Topology of symplectic torus actions with symplectic orbits

We give a concise overview of the classiﬁcation theory of symplectic manifolds equipped with torus actions for which the orbits are symplectic (this is equivalent to the existence of a symplectic principal orbit), and apply this theory to study the structure of the leaf space induced by the action. In particular we show that if M is a symplectic manifold on which a torus T acts effectively with symplectic orbits, then the leaf space M/T is a very good orbifold with ﬁrst Betti number b 1 (M/T ) = b 1 (M) − dim T .

T -actions are called symplectic, and (M, σ, T ) will be called a symplectic Tmanifold. Two symplectic T -manifolds (M, σ, T ) and (M , σ , T ) are called isomorphic if there exists a T -equivariant diffeomorphism from M onto M such that σ is equal to the pull-back * (σ ) of σ by the mapping .
A well studied type of symplectic torus actions are the so called Hamiltonian torus actions. A vector field v on M is called Hamiltonian if the contraction i v σ of σ with v is an exact one-form, that is, there exists a smooth real-valued function f on M such that Hamilton's equation i v σ = −df holds. For every element X of the Lie algebra t of T , the infinitesimal action X M of X on M is a smooth vector field on M. The T -action preserves the symplectic form if and only if for every X ∈ t the one-form i X M σ is closed. The T -action is called Hamiltonian if its infinitesimal action is Hamiltonian, where the Hamiltonian function f = μ X of X M can be chosen to depend linearly on X ∈ t. Then the equation X, μ = μ X , X ∈ t, defines a smooth mapping μ from M to the dual space t * of t, called the momentum mapping of the Hamiltonian T -action. The theorem of Atiyah [2, Theorem 1] and Guillemin-Sternberg theorem [13] says that the image μ(M) of the momentum mapping is equal to the convex hull in t * of the image under μ of the set M T of fixed points in M for the action of T , where the set μ(M T ) is finite and therefore μ(M) is a convex polytope. Note that this implies that M T = ∅. Delzant [8] proved that if dim(T ) = n, then μ(M) is a so called Delzant polytope, and μ(M) completely determines the Delzant space (M, σ, T ). Delzant [8] moreover proved that M is isomorphic to a smooth toric variety with σ equal to a Kähler form on it, and the action of T extends to a holomorphic action of the complexification T C of T . For this reason a Delzant space is also called a symplectic toric manifold. See also Guillemin [14] for a beautiful exposition of this subject.
If the first de Rham cohomology group of M is equal to zero, then every symplectic action on M is Hamiltonian. Nevertheless, in general the assumption that the symplectic torus action is Hamiltonian is very restrictive, as it implies that the action has fixed points and that all its orbits are isotropic submanifolds of M, that is, σ (X M , Y M ) = 0 for all X, Y ∈ t. Research on Hamiltonian and smooth torus actions has been extensive. Orlik-Raymond's [28,29] and Pao's [31,32] studied smooth actions of 2-tori on compact connected smooth 4-manifolds; Karshon and Tolman classified centered complexity one Hamiltonian torus actions in [18] and also studied Hamiltonian torus actions with 2-dimensional symplectic quotients in [17]; Kogan [20] worked on completely integrable systems with local torus actions; most recently, Pelayo and Vũ Ngo . c [34,35] have studied integrable systems on symplectic 4-manifolds in which one component of the integrable system comes from a Hamiltonian circle action. There are many other papers which relate integrable systems and Hamiltonian torus actions, for instance Duistermaat's paper on global action-angle coordinates [9] and Zung's work on the topology of integrable Hamiltonian systems [42,43].
Although Hamiltonian actions of n-dimensional tori on 2n-dimensional manifolds are present in many integrable systems in classical mechanics, non-Hamiltonian actions occur also in physics, cf. Novikov's article [27]. At the other extreme of a symplectic Hamiltonian T -action is the case of a symplectic T -action whose principal orbits are symplectic submanifolds of (M, σ ), in which case the action does not have any fixed points and the restriction of the symplectic form to the T -orbits is non-degenerate, which in particular implies that the action is never Hamiltonian. The classification of Pelayo [33], reviewed in the present paper, shows that there are lots of cases where this happens.
If one principal orbit is symplectic, then every orbit is symplectic, and the action is locally free in the sense that all the stabilizer groups are finite subgroups of T . We first describe in Sect. 3.2 the particular case when the action is free, and hence the orbit space M/T is a manifold; in this case the classification is more straightforward, see Proposition 3.2. If the action is not free, then the orbit space M/T is a good orbifold (proven in [33]), and the classification of Sect. 3.2 is generalized to this rather more delicate situation in Sect. 3.3. If dim M − dim T = 2, when the orbifold M/T is an orbisurface, the classification can be given in a stronger, more concrete fashion, see Sect. 5.
This paper contains the following new results: Theorem 1.1, Theorem 3.3, Proposition 4.1, Theorem 4.2, Proposition 4.3, Lemma 4.5 items (iii) and (iv) and Corollary 4.6. In particular, the following topological result is a consequence of Theorem 4.2 and Proposition 4.3.

Theorem 1.1 If M is compact, connected symplectic manifold on which a torus T acts effectively with symplectic orbits, then the leaf space M/T is a very good orbifold with Betti number
The above result may be considered a symplectic version of the classical work by Kirwan [19] on the computation of the Betti numbers of symplectic quotients in the Hamiltonian case.
There are a few results on non-Hamiltonian symplectic torus actions: McDuff [24] and McDuff and Salamon [25] studied non-Hamiltonian circle actions, and Ginzburg [15] non-Hamiltonian symplectic actions of compact groups under the assumption of a "Lefschetz condition". Benoist [3] proved a symplectic tube theorem for symplectic actions with coisotropic orbits and convexity result in the spirit of the of the Atiyah-Guillemin-Sternberg theorem [3]; Ortega-Ratiu [30] proved a local normal form theorem for symplectic torus actions with coisotropic orbits. These appear to be the most general results prior to the classification of symplectic torus actions with coisotropic principal orbits in Duistermaat-Pelayo [11] and Pelayo [33]. For a concise overview of the classification in [11] and an application to complex and Kähler geometry see [12].

Preliminaries
Let (M, σ, T ) be a symplectic T -manifold. For every x ∈ M the orbit T · x of the Taction containing x is a smooth manifold, and the mapping T → M : t → t ·x induces a diffeomorphism from T /T x onto T · x, where T x := {t ∈ T | t · x = x} denotes the stabilizer subgroup of x in T . The tangent mapping at 1T x of this diffeomorphism is a linear isomorphism from t/t x onto the tangent space T Here t and t x denote the respective Lie algebras of T and T x . That is, if X M (x) denotes the infinitesimal action at x of an element X of the Lie algebra t of T , then For an effective torus action the minimal stabilizer subgroups are the trivial ones T x = {1}, in which case the action of T is free at the point x, and the corresponding orbits are called the principal orbits. The set M reg of all x ∈ M such that T x = {1} is an open, dense, and T -invariant subset of M.
The orbit T · x is symplectic if, for every y ∈ T · x, restriction of σ y to the tangent space T y (T · x) of the orbit is a symplectic form. That is, if (T y (T · x)) σ y denotes the orthogonal complement of T y (T · x) in T y M with respect to the symplectic form σ y , then T y M is equal to the direct sum of T y (T · x) and its symplectic orthogonal complement.
Benoist [3,Lemme 2.1] observed that if u and v are smooth vector fields on M which preserve σ , then their Lie bracket [u, v] is Hamiltonian with Hamiltonian function equal to σ (u, v), that is, i [u,v] ). It therefore follows from the commutativity of T that if X, Y ∈ t, then d(σ (X M , Y M )) = 0, which means that there is a unique antisymmetric bilinear form σ t on t such that for every x ∈ M and X, if t h and l denote the sum of all t x 's and the kernel of σ t in t, respectively. Assume that for some x ∈ M the orbit T · x is dim T -dimensional and symplectic. Then σ t is nondegenerate, which in turn implies that every T -orbit is a symplectic submanifold of (M, σ ). Because t x ⊂ ker σ t = {0}, the closed subgroup T x of the compact group T is discrete, hence finite. Therefore the action is locally free and every T -orbit is dim T -dimensional.

Models for symplectic torus actions with symplectic principal orbits
We give a concise review of Pelayo [33, with some modifications in the exposition and present a new fact: Theorem 3.3.
We study symplectic actions of the torus T on the symplectic manifold (M, σ ) such that at least one T -orbit is a dim T -dimensional symplectic submanifold of (M, σ ). This condition means that there exists x ∈ M such that t x = {0} and the restriction of σ x to T x (T · x) is nondegenerate. It follows that the antisymmetric bilinear form σ t in (1) is nondegenerate, which in turn implies that for every x ∈ M we have t x = {0} and the restriction of σ x to T x (T · x) is nondegenerate. That is, the action of T on M is locally free, and all T -orbits are dim T -dimensional symplectic submanifolds of (M, σ ). We denote by σ T the unique invariant symplectic form σ on the Lie group T such that σ 1 = σ t on T 1 T = t.
It follows that for each x ∈ M the symplectic orthogonal complement x := (T x (T · x)) σ x of the tangent space of the T -orbit is a complementary linear subspace to T x (T · x) in T x M, and that the restriction to x of σ x is a symplectic form.
Furthermore the x depend smoothly on x ∈ M, and therefore define a distribution in M, a smooth vector subbundle of the tangent bundle TM of M.

Lemma 3.1 The distribution is T -invariant and integrable.
Proof The T -invariance of follows from the T -invariance of σ . There is a unique t-valued one-form θ on M, called the connection form, such that = ker θ and θ(X M ) = X for every X ∈ t.
is integrable if and only if θ is closed. Let X i , 1 ≤ i ≤ m := dim T be a basis of t, and let Y j be the σ t -dual basis of t, determined by the equations Hence θ is closed. Lemma 3.1 leads to the local models of the symplectic T -space described in the paragraph after Theorem 3.5. These local models can also be obtained by applying results of Benoist [3, Proposition 1.9] or Ortega and Ratiu [30,] to the case of a symplectic torus action with symplectic orbits. The proof of Lemma 3.1 in [33] uses these local models.

The model T × S I
Let I be a maximal connected integral manifold of the distribution , where σ I := ι * I σ is a symplectic form on I and ι I denotes the inclusion mapping from I into M. In other words, I is a leaf of the symplectic foliation in M of which the tangent bundle is equal to .
Let S := {s ∈ T | s · I = I}, which is a subgroup of T . If we provide I and S with the leaf topology and the discrete topology, respectively, then the action of S on I is proper. Because each other leaf is of the form t · I for some t ∈ T and T is commutative, the group S does not depend on the choice of the leaf I. Furthermore, because the leaves form a partition of M, we have t ∈ S if and only if t · I ∩ I = ∅, the mapping A : T × I → M : (t, x) → t · x is surjective, and A(t, x) = A(t , x ) if and only if there exists s ∈ S such that x = s · x and t = ts −1 . We let s ∈ S act on T × I by sending (t, x) to (ts −1 , s · x). Because S acts freely on T and properly on I, it acts freely and properly on T × I. Therefore the orbit space T × S I has a unique structure of a smooth manifold such that the canonical projection ψ : T × I → T × S I is a smooth covering map with S as its covering group. Moreover, the unique mapping α : the σ x -orthogonality of x and T x (T · x) implies that σ T × S I = α * (σ ). Finally, α intertwines the T -action on T × S I induced by the T -action (t , (t, x)) → (t t, x) on T × I with the T -action on M. We conclude that α is an isomorphism of symplectic T -spaces from (T × S I, σ T × S I , T ) onto (M, σ, T ).
The orbit space I/S for the action of S on I is provided with the finest topology on the orbit space I/S such that the canonical projection π I/S : I → I/S is continuous. Because the action of S on I is proper, the topology on I/S is Hausdorff.
The mapping ι : I/S → M/T induced by the inclusion maps I → M and S → T is bijective and continuous, and because M/T is compact, it follows that ι is a homeomorphism. This in turn implies that I/S is compact, that is, the action of S on I is cocompact.

When T acts freely
We assume in this subsection that the action of T on M is free. We will present a model of the symplectic T -space in which I and S are replaced by the universal covering of M/T and the monodromy homomorphism from the fundamental group of M/T to T , respectively.
The freeness of the T -action implies that M/T has a unique structure of a smooth The mapping ψ * : is an isomorphism of groups from π 1 (I, x 0 ) onto the kernel ker μ of the monodromy homomorphism μ : We are now ready to present the following model of our symplectic T -space (M, σ, T ).

Proposition 3.2 The composition
There is a unique symplectic form σ model of which the pullback by the canonical We conclude this subsection with a construction inspired by Kahn [ In other words, η = π * (β) for a unique closed t-valued one-form β on M/T , and θ = η + π * (β). If μ : π 1 (M/T , p 0 ) → T denotes the monodromy homomorphism defined by = ker, then Because T is commutative, the homomorphism μ : π 1 (M/T , p 0 ) → T is trivial on the commutator subgroup C of π 1 (M/T , p 0 ), the smallest normal subgroup of π 1 (O, p 0 ) which contains all commutators. It is a theorem of Hurewicz that C is equal to the kernel of the surjective homomorphism h 1 : π 1 (M/T , p 0 ) → H 1 (M/T , Z), called the Hurewicz homomorphism, which assigns to each [γ ] ∈ π 1 (M/T , p 0 ) its homology class. See Hu [22,Theorem 12.8]. It follows that there is a unique homomorphism μ h : The compactness of M/T implies that H 1 (M/T , Z) is a finitely generated commutative group, and therefore the group In contrast, the group H 1 (M/T , Z) tor is finite, isomorphic to the Cartesian product of finitely many cyclic finite groups. Let We therefore have proved the following theorem, in which the last statement is Pelayo [ It follows that the principal T -bundle π : If is as in Theorem 3.3, then there is a unique two-form σ on M such that σ = σ on the T -orbits and on the linear complements x of the tangent spaces to the T -orbits, but this time Then σ is a T -invariant symplectic form on M, the T -orbits are symplectic submanifolds of M, and is the distribution of the symplectic orthogonal complements to the tangent spaces of the orbits. The integral manifolds I of are compact if and only if the monodromy group μ (π 1 (M/T , p 0 )) of is finite, which according to Theorem 3.3 can always be arranged by means of a suitable choice of the T -invariant symplectic form σ on M, equal to σ on the T -orbits. We like to think of Theorem 3.3 as telling how the integral manifolds and the monodromy of can be changed when changing the symplectic form in the above manner, without changing the T -action.

Orbifolds
We return to the general case, when there may exist x ∈ M \ M reg , meaning that the stabilizer subgroup T x , which is finite, is nontrivial.
Let I denote the maximal integral manifold of such that In our good orbifold case the universality property of π O : O orb → O implies that there exists a unique orbifold covering π I : O orb → I such that π O = ψ • π I and π( p 0 ) = x 0 if x 0 is a base point in I such that ψ(x 0 ) = p 0 = π O ( p 0 ). Because I is a smooth manifold, O orb is a smooth manifold, diffeomorphic to the universal covering I of I, and acts on O orb by means of diffeomorphisms.
The mapping π I intertwines the action of on O orb with a unique action of on I, and there is a unique homomorphism μ : → S, called the orbifold monodromy homomorphism, such that c · x = μ(c) · x for every c ∈ and x ∈ I. We have μ( ) = S, and therefore the subgroup S of T is called the monodromy group. The homomorphism from to the group of deck transformations of π I : O orb → I has kernel equal to ker μ = ψ * ( ) and the image group is isomorphic to S /ψ * (π 1 (I, x 0 )). It also follows that O orb,reg : , there exists a c ∈ which maps p 0 to the endpoint of γ , where c is unique because acts freely on p 0 . Furthermore, any path in O orb from p 0 to c · p 0 is homotopic to γ , because O orb is simply connected. It follows that the mapping [γ ] → c is an isomorphism, from the group π orb 1 (O, p 0 ) of all orbifold homotopy classes of homotopy loops in O based at p 0 , onto . We use this isomorphism to identify the two groups, and write = π orb 1 (O, p 0 ) in the sequel.
With these notations and basic facts, we have the following model of (M, σ, T ), extending Proposition 3.2 to the case that the T -action is not free.  manifold (M, σ ) there a exist a G-invariant almost complex structure and Hermitian structure J and h on M, such that σ is equal to the imaginary part of h, see for instance [10,Sect. 15.5]. The same proof yields a T x -invariant complex structure and Hermitian structure h on x such that σ = Im h. In other words, T x acts on x by means of unitary complex linear transformations.
Because T x is commutative, there is an h-orthonormal basis of simultaneous eigenvectors for the T x -action in x . If z = (z 1 , . . . , z m ) ∈ C m denote the coordinates in x with respect to this basis, we have (t · z) j = λ j (t)z j for every 1 ≤ j ≤ m and t ∈ T x , where λ j is a homomorphism from T x to the multiplicative group C × of all nonzero complex numbers. Because T x is finite, there is a unique d j ∈ Z >0 and homomorphism l j : T x → Z/d j Z such that λ j (t) = e 2πil j (t)/d j for every 1 ≤ j ≤ m and every t ∈ T x . Because M reg is dense in M, T x acts effectively on x , which means that the homomorphism  (O, p 0 ). The boundary ∂N of N is a simplicial complex in the smooth manifold O reg . Because O reg \ N is a compact subset of O reg , it is a simplicial complex and therefore π 1 (O reg \ N, p 0 ) is finitely generated, hence π 1 (O reg , p 0 ) is finitely generated. Because ι reg * is surjective, we conclude: Let X be a path connected, simply connected metrizable locally compact topological space and a discrete group acting properly on X. Choose x 0 ∈ X and write p 0 = · x 0 ∈ X/ . For c ∈ , let δ be a path in X from x 0 to c · x 0 . If π : X → X/ denotes the canonical projection, then π • δ is a loop in X/ based at p 0 , and because X is simply connected, its homotopy class does not depend on the choice of γ and we can write [γ ] = ϕ(c) for a uniquely defined ϕ(c) ∈ π 1 (X/ , p 0 ). The theorem of Armstrong [1] says that ϕ : → π 1 (X/ , p 0 ) is a surjective homomorphism, with kernel equal to the smallest normal subgroup A of which contains all elements c ∈ which have a fixed point in X. If we apply this to X = O orb and = π orb 1 (O, p 0 ), then X/ is canonically identified with O and ϕ : π orb 1 (O, p 0 ) → π 1 (O, p 0 ) is the map obtained by forgetting the orbifold structure. It follows that this map ϕ is surjective from onto π 1 (O, p 0 ), and that its kernel is equal to the smallest normal subgroup A of which contains all c ∈ such that c · p = p for some p ∈ O orb . If c · p = p, then μ(c) · π I ( p) = π I (c · p) = π I ( p). That is, μ(c) ∈ T x if we write x = π I ( p). It follows that μ(ker ϕ) is contained in the product T • of all T x 's. Because the local normal form of the T -action, in combination with the compactness of M, implies that there are finitely many stabilizer subgroups T x , each of which is finite, T • is a finite subgroup of T .
We recall the surjective Hurewicz homomorphism h 1 : π 1 (O, p 0 ) → H 1 (O, Z) with kernel equal to the commutator subgroup of π 1 (O, p 0 ). Furthermore, an orbifold O is called very good if it is isomorphic, as an orbifold, to the orbit space of a finite group action on a smooth manifold. We now have the following orbifold version of Theorem 3.3. Because is finitely generated, H 1 (O, Z) = (h 1 • ϕ)( ) is a finitely generated commutative group. This can also be proved directly by observing that the orbit space stratification of M/T implies that the compact space M/T is homeomorphic to a simplicial complex. It follows that H 1

Theorem 4.2 Let σ be any T -invariant symplectic form on
Since the exponential mapping from t to T is surjective, we have μ(c j ) = exp(X j ) for suitable X j ∈ t.
Finally there exists an η as above such that For any c ∈ C there exist m j ∈ Z and f ∈ F such that In view of the Hurewicz theorem this is equivalent to  Proof Consider the action ((Y, γ ), (X, p)) → (X − Y, γ · p)) on t × O orb of the discrete subgroup := {(Y, γ ) ∈ t × | exp Y = μ(γ )} of t × . If (X − Y, γ · p) = (X, p), then Y = 0 hence μ(γ ) = exp Y = 1, and γ · p = p, and therefore γ = 1 because ker γ acts freely on O orb . It follows that the proper action of is also free, and Because t × O orb is simply connected, we conclude that the fundamental group of M is isomorphic to .
For any group G we denote by G/C(G) the abelianization of G, where C(G) is the smallest normal subgroup of G which contains all commutators of elements of G. The first homology group H 1 (M, Z) of M is isomorphic to the abelianization of the fundamental group of M, and therefore isomorphic to /C( ). Because μ is a homomorphism from to the commutative group T , we have μ = 1 on C( ). Therefore Recall the surjective homomorphism ϕ : = π orb 1 (O, p 0 ) → π 1 (O, p 0 ), of which the kernel is equal to the smallest normal subgroup A of which contains all γ ∈ such that γ · p = p for some p ∈ O orb . And the surjective homomorphism h 1 : π 1 (O, p 0 ) → H 1 (O, Z) with kernel equal to C(π 1 (O, p 0 )). It follows that the homomorphism h 1 • ϕ : → H 1 (O, Z) is surjective and has kernel equal to the smallest normal subgroup of which contains both A and C( ). If The surface O can be provided with the structure of a simplicial complex. Unless O is homeomorphic to a sphere, there exists a g ∈ Z >0 such that the surface can be obtained from the convex hull P in the plane of a regular 4g-gon in the following way. The boundary is viewed as a cycle of g quadruples of subsequent edges α j , β j , α j , β j , j ∈ Z/gZ, where α j and β j are positively oriented and α j and β j are negatively oriented. The surface O is obtained by identifying, for each j ∈ Z/gZ, α j with α j , where the identification respects the orientations. In the surface O all the vertices of the 4g-gon correspond to a single point p 0 ∈ O which is taken as the base point. The edges α j and β j define loops in O based at p 0 . If [α j , β j ] = α j β j α j −1 β j −1 denotes the commutator of α j and β j , then their concatenation for 1 ≤ j ≤ g corresponds to the loop which runs along the boundary of the polytope P at its interior side, and therefore is contractible. If we denote the homotopy classes of the loops α j and β j by the same letters, these considerations lead to the conclusion that the homomorphism, from the free group generated by the α j and β j to π 1 (O, p 0 ), is surjective, with kernel equal to the smallest normal subgroup which contains the cyclic concatenation of the commutators [α j , β j ], j ∈ Z/gZ. This is usually expressed by saying that π 1 (O, p 0 ) is generated by the α j , β j , subject to the single relation Because h 1 ([α j , β j ]) = 0, the commutative group H 1 (O, Z) is freely generated by the homology classes h 1 (α j ), h 1 (β j ), j ∈ Z/gZ. Therefore H 1 (O, Z) Z 2g , which implies that H 1 (O, Z) has no torsion. Therefore the positive integer g, called the genus of the surface O, has the topological interpretation that the first Betti number b = b 1 (O) is equal to 2g. With this interpretation, the two-dimensional sphere has genus g = 0. Any oriented compact connected surface of genus g > 0 is homeomorphic to a sphere with g handles.
It can be arranged that the singular points s i , 1 ≤ i ≤ n, lie in the interior of the polytope P . For each i, let γ i be a loop in O consisting of a path δ i from a vertex p 0 of P into the interior of P to a point close to s i , followed by a circle around s i in the positive direction and completed by the inverse of δ i . It can be arranged that the curves γ i don't intersect each other except at the base point and that the concatenation γ 1 . . . γ n is homotopic in O reg = O \ O sing to the cyclic concatenation of the commutators [α j , β j ], j ∈ Z/gZ. If we denote the homotopy classes of the γ i by the same letters, then this leads to an isomorphism from Q/R onto π 1 (O, p 0 ), where Q is the free group generated by the α j , β j , and γ i , and R is the smallest normal subgroup of Q which contains the product of the concatenation of the commutators [α j , β j ] with the inverse of the concatenation of the γ i . That is, π 1 (O reg , p 0 ) is generated by the α j , β j , and γ i , subject to the single relation where the left and/or the right hand side is equal to 1 if g = 0 and/or n = 0.
The surjective homomorphism ι reg * from π 1 (O reg , p 0 ) onto π orb 1 (O, p 0 ), discussed in the paragraphs preceding Proposition 4.1, has kernel equal to the [γ ] ∈ π 1 (O reg , p 0 ) such that there is an orbifold homotopy of γ to the trivial loop. The homotopy can be arranged to be transversal to the singular set, and it follows that γ is homotopic in O reg to a concatenation of conjugates of powers of the curves γ i introduced above. Let c i denote a small circle around s i in the positive direction and let c i denote its orbifold lift to the orbifold chart near s i . Then c i is equal to a rotation on a small circle around the origin about the angle 2π/o i , and c i k has an orbifold contraction in the chart around s i if and only if k ∈ Zo i . It follows that the kernel of ι reg * is equal to the smallest normal subgroup of π 1 (O reg , p 0 ) which contains the elements γ i o i , 1 ≤ i ≤ n. In other words, π orb 1 (O, p 0 ) is generated by the α j , β j , and γ i , subject to the relations cf. Scott [37, p. 424]. Note that the kernel of the surjective homomorphism ϕ : π orb 1 (O, p 0 ) → π 1 (O, p 0 ) is equal to the smallest normal subgroup containing the γ i 's. This is compatible with the aforementioned presentation of π 1 (O, p 0 ) without γ i 's.
The only bad compact connected oriented orbisurfaces O are the ones with g = 0, where O is homeomorphic to the two-sphere, and either n = 1 or n = 2 and o 1 = o 2 . See Scott [37,Theorem 2.3]. Every good compact connected oriented orbisurface is isomorphic to O orb / , where O orb is the two-sphere with the standard Riemannian structure, the Euclidean plane, or the hyperbolic plane, and is a discrete group of orientation preserving isometries acting on O orb . See Thurston [40,Sect. 5.5]. This description has been used to prove that every good compact connected oriented orbisurface is very good, see Scott [37,Theorem 2.5]. In the case of the hyperbolic plane = the complex upper half plane, is a cocompact discrete subgroup of PGL(2, R). That is, is a Fuchsian group of which the signature (g; o 1 , . . . , o n ) satisfies o i < ∞ for every 1 ≤ i ≤ n.
In each tangent space of O orb there is a unique rotation J about the angle π/2, with respect to the aforementioned Riemannian structure β and orientation. This defines an almost complex structure J , which is integrable because dim O orb = 2. The imaginary part of the Hermitian structure h defined by β and J is a two-form of which the exterior derivative is equal to zero, again because dim O orb = 2. It follows that h and Im h is a -invariant Kähler structure and symplectic form on O orb , and therefore defines an orbifold Kähler structure and orbifold symplectic form on O, respectively.
In the existence result Theorem 3.5 we need a compact and connected good orbisurface O provided with an orbifold smooth area form without zeros. The area form determines an orientation of O, and in the previous paragraphs we have described the compact and connected oriented good orbisurfaces. That is, for each i ∈ I , χ i is a non-negative orbifold smooth function on O with support in a U p i , the U p i form a locally finite covering and i∈I χ i = 1, where the left hand side is viewed as a locally finite sum. Write σ := i∈I χ i σ p i , a locally finite sum. Then σ is an orbifold smooth area form on O. If p ∈ O, i ∈ I , and p ∈ U i then, because σ p i and σ p both are orbifold smooth area forms without zeros on U p i ∩ U p compatible with the orientation of O, there exists an orbifold smooth function ϕ i on U p i ∩ U p such that σ p i = ϕ i σ p and ϕ i > 0 on U p i ∩ U p . It follows that σ = ( i∈I χ i ϕ i )σ p , where i∈I χ i ϕ i > 0 on a neighborhood of p, because χ i ≥ 0 for every i ∈ I and i∈I χ i = 1. This proves that σ has no zeros.
In Theorem 3.5 we also need a homomorphism μ : π orb 1 (O, p 0 ) → T such that the kernel of μ acts freely on the orbifold universal covering O orb of O. Proof (i) There is a unique homomorphism μ from the free group Q generated by the α j , β j , and γ i , such that μ(α j ) = a j , μ(β j ) = b j , and μ(γ j ) = c j for every 1 ≤ j ≤ g and 1 ≤ i ≤ n. (ii) An element γ ∈ does not act freely on O orb if and only if γ = 1 and γ ∈ p , the stabilizer subgroup of some p ∈ O orb . Because p is a singular point for the action of on O orb , its projection π O ( p) to O is one of the singular points s i of O. The description of the orbifold chart near s i implies that there exists an s i ∈ O orb such that π O ( s i ) = s i and γ i is the unique generator of s i Z/Zo i which acts near s i as a rotation about the angle 2π/o i . Because the fibers of π O are the -orbits, there exists c ∈ such that p = c · s i , hence p = c s i c −1 , and therefore γ = cγ i k c −1 for some k ∈ Z, where k / ∈ Zo i because γ = 1. Because T is commutative, we have  Proof Because H 1 (O, Z) Z 2g is torsionfree, U = ker(h 1 • ϕ) is equal to the normal subgroup of generated by the commutator subgroup C and Armstrong's subgroup A. Because T is commutative, μ = 1 on C, hence μ(U )μ(A) = μ(A) = T • , the subgroup of T generated by the T x 's = the subgroup of T generated by the c i 's. The conclusion therefore follows from Theorem 4.2 with R = I .
(β j ) = [β j ] + F of H 1 (O, Z) is a symplectic basis with respect to the intersection form, in the sense that (α j ) · (α k ) = 0, (β j ) · (β k ) = 0, and (α j ) · (β k ) = δ jk . The automorphism ϕ * /F of H 1 (O, Z) preserves the intersection form, and therefore is given on any symplectic Z-basis by a symplectic matrix in the sense that the g × g-matrices P , Q, R, S have integral entries and satisfy the equations P Q t − QP t = 0, RS t − SR t = 0, and P S t − QR t = I. It follows that for suitable u i j , v i j ∈ Z/Zo i . Dehn [7] proved that the mapping class group of the topological surface O, the group of isotopy classes of homeomorphisms of O, is generated by transformations which nowadays are called Dehn twists. These are diffeomorphisms equal to the identity outside a small annulus around a loop α without self-intersections, and act on H 1 (O, Z) by sending (β) to (β) + ((α) · (β))(α). As the latter transformations generate Sp(2g, Z) (see Magnus, Karass and Solitar [23, pp. 178, 355, 356]), every automorphism of H 1 (O, Z) which preserves the intersection form is equal to ϕ * /F , for an orbifold automorphism ϕ of O which leaves each singular point fixed.
It is shown in Pelayo [33, Sect. 6.4] that for each 1 ≤ i ≤ n there is an orbifold automorphism ϕ of O which leaves each singular point fixed, preserves all [α j ]'s and [β j ]'s except one of these, to which it adds [γ i ]. It follows that every automorphism of H orb 1 (O, Z) which is equal to the identity on the torsion subgroup F and preserves the intersection form is of the form ϕ * for an orbifold automorphism ϕ of O which leaves each singular point fixed.
Finally the proof of Moser [26] can be used to show that if σ and σ are two orbifold area forms on O, then there exists an orbifold automorphism ϕ of O such that σ = ϕ * (σ ) if and only if O σ = O σ . Moreover, if this is the case, then ϕ can be chosen to be orbifold isotopic to the identity, which implies that ϕ leaves each singular point of O fixed and acts as the identity on H orb 1 (O, Z). The unique homomorphisms μ h , μ h : H orb 1 (O, Z) → T such that μ = μ h • h 1 and μ = μ h • h 1 do not depend on the choice of p 0 , and we have μ h = μ h • ϕ * . If we write a j = μ(α j ), b j = μ(β j ), c i = μ(γ i ), a j = μ (α j ), b j = μ (β j ), and c i = μ (γ i ), then it follows from μ = μ • ϕ * and (4) that This leads to the following uniqueness theorem, which corresponds to [ (3) and u i j , v i j ∈ Z/Zo i , such that (5) holds for every 1 ≤ j ≤ g.
This completes the classification of compact connected symplectic T -spaces with symplectic principal orbits and for which the orbit space is 2-dimensional.