If $M$ is a differentiable manifold, De Rham's theorem gives for each positive integer $k$ an isomorphism $Rh^k : H^k_{DR}(M,\mathbb R) \to H^k_{singular}(M,\mathbb R)$. On the other hand, we have a canonical map $H^k_{sing}(M,\mathbb Z) \to H^k_{singular}(M,\mathbb R)$ . Allow me to denote (this is not standard) its image by $\tilde {H}^k_{singular }(M,\mathbb Z)$. My question is : how do you recognize if, given a closed differental $k$- form $\omega$ on $M$, its image $Rh^k([\omega]) \in H^k_{singular}(M,\mathbb R) $ is actually in $\tilde {H}^k_{singular}(M,\mathbb Z)$.
I would very much appreciate a concrete answer, ideally backed up by one or more explicit calculations. Thank you for your attention.