Thom class and Thom isomorphism: the AT interface

Assume AC as in The Axiom of Choice. Let $E\to B$ be an $R$-oriented numerable rank-$r$ real vector bundle over a CW complex, or over a paracompact Hausdorff base of CW type. Its AT Thom class is the unique $u_E\in H^r(D(E),S(E);R)$ of Thom class by fiberwise normalization, whose restriction to every oriented fiber disk pair is the supplied generator. In the based quotient model $D_h(E)\mathrm{/}{S}_h(E)$ of Disk, sphere, and Thom spaces of a metric vector bundle write the same class in ${\tilde{H}}^r({\mathrm{Th}}_h(E);R)\cong H^r(D(E),S(E);R)\mathrm{.}$ The isomorphism is the actual quotient-map pullback of The Thom quotient identifies relative and reduced cohomology, proved there from the exact cohomology excision and natural pair-sequence interfaces, including reduced degree0 and the empty sphere case. In rank zero it reads ${\tilde{H}}^k(B_{+};R)\cong H^k(B;R)$ in every degree, with the supplied rank-zero orientation normalization. The AT theorem Thom isomorphism for oriented vector bundles gives the isomorphism $a\mapsto {\pi }^{\ast }a⌣u_E$ from $H^k(B;R)$ onto $H^{k+r}(D(E),S(E);R)$ for all $k$, and Naturality and uniqueness of Thom classes gives uniqueness, oriented pullback naturality and integral sign reversal. For $R={\mathbb{F}}_2$ the orientation is automatic. No Thom class and no Thom isomorphism is constructed again in DT: the collapse and duality statements below consume exactly this interface. On compact smooth bases the finite-cover AT proof is available choice-free once the cover and its data are supplied; references to the general supplier retain its stated AC assumption.