Normal Stiefel-Whitney and Pontryagin classes of a closed manifold

Assume AC. Let $M$ be a closed smooth $m$-manifold and choose a stable normal inverse $(\nu ,\phi )$ of $M$, which exists by An embedding into Euclidean space gives a rank-(n-m) stable normal inverse ($\mathrm{A}\mathrm{C}$ implies the required countable choice by AC implies DC implies countable choice, and the embedding lemma applies to closed $M$). Define the total normal Stiefel-Whitney class and the total normal Pontryagin class by $\bar{w}(M):=w(\nu )\in H^{\ast }(M;{\mathbb{F}}_2),\bar{p}(M):=p(\nu )\in H^{\ast }(M;\mathbb{Q}),$ with components ${\bar{w}}_i(M)\in H^i(M;{\mathbb{F}}_2)$ and ${\bar{p}}_i(M)\in H^{4i}(M;\mathbb{Q})$ (Stiefel–Whitney classes from the projective-bundle relation, Pontryagin classes by complexification, Singular cohomology ring). By The normal Stiefel-Whitney class is the multiplicative inverse of the tangent class and The normal Pontryagin class is the rational inverse of the tangent Pontryagin class the classes ${\bar{w}}_i(M)$ and, for connected $M$, the classes ${\bar{p}}_i(M)$ are independent of the chosen inverse, so the definition is well posed; equivalently $\bar{w}(M)=w(TM{)}^{-1}$ and $\bar{p}(M)=p(TM{)}^{-1}$ are the explicit inverses realized by any inverse bundle. For a disconnected closed $M$ the Pontryagin definition is applied componentwise. These are the classes also called the normal, dual, or (in Skopenkov's terminology) Stiefel-Whitney and Pontryagin classes of the manifold; they are the classes read by the immersion and embedding tests of this page.