2021 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Geometry B1
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Kiyonori Gomi
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-6 Fri
- Class
- -
- Course Code
- MTH.B406
- Number of credits
- 100
- Course offered
- 2021
- Offered quarter
- 2Q
- Syllabus updated
- Jul 10, 2025
- Language
- English
Syllabus
Course overview and goals
Topological K-theory is one of the generalized cohomology theories, and roughly classifies vector bundles over topological spaces. In this lecture, the basic properties of topological K-theory including the Bott periodicity and the Thom isomorphism theorem will be explained. An application will also be provided at the end of the lecture.
Course description and aims
-to understand basic properties of topological K-theory.
-to understand an application of topological K-theory.
Keywords
vector bundles, K-theory, Bott periodicity, Thom isomorphism
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
A standard lecture course
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | The homotopy axiom and the excision axiom | Details will be provided during each class session |
| Class 2 | The exactness axiom | Details will be provided during each class session |
| Class 3 | The Bott periodicity, I | Details will be provided during each class session |
| Class 4 | The Bott periodicity, II | Details will be provided during each class session |
| Class 5 | The Thom isomorphism theorem, I | Details will be provided during each class session |
| Class 6 | The Thom isomorphism theorem, II | Details will be provided during each class session |
| Class 7 | Application | Details will be provided during each class session |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
No textbook is set.
Lecture note will be provided.
Reference books, course materials, etc.
M. F. Atiyah, K-theory. Lecture notes by D. W. Anderson W. A. Benjamin, Inc., New York-Amsterdam 1967
Evaluation methods and criteria
Assignments (100%).
Related courses
- MTH.B203 ： Introduction to Topology III
- MTH.B204 ： Introduction to Topology IV
- MTH.B341 ： Topology
- LAS.M106 ： Linear Algebra II
- MTH.A202 ： Introduction to Algebra II
- MTH.A203 ： Introduction to Algebra III
- MTH.A204 ： Introduction to Algebra IV
Prerequisites
require proficiency in basic topology (MTH.B203, MTH.B204, MTH.B341) and algebra (LAS.M106, MTH.A201, MTH.A202, MTH.A203, MTH.A204)