# Math 141: Differential Topology Essential Prerequisites:
Students should be very comfortable with the following concepts:
i. Continuity of functions (from R^n to R^n, and on general metric spaces)
ii. Metric spaces
iii. Topology: open and closed sets, compactness, covers, homeomorphisms.
iv. Multivariable calculus: Differentiability of functions from R^n to R^n. Linear maps.

Suggested reading for i-iii: A. Hatcher's Notes on introductory point-set topology or Chapter 2 of C. Pugh's book Real Mathematical Analysis.

## General information for students

#### Textbook

The textbook for this course is Differential Topology by Guillemin and Pollack. We will cover most of chapters 1, 2 and 3.

- Chapter 1 (two lectures on manifolds) from A mathematical gift III. Highly recommended as motivation for the content of this class.
- L. Christine Kinsey, Topology of Surfaces (alternate text suggested by a previous instructor)
- V. V. Prasolov, Intuitive topology. (not as closely related to this class, but a good perspective on aspects of topology)

#### Also of interest

You may enjoy doing an independent study through the Directed Reading Program .

• Week 1: Basic topology review.
Reading: A. Hatcher's Notes on introductory point-set topology, this is more than you need to know, but a good reference!
Homework: Problem set 1. Notes/solutions
Optional just-for-fun homework: from V. Praslov's "Intuitive topology"

• Week 2: Smooth maps, manifolds, diffeomorphisms
Reading: GP sections 1.1 and 1.2
Homework: Problem set 2. Some comments .
Information on bump functions (related to problem 18), written by J. Loftin, here.

• Week 3: The inverse function theorem, immersions and submersions.
Reading: GP sections 1.3 and 1.4.
Optional: notes on the implicit function theorem for R^n by T. Austin. Alternatively, you can refer to Chapter 5.5 in Pugh's book.
Homework: Problem set 3 . Some comments .

• Week 4: More on submersions, transversality
Reading: GP sections 1.4 and 1.5
No hand-in homework. Here is some information on the test next week, and extra problems

• Week 5: (30-minute test), Homotopy and stability
Homework: Problem set 5.
Here are SOLUTIONS to the test. The average was 10.4/16.

• Week 6: Measure and Sard's theorem
Reading: GP section 1.7 -- first part only. We will not discuss Morse theory yet. Beginning of 1.8.
Also, some notes on measure zero by H. Chan that you might find helpful.
Homework: Problem set 6.

• Week 7: Whitney embedding. Intro to manifolds with boundary
I will give you the idea of the proof of Whitney for noncompact manifolds, but you are not required to learn all the details.
Homework: Problem set 7.

• Week 8: Manifolds with boundary and related topics.
Suggested review problems and information on the midterm.
Extra office hours: I will have office hours 5-6pm on Thursday, and 3-4pm next Monday.

• Week 9: Midterm exam (Tuesday, in class). Thursday topic: the classification of 1-manifolds.
Homework: Problem set 8, due Tuesday, March 29.
(this is shorter than usual and certainly possible to finish before the spring break)

Just for fun: The classification of 1-manifolds, using only topological techniques not differential topology or smooth maps!
Presented as a series of exercises (a "take-home exam") by David Gale.
Those of you who feel very comfortable with topology might want to give this a try for fun.
(Note that you need to be on campus to access the link above.)

• Week 10: Transversaily and the epsilon-neighborhood theorem
Solutions to the midterm. The average on the midterm was 29.9/40
Homework: Problem set 9.

• Week 11: Intersection theory mod 2
Homework: Problem set 10.

• Week 12: Winding number, Jordan-Brouwer separation, and Borsuk-Ulam
Reading: GP section 2.5 and 2.6
Here is a solution to GP 2.3 problem 10 from the homework handed back this week!
Homework: Problem set 11. Here is the .tex file

Just for fun:
Outside in is a video that describes something like the winding number (here called ``turning number", not exactly the same thing, but close!), then describes how to turn a sphere inside out via a family of immersions.

• Week 13: End of Borsuk-Ulam, Orientation
Reading: GP section 3.1 and 3.2
Some remarks on the ham sandwich theorem from last week.
Final homework set: Problem set 12. Here is the .tex file
NOTE: this homework is due on the last day of class, Thursday April 28. (of course you can hand it in early if you want)

• Week 14: Oriented intersection number
Reading: GP section 3.2 and 3.3 (up to the fundamental theorem of algebra)

• Review materials and final exam info:
For reference: solutions to all the exercises for the proof of Jordan-Brouwer Separation, from Prof. Kozai's 2014 class.

Information on the final exam including suggested practice problems.
Practice final and solutions (to be posted)

I will have office hours at the following times: Friday, May 6, 3-5pm. Monday, May 9, 3-5pm,
You may pick up the last homework set on Monday.

Final Exam: Your final exam is Thursday, May 12, 8-11am in 234 HEARST GYM

Note: Unusual location!!! And extremely early start time!!

image credit: the illustration of topology at the top of the page is by Robert Ghrist