Leray-Hirsch theorem for cohomology

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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.