2026 (Current Year) Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Geometry A
- 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 (M-B43(H106))
- Class
- -
- Course Code
- MTH.B401
- Number of credits
- 100
- Course offered
- 2026
- Offered quarter
- 1Q
- Syllabus updated
- Mar 5, 2026
- Language
- English
Syllabus
Course overview and goals
This course is an introduction to the cohomology of groups and provides the necessary background for working in this field. Group cohomology has a long history and is utilized in various areas such as geometry, characteristic classes, and number theory. It is studied from both algebraic and topological perspectives, sometimes independently. In this course, after introducing definitions and examples, we will cover the fundamental results, including Hopf’s Theorem. The course also aims to cover the basics of covering spaces, fundamental groups, and CW complexes.
Course description and aims
Students will understand the fundamentals of group cohomology. The final goal of the course is to understand Hopf's Theorem and its applications.
Keywords
Group cohomology, Fundamental group, Covering spaces, Central extensions, Cup product
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 | Introduction: Low-dimensional cohomology and group extensions |
Details will be provided during each class session |
| Class 2 | Projective resolutions and examples |
Details will be provided during each class session |
| Class 3 | Fundamental groups, covering spaces, and CW complexes |
Details will be provided during each class session |
| Class 4 | Eilenberg-MacLane spaces and examples of calculations |
Details will be provided during each class session |
| Class 5 | Induced representations and Shapiro's lemma |
Details will be provided during each class session |
| Class 6 | The transfer map and its applications |
Details will be provided during each class session |
| Class 7 | Hopf’s Theorem and central extensions |
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
Prerequisites
Knowledge on topology (MTH.B341) and maniofolds (MTH.B301, MTH.B302) are required.
Other
A PDF handout for the class will be distributed to those who have registered for the class. Details will be explained in the first class.