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Daniel A. Ramras |
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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 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 |