Wedge product of closed form each with integral periods has integral period?

Question

Suppose $\alpha$ and $\beta$ are closed forms on $M$ which have integral periods, i.e. for all $[A] \in H_*(M, \mathbb{Z})$ represented by a smooth cycle $A$, we have $\int_A \alpha \in \mathbb{Z}$, and similarly for $\beta$. Does $\alpha \wedge \beta$ have integral period?

Answer 1

Having integral periods is equivalent to lying in the image of $H^*(M,\Bbb Z)$ under the coefficient change and de Rham maps $H^*(M,\Bbb Z) \to H_{dR}^*(M)$. These maps are homomorphisms, so yes.