Special and general types
The Poincaré Lemma asserts that if a smooth manifold is contractible, then its de Rham cohomology vanishes in positive degree.
In other words: if is contractible then for every closed differential form with there exists a differential form such that
Moreover, for a smooth smooth family of closed forms, there is a smooth family of 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.
Let be two smooth functions between smooth manifold and a (smooth) homotopy between them.
Then there is a chain homotopy between the induced morphisms
on the de Rham complexes of and .
In particular, the action on de Rham cohomology of and coincide,
Moreover, an explicit formula for the chain homotopy is given by
Here denotes contraction (see Cartan calculus). with the canonical vector field tangent to and the integration is that of functions with values in the vector space of differential forms.
where in the integral we used fist that the exterior differential commutes with pullback of differential forms, then Cartan's magic formula for the Lie derivative along the cylinder on and finally the Stokes theorem.
The Poincaré lemma proper is the special case of this statement for the case that is a function constant on a point :
If a smooth manifold admits a smooth contraction
then the de Rham cohomology of is concentrated on the ground field in degree 0. Moreover, for any closed form on in positive degree an explicit formula for a form with is given by
In the general situation discussed above we now have in positive degree.
A nice account collecting all the necessary background is in