In mathematics, the Leray–Hirsch theorem is a basic result on the algebraic topology of fiber bundles. It is named after Jean Leray and Guy Hirsch, who independently proved it in the late 1940s. It can be thought of as a mild generalization of the Künneth formula, which computes the cohomology of a product space as a tensor product of the cohomologies of the direct factors. It is a very special case of the Leray spectral sequence.

Statement

Setup

Let be a fibre bundle with fibre F. Assume that for each degree p, the singular cohomology rational vector space is finite-dimensional, and that the inclusion induces a surjection in rational cohomology Consider a section of this surjection by definition, this map satisfies

The Leray–Hirsch isomorphism

The Leray–Hirsch theorem states that the linear map is an isomorphism of H^*(B)-modules.

Statement in coordinates

In other words, if for every p, there exist classes 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. where {b_k} is a basis for H^(B) and thus, induces a basis for