1.Introduction. 2.The alternating algebra. 3.De Rham cohomology. 4.Chaim complexes and their cohomology. 5.The mayer-Vietoris sequences. 6.Homotopy. 7.Applications of de Rham Cohomology. 8.Smooth manifolds. 9.Differential forms on smoth manifolds. 10.Integration on manifolds. 11.Degree, Linking numbers and index of vector fields. 12.The pointcaré-Hopf theorem. 13.Poincare duality. 14.The complex projective space CP. 15.Fiber bundles and vector bundles. 16.Operations on vector bundles and their sections. 17.Connections and curvature. 18.Characteristic classes of complex vector bundles. 19.The Euler class. 20.Cohomology of projective and Grassmannian bundles. 21.Thom isomorphism and the general Gauss-Bonnet formula.