2026 (Current Year) Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Geometry B
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Takefumi Nosaka
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-6 Fri (W5-105)
- Class
- -
- Course Code
- MTH.B402
- Number of credits
- 100
- Course offered
- 2026
- Offered quarter
- 2Q
- Syllabus updated
- Mar 5, 2026
- Language
- English
Syllabus
Course overview and goals
This class introduces (higher) product structures in group cohomology (such as cup products and Massey products), and then shows research examples and applications of group cohomology. For example, we will mention applications to Fox dirivatives and 2- and 3-dimensional topology. We plan to cover one topic in about two sessions.This class is a continuation of the "Special Topics in Geometry" course taught in the first quarter.
Course description and aims
The goal of this class is to learn the usefulness and range of applications of group cohomology through cup products and application examples of group cohomology. Examples of the use and calculation of group cohomology will also be studied.
Keywords
Cohomology of groups, fundamental groups, Fox derivatives
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
This is a standard lecture course.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Diagonal approximation and cup products |
Details will be provided during each class session |
| Class 2 | Applications of cup products |
Details will be provided during each class session |
| Class 3 | Massey product and its application |
Details will be provided during each class session |
| Class 4 | Definition and some computation of Fox derivative |
Details will be provided during each class session |
| Class 5 | Fox derivatives, low-degree group cohomology, and topological meaning |
Details will be provided during each class session |
| Class 6 | Jacobian matrices of Fox derivatives and Groebner basis type invariants |
Details will be provided during each class session |
| Class 7 | Applications to low-dimensional topology |
Details will be provided during each class session |
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)
non required
Reference books, course materials, etc.
K. S. Brown 「Cohomology of groups 」
T. Satoh 「Cohomology of groups 」 publiched by Kindaisha
Evaluation methods and criteria
Assignments (100%).
Related courses
- MTH.B341 ： Topology
- MTH.B301 ： Geometry I
- MTH.B302 ： Geometry II
- MTH.B401 ： Advanced topics in Geometry A
- ZUA.B331 ： Advanced courses in Geometry A
Prerequisites
Knowledge on topology (MTH.B341) and maniofolds (MTH.B301, MTH.B302) are required. Also, students are supposed to have attended Advanced topics in Geometry A(MTH.B401) or Advanced courses in Geometry A(ZUA.B331).