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

- Integration on manifolds, Stokes formula.
- Homological algebra.
- De Rham and compact cohomologies. Mayer-Vietoris sequence and Poincare lemma.
- Degree of a proper map and Sard theorem. Finite-dimensionality of H
_{dR}. - Poincare duality (on orientable manifolds). Axiomatic definition of (co)homology theories.
- Kunneth and Leray-Hirsh formulas.
- Cohomologies of vector bundles. Thom isomorphism.
- Mayer-Vietoris sequence for countable cover.
- Cech cohomologies vs. de Rham.
- Singular homologies and cohomologies.
- De Rham theorem.
- Euler class. Euler characteristic and Hopf index theorem.
- Axiomatic definition and discussion of Stiefel-Witney classes.
- Chern classes and relation to other characteristic classes.
- Calculations and applications.