Daniel A. Ramras 


The most uptodate 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 35: Vector bundles and principal bundles Lectures 68: The classification of principal bundles; characteristic classes Lectures 911: Universal bundles over the Grassmannians Lecture 12: The long exact sequence in homotopy associated to a fiber bundle Lecture 1314: Definitions of Chern and StiefelWhitney classes Lectures 1516: Applications to immersions of real projective spaces Lecture 17: Constructing new bundles from old Lectures 1819: Orientability and the first StiefelWhitney class Lecture 20: Characteristic classes as obstructions Lecture 21: Cohomology of Projective Space Lecture 22: Proof of the Projective Bundle Theorem Lecture 2325: Verification of the axioms for Chern and StiefelWhitney classes Lecture 26: Final comments on characteristic classes Lecture 27: Ktheory and the Chern Character, Part I Lecture 28: Ktheory 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: Eigendecompositions of bundles [E, z+b] Bott Periodicity V: Continuity of the eigendecomposition; completion of the proof of periodicity 