Daniel A. Ramras 


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
12: Smooth manifolds
and their tangent bundles (updated 1/9/2017) Lecture 3: Examples: spheres and projective spaces (updated 1/24/2017) Lectures 34:
Clutching functions
and principal bundles (updated 1/24/2017) Lectures 57: 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
StiefelWhitney classes
Applications to immersions of real
projective spaces
Constructing new bundles from old
Orientability and the
first StiefelWhitney class
Characteristic classes as
obstructions
Cohomology of Projective Space
Proof of the Projective Bundle
Theorem
Verification of the axioms for Chern and StiefelWhitney classes Final comments on characteristic
classes The Euler and Thom classes, the
Thom Isomorphism Theorem, the Gysin
sequence, and embeddings of RP^n
Ktheory and the Chern Character,
Part I Ktheory 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: Eigendecompositions of bundles [E, z+b] Bott Periodicity V: Continuity of the eigendecomposition; completion of the proof of periodicity 