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2016　Advanced topics in Geometry A

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
Endo Hisaaki 
Class Format
Lecture     
Media-enhanced courses
Day/Period(Room No.)
Tue3-4(H115)  
Group
-
Course number
MTH.B401
Credits
1
Academic year
2016
Offered quarter
1Q
Syllabus updated
2016/12/14
Lecture notes updated
2016/4/4
Language used
Japanese
Access Index

Course description and aims

The main topics of this course are several basic concepts related to vector bundles. The instructor will first cover basic concepts related to sectioning, bundle mapping, induced vector bundles, Whitney sum and partial vector bundles. Using mesh functions, students are next given classifications of vector bundles on spheres. The instructor finally introduced Stiefel manifolds, Grassmann manifolds, and universal vector bundles, proving the classification theorem for vector bundles from classifying spaces and classifying mapping.
Vector bundles are one of the basic concepts of topology and differential geometry. This is an introductory course for vector bundles which provides some background knowledge for studying cutting edge geometry. This course is followed by "Advanced Topics in Geometry B (MTH.B402)", held in the second quarter.

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

  • 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.

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