Let's focus on the cohomology of a product manifold

Given two manifolds $M$ and $N$ of dimensions $m$ and $n$, the product $M\times N$ has a structure of manifold of dimension $m+n$. The product is projected over each factor, and much like in the Poincaré Lemma, forms in each factor induce forms in the product

The exterior product in forms allows us to define

and this exterior product works well with cohomology, so we also have

In fact, to obtain the maximum information about the cohomology of $M\times N$ of order $k$, we define the following map:

and we hope that it is at least surjective...