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Poincare lemma

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Idea

The Poincaré Lemma asserts that if a smooth manifold XX is contractible, then its de Rham cohomology vanishes in positive degree.

In other words: if XX is contractible then for every closed differential form ωΩ cl k(X)\omega \in \Omega^k_{cl}(X) with k1k \geq 1 there exists a differential form λΩ k1(X)\lambda \in \Omega^{k-1}(X) such that

ω=d dRλ. \omega = d_{dR} \lambda \,.

Moreover, for ω\omega a smooth smooth family of closed forms, there is a smooth family of λ\lambdas satisfying this condition.

The Poincaré lemma is a special case of the more general statement that the pullbacks of differential forms along homotopic smooth function related by a chain homotopy.

Statement

Theorem

Let f 1,f 2:XYf_1, f_2 : X \to Y be two smooth functions between smooth manifold and Ψ:[0,1]×XY\Psi : [0,1] \times X \to Y a (smooth) homotopy between them.

Then there is a chain homotopy between the induced morphisms

f 1 *,f 2 *:Ω (Y)Ω (X) f_1^*, f_2^* : \Omega^\bullet(Y) \to \Omega^\bullet(X)

on the de Rham complexes of XX and YY.

In particular, the action on de Rham cohomology of f 1 *f_1^* and f 2 *f_2^* coincide,

H dR (f 1 *)H dR (f 2 *). H_{dR}^\bullet(f_1^*) \simeq H_{dR}^\bullet(f_2^*) \,.

Moreover, an explicit formula for the chain homotopy ψ:f 1f 2\psi : f_1 \Rightarrow f_2 is given by

ψ:ω(x [0,1]ι t(Ψ t *ω)(x))dt. \psi : \omega \mapsto (x \mapsto \int_{[0,1]} \iota_{\partial_t} (\Psi_t^*\omega)(x) ) d t \,.

Here ι t\iota_{\partial t} denotes contraction (see Cartan calculus). with the canonical vector field tangent to [0,1][0,1] and the integration is that of functions with values in the vector space of differential forms.

Proof

We compute

d Yψ(ω)+ψ(d Xω) = [0,1]d Yι tΨ t *(ω)dt+ [0,1]ι tΨ t *(d Xω)dt = [0,1][d Y,ι t]Ψ t *(ω)dt = [0,1] tΨ t *(ω)dt = [0,1] tΨ t *(ω)dt = [0,1]d [0,1]Ψ t *(ω) =Ψ 1 *ωΨ 0 *ω =f 2 *ωf 1 *ω, \begin{aligned} d_{Y} \psi(\omega) + \psi( d_X \omega ) & = \int_{[0,1]} d_Y \iota_{\partial_t} \Psi_t^*(\omega) d t + \int_{[0,1]} \iota_{\partial_t} \Psi_t^*(d_X \omega) d t \\ & = \int_{[0,1]} [d_Y,\iota_{\partial_t}] \Psi_t^* (\omega) d t \\ & = \int_{[0,1]} \mathcal{L}_{t} \Psi_t^* (\omega) d t \\ & = \int_{[0,1]} \partial_t \Psi_t^* (\omega) d t \\ & = \int_{[0,1]} d_{[0,1]} \Psi_t^* (\omega) \\ & = \Psi_1^* \omega - \Psi_0^* \omega \\ & = f_2^* \omega - f_1^* \omega \end{aligned} \,,

where in the integral we used fist that the exterior differential commutes with pullback of differential forms, then Cartan's magic formula [d,ι t]= t[d,\iota_{\partial t}] = \mathcal{L}_t for the Lie derivative along the cylinder on XX and finally the Stokes theorem.

The Poincaré lemma proper is the special case of this statement for the case that f 1=const yf_1 = const_y is a function constant on a point yYy \in Y:

Corollary

If a smooth manifold XX admits a smooth contraction

X (id,0) id X×[0,1] Ψ X (id,1) const x X \array{ X \\ \downarrow^{\mathrlap{(id,0)}} & \searrow^{\mathrlap{id}} \\ X \times [0,1] & \stackrel{\Psi}{\to} & X \\ \uparrow^{\mathrlap{(id,1)}} & \nearrow_{\mathrlap{const_x}} \\ X }

then the de Rham cohomology of XX is concentrated on the ground field in degree 0. Moreover, for ω\omega any closed form on XX in positive degree an explicit formula for a form λ\lambda with dλ=ωd \lambda = \omega is given by

λ= [0,1]ι tΨ t *(ω)dt. \lambda = \int_{[0,1]} \iota_{\partial_t}\Psi_t^*(\omega) d t \,.
Proof

In the general situation discussed above we now have f 1 *=0f_1^* = 0 in positive degree.

References

A nice account collecting all the necessary background is in

Revised on August 5, 2013 01:05:51 by Urs Schreiber (89.204.137.139)