Integration Along the Fiber
Get Integration Along the Fiber essential facts below. View Videos or join the Integration Along the Fiber discussion. Add Integration Along the Fiber to your PopFlock.com topic list for future reference or share this resource on social media.
Integration Along the Fiber

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,

(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(i.e. ), 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 ? : E -> B over a manifold, we say a differential form ? on E has vertical-compact support if the restriction has compact support for each b in B. We write 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 ? on B and any form ? on E with vertical-compact support,
  2. If B is oriented as a manifold, then for any form ? on E with vertical compact support and any form ? on B with compact support,
    .

Proof: 1. Since the assertion is local, we can assume ? 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 ? 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 ?.
  2. ^ Bott-Tu 1982, Proposition 6.15.; note they use a different definition than the one here, resulting in change in sign.

References

  • Michele Audin, Torus actions on symplectic manifolds, Birkhauser, 2004
  • Bott, Raoul; Tu, Loring (1982), Differential Forms in Algebraic Topology, New York: Springer, ISBN 0-387-90613-4

  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Integration_along_the_fiber
 



 



 
Music Scenes