Here is something I lack intuition for. Let $M$ be a smooth manifold. Suppose $\alpha$ is a closed $p$-form and $\beta$ is a closed $q$-form on $M$. Why is it that, if $\alpha$ and $\beta$ are "integral" (meaning their integrals over appropriate-dimensional cycles are integers), then $\alpha \wedge \beta$ is also integral?
Of course, one explanation is that we have a ring homomorphism $H^*(M,\mathbb{Z}) \to H^*_\text{dR}(M)$ and the integral forms are just the closed forms representing classes in the image of this homomorphism. That's exactly what a user writes here: https://math.stackexchange.com/a/1683352/6608. But let's not appeal to this. I don't understand the cup product so well anyway.
Can someone pick apart the integral $\int_X \alpha \wedge \beta$, where $X\subseteq M$ is a closed $(p+q)$-dimensional submanifold (or simplicial complex or what have you) to give a more direct explanation of the integrality?
