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)