Singular homology. Geometric meaning of H0 e H1. Topological pairs and relative homology. The long exact sequence in relative homology. Mayer Vietoris exact sequence. Examples. Applications of the homology of spheres. The invariance of dimension and of the boundary. Generalized Jordan curve theorem. CW-complex of finite type. The cellular homology complex. Examples of cellular homology. Morse Functions. Morse Lemma. Morse theorems I and II. Reeb theorem. Differentiable compact manifolds and CW complexes . Morse's inequality and equality. Cup product. The cohomology ring. Examples. The universal coefficient theorem.