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". It is also called the fiber 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

$({\pi }_{*}\alpha {)}_b(w_1,\dots ,w_{k-m})=\underset{{\pi }^{-1}(b)}{\int }\beta$

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

$\beta (v_1,\dots ,v_m)=\alpha (v_1,\dots ,v_m,\widetilde{w_1},\dots ,\widetilde{w_{k-m}}).$

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

Then ${\pi }_{*}$ is a linear map ${\mathrm{\Omega }}^k(E)\to {\mathrm{\Omega }}^{k-m}(B)$. By Stokes' formula, if the fibers have no boundaries(i.e. $[d,\int ]=0$), the map descends to de Rham cohomology:

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

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 $0\to K\to {\mathrm{\Omega }}^{*}(E)\stackrel{{\pi }_{*}}{\to }{\mathrm{\Omega }}^{*}(B)\to 0$, K the kernel, which leads to a long exact sequence, dropping the coefficient $\mathbb{ℝ}$ and using ${\mathrm{H}}^k(B)\simeq {\mathrm{H}}^{k+m}(K)$:

$\cdots \to {\mathrm{H}}^k(B)\stackrel{\delta }{\to }{\mathrm{H}}^{k+m+1}(B)\stackrel{{\pi }^{*}}{\to }{\mathrm{H}}^{k+m+1}(E)\stackrel{{\pi }_{*}}{\to }{\mathrm{H}}^{k+1}(B)\to \cdots$,

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:

$\alpha =fd{x}_{i_1}\wedge \dots \wedge d{x}_{i_k}+gdt\wedge d{x}_{j_1}\wedge \dots \wedge d{x}_{j_{k-1}}.$

Then, at each point in M,

${\pi }_{*}(\alpha )={\pi }_{*}(gdt\wedge d{x}_{j_1}\wedge \dots \wedge d{x}_{j_{k-1}})=(\underset{0}{\overset{1}{\int }}g(\cdot ,t)dt)d{x}_{j_1}\wedge \dots \wedge d{x}_{j_{k-1}}.$

From this local calculation, the next formula follows easily (see Poincaré lemma): if $\alpha$ is any k-form on $M\times [0,1],$

${\pi }_{*}(d\alpha )={\alpha }_1-{\alpha }_0-d{\pi }_{*}(\alpha )$

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 (also called a chain homotopy):

$d\circ h+h\circ d=f_1^{*}-f_0^{*}:{\mathrm{\Omega }}^k(N)\to {\mathrm{\Omega }}^k(M),$

which implies $f_1,f_0$ induce 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 ${\mathrm{H}}^k(U;\mathbb{ℝ})={\mathrm{H}}^k(pt;\mathbb{ℝ})$, 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 $\alpha {|}_{{\pi }^{-1}(b)}$ has compact support for each b in B. We write ${\mathrm{\Omega }}_{vc}^{*}(E)$ 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:

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

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

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

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

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,

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

Proof: 1. Since the assertion is local, we can assume π is trivial: i.e., $\pi :E=B\times {\mathbb{ℝ}}^n\to B$ is a projection. Let $t_j$ be the coordinates on the fiber. If $\alpha =gd{t}_1\wedge \cdots \wedge d{t}_n\wedge {\pi }^{*}\eta$, then, since ${\pi }^{*}$ is a ring homomorphism,

${\pi }_{*}(\alpha \wedge {\pi }^{*}\beta )=(\underset{{\mathbb{ℝ}}^n}{\int }g(\cdot ,t_1,\dots ,t_n)d{t}_1\dots d{t}_n)\eta \wedge \beta ={\pi }_{*}(\alpha )\wedge \beta .$

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

See also

• Transgression map

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