Characteristic Classes
4 hours of lectures and seminars per week
Lecturer Johan Dupont
Content Characteristic classes are homological invariants associated to
geometric structures such as vector bundles on manifolds.
An important example is the Euler class of the tangent bundle
which is the main ingredient in both the Gauss-Bonnet theorem in
higher dimensions as well as in Hopf's theorem that the sum of local
indices of a vector field with finite singularities is the Euler
characteristic of the manifold.
The other classical characteristic classes are the Stiefel-Whitney
classes (in cohomology with coefficients in the integers modulo 2)
and the Pontrjagin classes for real vector bundles together with
the Chern classes for complex vector bundles (in integral
cohomology). In the course these classes are constructed and used
for various geometric applications.
Topics are: Real and complex vector bundles, Grassmann manifolds
and universal bundles. Stifel-Whitney classes, Oriented bundles, the
Thom isomorphism theorem and the Euler class, applications to the
(non-) existence of vector fields and immersions, complex vector
bundles and Chern classes, Pontrjagin classes, applications to the
cobordism ring and the Hirzebruch Signature theorem.
Prerequisites An introductory course in singular homology theory.
Text-books John W. Milnor and James D. Stasheff: "Characteristic Classes",
Princeton University Press, 1974.