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.
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.