Integration Along the Fiber
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Integration Along the Fiber

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".

Definition

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 for tangent vectors wi's at b, 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: with ${\displaystyle {\widetilde {w_{i}}}}$ lifts of ${\displaystyle w_{i}}$ 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 ${\displaystyle b\mapsto (\pi _{*}\alpha )_{b}}$ is smooth, work it out in coordinates; cf. an example below.)

Then ${\displaystyle \pi _{*}}$ is a linear map ${\displaystyle \Omega ^{k}(E)\to \Omega ^{k-m}(B)}$. By Stokes' formula, if the fibers have no boundaries(i.e. ${\displaystyle [d,\int ]=0}$), 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 ${\displaystyle \pi }$ 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, dropping the coefficient ${\displaystyle \mathbb {R} }$ and using ${\displaystyle \operatorname {H} ^{k}(B)\simeq \operatorname {H} ^{k+m}(K)}$:

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

called the Gysin sequence.

Example

Let ${\displaystyle \pi :M\times [0,1]\to M}$ be an obvious projection. First 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,

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

From this local calculation, 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 ${\displaystyle M\times \{i\}}$.

As an application of this formula, 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, 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 ${\displaystyle f_{t}:U\to U,x\mapsto tx}$. 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 ? : E -> B over a manifold, we say a differential form ? on E has vertical-compact support if the restriction ${\displaystyle \alpha |_{\pi ^{-1}(b)}}$ has compact support for each b in B. We write ${\displaystyle \Omega _{vc}^{*}(E)}$ 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 ${\displaystyle \pi :E\to B}$ be an oriented vector bundle over a manifold and ${\displaystyle \pi _{*}}$ the integration along the fiber. Then

1. ${\displaystyle \pi _{*}}$ is ${\displaystyle \Omega ^{*}(B)}$-linear; i.e., for any form ? on B and any form ? 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 ? on E with vertical compact support and any form ? 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 ? is trivial: i.e., ${\displaystyle \pi :E=B\times \mathbb {R} ^{n}\to B}$ is a projection. Let ${\displaystyle t_{j}}$ be the coordinates on the fiber. If ${\displaystyle \alpha =g\,dt_{1}\wedge \cdots \wedge dt_{n}\wedge \pi ^{*}\eta }$, then, since ${\displaystyle \pi ^{*}}$ 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 ? does not contain dt. The proof of 2. is similar. ${\displaystyle \square }$

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 ?.
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