At the end of the course the student will learn

- the definion of singular cohomology groups and relative cohomology groups of a space,
- Universal Coefficient Theorem for homology and cohomology,
- long exact sequence, homotopy invariance, excision for singular cohomology groups,
- cellular cohomology,
- the definition and basic properties of cup product and coholomogy ring of a space,
- computation of cohomology rings of real projective space and some familiar spaces,
- the homological definion of orientation on (topological) manifolds, fundamental class,
- the definition of cap product; Poincaré Duality,
- the definition of homotopy groups and basic constructions,
- the definition and basic properties of relative homotopy groups and homotopy exact sequence of a pair.