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.
be a fibre bundle with fibre . Assume that for each degree , 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 -modules.
Statement in coordinates
In other words, if for every , there exist classes
that restrict, on each fiber , to a basis of the cohomology in degree , the map given below is then an isomorphism of modules.
where is a basis for and thus, induces a basis for