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

• My office hours: Tuesday 12:30-1:30, Wednesday 1-2, Thursday 10-11.


• 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 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
  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 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
  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, 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!!

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