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Let $p\colon E\to B$ and $p'\colon E'\to B'$ be two fibrations. Assume for simplicity that $B,B'$ are simply connected. Let we have morphism of these fibrations, i.e. two continuous maps $$f\colon E\to E', g\colon B\to B'$$ such that $p'\circ f=g\circ p$. Let $F$ be the fiber of $p$ over some fixed point $b\in B$, and $F'$ be the fiber of $p'$ over $g(b)$.

Question. Does there exist a "natural" morphism of the Leray spectral sequences for the homology, which, in particular, satisfies

1) on the second terms this moprhism $H_*(F)\otimes H_*(B)\to H_*(F')\otimes H_*(B')$ coincides with $f_*\otimes g_*$.

2) $f_*\colon H_*(E)\to H_*(E')$ preserves the Leray filtrations and the associated graded map coincides with the required morphism on the $\infty$-terms.

A reference would be helpful.

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    $\begingroup$ Yes, this is true. My personal advice would be to just reread your favorite construction of the Serre spectral sequence and check that it is functorial in this sense (most proofs would give this). If I were reading a paper, I would be perfectly happy to see a reference to a cleanly presented proof of the Serre spectral sequence along with an assertion that the proof given there is functorial (though others might disagree). $\endgroup$
    – John Pardon
    Commented Mar 10, 2017 at 17:36
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    $\begingroup$ I should probably also point out that $E^2$-term is actually $H_\ast(B,H_\ast(F))$, which isn't the same as $H_\ast(B)\otimes H_\ast(F)$ in general (they often differ if $H_\ast(F)$ has torsion or if it forms a nontrivial local system over $B$). $\endgroup$
    – John Pardon
    Commented Mar 10, 2017 at 17:41
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    $\begingroup$ If you just need a reference, try Theorem 6.1 and 6.2 in Ch. 1 of Larry Smith's "Lectures on the Eilenberg-Moore spectral sequence" (LNM 134). I'm not claiming it's the original reference, but the theorem statements are exactly in the form you want, which is nice. $\endgroup$
    – Dylan Wilson
    Commented Mar 10, 2017 at 17:57

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