Daniel A. Ramras



Homework and Lecture Notes for Math 601: Vector Bundles

Homework 1

Homework 2

Homework 3

The most up-to-date version of these notes is available here. Various small errors in the notes below are corrected in the newer version. An older version of these notes, available here, contains 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