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.
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
4-manifold, intersection form, Whitehead's theorem
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||Practical and/or problem-solving skills|
Standard lecture course
|Course schedule||Required learning|
|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)|
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.
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.
Basic knowledge on topology (manifolds, homology groups) is required.