Università Roma Tre

1. Multilinear algebra. External algebra on a vector space, wedge product, standard basis and size of the q-forms space. 2. Differential forms inR. Smooth forms, external differential operator, de Rham's comology, orientation and integration, Poincar ́e lemma. Hodge inRn.3 operator. Elements of homological algebra. Complexes of chains and their comology, fundamental theorem of homological algebra (snake's lemma), lemma of five. 4. Integration on manifolds. Orientation on a manifold, integration of n-forms, Stokes' theorem. 5. De Rham comology. Mayer-Vietoris succession, sphere comology, domain invariance theorem. 6. Mayer-Vietoris argument.Existence of a good covering, finite-dimensionality of de Rham's comology, compact support comology, Poincar's duaity and for compact varieties, K ̈unneth formulation for the comology of a product. Fiber bundles and Leray-Hirsch theorem. The dual di Poincar is a closed oriented submanifold. 7. De Rham's theorem. Double complex, Cech comology of beams. Topological invariance of de Rham's comology.

[1]Raoul Bott, Loring W. Tu,Differential forms in algebraic topology.Springer, (1986).
[2]Marco Abate, Francesca Tovena,Geometria Differenziale.Springer, (2011).