Integration along fibers explained

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,} ... , 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,} ... , v _{m)} = \alpha(v _{1,} ... , v _{m,} \widetilde{w _{1},} ... , \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(i.e. [d,\int] =0 ), the map descends to de Rham cohomology:\pi _{*:} \operatorname{H} ^{k(E;} R ) \to \operatorname{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 \Omega ^{*(E)} \overset{\pi _{*}\to} \Omega ^{*(B)} \to 0 , K the kernel,which leads to a long exact sequence, dropping the coefficient R and using \operatorname{H} ^{k(B)} \simeq \operatorname{H} ^{k+m} (K) :… → \operatorname{H} ^{k(B)} \overset{\delta}\to \operatorname{H} ^{k+m+1} (B) \overset{\pi ^{*}} → \operatorname{H} ^{k+m+1} (E) \overset{\pi _{*}} → \operatorname{H} ^{k+1} (B) → … ,called the Gysin sequence.

Example

Let

\pi: M x [ 0 , 1 ] \to M be an obvious projection. First assume M = R ^{n} with coordinates x _{j} and consider a k-form:\alpha = f dx _{i _{1}} \wedge ... \wedge dx _{i _{k}} + g dt \wedge dx _{j _{1}} \wedge ... \wedge dx _{j _{k-1}} .

Then, at each point in M,

\pi _{*(\alpha)} = \pi _{*(g} dt \wedge dx _{j _{1}} \wedge ... \wedge dx _{j _{k-1}} ) = \left( \int ^{1} _{0} g( ⋅ , t) dt \right) {dx _{j _{1}} \wedge ... \wedge dx _{j _{k-1}} }.^[1]

From this local calculation, the next formula follows easily (see Poincaré_lemma#Direct_proof): if

\alpha is any k-form on M x [ 0 , 1 ], \pi _{*(d} \alpha) = \alpha _{1} - \alpha _{0} - d \pi _{*(\alpha)}

where

\alpha _{i} is the restriction of \alpha to M x \{i\} .

As an application of this formula, let

f: M x [ 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} \Omega ^{k(N)} \to \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 \operatorname{H} ^{k(U;} R ) = \operatorname{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 \Omega ^{*(E)} _{vc} 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 _{*:} \Omega ^{*(E)} _{vc} \to \Omega ^{*(B).}

The following is known as the projection formula.^[2] We make

\Omega ^{*(E)} _{vc} a right \Omega ^{*(B)} -module by setting \alpha ⋅ \beta = \alpha \wedge \pi ^{*} \beta .

Proof: 1. Since the assertion is local, we can assume π is trivial: i.e.,

\pi: E = B x R ^{n} \to B is a projection. Let t _{j} be the coordinates on the fiber. If \alpha = g dt _{1} \wedge … \wedge dt _{n} \wedge \pi ^{*} η , then, since \pi ^{*} is a ring homomorphism,\pi _{*(\alpha} \wedge \pi ^{*} \beta) = \left( \int _{R ^{n}} g( ⋅ , t _{1,} ... , t _{n)} dt _{1} ... dt _{n} \right) η \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

References

• Michele Audin, Torus actions on symplectic manifolds, Birkhauser, 2004

Notes and References

1. If \alpha = g dt \wedge d x _{j _{1}} \wedge … \wedge d x _{j _{k-1}} , then, at a point b of M, identifying \partial _{x _{j}} 's with their lifts, we have:\beta(\partial _{t)} = \alpha(\partial _{t,} \partial _{x _{j _{1}}} , ... , \partial _{x _{j _{k-1}}} ) = g(b, t) and so\pi _{*(\alpha)} _{b(\partial} ^{x _{j _{1}}} , ... , \partial _{x _{j _{k-1}}} ) = \int _{[0,} \beta = \int ^{1} _{0} g(b, t) dt. Hence, \pi _{*(\alpha)} _{b} = \left( \int ^{1} _{0} g(b, t) dt \right) d x _{j _{1}} \wedge … \wedge d x _{j _{k-1}} . By the same computation, \pi _{*(\alpha)} = 0 if dt does not appear in α.

2. note they use a different definition than the one here, resulting in change in sign.