# nLab Poincare lemma

cohomology

### Theorems

#### Differential geometry

differential geometry

synthetic differential geometry

# Contents

## Idea

The Poincaré Lemma in differential geometry and complex analytic geometry asserts that “every differential form $\omega$ which is closed, $d_{dR}\omega = 0$, is locally exact, $\omega|_U = d_{dR}\kappa$”.

More in detail: if $X$ is contractible then for every closed differential form $\omega \in \Omega^k_{cl}(X)$ with $k \geq 1$ there exists a differential form $\lambda \in \Omega^{k-1}(X)$ such that

$\omega = d_{dR} \lambda \,.$

Moreover, for $\omega$ a smooth smooth family of closed forms, there is a smooth family of $\lambda$s satisfying this condition.

This statement has several more abstract incarnations. One is that it says that on a Cartesian space (or a complex polydisc) the de Rham cohomology (the holomorphic de Rham cohomology) vanishes in positive degree.

Still more abstractly this says that the canonical morphisms of sheaves of chain complexes

$\mathbb{R} \to \Omega^\bullet_{dR}$
$\mathbb{C} \to \Omega^\bullet_{hol}$

from the locally constant sheaf on the real numbers (the complex numbers) to the de Rham complex (holomorphic de Rham complex) is a stalk-wise quasi-isomorphism – hence an equivalence in the derived category and hence induce an equivalence in hyper-abelian sheaf cohomology. (The latter statement fails in general in complex algebraic geometry, see (Illusie 12, 1.) ans see also at GAGA.)

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 $\Psi : [0,1] \times X \to Y$ a (smooth) homotopy between them.

Then there is a chain homotopy between the induced morphisms

$f_1^*, f_2^* : \Omega^\bullet(Y) \to \Omega^\bullet(X)$

on the de Rham complexes of $X$ and $Y$.

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

$H_{dR}^\bullet(f_1^*) \simeq H_{dR}^\bullet(f_2^*) \,.$

Moreover, an explicit formula for the chain homotopy $\psi : f_1 \Rightarrow f_2$ is given by

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

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

###### Proof

We compute

\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 $[d,\iota_{\partial t}] = \mathcal{L}_t$ 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 $y \in Y$:

###### Corollary

If a smooth manifold $X$ admits a smooth contraction

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

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

$\lambda = \int_{[0,1]} \iota_{\partial_t}\Psi_t^*(\omega) d t \,.$
###### 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 (in differential geometry) is in

Discussion in complex analytic geometry is in

• Luc Illusie, Around the Poincaré lemma, after Beilinson, talk notes 2012 (pdf)

following

• Alexander Beilinson, $p$-adic periods and de Rham cohomology, J. of the AMS 25 (2012), 715-738

Revised on June 6, 2014 08:37:17 by Urs Schreiber (89.204.139.83)