Daniel A. Ramras


Homework and Lecture Notes for Math 372B: Characteristic Classes

HW 1

HW 2

HW 3

HW 4

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.

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