Integration along fibers

In differential geometry, the integration along fibers of a k-form yields a $(k-m)$-form where m is the dimension of the fiber, via "integration".

Definition

Let $\pi :E\to B$ be a fiber bundle over a manifold with compact oriented fibers. If $\alpha$ is a k-form on E, then for tangent vectors w_i's at b, let

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

where $\beta$ is the induced top-form on the fiber $\pi ^{{-1}}(b)$; i.e., an $m$-form given by: with ${\displaystyle {\widetilde {w_{i}}}}$ the lifts of $w_{i}$ to E,

${\displaystyle \beta (v_{1},\dots ,v_{m})=\alpha (v_{1},\dots ,v_{m},{\widetilde {w_{1}}},\dots ,{\widetilde {w_{k-m}}}).}$

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

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

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

This is also called the fiber integration.

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

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

called the Gysin sequence.

Example

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

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

Then, at each point in M,

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

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

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

where $\alpha _{i}$ is the restriction of $\alpha$ to $M\times \{i\}$.

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

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

which implies $f_{1},f_{0}$ 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 $f_{t}:U\to U,x\mapsto tx$. Then ${\displaystyle \operatorname {H} ^{k}(U;\mathbb {R} )=\operatorname {H} ^{k}(pt;\mathbb {R} )}$, the fact known as the Poincaré lemma.

Projection formula

Given a vector bundle π : E → B over a manifold, we say a differential form α on E has vertical-compact support if the restriction ${\displaystyle \alpha |_{\pi ^{-1}(b)}}$ has compact support for each b in B. We written ${\displaystyle \Omega _{vc}^{*}(E)}$ 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:

${\displaystyle \pi _{*}:\Omega _{vc}^{*}(E)\to \Omega ^{*}(B).}$

The following is known as the projection formula.^[2] We make ${\displaystyle \Omega _{vc}^{*}(E)}$ a right ${\displaystyle \Omega ^{*}(B)}$-module by setting ${\displaystyle \alpha \cdot \beta =\alpha \wedge \pi ^{*}\beta }$.

Proposition — Let $\pi :E\to B$ be an oriented vector bundle over a manifold and $\pi _{*}$ the integration along the fiber. Then

1. $\pi _{*}$ is ${\displaystyle \Omega ^{*}(B)}$-linear; i.e., for any form β on B and any form α on E with vertical-compact support,

   ${\displaystyle \pi _{*}(\alpha \wedge \pi ^{*}\beta )=\pi _{*}\alpha \wedge \beta .}$

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,

   ${\displaystyle \int _{E}\alpha \wedge \pi ^{*}\beta =\int _{B}\pi _{*}\alpha \wedge \beta }$.

Proof: 1. Since the assertion is local, we can assume π is trivial: i.e., ${\displaystyle \pi :E=B\times \mathbb {R} ^{n}\to B}$ is a projection. Let $t_{j}$ be the coordinates on the fiber. If ${\displaystyle \alpha =g\,dt_{1}\wedge \cdots \wedge dt_{n}\wedge \pi ^{*}\eta }$, then, since $\pi^*$ is a ring homomorphism,

${\displaystyle \pi _{*}(\alpha \wedge \pi ^{*}\beta )=\left(\int _{\mathbb {R} ^{n}}g(\cdot ,t_{1},\dots ,t_{n})dt_{1}\dots dt_{n}\right)\eta \wedge \beta =\pi _{*}(\alpha )\wedge \beta .}$

Similarly, both sides are zero if α does not contain dt. The proof of 2. is similar. $\square$

See also

• Transgression map

Notes

1. If ${\displaystyle \alpha =g\,dt\wedge dx_{j_{1}}\wedge \cdots \wedge dx_{j_{k-1}}}$, then, at a point b of M, identifying ${\displaystyle \partial _{x_{j}}}$'s with their lifts, we have:

   ${\displaystyle \beta (\partial _{t})=\alpha (\partial _{t},\partial _{x_{j_{1}}},\dots ,\partial _{x_{j_{k-1}}})=g(b,t)}$

   and so

   ${\displaystyle \pi _{*}(\alpha )_{b}(\partial _{x_{j_{1}}},\dots ,\partial _{x_{j_{k-1}}})=\int _{[0,1]}\beta =\int _{0}^{1}g(b,t)\,dt.}$

   Hence, ${\displaystyle \pi _{*}(\alpha )_{b}=\left(\int _{0}^{1}g(b,t)\,dt\right)dx_{j_{1}}\wedge \cdots \wedge dx_{j_{k-1}}.}$ By the same computation, ${\displaystyle \pi _{*}(\alpha )=0}$ 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

• 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