Uniqueness of Stiefel–Whitney classes from normalization, naturality, and sum

The rule is determined on line bundles. Let $L\to B$ be a numerable real line bundle with classifying map $c$, so $c^{\ast }{\gamma }_1\cong L$. Then naturality of $w^{\mathrm{\prime }}$ and the normalization $w^{\mathrm{\prime }}({\gamma }_1)=1+a$ give $w^{\mathrm{\prime }}(L)=w^{\mathrm{\prime }}(c^{\ast }{\gamma }_1)=c^{\ast }{w}^{\mathrm{\prime }}({\gamma }_1)=c^{\ast }(1+a)=1+c^{\ast }a=1+w_1(L),$ where the last equality is [F2]. Hence $w^{\mathrm{\prime }}(L)=w(L)$ for every line bundle, including the trivial line, for which $c$ is nullhomotopic and $w_1=0$.

The rule is determined on every bundle. Let $E\to B$ have rank $n\ge 0$ and let $q:\mathrm{Fl}(E)\to B$ be its flag bundle, with $q^{\ast }E\cong L_1\oplus \cdots \oplus L_n$ and $q^{\ast }$ injective. Iterating multiplicativity of $w^{\mathrm{\prime }}$ over the successive summands gives $w^{\mathrm{\prime }}(q^{\ast }E)={\prod }_{j=1}^n{w}^{\mathrm{\prime }}(L_j)$, where the empty product for $n=0$ is $1$; by step 1.1 and [F1] this is ${\prod }_{j=1}^n(1+w_1(L_j))=w(L_1\oplus \cdots \oplus L_n)=w(q^{\ast }E)$, the last equality by the Whitney formula and [F1]. On the other hand naturality of both rules gives $w^{\mathrm{\prime }}(q^{\ast }E)=q^{\ast }{w}^{\mathrm{\prime }}(E)$ and $w(q^{\ast }E)=q^{\ast }w(E)$, so $q^{\ast }{w}^{\mathrm{\prime }}(E)=q^{\ast }w(E)$; injectivity of $q^{\ast }$ gives $w^{\mathrm{\prime }}(E)=w(E)$. For $n=0$ the bundle is the zero bundle, $\mathrm{Fl}(E)=B$, and both rules give $1$ by their degree-zero normalization, so the identity is literal.

The rank bound. Since $w^{\mathrm{\prime }}=w$ by step 2.1 and the classes $w_i(E)$ vanish for $i>n$ by the rank convention of their definition, also $w_i^{\mathrm{\prime }}(E)=0$ for $i>n$. Combined with clause 1 of the statement this shows the rule is exactly the total class computed from the projective-bundle relation, whose coefficients are the classes $w_i$.

Boundary cases. For rank $n=1$ the flag bundle is $B$ up to the identification $P(L)\cong B$, the splitting is $q^{\ast }L\cong L_1$ with $L_1\cong L$, and step 2.1 reduces to step 1.1. For the empty base all groups vanish and both rules give the zero class with degree-zero part the zero-ring unit. The normalization clause 4 is exactly the universal case of step 1.1 over ${\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$, and the tautological line is the line bundle with classifying map the identity. AC is used through the splitting principle of [F3], as recorded.