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- Academic unit or major
- Mathematics
- Instructor(s)
- Endo Hisaaki
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Tue3-4(H115)
- Group
- -
- Course number
- ZUA.B332
- Credits
- 1
- Academic year
- 2016
- Offered quarter
- 2Q
- Syllabus updated
- 2016/4/27
- Lecture notes updated
- 2016/4/4
- Language used
- Japanese
- 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 cobordism theory and the Hirzebruch signature theorem.
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 courses in Geometry A" (ZUA.B331) 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 | Details will be provided during each class session |
| Class 3 | cup and cross products, excision theorem | Details will be provided during each class session |
| Class 4 | the Thom isomorphism theorem, Thom class | Details will be provided during each class session |
| Class 5 | Euler class, Gysin exact sequence | Details will be provided during each class session |
| Class 6 | Stiefel-Whitney classes, existence of immersions of projective spaces into Euclidean spaces | Details will be provided during each class session |
| Class 7 | Chern classes, Pontrjagin classes | Details will be provided during each class session |
| Class 8 | cobordism theory, the Hirzebruch signature theorem | Details will be provided during each class session |
Textbook(s)
No textbook is set.
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
Assessment criteria and methods
Report submissions (100%)
Related courses
- ZUA.B331 ： Advanced courses in Geometry A
- MTH.B402 ： Advanced topics in Geometry B
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Students require the following knowledge: topological space, smooth manifold, homology theory. Enrollment in "Advanced courses in Geometry A" (ZUA.B331) is desirable.
