Naturality of the homological Serre spectral sequence

If $e\in E_a=p^{-1}(B^a)$, then $p^{\mathrm{\prime }}u(e)=fp(e)\in f(B^a)\subseteq (B^{\mathrm{\prime }}{)}^a$ because $f$ is cellular. Hence $u(E_a)\subseteq E_a^{\mathrm{\prime }}$, and the singular chain map $u_{\mathrm{\#}}$ preserves every filtration piece. It also induces $u_{\ast }(F_a{H}_n(E;R))\subseteq F_a{H}_n(E^{\mathrm{\prime }};R)$ by the commutative square formed by the two inclusions of $E_a$ and $E_a^{\mathrm{\prime }}$.

Apply [F2] to the filtered chain map of Step 1.1. It gives compatible maps $u_r:E_{a,b}^r(p)\to E_{a,b}^r(p^{\mathrm{\prime }})$ commuting with $d_r$ and with the specified homology-to-next-page isomorphisms. Filtered identity maps and composites induce the identity and composite page maps, so this construction is functorial.

Now let $U$ be a homotopy over the fixed map $f$. If a singular simplex has image in $E_a$, every prism simplex occurring in $P_U$ has image in $(p^{\mathrm{\prime }}{)}^{-1}(f(B^a))\subseteq E_a^{\mathrm{\prime }}$. Thus the prism operator $P=P_U$ preserves filtration and raises chain degree by one. By [F4], $u_{1\mathrm{\#}}-u_{0\mathrm{\#}}=\mathrm{\partial }P+P\mathrm{\partial }\mathrm{.}$ It follows at once on cycles that $u_{0\ast }=u_{1\ast }$ on homology; since both maps preserve the filtration, their filtered homology maps agree.

Restriction of the square to the strict fibers over $x$ gives $u_x:p^{-1}(x)\to (p^{\mathrm{\prime }}{)}^{-1}(f(x))$. Naturality of fiber transport makes the induced homology maps a coefficient morphism ${\mathcal{H}}_b(p;R)\to f^{\ast }{\mathcal{H}}_b(p^{\mathrm{\prime }};R)$. The relative-pair, excision, and transport maps used in the cellwise $E^1$ calculation commute with the maps induced by $(u,f)$; this is the naturality clause in [F1]. Taking homology of $d_1$ therefore gives precisely the local-coefficient map of [F3] on $E^2$.

Represent a stable class at $(a,n-a)$ by an actual cycle $z\in F_a{C}_n(E;R)$. Its page image is represented by $u_{\mathrm{\#}}z$, while its associated-graded image is the class of $u_{\ast }[z]$ modulo $F_{a-1}{H}_n(E^{\mathrm{\prime }};R)$. These are the same representative under the stable identifications in [F1]. Boundaries and lower-filtration cycles map to boundaries and lower-filtration cycles, so the comparison is well defined and proves that the stable map is the associated graded of the filtered homology map from Step 1.1.

Fix $r\ge 1$ and a representative $x\in Z_{a,b}^r$, so $x\in F_a{C}_n$ and $\mathrm{\partial }x\in F_{a-r}{C}_{n-1}$. The term $P\mathrm{\partial }x$ lies in $F_{a-r}{C}_n\subseteq F_{a-1}{C}_n$, and $\mathrm{\partial }(P\mathrm{\partial }x)=(u_{1\mathrm{\#}}-u_{0\mathrm{\#}})\mathrm{\partial }x\in F_{a-r}{C}_{n-1};$ hence $P\mathrm{\partial }x\in A_{a-1,n}^{r-1}$, the first boundary summand in [F4]. Also $Px\in F_a{C}_{n+1}\subseteq F_{a+r-1}{C}_{n+1}$ and $\mathrm{\partial }Px=(u_{1\mathrm{\#}}-u_{0\mathrm{\#}})x-P\mathrm{\partial }x\in F_a{C}_n,$ so $Px\in A_{a+r-1,n+1}^{r-1}$ and $\mathrm{\partial }Px$ is in the second boundary summand. The prism identity therefore makes $(u_{1\mathrm{\#}}-u_{0\mathrm{\#}})x$ zero in $E_{a,b}^r$. Thus the two page maps agree for every $r\ge 1$, which is stronger than the promised agreement from $E^2$.

If a base, total space, or strict fiber is empty, the corresponding chains and coefficient stalks are zero; path-connected nonempty bases supply all stated fibers. The zero ring, degree zero, filtration zero, axes, identity and constant maps, constant homotopies, and degenerate simplices obey the same formulas. Step 1.1 checks both filtration endpoints, Step 3.2 checks both stable/associated-graded directions, and Step 3.3 checks both summands in the $r$-boundary denominator, including $r=1$. Every construction is applied to supplied maps, chains, or one prism, so no AC is used. The theorem has no iff assertion.