Topological K-theory

This is the 2nd half of the course 'Algebraic Topology I&II' for gradute students at AMSS, and also for interested senior undergraduates. Basic homology theory is required. 

Each student should submit a report on a topic in homotopy theory, and gives a 20-minutes presentation. Presentations will be scheduled in the last few weeks of the course.

Lecture I: Basics on fibre bundles, Grothendieck construction, definition of K-groups and functorial properties

Lecture II: Geometric interpretation of reduced K-groups; principal bundles

Lecture III: Brown's representation theorem; universal bundles for linear groups

Lecture IV: Homotopy interpretation of K-groups; Adams' representation theorem

Lecture V: Homotopy interpretation of reduced K-groups, Bott periodicity, spectrum

Lecture VI: spectrum continued, homology and cohomology thoeries

Lecture VII: K-theory as a cohomology theory

Lecture VIII: ring structure of K-spectra, Hopf invaraint

Lecture IX: Hopf invariant one problem

Lecture X: Hopf invariant one problem continued; Chern character

Lecture XI: J-homomorphism and e-invariant

Lecture XII: J-homomorphism and e-invariant continued

Lecture XIII: Atiyah-Hirzebrch-Whitehead and Leray-Serre spectral sequences

Lecture XIV: Gysin and Wang Sequences

Students can choose report topics by themselves, or from the following list:

1. Brown's representation theory: proof and application 

2. Proof of Bott periodicty theorem

3. Homology and co-homology theory and spectra

4. Calculation of homotopy groups by using spectral sequences

5. Cohomology of linear Lie groups

6. Introduction to another generalized cohomology theory, such as cobordism

7. loop spaces of suspensions, the James construction and the James suspension splitting

8. The Hilton-Milnor theorem

9. EHP-sequences and calculations of homotopy groups of spheres

10. Cohomology of classifying spaces, characteristic classes and applications

11. Cobordism and genera, Atiyah-Singer index theorem

12. Introduction to simplical homology theory and simplicial homotopy theory