Integration along fibers


In differential geometry, the integration along fibers of a k-form yields a -form where m is the dimension of the fiber, via "integration".

Definition

Let be a fiber bundle over a manifold with compact oriented fibers. If is a k-form on E, then for tangent vectors wi's at b, let
where is the induced top-form on the fiber ; i.e., an -form given by: with lifts of to E,
Then is a linear map. By Stokes' formula, if the fibers have no boundaries, the map descends to de Rham cohomology:
This is also called the fiber integration.
Now, suppose is a sphere bundle; i.e., the typical fiber is a sphere. Then there is an exact sequence, K the kernel,
which leads to a long exact sequence, dropping the coefficient and using :
called the Gysin sequence.

Example

Let be an obvious projection. First assume with coordinates and consider a k-form:
Then, at each point in M,
From this local calculation, the next formula follows easily: if is any k-form on
where is the restriction of to.
As an application of this formula, let be a smooth map. Then the composition is a homotopy operator:
which implies induces the same map on cohomology, the fact known as the homotopy invariance of de Rham cohomology. As a corollary, for example, let U be an open ball in Rn with center at the origin and let. Then, the fact known as the Poincaré lemma.

Projection formula

Given a vector bundle π : E → B over a manifold, we say a differential form α on E has vertical-compact support if the restriction has compact support for each b in B. We written for the vector space of differential forms on E with vertical-compact support.
If E is oriented as a vector bundle, exactly as before, we can define the integration along the fiber:
The following is known as the projection formula. We make a right -module by setting.
Proof: 1. Since the assertion is local, we can assume π is trivial: i.e., is a projection. Let be the coordinates on the fiber. If, then, since is a ring homomorphism,
Similarly, both sides are zero if α does not contain dt. The proof of 2. is similar.