I'm wondering about a cross product for spectral sequences.  I've got an idea, and I wonder if it is written up anywhere, or if it even holds water.

Let's start with three spectral sequences, $E, F$ and $G$.  Assume that 
$G_1^{*,*} \cong E_1^{*,*}\otimes F_1^{*,*}$  as chain complexes.
Then the ordinary K\"unneth theorem gives us a map 
`\[
\times_2: E_2^{*,*} \otimes F_2^{*,*} \to G_2^{*,*} .
\]`


Now $E_2^{*,*} \otimes F_2^{*,*}$ has a differential -- the standard one for the 
tensor product of chain complexes, and I guess I have to hope that $\times_2$
is a chain map.   Given this, we apply K\"unneth again, and get 
`\[
\times_3: E_3^{*,*} \otimes F_3^{*,*} \to
H^{*,*}( E_2^{*,*} \otimes F_2^{*,*}) \to 
 G_3^{*,*} .
\]`
Repeating the process leads to cross products
`\[
\times_r :E_r^{*,*} \otimes F_r^{*,*} \to 
 G_r^{*,*} .
\]`
and presumably converging to the appropriate cross product at the end.