In differential geometry, the integration along fibers of a k-form yields a ( k − m ) {\displaystyle (k-m)} -form where m is the dimension of the fiber, via "integration". It is also called the fiber integration.
Definition Let π : E → B {\displaystyle \pi :E\to B} be a fiber bundle over a manifold with compact oriented fibers. If α {\displaystyle \alpha } is a k-form on E, then for tangent vectors wi's at b, let
( π ∗ α ) b ( w 1 , … , w k − m ) = ∫ π − 1 ( b ) β {\displaystyle (\pi _{*}\alpha )_{b}(w_{1},\dots ,w_{k-m})=\int _{\pi ^{-1}(b)}\beta }
where β {\displaystyle \beta } is the induced top-form on the fiber π − 1 ( b ) {\displaystyle \pi ^{-1}(b)} ; i.e., an m {\displaystyle m} -form given by: with w i ~ {\displaystyle {\widetilde {w_{i}}}} lifts of w i {\displaystyle w_{i}} to E {\displaystyle E} ,
β ( v 1 , … , v m ) = α ( v 1 , … , v m , w 1 ~ , … , w k − m ~ ) . {\displaystyle \beta (v_{1},\dots ,v_{m})=\alpha (v_{1},\dots ,v_{m},{\widetilde {w_{1}}},\dots ,{\widetilde {w_{k-m}}}).}
(To see b ↦ ( π ∗ α ) b {\displaystyle b\mapsto (\pi _{*}\alpha )_{b}} is smooth, work it out in coordinates; cf. an example below.) Then π ∗ {\displaystyle \pi _{*}} is a linear map Ω k ( E ) → Ω k − m ( B ) {\displaystyle \Omega ^{k}(E)\to \Omega ^{k-m}(B)} . By Stokes' formula, if the fibers have no boundaries(i.e. [ d , ∫ ] = 0 {\displaystyle [d,\int ]=0} ), the map descends to de Rham cohomology:
π ∗ : H k ( E ; R ) → H k − m ( B ; R ) . {\displaystyle \pi _{*}:\operatorname {H} ^{k}(E;\mathbb {R} )\to \operatorname {H} ^{k-m}(B;\mathbb {R} ).}
This is also called the fiber integration. Now, suppose π {\displaystyle \pi } is a sphere bundle; i.e., the typical fiber is a sphere. Then there is an exact sequence 0 → K → Ω ∗ ( E ) → π ∗ Ω ∗ ( B ) → 0 {\displaystyle 0\to K\to \Omega ^{*}(E){\overset {\pi _{*}}{\to }}\Omega ^{*}(B)\to 0} , K the kernel, which leads to a long exact sequence, dropping the coefficient R {\displaystyle \mathbb {R} } and using H k ( B ) ≃ H k + m ( K ) {\displaystyle \operatorname {H} ^{k}(B)\simeq \operatorname {H} ^{k+m}(K)} :
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