2021 Advanced topics in Geometry E1

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Academic unit or major
Graduate major in Mathematics
Endo Hisaaki 
Class Format
Media-enhanced courses
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Course description and aims

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.

Student learning outcomes

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

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

Standard lecture course

Course schedule/Required learning

  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)

Out-of-Class Study Time (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.



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.

Assessment criteria and methods

Assignments (100%).

Related courses

  • MTH.B506 : Advanced topics in Geometry F1

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Basic knowledge on topology (manifolds, homology groups) is required.



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