Spin Lefschetz fibrations are abundant

We prove that any finitely presented group can be realized as the fundamental group of a spin Lefschetz fibration over the 2-sphere. We moreover show that any admissible lattice point in the symplectic geography plane below the Noether line can be realized by a simply-connected spin Lefschetz fibration.


Introduction
Explicit constructions of Lefschetz fibrations with prescribed fundamental groups were given by Amorós, Bogomolov, Katzarkov and Pantev [1] and by Korkmaz [14]; also see [12].We show that the same result holds for a much smaller family of Lefschetz fibrations: Theorem A. Given any finitely presented group G, there exists a spin symplectic Lefschetz fibration X → S 2 with π 1 (X ) ∼ = G.These results were inspired by the pioneering work of Gompf, who proved that any finitely presented group G is the fundamental group of a closed symplectic 4-manifold [9], which can be assumed to be spin.By the existence of Lefschetz pencils on any symplectic 4-manifold due to Donaldson [6], it follows a priori that, after blowing up the base points of the pencil, one can realize G as the fundamental group of a symplectic Lefschetz fibration; however, these are never spin.
On the other hand, unlike Kähler surfaces, there are minimal symplectic 4manifolds of general type violating the Noether inequality, which was shown again by Gompf [9].More recently, Korkmaz, Simone, and Baykur showed that all the lattice points in the symplectic geography plane below the Noether line can be further realized by simply connected symplectic Lefschetz fibrations [4].We prove that a similar result holds in the spin case: Theorem B. For any pair of nonnegative integers (m, n) satisfying the inequalities n ≥ 0, n ≡ 8m (mod 16), n ≤ 8(m −6) and n ≤ 16  3 m, there exists a simply connected spin symplectic Lefschetz fibration X → S 2 such that χ h (X ) = m and c 2 1 (X ) = n.In particular, any admissible point in the symplectic geography plane below the Noether line is realized by a simply connected spin Lefschetz fibration.
Among the hypotheses in the theorem, the first inequality is due to a theorem of Taubes, who showed that c 2 1 (X ) ≥ 0 for any nonruled minimal symplectic 4manifold X , whereas the second equality follows from Rokhlin's theorem.A pair (m, n) ∈ ‫ގ‬ 2 satisfying this condition is called admissible.The first systematic production of spin symplectic 4-manifolds realizing the above admissible lattice points, but without the Lefschetz fibration structure we get, was first obtained by J. Park in [15].
Our examples are produced explicitly via positive Dehn twist factorizations in the mapping class group.The spin Lefschetz fibrations for Theorems A and B are obtained by adapting the strategies of [14] and [4], respectively, together with a subtle use of the breeding technique [2; 3] for the latter.The main challenge in producing the examples in either theorem is due to the fact that the monodromy of a spin Lefschetz fibration lies in a proper subgroup of the mapping class group (fixing a spin structure on the fiber), so throughout our work, we restrain ourselves to algebraic manipulations in this smaller mapping class group.

Preliminaries
We begin with a review of the concepts and background results underlying the rest of our article, along with our conventions.We refer the reader to [10] for more details and comprehensive references on Lefschetz fibrations, symplectic 4-manifolds, and monodromy factorizations, and to [3] for their interplay with spin structures.

Lefschetz fibrations and positive factorizations.
A Lefschetz fibration on a closed smooth oriented 4-manifold X is a smooth surjective map f : X → S 2 , a submersion on the complement of finitely many points { p i } ̸ = ∅ all in distinct fibers, around which f conforms (compatibly with fixed global orientations on X and S 2 ) to the local complex model of a nodal singularity (z 1 , z 2 ) → z 1 z 2 .We assume that there are no exceptional spheres contained in the fibers.Each nodal fiber of the Lefschetz fibration (X, f ) is obtained by crashing a simple closed curve, called a vanishing cycle, on a reference regular fiber F.
We We denote by t c ∈ Mod( b g ) the positive (right-handed) Dehn twist along the curve c ⊂ b g .For any ψ, φ ∈ Mod( b g ) we write the conjugate of ψ by φ as ψ φ = φψφ −1 .We act on any curve c in the order (ϕφ)(c) = ϕ(φ(c)).An elementary but crucial point is that t φ c = t φ(c) .For any product of Dehn twists W = ℓ i=1 t k i c i and φ in Mod( b g ), we denote the conjugated product by W φ = ℓ i=1 t k i φ(c i ) .Let {c i } be a nonempty collection of curves on b g which do not become nullhomotopic after an embedding b g → g .Let {δ j } be a collection of b curves parallel to distinct boundary components of b g .A relation of the form (1) in Mod( b g ) corresponds to a genus-g Lefschetz fibration (X, f ) with a reference regular fiber F identified with g , with vanishing cycles {c i } and b disjoint sections {S j } of selfintersections S j • S j = −k j .
The product on the left-hand side of the equality (1), the word P in positive Dehn twists, is called a positive factorization of the mapping class on the right-hand side that maps to the trivial word under the homomorphism induced by an embedding b g → g .We will often denote the corresponding Lefschetz fibration as X P .As shown by Gompf, every Lefschetz fibration (X, f ) admits a Thurston-type symplectic form with respect to which the fibers are symplectic.
2.2.Fiber sums and fundamental groups.A Lefschetz fibration X P corresponding to a positive factorization 1g ) (of some power of the boundary twist) has π 1 (X P ) ∼ = π 1 ( g )/N ({c i }), where N ({c i }) is the subgroup of π 1 ( g ) generated normally by collection of the vanishing cycles c i .
Given P 1 := t c 1 t c  with a section of self-intersection −(k 1 + k 2 ).This coincides with the well-known twisted fiber sum operation applied to the Lefschetz fibrations X P 1 and X P 2 .We have A neat trick of Korkmaz, applicable in the more special setting described in the next proposition, will come in handy for our arguments to follow: Proposition 1 (Korkmaz [14]).Let P = t c 1 t c 2 • • • t c ℓ be a positive factorization of (some power of) a boundary twist in Mod( 1 g ).Let d be a curve on g intersecting at least one c i transversally at one point.Then π 1 (X P P t d ) ∼ = π 1 ( g )/N ({c i } ∪ {d}).

Spin monodromies and fibrations.
A spin structure s on g is a cohomology class s ∈ H 1 (U T ( g ); ‫ޚ‬ 2 ) evaluating to 1 on a fiber of the unit tangent bundle U T ( g ).There is a bijection between the set of spin structures on g , which we denote by Spin( g ), and the set of quadratic forms on H 1 ( g ; ‫ޚ‬ 2 ) with respect to the intersection pairing.Recall that q : For a fixed spin structure s on g , the spin mapping class group Mod( g , s) is the stabilizer group of s, or equivalently that of the corresponding quadratic form q, in Mod( g ).For any nonseparating curve c ⊂ g , we have t c ∈ Mod( g , s) if and only if q(c) = 1.
The following, which is a reformulation of a theorem of Stipsicz, provides us with a criterion for the existence of a spin structure on a Lefschetz fibration: Theorem 2 (Stipsicz [16]).Let X P be the Lefschetz fibration prescribed by a positive factorization and let us denote the images of the twist curves under the embedding 1 g → g also by {c i }.Then, X P admits a spin structure with a quadratic form q if and only if k is even and q(c i ) = 1 for all i.

Spin Lefschetz fibrations with prescribed fundamental group
We prove Theorem A, adapting the strategy in [14], where Korkmaz takes twisted fiber sums of many copies of the same Lefschetz fibration (the building block) to obtain a new Lefschetz fibration whose fundamental group is the prescribed finitely presented group.To accomplish the same with spin fibrations, there are two essential refinements we will need to make.First is to identify a building block X P where the monodromy curves in the positive factorization P will satisfy the spin condition for some quadratic form we will describe.That is, we will show that for a carefully chosen spin structure s.Second is to make sure that when taking the twisted fiber sums to land on the desired fundamental group, in the corresponding positive factorization PP φ 1 • • • P φ m , we only use conjugations φ i ∈ Mod( g , s).

The building block.
A generalization of the monodromy factorization of the well-known genus-1 Lefschetz fibration on ‫ސރ‬ 2 #9 ‫ސރ‬ 2 ∼ = S 2 × S 2 #8 ‫ސރ‬ 2 to any odd genus g = 2n +1 Lefschetz fibration on S 2 × n #8 ‫ސރ‬ 2 was given by Korkmaz in [13], and by Cadavid in [5].It has the monodromy factorization where the curves B i , a, b are shown in the Figure 1.Capping off the boundary component of 1 g , we will regard the same curves also in g .Let us denote the above positive factorization by Clearly, π 1 (X P g ) ∼ = π 1 ( n ) will have larger number of generators we can work with as we increase g = 2n + 1.Let us first review the presentation for π 1 (X P g ).Consider the geometric basis {a i , b i } g i=1 for π 1 ( g ), where the based oriented curves a i , b i are as shown in Figure 2. We have Next, we will describe a spin structure for which the vanishing cycles of this Lefschetz fibration satisfy the monodromy condition. 2Forgetting the base point, the geometric basis {a j , b j } for π 1 ( g ) in Figure 2 becomes freely homotopic to a standard symplectic basis on g .We can then describe a quadratic form with respect to this basis and evaluate it on the mod-2 homology classes of the vanishing cycles described in this basis.The latter is easily derived from (2): The generators for π 1 ( g ) and the C i curves.
Set q : H 1 ( g , ‫)2/ޚ‬ → ‫2/ޚ‬ as q(a i ) = q(b i ) = 1.(There are in fact 2 n different spin structures that would work here; we are picking the one that will serve our needs the most in the next stages of the proof.)Note that for any ordered set of curves {d j } we have q n i=1 d i = n i=1 q(d i ) + i< j d i • d j .Thus, for each k as above, Hence all the monodromy curves of X P g satisfy the spin condition, which is all we needed at this point. 3To sum up, we have the following: Lemma 3. Let s ∈ Spin( g ) correspond to the quadratic form q that satisfies q(a i ) = q(b i ) = 1, for i = 1, . .., g, on the symplectic basis {a i , b i } above.We have , where B i , a, b are the curves on 1 g ⊂ g in Figure 1.

The construction.
In anticipation of a forthcoming issue, here we deviate a bit from Korkmaz's steps.In order to guarantee that we can represent the relators by embedded curves on g , we change the given presentation.Instead of reinventing the wheel here, we invoke the following result (cf.[8, Lemma 6.2]): Relator curves on g .
Lemma 4 (Ghiggini, Golla and Plamanevskaya [8]).For any finitely presented group G, there exists a presentation G ∼ = ⟨x 1 , . . ., x n | r 1 , . . ., r m ⟩ such that (i) each r j is a positive (no inverses) word in x 1 , . . ., x n ; (ii) each generator x i appears at most once in each r j ; (iii) the cyclic order (by index) of the generators x 1 , . . ., x n is preserved in each r j .
This means that if our generating set consists of only the curves {b i }, we can assume that all the relators in the generating set can be nicely represented by the embedded curves as in Figure 3, where R 1 represents x 2 x 3 , R 2 represents x 1 x 2 x 4 , and so on.
We are now ready to present our construction.
) 2 be the positive factorization in Mod( g , s) given in Lemma 3.Because q(a i ) = 1, we have t a i ∈ Mod( g , s) for all i.So we get a new spin factorization for each odd g ∈ ‫ޚ‬ + , which lifts to a positive factorization of t g+1 δ in Mod( 1 g ).From the expression of the monodromy curves of P g in the π 1 ( g ) basis {a i , b i } given in (2), one easily deduces that that is, we get a free group on n generators.For the first step, simply note that all a j and C j we had in (2) are trivial in this group.Now, identifying each generator x i with b i , for i = 1, . . ., n, we can represent each relator r j by an embedded curve on R j on g .(This is why we switched to this special presentation.)All {R j } can be contained on 1  n ⊂ g bounded by c n .It is possible that some q(R j ) = 0.If that is the case, we replace this R j with an embedded curve R ′ j representing R j a n+1 in π 1 ( g ).Such an embedded curve always exists; R j can be isotoped to meet a n+1 only at the base point and one can then resolve the intersection point compatibly with the orientations.So now q(R ′ j ) = 1.Otherwise we just take R ′ j := R j .We have t R ′ j ∈ Mod( g , s) for all j = 1, . . ., m.
It follows that we have a spin positive factorization which now lifts to a positive factorization of t g+m+1 δ in Mod( 1 g ).If m is odd, we add one more P g factor to the positive factorization above, so then its lift is a positive factorization of t g+m+2 δ .If m is even, leave it as it is.In either case let us denote this final positive factorization in Mod( g , s) simply by P. Let X P denote the corresponding Lefschetz fibration.By Theorem 2, X P is spin.By Proposition 1, and the above discussion, we have which is the presentation we had for G. □

Geography of spin Lefschetz fibrations
We prove Theorem B by a direct construction of a family of spin Lefschetz fibrations Z g,k populating the region below the Noether line in the geography plane.We prescribe these fibrations via new positive factorizations via algebraic manipulations in the mapping class group corresponding to twisted fiber sums and breedings [2; 3].We then verify how our careful choice of building blocks out of monodromy factorizations for Lefschetz pencils and fibrations indeed yields positive factorizations in spin mapping class groups.A somewhat longer calculation will show that our choices also guarantee that Z g,k are simply connected.We will then conclude by describing the portion of the geography plane spanned by our spin fibrations.While some of the particular choices we will make in the construction of Z g,k may look arbitrary at first, they are to achieve two somewhat competing properties simultaneously: the existence of a spin structure on Z g,k and the simple-connectivity of Z g,k .The latter calculation implies that the spin structure we describe on Z g,k is in fact unique.

4.1.
The construction.Our first building block is a positive factorization for a Lefschetz fibration on ‫ސރ‬ 2 #(4g + ‫ސރ)5‬ 2 given in [4].Taking p = q = 2g + 2 in 2 ) = 1 in Mod( g ), which is in fact Hurwitz equivalent to the square of the positive factorization of the hyperelliptic involution h : Here t i denotes a Dehn twist along the curve c i shown in Figure 4. We also assume that g ≥ 5 and is odd.Let also V be the following conjugate of U : We claim that φ(c 1 ) = a, φ(c 3 ) = b and ψ(c 1 ) = c, ψ(c 3 ) = d; see Figure 6.This can be easily verified because of the following elementary observation: whenever we have a k-chain of curves u 1 , . . .u k , Let us denote by Z g the Lefschetz fibration corresponding to the positive factorization P := V φ U ψ in Mod( g ), a twisted fiber sum of the Lefschetz fibration on ‫ސރ‬ 2 #(4g + ‫ސރ)5‬ 2 with itself.Note that we have ( 5) where V 1 , U 1 are the products of positive Dehn twists Vanishing cycles of the genus-2 pencil.
Our second building block is the following positive factorization by Hamada: for a genus-2 Lefschetz pencil on S 2 × T 2 , where the twist curves are as shown in Figure 5; see [3; 11].Since the curves {a, b, c, d} cobound a subsurface 4 2 of g , we can breed (see [2; 3]) the genus-2 pencil prescribed by ( 6) into the Lefschetz fibration prescribed by (5) for k times, for any k ≤ 2g + 2, and get a new positive factorization (7) where R is the image of the positive factorization Q under the homomorphism induced by a specific embedding 4  2 → g we describe below.We let Z g,k denote the Lefschetz fibration corresponding to the positive factorization P g,k .
The embedding 4 2 → g is described in Figure 6. Brown curves indicate where the boundary curves {δ i } of 4  2 in the positive factorization ( 6) are mapped to.Blue arrows illustrate how we isotope the boundaries of 4  2 ⊂ ‫ޒ‬ 3 before embedding it into g ⊂ ‫ޒ‬ 3 .Red curves constitute a geometric generating set for H 1 ( g ; ‫ޚ‬ 2 ).Red arcs are the parts of these curves contained in the image of the embedding 4  2 → g .

4.2.
The spin structure on Z g,k .We are going to invoke Theorem 2 to confirm that Z g,k admits a spin structure.The curves {x i , y i } in Figure 6 constitute a symplectic basis for H 1 ( g ; ‫ޚ‬ 2 ).Consider the quadratic form q for a spin structure s ∈ Spin( g ), where for any 1 ≤ i ≤ g, q(x i ) = 1 for all i, q(y i ) = 1 for i odd, q(y i ) = 0 for i even.
First of all, c 2i = x i , c 1 = y 1 , c 2g+1 = y g and c 2i+1 = y i − y i+1 .This means that q(c i ) = 1 for each i.Therefore, t i := t c i ∈ Mod( g , s) for all i and the positive factorizations given in ( 3) and ( 4) are in fact factorizations in Mod( g , s).
Thirdly, to check the spin condition for the new monodromy curves in R, we would like to express these curves in terms of the generators {x i , y i }. 4 We get in H 1 ( g , ‫)2/ޚ‬ the expressions So we have q(B 0 ) = q(x 1 +x 2 + y 3 + y 4 ) = q(x 1 )+q(x 2 )+q(y 3 )+q(y 4 ) = 1+1+1+0 = 1, q(B ′ 0 ) = q(x 1 +x 2 + y 4 + y 5 ) = q(x 1 )+q(x 2 )+q(y 4 )+q(y 5 ) = 1+1+0+1 = 1, 4 Let F denote the embedding 4  2 → g and let v j be a Dehn twist curve in R. Instead of F(v j ) • x i and F(v j ) • y i we can look at v j • F −1 (x i ) and v j • F −1 (y i ) to run the calculation here.Note that if x i or y i is only partially contained in the image of F, then we denote the arc in its preimage by x ′ i or y ′ i .
It is well known that the Lefschetz fibration with positive factorization U admits a (−1)-section; in fact this fibration is Hurwitz equivalent to a Lefschetz fibration obtained by blowing up all 4g + 4 base points of a genus-g pencil on S 2 × S 2 [17].Therefore U, V , and in turn U ψ , V φ , all lift to a positive factorization of t δ in Mod( 1g ), where δ is a boundary parallel curve on 1 g .We can pick a (−1)-section so that in the lift of U (and V ), the lifts of t c 1 , t c 3 are still along disjoint curves in 1 g . 5The same goes for t a , t b of V φ and t c , t d of U ψ .Let us continue denoting the twist curves in their lifts by a, b and c, d.After an isotopy, we can assume that P = V φ U ψ = 1 lifts to a positive factorization of t 2 δ in Mod( 1 g ) so that the boundary component is not contained in the subsurface 4  2 ⊂ g cobounded by {a, b, c, d}.
Therefore, for any k ≤ 2g + 2 we have a spin positive factorization which lifts to a positive factorization in Mod( 1 g ).Hence every Z g,k admits a spin structure by Theorem 2.

4.3.
The fundamental group.Let {x i , y i } be a geometric basis for π 1 ( g ) as shown in Figure 7. Since Z g,k has a section, we have G := π 1 (Z g,k ) ∼ = π 1 ( g )/N ({v j }), where v j are the Dehn twist curves in the positive factorization P g,k of Z g,k . Set ).
While the fundamental group of Z g,k can be calculated from the factorization The generators x i , y i for π 1 ( g ).
, it can also be calculated from the factorization . We will run our calculations for the latter.For k < 2g + 2 the Dehn twist curves of the latter factorization contain all the vanishing cycles {c i } in U , which we know kill all the generators of π 1 ( g ) to yield trivial the fundamental group, as P U has total space ‫ސރ‬ 2 #(4g + ‫ސރ)5‬ 2 , a simply connected space.Thus, π 1 (Z g,k ) = 1 for any k < 2g + 2.
For k = 2g + 2, first note that we can connect the vanishing cycles {c i } of U or V to the basepoint (where any two different paths connecting them to the base point will yield the same normal generating set) so that in π 1 ( g ) we have c 2i = x i , c 1 = y 1 , c 2g+1 = y g and c 2i+1 = x i x −1 i+1 for each i.It follows that π 1 ( g ) is generated by {c i }.To get G we quotient π 1 ( g ) by normally generated subgroup by relators coming from the Dehn twist curves in S (and not U ), which are of the form t i (c i−1 ) with 2 ≤ i ≤ 2g + 1 and t 3 (c 4 ), along with several other relations.We may assume that c i are oriented so that c i−1 • c i = +1 for all i.Then t i (c i−1 ) = c i−1 c i = 1 and t 3 (c 4 ) = c 4 c −1 3 .These relations imply that We thus see that 1 , rest of the relators coming from other vanishing cycles⟩ for our positive factorization S(R φ −1 ) 2g+2 S φ −1 ψ .At this point G is a quotient of the abelian group ‫ޚ‬ 2 generated by c 1 , so it is certainly an abelian group, and it suffices to show that H 1 (Z g,2g+2 ) = 0.
We will argue this by observing that the vanishing cycle coming from t φ −1 (B 2 ) induces a relator killing the homology class of c 1 .This is because it is homologous to an odd factor of c 1 .For this reason, it is in fact enough to consider t φ −1 (B 2 ) in denote by b g a compact connected oriented surface of genus g with b boundary components.Let Diff + ( b g ) denote the group of orientation-preserving diffeomorphisms of b g compactly supported away from the boundary.The mapping class group of b g is defined as Mod( b g ) := π 0 (Diff + ( b g )).When b = 0, we simply drop b from the above notation.Unless mentioned otherwise, by a curve c on b g we mean a smooth simple closed curve.

Figure 4 .
Figure 4. Dehn twist curves c i on g .