Contents

We classify finitely generated abelian groups and, using simplicial complex, describe various groups that can be associated to manifolds, such as homotopy, homology and cohomology. We present some theorems that are useful for calculating these groups, like the van Kampen theorem, the Mayer-Vietoris sequence, the universal coefficient theorem, Poincare ́ duality and the Künneth formula. We mention circle bundles and characteristic classes