Math 189P: Differential Topology

Spring 2016


Differential topology is the study of smooth spaces. We learn what it means for spaces to be smoothly equivalent. In calculus we learn how to differentiate and integrate in Euclidean space. In this course, we learn how to do calculus on smooth manifolds of any dimension. This is a beautiful and unifying subject, combining elements of differential geometry, topology, algebraic geometry, and analysis. It is a perfect gateway to those subjects, and is at the heart of some of the most important and interesting open problems in mathematics and physics. Specific topics may include manifolds in n-dimensional Euclidean space and smooth maps, Sard's Theorem, classification of compact one-manifolds, transversality and intersection modulo 2. We may cover additional topics as time permits, and depending on the interests of the class. Prerequisite: Mathematics 131 or permission of instructor.


Dagan Karp
Office: 3414 Shan
Office hours: Mon 4-5p.m., Tue 3-4pm, and open door.


This course will meet for lectures Monday at Wednesday, 1:15-2:30pm in Shanahan 3421.


Our required course text is Differential Topology, by Guillemin & Pollack. Unofficially, we'll also follow Topology from the Differential Viewpoint, by Milnor.


Grades will be based on homework (70%), one midterm examination (25%) and class participation (5%).


Homework is due in class on Wednesdays, and is posted below.

You are encouraged to discuss the homework with other members of the class, and it is appropriate to acknowledge the assistance of others. You will, however, be expected to write up your solutions on your own. The instructor will reserve the right to refuse to accept late homework. Please consult the HMC mathematics homework format guidelines for helpful tips on homework submission and formatting.


There will be one exam. It will be in-class on Wednesday, March 9.

Critical Readings

In addition to the written homework, suggested readings will be posted on the homework page in conjunction with upcoming lectures. Before each lecture, read the corresponding material. Class activities will depend on your study of this material in advance (which is a departure from most lower division courses).

The goals of the critical reading exercises are manifold: to better the student's independent intake of mathematical exposition, to train in independent learning, to increase the interactive nature of the course (by allowing the instructor to respond to questions and comments), and to make the course more tailored to the specific curiosities of the class as a whole.


Students interested in using LaTeX are encouraged to do so, but it is not required.


To request academic accommodations due to a disability, HMC students should contact Heidi Bird ( our Coordinator of Student Support. For students from the other Claremont Colleges, please contact your home college’s disability officer: CMC: Julia Easley ( CGU: Chris Bass ( Pitzer: Jill Hawthorne ( Pomona: Jan Collins-Eaglin ( Scripps: Sonia De La Torre-Iniguez (

Math is for all

My goal is to welcome everyone to mathematics. As an instructor, I hold the fundamental belief that everyone in the class is fully capable of engaging and mastering the material. My goal is to meet everyone at least halfway in the learning process. Our classroom should be an inclusive space, where ideas, questions, and misconceptions can be discussed with respect. There is usually more than one way to see and solve a problem and we will all be richer if we can be open to multiple paths to knowledge. I look forward to getting to know you all, as individuals and as a learning community.
Homework and Readings
Problems marked with R are to be read and not submitted.
    HW 1. Due Wed Jan 27.
    Reading: Section 1.1.
    Problems: (1.1) 3, 5, 6, 9, 10, 12.
    HW 2. Due Wed Feb 3.
    Reading: Section 1.2.
    Problems: (1.1) 16, 17, 18. (1.2) 2, 9, 10.
    HW 3. Due Wed Feb 10.
    Reading: Section 1.3.
    Problems: (1.2) 1, 3, 4, 5. (1.3) 1, 2.
    HW 4. Due Wed Feb 17
    Reading: Section 1.4.
    Problems: (1.3) 4, 7 (1.4) 7
    HW 5. Due Wed Mar 2.
    Reading: Sections 1.5 and 1.6
    Problems: (1.4) R10, R11, 12 (1.5) 2, R4, R5 (1.6) R4, R5, 6
    Wed Mar 9
    Midterm Exam
    In class
    HW 6. Due Wed Mar 30.
    Reading: 2.1, 2.2.
    Problems: (2.1) R3, 6, R7, 8 (2.2) 3, R6, R7.
    HW 7. Due Wed Apr 6.
    Reading: Western Mathematics, by Bishop
    and Chapter 1 of Radical Equations, by Moses.
    Problems: Midterm 1 re-writes.