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.
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
vector bundle, section, bundle map, Whitney sum, universal bundle
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||Practical and/or problem-solving skills|
Standard lecture course
|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|
No textbook is set.
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
Report submissions (100%)
Students require the following knowledge: topological space, smooth manifold, homology theory.