A note on signature of Lefschetz fibrations with planar fiber
Using theorems of Eliashberg and McDuff, Etnyre  proved that the intersection form of a symplectic filling of a contact 3-manifold supported by planar open book is negative definite. In this paper, we prove a signature formula for allowable Lefschetz fibrations over with planar fiber by computing Maslov index appearing in Wall’s non-additivity formula. The signature formula leads to an alternative proof of Etnyre’s theorem via works of Niederkrüger and Wendl  and Wendl . Conversely, Etnyre’s theorem, together with the existence theorem of Stein structures on Lefschetz fibrations over with bordered fiber by Loi and Piergallini , implies the formula.
Key words and phrases:4-manifold, Lefschetz fibration, signature
The signature of a smooth compact oriented -manifold is defined as the signature of the intersection form of . A signature formula for symplectic Lefschetz fibration over was studied by Smith . Endo  gave a local signature formula for hyperelliptic Lefschetz fibrations over closed surfaces. Endo and Nagami  defined the signature of relators in the mapping class group of a fiber to obtain a signature formula of Lefschetz fibrations over . An algorithm for computation of signature of Lefschetz fibrations over or was found by Ozbagci . Endo, Hasegawa, Kamada and Tanaka  showed that the signature of a Lefschetz fibration over a closed surface is equal to the signature of the corresponding chart. There are many results about signature of Lefschetz fibrations with closed fiber. However, there are few such results for Lefschetz fibrations with bordered fiber.
In this paper, we prove a signature formula for allowable Lefschetz fibrations over with planar fiber by computing the Maslov index appearing in Wall’s non-additivity formula . Let be a surface of genus with boundary components. Our main result is the following:
Let be an allowable Lefschetz fibration with fiber and vanishing cycles on . Then the following equality holds:
Here denotes the vector subspace of generated by the homology classes of .
Theorem 1.1 holds only in the case that the fiber of a Lefschetz fibration is planar.
Moreover, Theorem 1.1 implies the following corollaries.
For as in Theorem 1.1, the signature of is equal to .
The handle decomposition of consists of one 0-handle, 1-handles and 2-handles. The 1-handles correspond to the generators of and the boundary operator sends each 2-handle to the 1-chain of corresponding to the homology class of a vanishing cycle. Hence coincides with . Therefore we have from Theorem 1.1.
For as in Theorem 1.1, we have .
By Corollary 1.3 and the definition of signature, we get
Since the Euler characteristic of is related to the numbers of handles, we have
Hence we get
Since and are non-negative, the both of them are zero. ∎
Finally, we show that Corollary 1.4 implies the following theorem.
(Etnyre ) If is a symplectic filling of a contact 3-manifold supported by a planar open book, then .
Niederkrüger and Wendl  proved that any symplectic filling of a contact manifold supported by a planar open book is symplectically deformation equivalent to a blow up of a Stein filling. Moreover, Wendl  proved that any Stein filling of a contact manifold supported by such an open book admits an allowable Lefschetz fibration over with planar fiber. Since the blowing up preserves the negative definiteness, we get Theorem 1.5 from Corollary 1.4. ∎
Theorem 1.5 implies Theorem 1.1: By Loi and Piergallini , a Lefschetz fibration as in Theorem 1.1 admits a Stein structure. Therefore is a symplectic filling of the boundary . The restriction of an allowable Lefschetz fibration to the boundary is an open book of , then is a contact 3-manifold supported by a planar open book. Applying Theorem 1.5 to , we obtain Theorem 1.1.
2.1. Lefschetz fibration
Let be a smooth, compact and oriented 4-manifold.
A Lefschetz fibration is a smooth map such that:
are the critical values of , with a unique critical point of on , for each , and
about each and , there are local complex coordinate charts centered at and compatible with the orientations of and such that can be expressed as .
A Lefschetz fibration is called allowable if each vanishing cycle of the Lefschetz fibration represents a non-trivial homology class of the fiber.
In this paper, we always assume that given Lefschetz fibrations are allowable.
For a given oriented surface , we can construct a Lefschetz fibration over with fiber as follows. We start with a trivial -bundle . We choose a point and identify with . We choose distinct points and properly embedded simple closed curves such that is a curve on for each . Gluing and a 4-dimensional 2-handle along with framing relative to the product framing of , we obtain a 4-manifold . This manifold admits a Lefschetz fibration over with fiber , vanishing cycle and critical value . Continuing this process for with all of the framings relative to the product framings of 2-handles counter-clockwisely from , the resulting manifold admits a Lefschetz fibration with fiber , vanishing cycles and critical values . We can naturally consider the smooth fiber bundle
The fundamental group of punctured disk is generated by loops around each puncture and the local monodromy is defined as the monodromy along each of such loops. The local monodromy around is the right handed Dehn twist along (for more detail, see the book  by Gompf and Stipsicz).
By the above construction, we obtain a handle decomposition of a Lefschetz fibration. The unique 0-handle, some 1-handles (depending on the genus and the number of boundary components of the fiber) and some 2-handles. It is useful to compute the Euler charecteristic of the Lefschetz fibration and we use this in Section 3.
2.2. Wall’s non-additivity
Now we explain a formula called Wall’s non-additivity formula.
Let be -manifolds, be -manifolds and be a -manifold such that
induces the orientations of , and otherwise:
For each , let be the inclusion and be the induced map of to the first homology group. Then we define and as follows:
Let be the intersection form of and we consider the bilinear map . Here and are elements of and is an element of satisfying for some . One can see that induces a well-defined symmetric bilinear map . The signature of is denoted by . We also write the signature of a -manifold as . Wall  proved the following theorem:
We mainly use Wall’s formula to prove Theorem 1.1.
3. Proof of main theorem
In this section, we consider homology group with coefficient and denote the homolgy classes of the curves on surfaces by the same symbols of those curves.
Let be a Lefschetz fibration with fiber and vanishing cycles on . Since local monodromies of a Lefschetz fibration fix the boundaries of regular fiber point wise, the smooth fiber bundle
is a trivial
-bundle over . Here is a copy of for each .
We take a bundle isomorphism
and define and as follows:
where is the manifold obtained by gluing of and by .
Under the above assumption, the following two propositions hold.
Proof of Theorem 1.1.
By the above construction, can be naturally extended to a Lefschetz fibration over with regular fiber . More precisely, is diffeomorphic to . This implies that the signature of is . Since the signature of is obviously , we have
We choose and fix a point in for each . We define the homology classes and in as and , respectively, where is the point correspond to . Under this notation, we always assume that and are oriented such that is .
The vector space is spaned by
For this generating set, satisfies
Proof of Proposition 3.1.
We note that Proposition 3.1 is equivalent to the positive definiteness of . Let be the homology class and dim . Since generates , we can write the basis of as for some . We take an arbitrary and write it as . Then we get
Therefore, if we assume , is 0 for each . Then we get
This means that is positive definite.
Next, we prove Proposition 3.2.
Proof of Proposition 3.2.
We set the standard basis of and define the isomorphism Let and be the matrices defined by and , respectively. The statement that the dimension of is equal to the dimension of is equiverent to that the rank of is equal to the rank of . Now we write by , we have
Recall that, by definition, the intersection number of and in is . Then we get the expression for as with respect to the generating set and this implies that
On the other hand, this is written as
Next, we introduce the following lemma for the proof of Lemma 3.3.
Let be the surface for . The boundary of can be identified with the disjoint union of copies of . The boundary of can be also identified with . The point , defined in the first of this section, can be considered as a point of . We set proper embedded arcs which connect to , respectively. For a given orientation preserving diffeomorphism of , denotes the quotient space . We take a set of generators of such that and for and (Fig 3.1).
Let be simple closed curves in Int and the product of Dehn twists . We express the first homology group of as so that the inclusion map from to satisfying the following conditions:
The set is a basis of and satisfies ,
Then we have
If g is 0, this can be written as:
where denotes the induced map of the inclusion to the homology group.
For the case , the basis of and are and , respectively. Thus we can naturally think of as a subgroup of . Thus can be defined.
Proof of Lemma 3.3.
We already note that is generated by and in the first part of this section, and by the definitions, and is given by:
By these results, the subspaces and of are given by:
Therefore the vector space , the quotient of by , can be writen as that in the statement.
Next we consider the bilinear form . For each , we denote the element by . By the obvious facts that and , satisfies
We lastly prove Lemma 3.4.
Proof of Lemma 3.4.
The expression of can be easily induced by the long exact sequence proved in Hatcher’s book  at Example 2.48:
Let and be topological spaces and and maps from to . Define an equivalence relation on by and by the quotient of by . Then the following sequence is exact:
where denotes the induced map of .
We use this sequence for and . Consequently, becomes the mapping torus .
Next we prove the statement for fixed . We denote by the projection . Let be a simplicial 2-chain expressing and and simplicial 1-chains expressing and , respectively. We orient these chains so that (Fig 3.2).
Then the homology class is zero in . By the definition of , the class represented by is equal to the class represented by in . Here is the induced map of to the simplicial chain complex. Hence we get
For the case that is 0, we will prove
We take connected neighborhoods of in for . Attaching a -dimensional -handle along and together with and , we obtain a new surface diffeomorphic to (Fig 3.3).
Here denotes a simplicial 1-chain expressing the core of the in . We note that is spaned by and the intersection form satisfies
For a compact oriented suface , every orientation preserving diffeomorphism of is isotopic to a product of Dehn twists. Let be the right-handed Dehn twist along a simple closed curve . The action of on is given by the Picard-Lefschetz formula: for .
We will first prove the case that , and then the case of that is a product of Dehn twists. Let be a simple closed curve in . By the fact that is not in Int and the Picard-Lefschetz formula, we have
Then is a homology class in , it can be expressed by for some . Under this representation, it is not hard to see that equals to . Since and satisfy that equals to , is equal to . Hence we get
By the transposition of the term on the both sides of the above equality, this induces
Next, for any , we assume
We prove the statement by induction on . Let be and a simple closed curve in . The restrictions of these diffeomorphisms to are both the identity, so by the Picard-Lefschetz formula,
We can write as because of the Picard-Lefschetz formula, where is an element of . Then we get
By the transposition, it leads
On the other hand, is equal to and then, by the assumption, is equal to . Moreover, is 0, then vanishes. This implies that also vanishes. ∎
In this section, we give two examples of caluculation for the signature. Let be a positive integer and consider Lefschetz fibrations and with fiber . We denote the -th boundary component of by and the boundary of gluing region by Z.
The vanishing cycles of the first example are ordered set of the curves , where is a curve surrounding and for as in Fig 4.1.
The homology class of each vanishing cycle is and is equal to . We get: