Setup Let \pi\colon E\longrightarrow B be a
fibre bundle with fibre F. Assume that for each degree p, the
singular cohomology rational
vector space :H^p(F) = H^p(F; \mathbb{Q}) is finite-dimensional, and that the inclusion :\iota\colon F \longrightarrow E induces a
surjection in rational cohomology :\iota^* \colon H^*(E) \longrightarrow H^*(F). Consider a
section of this surjection : s\colon H^*(F) \longrightarrow H^*(E), by definition, this map satisfies :\iota^* \circ s = \mathrm {Id}.
The Leray–Hirsch isomorphism The Leray–Hirsch theorem states that the linear map :\begin{array}{ccc} H^* (F)\otimes H^*(B) & \longrightarrow & H^* (E) \\ \alpha \otimes \beta & \longmapsto & s (\alpha)\smallsmile \pi^*(\beta) \end{array} is an isomorphism of H^*(B)-modules.
Statement in coordinates In other words, if for every p, there exist classes :c_{1,p},\ldots,c_{m_p,p} \in H^p(E) that restrict, on each fiber F, to a basis of the cohomology in degree p, the map given below is then an
isomorphism of H^*(B)
modules. :\begin{array}{ccc} H^*(F)\otimes H^*(B) & \longrightarrow & H^*(E) \\ \sum_{i,j,k}a_{i,j,k}\iota^*(c_{i,j})\otimes b_k & \longmapsto & \sum_{i,j,k}a_{i,j,k}c_{i,j}\wedge\pi^*(b_k) \end{array} where \{b_k\} is a basis for H^*(B) and thus, induces a basis \{\iota^*(c_{i,j})\otimes b_k\} for H^*(F)\otimes H^*(B). ==Notes==