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

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

is then an isomorphism of $H * (Z)$ modules.