december 9th, 2014.
Masterclass: topological K-theory and applications
Strasbourg, February 21-27, 2015
Institut de Recherche de Mathématique Avancée
These lectures will cover the following notions of algebraic topology :
singular cohomology and its cup-product,
adjunction between suspension and the loop space,
unstable homotopy groups,
the basics of stable homotopy (Freudenthal's theorem).
Part 1: The aim of this part is the construction of orthogonal K-theory KO and unitary K-theory KU as generalized cohomology theories.
Bott periodicity theorem,
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.