Introduction: examples of topological spaces (manifolds, spaces of maps, CW spaces). Homotopy, homotopy equivalence, homotopy functor, homotopy groups. Fundamental group, van Kampen theorem, covering spaces, classification of covering spaces. Higher homotopy groups, fibred spaces, exact homotopy sequences, homotopy groups of spheres. Theorems of Freudenthal, Browder, Whitehead. Singular chain complex, singular homology, homotopy invariance, excision isomorphism. Exact homology sequences (pair, triple, Mayer-Vietoris). Homology of CW spaces. Homology and homotopy, theorems of Hurewicz and Whitehead. Homology with coefficients, cohomology, the universal coefficient theorem. The cup-, cap-, and cross-product, Kunneth theorem. Cohomology ring. Homology of manifolds, fundamental class, Poincare isomorphism and duality, Alexander-Pontryagin duality. Lefschetz formulas.

Prerequisites: Linear Algebra 1,2,  Topology, Differential Geometry, Calculus 1,2.

Bibliography:

. A. Fomenko, D. Fuchs, V. Gutenmacher. Homotopic topology.

. A. Hatcher. Algebraic topology.

. R. Switzer. Algebraic topology – Homology and homotopy.