= Leray-Hirsch theorem
{c}
{title2=$H^*(E;R)\cong\bigoplus_jH^{*-|e_j|}(B;R)e_j$}
For a <fiber bundle> $\pi:E\to B$ admitting a finite trivializing open cover, whose fiber has finite free <cohomology> over $R$, suppose classes $e_j\in H^*(E;R)$ restrict to a basis on every fiber. Then multiplication $a\otimes e_j\mapsto\pi^*a\smile e_j$ gives the indicated module isomorphism. A finite trivializing open cover suffices, by the local <Künneth theorem> and the <Mayer–Vietoris sequence>. This determines an additive module structure; multiplicative relations must still be computed. Powers of the hyperplane class on a complex <projective bundle> provide an important application.
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