MT 83101: Geometry/Topology III
Time/Location: MW 11-12:15, Carney 309
Instructor: John Baldwin
Office: Carney 254
Office Hours: By appointment
Course Description: This course is an introduction to vector bundles, fiber bundles and characteristic classes with a focus on bundles over smooth manifolds. We will begin with basic examples and constructions before moving on to the problem of classifying vector bundles. Along the way, we will construct Stiefel-Whitney, Euler, Chern and Pontryagin classes. This is a beautiful subject whose applications include a description of the oriented cobordism ring, the Hirzebruch signature formula and Milnor's construction of exotic 7-spheres. The hope is to discuss some of these. At the end, we will introduce connections and curvature for general vector bundles and talk about their relationship with characteristic classes.
Prerequisites: This course will be very challenging without a working knowledge of certain basic tools in algebraic topology like singular homology and cohomology and Poincare duality. You should also have an acquaintance with basic concepts in differential topology like smooth manifolds, tangent spaces and differential forms.
Textbook: We won't follow one book exclusively, but Milnor and Stasheff's Characteristic Classes is the best single resource for this course. We will also draw from include Hatcher's Vector Bundles and K-Theory and Bott and Tu's Differential Forms in Algebraic Topology. I've also found Johan Dupont's Fibre Bundles and Chern-Weil Theory helpful. For some of the more basic algebraic topology tools we'll use, I suggest Hatcher's Algebraic Topology.
Topics: The following is a tentative (and probably too ambitious) list of course topics in roughly the order we will cover them. The last three topics in particular are subject to change.
Assignments: There will be regular homework assignments (posted below) and a final exam. The homeworks will be collected (generally on Mondays) and looked at, and I will occasionally provide feedback, but they will not be assigned grades. However, doing the homework is essential to your getting a lot out of this course.
- Intro to vector and fiber bundles
- Classifying vector bundles
- Intro to Stiefel-Whitney classes
- The algebraic topology of Grassmannians
- Existence of Stiefel-Whitney classes
- Chern classes
- The Euler class
- Pontryagin classes
- Chern and Pontryagin numbers
- The oriented cobordism ring
- The Hirzebruch signature theorem
- Connections, curvature and Chern-Weil theory