Characteristic class as a universal natural bundle class

Every universal class gives a characteristic class. Let $u$ be a class of ${\gamma }_n$ on $BG$, where $BG$ is one of the three Grassmannian models. For a numerable bundle $E\to B$ over a base in both classes choose a classifying map $c_E$, that is, $c_E^{\ast }{\gamma }_n\cong E$; such a map exists by [F1] or [F2], since $B$ lies in the classification scope. Set $c_u(E)=c_E^{\ast }u$. If $c_E^{\mathrm{\prime }}$ is a second classifying map of the same bundle, both maps represent the same element of $[B,BG]$, because the bijection of [F1] or [F2] is defined on homotopy classes and has the same value on them; hence $c_E\simeq c_E^{\mathrm{\prime }}$ and [F3] gives $c_E^{\ast }u=c_E^{\mathrm{\prime }\ast }u$. Isomorphic bundles have the same classifying homotopy class by the same injectivity, so $c_u$ is well defined on isomorphism classes. For a map $f:B^{\mathrm{\prime }}\to B$ of such bases, $f^{\ast }E$ is classified by $c_{E}f$, since [F7] gives $(c_{E}f{)}^{\ast }{\gamma }_n\cong f^{\ast }{c}_E^{\ast }{\gamma }_n\cong f^{\ast }E$; therefore [F4] gives $c_u(f^{\ast }E)=(c_{E}f{)}^{\ast }u=f^{\ast }{c}_E^{\ast }u=f^{\ast }{c}_u(E)\mathrm{.}$ This clause uses AC exactly through [F1] and [F2].

Every characteristic class comes from a universal class. Let $c$ satisfy the two clauses of the Definition. For $BO(n)$ and $BU(n)$, [F5] supplies CW structures, so these models are paracompact Hausdorff CGWH and admissible. For $BSO(n)$ with $n\ge 1$, its double cover of $BO(n)$ has discrete two-point compact CW fiber. A subordinate partition on its trivializing cover makes it numerable by [F6]; the compact-fiber theorem in [F6] then makes its total space paracompact Hausdorff CGWH and of CW type. Thus the oriented model lies in both required classes without asserting that the Schubert theorem supplies its CW structure. For $n=0$ the model is a point. The tautological bundles have local linear charts from [F5] and are numerable by [F6] on these paracompact Hausdorff bases (the oriented one is also the pullback of the unoriented one). Hence $u:=c({\gamma }_n)$ is defined. For a numerable bundle $E\to B$ over a base in both classes with classifying map $c_E$ we have $c_E^{\ast }{\gamma }_n\cong E$, so isomorphism invariance and pullback naturality give $c(E)=c(c_E^{\ast }{\gamma }_n)=c_E^{\ast }c({\gamma }_n)=c_E^{\ast }u\mathrm{.}$ Thus $c$ is $c_u$ for the universal class $u=c({\gamma }_n)$.

The two passages are inverse and determine all values. Starting with $u$, the identity map classifies the universal bundle, so $c_u({\gamma }_n)={\mathrm{id}}^{\ast }u=u$ by [F4]. Starting with $c$, step 1.2 returns $c$ from $u=c({\gamma }_n)$. Hence the assignments $u\mapsto c_u$ and $c\mapsto c({\gamma }_n)$ are mutually inverse bijections between universal classes and characteristic classes, and a characteristic class is determined by its single value on the universal bundle. For $n=0$ all three Grassmannian models are points, so both sides are $H^k(\ast ;R)$: this group is zero for $k>0$ and is canonically $R$ for $k=0$. Thus the degree-zero rank-zero characteristic classes are the scalar classes $r\cdot 1$, one for each $r\in R$, rather than only the unit. The empty base carries the zero cohomology groups and the same formulas apply vacuously. Classifying-map ambiguity is absorbed in step 1.1. AC is inherited through classification and the admissibility/numeration arguments [F6]; it supplies DC where the partition theorem requires it.