Naturality, stability, and mod-two reduction of Pontryagin classes

Tensoring the transition functions of $E$ with $\mathbb{C}$ before or after pulling them back gives the same cocycle, using the same real transition matrices as complex matrices for complexification, so $(f^{\ast }E{)}_{\mathbb{C}}\cong f^{\ast }(E_{\mathbb{C}})$. Thus [F1] and naturality of Chern classes [F2] give $p_i(f^{\ast }E)=(-1{)}^i{c}_{2i}(f^{\ast }(E_{\mathbb{C}}))=(-1{)}^i{f}^{\ast }{c}_{2i}(E_{\mathbb{C}})=f^{\ast }{p}_i(E)$.

The fiberwise map obtained by distributing the tensor product gives $(E\oplus {\epsilon }^r{)}_{\mathbb{C}}\cong E_{\mathbb{C}}\oplus ({\epsilon }^r{)}_{\mathbb{C}}$ and is compatible with all transition functions. The Chern class of the trivial complex bundle is $1$ by [F2], so multiplicativity gives stability; the rank cutoff is part of [F1].

For each real fiber, $v\otimes (a+ib)\mapsto (av,bv)$ is a real-linear isomorphism from $(E_b{\otimes }_{\mathbb{R}}\mathbb{C}{)}_{\mathbb{R}}$ to $E_b\oplus E_b$; its inverse is $(x,y)\mapsto x\otimes 1+y\otimes i$, as is checked on simple tensors and their linear combinations; both formulas are continuous in every bundle chart and compatible with real transition functions and hence defines $(E_{\mathbb{C}}{)}_{\mathbb{R}}\cong E\oplus E$. For the complex bundle $E_{\mathbb{C}}$, [F3] now gives ${\rho }_2{c}_{2i}(E_{\mathbb{C}})=w_{4i}((E_{\mathbb{C}}{)}_{\mathbb{R}})$, while multiplicativity [F4] gives $w((E_{\mathbb{C}}{)}_{\mathbb{R}})=w(E{)}^2$. By [F6], in ${\mathbb{F}}_2$ coefficients all homogeneous classes commute (the sign becomes 1), so the cross terms in the finite square cancel in pairs and the square of a sum is the sum of squares, whose degree-$4i$ component is $w_{2i}(E{)}^2$.

Combining steps 1.2 and 1.3 with the definition [F1]: ${\rho }_2{p}_i(E)=(-1{)}^i{\rho }_2{c}_{2i}(E_{\mathbb{C}})=(-1{)}^i{w}_{2i}(E{)}^2=w_{2i}(E{)}^2$, because $(-1{)}^i$ is $\pm 1$ and the target has exponent two.

Boundary cases. For $i=0$ all three assertions read $1=1$; for the zero bundle $p(0)=1$, $w(0)=1$ and the identity is $1=1$. The rank cutoff makes $p_i(E)=0$ for $2i>n$ while the right side $w_{2i}(E{)}^2$ vanishes for $2i>n$ as well, so no mismatch occurs. The empty base is excluded explicitly and ${\mathbb{F}}_2$ is a field, so no zero-ring issue arises. AC is used only through [A1].