Faithful Semitoric Systems

This paper consists of two parts. The first provides a review of the basic properties of integrable and almost-toric systems, with a particular emphasis on the integral affine structure associated to an integrable system. The second part introduces faithful semitoric systems, a generalization of semitoric systems (introduced by Vu Ngoc and classified by Pelayo and Vu Ngoc) that provides the language to develop surgeries on almost-toric systems in dimension 4. We prove that faithful semitoric systems are natural building blocks of almost-toric systems. Moreover, we show that they enjoy many of the properties that their (proper) semitoric counterparts do.


Introduction
A driving problem in Hamiltonian mechanics and symplectic geometry is to classify integrable systems up to a suitable notion of equivalence. An integrable system is a triple (M, ω, Φ), where (M, ω) is a 2n-dimensional symplectic manifold and Φ : (M, ω) → R n is a smooth map whose components are in involution and functionally independent almost everywhere on M . This paper introduces vertical almost-toric systems, a category of integrable systems on 4-dimensional symplectic manifolds that generalizes toric and semi-toric systems and lays the foundation for studying almost-toric systems. A key feature of vertical almost-toric systems is that they behave well under a process of taking appropriate subsystems, a fact that facilitates development of precise language to define, for vertical almost-toric systems, integrable surgeries in the sense of Zung [51].
In general, classification of integrable systems becomes a tractable problem only under assumptions that restrict the topology of fibers of the system. Intuitively, the greatest challenge arises from noncompactness of the group action. Accordingly, full classifications were first established for toric systems in which the R n action is replaced by an T n -action. Building upon the foundational results of Atiyah [1] and Guillemin & Sternberg [14], Delzant [7] classified toric systems on closed manifolds. More recently, Karshon & Lerman [23] have extended Delzant's classification to non-compact toric manifolds, relying upon the local normal forms of Guillemin & Sternberg [15] and Marle [26].
Once one allows non-compactness of the group acting on the total space, complexity of both the fibers and of the total space can be reasonably controlled by restricting the singularities of the moment map. Symington [40] and Vũ Ngo . c [46] have proposed a notion of almosttoric systems on 4-dimensional symplectic manifolds that includes toric systems but also allows for so-called focus-focus fibers, which can be thought of as the Lagrangian analog of the nodal fibers that arise in Lefschetz fibrations. The diffeomorphism types of closed manifolds that support an almost-toric system has been determined (cf. Leung & Symington [25]), and recently, almost-toric systems have proved to be of independent interest in symplectic topology (cf. Vianna [41,42,43]).
While the classification problem of almost-toric systems has not been settled, even in the compact case, an important subclass of almosttoric systems has been completely understood: Pelayo & Vũ Ngo . c ([34, 35]) have classified semi-toric systems, which were initially introduced by Vũ Ngo . c in [46]. An integrable system (M, ω, Φ = (J, H)) is semi-toric if it is almost-toric and if J is a proper moment map of an effective Hamiltonian S 1 -action. Semi-toric systems, whose total spaces may be non-compact, share many fundamental properties with closed symplectic toric manifolds, like connectedness of the fibers of the moment map, but their classification is significantly more involved as the presence of focus-focus fibers introduces more data (see Pelayo & Vũ Ngo . c [34]). While semi-toric systems appear naturally both in symplectic topology and in (quantum) Hamiltonian mechanics (cf. Eliashberg [11,24,36]), the properness condition excludes some familiar almost-toric integrable systems, such as the spherical pendulum (cf. Duistermaat [9]). For this reason, Pelayo & Ratiu & Vũ Ngo . c ( [32,33]) introduce a family of almost-toric systems that share some of the main properties of semi-toric systems, like connectedness of the fibers of the moment map, while allowing enough freedom to include examples such as the spherical pendulum. These systems are called generalized semitoric. In such an almost-toric system (M, ω, Φ = (J, H)) the moment map Φ is proper and J is the moment map of an effective Hamiltonian S 1 -action that satisfies some constraints on the sets of singular points and values (see Definition 1.3 in Pelayo & Ratiu & Vũ Ngo . c [33]).
Vertical almost-toric systems, defined and studied in this paper, can be viewed as an extension of generalized semi-toric systems to the noncompact setting but were defined with different purposes in mind. For instance, vertical almost-toric systems provide a category that can accommodate non-compact systems that are convenient local models and building blocks for almost-toric systems. The essential difference between vertical almost-toric systems and generalized semi-toric systems is that the moment maps of the former are merely required to be proper onto their image while the moment map of a generalized semi-toric system must be proper. (The fibers of the moment maps of both types of systems are connected; for vertical almost-toric systems this connectedness is imposed by definition, while for semi-toric and generalized semi-toric systems it can be proven as a result of the other properties.) The definition of a vertical almost-toric system is crafted so that appropriately chosen subsystems are again vertical almost-toric. Specifically, given an open, connected subset U of the moment map image of a vertical-almost-toric system, if the intersection of U with any vertical line is either empty or connected then restricting the moment map to the preimage of U yields a vertical almost-toric system. In fact, such a subsystem of a (generalized) semi-toric system is vertical almost-toric.
In a forthcoming paper, [20], the process of taking such subsystems is an essential ingredient in the definition of surgeries in the category of vertical almost-toric systems. Those surgeries are going to be applied to the determination of which Hamiltonian S 1 -spaces underlie compact semi-toric systems ( [21], forthcoming), thus completing the work started in Hohloch & Sabatini & Sepe [19]. The language of vertical almost-toric systems also allows one to have a more conceptual understanding of the local-to-global arguments in the classification of semi-toric systems (cf. Pelayo & Vũ Ngo . c [34,35]); this is also going to be explored in a separate paper.
The main results of this paper are as follows: (A) A connected component of a fiber of an almost-toric system admits an open neighborhood that is isomorphic to a vertical almost-toric system (cf. Proposition 4.9); (B) Using terminology analogous to that for (generalized) semi-toric systems (cf. Pelayo & Ratiu & Vũ Ngo . c [33]), vertical almosttoric systems possess cartographic homeomorphisms (see Theorem 4.24). These are homeomorphisms of the moment map image onto subsets of R 2 that encode the induced Z-affine structures (cf. Section 2.3). In particular, the monodromy introduced by focus-focus fibers is encoded via vertical cuts. (C) The space of all cartographic homeomorphisms of a given vertical almost-toric system is described (cf. Theorem 4.36), generalizing the analogous result for semi-toric system (cf. Vũ Ngo . c [46]). This can be used to construct an invariant of the isomorphism class of a vertical almost-toric system analogous to the semi-toric polygon of Pelayo & Vũ Ngo . c [34,Definition 4.5]. (D) Given a vertical almost-toric system (M, ω, Φ = (J, H)) and a cartographic homeomorphism f : Φ(M ) → R 2 , the composition f • Φ may lack the smoothness required of a moment map. We provide a method for smoothing f •Φ to obtain a vertical almost-toric system isomorphic to (M, ω, Φ = (J, H)) whose moment map image equals f (Φ(M )) on the complement of arbitrarily small neighborhoods of the cuts used to define f (cf. Theorem 4.48).
Result (A) establishes vertical almost-toric systems as building blocks for almost-toric systems. While it is probably known to experts in the area, we could not find a complete, self-contained proof of this fact and decided to include it, along with proofs of basic topological facts leading up to it (cf. Section 3.2).
Results (B) and (C) are not surprising in light of the work in Pelayo & Ratiu & Vũ Ngo . c, Vũ Ngo . c [33,46] and, in fact, their proofs are heavily influenced by that work. However, we provide explicit proofs for several reasons: • There is ambiguity in the literature with regards to the notion of isomorphism in the category of semi-toric systems. It seems that, in spite of what is stated in Pelayo & Vũ Ngo . c [34,35], isomorphisms of semi-toric systems must induce orientation-preserving diffeomorphisms on their moment map images in order for the semi-toric invariants in Pelayo & Vũ Ngo . c [34,35] to be well-defined (cf. [37]). The restriction to orientation-preserving homeomorphisms has been made explicit in the definition of generalized semi-toric systems (cf. Pelayo & Ratiu & Vũ Ngo . c [33,Definition 2.4]). Deeming this restriction unnecessary, we allow orientation-reversing cartographic homeomorphisms, and hence provide a proof that accommodates such homeomorphisms. • Our alternative proof of the existence of cartographic homeomorphisms in the case in which the defining cuts disconnect the moment map image allows us to avoid the 'homotopy argument' of Pelayo & Ratiu & Vũ Ngo . c [33, Step 5 of the proof of Theorem B]. • The description of the set of cartographic homeomorphisms of a vertical almost-toric system is analogous to that of a semi-toric system. However, the potential for infinitely many focus-focus points in a vertical almost-toric system gives a richer behavior, as can be seen by comparing Section 4.4 with Vũ Ngo . c [46,Section 4]. Result (D), which ensures the existence of η-cartographic systems in the isomorphism class of a vertical almost-toric system, is tailored for applications. In particular, it plays an important role in defining surgeries of vertical almost-toric systems ( [20], forthcoming). Also, it allows one to make precise the notion that the image of a cartographic homeomorphism is a limit of moment map images (cf. Proposition 4.52).
Structure of the paper. Section 2 defines and explains notions that we use throughout the paper, while also establishing notation. While the section should serve as a self-contained primer to guide readers unfamiliar with the subject through this paper (and the forthcoming [20]), it may be of interest to experts in the field as well, for a few ideas which do not appear in many other places. For instance, we introduce the notion of a faithful integrable system, which, intuitively speaking, is one whose moment map image reflects the topology of the partition of the total space into the connected components of the fibers of the moment map (cf. Section 2.2). Moreover, in Section 2.7, we elaborate on the notion of cartographic homeomorphisms that is introduced in Pelayo & Ratiu & Vũ Ngo . c [33] for generalized semitoric systems and establish some general properties for these objects. Note that the section should not be taken as an exhaustive reference for either topological or symplectic aspects of integrable systems.
In Section 3, almost-toric systems are defined and their basic properties are explored. In particular, the neighborhood of a connected component of a fiber is described (cf. Section 3.2) and, in preparation for the next section on vertical almost-toric systems, we describe properties of systems that are both faithful and almost-toric (Section 3.3).
Section 4 is the heart of the paper: It contains the definition of vertical almost-toric systems as well as all the main results described above. Section 4.1 contains main result (A) as Proposition 4.9, characterizations of the moment map image of vertical almost-toric systems (Corollary 4.5), and a useful criterion to determine which saturated subsystems of vertical almost-toric systems are vertical almost-toric (Proposition 4.7). Section 4.2 explores the consequences of the presence of the S 1 -action. The existence of cartographic homeomorphisms (Theorem 4.24) is proved in Section 4.3, which also establishes some useful topological properties of the complements of the cuts needed to define cartographic homeomorphisms. Section 4.4 describes the set of cartographic homeomorphisms associated to a given vertical almosttoric system, paying particular attention to the subtleties that arise from allowing infinitely many focus-focus points (see Theorem 4.36). Finally, Section 4.5 proves that, in some sense, cartographic homeomorphisms can be made smooth everywhere by modifying them on arbitrarily small neighborhoods of the defining cuts. This is the content of Theorem 4.48, which can be used to establish the existence of η-cartographic vertical almost-toric systems in any given isomorphism class (Theorem 4.51).

Notation and conventions.
Topological conventions.
• A subset of a topological space is endowed with the subspace topology unless otherwise stated. • A pair of topological spaces (Y, Z) consists of a topological space Y together with a subset Z ⊂ Y endowed with the relative topology. A topological embedding of pairs of topological spaces (Y 1 , Z 1 ), (Y 2 , Z 2 ) is a topological embedding χ : Y 1 → Y 2 that restricts to a topological embedding of Z 1 into Z 2 . A homeomorphism between pairs of topological spaces is a topological embedding onto the target.
• A map f : X → Y between topological spaces X and Y is proper onto its image if the map f : X → f (X) is proper.
• Following Joyce [22], consider the subspace [0, +∞[ n ⊂ R n and let M be a topological space. An n-dimensional smooth atlas with corners on M is a set A : A smooth manifold with corners of dimension n is a Hausdorff, paracompact, second countable topological space together with an ndimensional smooth atlas with corners. A smooth structure with corners on a topological space is an equivalence class of smooth atlases with corners, where two atlases are deemed equivalent if their union is again a smooth atlas with corners of a given dimension.
for some (A, b) ∈ AGL(n; Z) = GL(n; Z) R n , where n is the dimension of the atlas. A Z-affine manifold with corners of dimension n is a Hausdorff, paracompact, second countable topological space together with an n-dimensional Z-affine atlas with corners. And a Zaffine structure with corners on a topological space is an equivalence class of Z-affine atlases with corners, where two atlases are deemed equivalent if their union is again a Z-affine atlas with corners of a given dimension.
Smooth atlases and Z-affine atlases, without corners, (and the corresponding manifold structures) are defined as above, with the stipulation that the images of the coordinate charts are subsets of R n . Also, note that, because the transition maps of are smooth, a Zaffine atlas (with or without corners) is also a smooth atlas, and hence defines a unique smooth structure.
for all x ∈ A there exists an open neighborhood U x ⊂ R n of x and a smooth map f x : U x → R m that is a local extension of f ; • A map f : A ⊂ R n → R m is a smooth embedding if it is a diffeomorphism onto its image.
• Manifolds are assumed to be without boundary or corners unless otherwise stated.
Boundary conventions. Two types of boundaries of subsets X ⊂ R n are dealt with in this paper whenever X is a smooth manifold with corners embedded in R n . The topological boundary, the closure of X ⊂ R n minus its interior, is denoted ∂X. Meanwhile, its boundary as a manifold with corners, X ∩ ∂X, is denoted ∂ ∞ X. For instance if X = {(x, y) | |x| < 1 and |y| ≤ 1}, then and Group conventions. Throughout the paper, the identification

Primer on integrable systems
This section introduces the basic notions regarding integrable systems that are used throughout the paper. Section 2.1 presents the category of integrable systems and defines the coarsest topological invariant: the leaf space of an integrable system (see Definition 2.4). Systems whose leaf spaces can be identified with the moment map images play an important role in this paper and are studied in Section 2.2; we call such systems faithful. In Sections 2.3 -2.6, we endow large subsets of the leaf space of an integrable system with an Z-affine structure. First, following Duistermaat [9], we show how the part of the leaf space corresponding to regular leaves inherits such a structure in Section 2.3. Second, we identify a class of systems that are isomorphic to systems equipped with Hamiltonian torus actions of maximal dimension: these are called weakly toric, are related to symplectic toric manifolds and are studied in Section 2.5. Third, Section 2.6 extends the Z-affine structure on the regular part of the leaf space to include singular leaves that admit a neighborhood supporting a Hamiltonian torus action of maximal dimension. In Section 2.4, we recall a fundamental property that Z-affine structures enjoy, namely that they can be developed. Finally, following Pelayo & Ratiu & Vũ Ngo . c [33], we introduce the notion of cartographic homeomorphisms, which, intuitively, can be thought of a way to encode the above Z-affine structure in a way that is compatible with singular orbits of the system.
is a diffeomorphism, and the following diagram commutes: Given Φ = (H 1 , . . . , H n ) : (M, ω) → R n denote by X i , for 1 ≤ i ≤ n, the Hamiltonian vector field associated to H i , defined implicitly by the equation ω(X i , ·) = dH i . If the flows of the vector fields X 1 , . . . , X n are complete, then there is a Hamiltonian R n -action on (M, ω), one of whose moment maps is precisely Φ.
Throughout this paper, integrable systems have compact fibers unless otherwise stated, so the above completeness assumption is satisfied and Φ is a moment map for a Hamiltonian R n -action (upon identifying the Lie algebra of R n with R n ). For this reason, it is referred to as the moment map of the system.
Many of the integrable systems considered in this paper arise from restricting a given system to a subset.
for some open subset U of Φ(M ), the subsystem of (M, ω, Φ) relative to V is also referred to as the subsystem of (M, ω, Φ) relative to U . Remark 2.3 Subsystems of integrable systems with compact fibers need not have compact fibers; moreover, subsystems of integrable systems supporting a Hamiltonian R n -action need not support a Hamiltonian R n -action.
The most accessible feature of an integrable system is the image of Φ, but, for arbitrary systems, there is no way to extract useful information from that image. However, the leaf space of the system reliably reflects some of the structure. and q belong to the same leaf, endowed with the quotient topology; • a subsystem (V, ω| V , Φ| V ) of (M, ω, Φ) is saturated if any leaf of (M, ω, Φ) that intersects V is contained in V .
Remark 2.5 Given an integrable system (M, ω, Φ) and an open subset U ⊂ Φ(M ), the leaf space of the subsystem relative to U is naturally included in the leaf space of (M, ω, Φ).
Given an integrable system (M, ω, Φ) with leaf space L, the moment map Φ factors through L, giving rise to the quotient map q : M → L and inducing a continuous map π : L → Φ(M ) that makes the following diagram commute: Remark 2.6 Isomorphic integrable systems possess homeomorphic leaf spaces.

Remark 2.7
The sets of connected components of the leaf space of an integrable system and of connected components of its total space are in bijection.
A natural way to enhance the topological data encoded in the leaf space of an integrable system is to identify the singular leaves of the system. Definition 2.8. Let (M, ω, Φ) be an integrable system with n degrees of freedom with leaf space L.
regular. • The subset L sing ⊂ L consisting of the image of singular leaves of Φ is said to be the singular part of L, while its complement L reg is said to be the regular part.
Definition 2.9. The pair (L, L reg ) associated to an integrable system (M, ω, Φ) is called the pair of leaf and regular leaf spaces of the system.
The above association descends to isomorphism classes of systems and behaves well with respect to saturated subsystems: Remark 2.10.
• Isomorphic integrable systems possess homeomorphic pairs of leaf and regular leaf spaces. • The pair of leaf and regular leaf spaces of saturated subsystems of (M, ω, Φ) naturally embed in the pair of leaf and regular leaf spaces of (M, ω, Φ).

2.2.
Faithful integrable systems. In light of Remark 2.6, it is helpful to distinguish those cases in which a moment map image at least carries the topological information of the leaf space.
Definition 2.11. An integrable system (M, ω, Φ) with leaf space L is said to be faithful if the induced map π : L → B = Φ(M ) is a homeomorphism. Here, B ⊂ R n is equipped with the subset topology.
Faithful integrable systems form a full subcategory of the category of all integrable systems.
Remark 2.12. If an integrable system (M, ω, Φ) is faithful, then so is every integrable system isomorphic to it, and every saturated subsystem. In particular, by Remark 2.7, if U ⊂ Φ(M ) is open, the set of connected components of U is in bijective correspondence with the set of connected components of the total space of the subsystem relative to U . Remark 2.13. Because the leaf space of a faithful integrable system is homeomorphic to a subset of R n , such a leaf space is second countable and Hausdorff. Example 2.14 illustrates two ways in which a failure of faithfulness can disrupt the relationship between the moment map image and the topology of the total space -first if the fibers of the moment map need not be connected, and second if the moment map need not be proper onto its image.
Example 2.14 Let A ⊂ R 2 be the closed annulus centered at the origin with inner and outer radii 1 and e. Let (M, ω, Φ) be the toric system (cf. Definition 2.26) that underlies the compact symplectic toric manifold whose moment map image is the rectangle R := [0, 1]×[0, 2π] ⊂ R 2 and define g k : R → A ⊂ R 2 by g k (x, y) = (e x cos(ky), e x sin(ky)) with k ∈ N.
1) For any k ∈ N, the integrable system (M, ω, g k • Φ) has fibers that are the disjoint union of • k tori or • 2 circles and k−1 tori or • 2 points and k−1 circles. 2) Let (M , ω , Φ ) be the subsystem of (M, ω, Φ) relative to the subset [0, 1] × [0, 2π[. Then the integrable system (M , ω , g 1 • Φ ) has the same (connected) fibers as (M , ω , Φ ), but unlike (M , ω , Φ ) -or any toric system defined on a compact manifold -has circle orbits in the preimage of some points on the interior of the moment map image A, cf. Figure 2.1. Notice however that, unlike Φ , the map g 1 • Φ is not proper onto its image: the preimage of the compact set A is not compact. Proof. Let L and B be the leaf space and moment map image, respectively, of (M, ω, Φ). Let q : M → L be the quotient map and let π : L → B be the induced map. Suppose first that Φ has connected fibers and is proper onto its image. By the definition of L, the continuous map π is a bijection because the fibers of Φ are connected. It remains to show that π is a closed map. In general, a continuous proper map to a metrizable space is closed (cf. Palais [31]). To see that π is proper, consider an arbitrary compact set K ⊂ B. The preimage Φ −1 (K) is compact because Φ is proper onto its image. Furthermore, π −1 (K) = q(Φ −1 (K)) because q is surjective, so π −1 (K) is compact because q is continuous and Φ −1 (K) is compact. Therefore π is proper. Then, since B ⊂ R 2 is metrizable, the map π is also closed. Consequently, π is a homeomorphism.
Conversely, suppose that π is a homeomorphism; in particular, it is a bijection, which implies that Φ has connected fibers. It remains to prove that Φ is proper onto its image; since π is a homeomorphism, it suffices to check that q is proper onto its image. Note that, since π is a homeomorphism, L is second countable, locally compact and Hausdorff.  When the fibers of an integrable system are compact, the regular leaf space inherits a geometric structure. This is a consequence of the Liouville-Arnol'd theorem, which provides a local normal form for a neighborhood of a regular leaf (cf. Cushman [38] for various versions of a proof). Let Ω be the canonical symplectic form on T * T n ∼ = T n × R n for which the projection pr 2 : (T * T n , Ω) → R n defines an integrable system. Theorem 2.16 (Liouville-Arnol'd). Let (M, ω, Φ) be an integrable system with n degrees of freedom and let F be a regular, compact leaf. Then there exist open neighborhoods V ⊂ (M, ω) of F and W ⊂ (T * T n , Ω) of T n × {0}, the latter saturated w.r.t. pr 2 , such that the subsystems of (M, ω, Φ) and of (T * T n , Ω, pr 2 ), relative to V and W , respectively, are isomorphic via a pair (Ψ, ψ) where ψ (Φ(F )) = 0.

Remark 2.17
Identify the Lie algebra of T n with R n . Using the notation of Theorem 2.16, the composition ψ • Φ| V is the moment map of a free, effective Hamiltonian T n -action, i.e. the Hamiltonian vector fields of its components have 2π-periodic flows. Moreover, if Φ = (H 1 , . . . , H n ) with H 1 being the moment map of an effective Hamiltonian S 1 -action, then the diffeomorphism ψ can be taken to be of the form where a ∈ R is a constant.
Corollary 2.18. Let (M, ω, Φ) be an integrable system with compact fibers. The regular leaf space L reg ⊂ L is open and it inherits a structure of smooth, Hausdorff manifold uniquely defined by requiring that the restriction of the quotient map q| q −1 (Lreg) be a submersion onto L reg . In particular, the restriction π| Lreg is smooth.
Proof. Moerdijk and Mrčun proved, in [29,Section 2.4], that the leaf space of a submersion whose fibers are compact can be endowed with the structure of a smooth manifold uniquely defined by demanding that the quotient map be a submersion. The version of the Liouville-Arnol'd theorem given in Theorem 2.16 implies that, for any point p ∈ L reg , the corresponding regular leaf F p has an open neighborhood that is saturated by regular leaves. Therefore, Φ| q −1 (Lreg) is a submersion whose leaf space is naturally isomorphic to L reg , an open subset of L. Since the fibers of Φ are compact by hypothesis, the result of Moerdijk and Mrčun implies the desired result.
A symplectomorphism ϕ of (T n × R n , Ω) that preserves the fibers of pr 2 must have the form ϕ = (ϕ (1) , ϕ (2) ) where ϕ (1) (t, x) = (A −1 ) T t + f (x) and ϕ (2) (t, x) = Ax + c for some A ∈ GL(n, Z), some c ∈ R n , and a smooth function f : R n → R n such that the matrix A −1 ∂f ∂x is symmetric (cf. Symington [40,Lemma 2.5].) This implies that maps of the form ψ • π, where ψ is as in Theorem 2.16, can be used to define a Z-affine atlas on L reg . Definition 2.19. For any n ≥ 1, the category of n-dimensional Zaffine manifolds, denoted by Aff Z (n), has objects and morphisms as follows: • Objects: Z-affine manifolds (with corners), as defined in Section 1.
• Morphisms: Z-affine maps, i.e. maps f : are local diffeomorphisms such that f * A 2 and A 1 define equivalent Z-affine atlases (cf. Section 1).
If (N 2 , A 2 ) is a Z-affine manifold and f : N 1 → N 2 is locally a homeomorphism then there exists a unique (up to isomorphism) Zaffine structure A 1 on N 1 that makes f into a Z-affine morphism. The structure A 1 is henceforth referred to as being induced by f . For Z-affine manifolds, it makes sense to consider (the sheaf of) Zaffine functions, i.e. (locally defined) smooth functions that, in local Z-affine coordinates (x 1 , . . . , x n ), are given by where k i ∈ Z and c ∈ R. The local normal form provided by the Liouville-Arnol'd Theorem (Theorem 2.16) implies that the regular leaf space of an integrable system with compact fibers can be characterized, as a Z-affine manifold, by the sheaf of functions that generate 2πperiodic flows tangent to the fibers of the quotient map.
Corollary 2.21. Let (M, ω, Φ) be an integrable system with compact fibers. Then the subset L reg ⊂ L inherits a Z-affine structure A reg , uniquely defined by the property that locally defined Z-affine functions from (L reg , A Lreg ) to (R, A 0 ) correspond, by taking the pull-back along the restriction to q −1 (L reg ) of the quotient map q : M → L, to functions on q −1 (L reg ) ⊂ M whose Hamiltonian vector fields are tangent to the fibers of q and have 2π-periodic flows.
Corollary 2.22. For each n ≥ 1, there is a functor IS(n) → Aff Z (n) that, on objects, is precisely the map (M, ω, Φ) → (L reg , A reg ) given by Corollary 2.21.
Proof. The above functor is completely determined by the following property, which can be checked directly: an isomorphism of integrable systems induces a Z-affine (iso)morphism between the associated Zaffine manifolds.
Furthermore, the correspondence between integrable systems and Zaffine manifolds given by Corollary 2.21 behaves well under restriction to saturated subsystems.
Corollary 2.23. Given an integrable system (M, ω, Φ), the natural inclusion of the leaf space of a saturated subsystem into the leaf space of (M, ω, Φ) corresponds to a Z-affine embedding of one Z-affine manifold into another.
Finally, the above discussion allows further refinement of the set of invariants that can be associated to an integrable system. Given (M, ω, Φ), associate the pair (L, (L reg , A reg )) to it, where (L, L reg ) is the pair of leaf and regular leaf spaces of (M, ω, Φ) and A reg is the Z-affine structure given by Corollary 2.21. This association descends to isomorphism classes of systems. In this case, isomorphisms of pairs are homeomorphisms of the underlying topological pairs that restrict to Z-affine isomorphisms on the Z-affine subspace.

Remark 2.24
For faithful integrable systems (M, ω, Φ), the leaf space L can be identified topologically with the moment map image B = Φ(M ). Under this correspondence, L reg is identified with the subset of regular values B reg ⊂ B. By Corollary 2.21, B reg inherits a Z-affine structure denoted by A reg which, in general, is not isomorphic to the standard one.
2.4. Developing maps. Given an n-dimensional Z-affine manifold (with corners) (N, A), letÑ denote its universal cover. The universal covering map q :Ñ → N induces a Z-affine structureÃ onÑ , making q : (Ñ,Ã) → (N, A) into a Z-affine morphism. Fix a basepoint x 0 ∈ N , a pointx 0 ∈Ñ with q(x 0 ) = x 0 , and a Z-affine coordinate chart φ 0 : U 0 → R n defined near x 0 . IdentifyÑ with the space of paths starting at x 0 , up to homotopy relative to endpoints. Then there is a Z-affine map dev : (Ñ,Ã) → (R n , A 0 ), uniquely determined by the property that, nearx 0 , it equals the restriction of φ 0 to a suitable neighborhood of x 0 , and a representation a : π 1 (N ; x 0 ) → AGL(n; Z), called the affine holonomy of (N, A), which is intertwined with dev as follows: for all γ ∈ π 1 (N ; x 0 ), the following diagram commutes where ·γ denotes the Z-affine isomorphism of (Ñ,Ã) induced by the natural action of π 1 (N ; x 0 ) onÑ .
The map dev :Ñ → R n is called the developing map of (N, A) (relative to the choices (x 0 ,x 0 , φ 0 )), cf. for details Goldman & Hirsch [13] and references therein. Note that, using the fundamental groupoid of N , the information of a developing map can be packaged and conveyed independent of choices (cf. Crainic & Fernandes & Martínez-Torres [4]).

Remark 2.25
• If dev, dev :Ñ → R n are developing maps for (N, A) constructed using different choices then there exists a unique element h ∈ AGL(n; Z) such that dev = h • dev. • If (N, A) is a Z-affine manifold with corners, then the image of any codimension-k face ofÑ (0 < k ≤ n) under a developing map is the intersection of k linear hyperplanes of R n whose normals can be chosen to span a unimodular sublattice of Z n , i.e. this span is a direct summand of Z n . • In general, developing maps need not be covering maps and their images can be rather complicated (cf. Sullivan & Thurston [39] for pathological examples).

Toric and Delzant systems.
This section describes the connection between integrable toric actions and integrable systems.
Henceforth, for each n ≥ 1, fix an isomorphism t * ∼ = R n so that the standard lattice in t * (dual to ker (exp : t → T n )) is mapped to Z n .
Note that the individual components of the moment map µ : (M, ω) → R n Poisson commute and µ is a submersion almost everywhere, due to the Marle-Guillemin-Sternberg local normal form for Hamiltonian actions of compact Lie groups (cf. Guillemin & Sternberg, Marle [15,26]). Therefore, taking Φ = µ, call (M, ω, Φ) the integrable system underlying (M, ω, µ). Because isomorphisms of symplectic toric manifolds induce isomorphisms of underlying integrable systems, for each n ≥ 1, the function from T M(2n) to IS(n) that maps a toric manifold to its underlying integrable system defines a 'forgetful functor' F : T M(2n) → IS(n).
It is useful to identify the integrable systems that underlie symplectic toric manifolds and systems that are isomorphic to such. Definition 2.27. An integrable system (M, ω, Φ) is toric if there exists a toric manifold (M, ω, µ) such that (M, ω, Φ) = F(M, ω, µ). An integrable system is weakly toric if it is isomorphic to a toric one.
Example 2.28. Let (M, ω, Φ) be an integrable system with compact fibers and q : M → L the quotient map to its leaf space. A Z-affine coordinate chart on L reg ⊂ L yields, pre-composing with the quotient map, the moment map of a locally defined effective (but not unique) Hamiltonian T n -action (cf. Corollary 2.21), and hence a toric system on the preimage by q of the domain of the coordinate chart.

Remark 2.29
The image of F is a subcategory of IS(n), but it is not full. For instance, if h ∈ AGL(n; Z) is an element different from the identity, the symplectic toric manifolds (M, ω, Φ) and (M, ω, h • Φ) are not isomorphic, but the underlying integrable systems are. In fact, the integral affine group AGL(n; Z) completely captures the failure of the functor to be full: Suppose that (M 1 , ω 1 , Φ 1 ) and (M 2 , ω 2 , Φ 2 ) are isomorphic toric systems with isomorphism denoted by (Ψ, ψ). Then , by the restriction of an element in AGL(n; Z).
The classification by Delzant [7] of compact symplectic toric manifolds (M, ω, µ) is well-known. In particular, the manifold M , the symplectic form ω and the moment map µ up to isomorphism are determined by the image of the moment map, µ(M ). Two properties of a compact symplectic toric manifold (M, ω, µ) that are important for the classification are that µ has connected fibers and that µ(M ) is a convex polytope. These properties follow from work of Atiyah [1] and Guillemin & Sternberg [14] and hold for the more general family of effective Hamiltonian torus actions on compact symplectic manifolds.
If the underlying symplectic manifold is not compact, neither of the above properties need hold (cf. Karshon & Lerman [23]). Nevertheless, Karshon & Lerman [23] achieve a classification of these objects. Among others, they use the following result: Orbits are tori that have a neighborhood that can be put in normal form (cf. Guillemin & Sternberg [15], Marle [26] Observe that W is saturated by the fibers of pr 2 ⊕ A • q.
The existence of these local normal forms implies that even singular orbits of the action make up whole connected components of the fibers of the moment map: this sets toric systems apart from general integrable systems.
In analogy with Corollary 2.21, the above local normal forms imply the following: Remark 2.30 Let (M, ω, µ) be a symplectic toric manifold with associated toric system (M, ω, Φ). Then the above local normal forms imply the following.
1) The orbit space M/T n of (M, ω, µ) is a Z-affine manifold with corners uniquely characterized as in Corollary 2.21. 2) M/T n is canonically homeomorphic to the leaf space L of (M, ω, Φ).
3) The image under π of a codimension-k face of L, where 0 < k ≤ n, is the intersection of k hyperplanes of R n whose normals can be chosen to span a unimodular sublattice of Z n (cf. Remark 2.25). 4) With the above identification, the Z-affine structure on L, denoted by Note that the developing map dev :L → R n for (L, A L ) makes the following diagram commuteL where q :L → L is the universal covering map from Karshon & Lerman [23, Proposition 1.1, Remarks 1.4 and 1.5] and π the so-called orbital moment map of (M, ω, µ) therein.
The orbital moment map is an essential invariant of symplectic toric manifolds; this is the content of the following result stated below without proof. Proof. This follows from the faithfulness of the moment map, connectedness of the moment map image, and the equivalence of the cardinality of the set of components of the leaf space and of the total space of an integrable system (Remark 2.7).
Under that equivalence, faithfulness of the moment maps Φ i corresponds to the orbital moment maps of the symplectic toric manifolds being embeddings. Therefore, since [23] implies the symplectic toric manifolds (M i , ω i , µ i ) are determined up to isomorphism by their moment map images. By the definition of T M(2n), the category of symplectic toric manifolds (Definition 2.26), (M i , ω i , µ i ), i = 1, 2 belong to the same isomorphism class if and only if µ 1 (M 1 ) and µ 2 (M 2 ) differ by a translation. The criterion for isomorphism of the toric systems then follows from Remark 2.29 and connectedness of the total space (Lemma 2.33).
Remark 2.35 Given a Delzant system (M, ω, Φ) with n degrees of freedom, identify its leaf space with its moment map image B. The above discussion, together with the fact that B is simply connected, implies that the inclusion B → R n is a developing map for the induced Z-affine structure on B. Thus the Z-affine structure on B coming from (M, ω, Φ) is the standard one.
Unlike compact symplectic toric manifolds (and their associated toric systems), the moment map image of a Delzant system need not be convex; however, the above local normal form, together with the defining properties of Delzant systems, imply the following result. 2.6. Locally weakly toric leaf spaces. Weakly toric provide examples of integrable systems whose leaf spaces are naturally endowed with the structure of a Z-affine manifold with corners. This is not a phenomenon to be expected in general. However it is natural to ask, what is the largest subset of the leaf space of an integrable system that does inherit the structure of a Z-affine manifold with corners?
That question motivates the following notions of a weakly toric leaf and the weakly locally toric leaf space of an integrable system. Definition 2.39. Given an integrable system (M, ω, Φ) with leaf space L, the subset L lt ⊂ L corresponding to weakly toric leaves is called the locally weakly toric leaf space associated to (M, ω, Φ).
A priori, if the fibers of an integrable system are not necessarily compact, the associated locally weakly toric leaf space may be empty. In contrast, when the fibers are required to be compact, the Liouville-Arnol'd Theorem (Theorem 2.16) implies L lt must be dense in L: In fact, if an integrable system has compact fibers, its locally weakly toric leaf space inherits a Z-affine structure.
Proposition 2.41. The locally weakly toric leaf space L lt of an integrable system, whenever non-empty, inherits a structure of Z-affine manifold with corners denoted by A lt . This structure is uniquely defined by the property that locally defined Z-affine functions from (L lt , A L lt ) correspond to functions on q −1 (L lt ) ⊂ M whose Hamiltonian vector fields are tangent to the fibers of q and have 2π-periodic flows.
Proof. Fix an integrable system (M, ω, Φ) with n degrees of freedom whose locally weakly toric leaf space L lt is not empty. To show that the locally weakly toric leaf space L lt is Hausdorff, it suffices to check that if L 1 , L 2 are weakly toric leaves with Φ(L 1 ) = Φ(L 2 ), then for i = 1, 2, there exists an open neighborhood V i of L i , saturated with respect to q and containing solely weakly toric leaves, with is open because the quotient maps in the local models for symplectic toric manifolds are open. Finally, since L lt is locally homeomorphic to a subset of Euclidean space, it is locally compact, thereby implying L lt is paracompact (by virtue of being a locally compact, Hausdorff, second countable space).
Next we define an open cover of L lt and coordinate charts whose codomain is [0, +∞[ n . Let L ⊂ M be a weakly toric leaf; by definition, there exists an open neighborhood V ⊂ M of L whose corresponding subsystem is weakly toric. By restricting V if necessary, it may be assumed that the corresponding subsystem is isomorphic to a local model for symplectic toric manifolds. Since is an open subset of L lt ; moreover, the above isomorphism implies that there is a map χ : q(V ) → [0, +∞[ n that is locally a homeomorphism. Since L ⊂ M is arbitrary, the above reasoning defines an open cover of L lt , denoted by {U i }, and, for each i, a map χ i : Then, unraveling the above definitions, we obtain an isomorphism (Ψ ij , ψ ij ) of saturated subsystems of local models of symplectic toric manifolds with ψ ij = χ j • χ −1 i ; observing that ψ ij is necessarily the restriction of an element in AGL(n; Z) shows the desired result. Finally, the defining property of A lt follows directly from its definition and the fact it holds for the (locally weakly toric) leaf spaces of symplectic toric manifolds.
Remark 2.42 Note that, by Proposition 2.41, the locally weakly toric leaf space of an integrable system is, tautologically, the largest subset of the leaf space of an integrable system that inherits the structure of a Z-affine manifold with corners. Moreover, if the system has compact fibers, the inclusion (L reg , As expected, the Z-affine manifold with corners (L lt , A lt ) associated to (M, ω, Φ) is an invariant of the isomorphism class of (M, ω, Φ) and behaves well with respect to restriction to saturated subsystems, i.e. statements analogous to Corollaries 2.22 and 2.23 hold for this (possibly larger) Z-affine manifold with corners. This allows one to associate to an integrable system (M, ω, Φ) the pair (L, (L lt , A lt )), the latter being an invariant of the isomorphism class of (M, ω, Φ).

Remark 2.43
If (M, ω, Φ) is faithful and has compact fibers, the locally weakly toric leaf space corresponds to an open, dense subset denoted by B lt ⊂ B = Φ(M ) and its boundary as a manifold with corners (corresponding to singular weakly toric leaves) satisfies ∂ ∞ B lt ⊂ B ∩ ∂B, where the inclusion may be strict.

2.7.
Cartographic maps. The geometry of the locally weakly toric leaf space of a faithful integrable system (M, ω, Φ) with compact fibers is captured by the moment map image whenever the inclusion B → R n is a Z-affine embedding when restricted to B lt ∼ = L lt , as is the case for faithful toric (and, in particular, Delzant) systems. In this case, the subsystem relative to B lt is toric; expanding the terminology of Pelayo & Ratiu & Vũ Ngo . c [33], we say that the moment map Φ is cartographic.
When the moment map of a faithful integrable system (M, ω, Φ) with compact fibers is not cartographic, one could ask whether there is an isomorphic system whose moment map is cartographic. Existence of such a system is tantamount to finding a smooth embedding of the moment map image B into R n whose restriction to the locally weakly toric leaf space B lt is a Z-affine embedding into (R n , A 0 ). A necessary condition is that the affine holonomy of (B lt , A lt ) be trivial; however, that condition is not sufficient, as the following example illustrates.
Consider the unique symplectic toric manifold (T 2 × R, ω, µ) defined by the orbital moment map W : R → R 2 given by g 2 (x, y) = (e x cos(2y), e x sin(2y)). Observe that ω is not the standard symplectic structure induced by inclusion of T 2 × R in (T 2 × R 2 , ω can ). Let Φ : T 2 × R → R ⊂ R 2 be the projection onto the second factor. Then (T 2 × R, ω, Φ) defines a faithful integrable system. Note that for any open simply connected subset , the subsystem of (T 2 × R, ω, µ) relative to U is the union of at least two disjoint subsystems, each of which is isomorphic to a subsystem of the toric system (T 2 × R, ω, µ). However, the integrable system (T 2 × R, ω, Φ) as a whole is not isomorphic to any system with a cartographic moment map as g 2 , which is not injective, is the unique map (up to composition on the right with an Z-affine diffeomorphism of (R 2 , A 0 )) such that g 2 • Φ generates an effective Hamiltonian T 2 -action on (T 2 × R, ω).
The faithful integrable systems with compact fibers considered in Sections 3 and 4 allow for singular fibers (focus-focus fibers, cf. Section 3.1.2) that induce non-trivial affine holonomy on the locally weakly toric leaf space (cf. Theorem 3.27). Thus for such an integrable system there is not necessarily a system in its isomorphism class that has a cartographic moment map. However, following the insight of Symington [40] and Vũ Ngo . c [46], for such systems it is reasonable to ask whether there is a homeomorphism of the moment map image that, when restricted to an open, dense subset of the locally weakly toric leaf space, is a Z-affine embedding. That question motivates introducing the following notion. is said to admit a cartographic homeomorphism whose image is said to be cartographic.

Remark 2.46
If (f, S) is a cartographic pair for (M, ω, Φ), there is no guarantee that S is maximal, i.e. that it is the largest open, dense subset of B lt on which f restricts to a Z-affine embedding.
The following lemma provides a simple but useful way to adjust a given cartographic homeomorphism.
Intuitively, cartographic homeomorphisms should be thought of as continuous extensions of restrictions of developing maps to suitable domains; for instance, if S is simply connected, S can be identified with a dense subset of a fundamental domain in the universal cover of B lt .
If S = B holds in Definition 2.45, then f •Φ is a cartographic moment map. If not, a cartographic homeomorphism at least provides a dense subset of the total space on which the system is isomorphic to a toric, and possibly Delzant system. More precisely: is toric. If, in addition, S is contractible, then the above system is Delzant.
Cartographic homeomorphisms restrict appropriately when taking saturated subsystems.
be an open subset. Then (f | U , S ∩ U ) is a cartographic pair for the subsystem relative to U .
Combining Corollaries 2.48 and 2.49, we obtain the following simple description of cartographic homeomorphisms when restricted to open subsets of the moment map image whose corresponding subsystems are Delzant.
Corollary 2.50. Let (M, ω, Φ) be a faithful integrable system with compact fibers with cartographic pair (f, S). Suppose that U ⊂ S is an open subset with the property that the subsystem relative to U is Delzant. Then f | U is the restriction of an element h U ∈ AGL(n; Z) and h −1 Delzant. In fact, the pair (id, f | U ) defines an isomorphism between these two systems. Because U is connected (by virtue of being contractible), there exists an element h U ∈ AGL(n; Finally, it is important to notice that the property of admitting a cartographic homeomorphism is independent of the choice of representative of the isomorphism class of a faithful integrable system. be faithful integrable systems with compact fibers isomorphic via (Ψ, ψ) and let (f 1 , S 1 ) be a cartographic pair for (M 1 , ω 1 , Φ 1 ). Then (f 2 := f 1 • ψ −1 , S 2 := ψ(S 1 )) is a cartographic pair for (M 2 , ω 2 , Φ 2 ).
In fact, cartographic images of isomorphic systems are homeomorphic via maps that extend Z-affine isomorphisms. More precisely, the following holds.
Corollary 2.52. Consider, for i = 1, 2, a faithful integrable system (M i , ω i , Φ i ) with n degrees of freedom and compact fibers with a cartographic pair (f i , S i ). Assume that there exists an isomorphism (Ψ, ψ) between (M 1 , ω 1 , Φ 1 ) and (M 2 , ω 2 , Φ 2 ). Then the cartographic images f 1 (B 1 ) and f 2 (B 2 ) are homeomorphic by a map that, when restricted to each connected component of f 1 (S 1 ∩ ψ −1 (S 2 )), is the restriction of an element of AGL(n; Z).
is a homeomorphism as it is the composition of homeomorphisms. In fact, we claim that it is the unique extension of a Z-affine isomorphism . To see this, begin by observing that S 1 ∩ ψ −1 (S 2 ) and ψ(S 1 ) ∩ S 2 are open and dense in B 1 and B 2 respectively. This implies that for each . This restriction is a Z-affine isomorphism, by the definition of cartographic homeomorphisms and Corollary 2.22. Since the Z-affine structures on f 1 (S 1 ∩ ψ −1 (S 2 )) and on f 2 (ψ(S 1 ) ∩ S 2 ) are isomorphic to the ones induced by inclusion into (R n , A 0 ), the restriction of the above Z-affine isomorphism to each connected component of f 1 (S 1 ∩ ψ −1 (S 2 )) is the restriction of an element of AGL(n; Z).

Almost-toric systems
Motivated by Symington [40] and Vũ Ngo . c [46], this section introduces and studies the fundamental properties of almost-toric systems, a category of integrable systems generalizing that of weakly toric systems (Definition 2.27) in dimension 4 by allowing for the presence of focus-focus fibers, which are the Lagrangian analog of nodal fibers in Lefschetz fibrations. To define this category formally, we first recall the notion of almost-toric singular orbits, which are a special class of nondegenerate singular orbits in dimension 4. This is achieved in Section 3.1. Section 3.2 defines almost-toric systems and establishes fundamental properties of leaves and their neighborhoods. Seeing as the vertical almost-toric systems of Section 4 are both faithful and almost-toric, Section 3.3 collects results about systems that satisfy both properties.

Singular orbits.
3.1.1. Non-degenerate singular orbits in arbitrary dimension. The types of singular orbits considered in Section 3.1.2 are a special case of nondegenerate singular orbits, a condition that is briefly recalled below and should be thought of as a 'symplectic' Morse-Bott condition. Throughout this subsection, let (M, ω, Φ) be an integrable system so that Φ is the moment map of an effective Hamiltonian R n -action; for any t ∈ R n , denote by φ t : (M, ω) → (M, ω) the symplectomorphism induced by acting via t. Moreover, for any p ∈ M , denote by O p the R n -orbit through p. If p is singular, then every point in O p is singular; thus the notion of singular orbit is well-defined. Next, we introduce the following useful notion. [49,Definition 3.4]. In particular, the rank and the degree of closedness of an orbit are well-defined notions.
Fix a singular orbit O ⊂ M of rank 0 ≤ k < n and let p ∈ O; since, for all t ∈ R n , φ t is a symplectomorphism sending O to itself, it follows that, for all t ∈ R n , D p φ t is a symplectomorphism of (( and Ω is the symplectic form induced by performing linear reduction. Thus we obtain a Lie algebra homomorphism R n → Sp ((T p O p ) ω /T p O p , Ω). In fact, this homomorphism only depends on the orbit and not on the choice of point; this is because the action is by an abelian Lie group. Choosing local Darboux coordinates, it is possible to identify Sp ((T p O p ) ω /T p O p , Ω) with Sp (2(n − k); R); therefore, by taking derivative at the identity, we obtain a Lie algebra homomorphism R n → sp (2(n − k); R) whose image is denoted by h O . Then, for all i = 1, . . . , n, the Hamiltonian vector field of H i vanishes at p; thus it makes sense to consider its linearization at p denoted by X Lin i (p) ∈ sp(2n; R). The point p is nondegenerate if X Lin 1 (p), . . . , X Lin n (p) are linearly independent and if there exists a linear combination λ 1 X Lin 1 (p)+. . .+λ n X Lin n (p) with 2n distinct, non-zero eigenvalues.
Cartan subalgebras of sp (2(n − k); R) have been classified up to conjugacy in Williamson [48] using the standard isomorphism sp (2(n − k); R) ∼ = Sym (2(n − k); R), where the latter is the Lie algebra of symmetric bilinear forms on the linear symplectic vector space R 2(n−k) with Lie bracket given by the commutator, and the isomorphism sends a quadratic polynomial to its Hamiltonian vector field. The classification of Williamson [48] is recalled below without proof.
Theorem 3.5. Fix a positive integer n and let h ⊂ Sym (2n; R) be a Cartan subalgebra. Then there exist canonical coordinates x i , y i of the linear symplectic vector space R 2n , a triple (k e , k h , k ff ) ∈ Z 3 ≥0 with k e + k h + 2k ff = n, and a basis H 1 , . . . , H n of h such that and, if i = k e +k h +1, k e +k h +3, . . . , k e +k h +2j −1, . . . , k e +k h +2k ff −1, then Moreover, the triple (k e , k h , k ff ) determines h up to conjugacy. Definition 3.6. Given a Cartan subalgebra h ⊂ sp (2n; R), the triple (k e , k h , k ff ) ∈ Z 3 ≥0 with k e + k h + 2k ff = n classifying it up to conjugacy is called the Williamson triple of h, where k e , k h , k ff are referred to as the number of elliptic, hyperbolic and focus-focus components.
Going back to non-degenerate singular orbits, adapting and following Zung [49,Definition 3.4], we introduce the following terminology.  [8,12,28] amongst others). While the various linearization results are beyond the scope of this article, it is worthwhile observing that, in the absence of hyperbolic blocks, i.e. if k h = 0, and when the orbits are compact, i.e. if k = c, the linearization result is stronger: there exist canonical coordinates which also put the moment map in standard form. This is part of the motivation for introducing the singular orbits studied in Section 3.1.2. To conclude this subsection, we state the following characterization of non-degenerate, compact singular orbits of purely elliptic type, i.e. whose Williamson types are given by elements of the form (k, k, n − k, 0, 0), relating them to singular toric leaves (cf. Dufour & Molino, Eliasson [8,12]). Such orbits are henceforth referred to as elliptic tori.
Observe that the flow of the Hamiltonian vector field X q 1 is periodic. In fact, W can be chosen to be saturated with respect to the effective Hamiltonian S 1 -action whose moment map is given by q 1 .
Remark 3.10 An immediate consequence of the above local normal form is that the set of focus-focus points is discrete.
Elliptic tori and focus-focus points differ significantly. An immediate topological difference is that the former are leaves of the system while for the latter are not. A crucial geometric difference lies in the fact that the former support a local effective Hamiltonian T 2 -action whose moment map has components that Poisson commute with the integrals of the system (we say that the action is system-preserving), while the latter possesses a unique (up to sign) system-preserving effective Hamiltonian S 1 -action (cf. Zung [50,Proposition 4]). • a fiber of Φ| V is either diffeomorphic to a cylinder (if it does not contain p) or given by the union of two Lagrangian planes intersecting transversally at p. In particular, if the latter fiber is denoted by L, then (V ∩ L) {p} consists of two connected components, each diffeomorphic to a cylinder; • there exist smooth sections σ 1 , σ 2 defined near Φ(p) whose image lies inside V with the property that σ 1 (Φ(p)) , σ 2 (Φ(p)) lie in distinct connected components of (V ∩ L) {p}, where L denotes the leaf through p.

3.2.
Definition and fundamental properties. Following Symington [40] and Vũ Ngo . c [46], we introduce a category of integrable systems of two degrees of freedom that generalize weakly toric systems on 4-dimensional manifolds while retaining significant similarities.
Definition 3.12. An integrable system (M, ω, Φ) on a 4-dimensional symplectic manifold is almost-toric if Φ is proper onto its image and all of its singular orbits are almost-toric.
The following result can be used to describe almost-toric systems in an equivalent fashion; seeing as it follows directly from Zung [49, Proposition 3.5], its proof is omitted.
Lemma 3.13. Let (M, ω, Φ) be an integrable system on a 4-dimensional symplectic manifold with compact fibers, all of whose singular orbits are non-degenerate without hyperbolic blocks. Then the singular orbits are compact and, in particular, almost-toric.
The above notion of almost-toric system differs slightly from that in Vũ Ngo . c [46], for Definition 3.12 only requires that Φ be proper onto its image, as opposed to being proper.
Remark 3.14 Almost-toric systems form a full subcategory of IS (2), henceforth referred to as the category of almost-toric systems and denoted by AT .
Remark 3.15 Note that saturated subsystems of almost-toric systems are almost-toric.
The restriction on the types of singular orbits in an almost-toric system (M, ω, Φ) implies the singular leaves of Φ are either elliptic tori or contain at least one focus-focus singular orbit. That dichotomy arises because the local normal form for elliptic singular orbits implies such orbits (which are elliptic tori) make up whole connected components of a fiber of Φ. Henceforth, leaves that contain focus-focus orbits are referred to as focus-focus leaves. Denote by L ff the set of points in the leaf space L corresponding to leaves containing focus-focus singular orbits. Then L sing = L e ∪ L ff , where L e is the elliptic part of L. Elements of L ff are called focus-focus values (in the leaf space). Focus-focus values, counted with multiplicity, are an invariant of the isomorphism class of an almost-toric system. Henceforth, all focusfocus values are counted with multiplicity unless otherwise stated.
The topology of focus-focus leaves is well-known and is completely determined by the finite number r ≥ 1 of focus-focus singular points contained in the corresponding leaf. In particular, each focus-focus leaf is homeomorphic to a torus with r homologous cycles, each collapsed to a point (cf. Bolsinov & Fomenko [2, Chapter 9.8]). Lying at the heart of this result is the existence of a vector field tangent to a focus-focus leaf, whose flow is periodic and whose fixed points are precisely the focus-focus points.
In fact, the S 1 -action on any focus-focus leaf can be extended to a suitable neighborhood of the leaf, one that is saturated with respect to the quotient map to the leaf space. To prove this, we start by establishing the existence of this suitable neighborhood. Proof. Fix a focus-focus leaf L. First, we show that L admits an open neighborhood Z that is saturated with respect to q and in which L is the only focus-focus leaf. Suppose not; then there exists a sequence of focus-focus points {p n } with the property that Φ(p n ) → Φ(L); since Φ is proper onto its image, there exists a convergent subsequence p n j → p. The limit point p is necessarily singular, but the local normal form for almost-toric orbits yields a contradiction.
Fix such a neighborhood Z and let p 1 , . . . , p r ∈ L denote the focusfocus points in L. For i = 1, . . . , r, let V i ⊂ Z be an open neighborhood of p i that can be put in local normal form (cf. Section 3.1.2). Consider the subsetV where, as in Section 3.1.2, φ t denotes the Hamiltonian action by t ∈ R 2 ; this is the union of the orbits that intersect at least one V i . Since, for i = 1, . . . , r, V i is open, so isV ; moreover,V contains L because Next we show thatV is also saturated with respect to q. To see this, observe that if p ∈ V i L, then the leaf passing through p is contained in Z and is not equal to L, thus implying that it is not a focus-focus leaf. Since p is regular (by the local normal form for focus-focus points), the leaf through p is regular and is, therefore, an orbit of the Hamiltonian R 2 -action, which is contained inV by construction.
The above construction does not necessarily guarantee thatV contains at most one leaf of any fiber of Φ. However, the local normal form for a focus-focus point and the R 2 action can be used to determine a possibly smaller neighborhood in which that property holds. Specifically, fix some i ∈ {1, . . . , r}. There exists a smooth section σ i of Φ defined near Φ(p i ) whose image is contained in V i ; this implies that Remark 3.11). The structure of the focus-focus leaf L implies that there exists t 0 ∈ R 2 with the following property • if r = 1, then σ i (Φ(p i )) and φ t 0 (σ i (Φ(p i ))) lie in different connected components of (V i ∩ L) {p i } (cf. Vũ Ngo . c [44]); . In other words, the section σ i flows out of V i and into V j (and if i = j, then it approaches V i from the 'opposite' side). Let c ∈ R 2 be sufficiently close to Φ(p i ) so that σ i (c) is defined; then the above properties show that . In the latter case, using again the structure of the focus-focus leaf L (cf. Bolsinov & Fomenko [2, Chapter 9.8]), we can iterate the above argument finitely many times to ensure that, for all c sufficiently close to Φ(L) (= Φ(p i ) for all i = 1, . . . , r), Φ −1 (c) ∩V is connected. This shows thatV can be shrunk as desired. Any open neighborhood of a focus-focus leaf as in Proposition 3.17 is henceforth referred to as a (q-)saturated regular neighborhood of a focus-focus leaf. A saturated regular neighborhood has the necessary S 1 -symmetry. In fact, a saturated regular neighborhood of a focus-focus leaf (of multiplicity one) is a singular Liouville foliation of (simple) focus-focus type in the sense of Vũ Ngo . c [45,Definition 2.4]. Moreover, saturated regular neighborhoods of focus-focus leaves ought to be thought of as analogous to the neighborhoods of elliptic tori that can be put in local normal form. For instance, the following result holds. To summarize the above results and motivate subsequent sections, we state the following description of regular neighborhoods of leaves in almost-toric systems.

3.3.
Faithful almost-toric systems. By Corollary 3.21, any leaf in an almost-toric system admits an open neighborhood whose corresponding subsystem is faithful almost-toric. Therefore it makes sense to view faithful almost-toric systems as building blocks of almost-toric systems. Throughout this subsection, let (M, ω, Φ) be a faithful almosttoric system. Moreover, fix the identification between L and B = Φ(M ) and denote by B ff the image of L ff under this identification.

Remark 3.22
The local normal form for focus-focus points implies that B ff ⊂ Int(B). Thus, by Corollary 3.18, B ff is discrete in Int(B).
The arguments in Vũ Ngo . c [46,Proposition 3.9] can be used to prove the following result, stated without proof. The restriction on the types of singular orbits, as well as faithfulness, imply the following useful fact for faithful almost-toric systems.  Example 3.26 Suppose (M, ω) is symplectomorphic to a K3 surface (for example, a smooth quartic hypersurface in CP 3 ). Such a symplectic manifold admits singular Lagrangian fibrations over S 2 in which each singular fiber has a neighborhood that, with respect to an appropriate coordinate chart on S 2 , defines a faithful almost-toric system that is a regular saturated neighborhood of a focus-focus fiber with one singular orbit. Suppose Π : (M, ω) → S 2 is such a fibration and let p ∈ S 2 be the image of a regular fiber. Let N = M Π −1 (p) and let φ : S 2 p → R 2 be an embedding. Then (N, ω| N , φ • Π| N ) defines a faithful almost-toric system with 24 focus-focus leaves.
A natural question arising from Remark 3.25 is whether the Z-affine structure A lt on B lt can be extended to B. The presence of focus-focus fibers prevents this from happening, as the Z-affine structure on any neighborhood of a focus-focus value has non-trivial affine holonomy.
where r c ≥ 1 is the multiplicity of c.

Remark 3.28
The eigenspace associated to the above representation reflects the uniqueness (up to sign) of the local effective systempreserving Hamiltonian S 1 -action in a neighborhood of a singular fiber containing focus-focus points. With respect to the local choice of basis in Theorem 3.27, this action is induced by the first integral of the moment map.
Another natural question is which almost-toric systems are isomorphic to toric ones? Certainly, the system must not contain focus-focus points. If the system is faithful, it suffices that there be a Z-affine immersion of the moment map image into (R 2 , A 0 ). However, as the next example illustrates, an absence of focus-focus points does not suffice. Then, up to isomorphism, M = S 2 × R × S 1 with ω = ω S 2 ⊕ da ∧ dθ, where ω S 2 is a suitable symplectic form on S 2 , and Φ(p, a, θ) = (h(p), a), where h : S 2 → R is a suitable height function. Let (M, ω,Φ) be the almosttoric system defined byΦ = g 1 • µ where g 1 (x, y) = (e x cos y, e x sin y) and consider the symplectic Z-action on M given by k · (p, a, θ) = (p, a + 2πk, θ). The quotient is M = S 2 × T 2 . Observe thatΦ is invariant under this Z-action. Therefore, taking the quotient yields an almost-toric system (M , ω , Φ ). This system has no focus-focus points but is not isomorphic to a toric system: it is faithful, and yet has a moment map image that is not simply connected and hence not homeomorphic to a polygon.

Remark 3.30
Consider a faithful almost-toric system (M, ω, Φ) with focus-focus points and suppose that (f, S) is a cartographic pair. The definition of a cartographic pair (Definition 2.45) and the local normal form for singular orbits of elliptic type together imply that the cartographic image of curved edges in S are line segments whose tangent vectors can be chosen to have coprime integer coefficients, and then those tangent vectors span the standard lattice Z 2 ⊂ R 2 whenever the edges are incident to a corner of S.
An important question is thus, when does a faithful almost-toric system admit a cartographic homeomorphism? Addressing this problem in full generality is beyond the scope of this paper. However, in light of Corollary 3.21, a natural family of faithful almost-toric systems to consider arises: namely, those admitting a system-preserving S 1 -action.

Vertical almost-toric systems
This section studies vertical almost-toric systems and proves the main results of the paper. As pointed out in the introduction, vertical almost-toric systems are closely related to (generalized) semi-toric systems (cf. Pelayo & Ratiu & Vũ Ngo . c, Vũ Ngo . c [33,46]). Many of the ideas and proofs that appear in this section are inspired by the work in op. cit. Section 4.1 introduces vertical almost-toric systems and establishes their basic properties, as well as proving that such systems can be viewed as 'building blocks' of almost-toric systems (see Proposition 4.9 for a formal statement). However, unlike their almost-toric counterparts, vertical almost-toric systems possess a global Hamiltonian S 1 -action, a fact that has some important geometric consequences which are investigated in Section 4.2. Section 4.3 is devoted to proving that vertical almost-toric systems admit cartographic homeomorphisms constructed by choosing suitable vertical cuts of their moment map images (Theorem 4.24). The set of all cartographic homeomorphisms of a given vertical almost-toric system is described in Section 4.4, which generalizes Vũ Ngo . c [46,Section 4]. Finally, Section 4.5 shows that cartographic homeomorphisms can be made smooth by modifying them on arbitrarily small neighborhoods of the corresponding cuts (Theorem 4.48). This result provides representatives (which we call η-cartographic) in the isomorphism class of any vertical almost-toric system, representatives that are particularly useful when defining surgeries on vertical almost-toric systems (cf. the forthcoming [20]).

Definition and basic properties.
To the best of our knowledge, there are no general results regarding the existence of cartographic homeomorphisms for faithful almost-toric systems, even if the total space is closed (cf. Leung & Symington [25] and Symington [40]). However, the existence results of Pelayo & Ratiu & Vũ Ngo . c [33] and Vũ Ngo . c [46] hint at the fact that if the first integral in an almost-toric system (M, ω, Φ = (J, H)) is the moment map for an effective Hamiltonian S 1 -action, then some control on J suffices.
Note that the leaves of a vertical almost-toric system are compact because the system is almost-toric and each fiber consists of a single leaf (and hence is connected) because the system is faithful. Moreover, property (V5) is equivalent to path-connectedness of the fibers of J by the local normal form for Hamiltonian S 1 -action (cf. Guillemin & Sternberg, Marle [15,26]).  Then the vertical segment joining c 1 to c 2 is contained in B by assumption; Lemma 3.24 then implies that there exists a path joining p 1 to p 2 contained in the preimage under Φ of that vertical segment. Since that preimage is contained in J −1 (x 0 ), this shows the desired result.
In fact, the path-connectedness of the intersection of B with any vertical line in turn implies that intersections of vertical lines with the interior of B are path-connected.  (Int(B)).
Proof. Suppose not, then there exists x 0 ∈ pr 1 (Int(B)) such that Int(B)∩{(x, y) | x = x 0 } is not path-connected. Since B∩{(x, y) | x = x 0 } is nonempty and path-connected, it follows that there exist y 1 < y 0 < y 2 such that (x 0 , y 0 ) ∈ ∂ ∞ B ⊂ ∂B and (x 0 , y i ) ∈ Int(B), for i = 1, 2. However this is impossible because it forces a disconnection of the intersection of B and a vertical line as follows.
For i = 1, 2, because the point (x 0 , y i ) is in Int(B), there is open disk of radius r i centered at (x 0 , y i ) that is a subset of Int(B). And because (x 0 , y 0 ) ∈ ∂ ∞ B ⊂ ∂B, there exists a sequence of points {(x n , y n )} such that each (x n , y n ) is in the open ball of radius 1 n centered at (x 0 , y 0 ) but (x n , y n ) / ∈ B. Consequently, for each n such that 1 n < min(r 1 , r 2 ) the intersection B ∩{(x, y) | x = x n } is disconnected because y 1 < y 0 − 1 n < y n < y 0 + 1 n < y 2 , with (x n , y 1 ), (x n , y 2 ) ∈ B and (x n , y n ) / ∈ B.
Corollary 4.5. The moment map image of a vertical almost-toric system is contractible.
Proof. Fix a vertical almost-toric system with moment map image B. As noted in Remark 3.25, B is a manifold with corners. The inclusion Int(B) = B ∂ ∞ B → B is a homotopy equivalence because B is homeomorphic to a smooth manifold with boundary, which is homotopy equivalent to the complement of its boundary Thus it suffices to prove that Int(B) is contractible. By Lemma 3.23, Int(B) is path-connected and so is pr 1 (Int(B)). Moreover, Int(B) ⊂ R 2 is open and pr 1 is an open map. Therefore pr 1 : Int(B) → pr 1 (Int(B)) is a surjective submersion whose fibers are diffeomorphic to R by Lemma 4.4. Thus it is a fiber bundle (cf. Meigniez [27,Page 3778]). Since pr 1 (Int(B)) is an interval, the bundle is trivial. Since both the base and the fiber of this trivial bundle are contractible, so is the total space.

Corollary 4.6. A vertical almost-toric system is Delzant if and only if it is toric.
Vertical almost-toric systems behave well with respect to taking certain 'vertical subsystems': Proposition 4.7. Let (M, ω, Φ) be a vertical almost-toric system and let U ⊂ B be open and path-connected. Then the subsystem of (M, ω, Φ) relative to U is a faithful almost-toric system that satisfies properties (V1) -(V4) of a vertical almost-toric system. Moreover, if for all x 0 ∈ R, {(x, y) | x = x 0 } ∩ U is either empty or path-connected, then the subsystem relative to U is, in fact, a vertical almost-toric system.
Proof. The subsystem relative to U is faithful by Remark 2.12 and almost-toric by Remark 3.15. The total space is connected because U is connected and the subsystem is faithful. Property (V2) is satisfied because Φ −1 (U ) is a union of orbits of its first integral. Finally, Properties (V3) and (V4) are preserved under taking subsystems. This proves the first assertion.
Assume that the intersection of U with any vertical line is either empty or path-connected. Because the subsystem relative to U is faithful almost-toric, Lemma 4.3 implies the fibers of J| U are pathconnected, as desired.  [32]). • On the other hand, as a result of being faithful, the moment map of a vertical almost-toric system is assumed to be proper onto its image and to have connected fibers. Accordingly, vertical almost-toric systems can be thought of as subsystems of generalized semi-toric systems. However, the notion of isomorphism of vertical almost-toric systems is strictly weaker than that of generalized semi-toric systems, as it allows for diffeomorphisms of the moment map images whose derivative has negative determinant (cf.  Proof. A leaf of an almost-toric system is either weakly toric or is a focus-focus leaf. In the former case, the local normal forms for elliptic tori yield the result. Thus, suppose that L is a focus-focus leaf and let V be a saturated regular neighborhood of L. By construction, V is connected; moreover, Proposition 3.20 gives that (V, ω| V , Φ| V ) is faithful almost-toric. Proposition 3.19 gives that there exists a systempreserving effective Hamiltonian S 1 -action on (V, ω| V , Φ| V ); in other words, that subsystem is isomorphic to an almost-toric system whose first integral is the moment map of an effective Hamiltonian S 1 -action. Thus, without loss of generality, it may be assumed that the first component of Φ| V is the moment map of an effective Hamiltonian S 1 -action. By construction of V , properties (V3) and (V4) of a vertical almosttoric system hold for be an open neighborhood of c with the property that its intersection with any vertical line is either empty or path-connected. Using Proposition 4.7, the subsystem relative to U of (V, ω| V , Φ| V ) is vertical almost-toric as desired.  H)) and denote the set of fixed points of the Hamiltonian S 1 -action, one of whose moment maps is J, by M S 1 . Its connected components are either isolated fixed points or symplectic fixed surfaces, i.e., symplectic submanifolds of dimension 2 that are fixed under the S 1 -action. This is a consequence of the Marle-Guillemin-Sternberg local normal form (cf. Guillemin & Sternberg [15], Marle [26]).
Proof. For i = 1, 2, 3, set c ∞ i = (x 0 , y i ) and assume, without loss of generality, that y 1 < y 2 < y 3 . Since the fibers of J are connected by property (V5), it follows that The local normal forms for almost-toric singular orbits in the presence of a system-preserving Hamiltonian S 1 -action (cf. Remark 2.38), together with faithfulness of (M, ω, Φ = (J, H)), force Φ −1 (c ∞ 2 ) to be a singular orbit of elliptic-regular type all of whose points are critical for J. Then Φ −1 (c ∞ 2 ) lies on a fixed surface and Proposition 4.11, together with connectedness of the fibers of J, completes the proof.
Combining Propositions 4.11 and 4.12, we obtain the following result.  • focus-focus singular orbits, or • elliptic-elliptic singular orbits whose image is not a corner adjacent to a vertical edge.

Cuts and cartographic homeomorphisms.
An important property of vertical almost-toric systems is that, as proved below, assuming the following mild restriction, they admit cartographic homeomorphisms (cf. Theorem 4.24).  H)) is said to be simple if the following property holds: (V6) Any focus-focus value has multiplicity 1.
The above condition is generic according to Zung [49]. Moreover, it is invariant under taking isomorphisms of vertical almost-toric systems and descends to saturated subsystems satisfying all the hypotheses of Proposition 4.7.
While the existence of cartographic homeomorphisms is expected to hold without imposing property (V6), proofs would require a more detailed understanding of neighborhoods of focus-focus fibers with more than one focus-focus point, which is beyond the scope of this paper (cf. Vũ Ngo . c [45,Section 7] for some sketched proofs in this direction). To the best of our knowledge, all existing proofs of the existence of cartographic homeomorphisms assume, either tacitly or explicitly, simplicity of the system (cf. Vũ Ngo . c [46, Step 4 of the Proof of  The aim of this section is to prove that any simple vertical almosttoric system admits a cartographic homeomorphism that, loosely speaking, encodes the affine holonomy of the Z-affine structure on the locally weakly toric part of the leaf space (cf. Theorem 4.24 for a precise statement). It is important to remark that there are proofs of this result for special families of vertical almost-toric systems (cf. Vũ Ngo . c [  In particular, (M, ω, f •Φ) is a Delzant system and f (B) ⊂ R 2 is locally convex.
Proof. The proof is analogous to Pelayo & Ratiu & Vũ Ngo . c [33, Step 2 of Theorem B], but is included in this paper for completeness.
The lack of focus-focus points implies B = B lt , thus B inherits a Z-affine structure. By Corollary 4.5, B is contractible, so there exists a developing map f : B ∼ =B → R 2 . By definition of the Z-affine structure on B, since the first integral J of (M, ω, Φ = (J, H)) is the moment map of an effective Hamiltonian S 1 -action, one can choose the above developing map to be of the form f (x, y) = x, f (2) (x, y) for some smooth function f (2) : B → R. Fix such a choice.
To show that f is the required cartographic homeomorphism, it suffices to show that f is injective. If f (x 0 , y 0 ) = f (x 1 , y 1 ), one gets immediately (2) ∂y does not vanish on B, because f is locally a diffeomorphism. This implies that y 0 = y 1 as required.
Since  Let (M, ω, Φ = (J, H)) be a simple vertical almost-toric system with B = Φ(M ) that contains at least one focus-focus point. An example is shown in Figure 4.1. We introduce vertical 'cuts' at the focus-focus values. This terminology is motivated by Vũ Ngo . c [46]. Symington [40] refers to these as 'eigenrays' in the more general context of (faithful) almost-toric systems.
where pr 1 : R 2 → R is projection onto the first component. By property (V3), this supremum x sup (respectively infimum x inf ) is either attained as a maximum (respectively as a minimum) or does not lie in J(M ). Set By construction, the cardinality of I equals that of B ff and thus we think of the elements of the latter as being indexed by I. Order the elements of B ff as follows. For i ∈ I, set c i = (x i , y i ). Then require that i < j implies either x i < x j , or x i = x j and y i < y j ; moreover, if 0, 1 ∈ I, require that x 0 < x 1 . (If I = Z, the above ordering is unique up to the choice of which focus-focus value is labeled with 0.) For each i ∈ I choose a sign ε i ∈ {+1, −1}, and denote the associated vertical cut in B at c i by When ε i = +1 (respectively −1), the cut l ε i is simply the intersection of B with the vertical half-line starting at c i going 'up' (respectively 'down'), cf. Figure 4.2. Therefore the former is referred to as being upward, while the latter as being downward. For a fixed ε ∈ {+1, −1} I , denote the union of the cuts by l ε and set S ε := B l ε . Moreover, to each element (x, y) ∈ B lt = B B ff , associate the integer with the convention that j ε (x, y) = 0 for (x, y) ∈ S ε . Finiteness of j ε (x, y) follows from property (V4) and Proposition 4.14.  Proof. Density of S ε in B is trivial, so it remains to prove its openness in B and, to this end, it suffices to prove that l ε is closed in B. Let {(x n , y n )} ⊂ l ε be a sequence which converges to (x 0 , y 0 ) ∈ B. This implies that the sequence {x n } = pr 1 ({(x n , y n )}) ⊂ J(M ) converges to x 0 ∈ J(M ). By construction and by Proposition 4.14, {x n } is contained in the subset of critical values of J, which does not contain any limit points in J(M ) by property (V3). Therefore, for all but finitely many n, x n = x 0 , which, in turn, implies that (x n , y n ) ∈ {(x, y) | x = x 0 } for all but finitely many n. By property (V4) and Proposition 4.14, the vertical line {(x, y) | x = x 0 } contains finitely many focus-focus values and, therefore, finitely many cuts. Seeing as each cut is a closed subset, then the union of all cuts contained on {(x, y) | x = x 0 } is closed. Therefore, (x 0 , y 0 ) ∈ l ε as required.
Remark 4.20 In general, it is not true that l ε is closed in R 2 , for x sup , x inf may belong to R J(M ).
The notation S ε is suggestive of the fact that there exists a cartographic homeomorphism f ε : B → R 2 such that (f ε , S ε ) is a cartographic pair for (M, ω, Φ = (J, H)). Before stating and proving the precise existence statement, we prove some further properties of S ε (cf.  Proof. Suppose first that S ε is path-connected and let i > j ∈ I be such that x i = x j . Let (x 1 , y 1 ), (x 2 , y 2 ) ∈ S ε be points with x 1 < x i < x 2 . Such points exist because focus-focus values are contained in Int(B). Since S ε is path-connected, there exists a path in S ε starting at (x 1 , y 1 ) and ending at (x 2 , y 2 ). Therefore, there exists a point (x i , y ) ∈ S ε and thus ε i ≥ ε j .
Conversely, suppose that ε satisfies the condition that ε i ≥ ε j for all i > j with x i = x j . First we show that, for all x 1 ∈ J(M ), {(x, y) | x = x 1 } ∩ S ε = ∅ and that the set is path-connected. If and the result follows from the fact that (M, ω, Φ = (J, H)) is a vertical almost-toric system. Suppose, therefore, that x 1 ∈ pr 1 (B ff ). By property (V4) and Proposition 4.14, there are finitely many focus-focus values (x 1 , y i 1 ), . . . , (x 1 , y i N ) lying on the vertical line {(x, y) | x = x 1 }. Set y + := inf{y i k | ε i k = +1}, Since ε satisfies the condition in the statement, it follows that y + > y − and therefore, which shows that {(x, y) | x = x 1 }∩S ε is path-connected. By Corollary 4.19, S ε is open in B. Thus S ε satisfies all the hypotheses of Proposition 4.7 and, therefore, the subsystem of (M, ω, Φ = (J, H)) relative to S ε is vertical almost-toric. By Corollary 4.5, S ε is contractible and, in particular, path-connected. Corollary 4.23. There exists a choice of ε making S ε path-connected.
Proof. The choice of ε i = +1 for all i ∈ I satisfies the condition of Lemma 4.21.
Having established the above preliminary results, we can state and prove existence of cartographic homeomorphisms for simple vertical almost-toric systems.   In particular, f ε (B) is locally convex.
Any cartographic homeomorphism f ε : B → R 2 satisfying the properties in Theorem 4.24 is said to be associated to ε.
The proof of Theorem 4.24 is split into two cases: when S ε is pathconnected and when it is not.
Proof of Theorem 4.24 if S ε is path-connected. Suppose that S ε is pathconnected; by Corollary 4.23, a choice of ε whose associated set S ε is path-connected exists. The idea is to argue as in Pelayo & Ratiu & Vũ Ngo . c [33, in the proof of Theorem B] recalling and adjusting it as much as necessary for our purposes.
By construction S ε ⊂ B lt . Now let q :B lt → B lt denote the universal covering. By Corollary 4.22, S ε is contractible and, in particular, simply connected. Therefore, there exists a smooth section σ : S ε →B lt of q. Consider a developing map dev :B lt → R 2 constructed by fixing basepoints x 0 ∈ S ε andx 0 ∈ σ (S ε ). Set f ε := dev • σ : S ε → R 2 . Arguing as in the proof of Lemma 4.17, it is possible to choose dev so that f ε (x, y) = x, f (2) ε (x, y) for any (x, y) ∈ S ε . Fix such a choice. Following the arguments in Pelayo & Ratiu & Vũ Ngo . c [33, Step 4 of the proof of Theorem B], f ε can be extended to an embedding B → R 2 which, by abuse of notation, is also denoted by f ε . By construction and by density of S ε ⊂ B, (f ε , S ε ) is a cartographic pair with f ε (x, y) = x, f (2) ε (x, y) . Thus for all (x, y) ∈ S ε , ∂f (2) ε ∂y (x, y) = 0. Since S ε is path-connected, property (C1) follows.
To complete the proof, there are two cases to consider, depending on whether sgn(f ε ) = +1 or sgn(f ε ) = −1. In the first case, property (C2) and local convexity of f ε (B) can be proved as in Vũ Ngo . c [46, Steps 5 and 6 of the proof of Theorem 3.8]. Thus suppose that sgn(f ε ) = −1.
Df ε (x, y) Df ε (x, y) This proves property (C2) in general. Finally, observe thatf ε (B) is locally convex and that Z-affine maps preserve this property. Thus f ε (B) is locally convex as required. Now we turn to the case of S ε not being path-connected. There exist proofs for such cases in the literature (cf. for instance Pelayo & Ratiu & Vũ Ngo . c [33, Step 5 of the proof of Theorem B]). The argument presented below, however, uses different techniques.
Before delving into the proof, we introduce some useful notions and notation. For any x ∈ J(M ), set (4.4) N x := |{i ∈ I | x i = x}| .
By property (V4) and Proposition 4.14, N x is finite for any x ∈ J(M ). Moreover, for a fixed ε ∈ {+1, −1} I , set (4.5) N ± x (ε) := ± |{i ∈ I | x i = x and ε i = ±1}| . Observe that, for any x ∈ J(M ) and any ε ∈ {+1, −1} I , . Moreover, for any (x, y) ∈ B lt and any ε ∈ {+1, −1} I , . Fix x ∈ J(M ) with N x = 0. Then, by property (V4) and Proposition 4.14, there exist finitely many indices i 1 < i 2 < . . . < i Nx in I with x i j = x. Observe that, by definition of the ordering on B ff , . . , N x − 1 and for all (x, y) ∈ (x, y) | x = x and y i k < y < y i k+1 ; • j ε (x, y) = N + x (ε) for all (x, y) ∈ (x, y) | x = x and y > y i Nx . Hereby, i 1 < i 2 < . . . < i Nx are the elements of I such that x i j = x. In particular, the function j ε (x, ·) only depends on N ± x (ε). Proof. Fix ε ∈ {+1, −1} I and consider (x, y) ∈ B lt such that N x = 0. Suppose first that y < y i 1 . This means that (x, y) is 'below' all focusfocus values on the vertical line {(x, y) | x = x}. By definition of the ordering on B ff and of the cuts associated to ε, for all k ∈ {1, . . . , N x }, Similarly, if y > y i Nx , then j ε (x, y) = N + x (ε), for (x, y) is 'above' all focus-focus values on the vertical line {(x, y) | x = x}.
It remains to prove the intermediate cases for which we proceed by induction on k. The base case is y < y i 1 , which has already been proved. Suppose that the required statement holds for all m < k and let y i k < y < y i k+1 . Set, for any (x,ȳ) ∈ B lt , Clearly, for any (x,ȳ) ∈ B lt , j ε (x,ȳ) = j + ε (x,ȳ) + j − ε (x,ȳ). Fix some (x, y ) ∈ B lt with y i k−1 < y < y i k . The inductive hypothesis implies j ε (x, y ) = N − x (ε) + k − 1. There are two cases to consider, depending on whether ε i k = +1 or ε i k = −1. In the former case, x (ε) + k as required. The latter case is proved analogously swapping the roles of j + ε (x, y) and j − ε (x, y). With the above results at hand, we finish the proof of Theorem 4.24.
Proof of Theorem 4.24 if S ε is not path-connected. Suppose that S ε is not path-connected. The idea is to reduce this situation to the pathconnected case by appealing to the following result. Lemma 4.26. There exists a uniqueε ∈ {+1, −1} I such that Assume Lemma 4.26, whose proof is below. Letε be as in Lemma 4.26 and let fε : B → R 2 be a cartographic homeomorphism associated tô ε. Set f ε := fε. Since S ε ⊂ Sε, Corollary 4.19 implies that (f ε , S ε ) is a cartographic pair; property (C1) holds by construction. Property (C2) holds because Lemma 4.26 implies that j ε (x, y) = jε(x, y) for all (x, y) ∈ B lt . Moreover, sgn(f ε ) = sgn(fε) holds by definition. Local convexity of f ε (B) = fε(B) is also true as fε is associated toε in the sense of Proposition 4.24. This finishes the proof of Theorem 4.24 for the case that S ε is not path-connected.
Suppose that there exist i > j with x i = x j . There are two cases to consider, depending on whether x i / ∈ Disc(ε) or x i ∈ Disc(ε). In the former, observe thatε i = ε i andε j = ε j . Since x i / ∈ Disc(ε), it follows thatε i = ε i ≥ ε j =ε j . On the other hand, if x i ∈ Disc(ε), then the definition ofε implies thatε i ≥ε j . Thus S ε is path-connected. Now, to show that S ε ⊂ Sε, consider (x, y) ∈ S ε . By definition of the cuts, if x = x i for any i ∈ I, then (x, y) ∈ Sε. Therefore, suppose that there exists i ∈ I with x = x i . By property (V4) and Proposition 4.14, there exist finitely many indices i 0 , . . . , i r ∈ I such that x i j = x for all j = 0, . . . , r. Then (x, y) ∈ S ε , implies that, for all j = 0, . . . , r, x i j / ∈ Disc(ε). Thusε i j = ε i j for all j = 0, . . . , r by definition, which implies (x, y) ∈ Sε.
Finally, to show that j ε (x, y) = jε(x, y) for all (x, y) ∈ B lt , consider some (x, y) ∈ B lt . If x = x i for all i ∈ I, the above equality is trivially true. Suppose, therefore, that there exists i ∈ I with x = x i . By Lemma 4.25, j ε (x, y) = jε(x, y) is equivalent to N ± x (ε) = N ± x (ε), but the latter equality follows from the way in whichε has been defined.
It remains to check thatε is the unique choice of signs satisfying the above properties. Suppose thatε is another such choice. The inclusion S ε ⊂ Sε impliesε i = ε i =ε i for all i ∈ I such that x i / ∈ Disc(ε). On the other hand, the condition that j ε (x, y) = jε(x, y) for all (x, y) ∈ B lt implies, using the above argument, thatε i =ε i for all i ∈ I such that x i ∈ Disc(ε). This shows thatε =ε as required.

Remark 4.27
The above argument for the case of S ε not being pathconnected only works because the system (M, ω, Φ = (J, H)) is assumed to be simple. If focus-focus values of higher multiplicity are allowed, then there may be no analogue ofε as in Lemma 4.26.

Remark 4.28
The argument proving the case of Theorem 4.24 when S ε is not path-connected shows that, for a given vertical almost-toric system, the set of all cartographic homeomorphisms that satisfy the hypotheses of Theorem 4.24 equals that of the cartographic homeomorphisms associated to those ε for which S ε is path-connected (cf. Vũ Ngo . c [46,Proposition 4.1]). First, we consider the set of all cartographic homeomorphisms associated to a given choice of signs; this is described in the following result, which is analogous to Remark 4.18.
Proof. It may be assumed without loss of generality that S ε is pathconnected since the not path-connected case can be reduced to the path-connected one as in the proof of Theorem 4.24. Let f ε ,f ε : B → R 2 be cartographic homeomorphisms associated to ε. Then their restrictions to S ε are developing maps for the induced Z-affine structure on S ε . Therefore, arguing as in Remark 4.18, there exists an element Conversely, the proof of Theorem 4.24 gives that composing a cartographic homeomorphism associated to ε on the left with an element of Vert(2; Z) yields another cartographic homeomorphism associated to ε.
Having established Corollary 4.29, we study the problem of relating cartographic homeomorphisms whose associated signs are not necessarily equal. Before stating the main result of this section we introduce some tools akin to those needed in Vũ Ngo . c [46,Section 4], but slightly more involved as vertical almost-toric systems allow for the presence of infinitely many focus-focus points (see Remarks 4.30,4.31 and 4.37 below).
As in Section 4.3, let I be the indexing set of the set of focus-focus values B ff defined in equation (4.2). Also, fix the ordering on B ff as in the paragraph following equation (4.2), so elements of B ff are denoted by c i = (x i , y i ) for i ∈ I. Henceforth, fix elements ε,ε ∈ {+1, −1} I and associated cartographic homeomorphisms f ε , fε : B → R 2 . Furthermore, fix a basepoint (x, y) ∈ B lt with the property that x 0 < x < x 1 .
(If x 0 or x 1 is not defined, then only the other inequality is required.) Moreover, for any i ∈ I define l i,ε,ε : R 2 → R 2 as follows: • If i ≤ 0, let l i,ε,ε be the identity.
• If i > 0, let l i,ε,ε be the piece-wise Z-affine transformation that acts as the identity on the half-space x < x i and as the shear Analogously, for any i ∈ I define r i,ε,ε : R 2 → R 2 as follows: • If i ≤ 0, let r i,ε,ε be the piece-wise Z-affine transformation that acts as the shear T −k i (ε,ε) on the half-space x < x i and as the identity on the half-space x ≥ x i . • If i > 0 let r i,ε,ε be the identity. Explicitly, we have that if i > 0 then (4.7) l i,ε,ε := The maps l i,ε,ε and r i,ε,ε are well-defined and satisfy the following properties, the proofs of which are left to the reader.
One ingredient in the construction of the homeomorphism that relates f ε and fε is the composition of the maps l i,ε,ε or r i,ε,ε as i range over all indices in I. Seeing as this may involve the composition of infinitely many maps different from the identity (as I may be infinite), some care is needed. To this end, we first introduce notation for the domains of these possibly infinite compositions. Set Next, we define the desired compositions. If the cardinality of I is finite, the situation is entirely analogous to the one considered in Vũ Ngo . c [46,Section 4]. For, with the above conventions, I being finite implies that x sup , x inf ∈ J(M ) and that I only contains positive elements. In this case, set l ε,ε : D sup = R 2 → R 2 and r ε,ε : D inf = R 2 → R 2 to be equal to the finite compositions l |I|,ε,ε • l |I|−1,ε,ε • . . . • l 1,ε,ε and r |I|,ε,ε • r |I|−1,ε,ε • . . . • r 1,ε,ε respectively. Observe that the latter is, by definition, equal to the identity. It remains to consider the case in which I is infinite. If I has infinitely many positive elements, set, for Analogously, if I has infinitely many non-positive elements, set, for any In the remaining cases, set l ε,ε = id = r ε,ε . Proof. The result follows immediately if I is finite, and for l ε,ε (respectively r ε,ε ) if I does not contain positive (respectively non-negative) elements. Suppose that I contains infinitely many positive elements. By definition of the maps l i,ε,ε , for all i 1 ≥ i 2 ≥ 1, the restrictions of This implies that l ε,ε is well-defined. The domain of l ε,ε is given by the subset of R 2 consisting of points (x, y) ∈ R 2 for which there exists a positive i ∈ I with x ≤ x i . By definition of x sup (see equation (4.1)), this is precisely D sup . Moreover, l ε,ε sends the first coordinate to itself, as do all l i,ε,ε . This means that the codomain of l ε,ε can be taken to be D sup . An analogous argument works for r ε,ε in the case of I containing infinitely many non-positive elements by reversing the above inequalities.
In fact, more is true.
Lemma 4.35. The maps l ε,ε : D sup → D sup and r ε,ε : D inf → D inf are homeomorphisms that are Z-affine isomorphisms away from the set Proof. As in the proof of Lemma 4.34, the only non-trivial cases to consider are those of l ε,ε if I contains infinitely many positive elements and of r ε,ε if I contains infinitely many non-positive elements. Consider the former case (the latter is entirely analogous). To see that l ε,ε is a homeomorphism, observe that the proof of Lemma 4.34 implies that, for all (x, y) ∈ D sup , l ε,ε (x, y) = x, l (2) ε,ε (x, y) , for some continuous function l (2) ε,ε : D sup → R. Moreover, it can be checked directly that, for any x < x sup , the function l (2) ε,ε (x, ·) is strictly increasing. Therefore, l ε,ε is a homeomorphism onto its image. Since for any i ≥ 1 the map l i,ε,ε • l i−1,ε,ε • . . . • l 1,ε,ε sends {(x, y) | x ≤ x i } onto itself, it follows that l ε,ε (D sup ) = D sup . The fact that it is a Z-affine isomorphism away from i∈I {(x, y) | x = x i } follows from the fact that, for any i ≥ 1, l i,ε,ε • l i−1,ε,ε • . . . • l 1,ε,ε also satisfies this property (see Corollary 4.32).
With the maps l ε,ε and r ε,ε at hand, we can state the main result of this section.  H)) be a simple vertical almosttoric system. Given any ε,ε ∈ {+1, −1} I and any two cartographic homeomorphisms f ε , fε : B → R 2 associated to ε,ε respectively, there exists a transformation h ε,ε ∈ Vert(2; Z) such that The main idea behind Theorem 4.36 is not new; it first appears in Vũ Ngo . c [46,Proposition 4.1] in the context of semi-toric systems. However, the more general context of vertical almost-toric systems, where there may be infinitely many focus-focus points, deserves to be dealt with carefully. For instance, the transformation r ε,ε , which is not needed in the study of semi-toric systems, is necessary in this context (cf. Remarks 4.30 and 4.31).

Remark 4.37
In fact, the statement and proof of Theorem 4.36 may be of use in the study of semi-toric systems as well. The main references for these systems make the underlying (tacit) assumption that the signs of the cartographic homeomorphisms, as in Theorem 4.24, are positive (cf. Pelayo & Vũ Ngo . c, Vũ Ngo . c [34,35,37,46]).
Proof of Theorem 4.36. The proof is split into three steps: • Construct the map h ε,ε ∈ Vert(2; Z).
• Reduce to the simpler case in which ε andε differ in precisely one component. • Prove the simpler case.
Step 1: constructing the map h ε,ε ∈ Vert(2; Z). As above, fix a basepoint (x, y) ∈ B lt with the property that x 0 < x < x 1 . (If x 0 or x 1 is not defined, then only the other inequality is required.) Denote by S the connected component of S ε ∩ Sε containing (x, y). Since both S ε and Sε are open, so is S. Moreover it is path-connected by definition. Furthermore, it can be checked that S intersects any vertical line either in an empty or in a connected set. Therefore, by Proposition 4.7, the subsystem of (M, ω, Φ = (J, H)) relative to S is vertical almost-toric and contains (x, y). Moreover, by construction, it contains no focus-focus value. By Corollary 2.49, the maps f ε | S and fε| S are cartographic homeomorphisms for the subsystem of (M, ω, Φ = (J, H)) relative to S. Therefore, by Remark 4.18, there exists h ε,ε ∈ Vert(2; Z) such that fε| S = h ε,ε | fε(S) • f ε | S . The map h ε,ε is the desired one.
Step 2: reducing to a simpler case. Observe that, by Corollary 4.29, the map h ε,ε • f ε is a cartographic homeomorphism associated to ε. Moreover, the above argument shows that fε| S = (h ε,ε • f ε ) | S . Thus, in order to prove the result in the statement of the theorem, it suffices to prove that, if fε| S = f ε | S , then fε = r ε,ε • l ε,ε • f ε . Henceforth, assume that fε| S = f ε | S , which implies sgn (fε) = sgn (f ε ). In fact, we can simplify the argument further: it suffices to prove the claimed result under the assumption that all but one of the components of ε,ε are equal. For, if the latter holds, we can argue as follows. Consider a sequence of choices of signs ε s , for s ∈ {1, 2, . . .} ∪ {∞}, such that ε 1 = ε, ε ∞ =ε, and, for any s ≥ 1, all but one component of ε s and ε s+1 are equal. Moreover, if ε,ε differ in finitely components, say in r components, choose the above sequence so that for all s ≥ r+1, ε s =ε. For each s ≥ 1, fix a choice of cartographic homeomorphism f εs with the property that f εs | S = f ε | S . Moreover, require that f ε 1 = f ε , that f ε∞ = fε, and that, if ε,ε differ in precisely r components, then for all s ≥ r + 1, f εs = fε . Using the above sequence of signs and associated cartographic homeomorphisms and the fact that the claimed result holds when all but one component of the signs are equal, we obtain, for all s ≥ 1, maps l εs,ε s+1 , r εs,ε s+1 satisfying (4.10) f ε s+1 = r εs,ε s+1 • l εs,ε s+1 • f εs .
Thus the result is proved if ε andε differ in finitely many components.
The case in which they differ by infinitely many components is entirely analogous, as we can consider the composite of infinitely many maps of the above form on, say, any compact subset of B (cf. the proof of Lemma 4.34) and use a compact exhaustion of B. Thus, assuming that the result holds when the choices of signs differ in precisely one component, the result holds in general.
Step 3: proving the simple case. Assume that fε| S = f ε | S and that ε andε differ in precisely one component. Under these assumptions the result can be proved exactly as in Vũ Ngo . c [46,Proposition 4.1], whose key ideas are explained below. Suppose that ε andε differ precisely in the ith component. By Corollary 2.52, fε • f −1 ε is a homeomorphism that is piecewise Z-affine. Using the proof of Theorem 4.24, it may be assumed without loss of generality that S ε and Sε are path-connected. As both sets are dense, it suffices to check the desired equality on S ε ∩ Sε. Since ε andε differ in precisely one component, it follows that S ε ∩ Sε has two connected components, S and S , which are open and satisfy the assumptions of Proposition 4.7. Thus the subsystems of (M, ω, Φ = (J, H)) relative to S and S are vertical almost-toric. By Corollary 2.49, the restrictions of f ε , fε to S and S are cartographic homeomorphisms for the respective subsystems of (M, ω, Φ = (J, H)). By construction, these subsystems contain no focus-focus points. Thus, Remark 4.18 implies that there exist h S , h S ∈ Vert(2; Z) with fε| S = h S • f ε | S and fε| S = h S • f ε | S .
By assumption, fε| S = f ε | S , so h S = id; on the other hand, the above assumptions imply that (r ε,ε • l ε,ε ) | S = id. Thus the desired equality holds on S. It remains to check that it does on S . Using property (C2) for f ε and fε and the fact that sgn (f ε ) = sgn (fε), it can be shown that the linear parts of h S and of (r ε,ε • l ε,ε ) | S are equal. To see that their translational components are equal, observe that the piecewise Z-affine transformation given on f ε (S) and on f ε (S ) by h S and h S , respectively, extends uniquely to a topological embedding of f ε (B) onto fε (B) (which equals fε • f −1 ε ). In particular, it acts as the identity on the vertical line containing the ith focus-focus value. This implies that the translational component of h S equals that of (r ε,ε • l ε,ε ) | S . 4.5. η-cartographic vertical almost-toric systems. Let (M, ω, Φ = (J, H)) be a simple vertical almost-toric system. The presence of a focusfocus point implies that no vertical almost-toric system isomorphic to (M, ω, Φ = (J, H)) has a cartographic moment map (cf. Section 2.7).
On the other hand, Theorem 4.24 provides cartographic homeomorphisms associated to choices of vertical cuts. Fix any such cartographic homeomorphism f ε ; while it is tempting to think of (M, ω, f ε • Φ) as an integrable system, the lack of smoothness of f ε prevents it from being one. (If we were to adopt the non-standard convention of Harada & Kaveh [17, Definition 2.1], (M, ω, f ε • Φ) would be an integrable system.) The aim of this section is to show that, in some sense, the next best scenario holds: Given a choice of signs ε with S ε connected, any cartographic homeomorphism associated to ε can be modified in an arbitrarily small neighborhood of the cuts associated to ε so that it becomes everywhere smooth (see Theorem 4.48 for a precise statement). This smoothing of cartographic homeomorphisms generates representatives in the isomorphism class of a simple vertical almost-toric system, which we call η-cartographic, that are particularly useful when defining surgeries on (isomorphism classes of) simple vertical almost-toric systems (cf. the forthcoming [20]). Moreover, we show that cartographic homeomorphisms are, in some sense, limits of what we call η-cartographic embeddings (see Proposition 4.52).
Throughout the rest of this section, a simple vertical almost-toric system (M, ω, Φ = (J, H)) containing at least one focus-focus singular point is fixed. As above, set B = Φ(M ), let {c i } ⊂ Int(B) denote the set of focus-focus values of (M, ω, Φ), while l ε denotes the union of the vertical cuts in B associated to a choice of signs ε ∈ {+1, −1} I . Also let S ε = B l ε denote the complement of those cuts. Finally, assume a cartographic homeomorphism has positive sign (cf. Theorem 4.24) unless otherwise stated. 4.5.1. Admissible half-strips for simple vertical almost-toric systems. First, we define the (closed) neighborhoods of vertical half-lines that we use to construct the smoothing of a given cartographic homeomorphism.
Definition 4.38. Fix ε ∈ {+1, −1}, (x 0 , y 0 ) ∈ R 2 , η > 0, and a continuous map γ : A half-strip centered at (x 0 , y 0 ) of sign ε and width η with bounding curve γ is the following closed subset of R 2 : and εy ≥ εγ(x) (see Figure 4.3). The vertical line {(x, y) | x = x 0 } is called the center line of the half-strip. When the center point (x 0 , y 0 ) and the bounding curve γ are not of particular concern, the half-strip is denoted by σ ε η .
The base of a half-strip σ ε η,γ (x 0 , y 0 ) is the subset Consider a choice of (countably many) points {(x i , y i )} i∈I , of signs ε ∈ {+1, −1} I , of positive numbers η ∈ {η i } i∈I I, and of continuous curves γ = {γ i } i∈I . Let σ ε i η i ,γ i (x i , y i ) be the half-strip centered at (x i , y i ) of sign ε i and width η i > 0 with bounding curve γ i . Moreover, set and denote the above choices of signs, widths and curves by the triple (ε, η, γ).
Definition 4.39. Suppose B ⊂ R 2 has the property that its intersection with any vertical line is either empty or path-connected, and consider a countable set of points {(x i , y i )} i∈I therein. A triple (ε, η, γ) as above is admissible for the subset B relative to the points {(x i , y i )} i∈I if it satisfies the following conditions: • For all i, the base of the half-strip • Whenever the half-strips σ ε i η i ,γ i (x i , y i ) and σ ε j η j ,γ j (x j , y j ) share the same center line, η i = η j .
• The intersection of any two distinct half-strips is either empty or equal to one of the half-strips. In this case, the corresponding half-strips are called admissible for B relative to the points {(x i , y i )} i∈I .
Examples of admissible half-strips are sketched in  Before establishing the existence of admissible half-strips for any simple vertical almost-toric system (Proposition 4.42), we derive the following necessary condition. Recall that the focus-focus values of a simple vertical almost-toric system are ordered as in Section 4.3.
intersect, but neither is contained in the other (cf.  Proof. Fix a choice of ε ∈ {+1, −1} I as above and let {c i = (x i , y i )} i∈I denote the set of focus-focus values of (M, ω, Φ = (J, H)). By Proposition 4.14, the set of first coordinates of focus-focus values is a subset of the set of critical values of J, which, by property (V3), does not contain any limit point in J(M ). Moreover, focus-focus values are discrete in B and, by property (V4), there are finitely many of them on a given vertical line. The above facts imply that there exists a choice of positive numbers η = {η i } i∈I such that • if x i = x j , η i = η j , • if, for i = j, x j ∈ x i − η i 2 , x i + η i 2 , then x i = x j , and • for all i, x i − η i 2 , x i + η i 2 × ε i y i − η i 2 , ε i y i + η i 2 is contained in Int(B). For each i ∈ I, define γ i : x i − η i 2 , x i + η i 2 → R to be γ i (x) := y i − ε i η 2 , and set γ = {γ i } i∈I . It can be checked that the triple (ε, η, γ) is admissible for (M, ω, Φ = (J, H)).

Remark 4.43
The choice of widths η in the proof of Proposition 4.42 can be made so that the following property also holds: If a half-strip σ ε i η i contains a corner of B, then it contains precisely one, and that corner lies on the center line of the half-strip. Henceforth, any admissible triple for a simple vertical almost-toric system is assumed to satisfy this property unless otherwise stated.
Fix an admissible triple (ε, η, γ) for (M, ω, Φ = (J, H)). The next results can be interpreted as showing that the complement of the corresponding half-strips in B behaves like S ε . is closed in B. Let {(x n , y n )} ⊂ σ ε η,γ be a sequence that converges to (x 0 , y 0 ) ∈ B and consider the sequence {x n } = pr 1 ({(x n , y n )}) which converges to x 0 ∈ J(M ). Since J(M ) is locally compact, there exists a compact neighborhood K ⊂ J(M ) of x 0 . Since x n → x 0 , it follows that all but finitely many of the x n are contained in K. Since K is compact and the critical values of J are discrete in J(M ) by property (V3), K contains at most finitely many critical values of J. Therefore, by property (V4) and Proposition 4.14, there are at most finitely many focus-focus values contained in pr −1 1 (K) ∩ B. Hence, all but finitely many of the (x n , y n ) are contained in the union of finitely many admissible half-strips, each of which is a closed subset of R 2 and, hence, of B. Thus (x 0 , y 0 ) belongs to this union of finitely many admissible half-strips and so to σ ε η,γ . Corollary 4.45. If (ε, η, γ) is admissible for a simple vertical almosttoric system (M, ω, Φ), then B σ ε η,γ is contractible. Proof. By Lemma 4.44 the subset B σ ε η,γ is open in B. If the intersection of B σ ε η,γ with every vertical line were either empty or connected, then Proposition 4.7 would imply that the subsystem relative to B σ ε η,γ would be vertical almost-toric, after which Corollary 4.5 would ensure that B σ ε η,γ , the moment map image of that vertical almost-toric subsystem, would be contractible. Thus it suffices to show that the intersection of B σ ε η,γ with every vertical line is either empty or connected. Fix x 0 ∈ pr 1 (B); if x 0 / ∈ pr 1 (σ ε η,γ ), B σ ε η,γ ∩ {(x, y) | x = x 0 } = B ∩ {(x, y) | x = x 0 } , and the result follows from the fact that (M, ω, Φ) is vertical almosttoric. Suppose that x 0 ∈ pr 1 (σ ε η,γ ) and call a half-strip σ ε k η k ,γ k (x k , y k ) maximal if it is not a proper subset of σ ε j η j ,γ j (x j , y j ) for any j = k. Because the triple (ε, η, γ) is admissible for (M, ω, Φ), there are at most two maximal half-strips σ ε i η i ,γ i (x i , y i ), σ ε j η j ,γ j (x j , y j ) with the property that for s = i, j, x 0 ∈ x s − ηs 2 , x s + ηs 2 . These half-strips are, by definition, disjoint and their bases are contained in Int(B). That property is sufficient to ensure that B σ ε η,γ ∩ {(x, y) | x = x 0 } is connected, as desired.
Finally, we note that admissible triples behave well under isomorphisms and taking saturated subsystems.

Remark 4.47
In fact, an admissible triple on the cartographic moment map image induces an admissible triple for the system by reversing the above construction.   H)) be a simple vertical almosttoric system and let ε be a choice of signs such that B l ε is connected.
Proof. Let ε be as in the statement and fix an admissible triple (ε, η, γ) for (M, ω, Φ). The map f ε is smooth on the complement of the cuts. As in the proof of Corollary 4.45, say that a half-strip is maximal if it is not a proper subset of any other half-strip. It is sufficient to modify f ε in the interior of maximal admissible half-strips, and since maximal half-strips are pairwise disjoint, it suffices to construct the modified map in the interior of each one separately.
Consider a maximal admissible half-strip, say σ ε j η j ,γ j (x j , y j ). Without loss of generality assume that ε j = +1 so as to drop the notational dependence of the half-strip on ε j . Moreover, fix an admissible triple (ε, η , γ ) for (M, ω, Φ = (J, H)), where, if i = j, η i = η i and γ i = γ i , and if i = j and η j < η j and γ j (x) > γ j (x) whenever both make sense.
There are two cases to consider, namely if η j can be chosen so that ∂ ∞ B ∩ σ η j ,γ j = ∅ or not. Suppose that the former holds; then the set W j = B ∩ Int(σ η j ,γ j ) is open in R 2 . The situation is sketched in Figure 4.4 (a). Let Γ j be an embedded curve in W j of the form Γ j (x) = (x, h j (x)), where h j is a smooth function, that is disjoint from the cut l ε j and is such that W j Γ j has two components. Let K j , L j be the closures in W j of the two components of W j Γ j , so K j ∩ L j = Γ j , and assume without loss of generality that the cut l ε j lies in K j .
Recall that the cartographic homeomorphism f ε is orientation-preserving and of the special form f ε (x, y) = (f (1) ε (x, y), f (2) ε (x, y)) = (x, f (2) ε (x, y)). Therefore, f • Γ j (x) = (x, f (2) ε (x, h j (x))) and, since ε j = +1, if (x, y) ∈ K j then y ≥ h j (x) and f (2) ε (x, y) ≥ f (2) ε (x, h j (x)), as f ε is orientationpreserving. Define g j : K j → R 2 by g j (x, y) := (x, y + f (2) ε (x, h j (x)) − h j (x)), which is an orientation-preserving diffeomorphism of K j onto its image that satisfies g j (x, h j (x)) = f ε (x, h j (x)). Now consider the map F ε j : Int(σ η j ,γ j ) ∩ B → R 2 , f ε (x, y) if (x, y) ∈ L j g j (x, y) if (x, y) ∈ K j . , which is a homeomorphism onto its image. Furthermore, because F ε j is a diffeomorphism on the complement of Γ j , which is a closed submanifold of W j , F ε j can be isotoped to be a diffeomorphism onto the image F ε j (W j ) via an isotopy that is supported in an arbitrarily small neighborhood of Γ j and is the identity in L j (cf. Hirsch [18,Chapter 8]). By construction, F ε j extends to all of σ η j ,γ j as a diffeomorphism, say F ε j , on σ η j ,γ j that agrees with f ε on σ η j ,γ j K j . The map F ε j is the desired smoothing. It remains to consider the case in which an admissible triple (ε, η , γ ) as above does not exist, i.e., for any choices of η j and γ j as above, the corresponding half-strip also intersects ∂ ∞ B (cf. Figure 4.4 (b)). In this case, modify the argument as follows. Let Γ j ⊂ B∩Int(σ η j ,γ j ) be chosen as above, and so that any boundary point p of Γ j also lies in pathconnected component of ∂ ∞ B 0 ∩Int(σ η j ,γ j ). Because f ε is, by definition, smooth at p, the map f ε and the smooth curve Γ j can be extended to a neighborhood of p ∈ R 2 . Make such an extension near the one or two boundary points of Γ, and let W j be an open tubular neighborhood of the extended curve Γ j . Let K j and L j be defined as in the first case (enlarged as per the extension just described), with the map F ε j defined as above. But to apply the smoothing argument, restrict attention to K j ∩ W j and L j ∩ W j so that Γ j is a closed submanifold of an open manifold, in this case the tubular neighborhood W j .  Given an admissible triple (ε, η, γ), a map F ε as constructed in Theorem 4.48 is referred to as an η-cartographic embedding. Moreover, if the dependence on η is to be remembered, an ηcartographic embedding is denoted by F ε,η .
Theorem 4.48 motivates introducing the following notion.
Definition 4.50. A simple vertical almost-toric system is η-cartographic if it admits an admissible triple (ε, η, γ) and a cartographic homeomorphism f ε whose restriction to the complement of the union of the corresponding admissible half-strips is the identity. If the choice of (ε, η, γ) is to be remembered, the system is said to be η-cartographic with respect to (ε, η, γ).
The first application of Theorem 4.51 is the following result.
Proposition 4.52. A cartographic moment map image of a simple vertical almost-toric system associated to a choice of signs whose corresponding cuts do not disconnect the moment map image is the direct limit of η-cartographic moment map images.