Leray–Hirsch theorem

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.

Let ${\displaystyle \pi :E\longrightarrow B}$ be a fibre bundle with fibre F. Assume that for each degree ${\displaystyle p}$, 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

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