Topology of spaces
of equivariant symplectic embeddings
Alvaro Pelayo
Abstract
We compute the homotopy type of the space of Tn-equivariant symplectic embeddings
from the standard 2n-dimensional ball of some fixed radius into a 2n-dimensional symplectic–
toric manifold (M, σ), and use this computation to define a Z≥0-valued step function on R≥0
which is an invariant of the symplectic–toric type of (M, σ). We conclude with a discussion
of the partially equivariant case of this result.
1 The main theorem
Let (M, σ) be a 2n-dimensional symplectic manifold and write Br for the compact 2n-ball of
radius r > 0 in the complex space Cn equipped with the restriction of the standard symplectic
form σ0 of C
n. (The proofs of the results in this paper hold verbatim for the open ball.) Recently a
lot of effort has been put into understanding the topological and geometric properties of the space
of symplectic embeddings from Br into M . This question is not only intriguing, but it is also very
fundamental because it acknowledges one of the main differences that exist between Riemannian
and symplectic geometry, e.g. Gromov’s non–squeezing theorem [12].
Figure 1: An equivariant and symplectic embedding B2r → S2s with r/s =
This question, posed with such generality, has proven to be extremely difficult to answer.
Significant progress has been made by McDuff [17], [18], Biran [3], [5] and most recently by
Lalonde–Pinsonnault [14], among other authors. One of the most general results is due to McDuff;
she showed the connectedness of the space of 4-balls into 4-manifolds with non-simple Seiberg–
Witten type, in particular rational or ruled surfaces. Recall that we say that a symplectic 4-manifold
Q has simple Seiberg–Witten type or just simple type if the only non–zero Gromov invariants of Q
occur in classes A ∈ H2(Q) for which k(A) = K · A + A2 = 0. It follows from work of Taubes
http://arxiv.org/abs/0704.1033v1
and Li–Liu that the symplectic 4-manifolds with non–simple type are blow–ups of (i) rational and
ruled manifolds; (ii) manifolds with b1 = 0, b
2 = 1, like the Enriques or Barlow surface; and
(iii) manifolds with b1 = 2 and (H1(X))
2 6= 0; examples (with K = 0) are hyperelliptic surfaces,
some non–Kähler T2-bundles over T2 and quotients T2 × Σg/G where Σg is a surface of genus g
greater than 1, and G is certain finite group. See [17] for further references and examples.
McDuff’s techniques are unique to dimension 4 and do not extend at all to higher dimensions—
this is also the case in the other authors’ work—existence of J-holomorphic curves with special
homological properties is essential in their proofs. Although J-holomorphic curves exist in all
even dimensions, it is only in dimension 4 where these homological properties hold.
In the present paper we study a special case of this question: M is a symplectic–toric manifold
of arbitrary dimension, and the symplectic embeddings that we consider preserve the toric struc-
ture, see Figure 1. Precisely this means that there exists an automorphism Λ of the n-torus Tn such
that the following diagram commutes:
n × Br
Λ×f //
f // M
, (1.1)
where ψ is a fixed effective and Hamiltonian Tn-action on M and · denotes the standard action by
rotations on Br (component by component). In this case we say that f is a Λ-equivariant mapping.
(1,0)(0,0)
(1,1)(0,1)(1,1)(0,1)
(2,0)(0,0)(1,0)(0,0)
(0,1)
Figure 2: The momentum polytope of CP2 and B21 (left), of a Hirzebruch surface (center) and of
(CP1)2 (right).
The feature that makes the study of symplectic manifolds equipped with Hamiltonian torus
actions richer than the study of generic symplectic manifolds is the presence of the smooth mo-
mentum map µM : M → Lie(Tn)∗, whose image ∆M is a convex polytope (called the momentum
polytope of M , cf. Figure 2) as shown independently by Atiyah and Guillemin–Sternberg [1], [8].
Here we are identifying the Lie algebra Lie(Tn) and its dual Lie(Tn)∗ with Rn. Since this identi-
fication is not canonical, we need to specify the convention we adopt in this paper. This amounts
to choosing an epimorphism R → T1 which we take to be x 7→ e2
−1x. This epimorphism in-
duces an isomorphism between Lie(T1) and R via ∂
7→ 1/2, giving rise to a new isomorphism
Lie(Tn) → Rn, ∂
7→ 1/2 ek, by canonically identifying Lie(Tn) with the product of n copies of
Lie(T1) (see [9] for more details).
For example, under the convention of the previous paragraph, the momentum map µBr of Br
is a mapping from Br into R
n with components µBrk (z) = |zk|2, for all integer k with 1 ≤ k ≤ n.
(There are a number of different conventions used in the literature, and our choice is intended to
give the simplest formula for the momentum map of Br.) The simplest symplectic manifolds which
admit Hamiltonian effective torus actions are called symplectic–toric.
Definition 1.1 A symplectic–toric manifold M , also called a Delzant manifold, is a compact
connected symplectic manifold equipped with an effective Hamiltonian action of a torus of dimen-
sion half of the dimension of the manifold. In this case the momentum polytope ∆M is called the
Delzant polytope of M . ⊘
Symplectic–toric manifolds were classified by Delzant in [7]. In particular, he showed that the
momentum image of such a manifold under the momentum map completely determines M up to
equivariant symplectomorphisms.
The main result of this paper, Theorem 1.2 below, describes the topology of the space of equiv-
ariant embeddings of symplectic balls into a symplectic–toric manifold. We denote by χ(M) the
Euler characteristic of M .
Theorem 1.2. For every symplectic–toric 2n-manifold M there is an associated Z-valued non–
increasing step function Emb(M,σ) : R≥0 → [0, n!χ(M)] such that for each r ≥ 0 the space of
equivariant symplectic embeddings from the 2n-ball Br into M is homotopically equivalent to a
disjoint union of Emb(M,σ)(r) subspaces, each of which is homeomorphic to the n-torus T
As a matter of fact we can explicitly and easily read Emb(M,σ) from the polytope ∆
Example 1.3 Let (M, σ) equal the blow–up of S2r0×S
with r0 = 1/
2 whose Delzant polytope
has vertices at (0, 0), (2, 0), (2, 1), (1, 2) and (0, 2) (see Figure 4). Then Emb(M,σ) = 10χ[0,1) +
2χ[1,
2), where χA denotes the characteristic function of A ⊂ R. We identify the 2-sphere of
radius r equipped with the standard area form with (CP1, 4r2 ·σFS), where σFS is the Fubini–Study
form. ⊘
Proposition 1.4. The function Emb(M,σ)(r) given in Theorem 1.2 is an invariant of the symplectic–
toric type of M and is given by the formula
Emb(M,σ)(r) = n!
p∈MTn
cp(r), (1.2)
where for each fixed point p ∈M , cp(r) = 1 if the infimum of the SL(n, Z)-lengths of the edges of
∆M meeting at µM(p) is strictly greater than r2, and cp(r) = 0 otherwise.
Example 1.5 Let σFS be the Fubini–Study form on CP
n and observe that Tn acts naturally on
(CPn, λ · σFS), λ > 0, with n + 1 fixed points. The momentum polytope is a tetrahedrum with
vertices at 0 and λ ei, where the ei are the canonical basis vectors in R
n. So if M = CPn × CPm,
the space of equivariant symplectic embeddings from Br into M is homotopically equivalent to
(n+m)!(n+1)(m+1)⊔
if r <
λ, and it is empty otherwise. ⊘
The study of the space of symplectic embeddings is directly related to the study of the sym-
plectic ball packing problem cf. [4], the equivariant version of which was treated in [19].
2 Proof of Theorem 1.2
In this section we prove Theorem 1.2. For clarity, the proof is divided in three steps, which we
describe next.
We start by introducing the notation and making the following observations:
i) Throughout the proof Rwill denote the space of rotations by matrices of the form (δi τ(i)θi j)ni, j=1
with τ ∈ Sn (the symmetric group) and θi j ∈ T1, and ET
x will denote the space of equivari-
ant symplectic embeddings f from Br into M such that f(0) = p and µ
M(p) = x ∈ ∆M ,
equipped with the Cm-Whitney topology (m ≥ 0). Throughout the present section we fix f .
Since each component of R is canonically identified with the n-torus Tn (cf. Corollary 2.5),
Theorem 1.2 amounts to prove that if p ∈MTn (MTn denotes the Tn-fixed point set) is such
that µM(p) = x, then the space ETnx gets identified with R via a homotopy equivalence.
ii) Secondly let BTnr denote the space of equivariant symplectomorphisms of Br (again with
respect to the Cm-Whitney topology).
Recall that, for example, the Cm-Whitney topology on BTnr is given by the well–known norm
‖ φ ‖Cm= max
0≤k≤m
‖ Tkφ ‖M(C),
where we are taking the norm ‖ · ‖M(C) on the right–hand side of this expression to be the
canonical Euclidean norm on the space of n× n matrices with complex entries.
iii) We identify the automorphism group Aut(Tn) with the matrix group
GL(n, Z).
iv) The elements αpi ’s, 1 ≤ i ≤ n, denote the weights of the isotropy representation of Tn on
TpM ; the canonical basis vectors ei ∈ Rn represent the weights of the isotropy representa-
tion of Tn on T0Br.
Step 1: Invariance of the image f(Br).
In this step we first show how to go from smooth maps on manifolds to affine maps on polytopes
(see diagram (2.3)), and secondly we use this to show the invariance of the image f(Br) ⊂ M .
Precisely, one can think of an embedding being equivariant in the sense of commuting with the
n-action, and it is when we reparametrize the torus that Λ appears.
Lemma 2.1. Let g be any Λ-equivariant and symplectic embedding such that the normalization
condition f(0) = g(0) = p holds. Then for all z ∈ Br, if Tn · z denotes the Tn-orbit that passes
through z, the identity f(Tn · z) = g(Tn · z) holds, and therefore f(Br) = g(Br).
Proof. Let f, g be Λ-equivariant and symplectic embeddings from Br into M with f(0) = g(0) =
p. Under the identifications described in Section 1, the following diagram commutes, where the
top arrow stands for the affine map with linear part (Λt)−1, which takes 0 to x:
(Λt)−1+x
// ∆M
. (2.3)
In order to prove the commutativity of diagram (2.3), we denote by ξM the vector field induced
by the element ξ ∈ Lie(Tn) via the exponential map and note that from the definition of the
momentum maps µM and µBr , the Λ-equivariance of f , and the fact that f ∗σ = σ0, we have
the following sequence of equalities, where Tf(z)µ
M and Tzµ
Br denote, respectively, the tangent
mapping of µM at f(z) and of µBr at z,
〈TzµBr(v), ξ〉z = (σ0)z(v, ξBr(z))
= σf(z)(Tzf(v), Λ(ξ)M(f(z)))
= 〈Tf(z)µM(Tzf(v)), Λ(ξ)〉f(z)
= 〈Λt ◦ Tf(z)µM(Tzf(v)), ξ〉z, (2.4)
where z ∈ Br, v ∈ TzBr and ξ ∈ Lie(Tn). Therefore by equation (2.4) and by using the chain rule
we obtain that for all z ∈ Br and v ∈ TzBr
Br(v) = Tz(Λ
t ◦ µM ◦ f)(v). (2.5)
Considering equation (2.5), f(0) = p and µM(p) = x and composing with (Λt)−1, after integration
we obtain the commutativity condition on diagram (2.3). Notice that diagram (2.3) also holds for
the embedding g.
Then it follows from the conjunction of diagram (2.3) and diagram (1.1) that for all t ∈ Tn the
following identities hold:
µM(ψ(Λ(t), f(z))) = µM(f(t · z))
= (Λt)−1 ◦ µBr(z) + x
= µM(g(t · z)) = µM(ψ(Λ(t), g(z))). (2.6)
Expression (2.6) is clearly equivalent to µM(Tn · f(z)) = µM(Tn · g(z)), since Λ is an auto-
morphism. Now since M is symplectic–toric, by the proof of the Atiyah–Guillemin–Sternberg
convexity theorem we know that each fiber of the momentum map µM consists of a single con-
nected orbit which together with the last equality implies that Tn · f(z) = Tn · g(z). Since f(Br)
is the union of the orbit images f(Tn · z), we immediately obtain that f(Br) = g(Br), which
concludes the proof.
It is possible to explicitly describe the momentum image µM(f(Br)), and for this purpose we
recall the notion of SL(n, Z)–length: if x, y ∈ Rn, we say that a segment line [x, y] joining x to y
(0,0)
(0,1) (1,1)
(1,0)
(0,1) (1,1)
(0,0) (1,0)
Figure 3: Equivariant symplectic ball embeddings in (CP1)2 (left); the triangle on the right does
not come from such embedding.
in Rn has SL(n, Z)-length d if there exists a matrix A ∈ SL(n, Z) such that A(d e1) = y − x (d
is not defined in general, only for segments of rational slope). The Euclidean length of a segment
line agrees with its SL(n, Z)-length if and only if the segment is parallel to one of the coordinate
axes in Rn.
In their article [16], Karshon and Tolman made the following two definitions. Let (Q, σQ) be
a connected symplectic 2m-dimensional manifold with momentum map µQ for an action of an
m-torus Tm on Q, and let Γ ⊂ (Lie(Tm))∗ be an open convex subset which contains the image
of Q under the momentum map µQ. The quadruple (Q, σQ, µQ, Γ) is a proper Hamiltonian Tm-
manifold if the momentum map µQ is proper as a map to Γ.
The proper Hamiltonian Tm-manifold (Q, σ, µQ, Γ) is said to be centered about a point α ∈ Γ
if α is contained in the momentum map image of every component of QK , for each K ⊂ Tm. Here
QK := {q ∈ Q |ψQ(a, q) = q, ∀a ∈ K},
where ψQ : T
m ×Q→ Q denotes the action of Tm on Q. The following lemma is Proposition 2.8
in [16].
Lemma 2.2 (Karshon–Tolman, [16]). Let the quadruple (Q, σQ, µQ, Γ) be a proper Hamiltonian
2m-dimensional Tm-manifold. Suppose that (Q, σQ, µQ, Γ) is centered about α ∈ Γ and that the
preimage (µQ)−1({α}) consists of a single fixed point q. Then Q is equivariantly symplectomor-
phic to
{z ∈ Cm |α+
|zj |2 ηqj ∈ Γ},
where ηq1, . . . , η
m are the weights of the isotropy representation of T
m on TqQ.
We use Lemma 2.2 in order to prove the following lemma.
Lemma 2.3. The momentum image µM(f(Br)) equals the subset of R
n given by the convex hull of
x and x+ r2 αpi , where 1 ≤ i ≤ n. Furthermore, the infimum of the SL(n, Z)-lengths of the edges
of ∆M meeting at x is greater than or equal to r2, if and only if for all 0 < s < r there exists an
embedding h : Bs → M which is Λ-equivariant and symplectic, satisfying h(0) = p.
Proof. The first observation is that ∆Br equals the convex hull in Rn of 0 and r2 e1, . . . , r
2 en.
Secondly, since µBr : Br → ∆Br is onto, it follows from diagram (2.3) that
µM(f(Br)) = (Λ
t)−1(∆Br) + x.
Since Λ is an automorphism, (Λt)−1 is an automorphism of the corresponding dual spaces and
therefore there exists a permutation τ ∈ Sn such that (Λt)−1(ei) = αpτ(i). Then the linearity of Λ
implies that µM(f(Br)) equals the the convex hull in R
n of the points x and x+ r2 αpi , 1 ≤ i ≤ n,
which proves the first claim.
Suppose that the infimum of the SL(n, Z)-lengths of the edges meeting at x is greater than or
equal to r2. Let Σ be the convex hull of x and x + r2 αpi , with 1 ≤ i ≤ n, and let Z be the convex
hull of x+r2 αpi , with 1 ≤ i ≤ n. Notice that Σ ⊂ ∆M , Σ\Z is open in ∆M , and let Γ ⊂ Rn be the
open half–space of Rn, whose closure’s boundary ∂(cl(Γ)) is the hyperplane of Rn that contains
Z, and such that Σ \ Z ⊂ Γ.
Let N := (µM)−1(Σ \ Z) and let σN be the symplectic form obtained by restricting σ to N .
The set N is open in M because it is the preimage of the open set Σ \Z under the momentum map
µM : M → ∆M . By the proof of Atiyah–Guillemin–Sternberg convexity theorem, cf. [1], [8], N
is a connected manifold. Since M is compact, the momentum map µM : M → ∆M is a proper
map and therefore its restriction µM : N → Σ \Z is a proper map, which means that µM : N → Γ
is proper, since (µM)−1(Γ \ (Σ \ Z)) = ∅. Therefore (N, σN , ψN) is a connected symplectic
manifold with momentum map µM , and the quadruple (N, σN , µM , Γ) is a proper Hamiltonian
n-space.
On the other hand, notice that the quadruple (N, σN , µM , Γ) is centered about the point x, and
(µM)−1({x}) = p, so we can apply Lemma 2.2, and conclude that N is equivariantly symplecto-
morphic to the submanifold X ⊂ Cn given by
X := {z ∈ Cn | x+
|zi|2 αpi ∈ Σ \ Z}
= {z ∈ Cn | x+
|zi|2 αpi ∈ Σ} \ {z ∈ Cn | x+
|zi|2 αpi ∈ Z}
= Br \ ∂Br = Int(Br).
Hence there exists an equivariant symplectomorphism φ : Int(Br) → N , and by letting j : Bs →
Int(Br) be the standard inclusion, if s < r, the map h := jN ◦φ◦ j : Bs → M , where jN : N →M
is the inclusion map, is an equivariant symplectic embedding for all s < r with h(0) = p. The
converse follows from the first statement of the lemma.
Note that µM(f(Br)) only depends on the fixed point p and the radius r (which was fixed a
priori) and not on f . In Figure 3 several momentum ball images are drawn using Lemma 2.3. Note
that the shaded triangle on the right picture is not a Delzant polytope since it fails to be smooth at
(0, 0). Delzant polytopes are simple, edge–rational and smooth polytopes, cf. Figure 2 (see [10]
or [6] for a definition of these notions).
Step 2: A deformation retraction on BTnr .
In this step we use Alexander’s trick to construct a deformation retraction from the space of
equivariant symplectomorphisms of the 2n-dimensional ball Br in C
n onto a disjoint union of
copies of Tn. The continuity of this deformation is standard and may be found in [13].
Lemma 2.4. The space BTnr of equivariant symplectomorphisms of the 2n-dimen-
sional ball Br in C
n, with respect to the standard symplectic form σ0 and the canonical action
of Tn by rotations, deformation retracts onto its subspace of linear, equivariant and symplectic
rotations given by matrices in R.
Proof. We define the transformationHT
from BTnr × [0, 1] into BT
r , by the formulaH
(φ, t) :=
φt, where φt is the composite map
φt := (mt)
−1 ◦ φ ◦mt, t 6= 0. (2.7)
The map mt in expression (2.7) denotes the linear contraction of factor 0 ≤ t ≤ 1 on Br, mt(z) =
t z; and when t = 0, φt = φ0 is defined to be the tangent mapping Tφ of the map φ, evaluated at 0.
(This expression for φt is known as Alexander’s trick.) It is easy to check that H
is continuous
and that the evaluation map [0, 1] × Br → Br given by (t, z) 7→ φt(z) is smooth. Since HT
the identity on linear maps, we conclude that it is a deformation retraction, not only a homotopy,
onto the space of ball rotations by matrices (δi τ(i)θi j)
i, j=1 with τ ∈ Sn (the symmetric group) and
θi j ∈ T1.
We have left to check that HT
is well defined, i.e. that φt ∈ BT
r . Indeed, the equivariance
of the mapping φt follows directly from formula (2.7); explicitely we have that if φ is equivariant
with respect to Λ ∈ Aut(Tn), then φt(s · z) = 1/t φ(s · tz) = Λ(s) · φt(z) for all s ∈ Tn. By
differentiating formula (2.7) we obtain that
T(φt) = (mt)
−1 ◦ Tφ ◦mt, (2.8)
and since mt is a linear isomorphism, the mapping φt is a diffeomorphism. Furthermore, since
the mapping φ is symplectic, it follows from expression (2.8) that for all z ∈ Br we have that
(φ∗tσ0)z(u, v) = (σ0)φ(z)(Tzφ(u), Tzφ(v)) = (σ0)z(u, v), for every pair of vectors u, v ∈ TzBr,
and hence φt is a symplectic mapping. Therefore φt is a diffeomorphism, which is equivariant and
symplectic, or equivalently φt ∈ BT
r . We have been assuming that t 6= 0, but if t = 0, it is trivial
that φ0 ∈ BT
Corollary 2.5. The space BTnr of equivariant symplectomorphisms of the 2n-dim-
ensional ball Br in C
n, with respect to the standard symplectic form σ0 and the canonical action
of Tn by rotations, is homotopically equivalent to a disjoint union of n! copies of Tn.
Proof. Apply Lemma 2.4 and observe that the space of ball rotations by matrices (δi τ(i)θi j)
i, j=1
with τ ∈ Sn and θi j ∈ T1 is homotopically equivalent to a disjoint union of n! copies of Tn.
We conclude the proof with Step 3, in which Lemma 2.1, Lemma 2.3 and Lemma 2.4 are
combined in order to prove Theorem 1.2. The proof of Proposition 1.4 will follow from the proof
of Theorem 1.2, since the function Emb(M,σ) will be explicitly computed.
Step 3: Lifting the deformation φt to ET
x and conclusion.
In this final step we show that ETnx is homotopically equivalent to a disjoint union of copies of
Lemma 2.6. Suppose that the infimum of the SL(n, Z)-lengths of the edges of ∆M meeting at x is
strictly greater than r2. Then there exists an equivariant and symplectic embedding u : Br → M
with u(0) = p such that if ρ is the identification map on ETnx which takes values on BT
r and is
given by formula ρ(h) := u−1 ◦ h, where h ∈ ETnx , the space ET
x is homotopically equivalent to
the space ρ−1(R).
Proof. The first observation is that by Lemma 2.3 there exists a Λ-equivariant and symplectic
embedding u from Br intoM with u(0) = p. In order to construct homotopy equivalences between
ETnx and R, we define ρ to be the identification map on ET
x which takes values in BT
r and is given
by formula ρ(h) := u−1◦h for every h ∈ ETnx . Now we claim that the mapHT
x from ET
x ×[0, 1] to
ETnx , given by the commutative diagram (2.9) below, is a well–defined and continuous homotopy
satisfying HT
x (ET
x × {0}) = ρ−1(R), while ET
x is preserved at time t = 1, i.e. we have that
x (ET
x × {1}) = ET
x . The diagram is the following:
ETnx × [0, 1]
BTnr × [0, 1]
Br // BTnr
. (2.9)
The mapping HT
x is well defined by Lemma 2.1. Note that H
x is continuous, since the identifi-
cations ρ and ρ−1 are obviously continuous and we showed in Lemma 2.4 that HT
is continuous.
We can therefore conclude, from the previous considerations and the fact that that HT
is a de-
formation retraction in the Cm-Whitney topology, that HT
x induces homotopy equivalences ρ and
δ(f) := HT
x (f, 0) between ET
x and ρ
−1(R), with ρ◦δ homotopic to idETn
and δ◦ρ = idρ−1(R).
In order to conclude the proof of Theorem 1.2 we simply make the following observations:
• First, the space described in it is precisely the disjoint union of the ETnx , x being a vertex of
∆M , because 0 is to be mapped to a fixed point of ψ.
• The number of Tn-fixed points, which is the same as the number of vertices of ∆M , is pre-
cisely χ(M). This follows from the analysis of the momentum map as in Atiyah–Delzant–
Guillemin–Sternberg theory (see for example [10], [11]).
• If we denote by Emb(M,σ)(r) the number of copies of Tn onto which the space considered in
Theorem 1.2 retracts (see formula (1.2)), Emb(M,σ)(r) is obtained by multiplying the number
of fixed points that admit such an embedding (see Lemma 2.3) by the number of copies of
n onto which ETnx (for the particular point) retracts; this latter number is n! (see Corollary
2.5), i.e. as many copies of Tn as possible ways that the canonical basis vectors ei may be
mapped onto the basis of weights αpi (for the particular point). Also, the former number is by
Lemma 2.3 controlled by the Boolean variable cp(r) defined in Proposition 1.4. Therefore
Emb(M,σ)(r) is given by Emb(M,σ)(r) = n!
p∈MTn cp(r), as we wanted to show.
• It is obviously true that if (M, σ) is equivariantly symplectomorphic to (M̃, σ̃), then Emb(M,σ)(r) =
(M̃, σ̃)
(r), so the integer Emb(M,σ)(r) is a symplectic–toric invariant.
(2,1)
(1,2)(0,2)
(2,0)(0,0)
(2,1)
(1,2)(0,2)
(2,0)(0,0)
Figure 4: Polytope corresponding to the Delzant manifold (M, σ) obtained by blowing up S2r0×S2r0
with r0 = 1/
2. Observe that Emb(M,σ)(
2) = 0 (proof in left figure) and Emb(M,σ)(1/
10 (proof in right figure). See Lemma 2.7.
As a final remark we observe that the invariant function Emb(M,σ) associated to the Delzant
manifold (M, σ) always reaches its minimum and maximum values on an interval of strictly posi-
tive length.
Lemma 2.7. There exist numbers r0, s0 > 0 such that if r ≤ r0, then the space of equivariant
symplectic embeddings from Br into M is homotopically equivalent to a disjoint union of n!χ(M)
copies of Tn, and if s ≥ s0, then it is empty.
Proof. It follows easily from Lemma 2.3, Corollary 2.5 and the previous observations.
This concludes the proof of Theorem 1.2 (and hence by construction the proof of Proposition
1.4).
3 Remarks on the partially equivariant case of Theorem 1.2
In this section we initiate a discussion on the topology of the space of partially equivariant sym-
plectic embeddings and sketch some suggestions to answer a question in this direction.
First the notion of Λ-equivariance (Λ ∈ Aut(Tn)) in Section 1 has a natural extension: we
say that an embedding from the 2n-ball Br into the 2n-dimensional Delzant manifold M is γ-
-equivariant with respect to a monomorphism γ : Tn−k → Tn, 1 ≤ k ≤ n − 1, if the following
diagram commutes:
n−k × Br
γ×f //
f // M
For example, Mγ is the set of p ∈ M such that ψ(γ(t), p) = p for all t ∈ Tn−k, and the rest of
terminology is also analogous. This definition extends naturally to the case when k = n, in which
the embeddings considered are purely symplectic, as well as to the case when k = 0, in which the
embeddings are fully equivariant, case which we treated previously in the paper. Unless otherwise
specified we do not consider these two cases in the discussion that follows. The question we would
like to address is the following:
Question 3.1 Let r be such that any connected component C ofMγ admits a Darboux–Weinstein
neighborhood of radius r, and by this we mean a neighborhood that is equivariantly symplecto-
morphic to a bundle over C with fiber the standard ball of radius r. Is the space of γ-equivariant
symplectic embeddings from Br intoM homotopically equivalent to the space of purely symplectic
embeddings from B2kr into M
γ up to reparametrization groups (as explained below)? ⊘
To analyze Question 3.1 first define B̂2kr to be the embedded 2k-ball in Br, i.e. the set of
points (z1, . . . , zk, 0) in Br so that
i=1 |zi|2 ≤ r2. The preimage under the momentum map of
the k-face corresponding to γ is the fixed point locus Mγ . Now consider any symplectic embed-
ding f : B̂2kr → Mγ . We want to find a canonical way to extend f to an equivariant symplectic
embedding can(f) : Br →M up to homotopy.
Here is an attempt to construct can(f): near the image of f , we can apply the equivariant ver-
sion of the Darboux–Weinstein’s theorem in order to find a neighborhood of Im(f) in M which
is symplectomorphic to Im(f)× B2(n−k), with the action of Tn−k given by the standard action on
2(n−k), and the symplectic form coinciding with the product symplectic form. Note that the sym-
plectic normal bundle toMγ is trivial over Im(f) because Im(f) is contractible, so a neighborhood
of Im(f) looks like Im(f)× B2(n−k) with a product symplectic form, and the action of Tn−k on it
is conjugate to the standard one. Using this identification M is described as a product, and we can
define can(f)(z) := (f(z1, . . . , zk), zk+1, . . . , zn). This expression for can(f) is clearly symplec-
tic and equivariant with respect to Tn−k-actions on the last n− k coordinates but is not canonical
because the local symplectomorphism given by Darboux–Weinstein’s theorem is not unique. We
cannot expect it to always be the same independently of f , because it is not true that globally the
normal bundle to Mγ is symplectically trivial, it only becomes true over a neighborhood of Im(f).
So this construction depends on choices of parameters.
Calling CAN the space of canonical embeddings can(f) : Br → ∆M , where f : B̂2kr → Mγ is
a symplectic embedding, observe that CAN is naturally identified with the space of purely sym-
plectic embeddings from the standard B2kr into M
γ , up to homotopy. The question then becomes
whether any γ-equivariant symplectic embedding f : B2kr → M may be deformed through a con-
tinuous family of equivariant symplectic embeddings to an embedding in CAN.
Equivalently, we ask the question: is the natural map between the space of partially equivariant
embeddings from Br into M and the space of symplectic embeddings from B
r into the fixed point
set Mγ (given by the restriction to the fixed ball B2kr ) a fibration? Note that the construction of
can(f) would give a section of this fibration.
Conjecture 3.1. Question 3.1 has an affirmative answer.
Example 3.2 If M = S2 × S2 with a product symplectic form and product T2-action (this space
has been carefully studied by Lalonde–Pinsonnault [14] and Anjos [2] among other authors), the
fixed point locus of the second S1 factor is S2×{a, b}, where a, b are the fixed points of the action
of S1 on S2. Now, given a symplectic embedding f of the ball B2 into S2, it is easy to build an
1-equivariant embedding of B4 into S2 × S2 canonically by (z1, z2) 7→ (f(z1), z2), where z2 is
taken to be a coordinate centered at the fixed point a. In this case the normal bundle to the fixed
point component S2 × {a} is globally trivial. ⊘
The combination of purely symplectic results of Biran, Lalonde–Pinsonnault and others and
an affirmative answer to Question 3.1 would give insight into the partially equivariant case in
higher dimensions; for example McDuff showed that if M is a symplectic 4-manifold with non-
simple Seiberg–Witten type, then the space of symplectic embeddings from Br into M is path
connected (which extends results of Biran). This is a consequence of the non–trivial result: any
two cohomologous and deformation equivalent symplectic forms on M are isotopic (proved in
[17]). Examples are known in dimensions 6 and above of cohomologous symplectic forms that are
deformation equivalent but not isotopic, so these techniques do not help to understand the topology
of the space of symplectic embeddings from Br into M . A positive answer to Question 3.1 would
give the first non–trivial result in dimension 6.
Another way of trying to generalize Theorem 1.2 is to consider embeddings equivariant with
respect to a complexity one action, that is, an action of Tn−1 on M2n. This is a hopeful approach
since a complete classification of complexity one actions has been recently achieved by Karshon
and Tolman [15].
Acknowledgments
The author is grateful to D. Auroux and V. Guillemin for discussions, and for hosting him at the
M.I.T. regularly during the Fall and Spring semesters of 2003 and 2004. He thanks D. Auroux,
J.J. Duistermaat, Y. Karshon and M. Pinsonnault for making comments on a preliminary version
of this paper. Finally, the author is grateful to an anonymous referee for helpful suggestions that
have shortened the proof in Section 2, as well as for his/her interesting comments on Section 3.
References
[1] M. Atiyah, Convexity and commuting Hamiltonians. Bull. London Math. Soc., 1 (1982) 1–15.
[2] S. Anjos, Homotopy type of symplectomorphism groups of S2 × S2. Geom. Topol. 6 (2002)
195–218.
[3] P. Biran, Connectedness of spaces of symplectic embeddings, Int. Math. Res. Lett., 10 (1996)
487–491.
[4] P. Biran, From symplectic packing to algebraic geometry and back, Proc. 3rd Europ. Congr.
of Math., Barcelona 2, Prog. Math., Birkhauser (2001) 507–524.
[5] P. Biran, Geometry of symplectic intersections, Proc. ICM Beijing III (2004) 1-3.
[6] A. Cannas da Silva, Lectures in symplectic geometry, Springer-Verlag, Berlin (2000).
[7] T. Delzant, Periodic Hamiltonians and convex images of the momentum mapping, Bull. Soc.
Math. France, 116 (1988) 315–339.
[8] V. Guillemin and S. Sternberg, Convexity properties of the momentum mapping, Invent.
Math., 67 (1982) 491–513.
[9] V. Guillemin and S. Sternberg, Symplectic techniques in physics, Cambr. Univ. Press (1984).
[10] V. Guillemin, Moment maps and combinatorial invariants of Hamiltonian Tn spaces, Prog.
Math., Birkhauser (1994).
[11] V. Guillemin, V. Ginzburg and Y. Karshon, Moment maps, cobordisms and Hamiltonian
group actions, Amer. Math. Soc. Monograph 98, (2002).
[12] M. Gromov, Pseudoholomorphic curves in symplectic geometry, Invent. Math., 82 (1985)
307–347.
[13] M. Hirsch, Differential topology. Springer-Verlag 33 (1980).
[14] F. Lalonde and M. Pinsonnault, The topology of the space of symplectic balls in rational
4-manifolds, Duke Math. J. 122 (2004), no. 2, 347–397.
[15] Y. Karshon and S. Tolman, Centered complexity one Hamiltonian torus actions, Trans. Amer.
Math. Soc. 353 (2001), no. 12, 4831–4861 (electronic).
[16] Y. Karshon and S. Tolman, The Gromov width of complex Grasmannians, Alg. and Geom.
Topol. 5 paper 38, 911–922.
[17] D. McDuff, From deformation to isotopy, Topics in Symplectic 4-manifolds, Internat. Press,
Cambridge, MA (1998).
[18] D. McDuff, The structure of rational and ruled symplectic 4-manifolds, J. Amer. Math. Soc.
3(3) (1990) 679–712.
[19] A. Pelayo, Toric symplectic ball packing, Top. and its Appl. 153 (2006) 3633–3644.
A. Pelayo
Department of Mathematics, University of Michigan
2074 East Hall, 530 Church Street, Ann Arbor, MI 48109–1043, USA
e-mail: apelayo@umich.edu
	The main theorem
	Proof of Theorem ??
	Remarks on the partially equivariant case of Theorem ??

  We compute the homotopy type of the space of T^n-equivariant symplectic
embeddings from the standard 2n-dimensional ball of some fixed radius into a
2n-dimensional symplectic-toric manifold M, and use this computation to define
a Z-valued step function on the positive real line which is an invariant of the
symplectic-toric type of M. We conclude with a discussion of the partially
equivariant case of this result.

Topology of spaces
of equivariant symplectic embeddings
Alvaro Pelayo
Abstract
We compute the homotopy type of the space of Tn-equivariant symplectic embeddings
from the standard 2n-dimensional ball of some fixed radius into a 2n-dimensional symplectic–
toric manifold (M, σ), and use this computation to define a Z≥0-valued step function on R≥0
which is an invariant of the symplectic–toric type of (M, σ). We conclude with a discussion
of the partially equivariant case of this result.
1 The main theorem
Let (M, σ) be a 2n-dimensional symplectic manifold and write Br for the compact 2n-ball of
radius r > 0 in the complex space Cn equipped with the restriction of the standard symplectic
form σ0 of C
n. (The proofs of the results in this paper hold verbatim for the open ball.) Recently a
lot of effort has been put into understanding the topological and geometric properties of the space
of symplectic embeddings from Br into M . This question is not only intriguing, but it is also very
fundamental because it acknowledges one of the main differences that exist between Riemannian
and symplectic geometry, e.g. Gromov’s non–squeezing theorem [12].
Figure 1: An equivariant and symplectic embedding B2r → S2s with r/s =
This question, posed with such generality, has proven to be extremely difficult to answer.
Significant progress has been made by McDuff [17], [18], Biran [3], [5] and most recently by
Lalonde–Pinsonnault [14], among other authors. One of the most general results is due to McDuff;
she showed the connectedness of the space of 4-balls into 4-manifolds with non-simple Seiberg–
Witten type, in particular rational or ruled surfaces. Recall that we say that a symplectic 4-manifold
Q has simple Seiberg–Witten type or just simple type if the only non–zero Gromov invariants of Q
occur in classes A ∈ H2(Q) for which k(A) = K · A + A2 = 0. It follows from work of Taubes
http://arxiv.org/abs/0704.1033v1
and Li–Liu that the symplectic 4-manifolds with non–simple type are blow–ups of (i) rational and
ruled manifolds; (ii) manifolds with b1 = 0, b
2 = 1, like the Enriques or Barlow surface; and
(iii) manifolds with b1 = 2 and (H1(X))
2 6= 0; examples (with K = 0) are hyperelliptic surfaces,
some non–Kähler T2-bundles over T2 and quotients T2 × Σg/G where Σg is a surface of genus g
greater than 1, and G is certain finite group. See [17] for further references and examples.
McDuff’s techniques are unique to dimension 4 and do not extend at all to higher dimensions—
this is also the case in the other authors’ work—existence of J-holomorphic curves with special
homological properties is essential in their proofs. Although J-holomorphic curves exist in all
even dimensions, it is only in dimension 4 where these homological properties hold.
In the present paper we study a special case of this question: M is a symplectic–toric manifold
of arbitrary dimension, and the symplectic embeddings that we consider preserve the toric struc-
ture, see Figure 1. Precisely this means that there exists an automorphism Λ of the n-torus Tn such
that the following diagram commutes:
n × Br
Λ×f //
f // M
, (1.1)
where ψ is a fixed effective and Hamiltonian Tn-action on M and · denotes the standard action by
rotations on Br (component by component). In this case we say that f is a Λ-equivariant mapping.
(1,0)(0,0)
(1,1)(0,1)(1,1)(0,1)
(2,0)(0,0)(1,0)(0,0)
(0,1)
Figure 2: The momentum polytope of CP2 and B21 (left), of a Hirzebruch surface (center) and of
(CP1)2 (right).
The feature that makes the study of symplectic manifolds equipped with Hamiltonian torus
actions richer than the study of generic symplectic manifolds is the presence of the smooth mo-
mentum map µM : M → Lie(Tn)∗, whose image ∆M is a convex polytope (called the momentum
polytope of M , cf. Figure 2) as shown independently by Atiyah and Guillemin–Sternberg [1], [8].
Here we are identifying the Lie algebra Lie(Tn) and its dual Lie(Tn)∗ with Rn. Since this identi-
fication is not canonical, we need to specify the convention we adopt in this paper. This amounts
to choosing an epimorphism R → T1 which we take to be x 7→ e2
−1x. This epimorphism in-
duces an isomorphism between Lie(T1) and R via ∂
7→ 1/2, giving rise to a new isomorphism
Lie(Tn) → Rn, ∂
7→ 1/2 ek, by canonically identifying Lie(Tn) with the product of n copies of
Lie(T1) (see [9] for more details).
For example, under the convention of the previous paragraph, the momentum map µBr of Br
is a mapping from Br into R
n with components µBrk (z) = |zk|2, for all integer k with 1 ≤ k ≤ n.
(There are a number of different conventions used in the literature, and our choice is intended to
give the simplest formula for the momentum map of Br.) The simplest symplectic manifolds which
admit Hamiltonian effective torus actions are called symplectic–toric.
Definition 1.1 A symplectic–toric manifold M , also called a Delzant manifold, is a compact
connected symplectic manifold equipped with an effective Hamiltonian action of a torus of dimen-
sion half of the dimension of the manifold. In this case the momentum polytope ∆M is called the
Delzant polytope of M . ⊘
Symplectic–toric manifolds were classified by Delzant in [7]. In particular, he showed that the
momentum image of such a manifold under the momentum map completely determines M up to
equivariant symplectomorphisms.
The main result of this paper, Theorem 1.2 below, describes the topology of the space of equiv-
ariant embeddings of symplectic balls into a symplectic–toric manifold. We denote by χ(M) the
Euler characteristic of M .
Theorem 1.2. For every symplectic–toric 2n-manifold M there is an associated Z-valued non–
increasing step function Emb(M,σ) : R≥0 → [0, n!χ(M)] such that for each r ≥ 0 the space of
equivariant symplectic embeddings from the 2n-ball Br into M is homotopically equivalent to a
disjoint union of Emb(M,σ)(r) subspaces, each of which is homeomorphic to the n-torus T
As a matter of fact we can explicitly and easily read Emb(M,σ) from the polytope ∆
Example 1.3 Let (M, σ) equal the blow–up of S2r0×S
with r0 = 1/
2 whose Delzant polytope
has vertices at (0, 0), (2, 0), (2, 1), (1, 2) and (0, 2) (see Figure 4). Then Emb(M,σ) = 10χ[0,1) +
2χ[1,
2), where χA denotes the characteristic function of A ⊂ R. We identify the 2-sphere of
radius r equipped with the standard area form with (CP1, 4r2 ·σFS), where σFS is the Fubini–Study
form. ⊘
Proposition 1.4. The function Emb(M,σ)(r) given in Theorem 1.2 is an invariant of the symplectic–
toric type of M and is given by the formula
Emb(M,σ)(r) = n!
p∈MTn
cp(r), (1.2)
where for each fixed point p ∈M , cp(r) = 1 if the infimum of the SL(n, Z)-lengths of the edges of
∆M meeting at µM(p) is strictly greater than r2, and cp(r) = 0 otherwise.
Example 1.5 Let σFS be the Fubini–Study form on CP
n and observe that Tn acts naturally on
(CPn, λ · σFS), λ > 0, with n + 1 fixed points. The momentum polytope is a tetrahedrum with
vertices at 0 and λ ei, where the ei are the canonical basis vectors in R
n. So if M = CPn × CPm,
the space of equivariant symplectic embeddings from Br into M is homotopically equivalent to
(n+m)!(n+1)(m+1)⊔
if r <
λ, and it is empty otherwise. ⊘
The study of the space of symplectic embeddings is directly related to the study of the sym-
plectic ball packing problem cf. [4], the equivariant version of which was treated in [19].
2 Proof of Theorem 1.2
In this section we prove Theorem 1.2. For clarity, the proof is divided in three steps, which we
describe next.
We start by introducing the notation and making the following observations:
i) Throughout the proof Rwill denote the space of rotations by matrices of the form (δi τ(i)θi j)ni, j=1
with τ ∈ Sn (the symmetric group) and θi j ∈ T1, and ET
x will denote the space of equivari-
ant symplectic embeddings f from Br into M such that f(0) = p and µ
M(p) = x ∈ ∆M ,
equipped with the Cm-Whitney topology (m ≥ 0). Throughout the present section we fix f .
Since each component of R is canonically identified with the n-torus Tn (cf. Corollary 2.5),
Theorem 1.2 amounts to prove that if p ∈MTn (MTn denotes the Tn-fixed point set) is such
that µM(p) = x, then the space ETnx gets identified with R via a homotopy equivalence.
ii) Secondly let BTnr denote the space of equivariant symplectomorphisms of Br (again with
respect to the Cm-Whitney topology).
Recall that, for example, the Cm-Whitney topology on BTnr is given by the well–known norm
‖ φ ‖Cm= max
0≤k≤m
‖ Tkφ ‖M(C),
where we are taking the norm ‖ · ‖M(C) on the right–hand side of this expression to be the
canonical Euclidean norm on the space of n× n matrices with complex entries.
iii) We identify the automorphism group Aut(Tn) with the matrix group
GL(n, Z).
iv) The elements αpi ’s, 1 ≤ i ≤ n, denote the weights of the isotropy representation of Tn on
TpM ; the canonical basis vectors ei ∈ Rn represent the weights of the isotropy representa-
tion of Tn on T0Br.
Step 1: Invariance of the image f(Br).
In this step we first show how to go from smooth maps on manifolds to affine maps on polytopes
(see diagram (2.3)), and secondly we use this to show the invariance of the image f(Br) ⊂ M .
Precisely, one can think of an embedding being equivariant in the sense of commuting with the
n-action, and it is when we reparametrize the torus that Λ appears.
Lemma 2.1. Let g be any Λ-equivariant and symplectic embedding such that the normalization
condition f(0) = g(0) = p holds. Then for all z ∈ Br, if Tn · z denotes the Tn-orbit that passes
through z, the identity f(Tn · z) = g(Tn · z) holds, and therefore f(Br) = g(Br).
Proof. Let f, g be Λ-equivariant and symplectic embeddings from Br into M with f(0) = g(0) =
p. Under the identifications described in Section 1, the following diagram commutes, where the
top arrow stands for the affine map with linear part (Λt)−1, which takes 0 to x:
(Λt)−1+x
// ∆M
. (2.3)
In order to prove the commutativity of diagram (2.3), we denote by ξM the vector field induced
by the element ξ ∈ Lie(Tn) via the exponential map and note that from the definition of the
momentum maps µM and µBr , the Λ-equivariance of f , and the fact that f ∗σ = σ0, we have
the following sequence of equalities, where Tf(z)µ
M and Tzµ
Br denote, respectively, the tangent
mapping of µM at f(z) and of µBr at z,
〈TzµBr(v), ξ〉z = (σ0)z(v, ξBr(z))
= σf(z)(Tzf(v), Λ(ξ)M(f(z)))
= 〈Tf(z)µM(Tzf(v)), Λ(ξ)〉f(z)
= 〈Λt ◦ Tf(z)µM(Tzf(v)), ξ〉z, (2.4)
where z ∈ Br, v ∈ TzBr and ξ ∈ Lie(Tn). Therefore by equation (2.4) and by using the chain rule
we obtain that for all z ∈ Br and v ∈ TzBr
Br(v) = Tz(Λ
t ◦ µM ◦ f)(v). (2.5)
Considering equation (2.5), f(0) = p and µM(p) = x and composing with (Λt)−1, after integration
we obtain the commutativity condition on diagram (2.3). Notice that diagram (2.3) also holds for
the embedding g.
Then it follows from the conjunction of diagram (2.3) and diagram (1.1) that for all t ∈ Tn the
following identities hold:
µM(ψ(Λ(t), f(z))) = µM(f(t · z))
= (Λt)−1 ◦ µBr(z) + x
= µM(g(t · z)) = µM(ψ(Λ(t), g(z))). (2.6)
Expression (2.6) is clearly equivalent to µM(Tn · f(z)) = µM(Tn · g(z)), since Λ is an auto-
morphism. Now since M is symplectic–toric, by the proof of the Atiyah–Guillemin–Sternberg
convexity theorem we know that each fiber of the momentum map µM consists of a single con-
nected orbit which together with the last equality implies that Tn · f(z) = Tn · g(z). Since f(Br)
is the union of the orbit images f(Tn · z), we immediately obtain that f(Br) = g(Br), which
concludes the proof.
It is possible to explicitly describe the momentum image µM(f(Br)), and for this purpose we
recall the notion of SL(n, Z)–length: if x, y ∈ Rn, we say that a segment line [x, y] joining x to y
(0,0)
(0,1) (1,1)
(1,0)
(0,1) (1,1)
(0,0) (1,0)
Figure 3: Equivariant symplectic ball embeddings in (CP1)2 (left); the triangle on the right does
not come from such embedding.
in Rn has SL(n, Z)-length d if there exists a matrix A ∈ SL(n, Z) such that A(d e1) = y − x (d
is not defined in general, only for segments of rational slope). The Euclidean length of a segment
line agrees with its SL(n, Z)-length if and only if the segment is parallel to one of the coordinate
axes in Rn.
In their article [16], Karshon and Tolman made the following two definitions. Let (Q, σQ) be
a connected symplectic 2m-dimensional manifold with momentum map µQ for an action of an
m-torus Tm on Q, and let Γ ⊂ (Lie(Tm))∗ be an open convex subset which contains the image
of Q under the momentum map µQ. The quadruple (Q, σQ, µQ, Γ) is a proper Hamiltonian Tm-
manifold if the momentum map µQ is proper as a map to Γ.
The proper Hamiltonian Tm-manifold (Q, σ, µQ, Γ) is said to be centered about a point α ∈ Γ
if α is contained in the momentum map image of every component of QK , for each K ⊂ Tm. Here
QK := {q ∈ Q |ψQ(a, q) = q, ∀a ∈ K},
where ψQ : T
m ×Q→ Q denotes the action of Tm on Q. The following lemma is Proposition 2.8
in [16].
Lemma 2.2 (Karshon–Tolman, [16]). Let the quadruple (Q, σQ, µQ, Γ) be a proper Hamiltonian
2m-dimensional Tm-manifold. Suppose that (Q, σQ, µQ, Γ) is centered about α ∈ Γ and that the
preimage (µQ)−1({α}) consists of a single fixed point q. Then Q is equivariantly symplectomor-
phic to
{z ∈ Cm |α+
|zj |2 ηqj ∈ Γ},
where ηq1, . . . , η
m are the weights of the isotropy representation of T
m on TqQ.
We use Lemma 2.2 in order to prove the following lemma.
Lemma 2.3. The momentum image µM(f(Br)) equals the subset of R
n given by the convex hull of
x and x+ r2 αpi , where 1 ≤ i ≤ n. Furthermore, the infimum of the SL(n, Z)-lengths of the edges
of ∆M meeting at x is greater than or equal to r2, if and only if for all 0 < s < r there exists an
embedding h : Bs → M which is Λ-equivariant and symplectic, satisfying h(0) = p.
Proof. The first observation is that ∆Br equals the convex hull in Rn of 0 and r2 e1, . . . , r
2 en.
Secondly, since µBr : Br → ∆Br is onto, it follows from diagram (2.3) that
µM(f(Br)) = (Λ
t)−1(∆Br) + x.
Since Λ is an automorphism, (Λt)−1 is an automorphism of the corresponding dual spaces and
therefore there exists a permutation τ ∈ Sn such that (Λt)−1(ei) = αpτ(i). Then the linearity of Λ
implies that µM(f(Br)) equals the the convex hull in R
n of the points x and x+ r2 αpi , 1 ≤ i ≤ n,
which proves the first claim.
Suppose that the infimum of the SL(n, Z)-lengths of the edges meeting at x is greater than or
equal to r2. Let Σ be the convex hull of x and x + r2 αpi , with 1 ≤ i ≤ n, and let Z be the convex
hull of x+r2 αpi , with 1 ≤ i ≤ n. Notice that Σ ⊂ ∆M , Σ\Z is open in ∆M , and let Γ ⊂ Rn be the
open half–space of Rn, whose closure’s boundary ∂(cl(Γ)) is the hyperplane of Rn that contains
Z, and such that Σ \ Z ⊂ Γ.
Let N := (µM)−1(Σ \ Z) and let σN be the symplectic form obtained by restricting σ to N .
The set N is open in M because it is the preimage of the open set Σ \Z under the momentum map
µM : M → ∆M . By the proof of Atiyah–Guillemin–Sternberg convexity theorem, cf. [1], [8], N
is a connected manifold. Since M is compact, the momentum map µM : M → ∆M is a proper
map and therefore its restriction µM : N → Σ \Z is a proper map, which means that µM : N → Γ
is proper, since (µM)−1(Γ \ (Σ \ Z)) = ∅. Therefore (N, σN , ψN) is a connected symplectic
manifold with momentum map µM , and the quadruple (N, σN , µM , Γ) is a proper Hamiltonian
n-space.
On the other hand, notice that the quadruple (N, σN , µM , Γ) is centered about the point x, and
(µM)−1({x}) = p, so we can apply Lemma 2.2, and conclude that N is equivariantly symplecto-
morphic to the submanifold X ⊂ Cn given by
X := {z ∈ Cn | x+
|zi|2 αpi ∈ Σ \ Z}
= {z ∈ Cn | x+
|zi|2 αpi ∈ Σ} \ {z ∈ Cn | x+
|zi|2 αpi ∈ Z}
= Br \ ∂Br = Int(Br).
Hence there exists an equivariant symplectomorphism φ : Int(Br) → N , and by letting j : Bs →
Int(Br) be the standard inclusion, if s < r, the map h := jN ◦φ◦ j : Bs → M , where jN : N →M
is the inclusion map, is an equivariant symplectic embedding for all s < r with h(0) = p. The
converse follows from the first statement of the lemma.
Note that µM(f(Br)) only depends on the fixed point p and the radius r (which was fixed a
priori) and not on f . In Figure 3 several momentum ball images are drawn using Lemma 2.3. Note
that the shaded triangle on the right picture is not a Delzant polytope since it fails to be smooth at
(0, 0). Delzant polytopes are simple, edge–rational and smooth polytopes, cf. Figure 2 (see [10]
or [6] for a definition of these notions).
Step 2: A deformation retraction on BTnr .
In this step we use Alexander’s trick to construct a deformation retraction from the space of
equivariant symplectomorphisms of the 2n-dimensional ball Br in C
n onto a disjoint union of
copies of Tn. The continuity of this deformation is standard and may be found in [13].
Lemma 2.4. The space BTnr of equivariant symplectomorphisms of the 2n-dimen-
sional ball Br in C
n, with respect to the standard symplectic form σ0 and the canonical action
of Tn by rotations, deformation retracts onto its subspace of linear, equivariant and symplectic
rotations given by matrices in R.
Proof. We define the transformationHT
from BTnr × [0, 1] into BT
r , by the formulaH
(φ, t) :=
φt, where φt is the composite map
φt := (mt)
−1 ◦ φ ◦mt, t 6= 0. (2.7)
The map mt in expression (2.7) denotes the linear contraction of factor 0 ≤ t ≤ 1 on Br, mt(z) =
t z; and when t = 0, φt = φ0 is defined to be the tangent mapping Tφ of the map φ, evaluated at 0.
(This expression for φt is known as Alexander’s trick.) It is easy to check that H
is continuous
and that the evaluation map [0, 1] × Br → Br given by (t, z) 7→ φt(z) is smooth. Since HT
the identity on linear maps, we conclude that it is a deformation retraction, not only a homotopy,
onto the space of ball rotations by matrices (δi τ(i)θi j)
i, j=1 with τ ∈ Sn (the symmetric group) and
θi j ∈ T1.
We have left to check that HT
is well defined, i.e. that φt ∈ BT
r . Indeed, the equivariance
of the mapping φt follows directly from formula (2.7); explicitely we have that if φ is equivariant
with respect to Λ ∈ Aut(Tn), then φt(s · z) = 1/t φ(s · tz) = Λ(s) · φt(z) for all s ∈ Tn. By
differentiating formula (2.7) we obtain that
T(φt) = (mt)
−1 ◦ Tφ ◦mt, (2.8)
and since mt is a linear isomorphism, the mapping φt is a diffeomorphism. Furthermore, since
the mapping φ is symplectic, it follows from expression (2.8) that for all z ∈ Br we have that
(φ∗tσ0)z(u, v) = (σ0)φ(z)(Tzφ(u), Tzφ(v)) = (σ0)z(u, v), for every pair of vectors u, v ∈ TzBr,
and hence φt is a symplectic mapping. Therefore φt is a diffeomorphism, which is equivariant and
symplectic, or equivalently φt ∈ BT
r . We have been assuming that t 6= 0, but if t = 0, it is trivial
that φ0 ∈ BT
Corollary 2.5. The space BTnr of equivariant symplectomorphisms of the 2n-dim-
ensional ball Br in C
n, with respect to the standard symplectic form σ0 and the canonical action
of Tn by rotations, is homotopically equivalent to a disjoint union of n! copies of Tn.
Proof. Apply Lemma 2.4 and observe that the space of ball rotations by matrices (δi τ(i)θi j)
i, j=1
with τ ∈ Sn and θi j ∈ T1 is homotopically equivalent to a disjoint union of n! copies of Tn.
We conclude the proof with Step 3, in which Lemma 2.1, Lemma 2.3 and Lemma 2.4 are
combined in order to prove Theorem 1.2. The proof of Proposition 1.4 will follow from the proof
of Theorem 1.2, since the function Emb(M,σ) will be explicitly computed.
Step 3: Lifting the deformation φt to ET
x and conclusion.
In this final step we show that ETnx is homotopically equivalent to a disjoint union of copies of
Lemma 2.6. Suppose that the infimum of the SL(n, Z)-lengths of the edges of ∆M meeting at x is
strictly greater than r2. Then there exists an equivariant and symplectic embedding u : Br → M
with u(0) = p such that if ρ is the identification map on ETnx which takes values on BT
r and is
given by formula ρ(h) := u−1 ◦ h, where h ∈ ETnx , the space ET
x is homotopically equivalent to
the space ρ−1(R).
Proof. The first observation is that by Lemma 2.3 there exists a Λ-equivariant and symplectic
embedding u from Br intoM with u(0) = p. In order to construct homotopy equivalences between
ETnx and R, we define ρ to be the identification map on ET
x which takes values in BT
r and is given
by formula ρ(h) := u−1◦h for every h ∈ ETnx . Now we claim that the mapHT
x from ET
x ×[0, 1] to
ETnx , given by the commutative diagram (2.9) below, is a well–defined and continuous homotopy
satisfying HT
x (ET
x × {0}) = ρ−1(R), while ET
x is preserved at time t = 1, i.e. we have that
x (ET
x × {1}) = ET
x . The diagram is the following:
ETnx × [0, 1]
BTnr × [0, 1]
Br // BTnr
. (2.9)
The mapping HT
x is well defined by Lemma 2.1. Note that H
x is continuous, since the identifi-
cations ρ and ρ−1 are obviously continuous and we showed in Lemma 2.4 that HT
is continuous.
We can therefore conclude, from the previous considerations and the fact that that HT
is a de-
formation retraction in the Cm-Whitney topology, that HT
x induces homotopy equivalences ρ and
δ(f) := HT
x (f, 0) between ET
x and ρ
−1(R), with ρ◦δ homotopic to idETn
and δ◦ρ = idρ−1(R).
In order to conclude the proof of Theorem 1.2 we simply make the following observations:
• First, the space described in it is precisely the disjoint union of the ETnx , x being a vertex of
∆M , because 0 is to be mapped to a fixed point of ψ.
• The number of Tn-fixed points, which is the same as the number of vertices of ∆M , is pre-
cisely χ(M). This follows from the analysis of the momentum map as in Atiyah–Delzant–
Guillemin–Sternberg theory (see for example [10], [11]).
• If we denote by Emb(M,σ)(r) the number of copies of Tn onto which the space considered in
Theorem 1.2 retracts (see formula (1.2)), Emb(M,σ)(r) is obtained by multiplying the number
of fixed points that admit such an embedding (see Lemma 2.3) by the number of copies of
n onto which ETnx (for the particular point) retracts; this latter number is n! (see Corollary
2.5), i.e. as many copies of Tn as possible ways that the canonical basis vectors ei may be
mapped onto the basis of weights αpi (for the particular point). Also, the former number is by
Lemma 2.3 controlled by the Boolean variable cp(r) defined in Proposition 1.4. Therefore
Emb(M,σ)(r) is given by Emb(M,σ)(r) = n!
p∈MTn cp(r), as we wanted to show.
• It is obviously true that if (M, σ) is equivariantly symplectomorphic to (M̃, σ̃), then Emb(M,σ)(r) =
(M̃, σ̃)
(r), so the integer Emb(M,σ)(r) is a symplectic–toric invariant.
(2,1)
(1,2)(0,2)
(2,0)(0,0)
(2,1)
(1,2)(0,2)
(2,0)(0,0)
Figure 4: Polytope corresponding to the Delzant manifold (M, σ) obtained by blowing up S2r0×S2r0
with r0 = 1/
2. Observe that Emb(M,σ)(
2) = 0 (proof in left figure) and Emb(M,σ)(1/
10 (proof in right figure). See Lemma 2.7.
As a final remark we observe that the invariant function Emb(M,σ) associated to the Delzant
manifold (M, σ) always reaches its minimum and maximum values on an interval of strictly posi-
tive length.
Lemma 2.7. There exist numbers r0, s0 > 0 such that if r ≤ r0, then the space of equivariant
symplectic embeddings from Br into M is homotopically equivalent to a disjoint union of n!χ(M)
copies of Tn, and if s ≥ s0, then it is empty.
Proof. It follows easily from Lemma 2.3, Corollary 2.5 and the previous observations.
This concludes the proof of Theorem 1.2 (and hence by construction the proof of Proposition
1.4).
3 Remarks on the partially equivariant case of Theorem 1.2
In this section we initiate a discussion on the topology of the space of partially equivariant sym-
plectic embeddings and sketch some suggestions to answer a question in this direction.
First the notion of Λ-equivariance (Λ ∈ Aut(Tn)) in Section 1 has a natural extension: we
say that an embedding from the 2n-ball Br into the 2n-dimensional Delzant manifold M is γ-
-equivariant with respect to a monomorphism γ : Tn−k → Tn, 1 ≤ k ≤ n − 1, if the following
diagram commutes:
n−k × Br
γ×f //
f // M
For example, Mγ is the set of p ∈ M such that ψ(γ(t), p) = p for all t ∈ Tn−k, and the rest of
terminology is also analogous. This definition extends naturally to the case when k = n, in which
the embeddings considered are purely symplectic, as well as to the case when k = 0, in which the
embeddings are fully equivariant, case which we treated previously in the paper. Unless otherwise
specified we do not consider these two cases in the discussion that follows. The question we would
like to address is the following:
Question 3.1 Let r be such that any connected component C ofMγ admits a Darboux–Weinstein
neighborhood of radius r, and by this we mean a neighborhood that is equivariantly symplecto-
morphic to a bundle over C with fiber the standard ball of radius r. Is the space of γ-equivariant
symplectic embeddings from Br intoM homotopically equivalent to the space of purely symplectic
embeddings from B2kr into M
γ up to reparametrization groups (as explained below)? ⊘
To analyze Question 3.1 first define B̂2kr to be the embedded 2k-ball in Br, i.e. the set of
points (z1, . . . , zk, 0) in Br so that
i=1 |zi|2 ≤ r2. The preimage under the momentum map of
the k-face corresponding to γ is the fixed point locus Mγ . Now consider any symplectic embed-
ding f : B̂2kr → Mγ . We want to find a canonical way to extend f to an equivariant symplectic
embedding can(f) : Br →M up to homotopy.
Here is an attempt to construct can(f): near the image of f , we can apply the equivariant ver-
sion of the Darboux–Weinstein’s theorem in order to find a neighborhood of Im(f) in M which
is symplectomorphic to Im(f)× B2(n−k), with the action of Tn−k given by the standard action on
2(n−k), and the symplectic form coinciding with the product symplectic form. Note that the sym-
plectic normal bundle toMγ is trivial over Im(f) because Im(f) is contractible, so a neighborhood
of Im(f) looks like Im(f)× B2(n−k) with a product symplectic form, and the action of Tn−k on it
is conjugate to the standard one. Using this identification M is described as a product, and we can
define can(f)(z) := (f(z1, . . . , zk), zk+1, . . . , zn). This expression for can(f) is clearly symplec-
tic and equivariant with respect to Tn−k-actions on the last n− k coordinates but is not canonical
because the local symplectomorphism given by Darboux–Weinstein’s theorem is not unique. We
cannot expect it to always be the same independently of f , because it is not true that globally the
normal bundle to Mγ is symplectically trivial, it only becomes true over a neighborhood of Im(f).
So this construction depends on choices of parameters.
Calling CAN the space of canonical embeddings can(f) : Br → ∆M , where f : B̂2kr → Mγ is
a symplectic embedding, observe that CAN is naturally identified with the space of purely sym-
plectic embeddings from the standard B2kr into M
γ , up to homotopy. The question then becomes
whether any γ-equivariant symplectic embedding f : B2kr → M may be deformed through a con-
tinuous family of equivariant symplectic embeddings to an embedding in CAN.
Equivalently, we ask the question: is the natural map between the space of partially equivariant
embeddings from Br into M and the space of symplectic embeddings from B
r into the fixed point
set Mγ (given by the restriction to the fixed ball B2kr ) a fibration? Note that the construction of
can(f) would give a section of this fibration.
Conjecture 3.1. Question 3.1 has an affirmative answer.
Example 3.2 If M = S2 × S2 with a product symplectic form and product T2-action (this space
has been carefully studied by Lalonde–Pinsonnault [14] and Anjos [2] among other authors), the
fixed point locus of the second S1 factor is S2×{a, b}, where a, b are the fixed points of the action
of S1 on S2. Now, given a symplectic embedding f of the ball B2 into S2, it is easy to build an
1-equivariant embedding of B4 into S2 × S2 canonically by (z1, z2) 7→ (f(z1), z2), where z2 is
taken to be a coordinate centered at the fixed point a. In this case the normal bundle to the fixed
point component S2 × {a} is globally trivial. ⊘
The combination of purely symplectic results of Biran, Lalonde–Pinsonnault and others and
an affirmative answer to Question 3.1 would give insight into the partially equivariant case in
higher dimensions; for example McDuff showed that if M is a symplectic 4-manifold with non-
simple Seiberg–Witten type, then the space of symplectic embeddings from Br into M is path
connected (which extends results of Biran). This is a consequence of the non–trivial result: any
two cohomologous and deformation equivalent symplectic forms on M are isotopic (proved in
[17]). Examples are known in dimensions 6 and above of cohomologous symplectic forms that are
deformation equivalent but not isotopic, so these techniques do not help to understand the topology
of the space of symplectic embeddings from Br into M . A positive answer to Question 3.1 would
give the first non–trivial result in dimension 6.
Another way of trying to generalize Theorem 1.2 is to consider embeddings equivariant with
respect to a complexity one action, that is, an action of Tn−1 on M2n. This is a hopeful approach
since a complete classification of complexity one actions has been recently achieved by Karshon
and Tolman [15].
Acknowledgments
The author is grateful to D. Auroux and V. Guillemin for discussions, and for hosting him at the
M.I.T. regularly during the Fall and Spring semesters of 2003 and 2004. He thanks D. Auroux,
J.J. Duistermaat, Y. Karshon and M. Pinsonnault for making comments on a preliminary version
of this paper. Finally, the author is grateful to an anonymous referee for helpful suggestions that
have shortened the proof in Section 2, as well as for his/her interesting comments on Section 3.
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A. Pelayo
Department of Mathematics, University of Michigan
2074 East Hall, 530 Church Street, Ann Arbor, MI 48109–1043, USA
e-mail: apelayo@umich.edu
	The main theorem
	Proof of Theorem ??
	Remarks on the partially equivariant case of Theorem ??
