This course covers the homology and cohomology groups of CW complexes and manifolds, important subjects of topology.
The instructor first discusses the CW complex and its properties, singular homology groups and cellular homology groups of CW complexes, and students then deepen their understanding of homology and cohomology groups of manifolds and their duality, as well as product structures of cohomology groups.
This course is followed by Advanced Topics in Geometry C, held in the third quarter.
CW complexes and manifolds are important subjects in geometry, and the general properties held by their homology and cohomology groups are very effective for calculating homology and cohomology groups for a specifically given space, to understand the properties of that space. The purpose of this course is to understand those properties.
By the end of this course, students will be able to:
・understand the definition of each term concernig 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.
cell complex, CW complex, homology theory, cohomology theory, cellular homology theory, cellular cohomology theory, manifold, duality, cup product
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Standard lecture course
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 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 |
Non required
None
Exams and reports. Details will be provided during class sessions.
Students are expected to have passed [Advanced topics in Geometry C]