It can be shown that $H_{dR}^*(X,\mathbb R)$ and $H^*(X,\mathbb R)$ (singular cohomology) are isomorphic for smooth manifolds. I was told that under that isomorphism closed differential forms whose integral over the generators of the homology of $X$ returns integer values, lets call them $H^*_{dR}(X,\mathbb Z)$, then correspond to $H^*(X, \mathbb Z)$.
But why does $H^*_{dR}(X,\mathbb Z)$ even carry a ring structure?
If $\alpha, \beta\in H^*_{dR}(X,\mathbb Z)\subset H^*_{dR}(X,\mathbb R)$, then why is $\alpha \wedge \beta \in H^*_{dR}(X,\mathbb Z)$, i.e. why is $\int_C \alpha \wedge \beta\in \mathbb Z$ for $C$ a generator of the homology group of appropriate degree?
