In Spring 2018 Lennart Meier and I are running a seminar for MSc students on topological K-theory. Here is the syllabus. We meet on Tuesdays, 13:15-15:00 in HFG 610.
The most convenient sources for the first part of the seminar are Atiyah's K-theory book and Hatcher's (partially written) book.
Schedule:
Feb 6, Gijs Heuts: Overview of topological K-theory.
Feb 13, Bjarne Kosmeijer: Vector bundles, basic constructions and homotopy invariance. Material: Hatcher 1.1 and a bit of 1.2, specifically Theorem 1.6. Bjarne's notes, including Exercises 1-5 to be handed in Feb 20.
Feb 20, Joost van Geffen: Classification of vector bundles. Material: Hatcher 1.2, specifically Theorem 1.16.
Feb 27, Thomas Blom: Real and complex K-theory, definitions and first examples. Material: Hatcher 2.1 and a bit of 2.2 (some exact sequences). Thomas' notes, including the exercises to be handed in on Mar 6.
Mar 6, Jorge Becerra: The product theorem, I. Material: First part of proof of the product theorem, roughly up to Proposition 2.6. These are Jorge's notes as well as some extra exercises on the classification of vector bundles, to be handed in on Mar 13.
Mar 13, Luka Zwaan: The product theorem, II. Material: Remainder of the proof of the product theorem, Bott periodicity as a consequence. The exercises of Jorge and Luka, to be handed in Mar 20.
Mar 20, Matthijs Lip: K-theory as a generalized cohomology theory. Material: The Eilenberg-Steenrod axioms, last part of Hatcher 2.2. Matthijs' notes, Exercises 1-3 to be handed in Mar 27.
Mar 27, Thomas Blom: Adams operations. Thomas' notes, including the homework exercises for Apr 3.
Apr 10, Jorge Becerra: The Hopf invariant one problem. Jorge's notes.
Apr 17, Joost van Geffen: Stiefel-Whitney classes and Chern classes. Axioms, first examples.
Apr 24, Matthijs Lip: Construction of characteristic classes and verification of the axioms. Matthijs' notes.
May 1, Bjarne Kosmeijer: Thom isomorphisms in K-theory and in cohomology. The Chern character and the Todd class. Bjarne's notes.
May 15, Luka Zwaan: Euler classes and Pontryagin classes.
May 29, Jorge Becerra: The space of Fredholm Operators.
June 19, Thomas Blom: The J-homomorphism, I.
June 26, Lennart Meier: The J-homomorphism, II.