2021 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Geometry F1
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Hisaaki Endo
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-4 Mon
- Class
- -
- Course Code
- MTH.B506
- Number of credits
- 100
- Course offered
- 2021
- Offered quarter
- 2Q
- Syllabus updated
- Jul 10, 2025
- Language
- English
Syllabus
Course overview and goals
The main subject of this course is several basic theorems on the topology of 4-manifolds.
After introducing some notions for handlebody theory, we prove two theorems of Wall: one on h-cobordism and the other on stabilization. We next prove Rochlin's theorem which states that the signature of a closed spin 4-manifold is divisible by 16. We finally prove Kervaire-Milnor theorem as an application of Rochlin's theorem. This course is a continuation of "Advanced topics in Geometry E1" held in 1st Quarter.
Course description and aims
Students are expected to:
- Understand the principle of handle decompositions of manifolds
- Understand statements and proofs of the theorems of Wall and Rochlin
- Be able to apply Rochlin's theorem to problems on representing homology classes
Keywords
4-manifold, intersection form, Wall's theorem, Rochlin's theorem
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Standard lecture course
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Handle decompositions and h-cobordism | Details will be provided during each class session. |
| Class 2 | Wall's theorem (1) | |
| Class 3 | Wall's theorem (2) | |
| Class 4 | The Arf invariant and characteristic surfaces | |
| Class 5 | Rochlin's theorem (1) | |
| Class 6 | Rochlin's theorem (2) | |
| Class 7 | Kervaire-Milnor theorem |
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)
none
Reference books, course materials, etc.
R. E. Gompf and A. I. Stipsicz, 4-Manifolds and Kirby Calculus, American Mathematical Society, 1999.
A. Scorpan, The Wild World of 4-Manifolds, American Mathematical Society, 2005.
R. C. Kirby, The Topology of 4-Manifolds, Lecture Notes in Mathematics, Vol. 1374, Springer, 1989.
Evaluation methods and criteria
Assignments (100%).
Related courses
- MTH.B505 ： Advanced topics in Geometry E1
Prerequisites
Basic knowledge on topology (manifolds, homology groups) is required.