nLab
Leray-Hirsch theorem

Contents

Context

Bundles

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

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

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

be a topological FF-fiber bundle and let RR be a commutative ring.

If there exists a finite set of elements

α iH (E,R)iI \alpha_i \in H^\bullet(E,R) \;\;\, i \in I

in the ordinary cohomology of EE with coefficients in RR, such that for each point xXx \in X the restriction of the α i\alpha_i to the fiber F xF_x is RR-linearly independent and their RR-linear span is isomorphic to the cohomology H (F,R)H^\bullet(F,R) of the fiber

H (F,R){ι x *α i} iI RRMod H^\bullet(F, R) \simeq \langle \{\iota_x^\ast \alpha_i\}_{i \in I} \rangle_R \;\;\;\; \in R Mod

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

H (E,R){α i} iI H (X,R)H (X,R)Mod 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 (X,R) RH (F,R)H (E,R) 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:

i,jc iι *(α j)i,jp *(c i)α j. \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.