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- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Murayama Mitsutaka
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Mon3-4(H115)
- Group
- -
- Course number
- MTH.B403
- Credits
- 1
- Academic year
- 2016
- Offered quarter
- 3Q
- Syllabus updated
- 2016/12/14
- Lecture notes updated
- 2016/11/25
- Language used
- Japanese
- Access Index
-
Course description and aims
This course covers homology theory for general coefficients as a generalization of homology with integer coefficients for a simplicial complex as well as cohomology theory, the dual concept of homology theory. The instructor will first review the homology groups (with integer coefficients) for a simplicial complex, then cover homology and cohomology theory for general coefficients of a simplicial complex, the definition of singular homology theory and cohomology theory with general coefficients, axiomatic systems of homology theory and cohomology theory, and applications. This course is followed by Advanced Topics in Geometry D.
In the context of modern mathematics, homology and cohomology groups are basic tools of not just geometry, but also algebra and analysis. In this course students will gain an understanding of the approaches to and properties of these tools.
Student learning outcomes
Students will learn the following variety of mathematical knowledge, ways of thinking, and calculation skills.
- Gain an understanding of definition of terminology related to keywords listed below
- Get a picture of the main ideas and viewpoints of homology and cohomology theories
- Learn to calculate homology and cohomology groups for several phase spaces
Keywords
homology and cohomology groups with coefficients in a module, chain complex, cochain complex, cycle, boundary, homotopy, exact sequence, excision
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 | review of homology groups of simplicial complexes, definition of homology and cohomology groups with general coeficients | Details will be provided during each class session |
| Class 2 | definitions of singular homology and cohomology groups of topological spaces, maps indeuced from continuous maps | Details will be provided during each class session |
| Class 3 | category and functor, functoriaility of homology and cohomology groups | Details will be provided during each class session |
| Class 4 | axiom of homology and cohomology theory | Details will be provided during each class session |
| Class 5 | homotopy axiom and its examples | Details will be provided during each class session |
| Class 6 | exact sequence and its examples | Details will be provided during each class session |
| Class 7 | excision | Details will be provided during each class session |
| Class 8 | Mayer-Vietoris exact sequences | Details will be provided during each class session |
Textbook(s)
Non required
Reference books, course materials, etc.
None
Assessment criteria and methods
Exams and reports. Details will be provided during class sessions.
Related courses
- MTH.B341 ： Topology
- MTH.B404 ： Advanced topics in Geometry D
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Students are expected to know the fundamental terms of (integral) homology groups of a simplicial complex.
Office hours
to be determined
