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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
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