aa r X i v : . [ m a t h . G T ] O c t A NOTE ON SECTIONS OF BROKEN LEFSCHETZ FIBRATIONS
KENTA HAYANO
Abstract.
We show that there exists a non-trivial simplified broken Lefschetz fibrationwhich has infinitely many homotopy classes of sections. We also construct a non-trivialsimplified broken Lefschetz fibration which has a section with non-negative square. It isknown that no Lefschetz fibration satisfies either of the above conditions. Smith proved thatevery Lefschetz fibration has only finitely many homotopy classes of sections, and Smith andStipsicz independently proved that a Lefschetz fibration is trivial if it has a section with non-negative square. So our results indicate that there are no generalizations of the above resultsto broken Lefschetz fibrations. We also give a necessary and sufficient condition for the totalspace of a simplified broken Lefschetz fibration with a section admitting a spin structure,which is a generalization of Stipsicz’s result on Lefschetz fibrations. Introduction
A broken Lefschetz fibration is a smooth map from a 4-manifold to a 2-manifold which hasat most two types of singularities, that is, Lefschetz singularity and indefinite fold singularity.Such a fibration was first introduced in [2] as a generalization of Lefschetz fibrations to near-symplectic setting. Broken Lefschetz fibrations have properties similar to those of Lefschetzfibrations in some aspects. So it is natural to try to study the former by the techniques used todevelop the latter and some of such attempts were successful (e.g. [5], [8] and [12]).On the other hand, there are also some crucial differences between two kinds of fibrations. Forexample, it is proved in [1], [3] and [11] that every closed oriented smooth 4-manifold admitsa broken Lefschetz fibration (furthermore, we can prove by using the results in [11] and [16]that every closed oriented smooth 4-manifold admits a simplified broken Lefschetz fibration).However, there exist a lot of 4-manifolds which never admits any Lefschetz fibrations since thetotal space of a Lefschetz fibration is symplectic [6]. So it is important to study how far brokenLefschetz fibrations are different from Lefschetz fibrations.Smith proved the following theorem as a generalization of Manin’s theorem.
Theorem 1.1 (Smith [13]) . Let f : M → S be a non-trivial relatively minimal Lefschetzfibration. Then f has only finitely many homotopy classes of sections. The following result implies that we cannot generalize Smith’s result to simplified brokenLefschetz fibrations.
Theorem 1.2.
For any g ≥ , there exists a non-trivial genus- g simplified broken Lefschetzfibration f : M → S such that no fiber of f contains ( − -sphere and f has infinitely manyhomotopy classes of sections. Smith and Stipsicz found a constraint on self-intersection numbers of sections of Lefschetzfibrations.
Theorem 1.3 (Smith [13], Stipsicz [14]) . Let f : M → S be a genus- g relatively minimalLefschetz fibration ( g ≥ ). If f has a section σ : S → M which satisfies [ σ ( S )] ≥ , then f is trivial. The following result indicates existence of non-trivial simplified broken Lefschetz fibrationshaving a section with non-negative square.
Theorem 1.4.
For any integer n ∈ Z and g ≥ , there exists a non-trivial genus- g simplifiedbroken Lefschetz fibration f : M → S such that f has a section σ : S → M with [ σ ( S )] = n . Remark 1.5.
Baykur had already proved in [4] that there exists a non-trivial genus-
Preliminaries
Broken Lefschetz fibrations.Definition 2.1.
Let M and B be compact oriented smooth manifolds of dimension 4 and 2,respectively. A smooth map f : M → B is called a broken Lefschetz fibration if it satisfies thefollowing conditions:(1) ∂M = f − ( ∂B );(2) f has at most the following types of singularities: • ( z , z ) ξ = z z , where ( z , z ) (resp. ξ ) is a complex local coordinate of M (resp. B ) compatible with its orientation; NOTE ON SECTIONS OF BROKEN LEFSCHETZ FIBRATIONS 3 • ( t, x , x , x ) ( y , y ) = ( t, x + x − x ), where ( t, x , x , x ) (resp. ( y , y )) is areal local coordinate of M (resp. B ).The singularities in the condition (2) of the definition are called a Lefschetz singularity andan indefinite fold singularity , respectively. For a broken Lefschetz fibration f , we denote by C f (resp. Z f ) the set of Lefschetz singularities (resp. indefinite fold singularities) of f . We call f a Lefschetz fibration if Z f = ∅ .In this paper, we will call broken Lefschetz fibrations (resp. Lefschetz fibrations) BLF (resp.LF), for short.Let f : M → S be a BLF. We assume that the restriction of f to the set of singularities isinjective, Z f is connected and all the fibers of f are connected. Then the set Z f is either theempty set or an embedded circle in M . If Z f is empty, f is an LF over S . If Z f is an embeddedcircle, the image f ( Z f ) divides the target 2-sphere into two 2-disks. We denote by νf ( Z f ) atubular neighborhood of f ( Z f ) and we put S \ int νf ( Z f ) = D a D , where D and D are 2-disks. It is easy to see that the genus of a regular fiber of the fibration f : f − ( D i ) → D i is just one higher than that of f : f − ( D j ) → D j . We call f − ( D i ) (resp. f − ( D j )) the higher side (resp. lower side ) of f and f − ( νf ( Z f )) the round cobordism of f . Definition 2.2.
A BLF f : M → S is said to be simplified if it satisfies the following conditions:(1) f | Z f ∪C f is injective;(2) Z f is connected and all the fibers of f are connected;(3) If Z f is not empty, C f is contained in the higher side of f .For a simplified BLF f , the genus of a regular fiber in the higher side of f is called the genus of f . The following lemma was proved by Baykur [4]: Lemma 2.3 (Baykur [4]) . Let f be a simplified BLF and we denote the higher side and theround cobordism of f by M h and M r , respectively. Then M h ∪ M r is obtained by -handleattachment to M h followed by -handle attachment. Moreover, the attaching circle of the -handle is a non-separating simple closed curve in a regular fiber of res f : M h → D and theframing of the -handle is along the regular fiber. We call an attaching circle of the 2-handle in the above lemma a vanishing cycle of theindefinite fold of f .2.2. Monodromy representations.
Let f : M → B be a genus- g LF and C f the set ofLefschetz singularities. We fix a point y ∈ B \ f ( C f ). Then a certain homomorphism ̺ f : π ( B \ f ( C f ) , y ) → M g , called a monodromy representation of f , is defined, where M g =Diff + Σ g / Diff +0 Σ g is the mapping class group of the genus- g closed oriented surface. (for theprecise definition of this homomorphism, see [7]). KENTA HAYANO
We assume that B = D and we put f ( C f ) = { y , . . . , y n } . We take embedded paths α , . . . , α n in D satisfying the following conditions: • each α i connects y to y i ; • if i = j , then α i ∩ α j = { y } ; • α , . . . , α n appear in this order when we travel counterclockwise around y .We obtain a i ∈ π ( D \{ y , . . . , y n } , y ) ( i = 1 , . . . , n ) by connecting a counterclockwise circlearound y i to y by using α i . We put W f = ( ̺ f ( a ) , . . . , ̺ f ( a n )) ∈ M gn . This sequence is calleda Hurwitz system of f . Kas proved in [10] that each ̺ f ( a i ) is the right-handed Dehn twist alonga simple closed curve c i in Σ g . c i is called a vanishing cycle of y i .Let f : M → S be a simplified BLF with Z f = ∅ and M h the higher side of f . Then therestriction of f to M h is an LF over D . So the monodromy representation and a Hurwitzsystem of this LF can be defined and are called the monodromy representation and a Hurwitzsystem of f , respectively. Lemma 2.4 (Baykur [4]) . Let f : M → S be a simplified BLF and ̺ f a monodromy represen-tation of f . Then a vanishing cycle c of the indefinite fold of f is preserved by ̺ f ([ ∂D ]) up toisotopy. We denote by M g ( γ ) the subgroup of M g which consists of elements represented by mapspreserving the simple closed curve γ in Σ g up to isotopy. The above lemma says that ̺ f ([ ∂D ])is in M g ( c ) for a vanishing cycle c of the indefinite fold of f . There is a natural homomorphism ϕ c : M g ( c ) → M g − defined by cutting the surface Σ g along c and pasting two 2-disks alongthe boundary. Lemma 2.5 (Baykur [4]) . The element ̺ f ([ ∂D ]) is in the kernel of ϕ c . Conversely, for asequence of simple closed curves c, c , . . . , c n in Σ g satisfying t c · · · · · t c n ∈ Ker ϕ c , there existsa simplified BLF f : M → S such that a Hurwitz system of f is ( t c , . . . , t c n ) and a vanishingcycle of the indefinite fold of f is c . Remark 2.6.
Such a simplified BLF f is not unique even up to diffeomorphism of the totalspace. Indeed, there exist infinitely many simplified BLFs such that Hurwitz systems of thesefibrations are all equivalent but the total spaces of these fibrations are mutually not diffeomorphic(see [5] or [8]). 3. Infinitely many homotopy classes of sections
To prove Theorem 1.2, we first give genus- g simplified BLF f g : M g → S and look at theset [ S , M g ]. We then construct a family of its sections and prove that any two sections in thefamily are not homotopic. (Proof of Theorem 1.2) : For g ≥
2, we denote by f g : M g → S a simplified BLF as shown inFigure 3.1. This diagram describes the total space of a simplified BLF whose Hurwitz systemis ( t µ , t µ ), where µ ⊂ Σ g is a simple closed curve described in Figure 3.2. NOTE ON SECTIONS OF BROKEN LEFSCHETZ FIBRATIONS 5
Figure 3.1.
The diagram of the total space M g of f g . 2 g Figure 3.2.
The simple closed curve µ in Σ g .We can change the diagram of M g as shown in Figure 3.3, and obtain: M g ∼ = S × Σ g − ♯S × S ♯ CP . To analyze the set [ S , M g ], we first look at the group π ( M g , p ) for a fixed point p ∈ M g .Let X g be a CW-complex obtained by attaching three 4-cells to M g along the three attachingregions of connected sum (see Figure 3.4).Let ι : M g → X g be the natural inclusion. By the cellular approximation theorem (for thistheorem, see [9]), the following map is isomorphism: ι ∗ : π ( M g , p ) → π ( X g , p ) . Since X g is homotopic to S × Σ g − ∨ S × S ∨ CP ∨ CP , we obtain: π ( X g , p ) ∼ = π ( S × Σ g − ∨ S × S ∨ CP ∨ CP , p ) ∼ = π ( S × Σ g − ∨ S ∨ S ∨ CP ∨ CP , p ) , where the second isomorphism is obtained by the cellular approximation theorem. We put Y g = S × Σ g − ∨ D ∨ S ∨ CP ∨ CP and denote by j : X g → Y g the inclusion map. Since D is contractible and S consists of the 0-cell and the 3-cell, we obtain: π ( Y g , p ) ∼ = π ( S × Σ g − ∨ CP ∨ CP , p ) . We denote by W g the universal cover of S × Σ g − ∨ CP ∨ CP . W g is obtained by attachingcountably many CP ∨ CP to S × D , where D is the universal cover of the closed surface, andis homotopic to S W µ ∈ π (Σ g − , q ) ( CP ∨ CP ) µ (see Figure 3.5). KENTA HAYANO
Figure 3.3.
The diagram of M g . Figure 3.4.
Left: the figure describing M g . Right: the figure describing X g .The shaded parts represent the attached 4-cells.In general, the second homotopy group of a CW-complex is isomorphic to that of the universalcover of the complex. Thus, we obtain: π ( S × Σ g − ∨ CP ∨ CP , p ) ∼ = π ( S _ µ ∈ π (Σ g − , q ) ( CP ∨ CP ) µ , p ) ∼ = Z M µ ∈ π (Σ g − , q ) ( Z ⊕ Z ) µ . NOTE ON SECTIONS OF BROKEN LEFSCHETZ FIBRATIONS 7
Figure 3.5.
Left: the figure describing W g . Right: the wedge sum of S andcountably many CP ∨ CP , which is obtained by collapsing D to a point.Eventually, we obtain the following group homomorphism:Φ = p − ∗ ◦ j ∗ ◦ ι ∗ : π ( M g , p ) → Z M µ ∈ π (Σ g − , q ) ( Z ⊕ Z ) µ . If Φ( s ) = ( l, ( m µ , n µ ) µ ) ∈ Z L µ ∈ π (Σ g − , q ) ( Z ⊕ Z ) µ for an element s ∈ π ( M g , p ), we haveΦ( γ · s ) = ( l, ( m µ , n µ ) λ · µ ) for an element γ = ( λ, z ) ∈ π (Σ g − , q ) ⊕ Z ∼ = π ( M g , p ).For an integer n , let σ n : S → M g be a section whose image intersects the boundary ofthe lower side of f g at the locus illustrated in Figure 3.6. Such a section exists since we cantrivialize the locus illustrated in Figure 3.6 in the boundary of a regular neighborhood of aregular fiber in the higher and lower side of f g . Figure 3.6. the bold curve represents the section σ n .We assume that the image of σ n contains the base point p , and regard σ n as an element in π ( M g , p ). By construction of σ n , there exists an element γ n such that Φ( γ n · σ n ) is equal to(1 , ( δ ,µ , δ µ n ,µ ) µ ), where δ ν,µ is equal to 1 if ν = µ and 0 otherwise, and µ ∈ π (Σ g − , q ) isthe element described in Figure 3.7 KENTA HAYANO
Figure 3.7. If n is not equal to m , (1 , ( δ ,µ , δ µ n ,µ ) µ ) is not equal to (1 , ( δ ,µ , δ µ m ,µ ) λ · µ ) for any elements λ ∈ π (Σ g − , q ) \ { } . This means that σ n is not homotopic to σ m if n is not equal to m . Thiscompletes the proof of Theorem 1.2. (cid:3) Self-intersection of sections
We denote by Σ g, the compact oriented surface with connected boundary and by δ a simpleclosed curve in Σ g, parallel to the boundary. Let M g, be the mapping class group of Σ g, . Itis known that there exists the natural surjective homomorphism ψ : M g, → M g induced bythe inclusion map i : Σ g, → Σ g . For a non-separating simple closed curve ˜ c in Σ g, , we define M g, (˜ c ) and ˜ ϕ ˜ c : M g, (˜ c ) → M g − , as we define M g ( c ) and ϕ c . Lemma 4.1.
Let ˜ d, ˜ d , . . . , ˜ d n be simple closed curves in Σ g, . Suppose that these simple closedcurves satisfy the following conditions:(1) ˜ d is non-separating;(2) t ˜ d · · · · · t ˜ d n ∈ M g, ( ˜ d ) ;(3) ϕ ˜ d ( t ˜ d · · · · · t ˜ d n ) = t δk , for some integer k .Then there exists a simplified BLF f : M → S such that f has a section σ with σ = − k .(Proof ) : We prove this lemma by constructing an explicit simplified BLF satisfying the desiredcondition. We take a 2-disk D in Σ g and we identify Σ g, with Σ g \ int D . We denote by A thecollar neighborhood of ∂ Σ g, in Σ g, . We fix an identification D ∼ = D and A ∼ = S × [1 ,
2] sothat ∂ Σ g, corresponds to S × { } in A . Then the map t δk is represented by the following map: x x ( x ∈ Σ g, \ A ) , (exp( √− θ + 2 πk √− − s )) , s ) ( x = (exp( √− θ ) , s ) ∈ A ∼ = S × [1 , . We first construct an LF over D by attaching n D × Σ g along i ( ˜ d ) , . . . , i ( ˜ d n )in a regular fiber of S × Σ g ⊂ D × Σ g with framing − t ˜ d · · · · · t ˜ d n ∈ M g, ( ˜ d ), wecan obtain a BLF over D by round 2-handle attachment (for details about this construction,see [4]).By the condition (3) in the statement, the boundary of the resulting BLF is described asfollows: Σ g, × I/ (( x, ∼ ( t δk ( x ) , ∪ D × I/ (( x, ∼ ( x, . Moreover, this BLF has a section ˜ σ whose boundary is { } × I/ (( x, ∼ ( x, ∈ D isthe center of the 2-disk. NOTE ON SECTIONS OF BROKEN LEFSCHETZ FIBRATIONS 9
To obtain a simplified BLF, we attach the trivial bundle Σ g × D to the above BLF by themap Φ : Σ g × I/ (( x, ∼ ( x, → Σ g, × I/ (( x, ∼ ( t δk ( x ) , ∪ D × I/ (( x, ∼ ( x, x, t ) = ( x, t ) ( x ∈ Σ g, \ A ) , ((exp( √− θ + 2 πk √− t ( s − , s ) , t ) ( x = (exp( √− θ ) , s ) ∈ A ) , ( r exp( √− θ − πk √− t ) , t ) ( x = r exp( √− θ ) ∈ D ) . The resulting simplified BLF has a section σ = { } × D ∪ Φ ˜ σ . By the construction, theself-intersection of σ is equal to − k . This completes the proof of Lemma 4.1. (cid:3) (Proof of Theorem 1.4) : We take simple closed curves ˜ c , . . . , ˜ c g , ˜ c g +1 , , ˜ c g +1 , in Σ g, as shownin Figure 4.1. Figure 4.1.
There exist the following relations in M g, for g ≥ t ˜ c · · · · · t ˜ c g − ) g − = t ξ ,(2) ( t ˜ c · · · · · t ˜ c g − ) g = t ˜ c g +1 , · t ˜ c g +1 , ,(3) ( t ˜ c · · · · · t ˜ c g ) g +1 = h ,where ξ is the simple closed curve described in Figure 4.2 and h is the element of M g, asshown in Figure 4.2. Figure 4.2. h twists the left side of the curve δ and fixes the right side of δ in the figure. By using these relations, we obtain the following relation: t ˜ c g · · · · · t ˜ c · t c · t ˜ c · · · · · t ˜ c g = t − c g +1 , · t − c g +1 , · h. Since h = t δ , we obtain: ( t ˜ c · · · · · t ˜ c g ) (4 g +2) n = t nδ , ( t ˜ c g · · · · · t ˜ c · t c · t ˜ c · · · · · t ˜ c g ) n = t − n ˜ c g +1 , · t − n ˜ c g +1 , · t nδ , ( t ˜ c g · · · · · t ˜ c · t c · t ˜ c · · · · · t ˜ c g ) n · ( t ˜ c · · · · · t ˜ c g − ) g − n = t − n ˜ c g +1 , · t − n ˜ c g +1 , · t nξ · t nδ , where n is a positive integer. The right side of the above equations are in M g, (˜ c g +1 , ). Since˜ ϕ ˜ c g +1 , ( t ˜ c g +1 , ) = 1, ˜ ϕ ˜ c g +1 , ( t ξ ) = ˜ ϕ ˜ c g +1 , ( t δ ) = ˜ ϕ ˜ c g +1 , ( t ˜ c g +1 , ) = t δ and ˜ ϕ ˜ c g +1 , ( h ) = h , weobtain: ˜ ϕ ˜ c g +1 , ( t nδ ) = t nδ , ˜ ϕ ˜ c g +1 , ( t − n ˜ c g +1 , · t − n ˜ c g +1 , · t nδ ) = t − nδ , ˜ ϕ ˜ c g +1 , ( t − n ˜ c g +1 , · t − n ˜ c g +1 , · t nξ · t nδ ) = 1 . Thus, the conclusion holds by Lemma 4.1. (cid:3) Spin structures
In this section, we discuss spin structures of total spaces of simplified BLFs.Let f : M → S be a simplified BLF. Denote by F ⊂ M a regular fiber in the lower side of f . A homology class S ∈ H ( M ; Z ) is called a dual of F if the intersection number S · [ F ] isequal to 1. If f has a section σ : S → M , the element [ σ ( S )] is a dual of F . It is also easyto see that a dual of F exists if the union of the higher side and the round cobordism of f issimply connected. Theorem 5.1.
Let f : M → S be a genus- g simplified BLF and F a regular fiber in the lowerside of f . We denote by d , . . . , d n ⊂ Σ g and d ⊂ Σ g vanishing cycles of Lefschetz singularitiesand the indefinite fold of f , respectively. Suppose that there exists a dual S ∈ H ( M ; Z ) of F .Then M admits a spin structure if and only if the following two conditions hold:(a) there exists a quadratic form q : H (Σ g ; Z / Z ) → Z / Z with respect to the intersection formof Σ g such that q ( d ) = 0 and q ( d i ) = 1 for all i ∈ { , . . . n } ;(b) the self-intersection of S is even.(Proof ) : We first prove that the condition (a) in the statement holds if and only if the union ofthe higher side and the round cobordism of f admits a spin structure. Let ˜ F be a regular fiberin the higher side of f and ν ˜ F ∼ = D × Σ g a regular neighborhood of ˜ F . It is known that ν ˜ F may admits exactly 2 g distinct spin structures and that there exists one to one correspondencebetween the set of equivalence classes of spin structures of ν ˜ F and the set of quadratic forms q : H (Σ g ; Z / Z ) → Z / Z . For a given spin structure s of ν ˜ F , the corresponding quadraticform q s is defined as follows: for an element γ ∈ H (Σ g ; Z / Z ), we take a simple closed curve c ⊂ Σ g ∼ = ˜ F which represents γ . Then q s ( γ ) is equal to 0 if the restriction of s to c can be NOTE ON SECTIONS OF BROKEN LEFSCHETZ FIBRATIONS 11 extended to the spin structure of the 2-disk whose boundary is c and is equal to 1 otherwise(the reader should turn to [15] for more details about this correspondence).By the argument in [15], the higher side of f admits a spin structure if and only if thereexists the quadratic form q : H (Σ g ; Z / Z ) → Z / Z such that q ( d i ) = 1 for all i = 1 , . . . , n .By Lemma 2.3, the union of the higher side and the round cobordism of f is obtained byattaching a 2-handle and a 3-handle to the higher side. Moreover, it is easy to see that theattaching map of the 2-handle preserves the spin structure obtained by restricting s to thesimple closed curve d . So we can extend s to the 2-handle if and only if s | d can be extendedto the bounding 2-disk. Since the attaching region of the 3-handle is diffeomorphic to S × D and has the unique spin structure, we can extend s to the round cobordism of f if and only if q ( d ) = 0. So the condition (a) is equivalent to the condition that the union of the higher sideand the round cobordism of f admits a spin structure.Now we are ready to prove Theorem 5.1. If M admits a spin structure, then the union ofthe higher side and the round cobordism of f also admits a spin structure. So the condition (a)holds. Since the intersection form of M is even, the self-intersection of S must be even.The converse direction is easily proved by the same argument as in [15]. (cid:3) Remark 5.2.
In [17], Williams introduced a surface diagram (Σ g , Γ) of a 4-manifold M , where g ≥ γ , . . . , γ k ) is a Z /k Z -indexed collection of simple closed curves in Σ g . Thisdiagram is defined by using a simplified purely wrinkled fibration f : M → S . The simple closedcurves in Γ represent the vanishing cycles of indefinite fold of f (for more details, see [17]).By using the modification defined by Lekili [11], we can change indefinite cusps into Lef-schetz singularities and indefinite folds and we obtain the simplified BLF h : M → S froma simplified purely wrinkled fibration f with the surface diagram (Σ g , Γ = ( γ , . . . , γ k )). Let W h = ( d , . . . , d k ) be the Hurwitz system of h , then the class [ d i ] ∈ H (Σ g ; Z / Z ) is equal to[ γ i ] + [ γ i +1 ]. For any quadratic form q : H (Σ g ; Z / Z ) → Z / Z , the following equation holds: q ([ d i ]) = q ([ γ i ]) + q ([ γ i +1 ]) + [ γ i ] · [ γ i +1 ]= q ([ γ i ]) + q ([ γ i +1 ]) + 1 . So q ( d i ) is equal to 1 if and only if q ( γ i ) = q ( γ i +1 ). Thus, we obtain the following corollary. Corollary 5.3.
Let f : M → S be a simplified purely wrinkled fibration and (Σ g , Γ) a surfacediagram of M induced by f . Denote by F a regular fiber of the lower side of f . We assumethat there exists a dual S ∈ H ( M ; Z ) of F . Then M admits a spin structure if and only if thefollowing conditions hold:(a) there exists a quadratic form q : H (Σ g ; Z / Z ) → Z / Z such that q ( d ) = 0 for all d ∈ Γ ;(b) the self-intersection of S is even. In the rest of this section, we will give some applications of Theorem 5.1.
Example 5.4.
For an integer n and a positive even integer g = 2 k , we denote by f g,n : M → S the genus- g simplified BLF constructed in the proof of Theorem 1.4 as a fibration with a section of square n . The Hurwitz system of f g,n is given as follows:( t c · · · · · t c g ) (4 g +2) | n | (if n is negative) , ( t c g · · · · · t c · t c · t c · · · · · t c g ) · ( t c · · · · · t c g − ) g − (if n is zero) , ( t c g · · · · · t c · t c · t c · · · · · t c g ) n (if n is positive) , where the simple closed curves c , . . . , c g +1 is described in Figure 5.1. The group H (Σ g ; Z / Z )is generated by the elements [ γ ] , . . . , [ γ g ], where γ , . . . , γ g ⊂ Σ g is simple closed curvesdescribed in Figure 5.1. Figure 5.1. simple closed curves on Σ g Let q : H (Σ g ; Z / Z ) → Z / Z be the quadratic form with respect to the intersection form ofΣ g such that q ([ γ i ]) = 1 for all i = 1 , . . . , g , q ([ γ j − ]) = 1 and q ([ γ j − ]) = 0 for all j = 1 , . . . , k .Since [ c ] = [ γ ], [ c g +1 ] = [ γ g − ], [ c i ] = [ γ i ] ( i = 1 , . . . , g ) and [ c j +1 ] = [ γ j − ] + [ γ j +1 ]( j = 1 , . . . , g − q ([ c i ]) as follows: q ([ c ]) = q ([ γ ]) = 1 ,q ([ c g +1 ]) = q ([ γ k − ]) = 0 ,q ([ c i ]) = q ([ γ i ]) = 1 ( i = 1 , . . . , g ) ,q ([ c j +1 ]) = q ([ γ j − ]) + q ([ γ j +1 ]) + [ γ j − ] · [ γ j +1 ]= 1 + 0 = 1 ( j = 1 , . . . , g − . So q satisfies the condition (a) of (ii) in Theorem 5.1 for f g,n . Moreover, f g,n has a section ofsquare n . Thus, the total space of f g,n admits a spin structure if n is even.We can completely classify spin genus-1 simplified BLF. NOTE ON SECTIONS OF BROKEN LEFSCHETZ FIBRATIONS 13
Proposition 5.5.
Let f : M → S be a genus- simplified BLF. We assume that M admitsa spin structure and that f has both Lefschetz and indefinite fold singularities. Then M isdiffeomorphic to ♯kS × S for some k ≥ . Remark 5.6.
Moishezon and Kas completely classified genus-1 LFs over S . Baykur, Kamada[5] and the author [8] classified genus-1 simplified BLFs without Lefschetz singularities. So allwe need to consider is the case f has both Lefschetz and indefinite fold singularities. (Proof of Proposition 5.5) : We take simple closed curves c , c ⊂ T so that the class [ c ] , [ c ] ∈ H ( T ; Z ) is a generator of H ( T ; Z ) and that c · c = 1. Denote by X i ∈ M = Diff + ( T ) / (isotopy)( i = 1 ,
2) the right-handed Dehn twist along c i . When we identify M with SL (2 , Z ) by a suit-able isomorphism, X and X correspond to the matrices ! and −
10 1 ! , respectively.We define the sequences of elements of SL (2 , Z ) S r and T ( n , . . . , n s ) as follows: S r = ( X , . . . , X ) ( r X stand in a line. ) ,T ( n , . . . , n s ) = ( X − n X X n , . . . , X − n s X X n s ) . By Theorem 3.11 in [8], we can assume that a Hurwitz system W f of f is equal to S r · T ( n , . . . , n s ) for some r, n , . . . , n s ∈ Z and that w ( W f ) corresponds to ± X n for n ∈ Z ,where w ( W f ) ∈ SL (2 , Z ) is the product of all elements in W f If r were not equal to 0, M would contain CP as a connected sum component and M wouldnot be spin. So we have r = 0.The vanishing cycles of Lefschetz singularities (resp. indefinite fold) of f are t n c ( c ) , . . . , t n s c ( c )(resp. c ). By the Picard-Lefschetz formula, we obtain:[ t n i c ( c )] = [ c ] + n i [ c ] ∈ H ( T ; Z ) . By Theorem 5.1, there exists a quadratic form q : H ( T ; Z / Z ) → Z / Z such that q ([ c ]) = 0and q ([ c ] + n i [ c ]) = 1. On the other hand, there exists exactly two quadratic forms q , q which satisfy q j ([ c ]) = 0 ( q ([ c ]) = 0, while q ([ c ]) = 1). q j ([ c ] + n i [ c ]) is calculated asfollows: q j ([ c ] + n i [ c ]) = q j ([ c ]) (if n i is even) ,q j ([ c ] + [ c ]) = q j ([ c ]) + 1 (if n i is odd) , = n i :even, j = 0 or n i :odd, j = 1) , n i :odd, j = 0 of n i :even, j = 1) . Eventually, the integers n , . . . , n s have same parity. In particular, the integers n − n , . . . , n s − − n s are all even.It is known that the group P SL (2 , Z ) has the following presentation: P SL (2 , Z ) = < a, b | a , b > ∼ = Z / Z ∗ Z / Z . Let p : SL (2 , Z ) → P SL (2 , Z ) be the natural projection. Then x = p ( X ) = aba and x = p ( X ) = ba . Since w ( W f ) = ± X n , we obtain: X − n X X n − n · · · · · X n s − − n s X X n s = ± X n , ⇒ x − n x x n − n · · · · · x n s − − n s x x n s = x n , ⇒ x x n − n · · · · · x n s − − n s x = x m , where m = n + n − n s . Lemma 5.7.
Suppose that n i − n i +1 = 2 for all i ∈ { , . . . , s − } . Then x x n − n · · · · · x n s − − n s x is equal to bS or a ba bS , where S = w · · · · · w k and ( w , . . . , w k ) is a reducedsequence (i.e. { w i , w i +1 } = { a, b } or { a , b } ) such that w = a or a .(Proof of Lemma 5.7) : We prove this statement by induction on s .We first look at the case s = 2. x x n − n x is calculated as follows: x x n − n x = ba · a ( ba ) n − n − ba · ba (if n − n ≥ ,ba · a ( ba ) − n + n − ba · ba (if n − n ≤ , = a ba ( ba ) n − n − baba (if n − n ≥ , ( ba ) − n + n ba ba (if n − n ≤ . So the statement holds.We then look at the general case. By the induction hypothesis, we obtain: x x n − n · · · · · x n s − − n s x = bS or a ba bS, where S is the product of a reduced sequence starting from a or a . We can calculate x x n − n as follows: x x n − n = ba · a ( ba ) n − n − ba (if n − n ≥ ,ba · a ( ba ) − n + n − ba (if n − n ≤ , = a ba ( ba ) n − n − ba ba (if n − n ≥ , ( ba ) − n + n ba (if n − n ≤ , Hence, we obtain: x x n − n · · · · · x n s − − n s x = a ba ( ba ) n − n − ba ba · bS (if n − n ≥ x x n − n · · · · · x n s − − n s x = bS ) ,a ba ( ba ) n − n − ba ba · a ba bS (if n − n ≥ x x n − n · · · · · x n s − − n s x = a ba bS ) , ( ba ) − n + n ba · bS (if n − n ≤ x x n − n · · · · · x n s − − n s x = bS ) , ( ba ) − n + n ba · a ba bS (if n − n ≤ x x n − n · · · · · x n s − − n s x = a ba bS ) , NOTE ON SECTIONS OF BROKEN LEFSCHETZ FIBRATIONS 15 = a ba ( ba ) n − n − ba babS (if n − n ≥ x x n − n · · · · · x n s − − n s x = bS ) ,a ba ( ba ) n − n − babS (if n − n ≥ x x n − n · · · · · x n s − − n s x = a ba bS ) , ( ba ) − n + n ba bS (if n − n ≤ x x n − n · · · · · x n s − − n s x = bS ) , ( ba ) − n + n baba bS (if n − n ≤ x x n − n · · · · · x n s − − n s x = a ba bS ) . This completes the proof of Lemma 5.7. (cid:3)
By Lemma 5.7, x x n − n · · · x n s − − n s x would not be equal to x m if n i − n i +1 = 2 for all i ∈ { , . . . , s − } . Hence, we have n i − n i +1 = 2 for some i ∈ { , . . . , s − } and M contains S × S as a connected sum component. By applying this argument successively, we can completethe proof of Proposition 5.5. (cid:3) Acknowledgments.
The author would like to thank Hisaaki Endo for his helpful commentsfor the draft of this paper. The author also wishes to express his gratitude to Naoyuki Monden forhis many useful suggestions, especially on self-intersection of sections. The author is supportedby Yoshida Scholarship ’Master21’ and he would like to thank Yoshida Scholarship Foundationfor their support.
References [1] S. Akbulut, C¸ . Karakurt, Every 4-manifold is BLF, J. G¨okova. Geom. Topol. (2008), 83–106[2] D. Auroux, S. K. Donaldson and L. Katzarkov, Singular Lefschetz pencils, Geom. Topol. (2005), 1043–1114[3] R. ˙I. Baykur, Existence of broken Lefschetz fibrations, Int. Math. Res. Not. (2008)[4] R. ˙I. Baykur, Topology of broken Lefschetz fibrations and near-symplectic 4-manifolds, Pacific J. Math. (2009), 201–230[5] R. ˙I. Baykur, S. Kamada, Classification of broken Lefschetz fibrations with small fiber genera, preprint, arXiv:math.GT/1010.5814 [6] R. E. Gompf, Toward a topological characterization of symplectic manifolds, J. Symplectic Geom. (2004),no.2, 177–206[7] R. E. Gompf, A.I.Stipsicz, , Graduate Studies in Mathematics , AmericanMathematical Society, 1999[8] K. Hayano, On genus-1 simplified broken Lefschetz fibrations, Algebr. Geom. Topol. (2011), 1267–1322[9] A. Hatcher, Algebraic Topology , Cambridge University Press, 2001[10] A. Kas, On the handlebody decomposition associated to a Lefschetz fibration, Pacific J. Math. (1980),89–104[11] Y. Lekili, Wrinkled fibrations on near-symplectic manifolds, Geom. Topol. (2009), 277–318[12] T. Perutz, Lagrangian matching invariants for fibred four-manifolds: I, Geom. Topol. (2007), 759–828[13] I. Smith, Geometric monodromy and the hyperbolic disc, Q. J. Math. (2001), 217–228[14] A. I. Stipsicz, Indecomposability of certain Lefschetz fibrations, Proc. Amer. Math. Soc. (2001), 1499–1502[15] A. I. Stipsicz, Spin structures on Lefschetz fibrations, Bull. London Math. Soc. (2001), 466–472[16] J. D. Williams, The h –principle for broken Lefschetz fibrations, Geom. Topol. (2010), no.2, 1015–1063[17] J. D. Williams, Uniqueness of surface diagrams of smooth 4-manifolds, arXiv:math.GT/1103.6263 Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka560-0043, Japan
E-mail address ::