Integration along fibers

In differential geometry, the '''integration along fibers''' of a ''k''-form yields a <math>(k-m)</math>-form where ''m'' is the dimension of the fiber, via "integration". It is also called the '''fiber integration'''.

== Definition == Let <math>\pi: E \to B</math> be a fiber bundle over a manifold with compact oriented fibers. If <math>\alpha</math> is a ''k''-form on ''E'', then for tangent vectors ''w''<sub>''i''</sub>'s at ''b'', let

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

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

:<math>\beta(v_1, \dots, v_m) = \alpha(v_1, \dots, v_m, \widetilde{w_1}, \dots, \widetilde{w_{k-m}}).</math>

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

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

:<math>\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>\pi</math> is a sphere bundle; i.e., the typical fiber is a sphere. Then there is an exact sequence <math>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>\mathbb{R}</math> and using <math>\operatorname{H}^k(B) \simeq \operatorname{H}^{k+m}(K)</math>: :<math>\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>\pi: M \times [0, 1] \to M</math> be an obvious projection. First assume <math>M = \mathbb{R}^n</math> with coordinates <math>x_j</math> and consider a ''k''-form:

:<math>\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>\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><ref>If <math>\alpha = g \, dt \wedge d x_{j_1} \wedge \cdots \wedge d x_{j_{k-1}}</math>, then, at a point ''b'' of ''M'', identifying <math>\partial_{x_j}</math>'s with their lifts, we have: :<math>\beta(\partial_t) = \alpha(\partial_t, \partial_{x_{j_1}}, \dots, \partial_{x_{j_{k-1}}}) = g(b, t)</math> and so :<math>\pi_*(\alpha)_b(\partial_{x_{j_1}}, \dots, \partial_{x_{j_{k-1}}}) = \int_{[0, 1]} \beta = \int_0^1 g(b, t) \, dt.</math> Hence, <math>\pi_*(\alpha)_b = \left( \int_0^1 g(b, t) \, dt \right) d x_{j_1} \wedge \cdots \wedge d x_{j_{k-1}}.</math> By the same computation, <math>\pi_*(\alpha) = 0</math> if ''dt'' does not appear in ''α''.</ref>

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

:<math>\pi_*(d \alpha) = \alpha_1 - \alpha_0 - d \pi_*(\alpha)</math>

where <math>\alpha_i</math> is the restriction of <math>\alpha</math> to <math>M \times \{i\}</math>.

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

:<math>d \circ h + h \circ d = f_1^* - f_0^*: \Omega^k(N) \to \Omega^k(M),</math>

which implies <math>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 '''R'''<sup>''n''</sup> with center at the origin and let <math>f_t: U \to U, x \mapsto tx</math>. Then <math>\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>\alpha|_{\pi^{-1}(b)}</math> has compact support for each ''b'' in ''B''. We write <math>\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>\pi_*: \Omega_{vc}^*(E) \to \Omega^*(B).</math>

The following is known as the projection formula.<ref>{{harvnb|Bott|Tu|1982|loc=Proposition 6.15.}}; note they use a different definition than the one here, resulting in change in sign.</ref> We make <math>\Omega_{vc}^*(E)</math> a right <math>\Omega^*(B)</math>-module by setting <math>\alpha \cdot \beta = \alpha \wedge \pi^* \beta</math>.

{{math_theorem|name=Proposition|math_statement=Let <math>\pi: E \to B</math> be an oriented vector bundle over a manifold and <math>\pi_*</math> the integration along the fiber. Then # <math>\pi_*</math> is <math>\Omega^*(B)</math>-linear; i.e., for any form ''β'' on ''B'' and any form ''α'' on ''E'' with vertical-compact support, #:<math>\pi_*(\alpha \wedge \pi^* \beta) = \pi_* \alpha \wedge \beta.</math> # 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>\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>\pi: E = B \times \mathbb{R}^n \to B</math> is a projection. Let <math>t_j</math> be the coordinates on the fiber. If <math>\alpha = g \, dt_1 \wedge \cdots \wedge dt_n \wedge \pi^* \eta</math>, then, since <math>\pi^*</math> is a ring homomorphism, :<math>\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>\square</math>

== See also == *Transgression map

== Notes == {{reflist}}

== 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}}

Category:Differential geometry