Integration along fibers

In differential geometry, the integration along fibers of a k-form yields a [math]\displaystyle{ (k-m) }[/math]-form where m is the dimension of the fiber, via "integration". It is also called the fiber integration.

Definition

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

[math]\displaystyle{ (\pi_* \alpha)_b(w_1, \dots, w_{k-m}) = \int_{\pi^{-1}(b)} \beta }[/math]

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

[math]\displaystyle{ \beta(v_1, \dots, v_m) = \alpha(v_1, \dots, v_m, \widetilde{w_1}, \dots, \widetilde{w_{k-m}}). }[/math]

(To see [math]\displaystyle{ b \mapsto (\pi_* \alpha)_b }[/math] is smooth, work it out in coordinates; cf. an example below.)

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

[math]\displaystyle{ \pi_*: \operatorname{H}^k(E; \mathbb{R}) \to \operatorname{H}^{k-m}(B; \mathbb{R}). }[/math]

This is also called the fiber integration.

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

[math]\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 }[/math],

called the Gysin sequence.

Example

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

[math]\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}}. }[/math]

Then, at each point in M,

[math]\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}}}. }[/math]

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

[math]\displaystyle{ \pi_*(d \alpha) = \alpha_1 - \alpha_0 - d \pi_*(\alpha) }[/math]

where [math]\displaystyle{ \alpha_i }[/math] is the restriction of [math]\displaystyle{ \alpha }[/math] to [math]\displaystyle{ M \times \{i\} }[/math].

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

[math]\displaystyle{ d \circ h + h \circ d = f_1^* - f_0^*: \Omega^k(N) \to \Omega^k(M), }[/math]

which implies [math]\displaystyle{ f_1, f_0 }[/math] 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 [math]\displaystyle{ f_t: U \to U, x \mapsto tx }[/math]. Then [math]\displaystyle{ \operatorname{H}^k(U; \mathbb{R}) = \operatorname{H}^k(pt; \mathbb{R}) }[/math], 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 [math]\displaystyle{ \alpha|_{\pi^{-1}(b)} }[/math] has compact support for each b in B. We write [math]\displaystyle{ \Omega_{vc}^*(E) }[/math] 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:

[math]\displaystyle{ \pi_*: \Omega_{vc}^*(E) \to \Omega^*(B). }[/math]

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

Proposition — Let [math]\displaystyle{ \pi: E \to B }[/math] be an oriented vector bundle over a manifold and [math]\displaystyle{ \pi_* }[/math] the integration along the fiber. Then

1. [math]\displaystyle{ \pi_* }[/math] is [math]\displaystyle{ \Omega^*(B) }[/math]-linear; i.e., for any form β on B and any form α on E with vertical-compact support,

   [math]\displaystyle{ \pi_*(\alpha \wedge \pi^* \beta) = \pi_* \alpha \wedge \beta. }[/math]

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,

   [math]\displaystyle{ \int_E \alpha \wedge \pi^* \beta = \int_B \pi_* \alpha \wedge \beta }[/math].

Proof: 1. Since the assertion is local, we can assume π is trivial: i.e., [math]\displaystyle{ \pi: E = B \times \mathbb{R}^n \to B }[/math] is a projection. Let [math]\displaystyle{ t_j }[/math] be the coordinates on the fiber. If [math]\displaystyle{ \alpha = g \, dt_1 \wedge \cdots \wedge dt_n \wedge \pi^* \eta }[/math], then, since [math]\displaystyle{ \pi^* }[/math] is a ring homomorphism,

[math]\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. }[/math]

Similarly, both sides are zero if α does not contain dt. The proof of 2. is similar. [math]\displaystyle{ \square }[/math]

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