Theorem

Leray-Hirsch Theorem

Topology

The Leray-Hirsch Theorem is a basic result on the algebraic topology of fiber bundles, giving conditions under which the cohomology of the total space of a fiber bundle can be computed from the cohomology of the base space together with the cohomology of the fiber. Named for Jean Leray and Guy Hirsch, who independently proved it in the late 1940s, it can be viewed as a mild generalization of the Kunneth Theorem for product spaces, and is itself a special case of the more general Leray spectral sequence.

Facts

Statement

For a fibre bundle E over B with fibre F such that each H^p(F;Q) is finite-dimensional and the inclusion of F into E induces a surjection in rational cohomology, choosing a section s of that surjection makes the map H*(F) tensor H*(B) to H*(E), sending alpha tensor beta to s(alpha) cup pi*(beta), an isomorphism of H*(B)-modules. 1

Classification

Statement Form

Characterization Theorem 1

Connections

Has Statement Form

Characterization

In Branch

Algebraic Topology

Sources

1. Leray-Hirsch theorem (Wikipedia)

Statement in coordinates
the map given below is then an isomorphism of
In Group: Algebraic Topology, Lead paragraph
In mathematics, the Leray-Hirsch theorem is a basic result on the algebraic topology of fiber bundles.

Switch Atlas

The Inner World

The Civil World

The Natural World

Leray-Hirsch Theorem

Has Statement Form

In Branch