The Serre Spectral Sequence and Applications

By [F1]–[F2], the displayed $j_b$ is the restriction of $j$ to the strict fiber. Let $a:(I^n,\mathrm{\partial }{I}^n)⟶({\mathrm{hofib}}_b(p),j_b(e_0))$ be a based cube, $n\ge 1$, and write $a(x)=(e(x),{\gamma }_x)$. The adjoint $H(x,s)={\gamma }_x(s)$ is a homotopy from $p(e(x))$ to the constant map $b$. Apply [F3] to the finite pair $(I^n,\mathrm{\partial }{I}^n)$, prescribing $e(x)$ at $s=0$ and the constant lift $e_0$ on $\mathrm{\partial }{I}^n\times I$. We obtain $\tilde{H}$ with $p\tilde{H}=H$. For $0\le t\le 1$ put $a_t(x)=(\tilde{H}(x,t),s⟼{\gamma }_x(t+(1-t)s\left)\right)\mathrm{.}$ The second coordinate starts at $p\tilde{H}(x,t)={\gamma }_x(t)$ and ends at $b$, so $a_t$ stays in the homotopy fiber. It is continuous by [F4], is based for every $t$, begins at $a$, and ends at $j_b(x\mapsto \tilde{H}(x,1))$. Thus $(j_b{)}_{\ast }$ is surjective on every positive homotopy group.

Let $(T,S)$ have every component of $T$ meeting $S$ and all positive relative homotopy sets trivial. For a finite relative $G$-cycle $c=\sum g_{\sigma }\sigma$, attach one simplex for every distinct iterated face of its finite support. By [F7] this produces a finite CW pair $(K,L)$, a map $v:(K,L)\to (T,S)$, and a relative cycle $\tilde{c}$ with $v_{\mathrm{\#}}\tilde{c}=c$. Compress the finitely many cells of $K\setminus L$ in increasing dimension: paths move zero-cells into $S$, [F6] compresses each later characteristic disk once its boundary lies in $S$, and the cofibration clause in [F6] extends each finite-stage homotopy. The endpoint $w$ maps $K$ into $S$. The [F8] prism identity says $w_{\mathrm{\#}}\tilde{c}-v_{\mathrm{\#}}\tilde{c}=\mathrm{\partial }P\tilde{c}+P\mathrm{\partial }\tilde{c};$ all terms except $c$ vanish in the relative complex. Its degree-zero clause handles $k=0$. Hence $H_k(T,S;G)=0$ for every $k$, using only finitely many choices for this one chain.

Suppose a based cube $c:(I^n,\mathrm{\partial }{I}^n)\to (F_b,e_0)$ becomes null after applying $j_b$. Represent the nullhomotopy by a map from $I^n\times I$ to the homotopy fiber, constant on $(\mathrm{\partial }{I}^n\times I)\cup (I^n\times \{1\})$ and equal to $j_{b}c$ on $I^n\times \{0\}$. Repeat step 1.1 with this finite cube as parameter space, but prescribe the evident strict-fiber lift on that whole boundary subcomplex. The straightening is relative there. At its endpoint it is a homotopy in $F_b$ from $c$ to the constant cube, so $(j_b{)}_{\ast }$ is injective. The same argument with parameter space $I$ and its two endpoints shows that a path between $j_b(e_0)$ and $j_b(e_1)$ straightens, relative to its endpoints, to a path from $e_0$ to $e_1$ in $F_b$. With a point as parameter, step 1.1 shows every homotopy-fiber component meets $j_b(F_b)$. Hence $j_b$ is also bijective on path components and is a weak homotopy equivalence.

For any weak equivalence $f:X\to Y$, [F5] gives the relative-homotopy hypotheses of step 1.2 for its mapping-cylinder pair $(M_f,jX)$. Hence $H_{\ast }(M_f,jX;G)=0$, and exactness in [F8] makes $j_{\ast }:H_{\ast }(X;G)\to H_{\ast }(M_f;G)$ an isomorphism. The standard quotient formulas retract $M_f$ onto $Y$ and deform the identity to that retraction by [F9]; [F8] makes the induced maps inverse on homology. Thus every weak equivalence induces $G$-homology isomorphisms without AC. Apply this to $j_b$ from step 2.1.

A strictly commuting square $vp=p^{\mathrm{\prime }}u$ induces the pointwise mapping-path map $U:E_p\to E_{p^{\mathrm{\prime }}},U(e,\gamma )=(u(e),v\gamma ),$ and the formulas give the literal equality $Uj=j^{\mathrm{\prime }}u$. Restricting to fibers therefore makes the $(j_b{)}_{\ast }$ natural before and after homology. For a base path $\gamma :b\to c$, the maps $U_c{T}_{\gamma }^{p_p}$ and $T_{v\gamma }^{p_{p^{\mathrm{\prime }}}}{U}_b$ are two endpoint maps obtained by lifting the same path $v\gamma$ with the same initial fiber map. Lifting-function independence in [F10] makes them homotopic in the target fiber. The prism identity [F8], tensored with the arbitrary abelian group $G$, therefore makes their induced $G$-homology maps equal. Define ${\tau }_{\gamma }=(j_c{)}_{\ast }^{-1}{H}_q(T_{\gamma }^{p_p};G)(j_b{)}_{\ast }\mathrm{.}$ Cancellation shows that $(j_b{)}_{\ast }$ commutes with transport, and the just-proved square together with $Uj=j^{\mathrm{\prime }}u$ proves naturality for $(u,v)$.

If $F_b=\mathrm{\varnothing }$ but $(e,\gamma )$ belonged to the homotopy fiber, path lifting for the single path $\gamma$ in [F3] would end at a point of $F_b$, a contradiction; hence both fibers are empty. A one-point fiber, constant paths, both path endpoints, $n=1$, $q=0$, $G=0$, and $G=\mathbb{Z}$ were included above. Every lift concerns one finite CW problem, and step 1.2 handles one finite chain at a time; no family of lifts, bases, or homology representatives is selected. Thus no form of AC is used.

Write $J_b:H_q(F_b;R)\to H_q(F_b^{\mathrm{\prime }};R)$ for the homology comparison. By [F1], ${\tau }_{\gamma }=J_c^{-1}{H}_q(T_{\gamma })J_b\mathrm{.}$ For the constant path, [F2]–[F3] give $H_q(T_{c_b})=\mathrm{id}$, so ${\tau }_{c_b}=\mathrm{id}$. Endpoint-fixed homotopic paths have equal middle maps. For composable $\gamma ,\eta$, the middle identity $H_q(T_{\gamma \ast \eta })=H_q(T_{\eta })H_q(T_{\gamma })$ makes the adjacent $J_c{J}_c^{-1}$ cancel and gives ${\tau }_{\gamma \ast \eta }={\tau }_{\eta }{\tau }_{\gamma }$. Thus [F4] applies, with no choice.

Under [F5], write $K_b=(j_b{)}^{\ast }:H^q(F_b^{\mathrm{\prime }};R)\to H^q(F_b;R)$. The formula in [F1] is $s_{\gamma }=K_c{H}^q(T_{\bar{\gamma }})K_b^{-1}\mathrm{.}$ Since $\overline{\gamma \ast \eta }=\bar{\eta }\ast \bar{\gamma }$, [F2]–[F3] give $T_{\overline{\gamma \ast \eta }}\simeq T_{\bar{\gamma }}{T}_{\bar{\eta }}$. Contravariance gives $H^q(T_{\overline{\gamma \ast \eta }})=H^q(T_{\bar{\eta }})H^q(T_{\bar{\gamma }})$. Cancelling $K_c{K}_c^{-1}$ now yields $s_{\gamma \ast \eta }=s_{\eta }{s}_{\gamma }$. Constants and endpoint-fixed homotopies are handled identically, so [F4] gives the cohomology local system.

If a component has empty fibers, all its stalks and maps are zero; a point fiber, the zero ring, and $q=0$ obey the same conjugation formulas. Reversal interchanges the two endpoints exactly as typed in step 1.2, and reversing twice returns the original class. The homology proof uses supplied maps and finite algebra only. The cohomology proof uses AC precisely through [F5] and does not spend it again. These checks establish every claimed functor law.

We first record the finite lifting calculation used repeatedly. Let $q:Y\to X$ be a Serre fibration and let $A\subseteq X$ be a specified strong deformation retract. Then $Y_A=q^{-1}(A)\hookrightarrow Y$ is a weak homotopy equivalence. Indeed, for a based cube in $Y$, lift the deformation of its projected cube, keeping its boundary at the chosen point in $Y_A$; the endpoint lies in $Y_A$. This proves surjectivity on positive homotopy groups. Apply the same relative lift to a cubical nullhomotopy, fixing its whole boundary, to prove injectivity. With a point and an interval as the finite parameter spaces, the same argument gives respectively surjectivity and injectivity on components. By [F3], the inclusion therefore induces an isomorphism on homology with the underlying additive group of $R$ as coefficients.

Pull $p$ back along ${\varphi }_e$ and denote by ${\tilde{D}}_e^p$ and ${\tilde{S}}_e^{p-1}$ the inverse images of the disk and its boundary. Fix, once for the standard disk, the usual positive and negative closed hemispheres, orient the positive hemisphere by the boundary-first convention, and iterate this convention down to a point $x_e\in S^{p-1}$. For the first stage put $X={\tilde{D}}_e^p,A={\tilde{S}}_e^{p-1},N={\tilde{D}}_{e,-}^{p-1}\mathrm{.}$ The disk strongly deformation retracts onto its negative hemisphere, so Step 1.1 gives $H_{\ast }(X,N;R)=0$. The degreewise short exact sequence $0\to C_{\ast }(A,N;R)\to C_{\ast }(X,N;R)\to C_{\ast }(X,A;R)\to 0$ and its elementary cycle-boundary exact sequence therefore make the connecting map $H_n(X,A;R)\stackrel{\cong }{\to }{H}_{n-1}(A,N;R)$ an isomorphism. Excision of a slightly shrunken negative hemisphere identifies the target with the relative homology over the positive hemisphere and its equator. Iterating $p$ times gives ${ϵ}_e:H_{p+q}({\tilde{D}}_e^p,{\tilde{S}}_e^{p-1};R)\stackrel{\cong }{\to }{H}_q(F_{{\varphi }_e(x_e)};R)\mathrm{.}$ The connecting maps use the displayed boundary convention, so reversing the orientation of $e$ multiplies ${ϵ}_e$ by $-1$. For $p=0$ there are no connecting maps and ${ϵ}_e$ is the identity on the fiber.

We now separate the cells. In each characteristic disk use the same radial coordinate. The union of $B^{p-1}$ with the outer radial collars of all $p$-cells is open by the weak topology and strongly deformation retracts onto $B^{p-1}$ by one cellwise radial formula. Step 1.1 applied to its inverse image shows that enlarging $E_{p-1}$ to this collar does not change the relative homology of $E_p$. Excise a smaller closed outer collar. What remains is the disjoint union, over the open $p$-cells, of pulled-back concentric disk pairs. Radial rescaling and another application of Step 1.1 identify each with $({\tilde{D}}_e^p,{\tilde{S}}_e^{p-1})$. Excision and arbitrary additivity therefore give $H_{p+q}(E_p,E_{p-1};R)\cong {⨁}_{e\in {\mathcal{E}}_p}{H}_{p+q}({\tilde{D}}_e^p,{\tilde{S}}_e^{p-1};R)\mathrm{.}$ Every singular chain has finite support, so the target is a direct sum even when there are infinitely many $p$-cells.

Let ${\rho }_e$ be the straight segment in $D^p$ from $x_e$ to $c_e$. Transport along ${\varphi }_e{\rho }_e$ followed by ${ϵ}_e$ identifies the last group in Step 2.1 with $H_q(F_{b_e};R)$. All hemisphere retractions, thickenings, and paths were fixed in the one standard disk. Naturality of connecting maps, excision, and transport shows that the result respects the characteristic map and its orientation; no family of unspecified lifts or paths is chosen.

Compose Step 2.2 with the maps of Step 3.1. By [F6], the resulting direct sum of the stalks $H_q(F_{b_e};R)$, indexed by the oriented $p$-cells, is exactly $C_p^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(B;{\mathcal{H}}_q(p;R))$. By [F1] the source is $E_{p,q}^1$. This proves both displayed homology identifications.

Finally, $R\{o_e\}[p]\otimes C_{\ast }(F_{b_e};R)$ has degree-$n$ term $C_{n-p}(F_{b_e};R)$ and differential $(-1{)}^p\mathrm{\partial }$, exactly the declared signed shift. The identity on the underlying modules is therefore a chain isomorphism. Its homology in total degree $p+q$ is $H_q(F_{b_e};R)$, the $e$-summand in Step 4.1. This verifies the stated chain-model claim without upgrading the excision map beyond what [F4] proves.

If $p=0$, the relative pair is the disjoint union of the fibers over the zero-cells and Step 2.1 has no suspension stage. If $q<0$, all fiber chain groups and both sides vanish; $q=0$ is included in the component argument of Step 1.1. An empty fiber contributes zero, as does the zero ring; no $p$-cells give the empty direct sum. One cell, $p=1$, constant or degenerate singular simplices, both collar endpoints, and either cell orientation are retained by the same relative complexes and signs. Each lift tests one finite cube, and every chain has finite support, so the construction uses no form of AC.

Put $n=p+q$. In the exact couple of the filtration, the first differential is $d_1=jk$: the connector $k:H_n(E_p,E_{p-1};R)⟶H_{n-1}(E_{p-1};R)$ is followed by the quotient map $j:H_{n-1}(E_{p-1};R)⟶H_{n-1}(E_{p-1},E_{p-2};R)\mathrm{.}$ By [F2] this exact-couple differential is the filtered-complex $d_1$ with no extra sign. Concretely, [F3] sends a relative cycle represented by $c$ to the class represented by $\mathrm{\partial }c$ in the preceding relative layer.

Fix oriented cells $e^p$ and $f^{p-1}$. Project the source and target of Step 1.1 to their $e$- and $f$-summands using the cell-excision maps of [F1]. Naturality of pair connectors and excision gives a commutative square whose upper map is the boundary contribution of the attaching map ${\varphi }_e{\mathrm{\mid }}_{S^{p-1}}$ to $f$, and whose vertical maps are the iterated hemisphere isomorphisms defining $\mathrm{\Psi }$. Thus the $(f,e)$ component of $\mathrm{\Psi }{d}_1{\mathrm{\Psi }}^{-1}$ is determined entirely by the oriented attaching incidence together with fiber transport along the path from $b_e$ through the attaching sphere to $b_f$.

To calculate that component, temporarily supply lifts of these two cells to a universal cover of their base component. Write $\mathrm{\partial }\tilde{e}={\sum }_f\tilde{f}{r}_{fe}$ with $r_{fe}$ the finite signed sum of deck transformations determined by the lifted attaching incidences. Finiteness follows from [F6]. For one orientation-preserving degree-one disk contribution, naturality of every connector and excision map in [F1] makes the fiber map exactly the transport along its incidence path. For an orientation-reversing contribution, reduce by naturality to a reflection of the one-dimensional disk. Its pair sequence has boundary kernel $\ker (H_q(F_{-};R)\oplus H_q(F_{+};R)\to H_q({\tilde{D}}^1;R))=\{(x,-x):x\in H_q(F_{-};R)\},$ after transport identifies the two endpoint groups. Interchanging the endpoints therefore multiplies the suspension isomorphism by $-1$. Additivity now makes a degree-$m$ incidence contribute $m$ times its transport, and summing the finitely many terms gives exactly the action of $r_{fe}$ fixed in [F4].

It follows for every supplied lift basis that $(\mathrm{\Psi }{d}_1{\mathrm{\Psi }}^{-1}{)}_f(x_e)={\sum }_e{r}_{fe}{x}_e,$ where the action notation is precisely the balanced tensor convention of [F4], hence precisely its signed transport formula. Changing lifts conjugates the incidence matrices and coefficient coordinates by the same diagonal deck transformations; changing orientations conjugates them by the same diagonal signs. Therefore [F4] descends this equality to the intrinsic cellular local complex, without any global choice of lifts. By [F5], the right side is ${\mathrm{\partial }}_{{\mathcal{H}}_q}^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}$. This proves the asserted intertwining identity.

The next page is the homology of $(E_{\ast ,q}^1,d_1)$. Step 4.1 identifies this chain complex with $C_{\ast }^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(B;{\mathcal{H}}_q(p;R))$, and [F5] identifies its homology with $H_{\ast }(B;{\mathcal{H}}_q(p;R))$. Hence $E_{p,q}^2\cong H_p(B;{\mathcal{H}}_q(p;R))$.

For $p=0$ both differentials out of the left edge are zero. For $p=1$ the reflection calculation in Step 3.1 is the whole sign calculation, and for $q=0$ the same formula acts on fiber components. An empty or zero stalk, the zero ring, no incident cells, and a zero incidence coefficient contribute zero. One incidence of either sign is explicit in Step 3.1; multiple and cancelling incidences use finite additivity. Degenerate singular simplices remain in the relative chain representative of Step 1.1. Both endpoints of every incidence path occur in the typed transport $b_e\to b_f$. A cellular chain has finite support, and [F6] makes each relevant attaching support finite, so only finite supplied lift data are ever used; the intrinsic conclusion uses no AC. The statement is not an iff.

Apply the filtered-complex constructions in [F2] to $C_{\ast }(E;R)$ with $F_a{C}_{\ast }=C_{\ast }(E_a;R)$. The resulting differentials have the displayed bidegree and the next page is their homology. By [F1], the initial page vanishes unless $a,b\ge 0$, its first differential is the cellular local boundary, and its next page is $H_a(B;{\mathcal{H}}_b(p;R))$. Since every later term is a subquotient of an earlier one, the first-quadrant vanishing persists.

It remains to prove the finite upper endpoint of the target filtration, which does not follow from chain-level degreewise finiteness. Let $[z]\in H_n(E;R)$ with $n\ge 0$, and write $z$ as a finite cycle. Attach one geometric simplex for each distinct iterated face in its finite support, identifying equally labelled faces. By [F4] this gives a finite CW complex $K$, a map $v:K\to E$, and a cellular $n$-cycle $\tilde{z}$ with $v_{\mathrm{\#}}\tilde{z}=z$. Apply the finite-source clause of [F5] to $pv:K\to B$. It gives a homotopy to a cellular map $g$, so $g(K^j)\subseteq B^j$. Lift this homotopy through $p$, starting at $v$; the endpoint $v^{\mathrm{\prime }}$ satisfies $p{v}^{\mathrm{\prime }}=g$. Every simplex of $\tilde{z}$ has dimension at most $n$, hence $v_{\mathrm{\#}}^{\mathrm{\prime }}\tilde{z}\in C_n(E_n;R)$. The prism identity in [F5] makes it homologous to $z$. Thus every class lies in $F_n{H}_n$, proving $F_n{H}_n=H_n$. Since $F_{-1}C=0$, also $F_{-1}{H}_n=0$. Negative-degree homology is zero.

Suppose now that $B$ is simply connected. For paths $\alpha ,\beta :x\to y$, the loop $\alpha \ast \bar{\beta }$ at $x$ represents the identity by [F6]. Concatenating its endpoint-fixed nullhomotopy with $\beta$ and applying the associativity, inverse, and identity path homotopies gives $[\alpha ]=[\beta ]$ in ${\mathrm{\Pi }}_1(B)$. Thus there is a unique path class between any two points. Functorial fiber transport is consequently path-independent. After fixing one stalk and its unique transport isomorphisms to the other stalks, ${\mathcal{H}}_b$ is the corresponding constant system, giving the untwisted $E^2$ formula.

Here “natural” has the following precise meaning. Given a commutative square of Serre fibrations with total-space map $u:E\to E^{\mathrm{\prime }}$ over a cellular map $f:B\to B^{\mathrm{\prime }}$, [F7] gives $u(E_a)\subseteq E_a^{\mathrm{\prime }}$ and hence a filtered singular-chain map. Its functorial page maps commute with every $d_r$. On $E^1$, the cellwise relative maps commute with the pair connectors, excision maps, and fiber transports used in [F1]; after taking $d_1$-homology this is exactly the $E^2$ map induced by $f$ and the fiber-homology coefficient morphism in [F7]. Thus the displayed spectral sequence is natural for these squares.

Fix a position $(a,b)$ in the first quadrant. The outgoing $d_r$ is zero for $r>a$, because its target has negative first coordinate. The incoming $d_r$ is zero for $r>b+1$, because its source has second coordinate $b-r+1<0$. Hence both incident differentials vanish once $r>\max (a,b+1)$, and the next-page isomorphism in [F2] makes this position stationary. The same bounds show two-sided regularity at every position.

We identify that stationary object. Put $n=a+b$ and $Z_a=F_a{C}_n\cap \ker \mathrm{\partial }$. From the numerator formula in [F2], for $r>a$ the $r$-cycles are exactly $Z_a$, since $F_{a-r}{C}_{n-1}=0$. The first denominator summand is then $F_{a-1}{C}_n\cap \ker \mathrm{\partial }$. Every element of the second summand is an actual boundary lying in $F_a{C}_n$. Conversely, let $x=\mathrm{\partial }y\in F_a{C}_n$. The filtration is exhaustive for this one finite chain, so $y\in F_s{C}_{n+1}$ for some integer $s$. For any $r$ with $a+r-1\ge s$, one has $y\in F_{a+r-1}{C}_{n+1}\cap {\mathrm{\partial }}^{-1}(F_a{C}_n),$ so $x$ belongs to the $r$-boundary denominator. The coherent class of $x$ therefore vanishes on some later page. Once Step 2.1 has reached its stationary range, every transition is an isomorphism, so that class was already zero at the first stationary page. Thus $E_{a,b}^{\mathrm{\infty }}\cong \frac{F_a{C}_n\cap \ker \mathrm{\partial }}{(F_{a-1}{C}_n\cap \ker \mathrm{\partial })+(F_a{C}_n\cap \mathrm{im}\mathrm{\partial })}\mathrm{.}$ This argument chooses a filtration bound only for the displayed $y$; it does not require a uniform bound for all $(n+1)$-chains.

The quotient in Step 3.1 is naturally $F_a{H}_n(E;R)\mathrm{/}{F}_{a-1}{H}_n(E;R)$. Send an actual filtered cycle to its homology class modulo the preceding image filtration. This is surjective by [F3]. If $z\in F_a{C}_n$ maps into $F_{a-1}{H}_n$, there is an actual cycle $w\in F_{a-1}{C}_n$ with $z-w=\mathrm{\partial }y$; hence $z$ lies in the displayed denominator. The converse is immediate. This proves both injectivity and surjectivity and gives the weak-convergence identifications required in [F3].

Step 1.2 makes the target filtration finite, hence exhaustive, separated, and complete by [F3]. Step 2.1 gives two-sided regularity, and Step 4.1 gives the specified weak-convergence isomorphisms. These are exactly the conditions for strong convergence in [F3]. No chain-level assertion $F_n{C}_n=C_n$ was used.

If $B=\mathrm{\varnothing }$, then $E=\mathrm{\varnothing }$ and every page and target is zero; otherwise path connectedness supplies paths used only one at a time. If $E$ or a fiber is empty, the corresponding chains and stalks are zero. The zero ring, $n=0$, $a=0$, $b=0$, the axes, a one-cell finite face complex, and constant or degenerate singular simplices are included in Steps 1.1–4.1. Step 2.1 checks both incoming and outgoing stationary bounds, and Step 4.1 checks both kernel and image directions. Identity and composite naturality follow from [F7]. Every filtration bound, cellular approximation, and lift is attached to one finite representative; no AC or simultaneous choice is used.

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.