 14 at 5pm.

Complete the following exercises in Spivak:

5-4, 5-9, 5-10, 5-12, 5-15, 5-19, 6-1, 6-2.

Due October 31 at 5pm. The assignment with the best costume wins first grading rights.

**10/7** - The N in 4-2 for Assignment #3 should be replaced by
R.

**10/7** - I will also mention this briefly on Monday, but here is a
patch for today's notes:

In the proof of Claim 2, we assumed WLOG that g had non-vanishing
derivative on all of U. This is justified by the same sort of covering
argument used
multiple times previously in the proof. g therefore has no critical
points, and every value is regular. In particular, V=g^{-1}(0) is the
level set of
a regular value and thus a smooth manifold of smaller dimension
(n-1).

**9/30** - Assignment #3 is due
October 17 at 5pm.

1) Every proper map is a closed map.

2) Every bijective closed map is a homeomorphism.

**9/21** - The Tuesday, September 27th office hour will be postponed to
Thursday, September 29th at 11am.

**9/18** - Assignment #2 is due October
3 at 5pm.

**8/29** - The polls have closed and office hours will be Tu 11-12, W 3-4, and F 12-1. I was able to reduce to only one hour with a known conflict, so
hopefully these times will work well for everyone.

**8/29** - Assignment #1 -
Due September 16 at 5pm. You may turn it in during class or slide it under my door (Evans 805).

**8/26** - Regarding the shortage of desks: 9/12 is the earliest a room
change can be requested. If there are still too many bodies for Evans 5 at
that point, we should be able to obtain a new room in about a week. For
now, unfortunately, the seats will go to the early arrivals unless some of
you can bring portable furniture.

**8/26** - S^n should be replaced by S^{n-1} in the obvious places in
your notes from today.