Counting Dirac braid relators and hyperelliptic Lefschetz fibrations

We define a new invariant $w$ for hyperelliptic Lefschetz fibrations over closed oriented surfaces, which counts the number of Dirac braids included intrinsically in the monodromy, by using chart description introduced by the second author. As an application, we prove that two hyperelliptic Lefschetz fibrations of genus $g$ over a given base space are stably isomorphic if and only if they have the same numbers of singular fibers of each type and they have the same value of $w$ if $g$ is odd. We also give examples of pair of hyperelliptic Lefschetz fibrations with the same numbers of singular fibers of each type which are not stably isomorphic.


Introduction
Since the seminal works of Donaldson [7] and Gompf [18] in 1998, Lefschetz fibrations have been investigated from various viewpoints. Several kinds of equivalence classes of a Lefschetz fibration such as isomorphism class, diffeomorphism type, homeomorphism type, and homotopy type, and their relations have been studied by many authors. The above work of Donaldson together with a theorem of Gompf [17] implies that there exists a Lefschetz fibration with prescribed fundamental group, which means the classification of all homotopy types of Lefschetz fibrations is not possible in principle. Baykur [4] proved that any symplectic 4-manifold which is not a rational or ruled surface, after sufficiently many blow-ups, admits an arbitrary number of nonisomorphic Lefschetz fibrations of the same genus (see also Park and Yun [32,33]). It clarified a significant difference between isomorphism classes and diffeomorphism types. Two Lefschetz fibrations of the same genus over a given base space are called stably isomorphic if they become isomorphic after fiber-summed with the same number of copies of a 'universal' Lefschetz fibration. Auroux [1] obtained a sufficient condition for two Lefschetz fibrations over the 2-sphere each of which admits a section to be stably isomorphic. Hasegawa, Tanaka, and the authors [10] relaxed Auroux's condition to obtain a necessary and sufficient condition for two Lefschetz fibrations over a closed surface to be stably isomorphic. Thus the stable isomorphism problem, in contrast to the isomorphism one, turned out to be within reach.
Hyperelliptic Lefschetz fibrations are Lefschetz fibrations for which the image of the monodromy is included in the hyperelliptic mapping class group. They are considered to be a natural generalization of elliptic surfaces because several properties are common to these two kinds of fibrations. For instance, many of fibrations can be obtained by branched covering construction, the signature of a fibration localizes on the singular fibers, typical fibrations are used as building blocks for constructions of more complicated fibrations and 4-manifolds, etc. (see Siebert and Tian [35], Fuller [15], Endo [8], and Endo and Nagami [12]). Although the classification of isomorphism classes of irreducible hyperelliptic Lefschetz fibrations was partially established in genus two case by Siebert and Tian [36], it seems there is little prospect for a complete classification of isomorphism classes of hyperelliptic Lefschetz fibrations. In fact, there are infinitely many distinct Lefschetz fibrations of genus two with the same numbers of singular fibers of each type (see Baykur and Korkmaz [5], cf. Ozbagci and Stipsicz [30] and Korkmaz [27]).
In the present paper, we define a Z 2 -valued invariant w for hyperelliptic Lefschetz fibrations over closed oriented surfaces by using chart description introduced by the second author [21] (Definition 5.1 and Proposition 5.3). This invariant counts the number of 'Dirac braids' included intrinsically in the monodromy representation of a hyperelliptic Lefschetz fibration of genus g. The Dirac braid is a full twist on all strands in the (2g + 2)-string braid group B 2g+2 (S 2 ) of a 2-sphere, which corresponds to a Dehn twist around all marked points in the mapping class group M 0,2g+2 of a 2-sphere with 2g + 2 marked points, and to a maximal chain relator in the hyperelliptic mapping class group H g of a connected closed oriented surface of genus g, under natural homomorphisms (see § 3.1 for details). The relator r 4 in M 0,2g+2 corresponding to the Dirac braid is called the Dirac braid relator (see § 3.2). The proof of the invariance of w under chart moves is the most technical part of this paper (Propositions 4.2 and 4.4). We expect the invariant w to coincide with a Z 2 -valued invariant for hyperelliptic Lefschetz fibrations of odd genus mentioned by Auroux and Smith [3] (Remark 5.8). Employing the invariant w, we prove that two hyperelliptic Lefschetz fibrations of genus g over a given base space are stably isomorphic if and only if they have the same numbers of singular fibers of each type and they have the same value of w if g is odd (Theorem 5.6). Two hyperelliptic Lefschetz fibrations of genus g over a given base space are called stably isomorphic if they become isomorphic after fiber-summed with the same number of copies of a hyperelliptic Lefschetz fibration on CP 2 #(4g + 5)CP 2 , which is a natural generalization of the rational elliptic surface E(1) (Definition 5.4). We also give examples of pair of hyperelliptic Lefschetz fibrations with the same numbers of singular fibers of each type which are not stably isomorphic (Examples 6.3 and 6.4).

Lefschetz fibrations and hyperelliptic structures
In this section, we review a precise definition and basic properties of Lefschetz fibrations and introduce a notion of hyperellipticity for Lefschetz fibrations. See also Matsumoto [29], Gompf and Stipsicz [18], and Endo and Kamada [11].

Lefschetz fibrations and their monodromies
We begin with a precise definition of Lefschetz fibration. Let Σ g be a connected closed oriented surface of genus g.
such that f is locally written as f (z 1 , z 2 ) = z 1 z 2 orz 1 z 2 with respect to some local complex coordinates around p and b which are compatible with orientations of M and B; (iii) no fiber contains a (±1)-sphere.
We call M the total space, B the base space, and f the projection. We call p a critical point of positive type (respectively of negative type) and F b a singular fiber of positive type (respectively of negative type) if f is locally written as f (z 1 , Let M g be the mapping class group of Σ g , namely the group of all isotopy classes of orientation preserving diffeomorphisms of Σ g . We assume that M g acts on the right: the symbol ϕψ means that we apply ϕ first and then ψ for ϕ, ψ ∈ M g . Let f : M → B be a Lefschetz fibration of genus g as in Definition 2.1. Take a base point b 0 ∈ B − Δ and an orientation preserving diffeomorphism Φ : Σ g → F 0 := f −1 (b 0 ). Since f restricted over B − Δ is a smooth fiber bundle with fiber Σ g , we can define a homomorphism called the monodromy representation of f with respect to Φ (see Matsumoto [29, § 2]). Let γ be the loop based at b 0 consisting of the boundary circle of a small disk neighborhood of b ∈ Δ oriented counterclockwise and a simple path connecting a point on the circle to b 0 in B − Δ. It is known that ρ([γ]) is a Dehn twist along some essential simple closed curve c on Σ g . The curve c is called the vanishing cycle of the critical point p on f −1 (b). If p is of positive type (respectively of negative type), then the Dehn twist is right-handed (respectively left-handed).
A singular fiber is said to be of type I if the vanishing cycle is non-separating and of type II h for h = 1, . . . , [g/2] if the vanishing cycle is separating and it bounds a genus-h subsurface of Σ g . A singular fiber is said to be of type I + (respectively type I − , type II + h , type II − h ) if it is of type I and of positive type (respectively of type I and of negative type, of type II h and of positive type, of type II h and of negative type). We denote by n + 0 (f ), n − 0 (f ), n + h (f ), and n − h (f ), the numbers of singular fibers of f of type I + , I − , II + h , and II − h , respectively. A Lefschetz fibration is called irreducible if every singular fiber is of type I. A Lefschetz fibration is called chiral if every singular fiber is of positive type.
Suppose that the cardinality of Δ is equal to n. A system A = (A 1 , . . . , A n ) of arcs on B is called a Hurwitz arc system for Δ with base point b 0 if each A i is an embedded arc connecting b 0 with a point of Δ in B such that A i ∩ A j = {b 0 } for i = j, and they appear in this order around b 0 (see Kamada [22]). When B is a 2-sphere, the system A determines a system of generators of π 1 (B − Δ, b 0 ), say (a 1 , . . . , a n ). We call (ρ(a 1 ), . . . , ρ(a n )) a Hurwitz system of f . It is easy to see that ρ is determined by (ρ(a 1 ), . . . , ρ(a n )).
is a hyperelliptic structure on f equivalent to Φ . If we can choose such an h isotopic to the identity relative to a given base point b 0 , we say that (f, Φ) is strictly H-isomorphic to (f , Φ ). If (f, Φ) is H-isomorphic (respectively strictly H-isomorphic) to (f , Φ ), then f is isomorphic (respectively strictly isomorphic) to f .
(2) (f, Φ) is strictly H-isomorphic to (f , Φ ) if and only if there exists an orientation preserving diffeomorphism h : B → B and an element α of H g , which satisfies the conditions (i), (ii), (iii), and (iv) h, is isotopic to the identity relative to a given base point b 0 .
Lemma 2.6. Suppose that g is greater than 1. Let ρ : π 1 (B − Δ, b 0 ) → H g be a homomorphism and A = (A 1 , . . . , A n ) a Hurwitz arc system for Δ with base point b 0 . We assume that ρ(a 1 ), . . . , ρ(a n ) are Dehn twists along simple closed curves on Σ g for the system of generators (a 1 , . . . , a n ) of π 1 (B − Δ, b 0 ) determined by A. Then there exists a hyperelliptic Lefschetz fibration (f, Φ) of genus g as in Definition 2.3 with monodromy representation ρ.
Two isomorphic hyperelliptic Lefschetz fibrations need not be H-isomorphic. We give a sufficient condition for isomorphic hyperelliptic Lefschetz fibrations to be H-isomorphic.
Let (f, Φ) and (f , Φ ) be hyperelliptic Lefschetz fibrations of genus g as in Definition 2.4, and ρ and ρ their monodromy representations as in Lemma 2.5.
Proof. The 'if' part is obvious. We show the 'only if' part. Since f is isomorphic to f , there exists an orientation preserving diffeomorphism h : B → B and an element α of M g which satisfies the following conditions (see Matsumoto [29,Theorem 2.4] . We will show that α belongs to H g , which implies that (f, Φ) is H-isomorphic to (f , Φ ) by Lemma 2.5.
For any i = 1, . . . , 2g + 1, there exists an element [14,Fact 3.6], where A is an orientation preserving diffeomorphism on Σ g representing α. Since c i is invariant under I, (c i )(AIA −1 I −1 ) is isotopic to c i . Hence there exists a diffeomorphism F on Σ g isotopic to the identity such that (c i )(AIA −1 I −1 F ) = c i for every i by virtue of [14,Lemma 2.9]. Since Σ g − (c 1 ∪ · · · ∪ c 2g+1 ) is a disjoint union of two open disks, AIA −1 I −1 F is isotopic to either the identity or I by [14,Proposition 2.8]. If AIA −1 I −1 F is isotopic to I, then we obtain ι = 1, which is a contradiction. Therefore AIA −1 I −1 F is isotopic to the identity and we have α ∈ H g . This completes the proof.

Chart descriptions
In this section, we introduce chart descriptions for hyperelliptic Lefschetz fibrations by employing finite presentations of hyperelliptic mapping class groups and two other groups. General theories of charts for presentations of groups were developed independently by Kamada [23] and Hasegawa [19]. We use the terminology of chart description in Kamada [23].

Three finite presentations
We first review finite presentations of three groups related with hyperelliptic Lefschetz fibrations.
Fadell and Van Buskirk [13] proved that the braid group of a 2-sphere is just the braid group of a 2-disk with a single additional relation.  [13]). Suppose that g is positive. The (2g + 2)-string braid group B 2g+2 (S 2 ) of a 2-sphere is generated by elements x 1 , x 2 , . . . , x 2g+1 and has defining relations: Magnus [28] obtained a presentation of the mapping class group of a 2-sphere with marked points, which is essentially the same as the following one.

Chart descriptions
We make use of the above presentations to introduce notions of chart which give graphic description of monodromy representations of Lefschetz fibrations. We first recall a general definition of chart given by Kamada [23].
Let X be a set and R and S sets of words in X ∪ X −1 . Let C := (X , R, S) be the triple consisting of X , R and S, and G the group with presentation X | R . Let B be a connected closed oriented surface and Γ a finite graph in B such that each edge of Γ is oriented and labeled an element of X . Choose a simple path γ which intersects with edges of Γ transversely and does not intersect with vertices of Γ. For such a path γ, we obtain a word w Γ (γ) in X ∪ X −1 by reading off the labels of intersecting edges along γ with exponents as in Figure 2(a). We call the word w Γ (γ) the intersection word of γ with respect to Γ. Conversely, we can specify the number, orientations, and labels of consecutive edges in Γ by indicating a (dashed) arrow intersecting the edges transversely together with the intersection word of the arrow with respect to Γ (see Figure 2(b) and (c)).
For a vertex v of Γ, a small simple closed curve surrounding v in the counterclockwise direction is called a meridian loop of v and denoted by m v . The vertex v is said to be marked if one of the regions around v is specified by an asterisk. If v is marked, the intersection word w Γ (m v ) of m v with respect to Γ is well defined. If not, it is determined up to cyclic permutation. See Kamada [23] for details.
is a finite graph Γ in B (possibly being empty or having hoops that are closed edges without vertices) whose edges are labeled an element of X , and oriented so that the following conditions are satisfied: (1) the vertices of Γ are classified into two families: white vertices and black vertices; is exactly an element of R ∪ R −1 (respectively of S). If a base point b 0 of B is specified, we always assume that a chart Γ is disjoint from b 0 . A chart consisting of two black vertices and one edge connecting them is called a free edge. A subchart of a C-chart Γ is the intersection of Γ with a compact two-dimensional submanifold of B.
Remark 3.5. It would be worth noting that the intersection word of a 'clockwise' meridian of a white vertex of type r is equal to r, while that of a 'counterclockwise' meridian of a black vertex of type s is equal to s in this paper (see also [10]).
Let Γ be a C-chart in B with base point b 0 and Δ Γ the set of black vertices of Γ.
Definition 3.6. For a loop η in B − Δ Γ based at b 0 , the element of G determined by the intersection word w Γ (η) of η with respect to Γ does not depend on a choice of representative of the homotopy class of η. Thus we obtain a homomorphism ρ Γ : Let Δ be a finite subset of B and b 0 a base point of B − Δ.
is conjugate to the element of G determined by an  Theorem 3.8 (Kamada [23], Hasegawa [19]). For any G-monodromy representation ρ : We next introduce several moves for charts. Let Γ and Γ be two C-charts on B and b 0 a base point of B.
Let D be a disk embedded in B − {b 0 }. Suppose that the boundary ∂D of D intersects Γ and Γ transversely. Let s and s be elements of S. Suppose that there exists a word w in X ∪ X −1 such that two words s and wsw −1 determine the same element of G.
Definition 3.10. If a C-chart Γ contains a black vertex of type s, then we can change a part of Γ near the vertex by using a local replacement depicted in Figure 4 to obtain another C-chart Γ . We say that Γ is obtained from Γ by a chart move of transition. Note that the box labeled T can be filled only with edges and white vertices.
Definition 3.11. We say that Γ is obtained from Γ by a chart move of conjugacy type if Γ is obtained from Γ by a local replacement depicted in Figure 5.
We say that ρ is equivalent to ρ if there exists an orientation preserving diffeomorphism h : B → B and an element α of G which satisfies the following conditions: . If we can choose such an h isotopic to the identity relative to a given base point b 0 , we say that ρ is strictly equivalent to ρ .
We state a classification theorem for G-monodromy representations in terms of charts and chart moves.
Theorem 3.13 (Kamada [23], Hasegawa [19]). Let Γ and Γ be C-charts in B, and (2) ρ Γ is strictly equivalent to ρ Γ if and only if Γ is transformed into Γ by a finite sequence of the moves (i), (ii), (iii), and (iv) provided that h is isotopic to the identity relative to a given base point b 0 .
We now define three explicit Cs corresponding to three groups M 0,2g+2 , B 2g+2 (S 2 ), and H g . For the group M 0,2g+2 , we set ). The relator r 4 corresponds to the Dirac braid and it is called the Dirac braid relator. Vertices of types 0 (i) ±1 , r 1 (i, j), r 2 (i), r 3 , r 4 , h in C 0 -charts are depicted in Figures 6 and 7, where the label ξ i is denoted by i for short. For the group B 2g+2 (S 2 ), we set C := (X ,R,S), Vertices of types˜ 0 (i) ±1 ,r 1 (i, j),r 2 (i),r 3 ,˜ h inC-charts are similar to those of types 0 (i) ±1 , r 1 (i, j), r 2 (i), r 3 , h in C 0 -charts (cf. Figures 6 and 7), respectively. For the group H g , we set C := (X ,R,Ŝ), Vertices of typesˆ 0 (i) ±1 ,r 1 (i, j),r 2 (i),r 4 ,ˆ h inĈ-charts are similar to those of types 0 (i) ±1 , Figures 6 and 7), respectively. Vertices of typesr 3 andr 5 inĈ-charts are depicted in Figure 8, where the label ζ i is denoted by i for short. Let B be a connected closed oriented surface and Γ a chart in B. We denote the number of white vertices of typer 1 (i, j) (respectivelyr 2 (i),r 3 ,r 4 ,r 5 ) minus the number of white vertices of typer 1 . Similarly, we denote the number of black vertices of typê

Charts and hyperelliptic Lefschetz fibrations
Combining Lemmas 2.5 and 2.6 with Theorems 3.8 and 3.13 forĈ-charts, we obtain classification theorems for hyperelliptic Lefschetz fibrations in terms ofĈ-charts.
Proposition 3.14. Suppose that g is greater than 1.
(1) Let (f, Φ) be a hyperelliptic Lefschetz fibration of genus g over B and ρ the monodromy representation of f with respect to Φ. Then there exists aĈ-chart Γ in B such that the homomorphism ρ Γ determined by Γ is equal to ρ.
(2) For everyĈ-chart Γ in B, there exists a hyperelliptic Lefschetz fibration (f, Φ) of genus g over B such that the monodromy representation of f with respect to Φ is equal to the homomorphism ρ Γ determined by Γ.
We call such Γ as in Proposition 3.14 (1) aĈ-chart corresponding to (f, Φ), and such (f, Φ) as in Proposition 3.14 (2) a hyperelliptic Lefschetz fibration described by Γ. (  If the word w is trivial, then the chart Γ# w Γ is denoted also by Γ ⊕ Γ , which is called a product of Γ and Γ .

Counting Dirac braid relators
In this section, we define a Z 2 -valued invariant for C 0 -charts in a given surface and prove its invariance under several chart moves.
We first give a precise definition of the invariant. Let B be a connected closed oriented surface and g an integer greater than 1.  We need a lemma.   Proof. Since the element of B 2g+2 (S 2 ) represented by (x 1 x 2 · · · x 2g+1 ) 2g+2 is included in the center of B 2g+2 (S 2 ), it commutes with each of x 1 , . . . , x 2g+1 in B 2g+2 (S 2 ). Thus the word [x k , (x 1 x 2 · · · x 2g+1 ) 2g+2 ] represents the identity element of B 2g+2 (S 2 ), and there exists a finite sequence of words inX ∪X −1 starting from the word [x k , (x 1 x 2 · · · x 2g+1 ) 2g+2 ] to the empty word such that each word is related to the previous one by one of the following transformations: (i) insertion or deletion of a trivial relator x ε x −ε for x ∈X and ε ∈ {+1, −1}; (ii) insertion ofr ε forr ∈R and ε ∈ {+1, −1}. We first consider aC-chart depicted in Figure 9. For each transformation (i) (respectively (ii)), we create an edge labeled x (respectively a vertex of typẽ r ε ). Repeating such creations, we can fill the box labeled T k with edges and white vertices of typesr 1 (i, j) ±1 ,r 2 (i) ±1 , andr Proof of Proposition 4.2. It suffices to show that w(Γ) = 0 for every C 0 -chart Γ in S 2 without black vertices. Let Γ be such a C 0 -chart in S 2 . We consider a chart move of type W depicted in Figure 10. Suppose that the box labeled T k is filled with edges and white vertices of types r 1 (i, j) ±1 , r 2 (i) ±1 , and r ±1 3 by Lemma 4.3. Using the sequence of chart moves repeatedly, we can make a C 0 -chart Γ 1 depicted in Figure 11(a) from Γ, where the box labeled Θ 1 is filled with edges and white vertices of types r 1 (i, j) ±1 , r 2 (i) ±1 , and r ±1 3 . The number of white vertices of type r 4 and that of white vertices of type r −1 4 included in Γ 1 are the same as those for Γ, respectively. We then apply deaths of pairs of white vertices of types r 4 and r −1 4 as in Figure 3(c) and sequences of chart moves of type W as in Figure 10 to Γ 1 repeatedly to obtain a C 0 -chart Γ 2 Figure 11. Charts Γ1, Γ2 and Γ3. depicted in Figure 11(b), where the box labeled Θ 2 is filled with edges and white vertices of types r 1 (i, j) ±1 , r 2 (i) ±1 , and r ±1 3 , and ε is equal to either +1 or −1. Let n be the number of white vertices of type r ε 4 included in Γ 2 . We replace all the white vertices of type r ε 4 included in Γ 2 with black vertices of types 0 (i) −ε to obtain a C 0 -chart Γ 3 depicted in Figure 11(c). The intersection word w of the dashed arrow with respect to Γ 3 in Figure 11(c) is equal to (ξ 1 ξ 2 · · · ξ 2g+1 ) ε(2g+2)n . Changing all labels ξ i of edges of Γ 3 into x i , we obtain aC-chart Γ 3 because Γ 3 does not contain white vertices of types r ±1 4 . The intersection word w of the dashed arrow with respect toΓ 3 in Figure 11(c) is equal to (x 1 x 2 · · · x 2g+1 ) ε(2g+2)n , which represents the identity element of B 2g+2 (S 2 ). Since the element of B 2g+2 (S 2 ) represented by (x 1 x 2 · · · x 2g+1 ) 2g+2 is of order 2 (see [14, § 9.1.4]), n must be even. Therefore we have and this completes the proof.

Definition 4.5. A chart move of transition as in
We first show a lemma for L 0 -moves.  Figure 4 without black vertices such that the number of white vertices of type r ±1 4 is even, that is, the filling consists of edges, white vertices of types r 1 (i, j) ±1 , r 2 (i) ±1 , r ±1 3 , and an even number of white vertices of types r ±1 4 .
Proof. We assume that ε = +1 for simplicity. Let ϕ and δ i be elements of M 0,2g+2 represented by w and ξ i , respectively. Since the intersection word of the boundary of the box labeled T = T 1 (k, ; +1) with respect to the C 0 -chart in Figure 4 Case 1: Suppose that k = . We consider a hyperelliptic involution I on Σ g and think of S 2 as the quotient of Σ g by the action of I. The image of the simple closed curve c i in Figure 1 under the double branched covering Σ g → S 2 is a simple arc a i on S 2 depicted in Figure 12, and a right-handed half twist D i about a i represents the mapping class δ i . We also consider an additional arc a 2g+2 on S 2 as in Figure 12, a right-handed half twist D 2g+2 about a 2g+2 ,   The relation ϕδ k ϕ −1 = δ k implies that (a k )F is isotopic to a k by [24,Lemma 4.1], where F is an orientation preserving diffeomorphism on S 2 representing ϕ. Since (a k )D k = a k and D k reverses the orientation of a k , we can assume that either F or F D k fixes a k pointwise. Cutting S 2 along a k , we obtain an orientation preserving diffeomorphism on a 2-disk which fixes the boundary pointwise. The mapping class group M 1 0,2g of the 2-disk with 2g marked points is generated by the half twists about arcs corresponding to a 1 , . . . , a 2g+2 except a k−1 and a k+1 (see [14, § 9.1.3]). Hence ϕ can be factorized as a product of δ ±1 1 , . . . , δ ±1 2g+2 except δ ±1 k−1 and δ ±1 k+1 , and it can be represented by a word w in ξ ±1 We divide the box labeled T 1 (k, k; +1) into three boxes labeled T 1 (k), Θ 1 (k), and Θ 1 (k) * as in Figure 13.
The box labeled Θ 1 (k) can be filled with edges and white vertices because both w and w represent the same mapping class ϕ. The box labeled Θ 1 (k) * can be filled with the mirror image of the subchart filling the box labeled Θ 1 (k) with edges orientation reversed. Since w is a word in ξ ±1 , the box labeled T 1 (k) can be filled with copies of three kinds of subcharts depicted in Figure 14. Note that the subcharts depicted in Figure 14 correspond to ξ k , ξ i (i = k − 1, k, k + 1, 2g + 2), and ξ 2g+2 , respectively. We also need subcharts corresponding to ξ −1 k , ξ −1 i (i = k − 1, k, k + 1, 2g + 2), and ξ −1 2g+2 . The box labeled T 1 (k) is filled with edges and white vertices of types r 1 (i, j) ±1 and r 2 (i) ±1 . The number of white vertices of type r ±1 4 included in the box labeled Θ 1 (k) * is equal to that  for the box labeled Θ 1 (k). Therefore the box labeled T 1 (k, k; +1) is filled with edges, white vertices of types r 1 (i, j) ±1 , r 2 (i) ±1 , r ±1 3 , and an even number of white vertices of types r ±1 4 .
By the proof of Case 1, the small box labeled T 1 (k, k; +1) is filled with edges, white vertices of types r 1 (i, j) ±1 , r 2 (i) ±1 , r ±1 3 , and an even number of white vertices of types r ±1 4 , so is the box labeled T 1 (k, ; +1). We next show a lemma for L + -moves.  Figure 4 without black vertices such that the number of white vertices of type r ±1 4 is even, that is, the filling consists of edges, white vertices of types 3 , and an even number of white vertices of types r ±1 4 .
We now prove the invariance of w under chart moves of transition. Proof of Proposition 4.4. Let Γ be a C 0 -chart Γ in B and Γ a C 0 -chart in B obtained from Γ by a chart move of transition. Suppose that the chart move of transition is an L 0 -move. Two C 0 -charts Γ and Γ are related by a chart move as in Figure 4 and the box is labeled T = T 1 (k, ) for some k, = 1, . . . , 2g + 1. Let Γ 1 be a C 0 -chart in B obtained from Γ by replacing the subchart inside the box labeled T 1 (k, ) with a subchart satisfying the condition given in Lemma 4.6. By virtue of Proposition 4.2 and Lemma 4.6, we have w(Γ) = w(Γ 1 ) = w(Γ ). It follows from a similar argument together with Proposition 4.2 and Lemma 4.7 that w(Γ) = w(Γ ) if g is odd and Γ is obtained from Γ by an L + -move. Thus we have proved the proposition.

Stable classification
In this section, we define a Z 2 -valued invariant of hyperelliptic Lefschetz fibrations of odd genus and show a stable classification theorem for hyperelliptic Lefschetz fibrations.
We first give a definition of the invariant. Let B be a connected closed oriented surface and g an integer greater than 1.
Definition 5.1. Let (f, Φ) be a hyperelliptic Lefschetz fibration of genus g over B and ρ : π 1 (B − Δ, b 0 ) → H g the monodromy representation with respect to Φ. Let Γ be a C 0 -chart in B such that the homomorphism determined by Γ is equal to ρ 0 := π • ρ : Remark 5.2. If we have aĈ-chartΓ corresponding to (f, Φ), a C 0 -chart Γ above is obtained fromΓ by changing the labels ζ i of all edges into ξ i and replacing all white vertices of typesr ±1    Proof. The statement follows from Propositions 3.14, 3.15, 4.2 and 4.4.
Thus w(f, Φ) turns out to be an invariant of the H-isomorphism class of (f, Φ). We next prove a stable classification theorem for hyperelliptic Lefschetz fibrations, which improves the stabilization theorem shown in [11]. Let (f, Φ) and (f , Φ ) be hyperelliptic Lefschetz fibrations of genus g over B. Let Γ 0 be aĈ-chart in S 2 depicted in Figure 22 and (f 0 , Φ 0 ) a hyperelliptic Lefschetz fibration described by Γ 0 .
Definition 5.4. We say that (f, Φ) is stably isomorphic to (f , Φ ) if there exists a nonnegative integer N such that an H-fiber sum (f #Nf 0 , Φ) is H-isomorphic to an H-fiber sum (f #Nf 0 , Φ ).  We give a complete classification of the stable isomorphism classes of hyperelliptic Lefschetz fibrations of genus g over B.
Theorem 5. 6. Let (f, Φ) and (f , Φ ) be hyperelliptic Lefschetz fibrations of genus g over B. Then (f, Φ) is stably isomorphic to (f , Φ ) if and only if the following conditions hold: Proof. We first prove the 'if' part. Assume that (f, Φ) and (f , Φ ) satisfy the conditions (i), (ii), and (iii). Let Γ and Γ beĈ-charts in B corresponding to (f, Φ) and (f , Φ ), respectively. Since every edge has two adjacent vertices, the sum of the signed numbers of adjacent edges for all vertices of Γ is equal to zero: A similar equality for Γ also holds. Interpreting the conditions (i) and (ii) as conditions on Γ and Γ , we have Let N be an integer larger than both of the number of edges of Γ and that of Γ . Choose a base point b 0 ∈ B − (Γ ∪ Γ ). The H-fiber sum (f #Nf 0 , Φ) is described by a chart (· · · ((Γ# w1 Γ 0 )# w2 Γ 0 ) · · · )# wN Γ 0 for some words w 1 , . . . , w N inX ∪X −1 . Since hoops surrounding Γ 0 can be removed by use of the edges of Γ 0 as in Figure 23, the chart is transformed into a product Γ ⊕ N Γ 0 by channel changes. Similarly, the H-fiber sum f #Nf 0 is described by a product Γ ⊕ N Γ 0 .
We choose and fix 2g + 1 edges of Γ 0 which are labeled with 1, 2, . . . , 2g + 1 and adjacent to black vertices. We apply chart moves only to these edges in the following. Since Γ 0 can pass through any edge of Γ as shown in Figure 24, we can move Γ 0 to any region of B − Γ by channel changes.
For each edge of Γ, we move a copy of Γ 0 to a region adjacent to the edge and apply a channel change to the edge and Γ 0 as in Figure 24(a) and (b). Applying chart moves of transition to each component of the chart as in Figure 25, we remove white vertices of typê r 1 (i, j) ±1 ,r 2 (i) ±1 ,r ±1 5 to obtain a union of copies of L 0 (i), L h , L h , R 3 , R 3 , R 4 , R 4 , Γ 0 shown in Figures 26 and 27, where we use a simplification of diagrams as in Figure 28. Although R 3 is the same chart as Γ 0 , we distinguish R 3 from Γ 0 s which do not come from vertices of type r 3 in Γ.
If there is a pair of R 3 and R 3 , we remove them by a death of a pair of white vertices to obtain 4(2g + 1) copies of Γ 0 . Similarly, we remove a pair of R 4 and R 4 to obtain 2(g + 1)(2g + 1)   copies of Γ 0 . Since there is at least one Γ 0 , any copy of L 0 (i) can be transformed into L 0 (1) as in Figure 29.
We can suppose that m 4 (Γ) m 4 (Γ ) without loss of generality. The number m 4 (Γ) − m 4 (Γ ) is even because of the equality ( * ) for even g and because of the condition (iii) for odd g. We put 2 := m 4 (Γ) − m 4 (Γ ). Taking N large enough, we can assume that k > 4 (g + 1)(2g + 1).   Applying births of 2 pairs of white vertices of typer ±1 4 to 4 (g + 1)(2g + 1) copies of Γ 0 included in Γ 1 , we obtain 2 pairs of R 4 and R 4 . Since 2R 4 is transformed into (g + 1)R 3 by a sequence of chart moves of type W and chart moves of transition (see [9,Lemma 4.1]), we obtain (g + 1) R 3 from 2 R 4 . We thus have a new chart Γ 2 . Removing pairs of R 3 and R 3 and those of R 4 and R 4 included in Γ 2 if necessary, Γ 2 and Γ 1 have the same numbers of copies of L 0 (1), L h , L h , R 3 (or R 3 ), R 4 (or R 4 ), and Γ 0 because of the equality ( * ). Then Γ 2 is transformed into Γ 1 by an ambient isotopy of B relative to b 0 , which means that Γ ⊕ N Γ 0 is transformed into Γ ⊕ N Γ 0 by chart moves of type W, chart moves of transition, and ambient isotopies of B relative to b 0 . Therefore f #Nf 0 is (strictly) H-isomorphic to f #Nf 0 by Proposition 3.15.
Remark 5.8. Auroux and Smith [3, § 2.1] remarked that the existence of a certain Z 2 -valued invariant and a stabilization theorem for hyperelliptic Lefschetz fibrations follow from a result of Kharlamov and Kulikov [26] about braid monodromy factorizations. Although they did not give any precise definition of the invariant, we expect that their invariant would be the same as the invariant w and their theorem would be similar to (a part of) Theorem 5.6.

Examples and remarks
In this section, we exhibit examples of stabilizations of hyperelliptic Lefschetz fibrations and examples of pairs of hyperelliptic Lefschetz fibrations which are not stably equivalent, and make a few remarks. We do not specify hyperelliptic structures of Lefschetz fibrations in the following because they are obvious from given Hurwitz systems.
We first show an example of non-isomorphic pair of irreducible chiral hyperelliptic Lefschetz fibrations which become isomorphic after one stabilization.
Example 6.1. Let g be an integer greater than 1. We consider the following Hurwitz systems: Let f CII : M CII → S 2 and f I 2g : M I 2g → S 2 be the hyperelliptic Lefschetz fibrations of genus g determined by C II and I 2g , respectively. f I 2g is nothing but the (untwisted) H-fiber sum of g copies of f 0 in Section 5. Both f CII and f I 2g are irreducible, chiral and have 4g(2g + 1) singular fibers. They are not H-isomorphic because the images of their monodromy representations are different (see [8,Example 4.14]). Moreover, the manifolds M CII and M I 2g are homeomorphic but not diffeomorphic by Freedman's theorem and a theorem of Usher [39] (see [9,Remark 4.9]). In particular f CII and f I 2g are not isomorphic. On the other hand, the (untwisted) H-fiber sums f CII #f 0 and f I 2g #f 0 are H-isomorphic because a Hurwitz system of f CII #f 0 is transformed into that of f I 2g #f 0 by elementary transformations and simultaneous conjugations (see [9,Lemma 4.1]). We next show an example of non-isomorphic pair of chiral hyperelliptic Lefschetz fibrations with singular fiber of type II g/2 which become isomorphic after one stabilization.
Let f P J : M P J → S 2 and f RI g−1 : M RI g−1 → S 2 be the hyperelliptic Lefschetz fibrations of genus g determined by P J and RI g−1 , respectively. Both P J and RI g−1 are chiral and have 6g 2 + 2g + 1 singular fibers. One singular fiber is of type II g/2 and the others are of type I. The manifolds M P J and M RI g−1 are homeomorphic but not diffeomorphic (see [9,Theorem 4.8] for g 4 and Sato [34, Answer to Question 5.1] for g = 2). In particular f P J and f RI g−1 are not isomorphic. On the other hand, the (untwisted) H-fiber sums f P J #f 0 and f RI g−1 #f 0 are H-isomorphic because a Hurwitz system of f P J #f 0 is transformed into that of f RI g−1 #f 0 by elementary transformations and simultaneous conjugations (see [9,Theorem 4.10]).
We then show an example of pair of irreducible chiral hyperelliptic Lefschetz fibrations with the same number of singular fibers which are not stably isomorphic. Example 6.3. Let g be an odd integer greater than 1. We consider the following Hurwitz systems: Let f CI : M CI → S 2 and f I g+1 : M I g+1 → S 2 be the hyperelliptic Lefschetz fibrations of genus g determined by C I and I g+1 , respectively. f I g+1 is nothing but the (untwisted) H-fiber sum of (g + 1)/2 copies of f 0 in Section 5. Both f CI and f I g+1 are irreducible, chiral and have 2(g + 1)(2g + 1) singular fibers.Ĉ-charts corresponding to f CI and f I g+1 are R 4 and ((g + 1)/2)R 3 (= ((g + 1)/2)Γ 0 ), respectively (see Figures 22 and 27). The values of the invariant w for f CI and f I g+1 are computed as follows (see Definitions 4.1 and 5.1): Therefore f CI and f I g+1 are not stably isomorphic by Theorem 5.6. Since the images of monodromy representations of the (untwisted) H-fiber sums f CI #Nf 0 and f I g+1 #Nf 0 coincide with H g for any positive integer N , they are not even isomorphic by virtue of Proposition 2.7. Both manifolds M CI and M I g+1 are simply-connected and have the Euler characteristic 2(2g 2 + g + 3) and signature −2(g + 1) 2 . If g ≡ 3 (mod 4), then M I g+1 is spin while M CI is not. Hence they are not homeomorphic. If g ≡ 1 (mod 4), then both M CI and M I g+1 are not spin, and they are homeomorphic by Freedman's theorem. Since M CI has a (−1)-section and M I g+1 is a non-trivial fiber sum, they are not diffeomorphic by a theorem of Usher [39]. The fact that f CI and f I g+1 are not isomorphic also follows from this observation, or from a result of Smith [37] and Stipsicz [38].
We lastly show an example of pair of chiral hyperelliptic Lefschetz fibrations with the same numbers of singular fibers of each type which have a singular fiber of type II (g−1)/2 and are not stably isomorphic.
Both manifolds M Q and M R are simply-connected, non-spin and have the Euler characteristic 2g 2 + 4g + 7 and signature −(g + 1) 2 . Hence they are homeomorphic by Freedman's theorem. It does not seem to be known whether they are diffeomorphic.
We end this section by making two remarks.
Remark 6.5. For a pair of stably isomorphic hyperelliptic Lefschetz fibrations, it is not easy to determine the minimal number of copies of f 0 which make them H-isomorphic. In particular, it does not seem to be known whether there exists a pair of stably isomorphic hyperelliptic Lefschetz fibrations which do not become H-isomorphic after taking an H-fiber sum with one copy of f 0 . On the other hand, the following observation shows that there are many examples of non-H-isomorphic hyperelliptic Lefschetz fibrations with the same base, the same fiber, and the same numbers of singular fibers of each type which become H-isomorphic after one stabilization.
Let g be an integer greater than 1 and B 1 , . . . , B r connected closed oriented surfaces. We consider a hyperelliptic Lefschetz fibration f i : M i → B i of genus g for each i ∈ {1, . . . , r}, and (possibly different) H-fiber sums f and f of f 1 , . . . , f r . It is shown by the same argument as the proof of [10, Proposition 4.10] that H-fiber sums f #f 0 and f #f 0 are H-isomorphic to each other. For example, various H-fiber sums of (generalizations of) Matsumoto's fibration studied by Ozbagci and Stipsicz [30] and Korkmaz [27] become H-isomorphic after one stabilization.
Since both Γ U and Γ V have a white vertex of typer 4 , we conclude that w(f U ) = w(f V ) = 1, and that f U and f V are stably isomorphic by Theorem 5.6. In fact it is not difficult to see that the (untwisted) H-fiber sums f U #f 0 and f V #f 0 are H-isomorphic (and hence isomorphic by Proposition 2.7) by applying chart moves to Γ U ⊕ Γ 0 and Γ V ⊕ Γ 0 . It is not clear whether f U and f V are H-isomorphic, and whether M U and M V are diffeomorphic. It would be worth noting that M U and M V cannot be distinguished by Usher's theorem [39] because both f U and f V have a (−1)-section. Two Hurwitz systems U and V are related by a 'partial twisting' operation, namely, V is obtained from U by replacing (ζ 1 , . . . , ζ 7 ) with (ζ 1 , . . . , ζ 7 ). Such pairs of Hurwitz systems often yield pairs of 4-manifolds with subtle difference (see works of Auroux [2] and Yasui [40]).