Contents

Contents

Idea

The Leray-Hirsch theorem states sufficient fiberwise condition for the ordinary cohomology of the total space of a fiber bundle with coefficients in a commutative ring to be free module over the cohomology ring of the base space.

An important consequence is the Thom isomorphism.

Statement

Let

$\array{ F &\stackrel{\iota}{\hookrightarrow}& E \\ && \downarrow^{\mathrlap{p}} \\ && X }$

be a topological $F$-fiber bundle and let $R$ be a commutative ring.

If there exists a finite set of elements

$\alpha_i \in H^\bullet(E,R) \;\;\, i \in I$

in the ordinary cohomology of $E$ with coefficients in $R$, such that for each point $x \in X$ the restriction of the $\alpha_i$ to the fiber $F_x$ is $R$-linearly independent and their $R$-linear span is isomorphic to the cohomology $H^\bullet(F,R)$ of the fiber

$H^\bullet(F, R) \simeq \langle \{\iota_x^\ast \alpha_i\}_{i \in I} \rangle_R \;\;\;\; \in R Mod$

then also the $\{\alpha_i\}$ are $H^\bullet(X,R)$-linearly independent and the cohomology of the total space is their $H^\bullet(X,R)$-linear span:

$H^\bullet(E,R) \simeq \langle \{\alpha_i\}_{i \in I} \rangle_{H^\bullet(X,R)} \;\;\;\;\; \in H^\bullet(X,R) Mod$

in fact there is an isomorphism

$H^\bullet(X,R) \otimes_R H^\bullet(F,R) \stackrel{\simeq}{\longrightarrow} H^\bullet(E,R)$

given by pulling back classes and forming their cup product:

$\underset{i,j}{\sum} c_i \otimes \iota^\ast(\alpha_j) \mapsto \underset{i,j}{\sum} p^\ast(c_i) \cup \alpha_j \,.$

References

e.g.

Last revised on July 11, 2017 at 13:54:44. See the history of this page for a list of all contributions to it.