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

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

where $\beta$ is the induced top-form on the fiber ${\pi }^{-1}(b)$; i.e., an $m$-form given by: with $\widetilde{w_i}$ 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 $b\mapsto ({\pi }_{\ast }\alpha {)}_b$ is smooth, work it out in coordinates; cf. an example below.)

Then ${\pi }_{\ast }$ 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:

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

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

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

called the Gysin sequence.

Example

Let $\pi :M\times [0,1]\to M$ be an obvious projection. First assume $M={ℝ}^n$ 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,

${\pi }_{\ast }(\alpha )={\pi }_{\ast }(gdt\wedge d{x}_{j_1}\wedge \dots \wedge d{x}_{j_{k-1}})=({\int }_0^{1}g(\cdot ,t)dt)d{x}_{j_1}\wedge \dots \wedge d{x}_{j_{k-1}}.$

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

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

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 $h={\pi }_{\ast }\circ f^{\ast }$ is a homotopy operator (also called a chain homotopy):

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

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 ${\mathrm{H}}^k(U;ℝ)={\mathrm{H}}^k(pt;ℝ)$, 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 ${\mathrm{\Omega }}_{vc}^{\ast }(E)$ 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:

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

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

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

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

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
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