nLab Poincaré lemma

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, $\omega|_U = d_{dR}\kappa$.”

In more 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 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.) and see also at GAGA.) (A variant of such resolutions of constant sheaves for the case over Klein geometries are BGG resolutions.)

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

Statement

Over contractible domains

Proposition

Let

• $f_0, f_1 \,\colon\, X \to Y$ be a pair of smooth functions between smooth manifolds

• $\Psi \colon [0,1] \times X \to Y$ a smooth homotopy between them.

Then there is a chain homotopy between the induced operations of pullback of differential forms:

$$f_0^*, f_1^* \;\colon\; \Omega^\bullet(Y) \to \Omega^\bullet(X)$$

on the de Rham complexes of $X$ and $Y$, an explicit formula for which is given by the following “homotopy operator”:

$$\psi \;\colon\; \omega \mapsto \Big( x \mapsto \textstyle{\int}_{[0,1]} \big( \iota_{\partial_t} \Psi^*\omega)(x) \big) d t \Big) \,.$$

In particular, $f_0^*$ and $f_1^*$ coincide on de Rham cohomology:

$$H_{dR}^\bullet(f_0^*) \,\simeq\, H_{dR}^\bullet(f_1^*) \,.$$

Here $\iota_{\partial t}$ denotes contraction (cf. 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 as follows:

$$\begin{array}{l} \mathrm{d}_{X} \psi(\omega) + \psi( \mathrm{d}_Y \omega ) \\ \;=\; \textstyle{\int}_{[0,1]} \mathrm{d}_{X} \iota_{\partial_t} \Psi^*(\omega) d t + \textstyle{\int}_{[0,1]} \iota_{\partial_t} \Psi^*(\mathrm{d}_Y \omega) d t \\ \;=\; \textstyle{\int}_{[0,1]} \mathrm{d}_{X} \iota_{\partial_t} \Psi^*(\omega) d t + \textstyle{\int}_{[0,1]} \iota_{\partial_t} \big(\mathrm{d}_X + \mathrm{d}_{[0,1]}\big) \Psi^*( \omega) d t \\ \;=\; \textstyle{\int}_{[0,1]} \iota_{\partial_t} \mathrm{d}_{[0,1]} \Psi^*(\omega) d t \\ \;=\; \textstyle{\int}_{[0,1]} \mathrm{d}_{[0,1]} \Psi^*(\omega) \\ \;=\; \Psi_1^\ast(\omega) - \Psi_0^\ast(\omega) \\ \;=\; f_1^\ast \omega - f_0^\ast \omega \,, \end{array}$$

where in the second step we used that exterior differential commutes with pullback of differential forms (this Prop.), and in the last step the Stokes theorem.

The Poincaré lemma proper is the special case of this statement for the case that $f_2 = const_y$ 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

$$\lambda \,=\, - \textstyle{\int}_{[0,1]} \iota_{\partial_t}\Psi^*(\omega) d t \,.$$

Proof

This is the special case of Prop. where $f_0^\ast = id$ and $f_1^* = 0$ in positive degree.

Over $n$-connected domains

More generally, the conclusion of the Poincaré lemma for differential forms of bounded degree $\leq n$ follows already on $n$-connected spaces (for instance by combining the Hurewicz theorem first with the universal coefficient theorem and then with the de Rham theorem).

Explicitly:

Proposition

On a simply-connected (i.e.: 1-connected) smooth manifold, a closed differential 1-form $\omega$ is exact, with potential function given at $x \in X$ by the integral of $\omega$ from any fixed base point along any smooth path to $x$.

This follows locally for instance by the fiberwise Stokes theorem (here) and then globally due to the independence of the choice of path, by the assumption of simple-connectivity and the plain Stokes theorem.

Textbook accounts which make this explicit include do Carmo 1994, Prop. 3 (p. 24) in §3. Exposition is also in Armstrong 2017.

Related concepts

• Poincaré lemma

  • Kähler potential

• Stokes theorem

• de Rham theorem

• Hodge theorem

• variational sequence

References

Textbook accounts:

• Dominic G. B. Edelen, Lem 4-1.2 and §5-3 in: Applied exterior calculus, Wiley (1985) [GoogleBooks]

• Manfredo P. do Carmo, §4.3 of Differential Forms and Applications, Springer (1994) [doi:10.1007/978-3-642-57951-6]

Course notes:

• Daniel Litt, The Poincaré lemma and de Rham cohomology, The Harvard College Math Review 1 2 (2007) [pdf]

  Litt: An expository account of differential forms and the Poincaré Lemma using modern methods, aimed at beginning undergraduates. Contains some minor errors and omissions (in the exterior power section).

Discussion in complex analytic geometry:

• 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

Generalization to supermanifolds

for plain differential forms (where the Lemma and its proof remain formally the same):

• Simone Noja, Thm. 3.9 in: On the Geometry of Forms on Supermanifolds, Differential Geometry and its Applications, 88 (2023) 101999 [arXiv:2111.12841, doi:10.1016/j.difgeo.2023.101999]

and for (compactly supported) integral forms:

• Konstantin Eder, John Huerta, Simone Noja, Thm. 4.13 in: Poincaré Duality for Supermanifolds, Higher Cartan Structures and Geometric Supergravity [arXiv:2312.05224]

Generalization to non-abelian Lie 2-algebra valued differential forms (local connections on 2-bundles):

• Getachew Alemu Demessie, Christian Saemann, Higher Poincaré Lemma and Integrability, J. Math. Phys. 56 082902 (2015) [doi:10.1063/1.4929537, arXiv:1406.5342]

Generalization to covariant derivatives:

• Radosław Antoni Kycia, Josef Šilhan: Inverting covariant exterior derivative [arXiv:2210.03663]