MTH 7375 - Section 2 - Topics in Topology: Fiber bundles and characteristic classes

Course description: Introduces fiber bundles and characteristic classes. Topics include the construction of universal bundles, homotopy classification of principal bundles, bundles over spheres, cohomology of classifying spaces, Stiefel-Whitney classes, Gysin and Wang sequences, Thom isomorphism, Euler class, obstructions, Chern classes, and Pontrjagin classes.

References:

  1. Required: The topology of fiber bundles – Lecture notes, Ralph Cohen. Available at https://math.stanford.edu/~ralph/fiber.pdf

  2. Optional: Fibre Bundles, Dave Husemoller

  3. Optional: Characteristic classes, John Milnor and James Stasheff

  4. Optional: From calculus to cohomology: De Rham cohomology and characteristic classes, Ib Madsen and Jxrgen Tornehave

Syllabus (pdf)

Lectures:

  • Lecture 1 [9/05/2024] pdf: Intro to fiber bundles, vector bundles and examples.

  • Lecture 2 [9/09/2024] pdf: Sections and principal G-bundles.

  • Lecture 3 [9/12/2024] pdf: Clutching functions, principal GL_n bundles vs vector bundles

  • Lecture 4 (zoom) [9/19/2024] pdf: