Undergraduate Seminar
The purpose of this seminar is to give an introduction to algebraic
topology through de Rham cohomology and characteristic classes
following the text listed below. We first define and study the de Rham
cohomology groups of open subsets of euclidean space. This makes it
possible to prove a number of classical theorems in algebraic
topology, including the Brouwer fixed point theorem and the invariance
of domain. We next define smooth manifolds and their de Rham
cohomology groups. We then introduce smooth vector bundles and vector
fields and prove the Poincare-Hopf theorem. We conclude with the
definition and study of characteristic classes.
Here is a syllabus in Japanese:
Text:
Ib Madsen and Jørgen Tornehave: From Calculus to
Cohomology: De Rham Cohomology and Characteristic Classes,
Cambridge University Press, 1997.
N. Bourbaki: Algebra I, Springer-Verlag, 1989.
Time and place: Wednesday from 1:00 p.m.
Tentative schedule:
April 17: Introduction.
April 24: The alternating algebra.
April 24: The alternating algebra, continued.
May 1: Modules, tensor products, the tensor algebra, and the alternating
algebra (following Bourbaki).
May 8: Differential forms and de Rham cohomology.
May 15: The Poincare lemma; cochain complexes and their
cohomology.
May 22: Smooth manifolds.
May 29: Differential forms on smooth manifolds.
June 12: Partition of unity and the Mayer-Vietoris exact
sequence.
June 19: Homotopy invariance.
June 26: Applications of de Rham cohomology.
July 3: Fiber bundles and vector bundles.
July 10: Operations on fiber bundles and their sections.
October 2: Associativity isomorphisms (following Bourbaki).
October 9: Duality and traces (following Bourbaki).
October 16: Connections and curvature.
October 23: Connections and curvature - continued.
October 30: Connections and curvature - continued.
November 6: Chern classes and Chern characters.
November 13: Chern classes and Chern characters - continued.
November 20: The projective bundle formula.