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". More precisely, let be a fiber bundle over a manifold with compact oriented fibers. If is a k-form on E, then let:

where is the induced top-form on the fiber ; i.e., an -form given by

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

is then a linear map , which is in fact surjective. 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, using :

,

called the Gysin sequence.

Example

Let be an obvious projection. For simplicity, assume with coordinates and consider a k-form:

Then, at each point in M,

From this the next formula follows easily: if is any k-form on

where is the restriction of to . This formula is a special case of Stokes' formula. As an application of this, 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. For example, let U be an open ball with center at the origin and let . Then , the fact known as the Poincaré lemma.

See also

References

  • Michele Audin, Torus actions on symplectic manifolds, Birkhauser, 2004