Last modifications:
december 9th, 2014.

## Masterclass: topological K-theory and applications

## Strasbourg, February 21-27, 2015

## Institut de Recherche de Mathématique Avancée

## Introductory lectures:

These lectures will cover the following notions of algebraic topology :
singular cohomology and its cup-product,
adjunction between suspension and the loop space,
fibrations,
unstable homotopy groups,
the basics of stable homotopy (Freudenthal's theorem).

## Main lectures:

**Part 1: **The aim of this part is the construction of orthogonal K-theory KO and unitary K-theory KU as generalized cohomology theories.
vector bundle,
classifying spaces,
Bott periodicity theorem,
Adams operations,
Chern character
Atiyah-Hirzebruch's spectral sequence.

**Part 2: ** The aim of this part is to illustrate the theory by examples:
relation between representations of a group G and K-theory of the classifying space BG established by Atiyah and Segal
applications of topological K-theory: in particular, Hopf invariant problem solved by Adams and Atiyah.
study of the image of J-homomorphism Im(J) which is a direct summand of the stable homotopy groups of spheres, which is detected in K-theory by the e-invariant.