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)

Homework:

  • Homework 1: pdf | Due 10/09/2024

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: Reduction of the structure group, Euclidean structures, orientations.

  • Lecture 5 [9/23/2024] pdf: Cohomological obstructions to orientability

  • Lecture 6 [9/26/2024] pdf: Homotopy invariance of fiber bundles, principal G-bundles over spheres

  • Lecture 7 [9/30/2024] pdf: The Milnor construction of BG and its universality

  • Lecture 8 [10/3/2024] pdf: Eilenberg-MacLane spaces, recognition of universal bundles, computations.

  • Lecture 9 [10/7/2024] pdf: Simplicial objects in a category, the simplicial construction of BG and K(G,n).

  • Lecture 10 [10/10/2024] pdf: Presheaves and their Cech cohomology, cohomological classification of Prin G bundles.

  • Lecture 11 [10/17//2024] pdf: Exact sequences in Cech cohomology, 1st Stiefel Whitney class, 1st Chern class

  • Lecture 12 [10/24/2024] pdf: What are characterstic classes, Axiomatic view of SW classes, applications.

  • Lecture 13 [10/28/2024] pdf: Parallelizability, immersions and cobordisms. Cohomology of Grassmannias part I.

  • Lecture 14 [10/31/2024] pdf: Cohomology of Grassmannians part II, uniqueness of Stiefel-Whitney classes.

  • Lecture 15 [11/04/2024] pdf: The Thom isomorphism theorem part I.

  • Lecture 16 [11/07/2024] pdf: The Thom isomorphism theorem part II, Steenrod squares, Construction of Stiefel Whitney classes via Thom’s identity