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Poincare lemma

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

In other words: if X is contractible then for every closed differential form ω∈Ω cl k(X) with k≥1 there exists a differential form λ∈Ω k−1(X) such that

ω=d dRλ.\omega = d_{dR} \lambda \,.

Moreover, for ω a smooth smooth family of closed forms, there is a smooth family of λs 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:X→Y be two smooth functions between smooth manifold and Ψ:[0,1]×X→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 X and Y.

In particular, the action on de Rham cohomology of f 1 * and 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 1⇒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 denotes contraction (see Cartan calculus). with the canonical vector field tangent to [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]ℒ tΨ t *(ω)dt =∫ [0,1]∂ tΨ t *(ω)dt =∫ [0,1]d [0,1]Ψ t *(ω) =Ψ 1 *ω−Ψ 0 *ω =f 2 *ω−f 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]=ℒ t for the Lie derivative along the cylinder on X and finally the Stokes theorem.

The Poincaré lemma proper is the special case of this statement for the case that f 1=const y is a function constant on a point y∈Y:

Corollary

If a smooth manifold X 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 X is concentrated on the ground field in degree 0. Moreover, for ω any closed form on X in positive degree an explicit formula for a form λ with dλ=ω 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 *=0 in positive degree.

References

A nice account collecting all the necessary background is in

Revised on December 19, 2012 20:39:13 by Urs Schreiber (131.174.41.124)
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