2022-present, spring semester
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:
Brown's representation theory: proof and application
Proof of Bott periodicty theorem
Homology and co-homology theory and spectra
Calculation of homotopy groups by using spectral sequences
Cohomology of linear Lie groups
Introduction to another generalized cohomology theory, such as cobordism
loop spaces of suspensions, the James construction and the James suspension splitting
The Hilton-Milnor theorem
EHP-sequences and calculations of homotopy groups of spheres
Cohomology of classifying spaces, characteristic classes and applications
Cobordism and genera, Atiyah-Singer index theorem
Introduction to simplical homology theory and simplicial homotopy theory
Primary on Algebraic K-theory
J-homomorphism and e-invariant
Iterated loop spaces and recognition principle
Serre's work on computation of homotopy groups by spectral sequences