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

: $(\pi_* \alpha)_b(w_1, \dots, w_) = \int_ \beta$

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

:$\beta(v_1, \dots, v_m) = \alpha(v_1, \dots, v_m, \widetilde, \dots, \widetilde).$

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

Then $\pi_*$ is a linear map $\Omega^k(E) \to \Omega^(B)$. By Stokes' formula, if the fibers have no boundaries(i.e. ,\int0), the map descends to de Rham cohomology:

:$\pi_*: \operatorname^k(E; \mathbb) \to \operatorname^(B; \mathbb).$

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 $0 \to K \to \Omega^*(E) \overset\to \Omega^*(B) \to 0$, ''K'' the kernel,
which leads to a long exact sequence, dropping the coefficient $\mathbb$ and using $\operatorname^k(B) \simeq \operatorname^(K)$:
:$\cdots \rightarrow \operatorname^k(B) \overset\to \operatorname^(B) \overset \rightarrow \operatorname^(E) \overset \rightarrow \operatorname^(B) \rightarrow \cdots$,
called the Gysin sequence.

Example

Let $\pi: M \times$, 1\to M be an obvious projection. First assume $M = \mathbb^n$ with coordinates $x_j$ and consider a ''k''-form:

:$\alpha = f \, dx_ \wedge \dots \wedge dx_ + g \, dt \wedge dx_ \wedge \dots \wedge dx_.$

Then, at each point in ''M'',

:$\pi_*(\alpha) = \pi_*(g \, dt \wedge dx_ \wedge \dots \wedge dx_) = \left( \int_0^1 g(\cdot, t) \, dt \right) \, .$

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

:$\pi_*(d \alpha) = \alpha_1 - \alpha_0 - d \pi_*(\alpha)$

where $\alpha_i$ is the restriction of $\alpha$ to $M \times \$.

As an application of this formula, let $f: M \times$, 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 R^''n'' with center at the origin and let $f_t: U \to U, x \mapsto tx$. Then $\operatorname^k(U; \mathbb) = \operatorname^k(pt; \mathbb)$, 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 $\alpha, _$ has compact support for each ''b'' in ''B''. We write $\Omega_^*(E)$ 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:
:$\pi_*: \Omega_^*(E) \to \Omega^*(B).$

The following is known as the projection formula.; note they use a different definition than the one here, resulting in change in sign. We make $\Omega_^*(E)$ a right $\Omega^*(B)$-module by setting $\alpha \cdot \beta = \alpha \wedge \pi^* \beta$.

Proof: 1. Since the assertion is local, we can assume ''π'' is trivial: i.e., $\pi: E = B \times \mathbb^n \to B$ is a projection. Let $t_j$ be the coordinates on the fiber. If $\alpha = g \, dt_1 \wedge \cdots \wedge dt_n \wedge \pi^* \eta$, then, since $\pi^*$ is a ring homomorphism,
:$\pi_*(\alpha \wedge \pi^* \beta) = \left( \int_ g(\cdot, t_1, \dots, t_n) dt_1 \dots dt_n \right) \eta \wedge \beta = \pi_*(\alpha) \wedge \beta.$
Similarly, both sides are zero if ''α'' does not contain ''dt''. The proof of 2. is similar. $\square$

See also

* Transgression map

Notes

References

* Michele Audin, Torus actions on symplectic manifolds, Birkhauser, 2004
*{{citation , last1 = Bott , first1 = Raoul , authorlink = Raoul Bott , last2=Tu , first2= Loring , title = Differential Forms in Algebraic Topology , year = 1982 , publisher = Springer , location = New York , isbn = 0-387-90613-4

Differential geometry