# Integration along fibers

In differential geometry, the integration along fibers of a k-form yields a ${\displaystyle (k-m)}$-form where m is the dimension of the fiber, via "integration". More precisely, let ${\displaystyle \pi :E\to B}$ be a fiber bundle over a manifold with compact oriented fibers. If ${\displaystyle \alpha }$ is a k-form on E, then let:

${\displaystyle (\pi _{*}\alpha )_{b}(w_{1},\dots ,w_{k-m})=\int _{\pi ^{-1}(b)}\beta }$

where ${\displaystyle \beta }$ is the induced top-form on the fiber ${\displaystyle \pi ^{-1}(b)}$; i.e., an ${\displaystyle m}$-form given by

${\displaystyle \beta (v_{1},\dots ,v_{m})=\alpha ({\widetilde {w_{1}}},\dots ,{\widetilde {w_{k-m}}},v_{1},\dots ,v_{m}),\quad {\widetilde {w_{i}}}{\text{ the lifts of }}w_{i}.}$

(To see ${\displaystyle b\mapsto (\pi _{*}\alpha )_{b}}$ is smooth, work it out in coordinates; cf. an example below.)

${\displaystyle \pi _{*}}$ is then a linear map ${\displaystyle \Omega ^{k}(E)\to \Omega ^{k-m}(B)}$, which is in fact surjective. By Stokes' formula, if the fibers have no boundaries, the map descends to de Rham cohomology:

${\displaystyle \pi _{*}:\operatorname {H} ^{k}(E)\to \operatorname {H} ^{k-m}(B).}$

This is also called the fiber integration. Now, suppose ${\displaystyle \pi }$ is a sphere bundle; i.e., the typical fiber is a sphere. Then there is an exact sequence ${\displaystyle 0\to K\to \Omega ^{*}(E){\overset {\pi _{*}}{\to }}\Omega ^{*}(B)\to 0}$, K the kernel, which leads to a long exact sequence, using ${\displaystyle \operatorname {H} ^{k}(B)\simeq \operatorname {H} ^{k+m}(K)}$:

${\displaystyle \dots \rightarrow \operatorname {H} ^{k}(B){\overset {\delta }{\to }}\operatorname {H} ^{k+m+1}(B){\overset {\pi ^{*}}{\rightarrow }}\operatorname {H} ^{k+m+1}(E){\overset {\pi _{*}}{\rightarrow }}\operatorname {H} ^{k+1}(B)\rightarrow \dots }$,

called the Gysin sequence.

## Example

Let ${\displaystyle \pi :M\times [0,1]\to M}$ be an obvious projection. For simplicity, assume ${\displaystyle M=\mathbb {R} ^{n}}$ with coordinates ${\displaystyle x_{j}}$ and consider a k-form:

${\displaystyle \alpha =f\,dx_{i_{1}}\wedge \dots \wedge dx_{i_{k}}+g\,dt\wedge dx_{j_{1}}\wedge \dots \wedge dx_{j_{k-1}}.}$

Then, at each point in M,

${\displaystyle \pi _{*}(\alpha )=\pi _{*}(g\,dt\wedge dx_{j_{1}}\wedge \dots \wedge dx_{j_{k-1}})=\left(\int _{0}^{1}g(\cdot ,t)\,dt\right)\,{dx_{j_{1}}\wedge \dots \wedge dx_{j_{k-1}}}.}$

From this the next formula follows easily: if ${\displaystyle \alpha }$ is any k-form on ${\displaystyle M\times I,}$

${\displaystyle \pi _{*}(d\alpha )=\alpha _{1}-\alpha _{0}-d\pi _{*}(\alpha )}$

where ${\displaystyle \alpha _{i}}$ is the restriction of ${\displaystyle \alpha }$ to ${\displaystyle M\times \{i\}}$. This formula is a special case of Stokes' formula. As an application of this, let ${\displaystyle f:M\times [0,1]\to N}$ be a smooth map (thought of as a homotopy). Then the composition ${\displaystyle h=\pi _{*}\circ f^{*}}$ is a homotopy operator:

${\displaystyle d\circ h+h\circ d=f_{1}^{*}-f_{0}^{*}:\Omega ^{k}(N)\to \Omega ^{k}(M),}$

which implies ${\displaystyle f_{1},f_{0}}$ induces the same map on cohomology. For example, let U be an open ball with center at the origin and let ${\displaystyle f_{t}:U\to U,x\mapsto tx}$. Then ${\displaystyle \operatorname {H} ^{k}(U)=\operatorname {H} ^{k}(pt)}$, the fact known as the Poincaré lemma.