The characteristic-class construction is cited, not rebuilt

Assume AC. Remark. This page computes with the characteristic classes but does not construct them. The exact interfaces are: the Stiefel-Whitney classes $w_i$ and the conventions $w_0=1$, $w_i=0$ for $i>\mathrm{rank}$ of Stiefel–Whitney classes from the projective-bundle relation together with the Whitney product and naturality of Whitney sum formula for Stiefel–Whitney classes and Naturality of Stiefel–Whitney classes; the Euler class as the zero-section pullback of the Thom class of Euler class by zero-section pullback of the Thom class; and the Pontryagin classes $p_i(E)=(-1{)}^i{c}_{2i}(E_{\mathbb{C}})$ of Pontryagin classes by complexification with the Whitney product away from two of Pontryagin Whitney product away from two and the odd-Chern two-torsion fact of Odd Chern classes of a complexified real bundle are two-torsion. Authoring must substitute these exact item ids and the two-torsion qualification before performing the calculations of The normal Pontryagin class is the rational inverse of the tangent Pontryagin class and High normal Pontryagin classes obstruct low-codimension immersions; no construction, normalization or product formula is minted here.