= Projective bundle definition of Chern classes
{title2=$h^r+\pi^*c_1(E)h^{r-1}+\cdots+\pi^*c_r(E)=0$}
For a rank-$r$ complex <vector bundle>, let $h=c_1(S^*)$ for the tautological line $S$ on its <projective bundle>. The <Leray-Hirsch theorem> gives the free basis $1,h,\ldots,h^{r-1}$ over base cohomology. The uniquely determined coefficients in the displayed monic relation define the integral <Chern classes>. The line normalization uses the <Euler class> of the canonically oriented underlying real two-plane bundle, and uniqueness makes this definition intrinsic and natural under pullback. No choice of a trivializing cover or classifying map remains in the definition.
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