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ELEMENTI DI TOPOLOGIA ALGEBRICA
|ELEMENTI DI TOPOLOGIA ALGEBRICA||MAT/03||LEZIONI||48|
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KnowledgeThe student who successfully completes the course will have the ability to compute the homology and cohomology of simplicial and cellular complexes; (s)he will be able to demonstrate a solid knowledge of the applications of homology theory; (s)he will display advanced knowledge of Poincaré duality and of the notions of orientation and degree of a map between manifolds. (s)he will master the algebraic notions of Tor and Ext and know the statements in homology and cohomology of the universal coefficients theorem.
Assessment criteria of knowledgeThe student will be asked to compute the homology and the cohomology ring of some topological spaces that can be realized as simplicial or cellular complexes. He will have to demonstrate a solid knowledge of homology and cohomology theory and their applications.
- Final oral exam
Delivery: face to face
- attending lectures
- individual study
- Bibliography search
SyllabusBasics of category theory. Simplicial complexes and their homology. First homology as Abelianization of the fundamental group. Dependence on the homotopy type only. Smooth and PL manifolds, with and without boundary. n-homology of an n-manifold. Degree and its applications. Relative homology and the axioms of homology. Homology of Delta-complexes. Singular homology. Homology with coefficients in a group. Tor and the universal coefficients theorem for homology. Cohomology. Ext and the universal coefficients theorem for cohomology. Cup product and Poincaré duality.
BibliographyTextbooks that will be used extensively: Hatcher - Algebraic topology Matveev - Lectures on algebraic topology Other reference textbooks Massey - A basic course in algebraic topology Munkres - Elements of algebraic topology Spanier - Algebraic topology Greenberg, Harper - Algebraic topology. A first course
Ultimo aggiornamento 14/11/2016 17:27