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- 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(H104)
- Group
- -
- Course number
- MTH.B506
- Credits
- 1
- Academic year
- 2019
- Offered quarter
- 2Q
- Syllabus updated
- 2019/3/18
- Lecture notes updated
- -
- Language used
- English
- Access Index
-
Course description and aims
The main subject of this course is basic concepts of characteristic classes of vector bundles. After introducing some notions for singular homology theory, we prove the Thom isomorphism theorem and give a definition of the Euler class of an oriented vector bundle. We next introduce other characteristic classes: Stiefel-Whitney, Chern, Pontrjagin classes and explain basic properties of them. We finally mention applications of the characteristic classes.
Characteristic classes of vector bundles are one of the fundamental notions in geometry and topology. This course is an introductory course on characteristic classes and offers several background knowledge to students who want to study advanced geometry and topology. This course is a continuation of "Advanced topics in Geometry E1" held in 1st Quarter.
Student learning outcomes
Students are expected to:
- Understand the principle of characteristic classes of vector bundles
- Understand the precise statement and importance of the Thom isomorphism theorem
- Be able to compute characteristic classes in easy cases
Keywords
characteristic class, Thom isomorphism, Euler class
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 | the concept of characteristic class | Details will be provided during each class session. |
| Class 2 | singular homology and cohomology | |
| Class 3 | cup and cross products, excision theorem | |
| Class 4 | the Thom isomorphism theorem, Thom class | |
| Class 5 | Euler class, Gysin exact sequence | |
| Class 6 | Stiefel-Whitney classes, existence of immersions of projective spaces into Euclidean spaces | |
| Class 7 | Chern classes, Pontrjagin classes | |
| Class 8 | Developments |
Textbook(s)
none
Reference books, course materials, etc.
Ichiro Tamura, "Differential Topology", Iwanami Shoten, 1991, (in Japanese), ISBN-13: 978-4007302350
J. W. Milnor and J. D. Stasheff, "Characteristic Classes", Princeton Univ. Press, 1974, ISBN-13: 978-0691081229
Husemoller. "Fibre Bundles". 3rd ed. 21
Assessment criteria and methods
Assignments (100%).
Related courses
- MTH.B505 ： Advanced topics in Geometry E1
- MTH.E532 ： Special lectures on advanced topics in Mathematics H
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Basic knowledge on geometry (general topology, manifolds, differential forms, homology group) is required.
