Suppose we have a product manifold $M = M_1 \times M_2$. Let $\omega$ be a closed but inexact form on $M_1$ and $\eta$ a closed but inexact form on $M_2$. Then the claim is that $$\omega \wedge \eta$$ is inexact on the manifold $M$. I $\require{enclose}\enclose{horizontalstrike}{\mathrm{suspect}}$ know that the general statement "a product of inexact forms is inexact" is false, but I don't know how to go about proving the above claim.

The context of this question is in de Rham cohomology: I'm trying to prove Künneth's formula, and as a prelude I want to show that differential forms of the form above correspond to non-trivial elements of the cohomology groups of $M$.

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    $\begingroup$ For a counterexample to the general case, it's not too hard to find inexact $\omega,\eta$ such that $\omega \wedge \eta = 0$. $\endgroup$ – Daniel Fischer May 6 '16 at 13:06
  • $\begingroup$ Ah that's obvious now, thank you $\endgroup$ – gj255 May 6 '16 at 13:21

I think the key here is de Rham's theorem, (part of) which can be pertinently paraphrased:

A closed $k$-form $\omega$ is exact if and only if $\int_c \omega = 0$ for every closed $k$-chain $c$.

The fact that

$$ \int_{c_1 \times c_2} \omega \wedge \eta = \int_{c_1} \omega \int_{c_2} \eta $$

is then all you need.


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