On Poincar\'e lemma or Volterra theorem about differential forms and cohomology groups

The Poincar\'{e} lemma (or Volterra theorem) is of utmost importance both in theory and in practice. It tells us every differential form which is closed, is locally exact. In other words, on a contractible manifold all closed forms are exact. The aim of this paper is to present some direct proofs of this lemma and explore some of its numerous consequences. Some connections with Cech-De Rham-Dolbeault cohomologies, $\overline{\partial}$-Poincar\'{e} lemma or Dolbeault-Grothendieck lemma are given.