MA-414s "Cohomology theory and topology of manifolds":

  1. Integration on manifolds, Stokes formula.
  2. Homological algebra.
  3. De Rham and compact cohomologies. Mayer-Vietoris sequence and Poincare lemma.
  4. Degree of a proper map and Sard theorem. Finite-dimensionality of HdR.
  5. Poincare duality (on orientable manifolds). Axiomatic definition of (co)homology theories.
  6. Kunneth and Leray-Hirsh formulas. 
  7. Cohomologies of vector bundles. Thom isomorphism.
  8. Mayer-Vietoris sequence for countable cover.
  9. Cech cohomologies vs. de Rham.
  10. Singular homologies and cohomologies.
  11. De Rham theorem.
  12. Euler class. Euler characteristic and Hopf index theorem.
  13. Axiomatic definition and discussion of Stiefel-Witney classes.
  14. Chern classes and relation to other characteristic classes.
  15. Calculations and applications.