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.
John W. Milnor and James D. Stasheff: Characteristic
Classes, Annals of Math. Studies, vol. 76, Princeton
University Press, 1974.
Time and place: Wednesday from 1:00 pm., Science Building A,
room 432.
Schedule:
April 14: Introduction. (Toda)
April 21: The alternating algebra. (Imahashi)
April 21: The alternating algebra, continued. (Toda)
May 12: Modules, tensor products, the tensor algebra, and the alternating
algebra. (Imahashi and Toda)
May 19: Differential forms and de Rham cohomology. (Toda and Imahashi)
June 16: The Poincare lemma; cochain complexes and their
cohomology. (Toda and Imahashi)
June 23: Smooth manifolds. (Toda and Imahashi)
June 30: Differential forms on smooth manifolds. (Toda and Imahashi)
July 7: Partition of unity and the Mayer-Vietoris exact
sequence. (Toda and Imahashi)
July 14: Homotopy invariance. (Toda and Imahashi)
July 28: Applications of de Rham cohomology. (Toda)
October 6: Fiber bundles and vector bundles. (Toda and
Imahashi)
October 13: Operations on fiber bundles and their sections. (Toda
and Imahashi)
October 20: Associativity isomorphisms (following Bourbaki). (Toda)
October 27: Duality and traces (following
Bourbaki). (Imahashi)
November 10: Connections and curvature. (Toda and Imahashi)
November 17: Connections and curvature - continued. (Toda and
Imahashi)
December 1: Connections and curvature - continued. (Toda and
Imahashi)
December 10: Chern classes and Chern characters. (Toda and
Imahashi)
January 12: Chern classes and Chern characters - continued. (Toda
and Imahashi)
January 19: The projective bundle formula. (Toda and Imahashi)