Constructing integrable systems of semitoric type

Alvaro Pelayo∗ and San Vu Ngo.c

Abstract

Let (M, ω) be a connected, symplectic4-manifold. A semitoric integrable system on(M, ω) es-sentially consists of a pair of independent, real-valued, smooth functionsJ andH onM , for whichJgenerates a Hamiltonian circle action under whichH is invariant. In this paper we give a general methodto construct, starting from a collection of five ingredients, a symplectic4-manifold equipped a semitoricintegrable system. Then we show that every semitoric integrable system on a symplectic4-manifold isobtained in this fashion. In conjunction with the uniqueness theorem proved recently by the authors, thisgives a classification of semitoric integrable systems on4-manifolds, in terms of five invariants.

1 Introduction

The present paper is motivated by some remarkable results proven in the 80s by Atiyah, Guillemin-Sternbergand Delzant, in the context of Hamiltonian torus actions. Indeed, Atiyah [1, Th. 1] and Guillemin-Sternberg[14] proved that if ann-dimensional torus acts on a compact, connected symplecticmanifold (M, ω) in aHamiltonian fashion, the imageµ(M) under the momentum mapµ := (µ1, . . . , µn) : M → R

n is a convexpolytope. Delzant [6] showed that if the dimensionn of the torus is half the dimension ofM , this polytope,which in this case is called aDelzant polytope(i.e. a convex polytope with the property that at each vertexofit there are preciselyn codimension one faces with normals which form aZ-basis of the integral latticeZn)determines the isomorphism type ofM , and moreover,M is a toric variety. He also showed that startingfrom any Delzant polytope one can construct a symplectic manifold with a Hamiltonian torus action forwhich its associated polytope is the one we started with.

From the viewpoint of symplectic geometry, the situation described by the momentum polytope is,nevertheless, very rigid. It is natural to wonder whether any of these striking results persist in the casewhere the torus is replaced by a non-compact group acting Hamiltonianly. The seemingly symplest casehappens when the group isRn, and the study of theseRn-actions is precisely the goal of the theory ofintegrable systems. Building on previous work of the authors, and of many other authors, we shall presenta “Delzant” type classification for integrable systems, forwhich one component of the system is generatedby a Hamiltonian circle action; these systems are called semitoric.

Let (M, ω) be a connected, symplectic4-dimensional manifold, where we do not assume thatM iscompact. Any smooth functionf onM induces a unique vector fieldXf onM which satisfiesω(Xf , ·) =−df . It is called theHamiltonian vector field induced byf . An integrable system onM is a pair of real-valued smooth functionsJ andH onM , for which the Poisson bracketJ, H := ω(XJ , XH) identicallyvanishes onM , and the differentialsdJ , dH are almost-everywhere linearly independendent. Of course,here(J, H) : M → R

2 is the analogue of the momentum map in the case of a torus action. In some local

∗Partially supported by an NSF Postdoctoral Fellowship.

1

symplectic coordinates ofM , (x, y, ξ, η), the symplectic formω is given bydξ ∧ dx +dη ∧ dy, and thevanishing of the Poisson bracketsJ, H amounts to the partial differential equation

∂J

∂ξ

∂H

∂x− ∂J

∂x

∂H

∂ξ+∂J

∂η

∂H

∂y− ∂J

∂y

∂H

∂η= 0.

This condition is equivalent toJ being constant along the integral curves ofXH (orH being constant alongthe integral curves ofXJ ).

A semitoric integrable systemon M is an integrable system for which the componentJ is a propermomentum map for a Hamiltonian circle action onM , and the associated mapF := (J, H) : M → R

2

has only non-degenerate singularities in the sense of Williamson, without real-hyperbolic blocks. We alsouse the term4-dimensional semitoric integrable systemto refer to the triple(M, ω, (J, H)). Recall that theproperness ofJ means that the preimage byJ of a compact set is compact inM (which is immediate ifMis compact), and the non-degeneracy hypothesis forF means that, ifp is a critical point ofF , then thereexists a 2 by 2 matrixB such that, if we denoteF = B (F − F (p)), one of the following situations holdsin some local symplectic coordinates(x, y, ξ, η) centered atp (meaning thatx = y = ξ = η = 0 atp) :

(1) F (x, y, ξ, η) = (η +O(η2), 12(x

2 + ξ2) +O((x, ξ)3),

(2) F (x, y, ξ, η) = 12(x

2 + ξ2, y2 + η2) +O((x, ξ, y, η)3),

(3) F (x, y, ξ, η) = (xξ + yη, xη − yξ) +O((x, ξ, y, η)3).

The first case is called atransversally— or codimension 1 —elliptic singularity; the second case is anelliptic-elliptic singularity; the last case is afocus-focus singularity. In [17, Th. 6.2] the authors constructed,starting from a given semitoric integrable system on a4-manifold, a collection of five symplectic invariantsassociated with it and proved that these completely determine the integrable system up to isomorphisms.The goal of the present is to complement that work, by providing a general method to constructany 4-dimensional semitoric integrable system starting from an abstract collection of ingredients. Both throughout[17] and the present paper we make a generic assumption on oursemitoric systems; this is explained inSection 2.1.

The symplectic invariants constructed in [17], for a given4-dimensional semitoric integrable system, arethe following: (i) the number of singularities invariant: an integermf counting the number of isolated sin-gularities; (ii)the singularity type invariant: a collection ofmf infinite Taylor series on two variables whichclassifies locally the type of singularity; (iii)the polygon invariant: the equivalence class of a weightedrational convex1 polygon (

∆, (ℓj)mf

j=1, (ǫj)mf

j=1

).

Here∆ is a convex polygon domain inR2, theℓj are vertical lines intersecting∆ and theǫj are±1 signsgiving each lineℓj an orientation; (iv)the volume invariant: mf numbers measuring volumes of certainsubmanifolds at the singularities; (v)the twisting index invariant: mf integers measuring how twisted thesystem is around singularities. This is a subtle invariant,which depends on the representative chosen in (iii).Here, we writemf to emphasize that the singularities thatmf counts are focus-focus singularities. We thenproved that two semitoric systems(M, ω1, (J1, H1)) and(M, ω2, (J2, H2)) are isomorphic if and only ifthey have the same invariants (i)–(v), where an isomorphismis a symplectomorphismϕ : M1 → M2 suchthatϕ∗(J2, H2) = (J1, f(J1, H1)) for some smooth functionf such that ∂f∂H1

nowhere vanishes.

1generalizing the Delzant polygon and which may be viewed as abifurcation diagram

2

(ℓ1, ǫ1 = 1) (ℓ2, ǫ2 = −1)y

x

Figure 1.1: Weighted polygon(∆, (ℓ1, ℓ2), (1, −1)).

We have found that some restrictions on these symplectic invariants must be imposed. Indeed, we call“semitoric list of ingredients” the following collection of items (i)-(v): (i) any integer number0 ≤ mf <∞;(ii) an mf -tuple of real formal power series in two variables, with vanishing constant term and first terms

σ1X + σ2 Y with σ2 ∈ [0, 2π); (iii) a Delzant weighted polygon(∆, (ℓj)

mf

j=1, (ǫj)mf

j=1

), of complexity

mf , where∆ is a polygon, theℓj are again vertical lines intersecting∆ and theǫj are±1 signs giving eachline ℓj an orientation; here the Delzant property for∆ is not the standard one for polygons, but rather a moredelicate one for weighted polygons which takes into accountthe presence of the linesℓj; (iv) anmf -tuple ofpositive real numbers(hi)

mf

i=1 such that0 < hi < length(∆ ∩ ℓi) for eachi ∈ 1, . . . ,mf. (v) an arbitrarycollection ofmf integers(ki)

mf

i=1. Our main theorem (Theorem 4.6) says that, starting from a semitoric listof ingredients one can construct a4-dimensional semitoric integrable system(M, ω, (J, H)) such that thelist of its invariants is equal to this semitoric list. Moreover,M is compact if and only the polygon in item(iii) is compact.

With this in mind we may formulate the uniqueness theorem in [17] as: two systems constructed inthis fashion are isomorphic if and only if ingredients (i), (ii) and (iv) are identical for both systems andingredients (iii) and (v) are related by some simple transformation. This is why, when we formulate theexistence theorem, ingredients (iii) and (v) are given by orbits of respectively weighted polygons and pon-dered weighted polygons, under the action of certain groups. Together with [17, Th. 6.2], this gives theaforementioned classification (Theorem 4.7) .

While the construction of semitoric systems in the present paper is relatively self-contained, we are in-debted to the articles of Delzant [6], Atiyah [1] and Guillemin-Sternberg [14], in the context of Hamiltoniantorus actions, which served as an inspiration to study the more general situation of integrable systems withcircular Hamiltonian symmetry. Furthermore, many works have played an important role in our investiga-tion of 4-dimensional semitoric systems, by serving as stepping stones to construct the symplectic invariantsin [20] associated with semitoric systems; notably we used work of Dufour-Molino [8], Duistermaat [7],Eliasson [9], Miranda-Zung [16] and Vu Ngo.c [19],[20].

In this work, we are in a situation where the moment map(J,H) is a “torus fibration” with singularities,and its base space becomes endowed with a singular integral affine structure. These structures have beenstudied in the context of integrable systems (in particularby Zung [23]), but also became a central conceptin the works by Symington [18], Symington-Leung [15] in the context of symplectic geometry and topol-ogy, and by Gross-Siebert [13], [11], [12] and [10], among others, in the context of mirror symmetry andalgebraic geometry. In fact, our ingredients (i), (iii) and(iv) could have been expressed in terms of this affinestructure. However ingredients (ii) and (v) do not appear inthe affine structure. Nevertheless it is expectedthat these ingredients play an important role in the quantumtheory of integrable systems. We hope to beable to explore these ideas in the future.

The paper is structured as follows: in Section 2 we recall howto construct a collection of symplectic

3

invariants for a semitoric system, and state more preciselythat two semitoric systems are isomorphic pre-cisely when they have the same invariants; this was done in [17], and we need to review it here in order tostate the existence theorem for semitoric systems. In Section 3 we explain the symplectic glueing construc-tion (i.e. how to glue symplectic manifolds equipped with momentum maps). The last two sections of thepaper are respectively devoted to state the main theorem andto prove it. One might argue that the proof ismore informative than the statement, as it gives anexplicit construction of all semitoric integrable systemsin dimension4.Acknowledgements. We are very grateful to an anonymous referee for several interesting comments andremarks which have led to improvements. We thank Denis Auroux for offering several comments, and forpointing out the papers of Gross and Siebert.

2 Review of the uniqueness theorem for semitoric systems

We recall the definition of the invariants that we assigned toa semitoric integrable system in our previouspaper [17], to which we refer to further details. Then we state the uniqueness theorem proved therein.

2.1 Taylor series invariant

It was proven in [20] that a semitoric system(M, ω, F := (J, H)) has finitely many focus-focus crit-ical valuesc1, . . . , cmf

, that if we writeB := F (M) then the set of regular values ofF is Int(B) \c1, . . . , cmf

, that the boundary ofB consists of all images of elliptic singularities, and that the fibers ofF are connected. The integermf was the first invariant that we associated with such a system.Let i be aninteger, with1 ≤ i ≤ mf .

We assume that the critical fiberFm := F−1(ci) contains only one critical pointm, which accordingto Zung [23] is a generic condition, and letF denote the associated singular foliation. Moreover, we makefor simplicity an even stronger generic assumption : ifm is a focus-focus critical point forF thenm is theunique critical point of the level setJ−1(J(m)). A semitoric system issimpleif this genericity assumptionis satisfied. These conditions imply that the valuesJ(c1), . . . , J(cmf

) are pairwise distinct. We assumethroughout the article that the critical valuesci’s areorderedby theirJ-values :J(c1) < J(c2) < · · · <J(cmf

). Vu Ngo.c proved in [19] that a semiglobal neighborhood of theith focus-focus is classified up tosymplectic equivalence by a Taylor series(Si)

∞ ∈ R[[X, Y ]] in two variables with vanishing constant term.The series(Si)∞ is called theTaylor series invariant of(M, ω, (J, H)) at the focus-focus pointci. For thepurpose of this paper we do not need to know to obtain this Taylor series from the integrable system; itsuffices to know that it completely classifies symplectically a neighborhood of the singular fiber above thefocus-focus point. A summary of the construction in [19] appeared in [17, Sec. 3.2].

2.2 Semitoric polygon invariant

The planeR2 is equipped with its standard affine structure with origin at(0, 0), and orientation. LetAff (2,R) := SL(2,R)⋉R

2 be the group of affine transformations ofR2. Let Aff(2,Z) := SL(2,Z)⋉R

2 bethe subgroup ofintegral-affinetransformations. LetI be the subgroup ofAff(2, Z) of those transformationswhich leave a vertical line invariant, or equivalently, an element ofI is a vertical translation composed witha matrixT k, wherek ∈ Z and

T k :=

(1 0k 1

)∈ SL(2, Z). (2.1)

4

Let ℓ ⊂ R2 be a vertical line in the plane, not necessarily through the origin, which splits it into two half-

-spaces, and letn ∈ Z. Fix an origin inℓ. Let tnℓ : R2 → R

2 be the identity on the left half-space, andT n on the right half-space. By definitiontnℓ is piecewise affine. Letℓi be a vertical line through the focus--focus valueci = (xi, yi), where1 ≤ i ≤ mf , and for any tuple~n := (n1, . . . , nmf

) ∈ Zmf we set

t~n := tn1

ℓ1 · · · tnmf

ℓmf. The mapt~n is piecewise affine.

Definition 2.1 A convex polygonal set∆ is the intersection inR2 of (finitely or infinitely many) closedhalf-planes such that on each compact subset of the intersection there is at most a finite number of cornerpoints. We say that∆ is rational if each edge is directed along a vector with rational coefficients. Forbrevity, in this paper we usually write“polygon” instead of“convex polygonal set”, see Remark 2.2. ⊘

Remark 2.2 The word “polygon” is commonly used to refer to the convex hull of a finite set of points inR2 which is a compact set (this is not necessarily the case in algebraic geometry, e.g. Newton polygons).

Notice (see Ziegler [24, Thm. 1.2]) that a convex polygonal set∆ which is not a half-space has exactly twoedges of infinite Euclidean length if and only if it is non-compact, and∆ has all of its edges of finite lengthif and only if it is compact.

Some authors call “2-dimensional convex polyhedron” what we call “convex polygonal domain”; weprefer not to use this terminology as historically the word “polyhedron” has referred to3-dimensional ob-jects. ⊘

Let Br := Int(B) \ c1, . . . , cmf, which is precisely the set of regular values ofF . Given a sign

ǫi ∈ −1,+1, let ℓǫii ⊂ ℓi be the vertical half-line starting atci at extending in the direction ofǫi :upwards ifǫi = 1, downwards ifǫi = −1. Let ℓ~ǫ :=

⋃mf

i=1 ℓǫii . In Th. 3.8 in [20] it was shown that for

~ǫ ∈ −1,+1mf there exists a homeomorphism onto its imagef = fǫ : B → f(B) ⊂ R2, modulo a left

composition by a transformation inI, such thatf |(B\ℓ~ǫ) is a diffeomorphism into its image∆ := f(B),which is arational convex polygon(or more precisely, a rational convex polygonal domain),f |(Br\ℓ~ǫ) isaffine (it sends the integral affine structure ofBr to the standard structure ofR2) andf preservesJ : i.e.f(x, y) = (x, f (2)(x, y)).

Let us recall what we mean by the affine structure ofBr [17]. An integral affine manifold is a smoothmanifold that admits an atlas where transition functions are in Aff(n, Z) := SL(n, Z)⋉R

n. On the tangentplane at any pointm of an integral affine manifoldM , there is a well-defined lattice, namely the preimageof Zn by the differentialdm ϕ of a chartϕ : U ⊂ M → R

n. Of courseRn itself is canonically an integralaffine manifold, endowed with the tangent latticeZ

n.It follows from the action–angle theorem [7] that any properLagrangian fibrationF :M → B naturally

defines an integral-affine structure on the baseB. This affine structure can be characterized by the followingfact : a local diffeomorphismg : (B, b) → (Rn, 0) is an integral-affine chart if and only if the Hamiltonianflows of then coordinate functions ofg F are periodic of primitive period equal to2π. Therefore anintegrable system with proper momentum mapF = (J, H) defines an integral-affine structure on the setBr of regular values ofF . In our case, this structure extends to the boundary ofBr in a natural way.

AlthoughBr is a subset ofR2, the integral-affine structure ofBr is in general different from the inducedcanonical integral-affine structure ofR2.

In order to arrive at∆ one cuts(J, H)(M) ⊂ R2 along each of the vertical half-linesℓǫii . Then the

resulting image becomes simply connected and thus there exists a global 2-torus action on the preimage ofthis set. The polygon∆ is just the closure of the image of a toric momentum map corresponding to thistorus action. Of course, if the system is toric (mf = 0), then∆ is the usual Delzant polygon. In the general

5

semitoric case, there are important differences to be noted. One of them is that theH-coordinates of thevertices are not all critical values ofH (while, as in the toric case, theJ-coordinates of the vertices areexactly the critical values ofJ). Another important fact is that this polygon is not unique.The choice of the“cut direction” is encoded in the signsǫj, and there remains some freedom for choosing the toric momentummap. Precisely, the choices and the corresponding homeomorphismsf are the following :

(a) an initial set of action variablesf0 of the form(J, K) near a regular Liouville torus in [20, Step 2,pf. of Th. 3.8]. If we choosef1 instead off0, we get a polygon∆′ obtained by left composition withan element ofI. Similarly, if we choosef1 instead off0, we obtainf composed on the left with anelement ofI;

(b) a tuple~ǫ of 1 and−1. If we choose~ǫ′ instead of~ǫ we get∆′ = t~u(∆) with ui = (ǫi − ǫ′i)/2, by [20,Prop. 4.1, expr. (11)]. Similarly instead off we obtainf ′ = t~u f .

Lemma 2.3. Oncef0 and~ǫ have been fixed as in (a) and (b), respectively, then there exists a unique toricmomentum mapµ onMr := F−1(IntB\(⋃ ℓ

ǫjj )) which preserves the foliationF , and coincides withf0F

where they are both defined. Then, necessarily, the first component ofµ is J , and we have

µ(Mr) = ∆. (2.2)

Proof. The uniqueness follows from the fact that IntB \ (⋃ℓǫjj ) is simply connected, and (2.2) follows

directly from the construction of∆ in [20], sinceµ = f F .

We sometimes callµ the (generalized) momentum map associated with the polytope∆. LetPolyg(R2)be the space of rational convex polygons inR

2. LetVert(R2) be the set of vertical lines inR2. A weighted

polygon of complexitys is a triple of the form∆w =(∆, (ℓλj

)sj=1, (ǫj)sj=1

)wheres is a non-negative

integer,∆ ∈ Polyg(R2), ℓλj∈ Vert(R2) for every j ∈ 1, . . . , s, and ǫj ∈ −1, 1 for every j ∈

1, . . . , s, mins∈∆ π1(s) < λ1 < . . . < λs < maxs∈∆ π1(s), whereπ1 : R2 → R is the canonicalprojectionπ1(x, y) = x andπ1(ℓλj

) = λj. For anys ∈ N, letGs := −1, +1s and letG := T k | k ∈Z. The groupG acts naturally onR2 by the affine transformationsT k. Obviously, it sends a rational convexpolygon to a rational convex polygon. It corresponds to the transformation described in (a). On the otherhand, the transformation described in (b) can be encoded by the groupGs acting on the triple∆w by theformula

(ǫ′j)sj=1 ·

(∆, (ℓλj

)sj=1, (ǫj)sj=1

)=(t~u(∆), (ℓλj

)sj=1, (ǫ′j ǫj)

sj=1

), (2.3)

where~u = ((ǫi − ǫ′i)/2)si=1. This, however, does not always preserve the convexity of∆, as is easily seen

when∆ is the unit square centered at the origin andλ1 = 0. However, when∆ comes from the constructiondescribed above for a semitoric system(J,H), the convexity is preserved. Thus, we say that

Definition 2.4 A weighted polygon isadmissiblewhen theGs-action preserves convexity. We denote byWPolygs(R

2) the space of all admissible weighted polygons of complexitys. ⊘

The setGs×G is an abelian group, with the natural product action. The action ofGs×G onWPolygs(R2)

is((ǫ′j)

sj=1, T

k) ·(∆, (ℓλj

)sj=1, (ǫj)sj=1

)=(t~u(T

k(∆)), (ℓλj)sj=1, (ǫ

′j ǫj)

sj=1

),

where~u = ((ǫi − ǫ′i)/2)si=1.

6

Definition 2.5 We call asemitoric polygonthe equivalence class of an admissible weighted polygon underthe(Gmf

× G)-action. ⊘

Let ∆ be a rational convex polygon obtained from the momentum image (J, H)(M) according to theabove construction of cutting along the half-linesℓǫ11 , . . . , ℓ

ǫmfmf

.

Definition 2.6 Thesemitoric polygon invariant of(M, ω, (J, H)) is the semitoric polygon equal to the(Gmf

× G)-orbit

(Gmf× G) ·

(∆, (ℓj)

mf

j=1, (ǫj)mf

j=1

)∈ WPolygmf

(R2)/(Gmf× G). (2.4)

⊘

2.3 The Volume Invariant

Consider a focus-focus critical pointmi whose image by(J, H) is ci, and let∆ be a rational convex polygoncorresponding to the system(M, ω, (J, H)). If µ is a toric momentum map for the system(M, ω, (J, H))corresponding to∆, then the imageµ(mi) is a point in the interior of∆, along the lineℓi. We proved in[17] that the vertical distance

hi := µ(mi)− mins∈ℓi∩∆

π2(s) > 0 (2.5)

is independent of the choice of momentum mapµ. Hereπ2 : R2 → R is π2(x, y) = y. See [17, Sec. 5.1]for an explanation of the usage of the word “volume”.

2.4 The Twisting-Index Invariant

The twisting-index at a focus-focus pointci is a dynamical invariant which expresses the fact in a neigh-bourhood ofci there is a privileged toric momentum mapν; this momentum map encodes the existence ofa unique hyperbolic radial vector field in a neighbourhood ofthe focus-focus fiber. We refer to [17, Sec. 5,Step 4] for a detailed construction of the privileged momentum map; what is most relevant in the proof ofthe existence result in this paper (Section 5, second stage)is the fact that we have theprivilegedchoice ofmomentum mapν. Since any semitoric polygon defines a (generalized) toric momentum mapµ, we definethe twisting-index atci as the integerki ∈ Z such thatdµ = T kidν. The twisting index invariant is es-sentially themf -tuple of valueski’s but one has to be more careful [17, Sec. 5, Step 5] because asit wasthe case with the semitoric polygon invariant, we need to factor out the actions ofG andGs in order to getthe actual well-defined invariant. For any integers, consider the action of the productGs × G on the spaceWPolygs(R

2)× Zs:

((ǫ′j)sj=1), T

k) ⋆(∆, (ℓλj

)sj=1, (ǫj)sj=1, (kj)

sj=1

)=(t~u(T

k(∆)), (ℓλj)sj=1, (ǫ

′j ǫj)

sj=1, (kj + k)sj=1

)

where~u = ((ǫj − ǫ′j)/2)sj=1, for all integerj, with j ∈ 1, . . . , s.

Definition 2.7 Thetwisting-index invariantof (M, ω, (J, H)) is the(Gmf×G)-orbit of weighted polygon

pondered by twisting indices at the focus-focus singularities of the system given by

(Gmf× G) ⋆

(∆, (ℓj)

mf

j=1, (ǫj)mf

j=1, (kj)mf

j=1

)∈ (WPolygmf

(R2)× Zmf )/(Gmf

× G). (2.6)

⊘

7

2.5 Uniqueness theorem

To a semitoric system we assign the above list of invariants and state the main theorem in [17].

Definition 2.8 Let (M, ω, (J, H)) be a4-dimensional simple semitoric integrable system. Thelist ofinvariants of(M, ω, (J, H)) consists of the following items.

(i) The integer number0 ≤ mf <∞ of focus-focus singular points.

(ii) Themf -tuple((Si)∞)mf

i=1, where(Si)∞ is the Taylor series of theith focus-focus point.

(iii) The semitoric polygon invariant, c.f. Definition 2.6.

(iv) The volume invariant, i.e. themf -tuple(hi)mf

i=1, wherehi is the height of theith focus-focus point.

(v) The twisting-index invariant, c.f. Definition 2.7.⊘

Theorem 2.9(Th. 6.2, [17]). The two4-dimensional simple semitoric integrable systems(M1, ω1, (J1, H1))and (M2, ω2, (J2, H2)) are isomorphic if and only if the list of invariants (i)-(v),as in Definition 2.8, of(M1, ω1, (J1, H1)) is equal to the list of invariants (i)-(v) of(M2, ω2, (J2, H2)).

3 The symplectic glueing construction

In this section we explain how to symplectically glue an arbitrary collection of symplectic manifolds(Mα)α∈Aequipped with continuous, proper mapsFα : Mα → R to form a new symplectic manifoldM equipped witha continuous, proper map which restricted toMα is equal toFα, c.f. Theorem 3.11. The results of thissection, while perhaps well-known among experts, we could not find in the literature.

3.1 Glueing maps, glueing groupoid

Let A be an arbitrary set of indices, and let(Mα)α∈A be a family of sets. Recall that thedisjoint union ofthe setsMα, α ∈ A is the subset of(

⋃α∈AMα)×A defined by

⊔

α∈A

Mα := (x, α) |x ∈Mα.

We denote byjα, α ∈ A, the natural inclusions :jα : Mα → ⊔α∈AMα, x 7→ (x, α). Notice that if

B ⊂ A then⊔

α∈BMα ⊂ ⊔α∈AMα. Of course, if allMα’s are pairwise disjoint, as sets, then there is a

natural bijection bewteen⊔

α∈AMα and the usual union⋃

α∈AMα.If theMα’s are topological spaces, the disjoint union

⊔α∈AMα is endowed with the final topology : the

finest topology that makes the inclusionsjα continuous. In particularjα(Mα) is an open set in⊔

α∈AMα.

Definition 3.1 A glueing map for the family(Mα)α∈A is a homeomorphismϕ : Uα → Uβ where(α, β) ∈A2, andUα ⊂Mα andUβ ⊂Mβ are open sets. ⊘

In this text we use the standard set-theoretical conventionthat the notationϕ includes the source andtarget setsUα andUβ ; in particular the notationϕ(x) impliesx ∈ Uα. When required, we use the notationU sϕ andU t

ϕ for the source and target sets ofϕ (assumingU tϕ = ϕ(U s

ϕ)).

8

Definition 3.2 Let G be a collection of glueing maps for(Mα)α∈A. The associatedglueing groupoidG isthe groupoid generated by the set of all restrictions of all glueing mapsϕ ∈ G to open subsets of the sourcesets, with the natural groupoid law :ϕ2 ϕ1 exists whenever the image of the source set ofϕ1 is includedin the source set ofϕ2. ⊘

Definition 3.3 We say thatG is freewhen there is no nontrivialϕ ∈ G with both source and target in thesame setMα. ⊘

3.2 Topological glueing

We define now the general patching construction. Throughoutthis section, and unless otherwise stated, wedo not require topological spaces to be paracompact or Hausdorff.

Definition 3.4 Let (Mα)α∈A be a collection of pairwise disjoint topological spaces, and G an associatedglueing groupoid. From this we define the setM , called theglueing of(Mα)α∈A along G, asM :=⊔

α∈AMα/ ∼ where∼ is the equivalence relation on⊔

α∈AMα defined by

(x, α) ∼ (x′, β) ⇐⇒(x = x′ or there existsϕ ∈ G with x′ = ϕ(x)

).

⊘

Let us check that∼ is indeed an equivalence relation. The reflexivity is obvious. If (x, α) ∼ (x′, β)and(x, α) 6= (x′, β) thenϕ(x) = x′ for someϕ ∈ G. ButG is a groupoid soϕ−1 ∈ G and of coursex = ϕ−1(x′), so(x′, β) ∼ (x, α), which proves the symmetry property. Finally, if(x, α) ∼ (x′, β) and(x′, β) ∼ (x′′, γ) then there existϕ anfϕ′ in G such thatϕ(x) = x′ andϕ′(x′) = x′′. Thereforeϕ′ ϕ iswell-defined on an open neighbourhood ofx, soϕ′ ϕ ∈ G, and(x, α) ∼ (x′′, γ), so we have shown thetransitivity property.

Here again we could have dropped the assumption that theMα’s are pairwise disjoint, or we could haveused a standard union instead of a disjoint union.

The following lemma follows from the definition of the equivalence relation.

Lemma 3.5. Letπ :⊔

α∈AMα →M be the quotient map. For any subsetK ⊂Mα, one has

π−1(yα(K)) = jα(K) ∪

⋃

ϕ∈G

jα(ϕ)(ϕ(K ∩ U sϕ))

,

where it is assumed that the union is over allϕ whose source setU sϕ intersectsK, andα(ϕ) is the element

in A such thatU tϕ ⊂Mα(ϕ).

Lemma 3.6. For the natural quotient topology onM , the mapsyα = π jα : Mα → M , α ∈ A are openand continuous. They are injective if and only ifG is free.

Proof. By definition of the quotient topology, the mapπ is continuous. Henceyα = π jα is continuous.Finally if U ⊂ Mα is open, then if follows from Lemma 3.5 thatπ−1(yα(U)) is open in

⊔α∈AMα. This

means thatyα(U) is open inM .Fix α ∈ A. Let x andx′ be elements ofMα. If yα(x) = yα(x

′) then eitherx = x′ or ϕ(x) = x′ forsomeϕ ∈ G. The latter is ruled out by the assumption that there is no nontrivial ϕ ∈ G with both sourceand target inMα. Thus in this caseyα is injective. If the condition is violated then there existx 6= x′ in Mα

with jα(x) ∼ jα(x′) soyα cannot be injective.

9

3.3 Smooth glueing

Lemma 3.7. If all Mα’s are smooth manifolds, allϕ ∈ G are diffeomorphisms andG is free then there existsa unique smooth structure onM for which the mapsyα, α ∈ A are embeddings.

Proof. Let U ⊂ Mα be open and letg : U → Rn be a homeomorphism. By Lemma 3.6,yα is a homeo-

morphism onto its image. LetU = yα(U) and g = g ((yα)|U )−1. ThenU is an open subset ofM andg : U → R

n is a homeomorphism. This shows that any chart ofMα descends onto a chart ofM . Obviouslythe union of a family of open covers ofMα for all α ∈ A descends to an open cover ofM . In order to getan atlas onM , it remains to check the compatibility condition when an open setVα coming from an atlasof Mα intersects an open setVβ coming from an atlas ofMβ . Thus, let(Vα, gα), Vα ⊂ Mα and(Vβ , gβ),Vβ ⊂ Mβ be local charts such thatyα(Vα) = yβ(Vβ) andα 6= β. Now consider the formula, given byLemma 3.5 :

jα(Vα) ∪

⋃

ϕ∈G

jα(ϕ)(ϕ(Vα ∩ U sϕ))

= jβ(Vβ) ∪

⋃

ϕ∈G

jα(ϕ)(ϕ(Vβ ∩ U sϕ))

.

BecauseG is free, anyϕ whose source set intersectsVα andwith α(ϕ) = α must be the identity. Hence, inthe lefthand side one can ommit allϕ’s such thatα(ϕ) = α. For the same reason, one can assume that allα(ϕ)’s are pairwise different. Of course the analogue observation holds for the righthand side. Hence wecan equate terms in the unions (up to permutation). In particular there must exist someϕ with α(ϕ) = βandjβ(ϕ(Vα∩U s

ϕ)) = jβ(Vβ). Sincejβ is injective,ϕ(Vα∩U sϕ) = Vβ . Letx ∈ Vβ andx′ = ϕ−1(x) ∈ Vα.

Thenyα(x′) = yβ(x), i.e. x′ = y−1α yβ(x). Thus((yα)|Vα)

−1 (yβ)|Vβ= (ϕ−1)|Vβ

. Hence the transitionmap for the chartsgu := gu ((yu)|Vu)

−1 (u = α, β) is equal to

gα g−1β = gα

(((yα)|Vα)

−1 (yβ)|Vβ

) g−1

β = gα ϕ−1 g−1β , (3.1)

which is indeed a composition of local diffeomorphisms. ThusM has a natural smooth structure.Consider now the mapyα : Mα → M . Read in a chart(Vα, gα) of M , with gα := gα ((yα)|Vα)

−1,for some chart(Vα, gα) onMα, it becomesgα yα = (gα)|Vα , which is a local diffeomorphism. Since wealready know thatyα is a homeomorphism onto its image, it is an embedding.

Conversely, ifyα, α ∈ A have to be embeddings for some smooth structure onM , then any local chartonMα is sent byyα to a local chart onM . Thus, necessarily, we obtain the same charts onM as the oneswe’ve just constructed.

Remark 3.8 The smooth manifoldM given in Lemma 3.7is not necessarily a Hausdorff space. The defi-nition of manifold in Bourbaki [3] does not requireM to be a Hausdorff topological space, or a paracomactspace. These are, however, conditions most frequently required. It follows from Bourbaki [3] thatM isHausdorff if, and only if, for any two smooth chartsϕ : U ⊂ M → R

n, ψ : V ⊂ M → Rn constructed

as in the proof of Lemma 3.7, we have that the graph ofψ ϕ−1 : ϕ(U ∩ V ) → ψ(U ∩ V ) is closed inϕ(U)× ψ(V ) ⊂ R

n × Rn. ⊘

3.4 Symplectic glueing

Unlike in the previous two sections, we shall be assuming that theMα, α ∈ A, are Haudorff, paracompactsmooth manifolds. Moreover, we will be assuming that there exist continuous, proper mapsFα : M → R

n

10

which can be glued together to give rise to a proper mapF : M → R. With the aid ofF we will show thatthe Hausdorff and paracompactness properties of theMα are inherited byM .

Lemma 3.9. If for each α ∈ A, Mα is symplectic with symplectic formωα, and if all ϕ ∈ G aresymplectomorphisms (andG is free) then there exists a unique symplectic structureω on M such thaty∗αω = ωi, α ∈ A.

Proof. Because (1) allyα’s are embeddings, (2)⋃

α∈A yα(Mα) =M , (3) whenyα(Mα) intersectsyβ(Mβ),α 6= β, theny−1

β (yα) = ϕ for someϕ ∈ G with ϕ∗ωβ = ωα, the formulay∗αω = ωα defines a uniquesymplectic formω onM .

We can finally apply this technique in our case :

Proposition 3.10. Let (Mα)α∈A be a collection of symplectic manifolds, each equipped witha mapFα :Mα → R

n. For anyα, β ∈ A letDαβ := Fα(Mα) ∩ Fβ(Mβ) and assume

1. Uα := F−1α (Dαβ) andUβ := F−1

β (Dαβ) are open.

2. ϕαβ : Uα → Uβ is a symplectomorphism such thatϕ∗αβFβ = Fα.

3. WhenDαβγ := Fα(Mα) ∩ Fβ(Mβ) ∩ Fγ(Mγ) 6= ∅, ϕβγ ϕαβ = ϕαγ (restricted toF−1α (Dαβγ)).

Then the smooth manifoldM obtained by glueing the collection(Mα)α∈A along the set of all(ϕαβ) issymplectic, and there exists a unique mapF : M → R

n verifyingFα = F yα, whereyα : Mα → M ,α ∈ A are the natural symplectic embeddings.

Proof. The third assumption (cocycle condition) implies that the corresponding glueing groupoid is free.

Theorem 3.11(Symplectic Glueing). Let (Mα)α∈A be a collection of symplectic manifolds, each equippedwith a continuous, proper mapFα : Mα → Vα ⊂ R

n, whereVα is open. For anyα, β ∈ A let Dαβ :=Vα ∩ Vβ and assume

1. ϕαβ : F−1α (Dαβ) → F−1

β (Dαβ) is a symplectomorphism such thatϕ∗αβFβ = Fα.

2. WhenVα ∩ Vβ ∩ Vγ 6= ∅, ϕβγ ϕαβ = ϕαγ .

Then the smooth manifoldM obtained by glueing the collection(Mα)α∈A along the set of all(ϕαβ) isHausdorff, paracompact (in other words, a smooth manifold in the usual sense) and symplectic, and thereexists a unique continuous, proper mapF : M → ⋃

α∈A Vα ⊂ Rn verifyingFα = F yα, whereyα :

Mα →M , α ∈ A, are the natural symplectic embeddings.

Proof. The main statement is a corollary of Proposition 3.10 sinceF−1(Vα∩Vβ) = F−1(F (Mα)∩F (Mβ))and thus the right handside is automatically open.

Next we show thatM is Hausdorff. Letz, w ∈M , wherez, w ∈ ⊔α∈AMα. There are two possibilities,thatF (z) = F (w) or thatF (z) 6= F (w). If F (z) = F (w), then by definition ofF (i.e.Fα = F yα), thereexistsα ∈ A such thatz ∈ Mα andw ∈ Mα. Here we are viewingMα as a subset of

⊔α∈AMα, under

the canonical identificationyα. BecauseMα is Hausdorff, there exist open setsUz ⊂ Mα, Uw ⊂ Mα, withz ∈ Uz, w ∈ Uw andUz ∩ Uw = ∅. BecauseMα is open in

⊔α∈AMα, by Lemma 3.6 we have thatπ(Uz)

andπ(Uw) are open subsets ofM . By construction,z ∈ π(Uz), w ∈ π(Uw). It follows from the definition ofπ as the quotient map

⊔α∈AMα →M =

⊔α∈AMα/ ∼, thatπ(Uz)∩π(Uw) = π(Uz ∩Uw) = π(∅) = ∅.

11

Suppose on the other hand thatF (z) 6= F (w). SinceF (z) ∈ Rn, F (w) ∈ R

n, andRn is Hausdorff,there exist open setsWz andWw in R

n such thatF (z) ∈Wz, F (w) ∈Ww andWz ∩Ww = ∅. SinceF iscontinuous,F−1(Wz) andF−1(Ww) are open. Also, by construction,z ∈ F−1(Wz) andw ∈ F−1(Ww).Of courseF−1(Wz) ∩ F−1(Ww) = F−1(Wz ∩Ww) = ∅.

Let us show thatF is proper. LetV :=⋃

α∈A Vα. LetK ⊂ V be compact inV . SinceK is compact,there exists a finite number of open ballsBi of radiusǫ > 0 that coverK and such that anyBi is includedin someVα(i), α(i) ∈ A. Let Oββ∈B be an open cover ofF−1(K). For anyi, the setBi is compactin Vα(i); henceF−1

α (Bi) is compact inMα. Thusyα(F−1α (Bi)) is compact inM , and hence there exists a

finite subsetBi ⊂ B such that⋃

β∈BiOβ ⊃ yα(F

−1α (Bi)). We can conclude, using the fact that

for all U ⊂ Vα, yα(F−1α (U)) = F−1(U), (3.2)

thatF−1(K) ⊂ ⋃i

⋃β∈Bi

Oβ, which shows thatF−1(K) is indeed compact.To complete the properness proof we must show that equality (3.2) holds. Indeed, the inclusion of sets

yα(F−1α (U)) ⊂ F−1(U) follows directly from the equalityF yα = Fα. For the converse, we come back

to the definition ofM . If z ∈ F−1(U) there must exist somezβ ∈ Mβ such thatπ(zβ) = z (π is thequotient map of Lemma 3.5). ThusFβ(zβ) = F (z). This means thatVα ∩ Vβ is not empty, and thereis a symplectomorphismϕβα such thatzα := ϕβα(zβ) ∈ Mα. This impliesπ(zα) = π(zβ) = z. ThusF (z) = Fα(zα) which proves the inclusionF−1(U) ⊂ yα(F

−1α (U)).

We have left to show thatM is a paracompact space. We have previously shown thatF : M → V isa proper map, so in particular, the fibers ofF are compact. On the other hand, for eachα ∈ A, Mα is amanifold in the usual sense, and hence it is locally compact,which then implies that

⊔α∈AMα is locally

compact. We claim thatM is locally compact. Indeed, letz ∈M , wherez ∈Mα for someα. BecauseMα

is locally compact, there is a compact neighborhoodKz of z inMα containing an open setUz, with z ∈ Uz.Sinceπ is continuous,π(Kz) is compact. Sinceπ is open,π(Uz) is open, and henceπ(Kz) is a compactneighborhood ofz, and we have shown thatM is locally compact.

On the other hand, a continuous, proper map between locally compact Hausdorff spaces is closed2 see [5,Prop. 3, p. 16]. We have already shown thatM is Hausdorff and locally compact. Hence, sinceF : M → Vis a proper map, it is a also a closed map.

Next we deduce the paracompactness ofM from the following result [21, 20G, p. 153], [4, Th. 1]: iff : X → Y is a continuous, closed surjective mapping between topological spaces with compact fibers, andY is paracompact, thenX is paracompact as well. We can apply this result withX equal toM , Y equal toF (M) ⊂ R

n, andf equal toF : M → F (M). The mapF : M → F (M) is continuous, closed, and it hascompact fibers, andF (M), as a subset ofRn, is paracompact. HenceM is paracompact. This concludesthe proof of the proposition.

4 Main Theorem: statement

Again we equip the planeR2 with its standard affine structure with origin at(0, 0), and orientation.

2Let f : X → Y be such a map. LetA be closed and lety ∈ F (A). SinceY is Hausdorffy is the intersection of closedneighborhoods ofy. SinceY is locally compact one can assume that one of these neighborhood is compact. Sincef is continuousand proper,A∩f−1(y) is a decreasing intersection of nonempty closed sets in a compact, and hence is not empty. Hencey ∈ f(A)andf(A) is closed.

12

4.1 Delzant semitoric polygons

Let∆ ∈Polyg(R2) be a rational convex polygon inR2, as in Definition 2.1. Recall that in our terminology,∆ is not necessarily compact. We call a vertex a point in the boundary∂∆ where the meeting edges are notcolinear. We shall make the following assumption

(a1) The intersection of∆ with a vertical line is either compact or empty.

Consider such a vertical line intersecting the polytope. Ifthe intersection is not just a point, then it is avertical segment. The top end of this segment is said to belong to thetop-boundaryof ∆.

To each vertexz of ∆ we associate a coupleBz of primitive integral vectors starting atz and extendingalong the direction of the edges meeting atz, in the order that makes them oriented. ThenBz defines aZ-basis ofZ2 ⊂ R

2 when, viewed as a2× 2 matrix, its determinant is equal to1.Let s ∈ N

∗ and let(λ1, . . . , λs) ∈ Rs with λ1 < · · · < λs. As beforeℓλj

is the vertical linex = λj.We are interested only in the following case

(a2) The vertical linesℓλj, j = 1, . . . , s intersect the top-boundary of∆.

Let T be the linear transformation acting as the matrix

T := T 1 =

(1 01 1

).

Definition 4.1 Let z be a vertex of the polygon∆ and(u, v) = Bz. The pointz is called

• aDelzant cornerwhen there is no vertical lineℓλjthrough it anddet(u, v) = 1,

• a hidden Delzant cornerwhen there is a vertical lineℓλjthrough it, it belongs to the top-boundary,

anddet(u, Tv) = 1.

• afake cornerwhen there is a vertical lineℓλjthrough it, it belongs to the top-boundary, anddet(u, Tv) =

0.⊘

For the following lemma recall the definition of admissible weighted polygon, c.f. Definition 2.4.

Lemma 4.2. Let∆ be a convex rational polygon equipped with a set of vertical lines(ℓλ1, . . . , ℓλs), such

that the assumptions (a1) and (a2) are satisfied. Suppose moreover that

• any point in the top-boundary that belongs to some vertical line ℓλjis either a hidden Delzant corner

or a fake corner;

• any other vertex of∆ is a Delzant corner.

Then the triple(∆, (ℓλj

)sj=1, (1, . . . , 1))

is an admissible weighted polygon.

Proof. We need to show that the convexity is preserved under theGs-action. This amounts to show that foranyj = 1, . . . , s, the polygont~ej(∆) is convex, where(~e1, . . . , ~es) is the canonical basis ofZs. Sincet~ej isaffine on both half-spaces delimited by the vertical lineℓλj

, it suffices to show thatt~ej (∆) is locally convexnear the points whereℓλj

meets the boundary∂∆.

13

We leta, z = ℓλj∩ ∂∆ and assumez lies on the top boundary. By assumption,z is either a hidden

Delzant corner or a fake corner. Let us consider the vectors(u, v) = Bz. Becausez belongs to the top--boundary, the vectoru must be directed to the lefthand side ofz andv to the righthand side. Since thetransformationt~ej acts only on the right half-space (and there it acts asT ), the transformed edges oft~ej (∆)at z are directed along(u, Tv). By assumptiondet(u, Tv) is either0 or 1, which implies local convexityatz.

Now consider the “bottom boundary” at the pointa. By assumption the polygon is already locallyconvex ata (which meansdet(u, v) > 0), and a quick calculation shows that the action oft~ej may onlymake it even “more” convex.

It is easy to see that the properties of the lemma are preserved by theG-action. Thus we can state thefollowing definition.

Definition 4.3 Let [∆w] be a semitoric polygon as in Definition 2.5, and suppose that∆w is a representative

of the form(∆, (ℓλj

)sj=1, (ǫj)sj=1

)with all ǫj ’s equal to+1. Then [∆w] is called aDelzant semitoric

polygon(of complexitys) if the polygon∆ equipped with the vertical linesℓλjsatisfies the hypothesis of

Lemma 4.2. ⊘

We denote byDPolygs(R2) ⊂ WPolygs(R

2)/Gs × G the space of Delzant semitoric polygons ofcomplexitys, wheres <∞.

The following observation is a consequence of the construction of the homeomorphismf in Section 2.2.

Lemma 4.4. The semitoric polygon in item (iii) of Definition 2.8 is a Delzant semitoric polygon.

In addition, note also that for any representative∆ of the semitoric polygon[∆w] in Definition 2.8, andfor eachi ∈ 1, . . . , mf as in item (iv) of Definition 2.8, the heighthi satisfies the inequality

0 < hi < length(∆ ∩ ℓi). (4.1)

This is because by (2.5) we havehi := µ(mi) − mins∈ℓi∩∆ π2(s), whereµ is a toric momentum map forthe system(M, ω, (J, H)) corresponding to∆. Now, sinceµ(mi) is a point in the interior of∆, along theline ℓi, expression (4.1) follows.

4.2 Main Theorem

The following definition describes a collection of abstractingredients. As we will see in the theorem fol-lowing the definition, each such a list of elements determines one, and one only one, integrable system ona symplectic4-manifold (which is not necessarily a compact manifold, butwe can characterize preciselywhen it is in terms of one of the ingredients of the list). Moreover, this integrable system is of semitorictype.

In the definition the termR[[X, Y ]] refers to the algebra of real formal power series in two variables,andR[[X, Y ]]0 is the subspace of such series with vanishing constant term,and first termσ1X+σ2 Y withσ2 ∈ [0, 2π).

Definition 4.5 A semitoric list of ingredientsconsists of the following items.

(i) An integer number0 ≤ mf <∞.

(ii) An mf -tuple of Taylor series((Si)∞)mf

i=1 ∈ (R[[X, Y ]]0)mf .

14

(iii) A Delzant semitoric polygon[∆w] of complexitymf , as in Definition 4.3.

We denote the representative∆w of [∆w] by(∆, (ℓλj

)mf

j=1, (ǫj)mf

j=1

).

(iv) An mf -tuple of numbers(hj)mf

j=1 such that0 < hj < length(∆ ∩ ℓi) for eachj ∈ 1, . . . ,mf.

(v) A (Gmf× G)-orbit of (∆w, (kj)

mf

j=1), where(kj)mf

j=1 is a collection of integers.⊘

Now we are ready to state the main theorem, the proof of which is contructive and, in view of Section 2and Lemma 4.4, gives a recipe to construct all semitoric integrable systems up to isomorphisms.

Theorem 4.6.For each semitoric list of ingredients, as in Definition 4.5,there exists a4-dimensional simplesemitoric integrable system(M, ω, (J, H)), such that the list of invariants (i)-(v) of(M, ω, (J, H)) as inDefinition 2.8 is equal to this list of ingredients. Moreover, M is compact if and only the polygon in (iii) iscompact.

4.3 Classification of4-dimensional semitoric systems

Consequently, putting Theorem 4.6 together with Theorem 2.9 proved in [17], we obtain the classificationof integrable systems in symplectic4-manifolds.

Theorem 4.7(Classification of4-dimensional semitoric integrable systems). For each semitoric list of in-gredients, as in Definition 4.5, there exists a4-dimensional simple semitoric integrable system with listofinvariants equal to this list of ingredients, c.f. Definition 2.8. Moreover, two4-dimensional simple semitoricintegrable systems are isomorphic if, and only if, they are constructed from the same list of ingredients.

5 Proof of Main Theorem

Let(∆, (ℓλj

)sj=1, (ǫj)sj=1

)be a representative of[∆w] with all ǫj ’s equal to+1. The strategy is to use the

glueing procedure of Section 3 in order to obtain a semitoricsystem by constructing a suitable singular torusfibration above∆ ⊂ R

2.For j = 1, . . . ,mf , let cj ∈ R

2 be the point with coordinates

cj = (λj , hj +min(π2(∆ ∩ ℓλj))). (5.1)

Because of the assumption onhj , all pointscj lie in the interior of the polygon∆. We call these pointsnodes. We denote byℓ+j the vertical half-line throughcj pointing upwards. We call these half-linescuts.

We have divided the proof of the theorem in a preliminary step, three intermediate steps and a conclusivestep. In the preliminary step we construct a convenient covering of the polygon∆.

Then we proceed as follows. First we construct a “semitoric system” over the part of the polygon awayfrom the sets in the covering that contain the cutsℓ+j ; then we attach to this “semitoric system” the focus-focus fibrations i.e. the models for the systems in a small neighborhood of the nodes. Third, we continue toglue the local models in a small neighborhood of the cuts. The“semitoric system” is given by a proper toricmap only in the preimage of the polygon away from the cuts. We use the results of Section 3 as a steppingstone throughout.

Finally we recover the smoothness of the system and observe that the invariants of the system are pre-cisely the ingredients we started with.

15

Preliminary stage. A convenient covering.— We construct an open cover of the polygon. Because of thediscreteness of the set of vertices of the polygon, and the local compactness ofR2, we one can find an opencover(Ωα)α∈A of ∆ such that the following three properties hold: there existsρ > 0 such that allΩα’sare integral-affine images of the open cubeC := I2 with I =:] − ρ, ρ[, i.e for everyα ∈ A there existsRα ∈ Aff (2,Z), such thatΩα = Rα(C); each vertex of the polygon, and each node, is contained in onlyone open setΩα; two open sets containing a vertex or a node never intersect each other. In fact, if

Ce := C ∩ y > 0, Cee := C ∩ x > 0 ∩ y > 0,

one can assume that, for anyα ∈ A, (1) if Ωα intersects∂∆ but does not contain any vertex thenΩα ∩∆ =Rα(Ce), and that(2) if Ωα contains a Delzant corner, thenΩα ∩ ∆ = Rα(Cee). The first case holds sincealong any edge one can find a primitive vector, and complete itto aZ-basis ofZ2. It remains to composeby a suitable translation to position the image ofCe at the right place. The second case is similar, since at aDelzant corner the primitive vectors of the meeting edges form aZ-basis ofZ2, c.f. Definition 4.1.

First stage.Away from the cuts.— LetA′ ⊂ A be the subset obtained by removing all indices intersectingthe cuts. We construct a semitoric system above

⋃α∈A′ Ωα, by glueing the following local models. LetD

be the open disk inT∗R = R

2 of radius√2ρ, centered at the origin. Consider the following models: the

regular model :Mr := T2 ×C ⊂ T ∗

T2 with momentum map

Fr(x1, x2, ξ1, ξ2) := (ξ1, ξ2);

the tranversally elliptic model :Me := (T1 × I)× D ⊂T∗T1 ×T∗

R, with momentum map

Fe(x1, ξ1, x2, ξ2) := (ξ1, (x22 + ξ22)/2);

and the elliptic-elliptic model :Mee := D× D ⊂T∗R×T∗

R, with momentum map

Fee(x1, ξ1, x2, ξ2) := ((x21 + ξ21)/2, (x22 + ξ22)/2).

Observe thatFr(Mr) = C, Fe(Me) = Ce, andFee(Mee) = Cee. Notice also that these models are all toric,in the sense that the momentum maps generate an effective hamiltonian T

2-action. What’s more, thesemomentum maps are proper for the topology induced on their images.

Given anyΩα,α ∈ A′, we obtain a (singular) proper Lagrangian momentum map overΩα, whose imageis preciselyΩα ∩∆, and which defines an effective HamiltonianT2-action, by the following simple rule :

(a) If Ωα contains no boundary point of∆ and no node, then we chooseMα := Mr, with momentummapFα := Rα Fr. ThenFα is a regular Lagrangian torus fibration.

(b) If Ωα intersects the boundary∂∆ but does not contain any vertex of∆, we chooseMα := Me, withmomentum mapFα := Rα Fe. The set of singular values ofFα is the bounded open line segmentRα(Ce∩y = 0). The two componentsJα, Hα of the momentum mapFα have colinear differentialson the preimageU := F−1

e (Ce ∩ y = 0) underFα of the set of singular values : if(a, b) ∈ R2 is a

vector colinear to this segment, then we have thata dHα = b dJα onU .

(c) If Ωα contains exactly one Delzant corner of∆, we chooseMα :=Mee, with momentum mapFα :=Rα Fee. Then the set of singular values ofFα is the “corner”Rα(∂Cee). If (a, b), (c, d) is an

16

orientedZ-basis ofR2, where(a, b) and (c, d) are directed along the respective directions of theedges of this corner, then the first component ofFα is

Jα = a(x21 + ξ21)/2 + c(x22 + ξ22)/2.

In other words, the numbersa, c are the isotropy weights of theS1-action induced byJα. This isuseful to notice, since this componentJα of Fα will not be modified, near the Delzant corner, by therest of the construction process.

We describe now the transition functions : when∆αβ := Ωα ∩ Ωβ 6= ∅, we want to define a symplec-tomorphism

ϕαβ : F−1α (∆αβ) → F−1

β (∆αβ) such that ϕ∗αβFβ = Fα. (5.2)

For this we use the following notation : whenR ∈ Aff (2, Z), we denote byR the symplectomorphismR : T2 × R

2(= T∗T2) → T

2 × R2 given by(x, ξ) 7→ ((tdR)−1x, Rξ), wheredR is the linear part ofR.

Remark thatξ R = R ξ.

Case 1.If both Fα andFβ are regular models, we let

ϕαβ := R−1β Rα. (5.3)

ThenFβ ϕαβ = Rβ Fr ϕαβ = Fr Rβ ϕαβ = Fr Rα = Fα, i.e. (5.2) holds.

Case 2.If Fα is regular andFβ is transversally elliptic, we introduce the symplectomorphism (symplecticpolar coordinates)

ϕre :Mr ∩ (T1 × R)× (T1 × R∗+) → (T1 × R)× (R2 \ 0) ∩Me

(x1, ξ1, x2, ξ2) 7→ (x1, ξ1,√

2 ξ2 cos(x2), −√

2 ξ2 sin(x2)).

Notice thatϕ∗reFe = Fr. Thus we can define

ϕαβ := ϕre R−1β Rα. (5.4)

We haveFβ ϕαβ = Rβ Fe ϕre R−1β Rα = Rβ Fr R−1

β Rα = Fr Rα = Fα, i.e. (5.2) holds.

Case 3. Similarly, if Fα is regular andFβ is elliptic-elliptic, we introduce the symplectomorphism

ϕree :Mr ∩ (T1 × R∗+)× (T1 × R

∗+) → (R2 \ 0) × (R2 \ 0) ∩Mee

(x1, ξ1, x2, ξ2) 7→(√

2 ξ1 cos(x1), −√

2 ξ1 sin(x1),√2 ξ2 cos(x2), −

√2 ξ2 sin(x2).

)

Againϕ∗reeFee= Fr, and if we define

ϕαβ := ϕree R−1β Rα, (5.5)

(5.2) holds.

Case 4. If both Fα andFβ are transversally elliptic models, then the affine mapRαβ := R−1β Rα is an

17

oriented transformation that preserves the upper half-plane. Thus the horizontal axis is globally preserved,and the vectore1 = (1, 0) is an eigenvector ofdRαβ . SincedRαβ ∈ SL(2, Z), it is of the form

Tk :=

(1 k0 1

)

for somek ∈ Z. HenceRαβ = τu Tk whereτu is the translation by a horizontal vectoru = (u1, 0).Consider the symplectomorphismRαβ(x1, ξ1, x2, ξ2) := (x′1, ξ

′1, x

′2, ξ

′2) ofT∗

T1 ×T∗

R given by

x′1 = x1

ξ′1 = ξ1 + k(x22 + ξ22)/2 + u1

(x′2 +i ξ′2) = ei kx1(x2 + iξ2).

Observe thatFe Rαβ = Rαβ Fe. Now we define

ϕαβ := Rαβ |F−1α (∆αβ)

, (5.6)

and we verifyFβ Rαβ = RβFe Rαβ = RβRαβFe = RαFe = Fα, hence (5.2) holds.

Case 5. If Fα is a transversally elliptic model, whileFβ is elliptic-elliptic, then, as in the previous case, theintersection∆αβ contains a portion of an edge, but not the vertex itself. Thisedge is mapped byRβ fromeither the horizontal or vertical positive axis. Suppose for simplicity that it is the horizontal axis. As before,the affine mapRαβ defined in Case 4 is an oriented transformation that either preserves the upper half-plane,and thus one can construct a symplectomorphismRαβ of T∗

T1 ×T∗

R such thatFe Rαβ = Rαβ Fe.Introduce the symplectomorphism

ϕeee :Me ∩ (T1 × R∗+)× R

2 → (R2 \ 0) × R2 ∩Mee

(x1, ξ1, x2, ξ2) 7→ (√

2 ξ1 cos(x1), −√2 ξ1 sin(x1), x2, ξ2).

Notice thatFee ϕeee= Fe and, whenever both are defined,ϕeee= ϕree ϕ−1re . We define

ϕαβ := ϕeee Rαβ , (5.7)

and verify now routinely thatFβ ϕαβ = Fα, i.e. (5.2) also holds in this case.We have defined the transition mapsϕαβ in the five cases (5.3), (5.4), (5.5), (5.6), and (5.7), and verified

that equation (5.2) holds for each of them. In fact one shouldalso mention that for the non-symmetriccases (5.4), (5.5), and (5.7), we letϕβα := ϕ−1

αβ (this is automatic for the symmetric cases (5.3) and (5.6)).Then it is easy to verify that the cocycle condition if fulfilled. Namely, when the triple intersectionΩαβ ∩Ωβγ ∩ Ωγα is not empty, then

ϕγα ϕβγ ϕαβ = Id.

Thus we can apply the glueing construction, c.f. Theorem 3.11, and obtain a symplectic manifoldMA′

with a surjective mapFA′ :MA′ →

⋃

α∈A′

Ωα ⊂ R2

and, for eachα ∈ A′ ⊂ A, there is a symplectic embeddingια : Mα → MA′ such thatι∗αFA′ = Fα. Sinceall Fα are proper smooth toric momentum maps, so isFA′ .

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Second stage. Attaching focus-focus fibrations.— Fix an integeri, with 1 ≤ i ≤ mf . Using theclassification result of [19], one can construct a focus-focus model associated with an arbitrary Taylor seriesinvariant. Precisely, for each nodeci, there exists a symplectic manifoldMi equipped with a smooth mapFi : Mi → C such that the symplectic invariant of the induced singular foliation is precisely the TaylorseriesS∞. Using the result of [20], one can construct a continuous mapµi : Mi → Di, whereDi ⊂ R

2

is some simply connected open set around the origin, that is asmooth proper toric momentum map outsideµ−1i (ℓ), whereℓ := (0, y) | y > 0. In factµi = gi Fi, for some homeomorphismgi : C → Di that is

smooth outsideℓ, and which preserves the first component : it is of the form

gi(x, y) = (x, fi(x, y)).

This construction depends on the choice of a local toric momentum map for the fibration overC \ ℓ. Herewe choose the privileged momentum map as defined in Section 2.4. We are now in position to add to theindex setA′ all the indicesα ∈ A corresponding to the nodes, and thus defining a new index setA′′. If Ωα

contains the nodeci, we letRα be the matrixTki left-composed by the translation from the origin to thenodeci. Herekj is the integer given as ingredient (v) in the list. We may assume thatΩα = Rα(Di). Thenwe chooseMα :=Mi with momentum mapFα := Rα µi.

Figure 5.1: The piecesMi and the chart diagrams forFα, Fi, gi andRα.

By makingρ small enough, one may assume that allΩβ, β ∈ A′, intersecting an open setΩα containing anode carry regular models. Thus we need to define transition functions between a regular model and a focus-focus model. On∆αβ := Ωα ∩ Ωβ, both momentum mapsFα andFβ are regular. Contrary to all previouscases, the focus-focus modelFα is not explicit, and we cannot simply provide an elementary formula forthe transition mapϕαβ . However, sinceC \ ℓ is simply connected and a set of regular values ofFi, we caninvoke the Liouville-Mineur-Arnold action-angle theoremand assert that there exists a symplectomorphismϕi : F

−1i (C \ ℓ) → T

2 × C ′ ⊂ T ∗T2 = (x, ξ) ∈ T

2 ×R2 such that

Fi = ϕ∗i (hi(ξ)) for some diffeomorphismhi : C

′ → C \ ℓ.

Thenµi = ϕ∗i (gi hi(ξ)). Since bothµi andξ are toric momentum maps for the same foliation, there exists

a transformationHi ∈ Aff (2,Z) such thatgi hi = Hi.

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Thus, ifFα is focus-focus andFβ is regular, we introduce the symplectomorphism

ϕαβ := R−1β RαHi ϕi : F−1

α (∆αβ) → F−1β (∆αβ). (5.8)

We verifyFβ ϕαβ = Fr Rβ ϕαβ = RαHiFr ϕi = Rαµi = Fα, so we have shown (5.2).We can now include these nodal pieces in the symplectic glueing construction using Theorem 3.11,

which defines a symplectic manifoldMA′′ and a proper map

FA′′ : MA′′ →⋃

α∈A′′

Ωα ⊂ R2.

HoweverFA′′ is not smooth everywhere, but it is a smooth toric momentum map outside the preimages ofthe cutsℓ+j (j = 1, . . . ,mf ).

Third stage. Filling in the gaps.— Here we add the open setsΩα that were covering the cutsℓi byswitching these lines on the other side. Letti := tℓλi as in Section 2.2. The cutℓ+i is invariant underti. Theopen setsti(Ωα), α ∈ A \ A′′ form a cover ofℓi ∩ ti(∆). Within the geometry of the new polygonti(∆),each of these open sets can be associated with either a regular model, a transversally elliptic model, or anelliptic-elliptic model (indeed, under the transformation ti, a fake corner disappears, and a hidden Delzantcorner unhides itself.)

Thus we can add these to our glueing data, which amounts to equip each such open setΩα with themodel(Mα, t

−1i Fα), where(Mα, Fα) is determined as before, but for the transformed polygonti(∆).

The transition maps are defined with the same formulas as before, taking into account that the mapRα

is now a piecewise affine transformation. The cocycle conditions remain valid as well.Doing this for all indicesi, because all theFα are continuous and proper, by Theorem 3.11, we obtain a

smooth symplectic manifoldM =MA equipped with a proper, continuous mapµ = FA

µ :M →⋃

α∈A

Ωα ⊂ R2, (5.9)

whose image is precisely∆.However, the mapµ is a proper toric momentum map only outside the cutsℓi. In other words,µ fails

to be smooth along the cutsℓi. (Note that in the symplectic glueing construction, Theorem 3.11, we did notmake any smoothness assumption on theFα, nor made any conclusion on the smoothness ofF ).

Fourth and final stage. Recovering smoothness.— In this step we compose the final momentum mapµ in (5.9) on the left by a suitable homeomorphism in order to make it smooth. LetΩα be the open setcontaining the nodeci. Let hi = g−1

i : Di → C. The maphi is a bilipschitz homeomorphism fixing theorigin and a smooth diffeomorphism outside the positive vertical axis. It is of the form

hi(x, y) = (x, ηi(x, y)).

Sincehi is orientation preserving,∂ηi∂y (x, y) > 0 for all (x, y) ∈ Di. Letδi > 0 be such that[−2δi, 2δi]2 ⊂

Di and consider the vertical half-stripSδi := [−δi, δi]× [−δi, ∞[.

Claim 5.1. There exists a functionηi : Di → C such that

(1) ηi(x, y) = ηi(x, y) for all (x, y) ∈ Di ∩ Sδi ;

20

(2) ηi(x, y) = y for all (x, y) ∈ Di \ S2δi ;

(3) ∂ηi∂y (x, y) > 0 for all (x, y) ∈ Di.

In order to show this recall that iff : A→ R is smooth andA ⊂ U ⊂ R2 is closed, thenf has a smooth

extension tof : U → R whereU is open, see for example [22, Lem. 5.58 and Rmk. below it]. Letus applythis fact in our situation. LetAδi := (Di ∩ Sδi) ∪ (Di \ Int(S 3δi

2

)), which is a closed subset ofDi ⊂ R2,

and letηi : Aδi → R be the smooth function given by

ηi(x, y) =

ηi(x, y) if (x, y) ∈ Di ∩ Sδi ;

y if (x, y) ∈ Di \ Int(S 3δi2

). (5.10)

Figure 5.2: The setAδi := (Di ∩ Sδi) ∪ (Di \ Int(S 3δi2

)), on whichηi is defined.

BecauseAδi ⊂ Di, andDi is bounded, there exists a constant0 < ci < 1 such that∂ηi∂y > c onAδi

and hence∂ηi∂y

> ci onAδi . Let ζi :=∂ηi∂y

− ci : Aδi → R, which by assumption is strictly positive. Bythe above factζi extends to a smooth functionGi : Di → R. Because the proof of the fact preserves non-negativity, andζi > 0, we have thatGi ≥ 0. By possibly shrinking the size ofDi we can assume thatDi isa disk of radiusri > 0 centered at the origin. LetXδi := [−ri, −3δi

2 ] ∪ [3δi2 , ri], Yδi := [−δi, δi], Zδi :=

[−3δi2 , −δi]∪ [δi,

3δi2 ] and letνi1 : Xδi → R andνi2 : Yδi → R be the functions given byνi1(x) := −ηi(x, 0)

and

νi2(x) := ηi(x, −3δi2)−

∫ −3δi2

0(Gi(x, t) + ci) dt,

where we are using the convention∫ b

ah = −

∫ a

bh whena > b. Becauseηi andGi are smooth functions,

νi1 andνi2 are also smooth. Letβi : [−ri, ri] → R be a smooth extension of the functionXδi ∪ Yδi → R

defined byνi1 onXδi and byνi2 onYδi , which again exists by a partitions of unity argument.

21

Consider the functionηi : Di → R given by

ηi(x, y) := βi(x) +

∫ y

0(Gi(x, t) + ci) dt.

Becauseβ is a smooth extension ofνi1 andνi2, andG is smooth,ηi is smooth. We claim thatηi|Aδi(x, y) =

ηi(x, y) if (x, y) ∈ Aδi . First assumex ∈ Yδi , and moreover that−ri ≤ y ≤ −3δi2 . BecauseGi is an

extension ofgi we have that

ηi|Aδi(x, y) = νi2(x) +

(∫ −3δi2

0(Gi(x, t) + ci) dt+

∫ y

−3δi2

∂ηi∂y

(x, t) dt),

and hence by the fundamental theorem of calculus, and using the definition ofνi2 we obtain that

ηi|Aδi(x, y) = νi2(x) +

( ∫ −3δi2

0(Gi(x, t) + ci) dt+ (ηi(x, y)− ηi(x, −

3δi2)))= ηi(x, y). (5.11)

The remaining subcases within the case ofx ∈ Yδi are when−δi ≤ y ≤ 0, which follows by the samereasoning as in (a) using the formula forνi1 instead ofνi2, the case of0 ≤ y ≤ ri, which is trivial because theextension is defined by the original function therein, and the case of−3δi

2 ≤ y ≤ −δi, in which(x, y) /∈ Aδi

so there is nothing to prove. The case ofx ∈ Xδi follows by the same type of argument as the case ofYδi .The case ofx ∈ Zδi is immediate because the extension is defined by the originalfunction therein.

Applying again the fundamental theorem of calculus, because the functionsνi1, νi2, β

i do not depend ony, we have that

∂ηi∂y

= Gi + ci, (5.12)

which is strictly positive sinceGi ≥ 0 andci > 0. Because (5.12) and (5.11) hold we in turn have, in viewof the definition (5.10) ofη, that properties 1, 2, 3 are satisfied. This concludes the proof of Claim 5.1

LetΩi := Di ∪ (x, y) | y < 2δi. Because of the properties 1, 2, 3 ofηi, the map

hi : (x, y) 7→ (x, ηi(x, y))

coincides withhi in Sδi , while it is equal to the identity outsideS2δi . Thus we can extend it toΩi by lettingit to be the identity outsideDi ∪ S2δi . We call this extensionhΩi

. Consider the map

hΩi:= hΩi

t−10 ,

wheret0 is the piecewise affine maptℓ with ℓ being the positive vertical axis. Int0(Ω ∩ Sδi), it it equal tohi t−1

0 , which is now smooth outside the negative vertical axis (this follows from [20, Thm. 3.8]; also fromthe fact that it is the homeomorphism that one obtains in the construction of the generalized momentum mapt0 gi Fi = t0 µi: this amounts to switching the cut downwards.) Using the claim at the beginning ofthis step upside-down we can modifyhΩi

in Ωi ∩ y > δi in such a way that we can then extend it to besmooth ont0(y > δi). We obtain a homeomorphism ofR2 that we call(hR2)i.

Define the mapϕi : R2 → R

2 by

ϕi := Rα (hR2)i t0 R−1α .

22

Becauseϕi is a composite of homeomorphisms, it is a homeomorphism. Moreover, outside ofS2δi we havethat

ϕi = Rα (hR2)i t0 R−1α = Rα (hΩi

t−10 ) t0 R−1

α ,

and sincehΩiis the identity outside ofS2δi we conclude thatϕi is the identity map outsideS2δi . Now let

ϕ : R2 → R2 be the piecewise defined map

ϕ(x, y) :=

ϕi(x, y) if (x, y) ∈ S2δi ;(x, y) otherwise.

(5.13)

Since eachϕi is a homeomorphism, and equal to the identity outside ofS2δi , the formula (5.13) defines ahomeomorphism.

Claim 5.2. The mapF : M → R2 defined byF := ϕ µ is proper, and smooth everywhere.

The properness claim is immediate sinceϕ is a homeomorphism andµ is proper.In order to show thatF is smooth, consider the mapFi : M → R

2 defined as a compositeFi := ϕi µ,where recallµ is the map (5.9). By definition ofϕ, we have thatF |Sδi

= Fi, and hence to prove the claim

it suffices to show that eachFi is smooth. To prove this, we distinguish three cases.

Case 1:in a neighborhood ofci. In the neighborhoodΩα of ci sent byR−1α into [−δ, δ]2, we have that

(hR2)i t0R−1α = hΩi

t0R−1α = hΩi

t−10 t0R

−1α = hiR

−1α .

Recall thaty∗αµ = Fα = Rα µi. Therefore one can write, in the preimage byµ of this neighbourhood,y∗α(Fi) = y∗α(hi µi) = Fi. SinceFi is smooth, it follows thatFi is smooth inΩα.

Case 2:away from the cutℓi. LetΛi :=⋃

j 6=i µ−1(ℓj) ⊂ R

2. We have that

(hR2)i t0R−1α = hΩ t0R

−1α = hΩR

−1α on the set(Rα t−1

0 )((x, y) | y < −δi/2),

which by construction is smooth on this set. ThusFi has the same degree of smoothness asµ on the setµ−1((Rα t−1

0 )((x, y) | y < −δ/2)). Note that the setµ−1((Rα t−1

0 )((x, y) | y < −δi/2))

doesnot containµ−1(ℓi). The same argument applies to the analogue subsets ofM corresponding to the re-gions (x, y) |x < −δi/2 and (x, y) |x > δi/2. On the subset ofM corresponding to the region(x, y) | y > δi/2, the map(hR2)i is smooth by construction. Hence the mapFi is smooth onM \ Λi.

Case 3:along the cutℓi, away fromci. Remark thatt0R−1α = R−1

α ti. By construction ofµ above theopen setsΩβ covering the cutℓi, we have thaty∗β µ = t−1

i Fβ . Hence

y∗β((hR2)i t0R−1α µ) = y∗β ((hR2)i Fβ) on the setµ−1(Ωβ),

and this expression defines a smooth map. ThusFi is smooth.

Hence putting cases 1, 2, 3 together we have shown thatFi is smooth onµ−1(Ωβ) for all Ωβ coveringthe cutℓi, and elsewhere,Fi is as smooth asµ. This concludes the proof of Claim 5.2.

Write F := (J, H). We then have the following conclusive claim.

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Claim 5.3. The symplectic manifold(M, ω) equipped withJ andH is a semitoric integrable system.Moreover, the list of invariants (i)-(v) of the semitoric integrable system(M, ω, (J, H)) is equal to the listof ingredients (i)-(v) that we started with. Finally,M is a compact manifold if and only∆ is compact.

Let us prove this claim. We know from Claim 5.2 thatF is smooth. Since the first componentJis obtained from glueing proper maps, it follows from Theorem 3.11 thatJ is proper. What’s more, theHamiltonian flow ofJ is everywhere periodic of period2π because it is true in any local pieceMα. ClearlyJ, H = 0, since it is a local property. It is also easy to see that the only singularities ofF come fromthe singularities of the modelsFα, for the glueing procedure does not create any additional singularities.Now, near any elliptic critical value, the homeomorphismµ is a local diffeomorphism, soF has the samesingularity type as the elliptic modelFα. Finally, near a node we have checked in the proof of Claim 5.2thatF is precisely equal to the modelFi, and hence possesses a focus-focus singularity. Thus, provided weshow thatM is connected,(J, H) is a semitoric system.

Let us now consider its invariants (the connectedness ofM will follow).

(i) As we mentioned, the singularities ofF are only elliptic, except for the nodesc1, . . . , cmfabove each

of which we have constructed a focus-focus singularity. Hence we havemf focus-focus singularities.

(ii) Each focus-focus singularity was constructed by glueing a semi-local model with prescribed Taylorseries invariant(Si)∞. Since this Taylor series is precisely a semi-local symplectic invariant, it isunchanged in the glued system(M, F ).

(iii) Thus we have a completely integrable system onM that defines an integral affine structure (withboundary) on the image ofF , except at the nodesci. For any choice of vertical half cuts(ℓi, ǫi), thegeneralized momentum polygon is the image of the affine developing map. But the momentum mapµ, outside the focus-focus fibres, is precisely such a developing map and its image, by the glueingprocedure, is the polygon∆. Hence the semitoric polygon invariant ofF is the orbit of∆w. (SeeLemma 2.3.)

Notice that this shows that the image ofµ is connected, which implies that the total spaceM , obtainedby glueing above the image ofµ, is connected as well.

(iv) It follows directly from (iii) above and the definition of the nodescj in (5.1) that the volume invariantdefined in (2.5) is equal to(h1, . . . , hmf

).

(v) We calculate the twisting indices of our semitoric system with respect to the fixed polygon∆ or,which amounts to the same, with respect to the toric momentummapµ. By definition, thejth twist isthe integerkj such that

dµ = T kjdµj,

whereµj is the privileged momentum map of the focus-focus fibration abovecj . From the secondstage of the construction, we know that

µ = Fα = Rα µj = τ T kj µj,

whereτ is some translation. Hencedµ = T kjdµj , and thuskj = kj.

Thus we see that we could prove the second part of the claim because our construction is by symplec-tically glueing local pieces with the appropriate ingredients as in Definition 4.5. This is an advantage ofconstructing by glueing local pieces rather than, for example, a global reduction on a larger space.

This concludes the proof of Claim 5.3, and hence the proof of the theorem.

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[18] Symington, M., Four dimensions from two in symplectic topology. pp. 153-208 inTopology andGeometry of Manifolds (Athens, GA, 2001). Proc. Sympos. Pure Math., 71, Amer. Math. Soc., Prov-idence, RI, 2003.

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Alvaro PelayoUniversity of California–BerkeleyMathematics Department, 970 Evans Hall# 3840Berkeley, CA 94720-3840, USA.E-mail: apelayo@math.berkeley.edu

Vu Ngo.c SanInstitut de Recherches Mathematiques de RennesUniversite de Rennes 1Campus de Beaulieu35042 Rennes cedex (France)E-mail: san.vu-ngoc@univ-rennes1.fr

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