My question is:



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> If $p\colon X \to B$, $q\colon Y \to B$ are proper submersions, is there a characterization of $H\_\*(X \times\_B Y)$ in terms of $H\_\*(X)$, $H\_\*(Y)$, $H\_\*(B)$ that is simpler than the Eilenberg-Moore spectral sequence?
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Ideally, what I would like is a chain complex $A$ built from $C\_\*(X)$, $C\_\*(Y)$, $C\_\*(B)$ with a map $A \to C\_\*(X \times\_B Y)$ which descends to an isomorphism.
But I'm willing to compromise -- it's OK if this map doesn't descend to an isomorphism.
Also, I don't care too much about what version of homology it is, and you can change the hypotheses on $p, q$ if you want (e.g. require them to be surjective).



Eilenberg-Zilber-type theorem for good fiber products?
There are fibrations
\begin{align\*}
 F \to X & \to B \\
 G \to Y & \to B \\
 F\times G \to X\times\_B Y & \to B \\
 F\times G \to X\times Y & \to B\times B \\
 F \to X\times\_BY &\to Y \\
 G \to X\times\_BY &\to X
\end{align\*}
and various comparison maps between them. You are probably better off reasoning indirectly from the associated Serre spectral sequences rather than using anything like the Eilenberg-Moore spectral sequence. Under mild assumptions the Serre spectral sequences will be finite-dimensional and have Poincaré duality on every page, and there will only be finitely many differentials. The Eilenberg-Moore spectral sequence will typically have an infinitely generated $E\_2$ page and infinitely many differentials, with no visible sign of Poincaré duality. It is hard to see how you could avoid similar issues unless you can come up with a version of Tor groups that takes account of Poincaré duality, which seems hard.


