|
|
|
|
These notes are a revised and
reorganized version of the notes available here
and here.
The first set of 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. Homework 1 due in class on Monday,
Feb. 13. Homework 2 due in class on Wednesday, March 8. Homework 3 due in class on
Wednesday, April 5. Homework 4 due on Wednesday, May 3.
Lecture
1-2: Smooth manifolds
and their tangent bundles (updated 1/9/2017) Lecture 3: Examples: spheres and projective spaces (updated 1/24/2017) Lectures 3-4:
Clutching functions
and principal bundles (updated 1/24/2017) Lectures 5-7: The classification of principal bundles;
characteristic classes (updated 2/1/17) Universal bundles over the Grassmannians The long exact
sequence in homotopy associated to a fiber
bundle Definitions of Chern and
Stiefel-Whitney classes
Applications to immersions of real
projective spaces
Constructing new bundles from old
Orientability and the
first Stiefel-Whitney class
Characteristic classes as
obstructions
Cohomology of Projective Space
Proof of the Projective Bundle
Theorem
Verification of the axioms for Chern and Stiefel-Whitney classes Final comments on characteristic
classes The Euler and Thom classes, the
Thom Isomorphism Theorem, the Gysin
sequence, and embeddings of RP^n
K-theory and the Chern Character,
Part I K-theory and the Chern Character,
Part II
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 |