Let $f:Y\to X$ be a smooth map of smooth manifolds, and let the same symbol $f^*$ denote the induced pullback homomorphism on cohomology.
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If $E\to X$ is a smooth complex vector bundle of finite rank, then its total Chern class satisfies
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\begin{align*}
c(f^*E)=f^*c(E)
\end{align*}
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in $H^{\mathrm{even}}(Y;\mathbb Z)$.
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If $E\to X$ is a smooth real vector bundle of finite rank, then its total Pontryagin class satisfies
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\begin{align*}
p(f^*E)=f^*p(E)
\end{align*}
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in $H^{4*}(Y;\mathbb Z)$.
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If $E\to X$ is a smooth oriented real vector bundle of finite rank, and $f^*E\to Y$ is equipped with the pulled-back orientation, then its Euler class satisfies