Let $f_1 : X_1 \to Y_1$ and $f_2 : X_2 \to Y_2$ be two continuous maps between topological spaces. Suppose that there exists a commutative diagram of singular cohomology groups (say with coefficients in $\mathbb{C}$):

$\require{AMScd}$ \begin{CD} H^*(Y_1) @>{f_1^*}>> H^*(X_1)\\ @VVV @VVV\\ H^*(Y_2) @>{f_2^*}>> H^*(X_2). \end{CD}

Is it enough to induce a map between the Leray spectral sequences of $f_1$ and $f_2$ ?

Note that I don't suppose that the above commutative diagram comes from topological maps, and therefore I have no way to define maps at the level of complexes preserving some filtration.

My question comes from the fact that, the $E_2$-term of the Leray spectral sequence of a map $f : X \to Y$ involves the presheaf $$U \to H^*(f^{-1}(U)),$$ and therefore only cohomological information on the map $f$ is required. Therefore, if we have a relation between two maps in cohomology, we should have a relation between their respective spectral sequences.

Thanks a lot !

wouldbe enough to construct a map in the derived category between the corresponding complexes $Rf_\ast \mathbb Z$. $\endgroup$ – Dan Petersen Jul 5 '18 at 5:37