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 the lifts of to E,

(To see is smooth, work it out in coordinates; cf. an example below.)

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,

[1]

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 (thought of as a homotopy). 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 : EB 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.[2] We make a right -module by setting .

Proposition  Let be an oriented vector bundle over a manifold and the integration along the fiber. Then

  1. is -linear; i.e., for any form β on B and any form α on E with vertical-compact support,
  2. If B is oriented as a manifold, then for any form α on E with vertical compact support and any form β on B with compact support,
    .

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.

See also

Notes

  1. If , then, at a point b of M, identifying 's with their lifts, we have:
    and so
    Hence, By the same computation, if dt does not appear in α.
  2. Bott−Tu 1982, Proposition 6.15.; note they use a different definition than the one here, resulting in change in sign.

References

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