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. The supplied orientation identifies that system with the constant system $R$. 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, $E_{\mathrm{\infty }}^{k,n}=H^k(B;R)$. Thus there is no additive extension to split.

In total degree $n$, the unit section $1\in H^0(B;R)=E_2^{0,n}$ survives and, through convergence, defines a class $u\in H^n(D(\xi ),S(\xi );R)$. The cellwise edge restriction sends it to the chosen generator in every fiber, so [F5] makes it a normalized Thom class. By [F2] and [F4], multiplication by this permanent edge class sends $a\in E_2^{k,0}(B)$ to $a\in E_2^{k,n}$ under the orientation identification. Since both source and target have a single filtration quotient, the abutment edge is exactly $a\mapsto {\pi }^{\ast }a⌣u$ and is an isomorphism.

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 1.1–3.1 to $f^{\ast }\xi$. 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 Thom maps and transport the normalized class and isomorphism back to $B$.

For $n=0$ the only row is $q=0$, $u=1$, and the edge is the identity. 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 and relative cell quotient add no choice.