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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.B334
- Credits
- 1
- Academic year
- 2016
- Offered quarter
- 4Q
- Syllabus updated
- 2017/1/11
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
-
Course description and aims
The main subject of this course is the homology and cohomology groups of CW complexes and manifolds. We introduce a CW complex, its properties, singular and cellular homology and cohomology groups of a CW complex. Next we explain the homology and cohomology groups of a manifold, their duality and products in cohomology groups. This course is a succession of “Advanced courses in Geometry C” in the third quarter.
CW complexes and manifolds are important objects of topology and geometry. The general properties of the homology and cohomology groups of CW complexes and manifolds play an important role to calculate the homology and cohomology groups of a given manifold or a CW complex, and to understand its properties. This course aims to grasp the properties of homology and cohomology groups of CW complexes and manifolds.
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 properties of homology and cohomology groups of CW complexes and manifolds
・calculate homology and cohomology groups of some CW complexes and manifolds.
Keywords
cell complex, CW complex, homology theory, cohomology theory, cellular homology theory, cellular cohomology theory, manifold, duality, cup product
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 | definition and property of a CW complex, examples | Details will be provided during each class session |
| Class 2 | cellular homology theory of CW complexes | Details will be provided during each class session |
| Class 3 | relation among simplicial homology, cellular homology and singular homology | Details will be provided during each class session |
| Class 4 | cellular cohomology theory of CW complexes, examples | Details will be provided during each class session |
| Class 5 | homology group of a manifold and a fundamental homology class | Details will be provided during each class session |
| Class 6 | cohomology group of a manifold and duality, examples | Details will be provided during each class session |
| Class 7 | tensor product of chain complexes | Details will be provided during each class session |
| Class 8 | cup product of cohomology classes | 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
- ZUA.B333 ： Advanced courses in Geometry C
- MTH.B404 ： Advanced topics in Geometry D
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Students are expected to have passed [Advanced courses in Geometry C]
