Daniel A. Ramras



Homework and Lecture Notes for Math 601: Vector Bundles

Homework 1

Homework 2

Homework 3

These notes are a revised and reorganized version of the notes available here. The older 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.

Lectures 1 and 2:
Smooth manifolds and their tangent bundles

Lectures 3-5: Vector bundles and principal bundles

Lectures 6-8: The classification of principal bundles; characteristic classes

Lectures 9-11: Universal bundles over the Grassmannians

Lecture 12: The long exact sequence in homotopy associated to a fiber bundle

Lecture 13-14: Definitions of Chern and Stiefel-Whitney classes

Lectures 15-16: Applications to immersions of real projective spaces

Lecture 17: Constructing new bundles from old

Lectures 18-19: Orientability and the first Stiefel-Whitney class

Lecture 20: Characteristic classes as obstructions

Lecture 21: Cohomology of Projective Space

Lecture 22: Proof of the Projective Bundle Theorem

Lecture 23-25: Verification of the axioms for Chern and Stiefel-Whitney classes

Lecture 26: Final comments on characteristic classes

Lecture 27: K-theory and the Chern Character, Part I

Lecture 28: K-theory and the Chern Character, Part II

Lecture 29: 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