Leray–Hirsch theorem

Statement

Setup

Let ${\displaystyle \pi \colon E\longrightarrow B}$ be a

${\displaystyle F}$

${\displaystyle p}$

${\displaystyle H^{p}(F)=H^{p}(F;\mathbb {Q} )}$

is finite-dimensional, and that the inclusion

${\displaystyle \iota \colon F\longrightarrow E}$

induces a surjection in rational cohomology

${\displaystyle \iota ^{*}\colon H^{*}(E)\longrightarrow H^{*}(F)}$.

Consider a section of this surjection

${\displaystyle s\colon H^{*}(F)\longrightarrow H^{*}(E)}$,

by definition, this map satisfies

${\displaystyle \iota ^{*}\circ s=\mathrm {Id} }$.

The Leray–Hirsch isomorphism

The Leray–Hirsch theorem states that the linear map

${\displaystyle {\begin{array}{ccc}H^{*}(F)\otimes H^{*}(B)&\longrightarrow &H^{*}(E)\\\alpha \otimes \beta &\longmapsto &s(\alpha )\smallsmile \pi ^{*}(\beta )\end{array}}}$

is an isomorphism of ${\displaystyle H^{*}(B)}$-modules.

Statement in coordinates

In other words, if for every ${\displaystyle p}$, there exist classes

${\displaystyle c_{1,p},\ldots ,c_{m_{p},p}\in H^{p}(E)}$