Daniel A. Ramras

---

Homework and Lecture Notes for Math 372B: Characteristic Classes HW 1 HW 2 HW 3 HW 4 Lecture 1: Smooth manifolds and their tangent bundles Lecture 2: Clutching functions and principal bundles Lecture 3: Homotopy theory of principal bundles Lecture 4: Proof of the bundle homotopy theorem Lecture 5: Characteristic classes and Stiefel manifolds Lecture 6: Universal vector bundles and the theory of fibrations Lecture 7: Axioms for Chern and Stiefel-Whitney classes; Grothendieck's definition in terms of projective bundles Lecture 8: Verification of the axioms Lecture 9: Calculation of c_1 and w_1 for tensor products Lecture 10: Cohomology of projective spaces and the Projective Bundle Theorem Lecture 11: Proof of the Projective Bundle Theorem Lecture 12: Bundles over paracompact spaces and important facts about characteristic classes Lecture 13: Applications of Stiefel-Whitney classes to immersions and parallelizability of real projective spaces Lecture 14: K-theory and the Chern character, Part I: definitions Lecture 15: Chern character, Part II: Long exact sequences in K-theory and Bott periodicity Lecture 16: The Chern character is a rational isomorphism Lecture 17: Bott Periodicity Part I: the Fundamental Product Theorem and clutching functions over XxS^2 Lecture 18: Bott Periodicity Part II: Fourier analysis and reduction to Laurent clutching functions Lecture 19: Bott Periodicity Part III: Linear clutching functions; surjectivity of the Bott map Lecture 20: Bott Periodicity Part IV: Injectivity of the Bott map via the Chern character; the classifying space for K-theory Lecture 21: Final comments on K-theory; Oriented bundles and the Euler class Lecture 22: Proof of the Thom Isomorphism Theorem; the Gysin Sequence; applications to embeddings of real projective spaces Lecture 23: The Euler characteristic and the Euler class Lecture 24: Relation between the Euler class and the top Stiefel-Whitney class Lecture 25: Characteristic classes as obstructions to sections Student Lectures: Hang Wang, Pontrjagin classes and the Hirzebruch Signature Theorem Sam Nolen, The Atiyah-Hirzebruch spectral sequence