Math 432 / 532 Winter 2017

Instructor: Robert Lipshitz

Basic course information

Time: MWF 1:00–1:50 p.m.
Place: 205 Deady Hall.
Textbook: John Lee, Introduction to Smooth Manifolds, second edition.
Office hours: Monday 3-4, Wednesday 11:30-12:30, Friday 3-4.
Teaching assistant: Nick Howell
Final exam: per Registrar's schedule.


Math 432 requires Math 431 (Introduction to Topology 1), Math 341 (Linear Algebra), and Math 281 (Several Variable Calculus). Math 342 and 282 are strongly recommended, and Math 413 is recommended. Math 532 is open to any beginning mathematics graduate student, but students in Math 532 are expected to learn any background they are missing on their own. Anyone except a math graduate student seeking to enroll in 532 should obtain the instructor's permission.

Description and goals

This course is an introduction to differential topology -- the study of smooth manifolds and smooth maps. We will approach the subject in the language and generality that modern differential topologists (and geometers) use; the goal is to get far enough in the subject to prove some interesting, or even surprising, topological results.

Math 432 and 532 are a bridge course, between the levels of undergraduate and graduate mathematics. In particular, students should expect to find Math 4/532 and other bridge courses substantially more challenging than earlier undergraduate courses.

Particular "learning objects" for the course include:

  • Mastering the total derivative, chain rule, inverse function theorem, and implicit differentiation in their modern forms.
  • Learning the definition of a smooth manifold and smooth maps between manifolds, and developing a library of examples of these.
  • Understanding tangent vectors and the tangent space to a smooth manifold, and how to work with them.
  • Being able to differentiate smooth maps between manifolds.
  • Recognizing special classes of smooth maps---immersions, embeddings, and submersions.
  • Understanding Sard's theorem and the Brouwer degree.
  • Understanding why it is hard to comb a cat.



Homework 35%
Midterm 25%
Final exam 40%



The textbook for the class is Introduction to Smooth Manifolds, second edition, by John Lee. Some of the material in the first two weeks is covered in more detail in Principles of Mathematical Analysis by Walter Rudin and Calculus on Manifolds by Michael Spivak. Material from the last few weeks is in John Milnor's Topology from the Differentiable Viewpoint.

Students are expected to read the sections in the textbook before coming to class each day. The relevant sections are listed in the syllabus below.


Homework is due at the beginning of class on Wednesdays most weeks. The first homework assignment is due on Friday of the first week. You may discuss the problems with other students, but must write up your solutions to the problems by yourself. Any resources you use other than the textbook must be cited in your homework. You may not use electronic resources (e.g., Google) other than the textbook. Failure to follow this policy constitutes cheating; if you are caught cheating on the homework you will receive a 0 for the homework portion of the class and will be reported to the administration.

Requirements for students in Math 432 and 532 are slightly different. In addition to the requirements for Math 432, students in Math 532 are expected to:

  • Solve at least one of the "challenge" homework problems per week.
  • Type the solutions to their homework assignments in LaTeX. (See below for some resources.)

Graduate students may skip the first non-challenge homework problem on each homework assignment.

Students with disabilities

I, and the University of Oregon in general, are committed to an inclusive learning environment. If you have a disability which may impact your performance on exams, please contact the Accessible Education Center to discuss appropriate accommodations. If there are other disability-related barriers to your participation in the course, please either discuss them with me directly or consult with the Accessible Education Center.

Assignments, handouts, and other resources


First day handout.



L: Lee's book. R: Rudin's book. M: Milnor's book. +: material beyond that covered in the textbooks.

Week Date Topic Textbook sections
1 1/9 No classes, because of snow.  
  1/11 Analysis in Rn: differentiability, total derivative. L: Appendix C.1
R: §9.1, 9.2
  1/13 Chain rule.
2 1/16 Martin Luther King Day (no class)

This is the last day to drop the class without a W.
  1/18 Inverse function theorem. Homework 1 due. L: Appendix C.4
R: §9.3, 9.4
  1/20 Implicit function theorem, implicit differentiation. Homework 2 due.
Review of topological manifolds.
R: §9.5, 9.6
L: §1.1
3 1/23 Smooth manifolds. L: §1.2
  1/25 Examples of smooth manifolds. L: §1.3
  1/27 Manifolds with boundary. Homework 3 due. L: §1.4
4 1/30 Smooth functions. L: §2.1
  2/1 Partitions of unity. L: §2.2
  2/3 Catching up. Homework 4 due.  
5 2/6 More partitions of unity. Tangent vectors. L: §3.1
  2/8 Derivative (differential). L: §3.2
  2/10 Computations in local coordinates. Homework 5 due. L: §3.3
6 2/13 Midterm exam.  
  2/15 Tangent bundle. L: §3.4
  2/17 Velocity vectors of curves. Homework 6 due. L: §3.5
7 2/20 Another definition of the tangent bundle. L: §3.6
  2/22 Immersions, Embeddings. L: §4.1, 4.2.
  2/24 Submersions. Homework 7 due.

The last day to withdraw from the class is 2/26.
L: §4.3.

8 2/27 Embedded / immersed submanifolds. L: §5.1, 5.2
  3/1 Restricting maps. The tangent space to a submanifold. L: §5.3, 5.4
  3/3 Oriented manifolds. Homework 8 due. L: §15.1, 15.2
9 3/6 Sets of measure 0. L: §6.1
  3/8 Sard's theorem. L: §6.2; M: Ch. 2
  3/10 Proof of Sard's theorem. Homework 9 due. M: Ch. 3
10 3/13 Whitney embedding theorem. L: §6.3
  3/15 Brouwer degree. M: Ch. 4, 5
  3/17 Brouwer fixed point theorem, hairy ball theorem. Homework 10 due.  



  • Read the sections in the textbook before class, and again after class. Read with a pencil and paper in hand. Note down points that confuse you, and come back to them later to make sure you understand them. Work through the examples. When the author says something is "clear" that means it is not obvious -- make sure you understand why it is true.
  • Get help as soon as you are confused. Your best options for help are my office hours, other students in the class, and other students not in the class.
  • Start on the homework problems as soon as we have covered the corresponding material in class.
  • Start working on the homework by yourself. After you have spent at least half an hour on each problem, try describing your solutions or where you are stuck to other students in the class. (Discussing the problems is helpful in both cases.) Then, go back and write (or type) your solutions nicely by yourself. If you work in a group, it is easy to think you've learned the material when you have not.
  • Do not look for hints or solutions on the internet. I guarantee you will not learn the material that way, even if you feel like you do.
  • Solve extra problems.
  • Think about the material, and how different parts of the material relate to each other, and examples of the different concepts, and counter-examples if you drop hypotheses from theorems, and how the material relates to material from other classes, constantly -- whenever there is a pause in your day.