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 iiii: A. Hatcher's Notes on introductory pointset topology or Chapter 2 of C. Pugh's book Real Mathematical Analysis.
General information for students
 My office hours: Tuesday 12:301:30, Wednesday 12, Thursday 1011.
 Course syllabus and rough schedule.
 DSP students should speak to the instructor as soon as possible, even if you don't have a letter yet.

Guidelines on what to do if you think you may have a conflict between this class and your extracurricular activities.
In particular, you must speak to the instructor before the end of the second week of classes.  Academic honesty in mathematics courses. A statement on cheating and plagiarism, courtesy of M. Hutchings.
 Policy on absences for tests and midterms.
 How to get an A in this class
Textbook
The textbook for this course is Differential Topology by Guillemin and Pollack. We will cover most of chapters 1, 2 and 3.Supplementary reading (not required)
 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 .Problem sets and reading
 Week 1: Basic topology review.
Reading: A. Hatcher's Notes on introductory pointset topology, this is more than you need to know, but a good reference!
Homework: Problem set 1. Notes/solutions
Optional justforfun 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 handin homework. Here is some information on the test next week, and extra problems
 Week 5: (30minute test), Homotopy and stability
Reading: GP section 1.6
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
Reading: GP section 1.8, 2.1
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.
Reading: GP section 2.1, 2.2.
Suggested review problems and information on the midterm.
Extra office hours: I will have office hours 56pm on Thursday, and 34pm next Monday.  Week 9: Midterm exam (Tuesday, in class). Thursday topic: the classification of 1manifolds.
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 1manifolds, using only topological techniques not differential topology or smooth maps!
Presented as a series of exercises (a "takehome 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 epsilonneighborhood theorem
Reading: GP section 2.3
Solutions to the midterm. The average on the midterm was 29.9/40
Homework: Problem set 9.
 Week 11: Intersection theory mod 2
Reading: GP section 2.4
Homework: Problem set 10.
 Week 12: Winding number, JordanBrouwer separation, and BorsukUlam
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 BorsukUlam, 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 JordanBrouwer 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, 35pm. Monday, May 9, 35pm,
You may pick up the last homework set on Monday.
Final Exam: Your final exam is Thursday, May 12, 811am 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