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In mathematics, the Leray–Hirsch theorem is a basic result on the
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classif ...
of
fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...
s. It is named after
Jean Leray Jean Leray (; 7 November 1906 – 10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology. Life and career He was born in Chantenay-sur-Loire (today part of Nantes). He studied at Éc ...
and Guy Hirsch, who independently proved it in the late 1940s. It can be thought of as a mild generalization of the Künneth formula, which computes the cohomology of a product space as a tensor product of the cohomologies of the direct factors. It is a very special case of the
Leray spectral sequence In mathematics, the Leray spectral sequence was a pioneering example in homological algebra, introduced in 1946 by Jean Leray. It is usually seen nowadays as a special case of the Grothendieck spectral sequence. Definition Let f:X\to Y be a cont ...
.


Statement


Setup

Let \pi\colon E\longrightarrow B be a
fibre bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...
with fibre F. Assume that for each degree p, the
singular cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
rational
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
:H^p(F) = H^p(F; \mathbb) is finite-dimensional, and that the inclusion :\iota\colon F \longrightarrow E induces a ''surjection'' in rational cohomology :\iota^* \colon H^*(E) \longrightarrow H^*(F). Consider a ''section'' of this surjection : s\colon H^*(F) \longrightarrow H^*(E), by definition, this map satisfies :\iota^* \circ s = \mathrm .


The Leray–Hirsch isomorphism

The Leray–Hirsch theorem states that the linear map :\begin H^* (F)\otimes H^*(B) & \longrightarrow & H^* (E) \\ \alpha \otimes \beta & \longmapsto & s (\alpha)\smallsmile \pi^*(\beta) \end is an isomorphism of H^*(B)-modules.


Statement in coordinates

In other words, if for every p, there exist classes :c_,\ldots,c_ \in H^p(E) that restrict, on each fiber F, to a basis of the cohomology in degree p, the map given below is then an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
of H^*(B) modules. :\begin H^*(F)\otimes H^*(B) & \longrightarrow & H^*(E) \\ \sum_a_\iota^*(c_)\otimes b_k & \longmapsto & \sum_a_c_\wedge\pi^*(b_k) \end where \ is a basis for H^*(B) and thus, induces a basis \ for H^*(F)\otimes H^*(B).


Notes

{{DEFAULTSORT:Leray-Hirsch theorem Fiber bundles Theorems in algebraic topology