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School of Environment and Society
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- Common courses
- 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.B505
- Credits
- 1
- Academic year
- 2019
- Offered quarter
- 1Q
- 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 vector bundles. We first explain basic notions for vector bundle, such as section, bundle map, induced bundle, Whitney sum, and subbundle. We next give a classification of the vector bundles over spheres by using clutching functions. We finally introduce Stiefel manifolds, Grassmann manifolds and universal bundles, and prove the classification theorem of vector bundles in terms of classifying spaces and classifying maps.
Vector bundles are one of the fundamental notions in geometry and topology. This course is an introductory course on vector bundles and offers several background knowledge to students who want to study advanced geometry and topology. "Advanced courses in Geometry E2" held in 2nd Quarter is a continuation of this course.
Student learning outcomes
Students are expected to:
- Be able to determine whether a given family of vector spaces is a vector bundle
- Understand precisely various constructions of vector bundles
- Be able to classify the vector bundles over a sphere in easy cases
- Understand the principle of classification of vector bundles in terms of classifying spaces
Keywords
vector bundle, section, bundle map, Whitney sum, universal bundle
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 | vector bundle, isomorphism of vector bundles, trivial bundle, tangent bundle | Details will be provided during each class session. |
| Class 2 | section of a vector bundle, linearly independent sections, bundle map, induced bundle | |
| Class 3 | restriction of a vector bundle, direct product, Whitney sum, tensor product, exterior product | |
| Class 4 | homotopy between maps and induced bundle | |
| Class 5 | classification of the vector bundles over a sphere, orientation of a vector bundle | |
| Class 6 | Stiefel manifold, Grassmann manifold, canonical bundle | |
| Class 7 | universal bundle, classifying space | |
| Class 8 | K group and principle G bundle |
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.B506 ： Advanced topics in Geometry F1
- 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.
Other
none
