Topological K-theory was introduced in algebraic geometry by Grothendieck, and, in algebraic topology, by Atiyah and Hirzebruch in the 1960's. The foundational ideas can be developed in many settings and, nowadays, K-theory comes in many flavours, such as algebraic K-theory, equivariant K-theory, Atiyah's real K-theory, and, more recently, twisted K-theory.

The aim of this Masterclass is to offer an introduction to K-theory, highlighting its significance in algebraic topology. Given a space X, one first defines an abelian group K⁰(X) in terms of vector bundles over X. The celebrated Bott periodicity theorem then allows this to be extended to a cohomology theory (in the sense of algebraic topology): it associates to any space X a graded abelian group K^*(X) and, to any continuous map from X to Y, a homomorphism from K^*(Y) to K^*(X).

The geometric origin of K-theory yields many applications, such as in the Atiyah-Singer index theorem in analysis. K-theory is equipped with cohomology operations induced by constructions on vector bundles; these are powerful, for example leading to the solution by Adams of the vector field problem for spheres.

In stable homotopy theory, topological K-theory detects chromatic type 1 phenomena; moreover, the action of complex conjugation on complex vector bundles underlies the relationship between real and complex K-theory. This is a motivating example in the study of higher chromatic phenomena, a central theme in modern algebraic topology. Some recent advances, such as the solution of the Kervaire invariant problem, are based on generalizations of this theory.

Main Lectures. (From Monday 23 to Friday 27 February 2015) :

• Christian Ausoni (Université Paris 13)

• Geoffrey Powell (CNRS - Université d'Angers)

• John Greenlees (University of Sheffield (UK))

• Niko Naumann (Universität Regensburg (Deutschland))

• Constanze Roitzheim (University of Kent (UK))