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SL2 can mean Situational Leadership II
 
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In differential geometry, the '''integration along fibers''' of a [[differential form|''k''-form]] yields a <math>(k-m)</math>-form where ''m'' is the dimension of the fiber, via "integration". More precisely, 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 let:
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: <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
 
:<math>\beta(v_1, \dots, v_m) = \alpha(\widetilde{w_1}, \dots, \widetilde{w_{k-m}}, v_1, \dots, v_m), \quad \widetilde{w_i}\text{ the lifts of } w_i.</math>
 
(To see <math>b \mapsto (\pi_* \alpha)_b</math> is smooth, work it out in coordinates; cf. an example below.)
 
<math>\pi_*</math> is then a linear map <math>\Omega^k(E) \to \Omega^{k-m}(B)</math>, which is in fact surjective. By Stokes' formula, if the fibers have no boundaries, the map descends to [[de Rham cohomology]]:
 
:<math>\pi_*: \operatorname{H}^k(E) \to \operatorname{H}^{k-m}(B).</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, using <math>\operatorname{H}^k(B) \simeq \operatorname{H}^{k+m}(K)</math>:
:<math>\dots \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 \dots</math>,
called the [[Gysin sequence]].
 
== Example ==
Let <math>\pi: M \times [0, 1] \to M</math> be an obvious projection. For simplicity, 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>
 
From this the next formula follows easily: if <math>\alpha</math> is any ''k''-form on <math>M \times I,</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>. This formula is a special case of Stokes' formula. As an application of this, 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]]:
 
:<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> induces the same map on cohomology. For example, let ''U'' be an open ball with center at the origin and let <math>f_t: U \to U, x \mapsto tx</math>. Then <math>\operatorname{H}^k(U) = \operatorname{H}^k(pt)</math>, the fact known as the [[Poincaré lemma]].
<!--
== Calculus formulae ==
The usual formulae from calculus (in particular, Stokes' formula) continue to hold for integration along fibers.
 
Let <math>\pi: E \to N</math> be a fiber bundle.
 
Stokes' formula says:
:<math>\pi_*(d \alpha) - d (\pi_* \alpha) = (-1)^{k+r+1} \pi'_* (i^* \alpha)</math>-->
 
== See also ==
*[[Gysin homomorphism]]
*[[Transgression map]]
 
== References ==
*[[Michele Audin]], Torus actions on symplectic manifolds, Birkhauser, 2004
 
[[Category:Differential geometry]]

Latest revision as of 16:56, 1 October 2014



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