Thom class by fiberwise normalization

Let $\xi \to B$ be an $R$-oriented metric rank-$n$ bundle with orientation section $o=(o_b)$. For the inclusion of pairs $i_b:(D({\xi }_b),S({\xi }_b))⟶(D(\xi ),S(\xi )),$ a Thom class normalized by $o$ is a class $u_{\xi }\in H^n(D(\xi ),S(\xi );R)$ such that $i_b^{\ast }{u}_{\xi }=o_b$ for every $b\in B$.

Relative singular cochains make every restriction $i_b^{\ast }$ well typed. This is a normalization condition only: the definition asserts neither existence nor uniqueness, which are proved later. For $B=\mathrm{\varnothing }$ the condition is vacuous and the unique class in the zero relative group is normalized. In rank zero, $D(\xi )=B$, $S(\xi )=\mathrm{\varnothing }$, and normalization says that the degree-zero class restricts to the chosen unit on every point. The zero ring, one-point bases, identity inclusions, and empty sphere fibers cause no exception. No representative or family of classes is selected, so the definition is choice-free.