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Annales Henri Lebesgue
4 (2021) 1521-1529
IOAN MĂRCUT,
POISSON STRUCTURES WHOSE
POISSON DIFFEOMORPHISM
GROUP IS NOT LOCALLY
PATH-CONNECTED
STRUCTURES DE POISSON DONT LE
GROUPE DE DIFFÉOMORPHISMES DE
POISSON N’EST PAS LOCALEMENT
CONNEXE PAR ARCS
Abstract. — We build examples of Poisson structure whose Poisson diffeomorphism group
is not locally path-connected.
Résumé. — Nous construisons des exemples de structures de Poisson dont le groupe de
difféomorphismes de Poisson n’est pas localement connexe par arcs.
Keywords: Poisson manifold, Poisson diffeomorphisms.
2020 Mathematics Subject Classification: 53D17, 58D05, 57S05.
DOI: https://doi.org/10.5802/ahl.108
1522 I. MĂRCUT,
Introduction
Let (M,π) be a Poisson manifold. The Poisson diffeomorphisms of (M,π)
Poiss(M,π) := {φ : M ∼−→M : φ∗(π) = π} ⊂ Diff(M)
form a C1-closed subgroup of the group of all diffeomorphisms. In analogy to Cartan’s
closed-subgroup theorem for finite-dimensional Lie groups, a natural question is
whether Poiss(M,π) is, in some sense, a smooth subgroup of Diff(M). To simplify
the discussion, we consider only the compact case; a convenient setting for dealing
with the non-compact case was developed in [KM97]. If M is compact, then Diff(M)
endowed with the C∞-topology has the structure of a Fréchet Lie group with Lie
algebra the space of all vector fields X(M) [Ham82, Mil84]. A suitable candidate for
the Lie algebra of Poiss(M,π) is formed by the Poisson vector fields:
poiss(M,π) := {X ∈ X(M) : LX(π) = 0} ⊂ X(M).
Very little is known about whether Poiss(M,π) is a Lie subgroup of Diff(M), but the
answer to this question must depend very much on the nature of the Poisson struc-
ture. In the symplectic case, it was proven by Weinstein that the symplectomorphism
group is a Lie subgroup, and the local charts were obtained as a consequence of his
Lagrangian tubular neighborhood theorem applied to the diagonal inM×M [Wei71].
For a regular Poisson manifold (M,π), using a foliated version of Weinstein’s argu-
ment, a Lie group structure can be constructed on the group Fol(M,π) consisting
of Poisson diffeomorphisms that send each symplectic leaf to itself [Ryb01] (see
also [Ban98] for a more restrictive version of this result). In contrast to the full
Poisson diffeomorphism group Poiss(M,π), the subgroup Fol(M,π) is in general not
closed (as an example, consider an invariant Poisson structure on the 3-torus with
dense leaves). In particular, the natural topology on Fol(M,π) from [Ryb01] can
be finer than the C∞-topology (compared to [Ban98]). For a Poisson manifold, one
can also consider the Lie algebra C∞(M) with the Poisson bracket, or the related
Lie algebra Ω1cl(M) of closed 1-forms with the Lie algebroid bracket. For integrable
Poisson manifolds, the latter Lie algebra can be integrated to the Lie group of La-
grangian bisections of the symlectic groupoid [Xu97] (see also [SW15] for Lie groups
of bisections). This much better behaved group maps into Fol(M,π), and the sub-
group corresponding to C∞(M) maps to the group of Hamiltonian diffeomorphisms.
In general this map is far from being an isomorphism.
In this note, we construct Poisson structures whose Poisson diffeomorphism group
is not locally path-connected, and so it is not a Lie group with the C∞-topology.
Our first example is compactly supported in a 2-disk, and can be grafted using a
local chart on any compact surface. By considering products with other Poisson
manifolds, one obtains compact examples in higher dimensions. The second example
is a regular Poisson structure given as a family of symplectic structures on a surface,
with parameter space a second surface. Also in this case, by applying standard
modifications, one can obtain compact regular examples in higher dimensions and
co-dimensions. However, we do not know whether there exist co-rank one Poisson
structures whose Poisson diffeomorphism group is not locally path-connected. Our
examples are not analytic. It would be interesting to investigate whether analytic
ANNALES HENRI LEBESGUE
Poisson structures whose Poisson diffeomorphism group is not locally path-connected 1523
Poisson structures can have a Poisson diffeomorphism group which is not locally
path-connected. Another interesting phenomenon that we observe in our examples
is that any neighborhood of the identity intersects an uncountable number of path-
components of the Poisson diffeomorphism group; it would be interesting to know
whether examples without this feature exist. Finally, it would be interesting to
understand whether there are natural topologies on Poiss(M,π) that make it into
Lie group with Lie algebra poiss(M,π); for example, a natural candidate is the
refinement of the C∞-topology obtained by declaring path-connected components
to be open.
Acknowledgements
I have learned about the question of whether the Poisson diffeomorphism group is
locally path-connected from Rui Loja Fernandes, and I have talked with João Nuno
Mestre about this problem and the specific construction presented in the paper.
I would like to thank both of them for these useful discussions.
1. The first example
Let Σ be a compact surface. Fix a closed disk B ⊂ Σ contained in a coordinate
chart x = (x1, x2) which identifies B with the closed disk of radius 1 around 0. We
construct a bivector π on Σ whose support is contained in the interior of B and
consists of a disjoint union of disks accumulating at 0. The centers of the disks are
arranged on circles of decreasing radii around 0. For each circle, we build a Poisson
diffeomorphism which rotates the disks on that circle, and fixes all other disks and
the exterior of B. We obtain a sequence of Poisson diffeomorphisms converging to
the identity, such that none of its elements can be joined continuously via Poisson
diffeomorphisms to the identity. The only cumbersome part of the construction is to
fix all coefficients such that everything converges in the C∞-topology.
For the analysis involved, we use the notation: for a = (a1, a2) ∈ N2, let
|a| := a1 + a2, Da :=
1
a1!
1
a2!
∂a1x1∂
a2
x2 .
Define the Ck-norm of a function f ∈ C∞(B) as:
‖f‖k := sup
{
|Daf(x)| : x ∈ B, a ∈ N2, |a| 6 k
}
.
The Ck-norms extend to vector-valued maps or more general tensors as the maximum
of the Ck-norms of the components.
We fix a smooth function χ : R→ [0, 1], which will be used in the construction of
the Poisson structure and of the diffeomorphisms, with the following properties:
χ(t) =

0, 1 6 |t|
> 0, 1/2 < |t| < 1
1, 0 6 |t| 6 1/2
.
TOME 4 (2021)
1524 I. MĂRCUT,
Since we work in dimension two, any smooth bivector field f(x1, x2)∂x1 ∧ ∂x2 is a
Poisson structure. For p ∈ R2 and δ > 0, define
πp, δ(x) := χ
(
|x− p|
δ
)
∂x1 ∧ ∂x2 .
Note that supp(πp, δ) = Bδ(p). We will construct the Poisson structure as an infinite
linear combination of bivectors of the form πp, δ. To ensure convergence, we will use
the following:
Lemma 1.1. — For each k > 0, there is Ck > 0 depending on χ, such that
‖πp, δ‖k 6 Ckδ−k, ∀ p ∈ R2, 0 < δ.
Proof. — We claim that, for any a ∈ N2 with |a| = k, there is Ck > 0 such that
(1.1) |Da(|x|)| 6 Ck|x|1−k.
For this, one shows inductively that there is are homogeneous polynomials of degree
k, denoted pa(x1, x2), such that
Da(|x|) = |x|1−2kpa(x1, x2),
and then one uses the obvious estimate:
|pa(x1, x2)| 6 Ck|x|k.
For k = 0, the statement is obvious. Since the Ck-norms are translation-invariant,
it suffices to prove the inequalities for π0, δ. Fix a ∈ N2 with |a| = k > 0. By the
general chain rule, we have:
Da (χ(|x|/δ)) =
∑
a1+ ...+ai = a
χ(i) (|x|/δ) δ−iDa1(|x|) . . . Dai(|x|),
where the sum runs over all decomposition of a into elements aj ∈ N2, with |aj| > 1.
Applying (1.1), we obtain
(1.2)
∣∣∣Da (χ(|x|/δ)) ∣∣∣ 6 Ca ∑
16 i6 k
∣∣∣χ(i)(|x|/δ)∣∣∣ δ−i|x|i−k,
for some constant Ca > 0. If |x| < δ/2 we have that χ(i)(|x|/δ) = 0, so all elements
vanish. For δ/2 6 |x|, (1.2) gives the estimate from the statement with Ck depending
on k and on the norm ‖χ‖k. 
For n > 4, denote the corners of the regular 2n-gon lying on the circle of radius
1/n around the origin as follows:
p(n, s) := 1
n
exp
(2πs
2n i
)
, 1 6 s 6 2n,
where we identify R2 = C, and let
δn :=
1
n2n .
The numbers δn were chosen such that the closed disks Bδn(p(n, s)), for distinct s,
do not intersect. This follows, because:∣∣∣p(n, s+ 1)− p(n, s)∣∣∣ = 2
n
sin
(
π
2n
)
>
2
n2n = 2δn.
ANNALES HENRI LEBESGUE
Poisson structures whose Poisson diffeomorphism group is not locally path-connected 1525
Consider also the closed annuli En ⊂ Fn given by:
En :=
{
x ∈ R2 : 1
n
− 14n2 6 |x| 6
1
n
+ 14n2
}
Fn :=
{
x ∈ R2 : 1
n
− 12n2 6 |x| 6
1
n
+ 12n2
}
.
The following is straightforward:
Lemma 1.2. — For all n,m > 4, with n 6= m, and all 1 6 s 6 2n, we have that:
Bδn (p(n, s)) ⊂ En and En ∩ Fm = ∅.
For n > 4, define
πn :=
1
n!
2n∑
s=1
πp(n, s), δn .
Using Lemma (1.1), and that the supports of the bivectors involved in the sum are
disjoint, we obtain for each k > 0 a constant Ck > 0, such that
‖πn‖k 6 Ck
nk2nk
n! .
This implies that the series ∑n πn converges uniformly in all Ck-norms. Consider
the Poisson structure π on Σ given by
π|B =
∞∑
n= 4
πn and π|Σ \B = 0.
The support of π is the compact set
K = {0} ∪
⋃
46n
⋃
16 s6 2n
Bδn (p(n, s)) .
Next, define φn ∈ Diff(Σ), n > 4, such that φn|Σ\B = id and for x ∈ B
φn(x) := x · exp
(2π
2n i χ
(
4n(n|x| − 1)
))
,
where we identify R2 = C.
The following implies that Poiss(Σ, π) is not locally path-connected.
Proposition 1.3. — The diffeomorphisms φn satisfy the following:
(a) φn preserves En and φn|En is the rotation through an angle of 2π2n .
(b) φn preserves Fn and φn is supported in Fn.
(c) φn ∈ Poiss(Σ, π).
(d) There exist no continuous family
φtn ∈ Poiss(Σ, π), 0 6 t 6 1,
such that φ0n = id and φ1n = φn.
(e) The sequence φn converges in all Ck-norms to the identity map:
lim
n→∞
φn = id .
TOME 4 (2021)
1526 I. MĂRCUT,
Proof. — The properties of χ imply immediately that φn satisfies (a) and (b).
Note that πn is invariant under the rotation through an angle of 2π2n . By Lemma 1.2,
supp(πn) ⊂ En, and so, by (a), (φn)∗(πn) = πn. On the other hand, for n 6= m,
also by Lemma 1.2 we have that supp(πm) ∩ Fn = ∅; thus (φn)∗(πm) = πm. These
properties imply item (c). For item (d), note that, for such a family φtn, the rank
of π along the curve φtn(p(n, 1)) would have be constant 2, and therefore this curve
cannot go out of Bδn(p(n, 1)) and reach p(n, 2) = φn(p(n, 1)). Item (e) is implied by
the following lemma. 
Lemma 1.4. — For each k > 0, there is Ck > 0 depending on χ, such that
‖φn − id ‖k 6 Ck
n2k
2n .
Proof. — Denote fn(x) := 2π2n i χ(4n(n|x| − 1)). First, we prove the following:
(1.3) ‖fn‖k 6 Ck
n2k
2n .
For k = 0, this is obvious. By the same steps as in the proof of Lemma 1.1, we obtain
for a ∈ N2, with |a| = k > 0, a constant Ca > 0 such that
(1.4)
∣∣∣Da (χ(4n(n|x| − 1)))∣∣∣ 6 Ca ∑
16 i6 k
∣∣∣χ(i)(4n(n|x| − 1))∣∣∣ n2i|x|i−k.
For |x| < 12n , we have that 4n(n|x| − 1) < −1, and so χ
(i)(4n(n|x| − 1)) = 0. For
|x| > 12n , (1.4) gives (1.3) with Ck depending on k and ‖χ‖k.
Next, we prove the inequality:
(1.5) ‖efn − 1‖k 6 Ck
n2k
2n .
Using that |efn(x)| = 1, we obtain the inequality for k = 0:∣∣∣efn(x) − 1∣∣∣ = ∣∣∣∣∫ 1
0
fn(x)esfn(x)ds
∣∣∣∣ 6 |fn(x)| 6 2π2n .
For |a| = k > 0, (1.5) follows from the general chain rule and (1.3):
∣∣∣Da(efn(x) − 1)∣∣∣ 6
∣∣∣∣∣∣
∑
a1+ ...+ai=a
efn(x)Da
1 (fn(x)) . . . Da
i (fn(x))
∣∣∣∣∣∣ 6 Ckn
2k
2n .
Finally, the inequality from the statement follows from (1.5), by writing
Da (φn(x)− x) =
∑
a1+a2=a
Da
1(x)Da2
(
efn(x) − 1
)
,
and using that because efn(x) − 1 has support in the disk B1(0), on which |Da
1(x)|
6 1. 
Corollary 1.5. — Every neighborhood of the identity in Poiss(Σ, π) intersects
an uncountable number of path-connected components of Poiss(Σ, π).
ANNALES HENRI LEBESGUE
Poisson structures whose Poisson diffeomorphism group is not locally path-connected 1527
Proof. — Let U be a neighborhood of the identity in Poiss(Σ, π). Then there is
ε > 0 and k > 0 such that any φ ∈ Poiss(Σ, π) with supp(φ) ⊂ B and ‖φ− id ‖k < ε
belongs to U . With the notation from Lemma 1.4, let nε > 4 be such that
(1.6) Ck
∑
n>nε
n2k
2n 6
ε
2 .
Consider any sequence u = {un}n>nε , with un ∈ {0, 1}. Since the supports of the
diffeomorphisms φn are disjoint, using Lemma 1.4, for any m > n > nε, and any
l > 0 the following holds:
(1.7) ‖φumm ◦ . . . ◦ φunn − id ‖l = ‖ (φumm − id) + . . . + (φunn − id) ‖l 6 Cl
∑
i>n
i2l
2i .
This shows that the sequence {φunn ◦ . . . ◦ φunεnε }n>nε converges to a Poisson dif-
feomorphism φu. By (1.6) and (1.7), we have that φu ∈ U . Consider two different
sequences u 6= v, say un = 1 and vn = 0, for some n. Then φu rotates the disks on
the circle of radius 1/n and φv does not. By the argument from the proof of item (d)
of Proposition 1.3, this implies that φu and φv belong to distinct path-components
of Poiss(Σ, π). 
2. The second example
Consider a connected, oriented surface Λ, and let ν be a volume form on Λ. Consider
a second compact manifold Σ and a positive smooth function f : Σ → (0,∞). We
endow the product Σ× Λ with the regular Poisson structure πν,f whose symplectic
leaves are (
{p} × Λ, f(p) ν
)
, p ∈ Σ.
Note that the group of diffeomorphisms of Σ which preserve f , denoted by
Diff(Σ, f) := {φ : Σ ∼−→ Σ : f ◦ φ = φ} ⊂ Diff(Σ),
can be embedded into the group of Poisson diffeomorphisms of πν,f :
i : Diff(Σ, f) ↪→ Poiss (Σ× Λ, πν, f ) , φ 7→ φ× id .
Moreover, i has a continuous right inverse, which is a group homomorphism
r : Poiss (Σ× Λ, πν, f )→ Diff (Σ, f) ,
where r(Φ)(p) = q if and only if Φ sends the leaf {p} × Λ to the leaf {q} × Λ.
That r(Φ) ∈ Diff(Σ, f) follows from the fact that Φ preserves the volume of the
symplectic leaves. Thus, if we build a function f such that Diff(Σ, f) is not locally
path-connected, then also Poiss(Σ× Λ, πν, f ) is not locally path connected.
Remark 2.1. — Let us remark that Poiss(Σ×Λ, πν,f ) is locally path-connected pre-
cisely when Diff(Σ, f) has this property. First note that the Poisson diffeomorphism
group is the semi-direct product:
Poiss (Σ× Λ, πν f ) = Fol (Σ× Λ, πν, f ) o i (Diff (Σ, f)) ,
TOME 4 (2021)
1528 I. MĂRCUT,
where Fol(Σ × Λ, πν, f) consists of Poisson diffeomorphisms that send each leaf to
itself. This group is a Lie group (hence, locally path-connected) when endowed with
the C∞-topology; this follows using [Ryb01], and that the foliation is simple.
To obtain an explicit example, we will rely on the work we have done so far.
As in the previous section, let Σ be a closed surface, and let B ⊂ Σ a closed
disk diffeomorphic to B1(0), and write the Poisson structure constructed there as
π|B = u(x1, x2)∂x1 ∧ ∂x2 . Extend u to Σ by 0 outside of B, and let f = u + 1. It is
easy to see that satisfy φn ∈ Diff(Σ, f). Also, note that φn permutes effectively some
connected components of the open set {f > 1}, and therefore it cannot be connected
to the identity through a continuous path in Diff(Σ, f). Since limn→∞ φn = id,
we obtain that Diff(Σ, f) is not locally path connected. Clearly, also a version
of Corollary 1.5 holds for Diff(Σ, f). We conclude that any neighborhood of the
identity in Poiss(Σ× Λ, πν, f ) intersects an uncountable number of path-connected
components.
BIBLIOGRAPHY
[Ban98] Augustin Banyaga, On Poisson Diffeomorphisms, Analysis on infinite dimensional Lie
groups and algebras (H. Heyer and J. Marion, eds.), World Scientific, 1998, pp. 1–8. ↑1522
[Ham82] Richard S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Am. Math.
Soc. 7 (1982), no. 1, 65–222. ↑1522
[KM97] Andreas Kriegl and Peter W. Michor, The Convenient Setting of Global Analysis, Mathe-
matical Surveys and Monographs, vol. 53, American Mathematical Society, 1997. ↑1522
[Mil84] John W. Milnor, Remarks on infinite-dimensional Lie groups, Relativity, groups and
topology, II (Les Houches, 1983), North-Holland, 1984, pp. 1007–1057. ↑1522
[Ryb01] Tomasz Rybicki, On foliated, Poisson and Hamiltonian diffeomorphisms, Diff. Geom.
Appl. 15 (2001), no. 1, 33–46. ↑1522, 1528
[SW15] Alexander Schmeding and Christoph Wockel, The Lie group of bisections of a Lie groupoid,
Ann. Global Anal. Geom. 48 (2015), no. 1, 87–123. ↑1522
[Wei71] Alan Weinstein, Symplectic manifolds and their lagrangian submanifolds, Adv. Math. 6
(1971), 329–346. ↑1522
[Xu97] Ping Xu, Flux homomorphism on symplectic groupoids, Math. Z. 226 (1997), no. 4, 575–
597. ↑1522
Manuscript received on 22nd April 2020,
revised on 22nd February 2021,
accepted on 1st April 2021.
Recommended by Editor A. Weinstein.
Published under license CC BY 4.0.
This journal is a member of Centre Mersenne.
ANNALES HENRI LEBESGUE
Poisson structures whose Poisson diffeomorphism group is not locally path-connected 1529
Ioan MĂRCUT,
Radboud University Nijmegen,
6500 GL Nijmegen, (The Netherlands)
ioan.marcut@ru.nl
TOME 4 (2021)


Poisson structures whose Poisson diffeomorphism group is not locally path-connected

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