Naturality of Stiefel–Whitney classes

Pulling back the defining relation. Assume first that $n\ge 1$. Let $p:P(E)\to B$ and $p^{\mathrm{\prime }}:P(f^{\ast }E)\to B^{\mathrm{\prime }}$ be the projections, and let $f^{\mathrm{\prime }}:P(f^{\ast }E)\to P(E)$ be the canonical map. In a chart $E{\mathrm{\mid }}_U\cong U\times {\mathbb{R}}^n$, it is $(b^{\mathrm{\prime }},\mathrm{\ell })\mapsto (f(b^{\mathrm{\prime }}),\mathrm{\ell })$. These formulas commute with transition matrices, so they glue to a continuous map and identify $P(f^{\ast }E)$ homeomorphically with $B^{\mathrm{\prime }}{\times }_{B}P(E)$, not generally with $P(E)$. The corresponding formulas on vectors in $\mathrm{\ell }$ give ${\gamma }_{f^{\ast }E}\cong f^{\mathrm{\prime }\ast }{\gamma }_E$ by [F1]. If $c$ classifies ${\gamma }_E$, then $c{f}^{\mathrm{\prime }}$ classifies ${\gamma }_{f^{\ast }E}$ by [F5], hence [F2] and [F4] give $x_{f^{\ast }E}=(c{f}^{\mathrm{\prime }}{)}^{\ast }a=f^{\mathrm{\prime }\ast }{x}_E$. Applying the ring homomorphism $f^{\mathrm{\prime }\ast }$ to the relation of [F3] and using [F4] together with $p{f}^{\mathrm{\prime }}=f{p}^{\mathrm{\prime }}$ gives $f^{\mathrm{\prime }\ast }(x_E^n+{\sum }_{i=1}^n{p}^{\ast }{w}_i(E)⌣x_E^{n-i})=x_{f^{\ast }E}^n+{\sum }_{i=1}^n{p}^{\mathrm{\prime }\ast }{f}^{\ast }{w}_i(E)⌣x_{f^{\ast }E}^{n-i}=0,$ where the equality uses the just-proved naturality of $x$. Thus the displayed class is a monic degree-$n$ relation for $x_{f^{\ast }E}$ over the base $B^{\mathrm{\prime }}$.

Comparing with the defining relation of $f^{\ast }E$. For $n\ge 1$, the pullback $f^{\ast }E$ is a numerable real rank-$n$ bundle over the admissible base $B^{\mathrm{\prime }}$, so [F3] provides its unique monic relation $x_{f^{\ast }E}^n+w_1(f^{\ast }E)x_{f^{\ast }E}^{n-1}+\cdots +w_n(f^{\ast }E)=0$. By the uniqueness in [F3], comparing with step 1.1, the coefficients agree: $w_i(f^{\ast }E)=f^{\ast }{w}_i(E)(1\le i\le n)\mathrm{.}$ For $i=0$ both sides are the unit $1$ by the conventions, and for $i>n$ both sides are $0$, since $f^{\ast }0=0$. Summing the finitely many nonzero terms gives $w(f^{\ast }E)={\sum }_i{f}^{\ast }{w}_i(E)=f^{\ast }w(E)$ by [F4]. When $n=0$, neither projective bundle nor $x_E$ is used: both $E$ and $f^{\ast }E$ are rank-zero bundles and the defining convention gives $w_0=1$ and $w_i=0$ for $i>0$, so the same conclusions hold directly.

Isomorphism invariance. Let $\phi :E\to E^{\mathrm{\prime }}$ be a bundle isomorphism over the identity of $B$. It induces a homeomorphism $P(E)\to P(E^{\mathrm{\prime }})$ over $B$ carrying tautological lines to tautological lines, write this homeomorphism as $P\phi$. The vector formula gives $(P\phi {)}^{\ast }{\gamma }_{E^{\mathrm{\prime }}}\cong {\gamma }_E$, so composition of a classifying map of ${\gamma }_{E^{\mathrm{\prime }}}$ with $P\phi$ gives $(P\phi {)}^{\ast }{x}_{E^{\mathrm{\prime }}}=x_E$, exactly as in step 1.1; the defining relation of $E$ is therefore carried to the defining relation of $E^{\mathrm{\prime }}$, and uniqueness of the monic relation gives $w_i(E)=w_i(E^{\mathrm{\prime }})$ for all $i$. When $n=0$ both bundles are the zero bundle, and the conventions give $w=1$ on both sides.

Boundary cases. For $n=0$ the bundle and its pullback are zero bundles, $w=1$ on both sides, and the identity $w(f^{\ast }0)=f^{\ast }w(0)=f^{\ast }1=1$ holds by [F4]. For $n=1$ the relation is $x+w_1=0$ and the argument is the displayed one with $i=1$. If $B^{\mathrm{\prime }}$ is empty, both sides of every naturality equality lie in its zero cohomology ring and vanish. If $B$ is empty, existence of $f$ forces $B^{\mathrm{\prime }}$ empty too. The identity map is the case $f=\mathrm{id}$, where [F5] identifies the pullback with $E$ itself. AC is inherited through the projective-space, classification and relation interfaces [F1]–[F3].