Daniel A. Ramras



Math 697: Topics in Topology - Characteristic Classes

These notes are a revised and reorganized version of the notes available here and here. The first set of notes contain some additional material: the Euler class and its relation to the Euler characteristic; the Thom isomorphism and the Gysin sequence; applications to embeddings of real projective spaces.

Homework 1 due in class on Monday, Feb. 13.

Homework 2 due in class on Wednesday, March 8.

Homework 3 due in class on Wednesday, April 5.

Homework 4 due on Wednesday, May 3.

Lecture 1-2: Smooth manifolds and their tangent bundles (updated 1/9/2017)

Lecture 3: Examples: spheres and projective spaces (updated 1/24/2017)

Lectures 3-4: Clutching functions and principal bundles (updated 1/24/2017)

Lectures 5-7: The classification of principal bundles; characteristic classes (updated 2/1/17)

Universal bundles over the Grassmannians

The long exact sequence in homotopy associated to a fiber bundle

Definitions of Chern and Stiefel-Whitney classes

Applications to immersions of real projective spaces

Constructing new bundles from old

Orientability and the first Stiefel-Whitney class

Characteristic classes as obstructions

Cohomology of Projective Space

Proof of the Projective Bundle Theorem

Verification of the axioms for Chern and Stiefel-Whitney classes 

Final comments on characteristic classes

The Euler and Thom classes, the Thom Isomorphism Theorem, the Gysin sequence, and embeddings of RP^n

Notes from older courses that we won't cover:

K-theory and the Chern Character, Part I

K-theory and the Chern Character, Part II

The Chern Character is a Rational Isomorphism

Construction of the Puppe Sequence

Bott Periodicity I: Clutching functions for bundles over X x S^2

Bott Periodicity II: Approximation by Laurent polynomial cluthings

Bott Periodicity III: Reduction to linear clutching functions and the computation of K(S^2)

Bott Periodicity IV: Eigen-decompositions of bundles [E, z+b]

Bott Periodicity V: Continuity of the eigen-decomposition; completion of the proof of periodicity