nLab Poincare lemma

Context

Differential geometry

differential geometry

synthetic differential geometry

Contents

Idea

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 $\omega \in {\Omega }_{\mathrm{cl}}^{k}\left(X\right)$ with $k\ge 1$ there exists a differential form $\lambda \in {\Omega }^{k-1}\left(X\right)$ such that

$\omega ={d}_{\mathrm{dR}}\lambda \phantom{\rule{thinmathspace}{0ex}}.$\omega = d_{dR} \lambda \,.

Moreover, for $\omega$ a smooth smooth family of closed forms, there is a smooth family of $\lambda$s 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 ${f}_{1},{f}_{2}:X\to Y$ be two smooth functions between smooth manifold and $\Psi :\left[0,1\right]×X\to Y$ a (smooth) homotopy between them.

Then there is a chain homotopy between the induced morphisms

${f}_{1}^{*},{f}_{2}^{*}:{\Omega }^{•}\left(Y\right)\to {\Omega }^{•}\left(X\right)$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,

${H}_{\mathrm{dR}}^{•}\left({f}_{1}^{*}\right)\simeq {H}_{\mathrm{dR}}^{•}\left({f}_{2}^{*}\right)\phantom{\rule{thinmathspace}{0ex}}.$H_{dR}^\bullet(f_1^*) \simeq H_{dR}^\bullet(f_2^*) \,.

Moreover, an explicit formula for the chain homotopy $\psi :{f}_{1}⇒{f}_{2}$ is given by

$\psi :\omega ↦\left(x↦{\int }_{\left[0,1\right]}{\iota }_{{\partial }_{t}}\left({\Psi }_{t}^{*}\omega \right)\left(x\right)\right)dt\phantom{\rule{thinmathspace}{0ex}}.$\psi : \omega \mapsto (x \mapsto \int_{[0,1]} \iota_{\partial_t} (\Psi_t^*\omega)(x) ) d t \,.

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

Proof

We compute

$\begin{array}{rl}{d}_{Y}\psi \left(\omega \right)+\psi \left({d}_{X}\omega \right)& ={\int }_{\left[0,1\right]}{d}_{Y}{\iota }_{{\partial }_{t}}{\Psi }_{t}^{*}\left(\omega \right)dt+{\int }_{\left[0,1\right]}{\iota }_{{\partial }_{t}}{\Psi }_{t}^{*}\left({d}_{X}\omega \right)dt\\ & ={\int }_{\left[0,1\right]}\left[{d}_{Y},{\iota }_{{\partial }_{t}}\right]{\Psi }_{t}^{*}\left(\omega \right)dt\\ & ={\int }_{\left[0,1\right]}{ℒ}_{t}{\Psi }_{t}^{*}\left(\omega \right)dt\\ & ={\int }_{\left[0,1\right]}{\partial }_{t}{\Psi }_{t}^{*}\left(\omega \right)dt\\ & ={\int }_{\left[0,1\right]}{d}_{\left[0,1\right]}{\Psi }_{t}^{*}\left(\omega \right)\\ & ={\Psi }_{1}^{*}\omega -{\Psi }_{0}^{*}\omega \\ & ={f}_{2}^{*}\omega -{f}_{1}^{*}\omega \end{array}\phantom{\rule{thinmathspace}{0ex}},$\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 $\left[d,{\iota }_{\partial t}\right]={ℒ}_{t}$ 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 ${f}_{1}={\mathrm{const}}_{y}$ is a function constant on a point $y\in Y$:

Corollary

If a smooth manifold $X$ admits a smooth contraction

$\begin{array}{c}X\\ {↓}^{\left(\mathrm{id},0\right)}& {↘}^{\mathrm{id}}\\ X×\left[0,1\right]& \stackrel{\Psi }{\to }& X\\ {↑}^{\left(\mathrm{id},1\right)}& {↗}_{{\mathrm{const}}_{x}}\\ X\end{array}$\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 $\omega$ any closed form on $X$ in positive degree an explicit formula for a form $\lambda$ with $d\lambda =\omega$ is given by

$\lambda ={\int }_{\left[0,1\right]}{\iota }_{{\partial }_{t}}{\Psi }_{t}^{*}\left(\omega \right)dt\phantom{\rule{thinmathspace}{0ex}}.$\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 August 5, 2013 01:05:51 by Urs Schreiber (89.204.137.139)