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School of Environment and Society
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- Common courses
- Liberal arts and basic science courses
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Nosaka Takefumi
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Fri5-6(H119A)
- Group
- -
- Course number
- MTH.B507
- Credits
- 1
- Academic year
- 2017
- Offered quarter
- 3Q
- Syllabus updated
- 2017/8/21
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
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Course description and aims
The course is an introduction to group cohomology, and gives the basic on group cohomology and its applications. The group cohomology has a long history and appear in some fields, and has studies from many viewpoints, including topology and algebra, number theory. In this course, I first give the definitions and examples of group cohomology, and give fundamental properties, and explain Hopf theorem. Furthermore, I also takes basics on covering, fundamental group, CW complexes with homology, and so on.
Student learning outcomes
The aim is to understand the basics of group cohomology. I fix the goal for understanding the Hopf theorem and its applications.
Keywords
Group cohomology, fundamental group, covering of CW complexes, central extension, cup products
Competencies that will be developed
| ✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Class flow
A common course
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | Introduction | Definitions and properties |
| Class 2 | Projective resolutions and exmaples | Definitions and properties |
| Class 3 | Fundamental group, covering, and CW complex | Definitions and properties |
| Class 4 | Eilenberg-MacLane spaces and some computations | Definitions and properties |
| Class 5 | Induced representaion, Shapiro lemma, and transfer | Definitions and properties |
| Class 6 | transfer and its applications Hopf's theorem and central extensions | Definitions and properties |
| Class 7 | Hopf theorem and central extensions | Definitions and properties |
| Class 8 | Cup products, the homology of abelian groups, and fox derivation | Definitions and properties |
Textbook(s)
No textbook
Reference books, course materials, etc.
K. S. Brown 「Cohomology of groups 」 (Springer-Verlag )
Assessment criteria and methods
Reporting assignments（100％）．
Related courses
- MTH.B202 ： Introduction to Topology II
- MTH.B301 ： Geometry I
- MTH.B302 ： Geometry II
Prerequisites (i.e., required knowledge, skills, courses, etc.)
I suppose elementary and basics on group and topology.
Other
I welcome any questions on this course
