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I have a concern regarding maps of serre fibrations and the induced map into the respective spectral sequence. The situation is the following.

Consider two serre fibrations $F \rightarrow E \rightarrow B$ and $F' \rightarrow E' \rightarrow B'$. Where $B$ and $B'$ are simply connected to make things easier.

Suppose that we have maps $f: E' \rightarrow E$, and $g: B' \rightarrow B$ that commute with the respective projections. (I couldn't figure out how to make a commutative diagram here) and let $h: F' \rightarrow F$ the map induced in the fibers.

If we consider cohomology with coefficients in a field for instance, the $E_2$ page decomposes as $$E_2 \cong H^*(B) \otimes H^*(F)$$

My first question is, do we have an induced map between the spectral sequences given by

$$g^* \otimes h^* : E_2 \rightarrow E'_2$$

if this is true, is also true that the map induced on the infinite term $E_\infty \rightarrow E'_\infty$ coincides with $f^*: H^*(E)\rightarrow H^*(E')$.

I appreciate any comment.

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You do get an induced map between the $E_r$ pages for $2 \leq r \leq \infty$, but a priori the map on the $E_\infty$ page coincides with $f^*$ only up to filtration.

Here's a (somewhat silly) example. Pick your favorite nontrivial space $X$, and consider the map of fibrations $$\begin{array}{ccc} * & \rightarrow & X \\ \downarrow & & \downarrow \\ X & \xrightarrow{=} & X \\ \downarrow & & \downarrow \\ X & \rightarrow & * \end{array}$$

The maps of fibers and base spaces are essentially trivial, so on each $E_r$ page we have an isomorphism of $E_r^{0,0}$ but otherwise the induced map is zero. But the actual map between total spaces is the identity. This is the maximally worst case: almost all of the information contained in $f^*$ is hidden in the filtration, and the spectral sequence exposes almost none of it.

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