The Converse of Poincare Lemma The Poincare lemma states that contractibility implies triviality of the de-Rham cohomology group. Does the converse still true? If the de-Rham cohomology is trivial, then the manifold is contractible?

No. For instance, $\mathbb{RP}^2$ has trivial de Rham cohomology, but it is not contractible (its fundamental group is nontrivial, for instance).