Daniel A. Ramras 


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 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 