# Leray-Hirsch theorem for cohomology

You might be looking for: Leray-Hirsch theorem for K-theory

## Statement

Let $p:E \to B$ be a fiber bundle with fiber space $F$, and $R$ be a commutative unital ring, such that the following hold:

• $H^n(F;R)$ is a finitely generated free $R$-module for every $n$
• There exist classes $c_j \in H^{k_j}(E;R)$ whose restrictions $i^*(c_j)$ form a basis for $H^*(F;R)$ in each fiber $F$, via the inclusion of the fiber in $E$ (the choice of these classes needs to be made independent of the fiber)

Then the map:

$\Phi: H^*(B;R) \otimes_R H^*(F;R) \to H^*(E;R)$

given by:

$\sum_{jk} b_j \otimes i^*(c_k) \mapsto \sum_{jk} p^*(b_j) \smile c_j$

is an isomorphism.