Leray–Hirsch theorem

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In mathematics, the Leray–Hirsch theorem[1] is a basic result on the algebraic topology of fiber bundles. It is named after Jean Leray 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.

Statement

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Setup

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Let   be a fibre bundle with fibre  . Assume that for each degree  , the singular cohomology rational vector space

 

is finite-dimensional, and that the inclusion

 

induces a surjection in rational cohomology

 .

Consider a section of this surjection

 ,

by definition, this map satisfies

 .

The Leray–Hirsch isomorphism

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The Leray–Hirsch theorem states that the linear map

 

is an isomorphism of  -modules.

Statement in coordinates

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In other words, if for every  , there exist classes

 

that restrict, on each fiber  , to a basis of the cohomology in degree  , the map given below is then an isomorphism of   modules.

 

where   is a basis for   and thus, induces a basis   for  

Notes

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  1. ^ Hatcher, Allen (2002), Algebraic Topology (PDF), Cambridge: Cambridge University Press, ISBN 0-521-79160-X