Leray–Hirsch theorem

Let ${\displaystyle \pi \colon E\longrightarrow B}$ be a fibre bundle with fibre ${\displaystyle F}$. Assume that for each degree ${\displaystyle p}$, the singular cohomology rational vector space

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

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 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.

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

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

that restrict, on each fiber ${\displaystyle F}$, to a basis of the cohomology in degree ${\displaystyle p}$, the map given below is then an isomorphism of ${\displaystyle H^{*}(B)}$ modules.

${\displaystyle {\begin{array}{ccc}H^{*}(F)\otimes H^{*}(B)&\longrightarrow &H^{*}(E)\\\sum _{i,j,k}a_{i,j,k}\iota ^{*}(c_{i,j})\otimes b_{k}&\longmapsto &\sum _{i,j,k}a_{i,j,k}c_{i,j}\wedge \pi ^{*}(b_{k})\end{array}}}$

where ${\displaystyle \{b_{k}\}}$ is a basis for ${\displaystyle H^{*}(B)}$ and thus, induces a basis ${\displaystyle \{\iota ^{*}(c_{i,j})\otimes b_{k}\}}$ for ${\displaystyle H^{*}(F)\otimes H^{*}(B).}$