Relates the homology of a fiber bundle with the homologies of its base and fiber
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.
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
The Leray–Hirsch theorem states that the linear map
is an isomorphism of -modules.
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