Towards a higher-dimensional construction of stable/unstable Lagrangian laminations

We generalize some properties of surface automorphisms of pseudo-Anosov type. First, we generalize the Penner construction of a pseudo-Anosov homeomorphism and show that a symplectic automorphism which is constructed by our generalized Penner construction has an invariant Lagrangian branched submanifold and an invariant Lagrangian lamination, which are higher-dimensional generalizations of a train track and a geodesic lamination in the surface case. Moreover, if a pair consisting of a symplectic automorphism $\psi$ and a Lagrangian branched surface $B_{\psi}$ satisfies some assumptions, we prove that there is an invariant Lagrangian lamination $\mathcal{L}$ which is a higher-dimensional generalization of a geodesic lamination.


I
By the Nielsen-Thurston classification of surface diffeomorphisms, an automorphism ψ : S ∼ → S of a compact oriented surface S is of one of three types: periodic, reducible or pseudo-Anosov [2], [10].A generic element of the mapping class group of S is of pseudo-Anosov type.
Let us assume that ψ is of pseudo-Anosov type.For any closed curve C ⊂ S, it is known that there is a sequence {L m } m∈N of closed geodesics such that L m is isotopic to ψ m (C) for all m ∈ N, and {L m } m∈N , as a sequence of closed subsets, converges to a closed subset L.Moreover, L is a geodesic lamination.The definitions of a lamination, a geodesic lamination and a Lagrangian lamination are the following: Definition 1.1.
(1) A k-dimensional lamination on an n-dimensional manifold M is a decomposition of a closed subset of M into k-dimensional submanifolds called leaves so that M is covered by charts of the form I k × I n−k where a leaf passing through a chart is a slice of the form I k × {pt}.(2) A 1-dimensional lamination L on a Riemannian 2-manifold (S, g) is a geodesic lamination if every leaf of L is geodesic.(3) A n-dimensional lamination L on a symplectic manifold (M 2n , ω) is a Lagrangian lamination if every leaf of L is a Lagrangian submanifold.
For more details, we refer the reader to [4,Chapter 15].In [3], Dimitrov, Haiden, Katzarkov, and Kontsevich defined the notion of a pseudo-Anosov functor of a category.A pseudo-Anosov map ψ on a compact oriented surface S induces a functor, also called ψ, on the derived Fukaya category D π F uk(S, ω), where ω is an area form of S. In [3], the authors showed that ψ is a pseudo-Anosov functor.
In [3,Section 4], the authors listed a number of open questions.One of them is to find a symplectic automorphism ψ on a symplectic manifold M of dimension greater than 2 which has invariant transversal stable/unstable Lagrangian measured foliations.A slightly weaker version of the question is to define a symplectic automorphism ψ with invariant stable/unstable Lagrangian laminations.
The goal of the present paper is to prove Theorems 1.2-1.5, which answer the latter question.
Theorem 1.2.Let M be a symplectic manifold and let ψ : M ∼ → M be a symplectic automorphism of generalized Penner type.Then, there exists a Lagrangian branched submanifold B ψ such that if L is a Lagrangian submanifold which is carried (resp.weakly carried) by B ψ , then ψ m (L) is carried (resp.weakly carried) by B ψ for all m ∈ N.
In Sections 2 and 3, we will explain the terminology that appears in the statement of Theorem 1.2, i.e., a symplectic automorphism of generalized Penner type, a Lagrangian branched submanifold, and the notion of "carried by".
We would like to remark that Theorem 1.2 is for ψ of generalized Penner type.However, there would be a generalized version of Theorem 1.2, which we do not prove in the current paper.
Theorem 1.3.Let M be a symplectic manifold and let ψ : M ∼ → M be a symplectic automorphism of generalized Penner type.Then, there is a Lagrangian lamination L such that if L is a Lagrangian submanifold of M which is carried by B ψ , then there is a Lagrangian submanifold L m for all m ∈ N, which is Hamiltonian isotopic to ψ m (L) and converges to L as closed sets as m → ∞.
We will also prove the following generalization of Theorem 1.3 Theorem 1.4.Let ψ : M ∼ → M be a symplectic automorphism and let B ψ be a Lagrangian branched submanifold such that ψ(B ψ ) is carried by B ψ .Moreover, if the associated branched manifold B ψ admits a decomposition into singular and regular disks, then there is a Lagrangian lamination L such that if L is a Lagrangian submanifold of M which is carried by B ψ , then there is a Lagrangian submanifold L m for all m ∈ N, which is Hamiltonian isotopic to ψ m (L) and converges to L as closed sets as m → ∞.
The associated branched manifold and singular/regular disks will be defined in Sections 3 and 4. Theorem 1.5.Let M be a plumbing space of Penner type and let η : M ∼ → M be the involution associated to M .Let assume that a transversal pair L 1 , L 2 ⊂ M of Lagrangian submanifolds satisfies the following: (1) η(L i ) = L i for i = 0, 1.
(2) Let Li = L i ∩ M i .Then, Li is a Lagrangian submanifold of M such that L0 and L1 are not isotopic to each other.
(3) L 0 ∩ L 1 = L0 ∩ L1 , (4) L 0 and L 1 are not isotopic to each other.Then, where HF k (L 1 , L 2 ) denotes Z/2-graded Lagrangian Floer homology over the Novikov ring of characteristic 2 and i( L1 , L2 ) denotes the geometric intersection number of L1 and L2 in the fixed surface M .
In Section 5, we will explain the terminology that appears in the statement of Theorem 1.5, i.e., a plumbing space M of Penner type, the involution η associated to M , and the fixed surface M of M .This paper consists of 5 sections.In Section 2, we review plumbing spaces and generalized Dehn twists.We will prove Theorem 1.2 in Section 3 and Theorems 1.3 and 1.4 in Section 4. In Section 5, we will prove Theorem 1.5.

P
In this section, we will review plumbings of cotangent bundles and generalized Dehn twists, partly to establish notation.
2.1.Plumbing spaces.Let α and β be oriented spheres S n .We describe how to plumb T * α and T * β at p ∈ α and q ∈ β.Let U ⊂ α and V ⊂ β be small disk neighborhoods of p and q.Then, we identify T * U and T * V so that the base U (resp.V ) of T * U (resp.T * V ) is identified with a fiber of T * V (resp.T * U ).
To do this rigorously, we fix coordinate charts ψ 1 : U → R n and ψ 2 : V → R n .Then, we obtain a compositions of symplectomorphisms A plumbing space P (α, β) of T * α and T * β is defined by is a symplectomorphism, P (α, β) has a natural symplectic structure induced by the standard symplectic structures of cotangent bundles.
Since the plumbing procedure is a local procedure, we can plumb a finite collection of cotangent bundles of the same dimension at finitely many points.For convenience, we plumb cotangent bundles of oriented manifolds.
Note that we can replace f by If we plumb T * α and T * β at one point using g, this plumbing space is symplectomorphic to the previous plumbing space P (α, β), which is plumbed using f .However, if we plumb at more than one point, then by replacing f with g at a plumbing point, the plumbing space will change.
(1) A plumbing data is a collection of pairs of non-negative integers (a i,j , b i,j ) for all 1 ≤ i ≤ j ≤ m and collections of distinct points (2) A plumbing space P (α 1 , • • • , α m ), with the given plumbing data, is given by where the equivalence relation ∼ is defined as follows: First, choose small disk neighborhoods U i,j k ⊂ α i of p i,j k and V i,j k ⊂ α j of q i,j k and orientationpreserving coordinate charts ψ i,j k : U i,j k ∼ → R n and φ i,j k : (3) A plumbing point is an identified point p i,j k ∼ q i,j k ∈ P (α 1 , • • • , α m ).

Generalized Dehn twist. Let
where (u; v) ∈ R n+1 × R n+1 and < u, v > is the standard inner product of u and v in R n+1 .Moreover, let 0 k be the origin in R k .We fix a Hamiltonian function µ(u; v) = v on T * S n \ S n .Then, µ induces a circle action on T * S n \ S n given by Let r : [0, ∞) → R be a smooth decreasing function such that r(0) = π and r(t) = 0 for all t ≥ for a small positive number .If ω 0 is the standard symplectic form of T * S n , we define a symplectic automorphism τ : (T * S n , ω 0 ) Let (M 2n , ω) be a symplectic manifold and let L S n be a Lagrangian sphere in M .By the Lagrangian neighborhood theorem [12], there is a neighborhood N (L) ⊃ L and a symplectomorphism φ : T * S n ∼ → N (L).We define a generalized Dehn twist τ L along L as follows: Note that the support of τ L is contained in N (L).From now on, a generalized Dehn twist will just be called a Dehn twist.Remark 2.2.In this paper, we will use two specific Dehn twists τ, τ : T * S n ∼ → T * S n which are defined by Equation (2.1) and two functions r, r : [0, ∞) → R. The function r (resp.r) defining τ (resp.τ ) satisfies the above conditions in addition to r(t) = π for all t ≤ 2 (resp.r (0) < 0).Two Dehn twists τ and τ are equivalent in the sense that τ • τ −1 is a Hamiltonian isotopy.
Dehn twists have been studied extensively by Seidel.For example, Seidel [9] proved the following theorem.
Theorem 2.3.Let α be a Lagrangian sphere and β be a Lagrangian submanifold of a symplectic manifold M .If α and β intersect transversally at only one point, β#α is Lagrangian isotopic to τ α (β) where β#α is a Lagrangian surgery of α and β.
Proof of Theorem 2.3.We use T * α and T * β to indicate neighborhoods of α and β inside M = P (α, β).Let p be the intersection point of α and β.Then, T * p α = β ∩ T * α.The closure of T * p α is denoted by D − p ; we use D to indicate that this is a disk and the subscript p means that p is the center of D − p .The meaning of the negative sign in D − p will be explained in the next section.Since τ α is supported on There exists φ : Then, Thus, τ α (D + p ) is given by spinning with respect to p and φ.Similarly, we can construct a Lagrangian isotopy connecting τ α (β) and β#α by spinning.This completes the proof.

L
In Section 3.1, we will define Lagrangian branched submanifolds.In Section 3.2, we will introduce a construction of a fibered neighborhood of a Lagrangian branched submanifolds.In Section 3.3, we will defined the notion of "carried by" by using a fibered neighborhood.In Section 3.4, we will introduce the generalized Penner construction.Finally, we will give a proof of Theorem 1.2 in Section 3.5.

3.1.
Lagrangian branched submanifolds.Thurston [11] used train tracks, which are 1-dimensional branched submanifolds of surfaces, and defined the notion of "carried by a train track".In this subsection, we generalize train tracks.
The generalization of a train track is an n-dimensional branched submanifold of a 2n-dimensional manifold.We define the n-dimensional branched submanifolds with local models, as Floyd and Oertel defined a branched surface in a 3dimensional manifold in [5], [7].For our definition, we need a smooth function s : R → R such that s(t) = 0 if t ≤ 0 and s(t) > 0 if t > 0. Definition 3.1.Let M 2n be a smooth manifold.
(1) A subset B ⊂ M is an n-dimensional branched submanifold if for every p ∈ B, there exists a chart φ p : (1) At every point p of a branched submanifold B, the tangent plane T p B is welldefined.Moreover, if B is Lagrangian, then T p B is a Lagrangian subspace of T p M .(2) A point on the branch locus is (a smooth version of) an arboreal singularity in the sense of Nadler [6].Example 3.3.
(1) Every train track of a surface equipped with an area form is a Lagrangian branched submanifold.( 2) Let (M, ω) be a symplectic manifold and let L 1 and L 2 be two Lagrangian submanifold of M such that The Lagrangian surgery of L 1 and L 2 at p will be denoted by In Section 3.3, we will define the notion of "carried by" which appears in Theorems 1.2 -1.4.In order to define the notion of carried by, we will construct a fibered neighborhood first in Section 3.2.

Construction of fibered neighborhoods.
Let B be a Lagrangian branched submanifold.A fibered neighborhood N (B) of B is, roughly speaking, a codimension zero compact submanifold with boundary and corners of M , which is foliated by Lagrangian closed disks which are called fibers.Definition 3.4.A fibered neighborhood of B is a union ∪ p∈B F p , where {F p | p ∈ B} is a family of Lagrangian disks satisfying (1) for any p ∈ B, F p B, (2) for any p, q ∈ B, either We will now give a specific construction of a fibered neighborhood N (B).Remark 3.5.By the Lagrangian neighborhood theorem [12], for any Lagrangian submanifold L of M , there exists a small neighborhood N (L) of the zero section of T * L such that a symplectic embedding i L : N (L) → M is defined on N (L).Without loss of generality, we assume that N (L) is a closed neighborhood.Than, Fibration over L( ).First, we will construct fibers near the branch locus.For each connected component of Locus(B), we choose a small closed Lagrangian neighborhood L( ) of .Then, by Remark 3.5, there exists a symplectic embedding By choosing a sufficiently small L( ), without loss of generality, the following hold: If p ∈ B is close to the branch locus, in other words, there is a connected component of Locus(B) such that p ∈ B ∩ U ( ), then there exists If p ∈ , then, Moreover, by choosing a sufficiently small L( ), for every p ∈ B ∩ U ( ), Equation (3.3) holds.
After possibly renaming U ( ), from now we assume that If p ∈ B ∩ U ( ), then there is a unique q ∈ L( ) such that p ∈ F q .We define F p := F q .Thus, for p ∈ B which is close to Locus(B), i.e., p ∈ U ( ) for some connected component of Locus(B), we can define a fiber F p at p. Fibration over S \ ∪ U ( ).If p ∈ B \ ∪ U ( ), then there is a sector S of B containing p. Since S is Lagrangian, there is an embedding i S : N (S) → M .We can assume that N (S) is small enough, so that For any sector S, let S • := S −∪ Int U ( ).Then, S • is a Lagrangian submanifold with boundary.The boundary of S • is a union of S( ) := S ∩ ∂ U ( ) .We fix a tubular neighborhood of S( ), which is contained in S • , and identify the tubular neighborhood with S( )×[0, 1).For convenience, we will pretend that S( )×[0, 1] ⊂ S and S( ) × {0} = S( ).
If p ∈ S • does not lie in any S( ) × (0, 1), then we set F p := B p .
To do this, we need the following facts: First, by the definition of F (p0,0) , F (p0,0) ∩ i S N (S) is a Lagrangian disk which contains (p 0 , 0), and is transversal to B at (p 0 , 0).Also, B (p0,0) is also a Lagrangian disk which contains (p 0 , 0), and is transversal to B.
From those two facts, we define a function f (p0,t) : B (p0,t) → R as follows: The first arrow comes from the parallel transport induced by g.
There is a map, It is easy to check that h(p 0 , t) = (p 0 , t).Moreover, h is the associated (time 1) flow of the Hamiltonian vector field of Finally, we construct F (p0,t0) by setting A fibered neighborhood N (B) is given by the union of fibers, i.e., N (B) = ∪ p∈B F p .Note that the construction of N (B) is not unique because the construction depends on some choices, including the choices of L( ) and a Riemannian metric g.

3.
3. Associated branched manifolds and the notion of "carried by".We constructed a fibered neighborhood N (B).From now on, we will define a projection map defined on N (B), in order to define the notion of "carried by".
First, we define the associated branched manifold B * of B.

F
. Black curves are part of a Lagrangian branched submanifold and marked points denote ; in (a), U ( ) is shaded blue, the vertical line segments are fibers; (b) fiber F p for p / ∈ S( ) × (0, 1] is in green; and in (c), fiber F p for p ∈ S( ) × (0, 1] is in red Let π : N (B) → B * denote the quotient map.Before defining the notion of "carried by", we note that B * is not contained in M .Moreover, since B * is a branched manifold, we can define the branch locus and sectors of B * as follows: Definition 3.7.
(1) A sector of B * is a connected component of {p ∈ B * | p has a neighborhood which is homeomorphic to R n }.
(2) A branch locus of B * is the complement of all the sectors.
(1) The construction of N (B) depends on the choices of a Riemannian metric, a closed neighborhood of Locus(B), and so on.Thus, fibered neighborhoods N (B) of B are not unique.However, B * is unique as a branched manifold since B and B * are equivalent as branched manifolds.
In the rest of this paper, when it comes to a Lagrangian branched submanifold B, we will consider a triple (B, N (B), B * ) with an arbitrary choice of N (B).Moreover, for any triple (B, N (B), B * ), the projection map is denoted by π for convenience.
(2) A fibered neighborhoodN (B) is a union of fibers, i.e., N (B) = ∪ p∈B F p .In the equation, B is an index set.However, there is a possibility of having In N (B), the blue, red, and green represent π −1 (S 0 ), π −1 (S 1 ), and π −1 (S 2 ), where S i is the corresponding sector of B * ; (b) represents F x where x is in the branch locus of B * in (a).two distinct points p, q ∈ B such that F p = F q .From now on, we will use B * as an index set.In other words, we replace F p by π −1 (π(p)).By abuse of notation, F x denotes π −1 (x) for all x ∈ B * .
(3) Let x be a branch point of B * .Then, there are sectors S 0 , S 1 , for some l such that x ∈ Si for every i = 0, 1, From now on, we define the notion of "carried by".If a Lagrangian submanifold L (resp.a Lagrangian branched submanifold L) is contained in N (B), there is a restriction of π to L (resp.L).For convenience, we will simply use π instead of π| L : L → B * .Definition 3.9.Let L be a Lagrangian submanifold (resp.L be a Lagrangian branched submanifold) of N (B).
( In the rest of this paper, if L is weakly carried by B, then we will assume that L ⊂ N (B) and L is minimally singular with respect to B.
Note that the notion of "carried by" used by Thurston in [10] is our notion of "strongly carried by".Thurston showed that for a pseudo-Anosov surface automorphism ψ : S ∼ → S, there is a 1-dimensional branched submanifold τ which is called a train track such that ψ(τ ) is strongly carried by τ .
Our higher-dimensional generalization is slightly weaker, i.e., for some symplectic automorphism ψ : (M, ω) ∼ → (M, ω), we construct a Lagrangian branched submanifold B ψ such that ψ(B ψ ) is weakly carried by B ψ .In other words, we allow non-transversality at countably many point p ∈ B ψ .However, we allow only one type of non-transversality.In the rest of the present subsection, we will describe the unique type of non-transversality.Definition 3.11.Let L be weakly carried by B. A singular component V of π : L → B is a connected component of the set of all singular points of π.Example 3.12.Let M * be a symplectic manifold T * R n R 2n equipped with the canonical symplectic form.The zero section is weakly carried by B * and π * has only one singular component there exists an open neighborhood U of V and a symplectomorphism φ : Generalized Penner construction : Let M be a symplectic manifold.A symplectic automorphism ψ : so that ψ is a product of positive powers of Dehn twists τ i along α i and negative powers of Dehn twists σ j along β j , subject to the condition that every sphere appear in the product.
A Lagrangian sphere α i (resp.β j ) is called a positive (resp.negative) sphere since only positive powers of τ i (resp.negative powers of σ j ) appear in ψ.Remark 3.15.
(1) In Theorems 1.2 and 1.3, we can assume that the symplectic manifold M is a plumbing space.Every τ i (resp.σ j ) is supported on a neighborhood of α i (resp.β j ), which is denoted by T * α i (resp.T * β j ).Thus, ψ is supported on the union of T * α i and T * β j .By the transversality condition α i β j , we can identify the union with a plumbing space Thus, it is suffices to prove Theorems 1.2 and 1.3 on the plumbing space P , which we take to be connected.(2) In [8], the Penner construction required that A and B fill the surface S, i.e., the complement of A ∪ B is a union of disks and annuli, one of whose boundary components is a component of ∂S.In the current paper, we do not require the analogue of the filling condition since we only construct an invariant Lagrangian branched submanifold and an invariant Lagrangian lamination, not an invariant singular foliation on all of M .
In the rest of this subsection, we define a set of Lagrangian branched submanifolds in a plumbing space We start from the simplest plumbing space, having one positive and one negative sphere intersecting at only one point.Example 3.16.Let α and β be n-dimensional spheres and let M be a plumbing P (α, β) which is plumbed at only one point p.Let β# p α be the Lagrangian surgery of α and β at p such that β# p α τ α (β) σ −1 β (α).See Figure 5, which represents the case n = 1.The cross-shape is the plumbing space P (α, β), where α is the horizontal line and β is the vertical line.
The neck N p at p connecting α and β is the closure of (β# p α) − (α ∪ β).In On a general plumbing space with positive spheres α i and negative spheres β j , we similarly construct Lagrangian branched submanifolds.More precisely, given a plumbing point p, N p , D + p , D − p are the closures of Then, we construct a Lagrangian branched submanifold B by setting There are 2 N possible choices of B, where N is the number of plumbing points.Let B be the set of all 2 N Lagrangian branched submanifolds constructed above.

Proof of Theorem 1.2. In this subsection, let
let τ i (resp.σ j ) be a Dehn twist along α i (resp.β j ), and let ψ be of generalized Penner type.
In the rest of the paper, we assume that every Dehn twist τ i and σ j satisfies the following: (1) τ i (resp.σ j ) is supported on a small neighborhood T * α i (resp.T * β j ) of α i (resp.β j ).(2) τ i (resp.σ j ) agrees with the antipodal map on α i (resp.β j ).
We define the following: In words, D+ p (resp.D− p ) is a neighborhood of an antipodal point of p in α i (resp.β j ).We are assuming that D ± p and D± p are sufficiently small so that they are disjoint to each other.
Recall that B is the set of Lagrangian branched submanifolds defined in Section 3.2.Lemma 3.17.For all k, there exists a function Proof.In this proof, τ k is given by Equation (2.2) and τ : where φ is an identification of T * S n and a neighborhood of α k .
Given B ∈ B, B admits the following decomposition: where D p is either D + p or D − p .This follows from Equations (3.4) and (3.5).We prove the first statement for τ k ; the proof for σ −1 j is analogous.Our strategy is to apply τ k to α i , β j , N p , D± p , and D ± p .We claim the following: where Dp is (i) Since τ k agrees with the antipodal map on α k , τ k (α k ) = α k and τ k (α k ) is strongly carried by α k .Moreover, since τ k is supported on T * α k , α i does not intersect the support of τ k for all i = k.Thus, τ k (α i ) agrees with α i and τ k (α i ) is strongly carried by itself.The same proof applies to τ k (β j ).
(ii) and (iii) are proved in the same way.
(iv) We compute τ k (D − p ) and τ k (N p ) by spinning with respect to p and φ.We assume φ((1, 0 n ; 0 n+1 )) = p without loss of generality.Using the notation from Section 2, D − p and N p are contained in ∪ y∈S n−1 φ(W y ).Thus, The restriction τ | Wy is a Dehn twist on W y T * S 1 along the zero section.Thus, we obtain Figure 6, which represents intersections , and φ(W y ) ∩ τ k (N p ). Equation (3.9) and Figure 6 imply that F .In the left picture, the blue curve represents D − p and the red curve represents N p ; in the middle picture, the red curve represents τ k (N p ); and in the right picture, the blue curve represents τ k (D − p ). Proof.We can assume that L is contained in an arbitrary small neighborhood of B. Then, we apply a Dehn twist τ k as we did in the proof of Lemma 3.17.The details are similar to the proof of Lemma 3.17.(1) Note that a singular value of π : ψ m (L) → B * , which is defined in Section 3.1, can be moved by isotoping ψ m (L).(2) We observe that every singular value of π :

C L
In this section, we will prove Theorems 1.3 and 1.4..

Singular and regular disks.
In order to prove Theorems 1.3 and 1.4, we would like to construct a stable Lagrangian lamination L of a symplectic automorphism ψ from a Lagrangian branched submanifold B ψ .One of the difficulties is that singular components occur naturally.In order to control the singularities, we introduce singular and regular disks.
In general, we assume that B * ψ , the associated branched manifold, can be decomposed into the union of a finite number of disks S i D n , which are called singular disks, and R j D n , which are called regular disks, i.e., (1) each singular disk S i is either a closed disk contained in the interior of a sector of B * ψ or a closure of a sector, (2) S i ∩ S j = ∅ for any i = j, (3) every singular value of π : ψ m (B ψ ) → B ψ after weakly fibered isotopy lies in ∪ i Si for all m ∈ N, where Si is the interior of S i , (4) each regular disk R j is obtained by cutting up a closure of a sector minus ∪ i Si , (5) S i and R j (resp.R i and R j for i = j) meet only along their boundaries.
In Section 4.2, we will define braids b(L, S i ) for a Lagrangian L, which is carried by B ψ , and a singular disk S i .By Theorem 1.2, there exist sequences of braids b(ψ m (L), S i ) mıN , and we will construct limits of those braid sequences as m → ∞.We then extend the limit lamination to a Lagrangian lamination of π −1 (S i ) in Section 4.3, and a Lagrangian lamination of π −1 (R j ) in Section 4.4.

Remark 4.2.
(1) In Section 4.3 (resp.Section 4.4), we will construct a Lagrangian lamination Figure 4 is an example.If S 1 in Figure 4 is a singular disk, then π −1 (S 1 ) is the union of the red box in Figure 4 (a) and F x .
In order to construct a symplectomorphism, we will consider the following: Let D be a largest Lagrangian disk such that By Remark 3.5, there exists a symplectic embedding i D : N (D) → M .It is easy to construct a vector field on i D (N (D)), whose (time 1) flow moves i D (N (D) ∩ T * p D) to F π(p) for any p ∈ Int(D).Moreover, the vector field is a symplectic vector field, i.e., the flow is a symplectomorphism, and .
From now on, we assume that a symplectic automorphism ψ is of generalized Penner type until the end of Section 4.3.

Decomposition of B *
ψ for ψ of generalized Penner type.We will now explain how to decompose B * , the associated branched manifold of B ∈ B, into the union of specific singular and regular disks.Note that B is defined in Section 3. 4 into regular disks.In order to do this, we use a symplectic submanifold W 2n−2 ⊂ M 2n , which is defined as follows: For each α i (resp.β j ), there is an equator C αi (resp.C βj ) S n−1 such that (1) for any plumbing point p ∈ α i (resp.β j ), p lies on C αi (resp.C βj ), (2) if p ∈ α i ∩ β j , then T * C αi ≡ T * C βj near p.Note that the equators on a Lagrangian sphere α i (resp.β j ) are defined using an identification φ αi : α i ∼ → S n (resp.φ βj : β j ∼ → S n ).Thus, by choosing proper identification φ αi and φ βj , we can assume the existence of C αi and C βj .Then, We cut (4.11)along π(W ).These are the regular disks R k .Each R k is a manifold with corners, where the corners are at R k ∩ π(W ) ∩ S l .

Braids. Consider the decomposition of B *
ψ into specific singular and regular disks as in the previous subsection.In this subsection, for a given compact Lagrangian submanifold L which is carried by B ψ , we define a sequence of braids b(ψ m (L), S i ) corresponding to ψ m (L) over the boundary of each singular disk S i of B * ψ .Lemma 4.4 gives an inductive description of the sequences b(ψ m (L), S i ).We will end this subsection by constructing limits of b(ψ m (L), S i ) as m → ∞.
For a singular disk S, π −1 (∂S) = ∪ p∈∂S F p is a D n -bundle over ∂S S n−1 .Note that we use D n to indicate a closed disk, and we will use Dn to indicate an open disk.
Let ϕ : π −1 (∂S) ∼ → S n−1 × D n be a bundle map.If L is a Lagrangian submanifold which is carried by B ψ , then, for all p ∈ ∂S, ϕ(L ∩ F p ) is a finite collection of isolated points in D n ; recall that π : L → B * has no singular value on ∂S.Thus, ϕ(L ∩ π −1 (∂S)) can be identified with a map from ∂S S n−1 to the configuration space Conf l (D n ) of l points on D n where l = l(L, S), i.e., a braid.
We explained that L ∩ π −1 (∂S) could be identified with a braid.Since L is a Lagrangian submanifold of M , the braid corresponding to L ∩ π −1 (∂S) satisfies a symplectic property.The symplectic property is the following: For the bundle map From now on, we will define the braids on the boundary of a singular disk S. Let f : S n−1 → Conf l (D n ) for some l.In other words, there are maps Note that Br ∂S is a set of closed subsets of π −1 (∂S) and independent of ϕ.
We define an equivalence relation on Br Recall the functions F τi and F σ −1 j in Lemma 4.4 are defined in Lemma 3.17.
Proof of Lemma 4.4.In Steps 1-3, we prove Lemma 4.4 for a particular example; this is just for notational simplicity.In Step 4, we briefly describe how to prove the general case.
(resp.φ βj ) is a symplectomorphism from T * S 2 to a neighborhood of α (resp.β j ).The neighborhood of α (resp.β j ) will be denoted by T * α (resp.T * β j ).Remark 4.5.Recall that τ is a Dehn twist on T * S n which agrees with the antipodal map on a neighborhood of the zero section S n .
Note that the singular disk for b 4 is S + q and the singular disk for b 4 is S − q , i.e., two singular disks have the same center but different sign.However, for i = 4, the singular disks for b i and b i have the same center and the same sign.
For convenience, the singular disk of B ψ (resp.
Since σ −1 2 is supported on T * β 2 , a small neighborhood of β 2 , b i and b i are the same braid in Br ∂Si for i = 1, 2, 3, and 5.We will explain how b 6 is constructed.
We can obtain σ −1 2 (B ψ ) by spinning with respect to q in T * β 2 , i.e., σ −1 2 (B ψ ) is the union of curves in 2-dimensional submanifold φ β2 (W y ) over y ∈ S 1 .Recall that the spinning and W y are defined in Section 2.2.
By spinning blue, red, and green points in Figure 7, we obtain σ −1 2 (B ψ )∩π −1 (∂S 6 ).Let B, R, and G be the circles obtained by spinning blue, red, and green points respectively. Since ) consists of three connected components, which are neighborhoods of B, R, and G.Each connected component will be called N (B), N (R), and N (G).
The group of strands in N (B) is given by σ −1 2 (L)∩N (B).we will consider . One of the main difficulties is that the action of σ −1 2 on σ 2 N (B) is not simple.To make it simpler, we will construct a Hamiltonian isotopy Φ t , so that there is a disk ∩ L corresponds to the group of strands in N (B).We construct Φ t as follow: Let H t : R 4 → R 4 be a Hamiltonian isotopy given by and let δ : [0, ∞) → R be a smooth decreasing function such that δ(x) = π 2 for all x < 1 and δ(x) = 0 for all x > 2. We choose a neighborhood U ⊂ β 2 of σ −1 2 (q) and a Darboux chart φ q : T * U ∼ → R 4 such that φ q (σ −1 2 (q)) is the origin.We remark that T * β 2 denotes a neighborhood of β 2 in M , which is symplectomorphic to the cotangent bundle of β 2 .Thus, for a subset U of β 2 , one can assume that T * U is a subset of M .
For convenience, let φ q (x) = (x 1 ; x 2 ) where x i ∈ R 2 .Then, there is a Hamiltonian isotopy where c i is a positive constant and • is the standard norm on R 2 .
To visualize, we use D + q and D− q instead of S 4 and S 6 in Figure 8. Figure 8 ) in the left, middle, and right pictures respectively.By choosing proper c i , we obtain a small disk D ).More precisely, we obtain a disk D ⊂ D + q which is in blue in the left of Figure 8. Blue curves in the middle and right of Figure 8 represent On a small neighborhood of D B , σ −1 2 agrees with the antipodal map of φ β2 (T * β 2 ) T * S 2 , as we mentioned in Remark 4.5.Then, we obtain a map The first identification φ4 is the restriction of ϕ 4 : Remark 4.6.
(1) Note that ϕ 6 Im(f Similarly, for the groups of strands in N (R) and N (G), one can obtain two functions f 2 and f 3 on 1 × D 2 in the same way.Then, the images Im(f 2 ) and Im(f 3 ) correspond to Thus, f 1 explains the contribution of b4 , and f 2 and f 3 explain the contribution of b4 on the construction of b 6 .(2) The constant r 1 is determined by specific choices of an identification φ β2 : T * S 2 ∼ → T * β 2 , the fixed Dehn twist τ in Remark 2.2, and so on.However, r 1 has to be smaller than 1.This is because Im(f 1 ), Im(f 2 ), and Im(f 3 ) are mutually disjoint, since they corresponds to N (B), N (R), and N (G) respectively.
The strands of b 6 which are contained in N (B) correspond to We will prove that L∩π −1 (∂D B ) represents the same braid with b4 .We can assume that there is no singular value of π on S 4 \ D B .Then, ϕ 4 ( b4 ) and φ4 (L ∩ π −1 (∂D B )) represent the same braid in S 1 ×D 2 because of non-singularity on S 4 \D B .Thus, in S 1 × D 2 , f 1 (ϕ 4 ( b4 )) and f 1 ( φ4 (L ∩ π −1 (∂D B ))) represent the same braid.It proves that ϕ −1 6 (f 1 (ϕ 4 ( b4 ))) and the group of strands in N (B) represent the same braid in Br ∂S 6 .
For the groups of strands in N (R) and N (G), we obtain the following maps f 2 and f 3 in the same way, (θ, x, y) → (θ + π, r 0 cos θ + r 2 x, r 0 sin θ + r 2 y), where r 0 and r 2 are positive constants which are smaller than 1.Remark 4.8.
In the same way that we proved that f 1 ( b4 ) and the group of strands in N (B) represent the same braid in Br ∂S 6 , we can prove that f 2 ( b4 ) (resp.f 3 ( b4 )) and the group of strand in N (R) (resp.N (G)) represent the same braid in Br ∂S 6 .Then, b 6 is represented by f 1 ( b4 ) f 2 ( b4 ) f 3 ( b4 ).Note that we are abusing notation for convenience as we mentioned in Remark 4.7.
The situation for b 4 is analogous.We obtain three maps g 1 , g 2 and g 3 in the same way.At the end, b 4 is represented by g 1 ( b4 ) g 2 ( b4 ) g 3 (b 6 ).This proves Lemma 4.4 for the case of σ −1 2 .Note that maps f i and g j are given by specific maps acting on S 1 × D 2 , but we would like to consider them as maps on Br ∂S k for some k.Then, we summarize the effect of σ −1 2 as a matrix . Remark 4.9.We remark that in surface theory, we can do linear algebra on weights, but in a higher-dimensional case, we cannot do linear algebra with the matrix Σ 2,B ψ , because there is no module structure on Br ∂Si .
Step 3 (Effects of τ 0 on B ψ ).We use the same notation, i.e., b 1 , • • • , b 6 denote the braids on singular disks S − i of B * ψ , and so that the singular disk corresponding to b i has the same center as the singular disk corresponding to b i .We also use bi and b i , S i and S i , ϕ i and ϕ i to indicate representatives of braids, singular disks in B ψ and F τ0 (B ψ ), identifications induced by fixed coordinate charts.
If a map is of the same type to f 1 , in other words, if the map is given by a translation on S 1 and a scaling on D 2 , let the map be of scaling type.This is because the formula defining the map is given by a scaling on fibers.The maps of scaling type explain how braids b(L, S ± p ) or b(L, S± p ) contribute on the construction of braids b(δ(L), S ± δ(p) ) or b(δ(L), S± δ(p) ) through δ π −1 (S ± p ) , where δ is a Dehn twist.If a map is of the same type to f 2 (resp.f 3 ), let the map be of the first (resp.second) singular type.This is because they are related to a creation of new singular component.The maps of the first and second singular types explain how the braid b(L, δ(S p )) contributes on the construction of braid b(δ(L), S± δ(p) ).To summarize, if b i contributes the construction of b j and if the center of a singular disk corresponding to b i is either the same point or the antipodal point of the center of the singular disk corresponding to b j , maps of these three types explain the contribution of b i on the construction of b j .Note that the center of a singular disk is defined in Remark 3.19.
The maps of these three types explain the effects of σ −1 2 on B. However, to explain the effects of τ 0 on B ψ , we need maps of one more type.This is because α has two plumbing points, unlike β i has only one plumbing point.Thus, when we apply τ 0 , b i can contribute on b j even if the centers of singular disks corresponding to b i and b j are neither the same nor antipodals of each other.For example, L ∩ π −1 π(N p ) is stretched by τ 0 .The stretched part τ 0 L ∩ π −1 (π(N p )) has intersection with π −1 (S 4 ) and π −1 (S 5 ).Thus, b 4 has some strands corresponding to τ 0 (L∩π −1 (π(N p )))∩π −1 (∂S 4 ) These strands are the contribution of b1 on the construction of b 4 .Similarly, b1 contributes on the construction of b 5 , and b4 contributes on the constructions of b 1 and b 2 .
To describe the contribution of b1 on b 4 , without loss of generality, we assume that there is no singular value for grangian disks on π −1 ( S4 ) and b1 contributes on b 4 by adding strands near τ 0 (N p )∩ π −1 (∂S 4 ) which are not braided to each other.The number of the added strands is the same as the number of strands of b1 .In the same way, b1 contributes on the construction of b 5 .
To describe the contribution of b1 on b 4 as a map acting on which represents a trivial braid having the same number of strands with b1 .This is because we only need the number of the strands in b1 , not the way b1 is braided.We construct a Hamiltonian isotopy Φ t by extending a Lagrangian isotopy connecting τ 0 (N p ) ∩ π −1 (∂S 4 ) and as we did in Remark 4.8.Then, one obtains where r 0 is a positive constant number less than 1.Then, h t ( b• 1 ) represents the same braid to the strands in b 4 , which correspond to τ 0 (L ∩ π −1 (π(N p ))).We recall that we are abusing notation as mentioned in Remark 4.7.
Similarly, if b i contributes the construction of b j and if the center of a singular disk corresponding to b i is neither the same point nor the antipodal point of the center of the singular disk corresponding to b j , then the contribution of b i on b j can be described by a map like h t .If a map is of the same type with h t , let the map be of trivial type, because a map of trivial type adds strands which are not braided with each other.
Then, we can describe the effect of τ 0 on B ψ as a matrix Among the entries, h 1 , i, and id are of scaling type, h 2 and h 3 are of the first and second singular types, and h t and i t are of trivial type.
Step 4 (General case).A ψ of generalized Penner type is a product of Dehn twists.In the general case, when we apply ψ, each Dehn twist is followed by a Hamiltonian isotopy as σ −1 2 is followed by is a Dehn twist, and Φ i,t is a Hamiltonian isotopy which follows δ i .
After applying the Hamiltonian isotopy, the effect of a Dehn twist τ i (resp.σ −1 j ) on B ∈ B is described by a matrix T i,B (resp.Σ j,B ), whose entries are sums of maps of four types.As we mentioned in Step  Taking the limit of a braid sequence.We have obtained braid sequences {b(ψ m (L), S i )} m∈N , where L is carried by B ψ , and S i is a singular disk of B * ψ .In the rest of this subsection, we construct a limit of {b(ψ m (L), S i )} m∈N as m → ∞.
We argue with the above example, i.e., For convenience, let and let singular disks S + p , S+ p , S− p , S + q , S+ q , and S− q of B be S 1 , • • • , S 6 .Using notation from the proof of Lemma 4.4, we have matrices T 0,B , Σ 1,B , and Σ 2,B .Then, we obtain Ψ = T 0,B • Σ 1,B • Σ 2,B by defining a multiplication of maps as a composition of them.Note that a product of two arbitrary matrices is not defined.For example, an input of Σ 2,B and an output of T 0,B are tuples of braids on singular disks of B * .Thus, Σ 2,B • T 0,B is defined.However, T B • Σ 2,B is not defined since an input of T 0,B is a tuple of braids on singular disks of B * , but an output of Σ 2,B is a tuple of braids on singular disks of B * .

Let bi be a representative of b
Then, B i,m is the disjoint union of solid tori.More precisely, each solid torus in B i,m is the image of a composition of 3m-maps, appearing in Ψ m .Conversely, for each composition of 3m-maps appearing in Ψ m , the image is a solid torus contained in B i,m .The radii of solid tori in B i,m are decreasing exponentially and are converging to zero as m → ∞.
Since bi,m ⊂ B i,m and B i+1,m ⊂ B i,m for all m ∈ N, there is a limit Thus, B i,∞ is the union of infinite strands as a subset of π −1 (∂S i ) and as a sequence of closed sets in π −1 (∂S i ).
(1) We have constructed a sequence of specific representatives { bi,m } m∈N such that lim m→∞ bi,m = B i,∞ .For the purposes of extending the lamination to the singular and regular disks in Sections 4.3 and 4.4, we assume that the limit B i,∞ is a specific closed subset in π −1 (∂S i ).
(2) Each strand of B i,∞ corresponds to an infinite sequence {f m } m∈N such that

4.3.
Lagrangian lamination on a singular disk.Let ψ be of generalized Penner type and let L be a Lagrangian submanifold which is carried by B ψ .In Section 4.2, on each singular disk S i , we gave an inductive description of a sequence {b(ψ m (L), S i )} m∈N .There is a limit B i,∞ of the sequence.Moreover, the limit B i,∞ depends only on ψ and B i,∞ is independent to L. In this present subsection, we will construct a Lagrangian lamination In this case, we will construct a Lagrangian lamination from B i,∞ = B i,∞ ∩ π −1 ( Si ).Thus, we will simply say that the Lagrangian lamination is constructed from B i,∞ ∩ π −1 ( Si ).Lemma 4.12.Let ψ be of generalized Penner type.For each singular disk S i of B ψ , there is a Lagrangian lamination , where B i,∞ is the limit of a braid sequence, which depends only on ψ. (2) If L is a Lagrangian submanifold of M which is carried by B ψ , then for every m ∈ N, there is a Lagrangian submanifold L m which is Hamiltonian isotopic to ψ m (L) and L m ∩ π −1 ( Si ) converges to L i as a sequence of closed subsets.
Proof.Let ψ be of generalized Penner type, i.e., Dehn twist τ i or σ −1 j .We will use similar notation with the previous subsection, for example, S i denotes a singular disk of B ψ , Ψ denotes a matrix corresponding to ψ, ϕ i : π −1 (∂S i ) ∼ → S n−1 × D n denotes the identification induced from the fixed coordinate chart on S i , and so on.
In this proof, first, we will construct L i ⊂ π −1 ( Si ) satisfying the first condition, i.e., L i ∩ π −1 (∂S i ) = B i,∞ ∩ π( Si ).Then, we will show that the constructed L i satisfies the second condition.
Construction of L i .As we mentioned in Remark 4.10, a strand of Note that we are assuming that ψ = δ 1 • • • • • δ l for some positive number l.For each strand {f m } m∈N of B i,∞ ∩ π −1 ( Si ), we will construct a Lagrangian submanifold of π −1 ( Si ) whose boundary agrees with the strand {f m } m∈N .First, for a given strand {f m } m∈N , let us assume that f 1 is of trivial type.Then, the strand is identified with a straight curve where x i is a constant.A subsequence {f m } m≥2 determines constants x i .Let Then, ϕ i (D) is a Lagrangian disk in π −1 ( Si ), whose boundary agrees with the strands {f m } m∈N .Second, let us assume that f 1 is not of trivial type, but there exists m ∈ N such that f m is of trivial type.Let k > 1 be the smallest number such that f k is of trivial type appearing in {f m } m∈N .Then, where k 0 ∼ = k(mod l), is of generalized Penner type such that B ψ has a singular disk Sj , so that Bj,∞ , the limit of the braid sequence corresponding to ψ and Sj , has a strand identified with {f m } m≥k .Thus, there is a Lagrangian disk in π −1 ( Sj ) whose boundary agrees with {f m } m≥k .Let D denote the Lagrangian disk in π −1 ( Sj ).Then, there is a connected component of whose boundary is {f m } m∈N , where Φ i,t is a Hamiltonian isotopy mentioned mentioned in Section 4.1.
To summarize, if there is at least one map of trivial type in {f m } m∈N , then we have a Lagrangian submanifold in π −1 ( Si ), whose boundary agrees with {f m } m∈N .Let L i,∞ be the union of those Lagrangian submanifolds.Finally, let us assume that for all m ∈ N, f m is not of trivial type.Then, for all k ∈ N, we will construct a sequence {f k m } m∈N for each k ∈ N, satisfying If there is a sphere having 2 or more plumbing points, there exists a sequence {f k m } m∈N for all k ∈ N.This is because of the following: We note that the finite sequence {f t } 1≤t≤kl explains a contribution of the braid on a singular disk S i0 on the construction of the braid on a singular disk S j0 when one applies ψ k .In other words, from the view point of Remark 4.6, there is a con- ), where p is the center of S i0 and N p is the neck at p, such that the boundary of the connected component is the image of If there exists a sphere having 2 or more plumbing points, the Dehn twist along the sphere appears in ψ, because of our assumption that every Dehn twist appears in ψ.Let δ i be the Dehn twist.For any plumbing points p and q of the sphere, δ i (π −1 (π(N p ))) intersects π −1 (S + q ), if the sphere is positive, or π −1 (S − q ), otherwise.Thus, there is a map of trivial type in ∆ i , the matrix corresponding to δ i .
For a sufficiently large N , (ψ ).We can prove this by observing that (ψ ) = ∅ for some sufficiently large N .Thus, there is a finite sequence of functions {g j } 1≤j≤N l+i such that g j is an entry of ∆ j , the matrix corresponding to δ j , where j ∼ = j(mod l), and the image of ).Moreover, we can extend the finite sequence {g j } 1≤j≤N l+i to an infinite sequence {g j } j∈N such that {g j } j∈N appears in B i,∞ .Then, by setting f k kl+j = g j , we prove the existence of {f k m } m∈N .We obtain a strand {f k m } k∈N for each k ∈ N.These strands converge to {f m } m∈N as k → ∞.Moreover, by definition of L i,∞ , the boundary of L i,∞ contains strands {f k m } m∈N for all k ∈ N. Thus, the strand {f m } m∈N is contained in the boundary of L i , where If there is no sphere with 2 or more plumbing points, then there is only one positive and one negative sphere intersecting at only one point because we are working on a connected plumbing space.For the case, we can construct a Lagrangian lamination L on M by spinning.Then, L i := L ∩ π −1 (S i ) is a Lagrangian lamination which we want to construct.Remark 4.13.We note that, if there is no sphere with 2 or more plumbing points, then we can construct L without using singular and regular disks.
Convergence to L i .Let L m := ψ m H (L). We defined ψ H in Step 4 of the proof of Lemma 4.4.We will prove that L m ∩ π −1 ( Si ) converges to L i .
First, we will show that . Thus, there exists the limit If we equip a Riemannian metric g on M , then d , where d H is the Hausdorff metric induced from g, converges to zero as m → ∞ because of the same reason that Since for a large N 0 , L N0 intersects π −1 (S j ) for any singular disk This proves Equation (4.13).Let L i be the limit in Equation (4.13).Second, we show that L i is L i .By the construction of L i , we know that Because every connected component of L i has a boundary on ∂S i , this shows L i = L i .4.4.Lagrangian lamination on a regular disk.In the previous subsection, we constructed Lagrangian laminations on singular disks, when boundary data for singular disks are given.In the present subsection, first, we will define boundary data for a regular disk.Then, second, we will construct Lagrangian laminations on regular disks from the given data.Finally, we will prove Theorem 1.3 as a corollary of Lemmas 4.12 and 4.15.
Before defining the boundary data, we remark that, by Remark 4.2, π −1 ( Ri ) is symplectomorphic to DT * D, where D is a disk.
We define a data c j,m on the boundary of a regular disk R j for ψ m (L), by setting c j,m := L m ∩ π −1 (∂R j ).
We defined L m in the proof of Lemma 4.12.Note that c j,m is a closed subset, not a class of a closed subset.
To obtain a limit of c j,m , we consider as we did in Section 4.2.Since ψ . Moreover, C j,m is the union of solid tori (resp.S n−1 ×D n ) in π −1 (∂R j ) for the case n = 2 (resp. of general n).If a symplectic manifold M is equipped with a Riemannian metric g, we can measure the radii of solid tori in C j,m .The radii decrease exponentially and converge to zero as m → ∞, because of the same reason that radii of solid tori comprising B i,m decrease exponentially and converge to zero as m → ∞ in Section 4.2.Then, the limit of c j,m is given by The next step is to smooth R j .A regular disk R j has corners.We will replace R j with a smooth disk R j .This is because, at the end, a Lagrangian lamination will be given as graphs of closed sections.By smoothing R j , it will be easier to handle closed sections.
To smooth R j , we subtract a tubular neighborhood N (∂R j ) ⊂ R j from R j .Let R j := R j \ N (∂R j ).Then, R j is a smooth disk.We replace R j with R j .To finish smoothing, we need to determine boundary data for R j from c j,m .
Each connected component of c j,m can be identified wit a section of a bundle π −1 (∂R j ) over ∂R j .We can extend this section to a closed section of a bundle π −1 (N (∂R j )) over N (∂R j ) by computations.Then, the graph of the extended section is a Lagrangian submanifold of π −1 (N (∂R j )).The boundary of the Lagrangian submanifold on ∂R j makes up the boundary data for R j .
From now, we assume that a regular disk R j is a smoothed disk.Lemma 4.14 claims that for a given data c j,m on a smoothed regular disk R j , we can construct a Lagrangian submanifold N j,m ⊂ π −1 ( Ri ) such that ∂N j,m = c j,m ∩ π −1 ( Ri ).Lemma 4.14.Let Q be a closed subset of ∂T * D n such that there exists a Lagrangian submanifold L ⊂ T * D n so that L ∩ ∂T * D n = Q and L is a union of Lagrangian disks transverse to fibers.Then, we can construct a Lagrangian submanifold L uniquely up to Hamiltonian isotopy through Lagrangians transverse to the fibers.
To prove Lemma 4.14, we will use the following: in Lemma 4.14, if an identification ϕ : Proof of Lemma 4.14.The proof of Lemma 4.14 consists of two parts, the construction of L and the uniqueness of L.
Construction.We start the proof with the simplest case, i.e., when Q is connected.In other words, Q represents the braid with only one strand.
By fixing coordinate charts on D n , we can write down Q as a section of a disk bundle ∂T * D n over ∂D n , i.e., Then, the simplest case is proved by determining a function φ : The graph of dφ is a Lagrangian submanifold which we would like to find.Note that there are infinitely many φ satisfying the conditions, but the Hamiltonian isotopy class of the graph of dφ is unique through Lagrangians transverse to the fibers.
If Q has 2 or more connected components l i , then we can write l i as a section over ∂D n .For each i, we need to determine functions φ i : D n → R such that dφ i agrees with l i on ∂D n .Moreover, to avoid self-intersection, we need dφ i = dφ j for all i = j everywhere.Then, the union of graphs of dφ i on T * D n is a Lagrangian submanifold L which we want to construct.
We discuss with the simplest non-trivial case, i.e., Q has two connected components l 0 and l 1 , and the dimension 2n = 4. Without loss of generality, we assume that l 0 is the zero section.Then, we can assume that φ 0 ≡ 0. We only need to determine φ 1 such that dφ 1 does not vanish everywhere.
We assume that there exists φ 1 satisfying the conditions.Then, we will collect combinatorial data from φ 1 , and we will construct a function φ1 satisfying conditions, from the combinatorial data.Through this, we will see what combinatorial data we need.We will end the construction part by obtaining the combinatorial data from the given Q.
We mark (r * , θ * ) as a red (resp.blue) point if the above function has a local maxima (resp.minima) at θ * .If r * = 1, there are same number of red/blue marked points on {1} × S 1 , and there are only one red/blue marked point on {r 0 } × S 1 .On [r 0 , 1] × S 1 , we have a collection C of curves shaded red and blue.If a curve in C is not a circle, then the curve has two end points on the boundary of [r 0 , 1] × S 1 .There are exactly two curves connecting both boundary components of [r 0 , 1] × S 1 , and those two curves have end points of the same color.
If we write dφ 1 = f dθ + gdr, then f is zero on curves in C. Since dφ 1 does not vanish, g cannot be zero on the curves.Thus, we can assign the sign of g for each curve.Figure 9 is an example of a collection C.
Conversely, if we have a collection C of curves such that each curve is shaded red and blue and is equipped with a sign, then we can draw a graph of φ1 roughly.This is because, the collection C determines the sign of horizontal directional derivative of φ1 , i.e., d φ1 (∂ θ ) on every point of [r 0 , 1] × S 1 , and vertical directional derivative of φ1 , i.e., d φ1 (∂ r ) on the curves.From these, one obtains a (rough) graph of φ1 .Thus, in order to determine a function φ 1 , it is enough to determine a collection C of curves in [r 0 , 1] × S 1 from the given Q.
For the given Q, we assume that a connected component l 0 of Q is the zero section without loss of generality.For the other connected component l 1 , one has f 1 , g 1 : S 1 → R such that l 1 is the graph of f 1 dθ +g 1 dr on {1}×S 1 = ∂D 2 .We know that Q represents the trivial braid with respect to the standard (x, y)-coordinate of D 2 .Thus, there is an isotopy Γ : where A is a constant.
For every r ∈ [r 0 , 1], let γ r (θ) = Γ(r, θ).Then, γ r is a closed curve in D 2 , for all r.Moreover, Γ is a path connecting γ 1 and γ r0 in the loop space of D2 without touching the origin.
We mark (r, θ) on [r 0 , 1] × S 1 as a red (resp.blue) point if γ r (θ) intersects dr-axis from right to left (resp.from left to right).These marked points comprise curves in [r 0 , 1]×S 1 , and we have a collection C of curves, shaded red and blue, in [r 0 , 1]×S 1 .We know that γ 1 has an even number of intersection points.When r decreases, there is a series of creations/removes of intersection points, which are given by finger moves along dr-axis.Each finger move does not touch the origin.Thus, for a curve in C, every intersection point composing the curve lies on either the positive dr-axis or the negative dr-axis.Then, we can assign a sign for each curve in C. Figure 10 is an example of Γ, corresponding to the case described by Figure 9.The upper left of Figure 10 is γ 1 and the upper right is γ r0 .Through the first arrow, we observe a finger move removing two intersection points.Those two intersection points correspond to m 2 , a local maxima shaded red, and n 2 , a local minima shaded blue.Thus, we obtain a curve connecting m 2 and n 2 in Figure 9.Moreover, the intersection points lie in the negative part of the dr-axis.Thus, we assign a negative sign to the curve.Similarly, we observe there are finger moves removing intersection points.We obtain curves connecting m i and n i for i = 1, 2, and 3 in Figure 9.After the finger moves, there are only two intersection points corresponding to m * and n * , and we obtain curves connecting m 4 (resp.n 4 ) and m * (resp.n * ).
We have constructed a collection C of curves on [r 0 , 1] × S 1 from an isotopy Γ.Thus, we can obtain a function φ 1 : [r 0 , 1]×S 1 → R. In order to complete the proof, we need to extend φ 1 into a small disk with radius r 0 .We have φ 1 (x, y) = Ar sin θ = Ay on the small disk.
The situation for the general case is analogous.If Q has more connected components l i for i = 0, • • • , k, then we have to determine φ i : D 2 → R such that dφ i = l i on ∂D 2 , and dφ i = dφ j for all i = j.We fix an isotopy Γ, and obtain a collection C of curves on [r 0 , 1] × S 1 from Γ.Each curve in C encodes restrictions on dφ i − dφ j for some i and j.More precisely, (φ i − φ j ) has a local maxima (resp.minima) in the horizontal direction, only at a point of a curve shaded red (resp.blue), and (dφ i − dφ j )(∂ r ) has the sign assigned on the curve.For the case of general dimension 2n, we obtain combinatorial data from Q, i.e., a collection of curves on [r 0 , 1]×S n−1 assigned a sign, and construct functions on D n from the combinatorial data.
Uniqueness.Recall that the construction consists of three steps.First, we choose an isotopy Γ connecting Q and the trivial representative of the trivial braid.Then, we obtained a collection C of curves from Γ, such that each curve encodes restrictions on dφ i − dφ j .The last step is to construct a set of functions {φ i : D n → R}.
The construction depends on choices in the first and last steps.More precisely, for the first step, the choice of isotopy Γ is not unique.If we choose an isotopy Γ, then there is a unique collection C.However, a set {φ i } of functions, which is constructed from the collection C, is not unique.We will show that the Hamiltonian isotopy class of L, through Lagrangians transverse to the fibers, is independent to those choices.
First, we discuss the choice in the third step.Let us assume that we have a collection C of curves in [r 0 , 1] × S n−1 and two sets of functions {φ i } i and {ζ i } i satisfying the restrictions encoded by C.Then, by setting η i,t := (1 − t)φ i + tζ i , we obtain a family of sets of functions such that every member of the family satisfies the restrictions encoded by C.
Let L t be the Lagrangian submanifold corresponding to {η i,t } for a fixed t.Then, L t is a Lagrangian isotopy connecting L 0 , corresponding to {φ i }, and L 1 , corresponding to {ζ i }.Since L t is a disjoint union of Lagrangian disks in T * D n , L 0 and L 1 are Hamiltonian isotopic.Thus, the Hamiltonian class of L through Lagrangians transverse to the fibers is independent of the choice of functions for the third step of the construction.
Before discussing the choice of the first step, note that a continuous change on a collection C does not make a change on the Hamiltonian isotopy class.More precisely, let C 0 = {γ 1 , • • • , γ N } be a collection of curves and let {φ i } be a set of functions corresponding to C 0 .If {γ k,t } is a continuous family of curves with respect to t such that γ k,0 = γ k for all k, then we can obtain a continuous family {φ Then, it is easy to check that the Hamiltonian isotopy class of the union of graphs of dφ i,t in T * D n , through Lagrangians transverse to the fibers, is independent to t.
Finally, we will discuss the choice of Γ.Let Γ 0 and Γ 1 be two isotopies obtained from the given Q in the first step.Then, we can understand Γ 0 and Γ 1 as paths on the loop space of the configuration space of Dn .Since the loop space is simply connected, there is a continuous family {Γ t } t∈[0,1] connecting γ 0 and γ 1 .
Let C t be the collection of curves obtained from Γ t and let {φ i } be a set of functions constructed from C 0 .There is {φ i,t } corresponding to C t such that φ i,0 = φ i .Then, if L t is the union of graphs of dφ i,t , then the Hamiltonian class of L t is independent to t.This shows the uniqueness of L, up to Hamiltonian isotopy, through Lagrangians transverse to the fibers.
For a smoothed regular disk R j , there is a sequence of data c j,m for each m ∈ N.
Then, we can construct a sequence of Lagrangian submanifolds N j,m ⊂ π −1 ( Rj ) such that N j,m ∩ ∂π −1 ( Rj ) = c j,m .The following lemma, Lemma 4.15, claims that we can construct N j,m wisely, so that N j,m converges to a Lagrangian lamination N j as m goes to ∞. Lemma 4.15.It is possible to construct N j,m ⊂ π −1 ( Rj ) so that the sequence N j,m converges to a Lagrangian lamination N j ⊂ π −1 ( Rj ) as m → ∞.
Proof.Let the boundary condition c j,m be the set {l 1,m , • • • , l Nm,m }, where l i,m is a connected component of c j,m , or equivalently, l i,m is a strand of the braid represented by c j,m .We defined C j,m as a disjoint union of solid tori in π −1 (∂R j ) at the beginning of the present subsection.Then, we can divide c j,m into a partition, so that l i,m and l j,m are in the same subset if and only if l i,m and l j,m are in the same solid torus (resp.S n−1 × D n for a higher dimensional case) in C j,m .After that, we randomly choose a connected component l s,m from each subset of the partition.By Lemma 4.14, there is φ s,m : R j → R such that dφ s,m = l s,m on ∂R j .Then, the graph of dφ s,m is a Lagrangian disk in π −1 ( Ri ).We can choose a neighborhood N (φ s,m ) of the graph of dφ s,m in π −1 ( Ri ), such that N (φ s,m ) T * D n and N (φ s,m )∩ π −1 (∂R j ) is the torus in C j,m containing l s,m .Moreover, we can assume that d H (N (φ s,m ), the graph of dφ s,m ) < 2r m , where d H is the Hausdorff metric induced by a fixed Riemannian metric.
Let l be a strand of C j,∞ .Then, there is a sequence l im,m ∈ c j,m such that l im,m converges to l.If we construct φ i,m by repeating the above procedure, we know that d H (dφ im,m , dφ in,n ) < 4r max(m,n) .
Thus, dφ im,m converges.Moreover, by assuming that φ i,m (p) = 0 for every i and m, where p is a center of R j , φ im,m converges to a function φ.The graph of dφ is a Lagrangian disk in π −1 ( Rj ) such that whose boundary is l, the stand of C j,∞ .The union of graphs of dφ is the Lagrangian lamination N j which N j,m converges to.
Proof of Theorem 1.3.By Lemma 4.12, there is a Lagrangian lamination L i in π −1 ( Si ) and by Lemma 4.15, there is a Lagrangian lamination N j in π −1 ( Rj ).Moreover, every Lagrangian lamination agrees with each other along boundaries, thus we can glue them.Then we obtain a Lagrangian lamination L in M .

A generalization.
In the previous sections, we assumed that ψ is of generalized Penner type.In the present subsection, we discuss a symplectic automorphism ψ : (M, ω) → (M, ω), not necessarily to be of generalized Penner type, with some assumptions.First, we assume that there is a Lagrangian branched submanifold B ψ such that ψ(B ψ ) is (weakly) carried by B ψ .The proof of Lemma 3.17 carries over with no change.Thus, if a Lagrangian submanifold L is (weakly) carried by B ψ , then ψ(L) is carried by B ψ .
As mentioned in Section 4.1, we assume that B * ψ admits a decomposition into a union of finite number of singular disks S i D n and regular disks R j D n .
Proof of Theorem 1.4.First, we define data on the boundary of each singular and regular disk, in the same way we did for the case of ψ of generalized Penner type.Then, on a regular disk R j , the proofs of Lemma 4.14 and Lemma 4.15 carry over with no change.Thus, we can construct a Lagrangian lamination on π −1 (R j ).
On a singular disk S i , we define the boundary data in the same way.In other words, the boundary data is defined by the isotopy class of ψ m (L) ∩ π −1 (∂S i ).We also can obtain a matrix Ψ, which explains how the sequences of braids are constructed inductively.However, the rest of the proof of Lemma 4.12 does not carry over.This is because in the proof of Lemma 4.12, functions of trivial type have a key role.To use the same proof, we need to show that there are enough functions of trivial type.However, the assumptions cannot imply the existence of enough functions of trivial type.For a singular disk S i , let {f m } m∈N be a strand of the limit braid on S i .We note that each strand can be identified to an infinite sequence of functions.We forget specific functions f m , but remember their types.Then, we obtain a sequence of types.The sequence of types determines the "shape" of strand, for example, how many times the strand is rotated.
We can construct a symplectomorphism φ which is of generalized Penner type such that B φ has a singular disk S so that the limit braid assigned on S has a strand of the same shape.In Section 4.3, we constructed a Lagrangian submanifold L 0 ⊂ π −1 ( S) such that ∂L 0 is the strand.Since π −1 ( S) π −1 ( Si ), we assume that L 0 is a Lagrangian submanifold in π −1 ( Si ) and ∂L 0 has the same shape to the strand which we choose.By scaling and translating L 0 inside π −1 ( Si ), we obtain a Lagrangian submanifold whose boundary agrees with the strand.
The rest of the proof is the same as the proof of Theorem 1.3.

A L F
In this section, we will give an application of the previous sections on Lagrangian Floer homology.More precisely, we will prove Theorem 1.5 and give an example in Section 5.3.

5.1.
Setting.In the present subsection, we will explain terminology in Theorem 1.5.
In Section 5, we assume that our symplectic manifold M is a plumbing space ) α i ∩ α j = ∅, and β i ∩ β j = ∅, for all i = j.

Note that P
From now on, we will define an involution η : M ∼ → M , which is associated to M .
Involution η 0 on T * S n : First, we will define an involution η 0 on T * S n .Let Then, we define η 0 : T * S n ∼ → T * S n as follow: Then, it is easy to check that T * S is the set of fixed points of η 0 , equivalently, η f ixed 0 = T * S.
Involution η associated to M : First, we will construct an involution η αi and η βj on T * α i and T * β j for every i and j.Note that T * α i , T * β j ⊂ M .For each α i , we will choose an embedded circle S αi ⊂ α i such that S αi contains every plumbing point of α i .Then, there is a symplectic isomorphism φ αi : T * S n ∼ → T * α i such that φ αi (S n ) = α i and φ αi (S) = S αi .One obtains an involution η αi : T * α i ∼ → T * α i by setting Similarly, one obtains an involution η βj : T * β j ∼ → T * β j in the same way.Without loss of generality, one can assume that η αi (x) = η βj (x) for every x ∈ T * α i ∩ T * β j .Finally, the involution η : M ∼ → M is defined as follows: We will call η the involution associated to M .(4) L 0 and L 1 are not isotopic to each other.We will compute Z/2-graded Lagrangian Floer homology HF * (L 0 , L 1 ) over the Novikov field Λ of characteristic 2. To do this, we will prove that chain complexes CF * (L 0 , L 1 ) and CF * ( L0 , L1 ) are the same chain complexes.More precisely, we will show that those two chain complexes have the same generators and the same differential maps.
Second, let ∂ (resp.∂) denote the differential map on CF * (L 0 , L 1 ) (resp.CF * ( L0 , L1 )).Then, (#M(p, q; [u], J))T ω([u]) q, where J is an almost complex structure on M , u is a holomorphic strip connecting p and q, and M(p, q; [u], J) is the moduli space of holomorphic strips.We skip the foundational details of the definition of ∂.
One can easily check that η • u is also another holomorphic strip connecting p and q.Let assume that for a holomorphic strip u, the image of u is not contained in M .Then, u and η • u will be canceled together in ∂(p), since the Novikov field Λ is of characteristic 2. Thus, in order to define the differential map ∂, it is enough to count holomorphic strips u such that the image of u is contained in M .
Under the assumptions, HF * (L 0 , L 1 ) = HF * ( L0 , L1 ).Note that the former is defined on M 2n , but the latter is defined on a surface M .Then, Lemma 2.18 of [3] completes the proof.5.3.Example 5.5.In the present subsection, we will prove Lemmas 5.3 and 5.4 in order to slightly weaken the difficulty of applying Theorem 1.5.Then, we will give the Example 5.5.
Before giving the statement of Lemmas 5.3 and 5.4, we will establish notation.In Section 5, M = P (α 1 , • • • , α m , β 1 , • • • , β l ) is a plumbing space of Penner type.Then, as we did in Section 3.4, we can constructed a set B of Lagrangian branched submanifolds of M .
Every Lagrangian branched submanifold B ∈ B is a union of (parts of) α i and β j and Lagrangian connected sums α i and β j .However, there are two possible Lagrangian connect sums of α i and β j at each plumbing point p ∈ α i ∩ β j .They are α i # p β j and β j # p α i .By assuming that α i is a positive sphere and β j is a negative sphere, one considers the Lagrangian connected sum β j # p α i , not α i # p β j .Similarly, by assuming that α i is negative and β j is positive, one can construct another set B op of Lagrangian branched submanifolds.(3) for every q ∈ B 0 ∩ Φ 1 (B 1 ), q is not a plumbing point or the antipodal point of a plumbing point.
Proof.Since B 1 is a union of (parts of) compact cores and their Lagrangian connected sums, we will construct Hailtonian isotopies perturbing each compact cores α i and β j .Then, one obtains a perturbation of B 1 as a union of (parts of) perturbations of α i , β j and their Lagrangian connected sums.First, we choose a smooth function f i : α i → R with isolated critical points such that (1) every plumbing point p ∈ α i , f i (p) = f i (−p) = 0, where −p is the antipodal point of p on α i , (2) every critical point q of f i lies on S αi and q = p, −p for any plumbing point p ∈ α i , (3) |df i (x)| < for all x ∈ α i and for a sufficiently small fixed positive number , (4) f i • η αi = f i , where η αi is the involution on T * α i defined in Section 5.1.We remark that where φ αi : T * S n ∼ → T * α i is the identification which we used in Section 5.1.Also, we remark that in (3), |df i (x)| is given by the standard metric on R 2n+2 .Then, we can extend f i to fi : T * α i → R as follows.Let δ : [0, ∞) → R be a smooth decreasing function such that δ([0, ]) = 1, δ([2 , ∞)) = 0.
Similarly, we can get gj : T * β j → R in the same way.These Hamiltonian functions fi and gj induce Hamiltonian isotopies on T * α i and T * β j .Moreover, these Hamiltonian isotopies could be extended on the plumbing space M since the Hamiltonian isotopies have compact supports on T * α i and T * β j .
Let Φ αi,t : M ∼ → M be the (extended) Hamiltonian isotopy associated to fi .Then, it is easy to check that Φ αi,t • η = η • Φ αi,t , Φ αi,t (β j ) = β j for all j, where Γ(df i ) is the graph of df i in T * α i ⊂ M .Similarly, one can obtain a Hamiltonian isotopy Φ βj ,t : M ∼ → M for each β j in the same way.
Then, it is easy to check that Φ t satisfies the first condition of Lemma 5.3.Moreover, one can assume that Φ 1 (B 1 ) is constructed from Φ 1 (α i ) and Φ 1 (β j ).Thus, it is easy to prove that B 0 and Φ 1 (B 1 ) satisfy the second and the last conditions of Lemma 5.3.
First, we remark that L0 (resp.Φ 1 ( L1 )) is a curve which is carried by a train track B 0 ∩ M (resp.Φ 1 (B 1 ) ∩ M ).Then, L0 (resp.Φ 1 ( L1 )) has weights on the train track B 0 ∩ M (resp.Φ 1 (B 1 ) ∩ M ).Moreover, the number of L0 ∩ Φ 1 ( L1 ) is the following: To count the number of L 0 ∩Φ 1 (L 1 ), we can assume that L 0 ∩Φ 1 (L 1 ) is contained in a small neighborhood of B 0 ∩ Φ 1 (B 1 ).Since L 0 is carried by B 0 , not strongly carried by, L 0 can have singular points.However, the singular points are lying near plumbing points or the antipodal of plumbing points.Since the intersection points of B 0 and Φ 1 (B 1 ) are not plumbing points or their antipodals, every x ∈ L 0 ∩Φ 1 (L 1 )

Figure 1
Figure 1 is examples of plumbing spaces.If α i is of dimension n ≥ 2, then specific choices of plumbing points do not change the symplectic topology of P (α 1 , • • • , α m ).

( 2 )
A sector of B is a connected component of the set of all points in B that are locally modeled by L 0 , i.e., k = 0. (3) A branch locus Locus(B) of B is the complement of all the sectors.(4) Let (M 2n , ω) be a symplectic manifold.A subset B ⊂ M is a Lagrangian branched submanifold if for every p ∈ B, there exists a Darboux chart φ p : (U p , ω| Up ) ∼ → (R 2n , ω 0 ) about p, satisfying the conditions of an n-dimensional branched submanifold.Remark 3.2.

F.
Black curves are part of a Lagrangian branched submanifold and the black marked points denote a connected component of Locus(B).in (a), L( ) is in red, and the fibers F p , for p ∈ B∩U ( ), are in blue; (b) and (c) are not allowed by Equation (3.3); and in (d), the red and green boxes are examples of N (S).

Figure 2 (
Figure 2 (d) represents examples of N (S).We define B p for all p ∈ S by setting

Definition 3 . 6 .
Let B be a Lagrangian branched submanifold of M and let N (B) be a fibered neighborhood of B. Then, the associated branched submanifold B * is defined by setting B * := N (B)/ ∼, x ∼ y if ∃F p such that x, y ∈ F p .(a) U ( ).(b) F p for p / ∈ S( ) × (0, 1].(c) F p for p ∈ S( ) × (0, 1].

Figure 4
Figure 4 represents this.

3 . 4 .
where M * , B * , V * , and π * are defined in Example 3.12.Definition 3.14.A Lagrangian submanifold L (resp.a Lagrangian branched submanifold L) is carried by a Lagrangian branched submanifold B if L (resp.L) is weakly carried by B and every singular component of π (resp.π) is a singular component of real blow-up type.The generalized Penner construction.In this subsection, we give a higherdimensional generalization of Penner construction [8] of pseudo-Anosov surface automorphisms.The generalization replaces Dehn twists by generalized Dehn twists along Lagrangian spheres.

Figure 5 ,
N p is drawn in red.The positive disk D + p at p is the closure of α − (β# p α) and the negative disk D − p at p is the closure of β −(β# p α).The disks D ± p are drawn in blue in Figure 5.Then, by attaching D + p or D − p to β# p α, we obtain Lagrangian branched submanifolds (β# p α) ∪ α and (β# p α) ∪ β. p β) ∪ α (α# p β) ∪ β F .The blue curves represent D + p in the left hand picture and D − p in the right hand picture, the red curves represent N p in both.

1 j
, we have specific functions F τi and F σ −1 j acting on B. We claim that F ψ is a constant map, i.e., Im(F ψ ) is a point B ψ , which we define as follows: in Equation (3.4), we set D p = D + p for p ∈ α i ∩ β j if the last τ i in ψ appears later than the last σ −1 j , and D p = D − p otherwise.Note that every Dehn twist τ i and σ −appears in ψ, thus B ψ is well-defined.By Equation (3.7), F ψ (B) = B ψ for all B ∈ B. Remark 3.19.

Remark 4 . 1 .
From now on, for any compact Lagrangian submanifold L which is carried by B ψ , we will assume that every singular value of π : L → B ψ lies in the interior of a singular disk by Remark 3.19.If B * admits Equation (4.10), then one obtains a decomposition of N (B) as follows:

Definition 4 . 3 .Lemma 4 . 4 .
∂S as follows: b 0 ∼ b 1 for b i ∈ Br ∂S if there exists a smooth 1-parameter family b t ∈ Br ∂S connecting b 0 and b 1 .Let Br ∂S := Br ∂S / ∼.Let B ∈ B and let S be a singular disk of B. If L is a Lagrangian submanifold which is carried by B, then the braid b(L, S) of L on S is the braid isotopy class of Br ∂S which is given by b(L, S) = L ∩ π −1 (∂S) ∈ Br ∂S .Recall that B is a set of Lagrangian branched submanifold defined in Section 3.4 and for any B ∈ B, we decompose B into the union of specific singular disks and regular disks, introduced in Section 4.1.Let L be a Lagrangian submanifold of M which is carried by B. For a given singular disk S of F τi (B) (resp.F σ −1 j (B)), there exist maps f k from Br Si k to Br S , where S i k is a singular disk of B, and there exist closed sets bi 3, the maps of scaling type explain how braids b(L, S ± p ) or b(L, S± p ) contribute on the construction of braids b(δ(L), S ± δ(p) ) or b(δ(L), S± δ(p) ), where δ is a Dehn twist.Similarly, the maps of the first and second singular types explain how braids b(L, δ(S p )) contribute on the construction of braid to b(δ(L), S± δ(p) ).Finally, the maps of trivial type explain the other cases.This completes the proof of Lemma 4.4.

Remark 4 . 11 .
If ∂S i is contained in the branch locus of B * ψ , B i,∞ can be divided into two groups, as a braid b was divided into b and b in the Step 1 of the proof of Lemma 4.4.We will construct

Definition 5 . 1 .
Penner type defined as follows: A plumbing space M

Remark 5 . 2 . 5 . 2 .
Let M be the set of fixed points of η, i.e.,M = {x ∈ M | η(x) = x}.It is easy to check that M is a 2-dimensional symplectic submanifold of M .Moreover, M is symplectomorphic to a plumbing space P (S α1 , • • • , S αm , S β1 , • • • , S β l )of Penner type.Note that S αi and S βj are embedded circles in α i and β j .We call M the fixed surface of M .Proof of Theorem 1.5.Let M be a plumbing space of Penner type.Let η be the associated involution of M .Let L 0 and L 1 be a transversal pair of Lagrangian submanifolds such that (1) η(L i ) = L i .(2)Let Li = L i ∩ M i .Then, Li is a Lagrangian submanifold of M .(3)L 0 ∩ L 1 = L0 ∩ L1 .

Definition 3.10.
∈ B * is a singular value of π if there is a singular point x of π such that π(x) = y.(3) L is minimally singular with respect to B if π : L → B * has no singular value on the branch locus of B * and |F x ∩L| = |F y ∩L|, for any non-singular value x and y which lie in the same sector of B * , where | • | means the cardinality of a set.Two Lagrangian submanifolds L and L that are weakly carried by B are weakly fiber isotopic if there exists an isotopy for L and L through Lagrangians that are weakly carried by B.
) x ∈ L (resp.L) is a singular point of π if x is not regular point of π : L → B * .Moreover, values of π at singular points are called singular values of π.y every singular value of π : ψ n (B ψ ) → B * lies in the interiors of S ± p,B ψ or S± p,B ψ for some plumbing point p.
by isotoping, where p is a plumbing point.More precisely, let S + p,B ψ (resp.S − p,B ψ ) be the sector of B * ψ containing π(p) if D p = D + p (resp.D − p ), where D p , D + p and D − p are defined in Section 3.4.Similarly, let S± p,B ψ be π( D± p ), where D± p is defined in Equation (3.5).Then, by isotoping ψ n (B ψ ), p .
Thus, in order to keep track of braid sequences{b i,m } m∈N , it is enough to keep track of Ψ m .Every entry of Ψ m is a sum of compositions of 3m-maps.The image of a composition of 3m-maps is a solid torus.By Remarks 4.6 and 4.8, the radius of each solid torus appearing in Ψ m decreases exponentially and converges to zero as m → ∞.From another view points, we consider ψ H .Note that ψ H is defined in step 4 of the proof of Lemma 4.4.The proof of Lemma 4.4 implies that bi,m ⊂ ψ m H