General Thom isomorphism from the relative Serre spectral sequence

Choose the metric from [F0]. Over a CW base, filter the relative cochain complex $C^{\ast }(D(\xi ),S(\xi );R)$ by inverse images of the base skeleta. In the cellwise calculation in [F2], quotient every disk-bundle chain group by its sphere-bundle subcomplex. Subdivision and fibration lifting in [F1] preserve that subcomplex, so the identical exact-couple argument has fiber term $H^q(D^n,S^{n-1};R)$ and yields the displayed relative $E_2$ page with the same convergence bounds.

By [F3], those fiber groups vanish unless $q=n$, where they form the orientation local system ${\mathcal{O}}_R(\xi )$. Hence every differential has a zero source or target, $E_2=E_{\mathrm{\infty }}$, and in total degree $k+n$ there is exactly one filtration quotient, $E_{\mathrm{\infty }}^{k,n}=H^k(B;{\mathcal{O}}_R(\xi ))$. The strong convergence in [F2] therefore identifies it with $H^{k+n}(D(\xi ),S(\xi );R)$, giving the canonical additive isomorphism; there is no additive extension to split. No orientation, Thom class, or oriented statement is used here.

Set $D_a={\pi }^{-1}(B^{(a)})$ for the metric skeleta of step 1.1 and write $S=S(\xi )$. Give the relative cochain complex of [F7], namely the cochains vanishing on every singular simplex with image in $S$, the relative disk/sphere filtration $F^a{C}^m(D,S;R)=\{\phi :\phi (\sigma )=0\text{ for every singular simplex }\sigma \text{ with image in }{D}_{a-1}\},$ and give the absolute cochain complex $C^{\ast }(B;R)$ the base skeletal filtration $F^a{C}^m(B;R)=\{\alpha :\alpha (\tau )=0\text{ for every singular simplex }\tau \text{ with image in }{B}^{(a-1)}\}\mathrm{.}$ For $\alpha \in C^p(B;R)$ and $\phi \in C^q(D,S;R)$ put $\alpha \star \phi ={\pi }^{\ast }\alpha ⌣\phi$, formed by the front/back formula of [F7]. If a simplex has image in $S$, then so does its back face, on which $\phi$ vanishes; hence $\alpha \star \phi$ vanishes on $C_{\ast }(S;R)$ and $\star$ takes values in $C^{p+q}(D,S;R)$. If $\sigma$ has image in $D_{a-1}$ and $\alpha \in F^a{C}^p(B;R)$, then every face of $\sigma$ has image in $D_{a-1}$, so the front face projects into $B^{(a-1)}$ and $\alpha$ vanishes on it; therefore ${\pi }^{\ast }(F^a{C}^p(B;R))\star C^q(D,S;R)\subseteq F^a{C}^{p+q}(D,S;R)\mathrm{.}$ By the Leibniz identity of [F7], $\delta ({\pi }^{\ast }\alpha ⌣\upsilon )={\pi }^{\ast }\delta \alpha ⌣\upsilon$ whenever $\delta \upsilon =0$. Thus every cocycle $\upsilon \in C^n(D,S;R)$ makes ${\mathrm{\Phi }}_{\upsilon }^{\bullet }(\alpha )={\pi }^{\ast }\alpha ⌣\upsilon$ a filtration-preserving cochain map $C^{\bullet }(B;R)\to C^{\bullet +n}(D,S;R)$, and the induced map ${\mathrm{\Phi }}_{\upsilon }:H^m(B;R)\to H^{m+n}(D,S;R)$ sends $[\alpha ]$ to the relative cup product of $[\alpha ]$ and $[\upsilon ]$, which is natural in the base and the bundle by the same cochain formula and [F4].

By [F8] the two filtrations of step 2.2 present these complexes with layers $E_0^{a,q}={\mathrm{gr}}^a$ and pages $E_1^{a,q}\cong H^{a+q}({\mathrm{gr}}^a)$, and a filtration-preserving cochain map induces a map of the layer complexes, hence maps ${\mathrm{\Phi }}_r:E_r^{a,q}(B;R)\to E_r^{a,q+n}(D,S;R)$ commuting with the differentials; the map induced on the abutment is ${\mathrm{\Phi }}_{\upsilon }$, whose associated graded map is the one induced by ${\mathrm{\Phi }}_1$. On the base side $E_1^{a,q}(B;R)=0$ for $q\mathrm{\ne }0$, because $B^{(a)}\mathrm{/}{B}^{(a-1)}$ is a wedge of $a$-spheres, and $E_1^{a,0}(B;R)$ is the cellular cochain group of the $a$-cells with $d_1$ the cellular coboundary, so $E_2^{a,0}(B;R)=H^a(B;R)$ and the base sequence has one row. On the relative side, the cellwise exact-couple identification of step 1.1 identifies $E_1^{a,q}(D,S)$ with the $a$-cell cochains of $B$ with values in the fiber system ${\mathcal{H}}^q(D^n,S^{n-1};R)$, with $d_1$ its cellular coboundary and $E_2^{a,q}(D,S)=H^a(B;{\mathcal{H}}^q(D^n,S^{n-1};R))$; in particular $E_2^{a,n}(D,S)=H^a(B;{\mathcal{O}}_R(\xi ))$ by [F3]. Evaluating ${\pi }^{\ast }\alpha ⌣\upsilon$ on a simplex of $D$, the front/back formula returns the value of $\alpha$ on the front face over the base together with the value of $\upsilon$ on the back face, transported to the initial vertex; with the cellular identification this is exactly the local-coefficient cup formula of [F9] for the coefficient pairing ${\mathcal{H}}^0\otimes {\mathcal{H}}^n\to {\mathcal{H}}^n$ given by the fiber cup with the top class, and its sign is $(-1{)}^{bc}=1$ because the first factor has fiber degree $b=0$, matching the page-product convention of [F2]. Therefore, in the cell identifications, ${\mathrm{\Phi }}_1$ is cellular local-coefficient cup product with the section $s_{\upsilon }$ of ${\mathcal{O}}_R(\xi )$ carried by the row-$n$ class of $\upsilon$: that class is closed, since it is the image of the cocycle class $[\upsilon ]$ under the edge map into the row of [F8], so it defines $s_{\upsilon }\in H^0(B;{\mathcal{O}}_R(\xi ))=E_2^{0,n}$ and ${\mathrm{\Phi }}_2:H^a(B;R)\to H^a(B;{\mathcal{O}}_R(\xi ))$ is cup product with $s_{\upsilon }$. When $\upsilon$ represents the class normalized by a supplied orientation, $s_{\upsilon }$ is the orientation section itself, and ${\mathrm{\Phi }}_2$ is the corresponding trivialization of the rank-one orientation local system, hence an isomorphism.

Now suppose an $R$-orientation $o$ is supplied. Its nowhere-zero section trivializes the rank-one local system ${\mathcal{O}}_R(\xi )$, so the orientation section $o\in H^0(B;{\mathcal{O}}_R(\xi ))=E_2^{0,n}$ corresponds through the collapse of step 2.1 to a class $u\in H^n(D(\xi ),S(\xi );R)$ whose fiber restriction is $o_b$ over every $b$, and [F5] makes $u$ the normalized Thom class. Choose a cocycle representative $\upsilon$ of $u$ and apply steps 2.2 and 3.1: the map $\mathrm{\Phi }(a)={\pi }^{\ast }a⌣u$ has associated graded map ${\mathrm{\Phi }}_2$, the trivialization $\alpha \mapsto \alpha ⌣o$ of ${\mathcal{O}}_R(\xi )$. The source filtration of $\mathrm{\Phi }$ is concentrated on ${\mathrm{gr}}^a{H}^a(B;R)$ and, by the one-row collapse of step 2.1, the target filtration is concentrated on ${\mathrm{gr}}^a{H}^{a+n}(D,S;R)$, so ${\mathrm{gr}}^a\mathrm{\Phi }$ is that isomorphism for every $a$ and the remaining graded pieces are maps between zero groups. Both image filtrations are exhaustive, separated and complete, with $F^{n+1}{H}^n=0$, by [F2] and [F8], so [F10] makes $\mathrm{\Phi }$ an isomorphism and its inverse preserves each filtration piece. Let $\mathrm{\Psi }$ be the composite of the trivialization $\alpha \mapsto \alpha ⌣o$ of ${\mathcal{O}}_R(\xi )$ with the collapse isomorphism of step 2.1. Both $\mathrm{\Phi }$ and $\mathrm{\Psi }$ induce the same map on associated graded objects, namely that trivialization on $E_2^{a,n}$: for $\mathrm{\Phi }$ this is step 3.1 applied to a representative $\upsilon$ of the normalized class, and for $\mathrm{\Psi }$ it is the definition of the associated graded together with the trivialization. The edge map $H^{a+n}(D(\xi ),S(\xi );R)\to E_{\mathrm{\infty }}^{a,n}$ is injective, because its kernel is $F^{a+1}{H}^{a+n}(D(\xi ),S(\xi );R)$, which vanishes: by the one-row collapse of step 2.1 every associated graded piece above degree $a$ is zero, and $F^{a+n+1}{H}^{a+n}=0$ by the finite image filtration of [F2]. Hence $\mathrm{\Phi }=\mathrm{\Psi }$, so the edge of step 2.1 is exactly $a\mapsto {\pi }^{\ast }a⌣u$, with $u$ the normalized Thom class.

Now let $B$ be paracompact Hausdorff of CW type and choose a homotopy equivalence $f:K\to B$ from a CW complex, part of the CW-type hypothesis. Apply steps 1.1–4.1 to $f^{\ast }\xi$; a supplied orientation pulls back, and naturality carries the twisted row, the filtered module action and both isomorphisms. A homotopy inverse and [F6] identify the iterated pullbacks with the original bundle; the induced radial bundle maps are homotopy inverse maps of disk/sphere pairs. Ordinary and relative homotopy invariance therefore identify the two additive twisted isomorphisms and, when an orientation is supplied, transport the normalized class and the oriented cup-product isomorphism back to $B$.

At rank zero, $D(\xi )=B$, $S(\xi )=\mathrm{\varnothing }$, and every transition unit on $H^0$ is trivial, so ${\mathcal{O}}_R(\xi )$ is the constant system $R$ and the twisted isomorphism of step 2.1 is the identity identification $H^k(B;R)\cong H^k(B;R)$. A supplied rank-zero orientation is the unit-valued class $o\in H^0(B;R)$; normalization gives $u=o$, and the oriented edge is multiplication by $o$, inverted by the componentwise inverse $o^{-1}$. This is the identity exactly for the standard unit orientation $o=1$, while the reversed integral orientation on a point gives multiplication by $-1$. In rank zero the relative complex of step 2.2 is the absolute complex of $B$, its relative filtration is the base filtration, and the module action of steps 2.2–4.1 reduces to cup product with the unit-valued class $o$, whose associated graded map is multiplication by $o$. Empty bases are handled componentwise by zero groups; disconnected bases use the componentwise construction and AC already assumed in [A1] for the cohomological comparison. The zero ring, a point base, the first and last filtration pieces, identity pullback, and both homotopy-equivalence composites are included. AC is used exactly through [F0], [F2], and [F6]; the one-row collapse, the filtered module action and the relative cell quotient add no choice.