(see also Chern-Weil theory, parameterized homotopy theory)
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The Leray-Hirsch theorem states sufficient fiberwise condition for the ordinary cohomology of the total space of a fiber bundle with coefficients in a commutative ring to be free module over the cohomology ring of the base space.
An important consequence is the Thom isomorphism.
Let
be a topological $F$-fiber bundle and let $R$ be a commutative ring.
If there exists a finite set of elements
in the ordinary cohomology of $E$ with coefficients in $R$, such that for each point $x \in X$ the restriction of the $\alpha_i$ to the fiber $F_x$ is $R$-linearly independent and their $R$-linear span is isomorphic to the cohomology $H^\bullet(F,R)$ of the fiber
then also the $\{\alpha_i\}$ are $H^\bullet(X,R)$-linearly independent and the cohomology of the total space is their $H^\bullet(X,R)$-linear span:
in fact there is an isomorphism
given by pulling back classes and forming their cup product:
e.g.
Alan Hatcher, Algebraic topology, 2002, theorem 4D.1 on p. 432 (pdf)
Johannes Ebert, section 2.3 of A lecture course on Cobordism Theory, 2012 (pdf)
Last revised on July 11, 2017 at 13:54:44. See the history of this page for a list of all contributions to it.