2021 Advanced topics in Geometry F1

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Academic unit or major
Graduate major in Mathematics
Endo Hisaaki 
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Course description and aims

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.

Student learning outcomes

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


4-manifold, intersection form, Wall's theorem, Rochlin'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 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

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.B505 : Advanced topics in Geometry E1

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

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

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