### 2016　Advanced courses in Geometry A

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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.B331
Credits
1
2016
Offered quarter
1Q
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 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 B" (ZUA.B332) 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 Details will be provided during each class session
Class 3 restriction of a vector bundle, direct product, Whitney sum, tensor product, exterior product Details will be provided during each class session
Class 4 subbundle, inner product, decomposition into Whitney sum Details will be provided during each class session
Class 5 homotopy between maps and induced bundle Details will be provided during each class session
Class 6 classification of the vector bundles over a sphere, orientation of a vector bundle Details will be provided during each class session
Class 7 Stiefel manifold, Grassmann manifold, canonical bundle Details will be provided during each class session
Class 8 universal bundle, classifying space 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.B332 ： Advanced courses in Geometry B
• MTH.B401 ： Advanced topics in Geometry A

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

Students require the following knowledge: topological space, smooth manifold, homology theory.