2021 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Geometry E1
- 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.B505
- Number of credits
- 100
- Course offered
- 2021
- Offered quarter
- 1Q
- Syllabus updated
- Jul 10, 2025
- Language
- English
Syllabus
Course overview and goals
The main subject of this course is basic concepts of intersection forms of 4-manifolds. We first explain basic notions for intersection form, such as symmetric bilinear form, rank, signature, parity, direct sum, characteristic element, and unimodularity. We next exhibit examples of simply-connected 4-manifolds including the complex projective plane, the product of 2-spheres, and K3 surfaces. We finally prove Whitehaed's theorem which states that the homotopy type of a simply-connected 4-manifold is determined by its intersection form. "Advanced courses in Geometry F1" held in 2nd Quarter is a continuation of this course.
Course description and aims
Students are expected to:
- Understand precisely various properties of symmetric bilinear forms
- Be able to determine intersection forms of basic 4-manifolds
- Understand an outline of the proof of Whitehead's theorem
Keywords
4-manifold, intersection form, Whitehead'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 | Intersection forms of 4-manifolds | Details will be provided during each class session. |
| Class 2 | Symmetric bilinear forms and their classification (1) | |
| Class 3 | Symmetric bilinear forms and their classification (2) | |
| Class 4 | Fundamental theorems and examples of 4-manifolds | |
| Class 5 | Invariants of K3 surfaces | |
| Class 6 | Whitehead's theorem (1) | |
| Class 7 | Whitehead's theorem (2) |
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.B506 ： Advanced topics in Geometry F1
Prerequisites
Basic knowledge on topology (manifolds, homology groups) is required.
Other
none