Leray-Hirsch theorem

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In mathematics, the Leray-Hirsch theorem^[1] 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.

The statement is as follows: let

$\pi:Y\longrightarrow Z$

be a fibre bundle with fibre $X$ . Assume that for each degree $p$ , the vector space

$H^p(X) = H^p(X, \mathbb{Q})$

of singular cohomology has finite dimension $m p$ . Finally, assume that, for every $p$ , there exist classes

$c_{1,p},\ldots,c_{m_p,p} \in H^p(Y)$

that restrict, on each fiber $X$ , to a basis of the cohomology in degree $p$ . Let $\iota: X\longrightarrow Y$ the inclusion of a fibre. The map given below, is then an isomorphism of $H * (Z)$ modules.

$\begin{array}{ccc} H^*(X)\otimes H^*(Z) & \longrightarrow & H^*(Y) \\ \sum_{i,j,k}a_{i,j,k}\iota^*(c_{i,j})\otimes b_k & \longmapsto & \sum_{i,j,k}a_{i,j,k}c_{i,j}\wedge\pi^*(b_k) \end{array}$

where ${b k}$ is a basis for $H * (Z)$ and thus, induces a basis $\{\iota^*(c_{i,j})\otimes b_k\}$ for $H^*(X)\otimes H^*(Z)$