University at Buffalo 3 credits Mathematics
General topology: topological spaces, continuous maps, connected spaces, compact spaces. Homotopy theory: homotopy classes of maps, fundamental groups, Van Kampen’s theorem, covering spaces, classification of covering spaces. Elementary manifold theory: tangent vectors, derivative of maps, transversality, Sard’s theorem, differential forms, exterior derivative, de Rham cohomology. Singular homology theory: chain complex, relative homology, long exact sequence, excision, Mayer-Vietoris exact sequence. Homology of manifold: cohomology, cup and cap products, Poincaré duality, Lefschetz fixed point theorem.
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