Math 8230: Topics in Topology and Geometry

Time and Place: Tuesday-Thursday: 12:45pm--2pm at Boyd 302
Course webpage:
Office hours: By appointment


Prerequisite: The course will assume a basic understanding of smooth manifolds (smooth maps, derivatives, differential forms) and algebraic topology (homology, cohomology).

Description: This course is about characteristic classes, which are cohomology classes naturally associated to vector bundles or, more generally, principal bundles. They are a key tool in modern {algebraic, differential}×{topology, geometry}. The course starts with an introduction to vector bundles and principal bundles. It then discusses their main characteristic classes—the Euler class, Stiefel-Whitney classes, Chern classes, and Pontrjagin classes. The last part of the class discusses some applications of characteristic classes to bordisms. In the process, we will see some nice applications (e.g., to immersions) and review some important parts of algebraic topology (e.g., obstruction theory).

Homework: Registered students are expectedly to regularly attend class. Suggested homework will be given roughly biweekly. Students are required to turn in the solution to one homework problem by the middle of the semester and a second homework problem by the end of the semester.

Homework 1

Homework 2

Exam: This course does not have a final exam.

Academic honesty: As a University of Georgia student, you have agreed to abide by the Universitys academic honesty policy, A Culture of Honesty, and the Student Honor Code. All academic work must meet the standards described in A Culture of Honesty found at: Lack of knowledge of the academic honesty policy is not a reasonable explanation for a violation. Questions related to course assignments and the academic honesty policy should be directed to the instructor.

Tentetive Schedule:

Week Topics References
1 Course overview. Vector bundles, fiber bundles.
2 Fiber bundles with structure, principal G-bundles, Clutching, G-bundle isomorphism, mixing.
3 Equivalence between bundles with structure and principal G-bundles, Bundle homotopy lemma and applications.
4 Universal bundles and classifying spaces, Existence of universal bundles.
5 Explicit examples of universal bundles. Definition of characteristic classes, and interpretation in terms of classifying spaces.
6 Thom isomorphism theorem, Euler class
7 First properties of the Euler class.
8 Obstruction theory and the Euler class.
9 Stiefel-Whitney classes and Chern classes: properties and existance
10 Continue with properties and applications of Stiefel-Whitney and Chern classes.
11 Cohomology of BO(n)
12 Pontrjagin classes, Cohomology of BSO(n)
13 Continue with Pontrjagin classes
14 Catch up
15 Bordism, Thanksgiving
16 Hirzebruch signature theorem and differentiable structures on spheres.