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
StatementFor 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 FormCharacterization Theorem 1 Connections
Has Statement Form
In Branch
Principal Result Of
Source Guy Hirsch (Wikipedia)
Source Leray-Hirsch theorem (Wikipedia)
Proved By
Source Leray-Hirsch theorem (Wikipedia)
Source Leray-Hirsch theorem (Wikipedia)
Sources
1. Leray-Hirsch theorem (Wikipedia)
Statement in coordinates
the map given below is then an isomorphism of
Proved By: Jean Leray, Lead sentence
It is named after Jean Leray and Guy Hirsch, who independently proved it in the late 1940s.
Principal Result Of: Jean Leray, Lead sentence of the article on the result
It is named after Jean Leray and Guy Hirsch, who independently proved it in the late 1940s.
Proved By: Guy Hirsch, Lead sentence
It is named after Jean Leray and Guy Hirsch, who independently proved it in the late 1940s.
In Group: Algebraic Topology, Lead paragraph
In mathematics, the Leray-Hirsch theorem is a basic result on the algebraic topology of fiber bundles.
View the SourceGuy Hirsch (Wikipedia)
Principal Result Of: Guy Hirsch, Lead, known forQuote, Principal Result Of: Guy Hirsch, Lead, known for
He is known for the Leray
Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.