Masters Course (Topological K-theory)

Contact Information

Contact: Chris Bourne   cbourne [at] nagoya-u.jp,　bourne.christopher.jack.x7 [at] f.mail.nagoya-u.ac.jp

Office Hours: To be announced
Location: Room 419, Humanities Common Facility Building (A4② on the Campus Map).
I may also be available in Science Bldg A, Rm 353 on Friday afternoons.

Overview:

Topological K-theory was first considered in the 1960s by Atiyah and Hirzebruch and is a cohomology theory built from vector bundles over topological spaces. The theory has become an important tool in topology, geometry and index theory. The goal of the course is to give an introduction to topological K-theory and study its fundamental properties. We will also compare the topological approach to K-theory with an algebraic approach described in terms of idempotents and invertible matrices over Banach algebras.

位相的K理論は1960年代Atiyah-Hirzebruchによって定義され，ベクトル束から構築されたコホモロジー理論です．この理論は幾何学，位相幾何学，指数理論といった分野において重要なツールとして発展してきました．本講義の目的は位相的K理論を入門し，基礎の性質を勉強することです．またバナッハ環のベキ等元と可逆元を用いるK理論への代数的なアプローチを比較します．

Course outline (tentative):

The course will be conducted in English. この講義を英語で行います．

- Vector bundles and their operations
- Idempotents, invertible elements and other preliminaries
- The K-theory groups and basic properties
- Bott periodicity and applications
- Further structure in K-theory
- Characteristic classes and the Chern character (if time)

References

M. F. Atiyah, K-Theory. W. A. Benjamin, 1967.
E. Park, Complex Topological K-Theory. Cambridge University Press, 2008.
M. F. アティヤ, 松尾信一郎 (監訳), K理論. 岩波書店, 2022.

Additional materials and resources will be distributed where necessary.

Homework

In preparation