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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

$$ \require{extpfeil} \Newextarrow{\xleftarrow}{25,25}{0x2190} \Newextarrow{\xrightarrow}{25,25}{0x2192} M\xleftarrow{\pi_1}M\times N \xrightarrow{\pi_2}N\qquad\qquad \Omega^*(M) \xrightarrow{\pi_1^*}\Omega^*(M\times N)\xleftarrow{\pi_2^*}\Omega^*(N) $$

The exterior product in forms allows us to define

$$ \require{extpfeil} \Newextarrow{\xrightarrow}{25,25}{0x2192} \Newextarrow{\xrightmapsto}{25,25}{0x21A6} \Omega^a(M)\otimes\Omega^b(N)\xrightarrow{}\Omega^{a+b}(M\times N )\qquad\qquad\alpha\otimes\beta\xrightmapsto{}\pi_1^*(\alpha)\wedge\pi_2^*(\beta) $$

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

$$ \require{extpfeil} \Newextarrow{\xrightarrow}{25,25}{0x2192} \Newextarrow{\xrightmapsto}{25,25}{0x21A6} H^a(M)\otimes H^b(N)\xrightarrow{}H^{a+b}(M\times N)\qquad\qquad [\alpha]\otimes[\beta]\xrightmapsto{}[\pi_1^*(\alpha)\wedge\pi_2^*(\beta)] $$

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

$$ \require{extpfeil} \Newextarrow{\xrightarrow}{25,25}{0x2192} \Newextarrow{\xrightmapsto}{25,25}{0x21A6} \bigoplus_{a+b=k} H^a(M)\otimes H^b(N)\xrightarrow{}H^k(M\times N)\qquad\qquad \sum[\alpha]\otimes[\beta]\xrightmapsto{}\sum[\pi_1^*(\alpha)\wedge\pi_2^*(\beta)] $$

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