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- Academic unit or major
- Mathematics
- Instructor(s)
- Murayama Mitsutaka
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Mon3-4(H115)
- Group
- -
- Course number
- ZUA.B333
- Credits
- 1
- Academic year
- 2016
- Offered quarter
- 3Q
- Syllabus updated
- 2017/1/11
- Lecture notes updated
- 2016/11/25
- Language used
- Japanese
- Access Index
-
Course description and aims
The main subject of this course is the homology and cohomology theories of topological spaces with coefficients in a general module. After reviewing integral homology groups of a simplicial complex, we introduce homology and cohomology groups with coefficients in a module, singular homology and cohomology groups with coefficients in a module, its axioms, and its application. This course will be succeeded by “Advanced courses in Geometry D” in the fourth quarter.
Homology and cohomology groups are basic tools among topology, geometry, algebra, and analysis. This course aims to grasp the fundamental concept and properties of homology and cohomology theories.
Student learning outcomes
By the end of this course, students will be able to:
・understand the definition of each term concerning the following "keywords"
・grasp the main ideas of the homology and cohomology theories
・calculate homology and cohomology groups of some 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 complices, 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
- ZUA.B334 ： Advanced courses in Geometry D
- MTH.B403 ： Advanced topics in Geometry C
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Students are expected to know the fundamental terms of (integral) homology groups of a simplicial complex.
