# Vector bundles and topological K-theory

## Lecture Notes

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

Characteristic classes

Milnor and Stasheff, 1974.

# Additional Resources by Topic

## Topological K-theory and Hopf invariant 1 problem

K-theory

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 Adamsâ€™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.