Partial fiber sum decompositions and signatures of Lefschetz fibrations
Adalet Çengel Çağrı Karakurt
math.GT



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  Signatures of Lefschetz Fibrations]
Partial Fiber Sum Decompositions and Signatures of Lefschetz Fibrations


 Department of Mathematics, Boğaziçi University, Bebek 34342
 School of Mathematics, University of Minnesota, Minneapolis,MN, 55455
   
cenge002@umn.edu
 

 


 Department of Mathematics, Boğaziçi University, Bebek 34342
   
cagri.karakurt@boun.edu.tr


 
 



In his Ph.D. thesis, Burak Ozbagci described an algorithm for computing signatures of Lefschetz fibrations where the input is a factorization of the monodromy into a product of Dehn twists. In this note, we give a reformulation of Ozbagci's algorithm which becomes much easier to implement. Our main tool  is Wall's non-additivity formula applied to what we call partial fiber sum decomposition of a Lefschetz fibration over the 2-disk. We show that our algorithm works for bordered Lefschetz fibrations over disk and it yields a formula for the signature of branched covers where the branched loci are regular fibers. As an application, we give the explicit monodromy factorization of a Lefschetz fibration over disk whose total space has arbitrarily large positive signature for any positive fiber genus.    


 [
    Çağri Karakurt
    December 30, 2023
=====================



  
 



§ INTRODUCTION


Donaldson's ground breaking result says that the study of symplectic 4-manifolds up to blow-up is equivalent to that of Lefschetz fibrations over the 2-sphere.  This relationship has been extended to the relative case by Akbulut-Ozbagci <cit.> and Loi-Piergallini  <cit.> who established a correspondence  between Stein manifolds and positive allowable Lefschetz fibrations. Working with Lefschetz  fibrations is more preferable as the topology of the total space, which is a  4-dimensional manifold, is completely determined by a monodromy factorization in mapping class groups, which can be understood through 2-dimensional techniques. It remains a good challenge for 4-dimensional topologist to compute invariants of a Lefschetz fibration out of its monodromy factorization. Signature is perhaps the simplest non-trivial  invariant among them. Recall that the signature σ(Y) of a compact oriented 4-manifold Y is the signature of the intersection form on the second homology group H_2(Y;).


Several computation techniques and formulas for signature of Lefschetz fibrations on 4-manifolds exist in the literature. For Lefschetz fibrations over ^2 with fiber genus g=1,2 Matsumoto <cit.>; for hyperelliptic Lefschetz fibrations over surfaces with fiber genus g ≥ 3, Endo <cit.> gave signature formulas. On the other hand, Endo and Nagami <cit.> showed that the signature of a Lefschetz fibration over ^2 can be calculated by using the signatures of relations contained in its monodromy. Ozbagci gave an algorithm which works for Lefschetz fibrations over ^2 or ^2 with closed fibers <cit.>. Recently, Miyamura <cit.> presented a formula for  Lefschetz fibrations over ^2 with planar fibers. In this paper, we present an algorithm for computing the signature of Lefschetz fibrations over ^2 of any genus g fibers which can be closed or bordered.

A Lefschetz fibration on a smooth 4-manifold Y has a handlebody decomposition determined by a sequence of vanishing cycles <cit.>. Ozbagci used this description and Wall non-additivity formula in his algorithm <cit.>. The singular fibers of a Lefschetz fibration are obtained by attaching 2-handles along the correspondig vanishing cycles in different regular fibers Σ_g ×^2. For each attachment there is a signature contribution in the set {-1,0,+1} and their sum  gives the signature of the total space. Computing Ozbagci's local signature contribution is not a straightforward matter, as it is necessary to understand a presentation of the homology of the complement of the vanishing cycle in the boundary at each step.  The purpose of the present paper is to give an alternative method for computing these local signature using elementary linear algebra only. 

To state the main result, we fix our terminology first. Let Σ_g^b be a compact, connected, oriented surface of genus g with b boundary components and Mod(Σ_g^b) denote its mapping class group, the group consisting of isotopy classes of orientation-preserving self-diffeomorphisms of Σ_g^b which restrict to the identity on ∂Σ_g^b. We write Σ_g=Σ_g^0 and Mod(Σ_g)= Mod(Σ_g^0) for simplicity.

It is well-known that when the surface has no boundary components, the group Mod(Σ_g) naturally  acts on _1 (Σ_g;) preserving the algebraic intersection form Q. Hence we have a canonical homomorphism from  Mod(Σ_g) to  (2g, ) called the symplectic representation of Mod(Σ_g).  When b≥ 1 we pick a point  p_j from each boundary component and  denote the set {p_1,…,p_b} of distinguished points by 𝔡. The relative version of the symplectic representation for Mod(Σ_g^b) by   its action on _1 (Σ_g^b;𝔡;) yields a homomorphism  Mod(Σ_g^b)→(2(g+b-1),).
It is an easy exercise to see that the above representation is obtained by connecting a distinguished boundary component  of Σ_g^b to the others by adding one handles reducing the total number of boundary components to one and then capping off with a disk to obtain a closed surface of genus g+b-1  and seeing Mod(Σ_g^b) as a subgroup of Mod(Σ_g+b-1). 


Let γ be a simple closed curve on Σ_g^b. We denote by t_γ, and t_γ^* the Dehn twist about γ and respectively the image of the Dehn twist t_γ under this symplectic representation. The sign of a  real number x is denoted by sign(x)∈{-1,0,1}.   Our main result is the following.



Given a Lefschetz fibration f_n:Y_n→^2 with regular fiber Σ_g^b and monodromy factorization ϕ=t_γ_n⋯ t_γ_1; the signature σ(Y_n) of the total space Y_n is given by the algorithm below.

For every k∈{1,…, n},   we determine σ_k ∈{ -1,0,1} as follows. If [γ_k]=0  in _1 (Σ_g^b,𝔡 ; ),  then let σ_k=0. If [γ_k] ≠ 0 then check if there exists a homology class [x_k] ∈_1 (Σ_g^b,𝔡; ) solving the following linear equation:

    (Id-t_γ_k^* ⋯ t_γ_1^*)  ([x_k]) =[γ_k]

where Id is the identity element in the corresponding symplectic group.  If no such solution exists, let σ_k=0. If there is a solution, let 

    σ_k= sign(1+Q([γ_k], [x_k])).



Then the signature is given by

    σ(Y_n)=-∑_k=1^nσ_k - 
    #{ k ∈{1,…, n}; [γ_k]=0    in _1 (Σ_g^b,𝔡; )}.








  * During the course of our proof, we will show that any solution for the linear equation (<ref>) gives the same  σ_k. 

  * When b=0 the above algorithm is in fact a reformulation of Ozbagci's algorithm. Our σ_k agrees with Ozbagci's local signature contributions for each vanishing cycle. A computer program written in SAGE implementing the above algorithm is available at our web sites. 

  * The above theorem can be modified to compute signatures of achiral Lefschetz fibrations. One just needs to change the sign of  σ_k's whenever the vanishing cycle  γ_k corresponds to an achiral singularity. 

  * We also  interpret σ_k as the   Maslov ternary index of some naturally occurring Lagrangian subspaces which are graphs of some symplectic maps associated with the monodromy of the Lefschetz fibration, <cit.>.   Equivalently, in the spirit of  <cit.>,  we show that σ_k corresponds to a special value of the Meyer's cocycle .

  * A systematic study of signatures of bordered Lefschetz fibrations is initiated by Miyamura, <cit.>. When we extend our results to relative case we benefited a lot from his ideas. 



Next we give a formula that computes the signatures of finite cyclic branched covers of Lefschetz fibrations over disk where the branch locus is a regular fiber. 

 For any linear automorphism φ, the eigenspace corresponding to a real number λ is denoted by E_λ(φ), which could be the trivial vector space if λ is not an eigenvalue. Consider a mapping class ϕ∈ Mod(Σ_g^b) of the surface. For any positive integer m, we can define a symmetric bilinear pairing 

    I_ϕ^m+1: _1(Σ_g^b, 𝔡;)/E_1(ϕ^m+1_*)×_1(Σ_g^b, 𝔡;)/E_1(ϕ^m+1_*)→


    I_ϕ^m+1(z_1,z_2)   =Q(ϕ^m_*(z_1),z_2)+Q(ϕ^m-1_*(z_1),z_2)+…+Q(ϕ_*(z_1),z_2)
       -Q(z_1,ϕ_*^m (z_2))-Q(z_1,ϕ_*^m-1 (z_2))-…-Q(z_1,ϕ_*(z_2))

for z_1,z_2 ∈_1(Σ_g^b, 𝔡;)/E_1(ϕ^m+1_*).
Denote the signature of I_ϕ^m+1 by σ_ϕ^m+1_c.
Let f:Y→^2 be the Lefschetz fibration with regular fiber Σ_g^b given by the monodromy factorization ϕ=t_l… t_1. Let f:Y→^2 be the  Lefschetz fibration with regular fiber Σ_g^b given by the monodromy factorization ϕ^n=(t_l… t_1)^n. Note that Y is the n-fold cyclic branched cover of Y branched along a regular fiber.

We have

    σ(Y)=nσ(Y)-∑_m=1^n-1σ_ϕ^m+1_c

  
Notice that when b≠ 0, the  sum on right hand side gives signature invariant of the fibered link ∂Σ_g^b due to Gordon, Litherland and Murasugi <cit.>. One can compute this invariant  from the Seifert matrix using a formula similar to above <cit.>. It is interesting to observe that the same can be done using the homological monodromy.   

Finally we give an example of Lefschetz fibration with positive signature. Such examples can easily be constructed from Stein manifolds whose handlebody diagrams are on  Legendrian knots with large positive Thurston-Bennequin number and by taking the corresponding positive allowable Lefschetz fibration, but it is not clear whether this approach yields small fiber genus. The important property of our example is that we can achieve arbitrarily large positive signature with fiber genus one. Our result is optimal in the sense that Lefschetz fibrations with planar fibers are known to have non-positive signature <cit.>. 


Given any g ≥ 1, b ≥ 0, n > 0, there exists a Lefschetz fibration over 𝔻^2 with fiber genus g and b boundary components, having 3n singular fibers,  whose  total space has signature  n. 




The first example of a positive signature Lefschetz fibration over 𝔻^2 with fiber genus one with one boundary component was found by Ozbagci which appeared in his unpublished notes. With our techniques, we are able to promote Ozbagci's example to arbitrary genus and boundary components. It remains an open question whether there exists a Lefschetz fibration over ^2 with positive signature. Thanks to Endo's result, if such a Lefschetz fibration exists then it  cannot be hyperelliptic, hence its fiber genus must be at least three. Another open problem is  whether the number of vanishing cycles in our result is optimal. More precisely, can one have a Lefschetz fibration over 𝔻^2 with signature n>0 having k<3n singular fibers? An easy argument due to Ozbagci shows this is impossible for n=1, but the other cases are still open.  










§ PRELIMINARIES




 §.§ Symplectic Vector Spaces



Let V be a real 2g-dimensional vector space. A skew-symmetric, non-degenerate bilinear form Q on V is called 
a symplectic form, and the pair (V, Q) is called a symplectic vector space. A basis {a_1,b_1, a_2, b_2 …, a_g, b_g,} of V is called a symplectic basis if it satisfies
Q(a_i,a_j)= Q(b_i,b_j)=Q(a_i,b_j)=δ_ij for 1 ≤ i,j ≤ g.

Consider ^2g with basis { x_1,y_1,…, x_g,y_g}.  Then with respect to the  standard symplectic form 

    Q_std=∑_n=1^g dx_i∧ dy_i,

(^2g,Q_std) is a symplectic vector space and { x_1,y_1,…, x_g,y_g} is a symplectic basis. Conversely any symplectic vector space (V, Q) has a symplectic basis giving an isomorphism  (V, Q)≅ (^2g,Q_std) for some g.
The linear symplectic group (2g,) is defined to be the group of linear automorphisms of ^2g preserving the symplectic form Q_std. In terms of matrices 

    (2g,)={ A ∈ GL(2g,) :  A^TJA=J}

where J is the 2g × 2g matrix: 
 

    J = 
    [         0         1         0 0   …   0         0;        -1         0         0 0   …   0         0;         0         0         0 1   …   0         0;         0         0        -1 0   …   0         0;         ⋮;         0         0         0 0   …   0         1;         0         0         0 0   …  -1         0 ].

The same definition can be made for any symplectic vector space (V,Q): The set of all automorphisms φ:V→ V satisfiying Q(φ(x), φ(y))=Q(x, y) is called the symplectic group of (V,Q) and is denoted by (V,Q). The symplectic basis theorem gives rise to an isomorphism  (V,Q) ≅(2g,) where g=dim(V)/2.






 A subspace L of a symplectic vector space (V,Q) is called Lagrangian if Q(x,y)=0 for every x,y ∈ L and dim(L)=dim(V)/2. Symplectic automorphisms naturally give rise to Lagrangian subspaces. 

For a given symplectic automorphism φ∈(V,Q) of the symplectic vector space (V,Q), the graph and respectively conjugate graph of φ are subspaces  defined by

    graph(φ)    ={(x,φ(x))   |   x∈ V },
    graph(φ)   ={(φ (x),x)   |   x∈ V }.




Both graph (φ)  and graph(φ) are  Lagragian subspaces of ^2g⊕^2g with the symplectic form Q ⊕ -Q

    ((x,y),(x', y') ↦ Q( x,x') - Q( y,y').


In our context, V will mostly refer to the first homology group with real coefficients of a (possibly disconnected) surface, or the relative homology group of a surface with boundary.   The intersection form on the homology with  coeffients induces a symplectic form on V. Note that reversing the orientation of the surface changes the sign of the symplectic form. 





 


 §.§ Maslov Triple (Ternary) Index



 <cit.>
For a given three Lagrangians A,B,C in a symplectic vector space (V,Q), the Maslov ternary index τ_V(A,B,C) is characterized by the following properties.



	
  *  Skew Symmetry: 
	For a permutation p of the three letters,
	
    τ_V (p(A), p(B), p(C))=Sign(p) ·τ_V(A,B,C),

	where Sign(p) is the sign of the permutation p.
	
  * Symplectic Additivity: For three Lagrangians A,B,C ⊂ V, and three Lagrangians A^',B^',C^'⊂ W,
	
    τ_V⊕ W(A ⊕ A^',B ⊕ B^',C ⊕ C^')=τ_V(A,B,C)+ τ_W(A^',B^',C^').

	
	
  * Symplectic Invariance:
	For a symplectic automorphism φ∈(V,Q),
	
    τ_V(A,B,C)=τ_V(φ(A),φ(B),φ(C)).

	
	
  * Normalization: 
	For the Lagrangians , (1+i),(i) in ^2= with the standard skew symplectic form Q_std in ,
	
    τ_(,(1+i),(i))=-1.

	
	


By <cit.>, there exists a unique system of functions τ_V(A,B,C) which satisfies the above Properties (<ref>) through (<ref>), so the  definition makes sense. One can think of the Maslov ternary index as a kind of “cross ratio” of triplets of Lagrangians in a symplectic vector space.   If (A,B) and (A',B') are two pairs of transverse Lagrangians in a symplectic vector space one can find a symplectic automorphism  φ such that φ(A)=A^' and φ(B)=B^'. However, the symplectic group does not act transitively on triples of Lagrangians. In fact The configuration of three transverse Lagrangians is completely determined by their index.  


It was also shown in <cit.> that the Maslov ternary index is the same as Wall's signature defect which we review now. Let  

    W=W_V(A,B,C)= B∩(C+A)/(B∩ C)+ (B∩ A).

Then a bilinear map Ψ^':B∩(C+A) × B∩(C+A) → is defined by Ψ^'(b,b^')=Q(b,c^') where a^'+b^'+c^'=0 for some a^'∈ A, c^'∈ C. Moreover any other choice of a^' and c^' yields the same value, so Ψ^' is a well-defined bilinear map on B∩(C+A). It is easy to see that Ψ^'(b,b^')=0 if b or b^' is in B∩ C+B∩ A. Hence Ψ^' gives rise to a symmetric nonsingular bilinear map Ψ on W.  Then the signature of Ψ equals the Maslov ternary index τ_V(A,B,C).  Note that the τ_V(A,B,C)=0 whenever dim W=0.  This simple observation  will be very useful throughout the paper. 








 §.§ Lefschetz fibrations

Let Y be a compact, oriented smooth 4-manifold. A Lefschetz fibration on Y is a smooth surjective map f:Y→ B such that:

	
  * { b_1,b_2, …, b_n } are the critical values of f inside B with p_i ∈ f^-1(b_i) a unique critical point of f  for each i, and
	
  * about each b_i and p_i, there are local complex coordinate charts agreeing with the orientations of Y and B such that locally f can be expressed as f(z_1,z_2)=z_1^2+z_2^2.

A bordered Lefschetz fibration is a Lefschetz fibration over ^2 where the fibers have non-empty boundary.  The boundary of a  bordered Lefschetz fibration defines an open book structure.

 
An open book structure (or decomposition) of a 3-manifold X is a surjective map  π: X →^2 such that π^-1(int) is a disjoint union of solid tori and 

	
  * π is a fibration over ^1=∂^2.
	
  * On each of the solid tori, π is the projection map ^1 ×^2 →^2. The centers of these solid tori are called binding.

	So in the complement of the binding π is a fibration over ^1, with fibers surfaces with boundary. Alternatively, we can define an open book structure on X just in terms of its pages and the monodromy (Σ_g^b, ϕ) of the fibration π: X→^1. Here Σ_g^b is an oriented surface with non-empty boundary and ϕ:Σ_g^b→Σ_g^b is a diffeomorphism which is identity on ∂Σ_g^b. Then X is the union of the mapping torus of ϕ and solid tori glued along their boundaries in the obvious way. The monodromy of the open books arising from the boundary of bordered Lefschetz fibrations  is a composition of right handed Dehn twists (coming from the vanishing cycles).

 

 Given any bordered Lefschetz  fibration f:Y→𝔻^2 with fiber Σ_g^b with b≥ 1, we  define its closure,  which is a new Lefschetz fibration f:Y→𝔻^2  with closed fibers of genus g+b-1, as follows. We first fiberwise attach 2-dimensional 1-handles along different components ∂Σ_g^b reducing the number of boundary components to one, trivially extending the monodromy over the one-handles.   Then cap off the remaining boundary component by a disk. 

  
 Notice that  when b≥ 1 the isomorphism _1(Σ_g^b,𝔡;ℝ)≅_1(Σ_g+b-1;) and the natural homomorphism Mod(Σ_g^b)→Mod(Σ _g+b-1) identifies  the symplectic representations of the monodromies of the original Lefschetz fibration and its closure. More precisely the basis of  _1(Σ_g^b,𝔡;ℝ) given on the left side of Figure <ref>  naturally corresponds to the basis  of _1(Σ_g+b-1;) and the homological actions of any element in Mod(Σ_g^b) to these basis are the same.









 §.§ Signature and Wall's formula
 

Let Y_1 and Y_2 be two compact oriented 4-manifolds. If a closed manifold Y is obtained by gluing along the whole boundaries of Y_1 and Y_2 via an orientation reversing diffeomorphism, then the signature of Y is equal to sum of the signatures of Y_1 and Y_2. This is Novikov additivity <cit.>.  On the other hand it is also possible to glue manifolds Y_1 and Y_2 along  common submanifold X_0 in their boundary where X_0 may itself have boundary. In this case, the resulting manifold Y is a 4-manifold with boundary and its signature is not the sum of the signatures of Y_1 and Y_2, the defect in this argument is a Maslov ternary index<cit.>. 









 §.§ Wall's Non-additivity Formula
 

Let Y, Y^-, Y^+ be 4-manifolds, X^0,X^-, X^+ be 3-manifolds and Z be a 2-manifold such that Y=Y^-∪ Y^+, and Y^-∩ Y^+=X^0. Moreover

    ∂ Y^-   =X^-∪ X^0, 
    ∂ Y^+   =X^0∪ X^+,
    ∂ X^-   =∂ X^0=∂ X^+=Z.


Suppose Y oriented, inducing orientations on Y^- and Y^+.  The rest is oriented as follows:

    ∂_* [Y^-]   =[X^0]-[X^-],
    ∂_* [Y^+]   =[X^+]-[X^0],
    ∂_* [X^-]   =∂_*[X^0]=∂_*[X^+]=[Z].





Let V=_1(Z;).  Define the following subspaces of V

    A   =ker (i_*: _1(Z;) →_1(X^-;),  
    	B   =ker (i_*:_1(Z;) →_1(X^0;),  
    	C   =ker (i_*:_1(Z;) →_1(X^+;),

Then dim  A= dim  B =dim  C= 1/2dim  V. Denote the  symplectic intersection form of V by Q and Q(A× A)=Q(B× B)=Q(C× C)=0. The subspaces A,B and C are Lagrangian subspaces  for Q. Then we have 


<cit.>
σ(Y)=σ(Y^-)+σ(Y^+)-τ_V(A,B,C).




§ SIGNATURE OF CLOSURE


The following statement allows us to reduce the proofs of our result to their respective special cases where the fibers are closed.
 
Signature of the total space of a bordered Lefschetz fibration is equal to that of its closure. 


We will use standard topological arguments about Lefschetz fibrations. For details, the reader can consult <cit.> where our method already appeared in.   Attaching  2-dimensional 1-handles to fibers corresponds to attaching 4-dimensional 1-handles to the total space, so does not change the signature. Hence it suffices to prove the statement in the case where fibers have connected boundary. 


Suppose f^+:Y^+→^2 is a bordered Lefschetz fibration with regular fiber Σ_g^1 which is a compact genus g surface with one boundary component. Our aim is to compute σ(Y^+). We study the construction of the closure f of f^+  carefully. 
The restriction of f^+ to ∂ Y^+ is an open book decomposition with a connected binding K which is just a copy of  ∂Σ_g^1 in ∂ Y^+. The complement of a neighborhood of 𝒩(K) is a surface bundle over ^1 with fibers Σ_g^1.
We can cap off the fibers of f^+ by attaching a 2-handle Y^-=^2×^2 attached along ∂^2 ×^2 to 𝒩(K)⊆∂ Y^+ in such a way that the circles ∂^2×{p} are identified with the boundaries of the fibers Σ_g^1. In other words the 2-handle is attached along the binding with the page framing. The result is a Lefschetz fibration f:Y→^2 where the fibers Σ_g are now closed surfaces of genus g. We write Y=Y^+ ∪ Y^-. Let 

    X^0   =𝒩(K)=∂^2 ×^2 
    
    X^+   =(∂ Y^+) ∖𝒩(K) 
    
    X^-   = (∂ Y^-) ∖ (∂^2×^2) 
    
    Z   =∂𝒩(K) = ∂^2×∂^2

Note that Z is a torus. We refer to homology generators μ=({p}×∂^2) and ℓ=(∂^2 ×{p}) in V=_1(Z;) the meridian and the longitude respectively of the torus Z.
Apply Wall's formula to the above setting, equations (<ref>), (<ref>), and (<ref>) give A=⟨ℓ⟩, B=⟨μ⟩,   C=⟨ℓ⟩. Then B∩(C+C)={0} implying dim  W=0. Hence the Maslov  index τ_V(A,B,C) vanishes. Since σ(Y^-)=0, Theorem <ref> gives σ(Y)=σ(Y^+).







§ WALL'S FORMULA AND PARTIAL FIBER SUM DECOMPOSITIONS

 Suppose we are given a (bordered) Lefschetz fibration f:Y→^2  with  regular fiber Σ_g^b and monodromy ϕ. If necessary by postcomposing f with a diffeomorphism of a disk,
we can always identify the base disk ^2 with the unit disk  {(x,y) :  x^2+y^2≤ 1} in ^2. Without loss of generality, we assume there exists no singular value of f on the line segment [-1,1]×{0}. 
Define


    D^+   =   {(x,y)∈^2 : y≥ 0 }
    
    D^-   =   {(x,y)∈^2 : y≤ 0 }


so that ^2=D^+∪ D^- and Y=Y^+∪ Y^- where  
f^±=f|_Y^± and Y^±=f^±^-1(D^±). The pair (f^+,f^-) is called a partial fiber sum decomposition of the Lefschetz fibration f. We denote by ϕ^+ and ϕ^- the monodromies of f^+ and f^- respectively. Let 
 
    V   =_1(Σ_g^b,𝔡;) ⊕_1( Σ_g^b,𝔡;),

equipped with the intersection form Q⊕-Q  where Q is the intersection form on _1(Σ_g^b,𝔡;).  We define three Lagrangian subspaces of V by

    A   =graph(ϕ_*^-),
    
    	B   =graph (id), 
    
    	C    = graph(ϕ_*^+).


 For the partial fiber sum decomposition f=(f^+,f^-),
	
    σ(Y)=σ(Y^+)  + σ(Y^-)- τ_V(graph(ϕ_*^-), graph (id), graph(ϕ_*^+)).

	
	



We will use Wall's non-addivity formula discussed in Section <ref> to understand how signature behaves under partial fiber sum decompositions.  First  assume that fibers are closed, so b=0.   Define 

    X^0   =f^-1([-1,1]×{0}), 
    
    X^-   =f^-1(D^-∩∂^2), 
    
    X^+   =f^-1(D^+∩∂^2), 
    
    Z   = ∂ X^-= ∂ X^0= ∂ X^+=f^-1({-1,1}×{0})=Σ_g∐ -Σ_g.

 When we choose an orientation on Y, it induces orientations on Y^+ and Y^-. Then orient X^0,X^+ and X^-  as in Wall's formula.  According to the above setting V, A, B, and  C are given as in (<ref>),(<ref>), (<ref>), and (<ref>) respectively. The result follows from Theorem <ref>.

To prove the theorem when the fibers Σ_g^b are bordered, b≥ 1, we first take the closure of our Lefschetz fibration and apply partial fiber sum decomposition the same way we do the bordered Lefschetz fibration. Noticing that  (f)^±=f^±, the result follows from Theorem <ref> and the case b=0.


Even though we do not need this for the rest of the paper, we would like to point out the relationship between our  signature defect corresponds to Meyer's 2-cocycle Mey:Sp(2g,ℤ)× Sp(2g,ℤ)→ℤ. It is known that 

    Mey(ϕ^-_*,ϕ^+_*)=-τ_V(graph(ϕ_*^-), graph (id), graph(ϕ_*^+)).
 



§ PROOF OF THE MAIN THEOREM



In this section we prove our main theorem assuming that fibers are closed. This is sufficient by Theorem <ref>. The proof follows by induction on number of vanishing cycles. 



 §.§ Base step
 
This is the case when a Lefschetz fibration has only one singular fiber. It is well known that the signature of the total space of this Lefschetz fibration is 0 if its unique vanishing cycle is non-separating and is -1 if the vanishing cycle is separating. See for example Ozbagci in <cit.>. For convenience of the reader, we include the proof here:
When a Lefschetz fibration over disk has no singular fiber, then its total space is Σ_g×^2 which has signature 0, because the second homology is generated by a regular fiber which has self-intersection 0. To get a singular fiber we attach a -1-framed 2-handle to a curve γ on the fiber. If γ is non-separating, then the homotopy type of the total space is Σ_g-1⋁^1 and its second homology is again generated by homology class of a regular fiber which has self intersection 0, hence the total space has signature 0. When γ is separating, the homotopy type of the total space is Σ_k ⋁Σ_g-k for some 1≤ k≤ g-1, so the second homology has rank 2. In addition to the homology class of a regular fiber, a second generator comes from a subsurface bounded by γ in a regular fiber and the core of the 2-handle. Since the 2-handle has framing -1, the resulting class has self intersection -1. The intersection form in this basis is 
[  0  0;  0 -1 ],
so signature is -1.





 §.§ Inductive step
 In this step we use a partial fiber sum decomposition to reduce the number of Dehn twists appearing in the monodromy factorization of Lefschetz fibrations. For every k ∈{1, …, n} we are given a Lefschetz fibration f_k:Y_k→^2  with regular fiber Σ_g and monodromy t_γ_k⋯ t_γ_1.  In what follows we will consider a special kind of decomposition so that D^- contains only one Lefschetz critical value corresponding to γ_k and D^+ contains all the others, i.e., the γ_i's where 1 ≤ i ≤ k-1. Hence Y_k^+ is diffeomorphic to Y_k-1, for all k=2,…, n.

 Suppose ϕ_k^±:Σ_g →Σ_g is the monodromy of f_k^±. From our choices ϕ_k^-=t_γ_k and ϕ_k^+=t_γ_k-1⋯ t_γ_1. Let V=H_1(Σ_g;)⊕  H_1(Σ_g;) equipped with the symplectic form Q⊕ -Q where Q is intersection form on Σ_g. Let A_k, B_k and C_k denote the subspaces graph(ϕ_k *^-), graph (id), and graph(ϕ_k *^+) respectively.  We will show that σ_k given in the statement of the theorem is equal to  τ_V(A_k, B_k, C_k), for all k=2,…, n. This finishes the proof because the  signature formula for partial fiber sums implies

    σ(Y_n)=σ(Y_n-1)+σ(Y_n^-)-σ_n,

by base step 

    σ(Y_n^-)=
     
          0    if γ_n  is non-separating
    
          -1    if γ_n  is separating,

and by inductive step assumption we have

    σ(Y_n-1)=-∑_j=1^n-1σ_j -#{ j ∈{1,…, n-1}; [γ_j]=0    in _1(Σ_g;)}.









 §.§ Identifying the local signatures
 It remains to prove σ_k=τ_V(A_k,B_k,C_k). 

Assume γ_k is a nonseparating curve then dim(E_1(ϕ^-_k *))=2g-1.



The surface Σ_g ∖γ_k is of genus g-1 and has two boundary components. Let α_1,…,α_2g-2 be a homology basis for _1(Σ_g ∖γ_k;), then they are linearly independent in _1(Σ_g ∖γ_k;) as well. Clearly ϕ^-_k *(α_j)=α_j for ∀ j=1,…, 2g-2. Also ϕ^-_k *(γ_k)=γ_k. So γ_k,α_1,…,α_2g-2 is a basis for E_1(ϕ^-_k *).



Let W_k=W_V(A_k,B_k,C_k) as Equation (<ref>), and let γ_k be a dual of γ_k. i.e., Q(γ_k, γ_k)=1. Then

    ϕ_k *^-(γ_k)=γ_k+γ_k

which implies dim(_1(Σ_g;)/E_1(ϕ^-_k *))=1 and the coset containing γ_k is a basis for _1(Σ_g;)/E_1(ϕ^-_k *).
The map x→ (x,x) gives rise to an isomorphism between _1(Σ_g;)/E_1(ϕ^-_k *) and B_k/B_k∩ A_k. Since W_k is isomorphic to a subspace of a quotient space of B_k/B_k∩ A_k, the space W_k is at most 1-dimensional and is generated by the coset containing (γ_k,γ_k). 


Therefore it sufficies to figure out Ψ_W_k((γ_k,γ_k), (γ_k,γ_k)).
From the description of  Ψ_W_k in Section <ref>, first look for z_k and x_k∈_1(Σ_g;) solving   

    (z_k,ϕ^-_k *z_k)+(γ_k, γ_k)+(ϕ^+_k *x_k,x_k)=0

where (z_k,ϕ^-_k *z_k)∈ A_k, (γ_k, γ_k) ∈ B_k and (ϕ^+_k *x_k,x_k)∈ C_k. Then

    z_k+γ_k+ϕ^+_k *x_k   =0 ,  
    ϕ^-_k *z_k+γ_k+x_k   =0 .


In the equation (<ref>) if we substitute z_k from the equation (<ref>), we obtain

    (Id-ϕ^-_k *ϕ^+_k *)· x_k=γ_k.

If no such x_k exists, then this means (γ_k, γ_k) is not in B_k∩(C_k+A_k). So dim W_k=0 implying τ_V(A_k,B_k,C_k)=0. Conversely if x_k is any solution to the above equation then we let z_k=-γ_k-ϕ^+_k *x_k from the equation (<ref>), and see that equation (<ref>) has a solution.  Then by (<ref>),

    Ψ_W_k((γ_k, γ_k),(γ_k, γ_k))    = Q(γ_k,ϕ^+_k *x_k)-Q(γ_k,x_k).
       = Q(ϕ^-_k *γ_k,ϕ^-_k *ϕ^+_k *x_k) - Q(γ_k,x_k).
       = Q(γ_k+γ_k,x_k-γ_k) - Q(γ_k,x_k)
       = Q(γ_k,x_k)+Q(γ_k,x_k)-Q(γ_k,γ_k)-Q(γ_k,x_k)
       =Q(γ_k,x_k)+1

 which shows that τ_V(A_k, B_k, C_k)=σ_k as given in Equation (<ref>). Here we used the fact that  ϕ^-_k *∈(2g;) (so it preserves the intersection form Q), as well as  Equations (<ref>) and (<ref>).


















§ SOME COMPUTATIONAL SHORTCUTS

In this section, we will prove some lemmas which will be useful in our computations. Throughout we assume the fibers of our Lefschetz fibrations are closed. For the following lemma, we continue using the notation in the previous section.  


Suppose γ_k is non-separating. If ϕ^+_k *γ_k=γ_k  then σ_k=0. i.e., if the monodromy of a Lefschetz fibration preserves the dual curve of γ_k, adding a vanishing cycle on γ_k does not change the signature.



We know B_k∩ A_k is isomorphic to E_1(ϕ^-_k *) which has codimesion one, and the quotient space _1(Σ_g;)/E_1(ϕ^-_k *) is generated by the coset containing γ_k. We also know that B_k∩ C_k is isomorphic to E_1(ϕ^+_k *). The latter contains the homology class of γ_k by our assumption. Hence 

    E_1(ϕ^+_k *)+E_1(ϕ^-_k *)=_1(Σ_g;)

and W_k is isomorphic to a subspace of the 0-dimensional space _1(Σ_g;)/E_1(ϕ^+_k *)+E_1(ϕ^-_k *).
Therefore W_k is itself 0-dimensional implying σ_k=0.



The next lemma considers an arbitrary partial fiber sum decomposition of a Lefschetz fibration whose monodromy is homologically trivial. 



Let f:Y →^2 be a Lefschetz fibration with monodromy acting trivially on homology of Σ_g. Suppose we applied a partial fiber sum decomposition f=(f_+,f_-) with the corresponding total space Y^+ and Y^-, then
σ(Y)=σ(Y^-)+σ(Y^+). 


Let ϕ^± denote the monodromy of f^±. Then ϕ^-_*ϕ^+_*=Id. By permuting the roles of A,B and C, we will show that W=W_V(B,A,C) is 0-dimensional.  Suppose (x,ϕ^-_*(x))∈ A and (ϕ^+_*(z), z)∈ C then ϕ^-(ϕ^+z)=z which implies z∈ E_1(ϕ^-_*ϕ^+_*). Then A∩ C= {(ϕ^+_*z,z) : z∈ E_1(ϕ^-_*ϕ^+_*)}.
Since ϕ^-_*ϕ^+_*=Id and A∩ C is 2g dimensional, from the definition A∩ (B+C) is at most 2g dimension. Hence W is 0 dimensional, so the Maslov index τ_V(graph(ϕ_*^-), graph (id), graph(ϕ_*^+)) vanishes.





If the monodromy is identically trivial, the above lemma can also be proved using Novikov additivity.













§ TAKING EXPONENTS

In this section we shall prove Theorem <ref>. Let f:Y→^2 be a Lefschetz fibration with regular fiber Σ_g^b and  monodromy ϕ^m+1. Consider a partial fiber sum decomposition (f^+,f^-) where f^+ and f^- are Lefschetz fibrations with monodromy ϕ^-=ϕ and ϕ^+=ϕ^m. The manifolds Y^± corresponds to (f^±)^-1(D^±) as before.

We have 

    σ(Y)=σ(Y^+)+σ(Y^-)-σ_ϕ^m+1_c.



We apply Theorem <ref> to the aforementioned partial fiber sum decomposition. We have V=H_1(Σ_g^b,𝔡;)⊕  H_1(Σ_g^b,𝔡;) equipped with the symplectic form Q⊕ -Q where Q is intersection form on Σ_g^b, A =graph(ϕ_*),  B =graph(Id),  C =graph(ϕ^m_*), and W=W_V(A,B,C).


We must show that σ_ϕ^m+1_c=τ_V(A,B,C). From the definitions, it is clear that 

    B∩ A={(x,x):x ∈ E_1(ϕ_*)}⊆ B∩ C={(x,x):x ∈ E_1(ϕ_*^m)}.


    B∩ (C+A)   = {(ϕ_*(x)+z,  ϕ_*(x)+z) : ϕ_*^m(z)-z=ϕ_*(x)-x}.

Letting y=(Id+ϕ_*+… +ϕ_*^m-1)(z)-x in the above description, we see that 

    ϕ_*(y)   =   ϕ_*(z)+ϕ_*^2(z)+… +ϕ_*^m(z)-ϕ_*(x)
       =   ϕ_*(z)+ϕ_*^2(z)+… +ϕ_*^m-1(z)+z-x 
       =   y.

Hence y ∈ E_1(ϕ_*), and we have

    B∩ (C+A)   = {(z+ϕ_*(z)+…+ϕ_*^m(z)-y, z+ϕ_*(z)+…+ϕ_*^m(z)-y) :
       y,z ∈_1(Σ_g^b,𝔡;)  and  y ∈ E_1(ϕ_*)}.

Hence the map z → (z+ϕ_*(z)+…+ϕ_*^m(z),   z+ϕ_*(z)+…+ϕ_*^m(z)) gives rise to an isomorphism from _1(Σ_g^b,𝔡;)/E_1(ϕ_*^m+1) to W.
It remains to identify the symmetric bilinear pairings I_ϕ^m+1 and Ψ under the above isomorphism.
We have
 
    Ψ^'((z_1+… +ϕ_*^m z_1, z_1+…+ϕ_*^m z_1), (z_2+… +ϕ_*^m z_2, z_2+…+ϕ_*^m z_2))
    
       =-Q(z_1+…+ϕ_*^m z_1,ϕ_*^m z_2)+Q(z_1+…+ϕ_*^m z_1, z_2)
       =Q(z_1,z_2)+…+Q(ϕ_*^m(z_1),z_2)-Q(z_1, ϕ_*^m(z_2))-…-Q(ϕ_*^m(z_1),ϕ_*^m(z_2)) 
       =I_ϕ^m+1(z_1,z_2).


Here we use the fact that  Q(z_1,z_2)=Q(ϕ_*^m (z_1), ϕ_*^m(z_2)) as ϕ_*^m∈(2g;). Since the forms are identical, the result follows.


For our computations we also need a matrix representation for the symmetric bilinear form I_ϕ^k+1. Take a symplectic basis so that the intersection form on H_1(Σ,ℝ) is represented by the matrix J. We represent the linear map ϕ_* as a matrix using the same basis. Then by Equation (<ref>), 

    σ_ϕ^k+1_c= sign( (ϕ^T)^k · J+ (ϕ^T)^k-1· J+ ⋯ + ϕ^T · J - J·ϕ⋯ -J·ϕ^k ).






 Let f:Y→^2 be a Lefschetz fibration with monodromy (C_4C_3C_2C_1)^10 =Id where each c_i   (i=1,2,3,4) is a curve on genus-2 surface as in Figure <ref>. It is known that the total space  of this Lefschetz fibration is homeomorphic (but not diffeomorphic) to 5ℂℙ^2♯ 29ℂℙ^2 <cit.>, so it has signature -24. We will verify this using our techniques. Let C_i denote positive Dehn twist along each c_i. We compute the signature of Y by using decomposition of f. Let J be the 4× 4 matrix:

    J = 
    [  0  1  0  0; -1  0  0  0;  0  0  0  1;  0  0 -1  0 ]
 and denote the product (C_4C_3C_2C_1) by ϕ. The matrix representation of ϕ is

    [  0  1  0 -1; -1  0  0  0;  1  0  1  1; -1  0 -1  0 ].
 
Since each vanishing cycle is nonseparating and dual curve of each can be chosen disjoint from monodromy, by Lemma <ref> and Theorem <ref>, the signature of Y; σ_ϕ=0. Now consider the Lefschetz fibration with monodromy ϕ^2. Again by Wall's formula 

    σ_ϕ^2=σ_ϕ+σ_ϕ-σ_ϕ^2_c

and by a result of Theorem <ref>, 

    σ_ϕ^2_c=sign(ϕ^TJ-Jϕ)=sign (
    [  2  0  1  1;  0  2  0 -1;  1  0  2  1;  1 -1  1  2 ] ).

Thus σ_ϕ^2=-4. By squaring the ϕ^2, we have a new Lefschetz fibration with monodromy ϕ^4 and its signature is 
    σ_ϕ^4=σ_ϕ^2+σ_ϕ^2-σ_ϕ^4_c
 where 

    σ_ϕ^4_c=sign((ϕ^2)^TJ-J(ϕ^2))=sign (
    [  0  1  1 -1;  1  0  2  1;  1  2  2  0; -1  1  0  0 ] ).
 
This gives σ_ϕ^4=-8. Now consider the Lefschetz fibration with mondromy ϕ^5. At this step, decomposition is applied with ϕ^4 and ϕ. By Wall's non-additivity formula 



    σ_ϕ^5=σ_ϕ^4+σ_ϕ-σ_ϕ^5_c
 
where  

    σ_ϕ^5_c=sign [(ϕ^4)^TJ+(ϕ^3)^TJ+(ϕ^2)^TJ+(ϕ)^TJ 
       -J(ϕ)-J(ϕ^2)-J(ϕ^3)-J(ϕ^4)] .

Since sign (
[  0  1  1 -1;  1  0  2  1;  1  2  2  0; -1  1  0  0 ] )=4, σ_ϕ^5=-12.
By squaring the monodromy ϕ^5, one can easily have a Lefschetz fibration with monodromy ϕ^10 whose signature is -24. 























Consider the Lefschetz fibration with monodromy 
    (C_3C_2C_1)^4 =δ_1δ_2.
  We will show that  the signatures of the Lefschetz fibrations corresponding to the left hand side and right hand side of this equation are -7 and -1 respectively.  Our computation is consistent with a result of Endo and Nagami <cit.>  where it was proven  that the difference of the signatures of these two Lefschetz fibrations is -6. To compute σ_δ_1δ_2 and σ_(C_3C_2C_1)^4, attach a 1-handle connecting the two boundary components and then cap off the boundary. By above discussion, the signature is unchanged. Note that δ_1=δ_2=δ
in _1(Σ;) where Σ is a surface below. 



By the Wall non-additivity formula,

    σ_δ_1δ_2=2σ_δ-σ_δ_c^2

where σ_δ_c^2=sign (δ_*^TJ-Jδ_*) and δ_*=
 [ 1 0 0 0; 0 1 0 0; 0 0 1 1; 0 0 0 1 ]. Since δ is non-separating in  Σ, σ_δ=0 and a direct computation shows σ_δ_c^2=1. Hence σ_δ_1δ_2=-1. Now let ϕ=C_3C_2C_1. By Lemma <ref>, σ_ϕ=0.


    σ_ϕ^2   =2σ_ϕ-sign(ϕ^TJ-Jϕ)=-3,
    σ_ϕ^4   =2σ_ϕ^2-sign((ϕ^2)^TJ-Jϕ^2)=-7.











































§ POSITIVE SIGNATURE

In this section, we will prove Theorem <ref>. First We need a simple obsevation about the signatures of a special class of 2× 2 matrices. Consider the following subset of 2 × 2 matrices:

    𝒜= {A ∈ M_2 × 2 : A_12=A_21,   A_11 < 0, A_22> 0}.


Under the matrix addition 𝒜 is a monoid. Note that since each A ∈𝒜 is a symmetric matrix and diagonalizable with real eigenvalues, det A < 0. Therefore sign (A)=0.

 
Let B be a matrix with positive entries. Then for k∈{1,…, n}, 

    (B^T)^kJ-JB^k ∈𝒜.

Consequently, ∑_k=1^n ((B^T)^kJ-JB^k) ∈𝒜. 



Let B=
[ B_11 B_12; B_21 B_22 ]
in ℬ. Then
B^TJ-JB=
[ -B_21-B_21  B_11-B_22; -B_22+B_11  B_12+B_12 ]
which is in 𝒜. Then for each k, ((B^T)^kJ-JB^k) ∈𝒜 also holds since power of positive matrices are again positive. As 𝒜 is a monoid, the sum ∑_k=1^n ((B^T)^kJ-JB^k) is in 𝒜 .




We will also need the following example which is due to Ozbagci. 

(Ozbagci) There exists a Lefschetz fibration over 𝔻^2 with fiber genus one and one boundary component, and the signature of the total space is +1.


 Let Σ=Σ_1^1 be a surface of genus one with one boundary component. As H_1(Σ;ℤ)=ℤ×ℤ, we can denote its homology classses by row vectors [a b] where a and b are integers. 
Let γ_1, γ_2 and γ_3 be simple closed curves representing the homology classes γ_3=[1   5], γ_2=[2   5] and γ_1=[1   0] respectively. The homology action of Dehn twist along γ_3 is C_3=
[  -4   1; -25   6 ], along γ_2 is C_2=
[  -9   4; -25  11 ], and along γ_1 is C_1=
[ 1 1; 0 1 ].
 
 
Let f:Y→^2 be a Lefschetz fibration with three vanishing cycles γ_1,γ_2 and γ_3 on the regular fiber Σ_1^1 and the monodromy factorization of f is ϕ=C_3C_2C_1=
[ 11  6; 75 41 ].

By using the formula in Theorem <ref>, it can be shown that the signature of the total space is 1. 


Now we will construct a Lefschetz fibration with signature n.


Let f:Y→^2 be the  Lefschetz fibration with regular fiber Σ_1^1 given by the monodromy factorization ϕ^n=(C_3C_2C_1)^n and Y is the n-fold cyclic branched cover of Y branched along a regular fiber. By the Theorem <ref>, we have

    σ(Y)=nσ(Y)-∑_k=1^n-1σ_ϕ^k+1_c.

We will show that ∑_k=1^n-1σ_ϕ^k+1_c=0.
For each k, σ_ϕ^k+1_c=0 implies that  the signature of
 ∑_i=1^k ((ϕ^T)^iJ-Jϕ^i) ∈𝒜 is zero by the Equation (<ref>). It sufficies to show that the determinant of this 2× 2 matrices summation is negative.
By Lemma <ref>, the matrices in the equation belong to 𝒜. The result follows for g=b=1.


For g> 1, b≥ 1, we attach one handles as required to make fiber Σ_g^b without changing the signature . Finally if b=0, we first do the construction for b=1 and cap off the boundary as in Theorem <ref>.
























§ ACKNOWLEDGEMENTS
 We are grateful to Burak Ozbagci for sharing his old notes with us. A special thanks goes to Ferit Öztürk for noticing a mistake in the earlier version.  While working on this project AÇ was supported by TUBITAK postdoctoral fellowship BIDEB-2218, No:1929B011700264 (2017/2) and ÇK was supported by BAGEP award of the Science Academy and  Boğaziçi University Research Fund Grant Number 12482.


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