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Concept
Poincaré lemma
Summary
In mathematics, the Poincaré lemma gives a sufficient condition for a closed differential form to be exact (while an exact form is necessarily closed). Precisely, it states that every closed p-form on an open ball in Rn is exact for p with 1 ≤ p ≤ n. The lemma was introduced by Henri Poincaré in 1886.
Especially in calculus, the Poincaré lemma also says that every closed 1-form on a simply connected open subset in is exact.
In the language of cohomology, the Poincaré lemma says that the k-th de Rham cohomology group of a contractible open subset of a manifold M (e.g., ) vanishes for . In particular, it implies that the de Rham complex yields a resolution of the constant sheaf on M. The singular cohomology of a contractible space vanishes in positive degree, but the Poincaré lemma does not follow from this, since the fact that the singular cohomology of a manifold can be computed as the de Rham cohomology of it, that is, the de Rham theorem, relies on the Poincaré lemma. It does, however, mean that it is enough to prove the Poincaré lemma for open balls; the version for contractible manifolds then follows from the topological consideration.
The Poincaré lemma is also a special case of the homotopy invariance of de Rham cohomology; in fact, it is common to establish the lemma by showing the homotopy invariance or at least a version of it.
We shall prove the lemma for an open subset that is star-shaped or a cone over ; i.e., if is in , then is in for . This case in particular covers the open ball case, since an open ball can be assumed to centered at the origin without loss of generality.
The trick is to consider differential forms on (we use for the coordinate on ). First define the operator (called the fiber integration) for k-forms on by
where , and similarly for and . Now, for , since , using the differentiation under the integral sign, we have:
where denote the restrictions of to the hyperplanes and they are zero since is zero there. If , then a similar computation gives
Thus, the above formula holds for any -form on .
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