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.