Integration along fibers

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

Definition

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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  ,

 

(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

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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 (see Poincaré_lemma#Direct_proof): 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 (also called a chain homotopy):

 

which implies   induce 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

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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 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 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

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Notes

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  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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  • 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