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Poincaré lemma

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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 ≤ pn.[1] The lemma was introduced by Henri Poincaré in 1886.[2][3]

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.

Proofs

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A standard proof of the Poincaré lemma uses the homotopy invariance formula (cf. see the proofs below as well as Integration along fibers#Example).[4][5][6][7] The local form of the homotopy operator is described in Edelen (2005) and the connection of the lemma with the Maurer-Cartan form is explained in Sharpe (1997).[8][9]

Direct proof[10]

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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 . Finally, let and then set . Then, with the notation , we get: for any -form on ,

the formula known as the homotopy formula. The operator is called the homotopy operator (also called a chain homotopy). Now, if is closed, . On the other hand, and . Hence,

which proves the Poincaré lemma.

The same proof in fact shows the Poincaré lemma for any contractible open subset U of a manifold. Indeed, given such a U, we have the homotopy with the identity and a point. Approximating such , we can assume is in fact smooth. The fiber integration is also defined for . Hence, the same argument goes through.

Proof using Lie derivatives

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Cartan's magic formula for Lie derivatives can be used to give a short proof of the Poincaré lemma. The formula states that the Lie derivative along a vector field is given as: [11]

where denotes the interior product; i.e., .

Let be a smooth family of smooth maps for some open subset U of such that is defined for t in some closed interval I and is a diffeomorphism for t in the interior of I. Let denote the tangent vectors to the curve ; i.e., . For a fixed t in the interior of I, let . Then . Thus, by the definition of a Lie derivative,

.

That is,

Assume . Then, integrating both sides of the above and then using Cartan's formula and the differentiation under the integral sign, we get: for ,

where the integration means the integration of each coefficient in a differential form. Letting , we then have:

with the notation

Now, assume is an open ball with center ; then we can take . Then the above formula becomes:

,

which proves the Poincaré lemma when is closed.


Proof in the two-dimensional case

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In two dimensions the Poincaré lemma can be proved directly for closed 1-forms and 2-forms as follows.[12]

If ω = p dx + q dy is a closed 1-form on (a, b) × (c, d), then py = qx. If ω = df then p = fx and q = fy. Set

so that gx = p. Then h = fg must satisfy hx = 0 and hy = qgy. The right hand side here is independent of x since its partial derivative with respect to x is 0. So

and hence

Similarly, if Ω = r dxdy then Ω = d(a dx + b dy) with bxay = r. Thus a solution is given by a = 0 and

Implication for de Rham cohomology

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By definition, the k-th de Rham cohomology group of an open subset U of a manifold M is defined as the quotient vector space

Hence, the conclusion of the Poincaré lemma is precisely that for . Now, differential forms determine a cochain complex called the de Rham complex:

where n = the dimension of M and denotes the sheaf of differential k-forms; i.e., consists of k-forms on U for each open subset U of M. It then gives rise to the complex (the augmented complex)

where is the constant sheaf with values in ; i.e., it is the sheaf of locally constant real-valued functions and the inclusion.

The kernel of is , since the smooth functions with zero derivatives are locally constant. Also, a sequence of sheaves is exact if and only if it is so locally. The Poincaré lemma thus says the rest of the sequence is exact too (since each point has an open ball as a neighborhood). In the language of homological algebra, it means that the de Rham complex determines a resolution of the constant sheaf . This then implies the de Rham theorem; i.e., the de Rham cohomology of a manifold coincides with the singular cohomology of it (in short, because the singular cohomology can be viewed as a sheaf cohomology.)

Once one knows the de Rham theorem, the conclusion of the Poincaré lemma can then be obtained purely topologically. For example, it implies a version of the Poincaré lemma for simply connected open sets (see §Simply connected case).

Simply connected case

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Especially in calculus, the Poincaré lemma is stated for a simply connected open subset . In that case, the lemma says that each closed 1-form on U is exact. This version can be seen using algebraic topology as follows. The rational Hurewicz theorem (or rather the real analog of that) says that since U is simply connected. Since is a field, the k-th cohomology is the dual vector space of the k-th homology . In particular, By the de Rham theorem (which follows from the Poincaré lemma for open balls), is the same as the first de Rham cohomology group (see §Implication to de Rham cohomology). Hence, each closed 1-form on U is exact.

Complex-geometry analog

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On complex manifolds, the use of the Dolbeault operators and for complex differential forms, which refine the exterior derivative by the formula , lead to the notion of -closed and -exact differential forms. The local exactness result for such closed forms is known as the Dolbeault–Grothendieck lemma (or -Poincaré lemma). Importantly, the geometry of the domain on which a -closed differential form is -exact is more restricted than for the Poincaré lemma, since the proof of the Dolbeault–Grothendieck lemma holds on a polydisk (a product of disks in the complex plane, on which the multidimensional Cauchy's integral formula may be applied) and there exist counterexamples to the lemma even on contractible domains.[Note 1] The -Poincaré lemma holds in more generality for pseudoconvex domains.[13]

Using both the Poincaré lemma and the -Poincaré lemma, a refined local -Poincaré lemma can be proven, which is valid on domains upon which both the aforementioned lemmas are applicable. This lemma states that -closed complex differential forms are actually locally -exact (rather than just or -exact, as implied by the above lemmas).

Relative Poincaré lemma

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The relative Poincaré lemma generalizes Poincaré lemma from a point to a submanifold (or some more general locally closed subset). It states: let V be a submanifold of a manifold M and U a tubular neighborhood of V. If is a closed k-form on U, k ≥ 1, that vanishes on V, then there exists a (k-1)-form on U such that and vanishes on V.[14]

The relative Poincaré lemma can be proved in the same way the original Poincaré lemma is proved. Indeed, since U is a tubular neighborhood, there is a smooth strong deformation retract from U to V; i.e., there is a smooth homotopy from the projection to the identity such that is the identity on V. Then we have the homotopy formula on U:

where is the homotopy operator given by either Lie derivatives or integration along fibers. Now, and so . Since and , we get ; take . That vanishes on V follows from the definition of J and the fact . (So the proof actually goes through if U is not a tubular neighborhood but if U deformation-retracts to V with homotopy relative to V.)

On singular spaces

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The Poincaré lemma generally fails for singular spaces. For example, if one considers algebraic differential forms on a complex algebraic variety (in the Zariski topology), the lemma is not true for those differential forms.[15]

However, the variants of the lemma still likely hold for some singular spaces (precise formulation and proof depend on the definitions of such spaces and non-smooth differential forms on them.) For example, Kontsevich and Soibelman claim the lemma holds for certain variants of different forms (called PA forms) on their piecewise algebraic spaces.[16]

Notes

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  1. ^ For counterexamples on contractible domains which have non-vanishing first Dolbeault cohomology, see the post https://mathoverflow.net/a/59554.
  1. ^ Warner 1983, pp. 155–156
  2. ^ Ciliberto, Ciro (2013). "Henri Poincaré and algebraic geometry". Lettera Matematica. 1 (1–2): 23–31. doi:10.1007/s40329-013-0003-3. S2CID 122614329.
  3. ^ Poincaré, H. (1886). "Sur les résidus des intégrales doubles". Comptes rendus hebdomadaires des séances de l'Académie des sciences. 102: 202–204.
  4. ^ Lee (2012), Tu (2011) and Bott & Tu (1982).
  5. ^ Lee, John M. (2012). Introduction to smooth manifolds (2nd ed.). New York: Springer. ISBN 978-1-4419-9982-5. OCLC 808682771.
  6. ^ Tu, Loring W. (2011). An introduction to manifolds (2nd ed.). New York: Springer. ISBN 978-1-4419-7400-6. OCLC 682907530.
  7. ^ Bott, Raoul; Tu, Loring W. (1982). Differential Forms in Algebraic Topology. Graduate Texts in Mathematics. Vol. 82. New York, NY: Springer New York. doi:10.1007/978-1-4757-3951-0. ISBN 978-1-4419-2815-3.
  8. ^ Edelen, Dominic G. B. (2005). Applied exterior calculus (Rev ed.). Mineola, N.Y.: Dover Publications. ISBN 0-486-43871-6. OCLC 56347718.
  9. ^ Sharpe, R. W. (1997). Differential geometry : Cartan's generalization of Klein's Erlangen program. New York: Springer. ISBN 0-387-94732-9. OCLC 34356972.
  10. ^ https://www.math.brown.edu/reschwar/M114/notes7.pdf
  11. ^ Warner 1983, pp. 69–72
  12. ^ Napier & Ramachandran 2011, pp. 443–444
  13. ^ Aeppli, A. (1965). "On the Cohomology Structure of Stein Manifolds". Proceedings of the Conference on Complex Analysis. pp. 58–70. doi:10.1007/978-3-642-48016-4_7. ISBN 978-3-642-48018-8.
  14. ^ Domitrz, W.; Janeczko, S.; Zhitomirskii, M. (2004). "Relative Poincaré lemma, contractibility, quasi-homogeneity and vector fields tangent to a singular variety § 2. Relative Poincare lemma and contractibility". Illinois Journal of Mathematics. 48 (3). doi:10.1215/IJM/1258131054. S2CID 51762845.
  15. ^ Illusie 2012, § 1.
  16. ^ Kontsevich, Maxim; Soibelman, Yan (2000). "Deformations of algebras over operads and Deligne's conjecture". Conférence Moshé Flato 1999: Quantization, Deformations, and Symmetries I. pp. 255–307. arXiv:math/0001151. ISBN 9780792365402.

References

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Further reading

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