Professor Shelly Harvey
Herman Brown 446
Phone: x3659
shelly at rice dot edu

Course Information
Class meets: MWF 11-11:50pm in HB 427
Office Hours: Mondays 10-10:45am, Thursdays 10-11am, Fridays 10-10:45am, or by appointment

Required textbook
Characteristic Classes by Milnor and Stasheff
Free pdf from
Link to physical copy at amazon

Link to Lecture Notes

You should have taken Math 444/539. Specifically, knowledge of smooth manifolds and tangent spaces will be mostly assumed (though there will be a brief recap at the beginning of the semester). Milnor’s Topology from a Differentiable Viewpoint is a good reference for this material. You should have taken or be concurrently enrolled in Math 445/540.

Course objectives and learning outcomes
This class is an important foundational course in bundles and characteristic classes. Any graduate student who is interested in topology, geometry, algebraic geometry, or number theory should take it. I would encourage first year students to take the course as it is only offered every other year.

In the smooth and other categories, much of the fine structure of manifolds and varieties is encoded in the tangent bundle or other associated bundles. In this course, we will cover the following topics.

1. A brief summary of smooth manifolds and tangent spaces.

2. Basics of vector (and fibre) bundles, building towards characteristic classes (Steifel–Whitney, Euler, Chern, Pontryagin).

3. Existence of exotic smooth structures on S7 (time-permitting).

By the end of the course, the student will have a thorough understanding of the material presented in class.

Homework and in class participation
I will assign homework problems on occasion. You are encouraged to do the problems that you can and turn them in; this is the best way to learn a subject. If you turn them in, I will look at them and give you feedback. In addition, we will have in class activities. You are required to participate in these as much as you can with your mathematical background (there may some homework problems or activities that first year students may not have the background to do). There will be no exams.

There will be no exams.

Students are required to attend every lecture and participate in the in class activities (unless they have permission from the instructor to miss). Your grade is completely determined by attendance. In order to get an A, you must miss fewer than 4 lectures. If you miss 4-6 lectures, you get a B. If you miss 7-9 lectures, you get a C. If you miss 10-12 lectures, you get a D. If you miss more than 12 lectures, you get an F.

Disability Support
Any student with a documented disability seeking academic adjustments or accommodations is requested to speak with me during the first two weeks of class. All discussions will remain as confidential as possible. Students with disabilities will also need to contact the Disability Support Services Office in the Ley Student Center.