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.