Vector bundles and topological K-theory
Vector Bundles and K-Theory
Allen Hatcher, 2003.
K-theory and Characteristic Classes
Inna Zakharevich, 2018.
The Topology of Fiber Bundles
Ralph Cohen, 1998.
Papers and Books
Milnor and Stasheff, 1974.
Additional Resources by Topic
Topological K-theory and Hopf invariant 1 problem
M. F Atiyah, D.W. Anderson, 1967.
K-theory and the Hopf invariant
J. F. Adams, M. F. Atiyah, 1966.
This gives a short proof to Adams’s long Annals paper result on the Hopf invariant 1 using the Adams spectral sequence.
Vector fields on spheres (TeX notes), original is Part 1 and Part 2
Course taught by Haynes Miller, 1989.
Topological K-theory and some of its applications
Saad Slaoui, 2019.
Blog post on K-theory and the Hopf invariant
Akhil Mathew, 2012.
18 Lectures on K-theory
Ioannis P. Zois, 2010.
Lecture notes from an Introduction to K-theory course covering topological K-theory, analytic K-theory and higher algebraic K-theory.
Atiyah-Singer index theorem
The Index of Elliptic Operators, I
Atiyah and Singer, 1968.
This paper gives a proof of the Atiyah-Singer index using K-theory.
The Index of Elliptic Operators: II
Michael Atiyah, Graeme Segal, 1968.
This reformulates the result as a sort of Lefschetz fixed point theorem, using equivariant K-theory. This makes it more challenging on first approach.
K-theory and elliptic operators
Gregory D. Landweber, 2005.
An expository account.