Spine removal surgery and the geography of symplectic fillings

TL;DR

This paper establishes universal bounds on the Euler characteristic and signature of minimal symplectic fillings for certain contact 3-manifolds using a novel spine removal surgery technique.

Contribution

It introduces a new application of spine removal surgery to bound topological invariants of symplectic fillings in contact geometry.

Findings

Universal bounds on Euler characteristic and signature for specified contact 3-manifolds.

Application of spine removal surgery to symplectic topology.

Demonstrates the effectiveness of the new surgery technique in bounding invariants.

Abstract

We prove that for any contact 3-manifold supported by a spinal open book decomposition with planar pages, there is a universal bound on the Euler characteristic and signature of its minimal symplectic fillings. The proof is an application of the spine removal surgery operation recently introduced in joint work of the authors with Van Horn-Morris.

Equations

$$|\chi (W)|\le N,\text{ and }|\sigma (W)|\le N.$$

$$\boldsymbol{\pi}:=\Big{(}\pi_{\Sigma}:M_{\Sigma}\to\Sigma,\ \pi_{P}:M_{P}\to S^{1}\Big{)}.$$

$$M_{\Sigma }={\Sigma }_1\times S^1⨿\dots ⨿{\Sigma }_N\times S^1.$$

$$Z:=W{\cup }_{M}X\stackrel{\Pi }{⟶}{\Sigma }_0$$

$$C_1\cup \dots \cup C_N\subset X\subset Z$$

$$\check{Z}\cong {\Sigma }_0\times S^2\stackrel{\check{\Pi }}{⟶}{\Sigma }_0,$$

$${\check{C}}_1,\dots ,{\check{C}}_N\subset {\Sigma }_0\times S^2.$$

$$\boldsymbol{\pi}:=\Big{(}\pi_{\Sigma}:M_{\Sigma}\to\Sigma,\ \pi_{P}:M_{P}\to S^{1}\Big{)}.$$

$$X\cong \left([0,1]\times M\right){\cup }_{\{1\}\times M_{\Sigma }}\coprod _{i=1}^N{\Sigma }_i\times {\mathbb{D}}^2$$

$$\partial X=(-{\partial }_{-}X)⨿{\partial }_{+}X,$$

$$C_i:={\Sigma }_i\times \{0\}\subset {\Sigma }_i\times {\mathbb{D}}^2\subset X$$

$$\partial C_i=\partial {\Sigma }_i\times \{0\}\subset {\partial }_{+}X$$

$$(Z,\omega ):=(W,{\omega }_W){\cup }_M(X,{\omega }_X)$$

$$\Pi :Z\to {\Sigma }_0,$$

$$[F]\cdot [C_i]=k_i.$$

$$\left\{(z^{2},az)\in{\mathbb{D}}^{2}\times S^{2}\ \big{|}\ z\in{\mathbb{D}}^{2}\right\},\qquad a\in{\mathbb{C}}\setminus\{0\}$$

$$\left\{(z,b)\in{\mathbb{D}}^{2}\times S^{2}\ \big{|}\ z\in{\mathbb{D}}^{2}\right\}\qquad b\in{\mathbb{C}}.$$

$$u:{\mathbb{D}}^2\hookrightarrow {\mathbb{D}}^2\times S^2:z\mapsto (z^2+R(z),az)$$

$$u_{ϵ}(z):=(z^2+\beta (|z|/ϵ)R(z),az),$$

$${\phi }_i:=\Pi {|}_{C_i}:C_i\to {\Sigma }_0$$

$$k_i\chi ({\Sigma }_0)=\chi (C_i)+b_i$$

$$\chi ({\Sigma }_0)\ge \frac{1}{k_i}\chi (C_i)\text{ for each i=1,…,N}.$$

$$Z\cong ({\Sigma }_0\times S^2)\#m{\overline{\mathbb{C}P}}^2.$$

$$E_1,\dots ,E_m\subset Z$$

$$\beta :(Z,J)\to (\check{Z},\check{J}),$$

$$\check{\Pi }:\check{Z}\to {\Sigma }_0$$

$${\check{C}}_i\looparrowright \check{Z}$$

$$c_1^{\tau }(N_{{\check{C}}_i})=\sum _{j=1}^m[C_i]\cdot [E_j]\ge 0.$$

$$c_1^{\tau }(N_{{\check{C}}_i})=c_1^{\tau }({\check{C}}_i)-\chi (C_i),$$

$$\partial Z=\partial \check{Z}\stackrel{\check{\Pi }}{⟶}\partial {\Sigma }_0,$$

Full text

S.L. was partially supported during this project by the ERC Starting Grant of Frédéric Bourgeois StG-239781-ContactMath, Vincent Colin’s ERC Grant geodycon, and by a University of Mississippi CLA SRG. C.W. was partially supported by a Royal Society University Research Fellowship, EPSRC grant EP/K011588/1, and the ERC Grant TRANSHOLOMORPHIC

1. Introduction

It was conjectured in 2002 by Stipsicz [Stipsicz:gaugeStein] that every closed contact $3$-manifold admits a universal bound on the signatures and Euler characteristics of its possible Stein fillings. Counterexamples to this conjecture were found a few years ago by Baykur and Van Horn-Morris [BaykurVanhorn:large], but it was also shown by Kaloti [Kaloti:Stein] that the conjecture holds for the special class of planar contact manifolds, i.e. those that are supported (in the sense of Giroux [Giroux:ICM]) by a planar open book decomposition. Kaloti’s proof was based on relations in the mapping class groups of compact planar surfaces with boundary, using a theorem of the second author [Wendl:fillable] that describes fillings of planar contact manifolds in terms of Lefschetz fibrations, together with methods of Plamenevskaya and Van Horn-Morris [PlamenevskayaVanHorn] to achieve bounds on numbers of positive factorizations. In this note, we shall use completely different methods to prove the following generalization of Kaloti’s result:

Theorem 1**.**

Suppose $(M,\xi)$ is a closed contact $3$-manifold with a supporting spinal open book whose pages are planar. Then there exists a finite list of $4$-manifolds $W_{1}^{\prime},\ldots,W_{n}^{\prime}$ with boundary, and a compact $4$-manifold $X$ that has $-M$ as a boundary component, such that if $(W,\omega)$ is any minimal strong filling of $(M,\xi)$, then $W\cup_{M}X\cong W_{j}^{\prime}$ for some $j\in\{1,\ldots,n\}$.

Corollary 2**.**

Under the hypotheses of Theorem 1, there exists a number $N\in{\mathbb{N}}$ such that for all minimal strong fillings $(W,\omega)$ of $(M,\xi)$,

[TABLE]

Proof.

If $W\cup_{M}X=W_{j}^{\prime}$, then $\chi(W)=\chi(W_{j}^{\prime})-\chi(X)$ since $M$ has vanishing Euler characteristic, so this gives the bound on $\chi(W)$. A bound on $\sigma(W)$ follows immediately from Novikov additivity; in more elementary terms, one can also see it from the observation that the inclusion $W\hookrightarrow W_{j}^{\prime}$ induces an injection of any positive/negative-definite subspace of $H_{2}(W;{\mathbb{Q}})$ into $H_{2}(W_{j}^{\prime};{\mathbb{Q}})$, implying $b_{2}^{\pm}(W)\leq b_{2}^{\pm}(W_{j}^{\prime})$. ∎

Spinal open book decompositions were recently introduced in joint work of the authors with Jeremy Van Horn-Morris [LisiVanhornWendl1]. The main motivation behind them is that they are the natural structure one obtains on the boundary of any Lefschetz fibration whose fibers and base are both compact oriented surfaces with nonempty boundary. As with ordinary open books, a spinal open book on $(M,\xi)$ with pages of genus zero gives rise to a well-behaved family of pseudoholomorphic curves in the symplectization of $(M,\xi)$, and the followup paper [LisiVanhornWendl2] carries out the program of using this technology to classify fillings of $(M,\xi)$. The present paper, however, does not rely on those results: our proof of Theorem 1 is comparatively low-tech. Its main ingredients are the theory of closed $J$-holomorphic spheres in the spirit of Gromov/McDuff (cf. [McDuff:rationalRuled]), and a surgical operation on spinal open books that was introduced in [LisiVanhornWendl1], known as spine removal surgery. This operation is a generalization of several previous symplectic cobordism constructions that were inspired by the $2$-handle attachment in Eliashberg’s symplectic capping argument for contact $3$-manifolds [Eliashberg:cap].

For the convenience of the reader, we recall from [LisiVanhornWendl1] the definition of a spinal open book decomposition for a closed $3$-manifold. The manifold $M$ is decomposed into two compact regions with matching boundary, the spine $M_{\Sigma}$ and the paper $M_{P}$. These are equipped with fibrations $\pi_{\Sigma}\colon M_{\Sigma}\to\Sigma$ and $\pi_{P}\colon M_{P}\to S^{1}$, where $\Sigma$ is a (possibly disconnected) oriented surface, each of whose connected components has non-empty boundary, and the fibers of $\pi_{\Sigma}$ are assumed connected. The connected components of the fibers of $\pi_{P}$ are called the pages: they also have nonempty boundaries, which are disjoint unions of fibers of $\pi_{\Sigma}$. The spinal open book decomposition is planar if the pages are surfaces of genus 0.

The spinal open book decomposition is then the data

[TABLE]

A contact structure is supported by this spinal open book decomposition if it admits a contact form that restricts to each fiber of $\pi_{P}$ as a Liouville form and each fiber of $\pi_{\Sigma}$ is a closed Reeb orbit.

Our conditions give that the components of $M_{\Sigma}$ are each $S^{1}$-bundles over compact oriented surfaces $\Sigma_{1},\ldots,\Sigma_{N}$ with nonempty boundary. We fix a trivialization of each so as to identify the spine with

[TABLE]

This choice of trivializations will be referred to in the following as a framing of the spinal open book, and several details will depend on this choice, but the important point is that it can be fixed in advance, with no knowledge of the fillings of $(M,\xi)$.

According to the results of [LisiVanhornWendl1], if $(M,\xi)$ is supported by a planar spinal open book decomposition, it can only be strongly fillable if $\boldsymbol{\pi}$ satisfies a condition known as symmetry, which implies that all pages have the same topology and there exist numbers $k_{i}\in{\mathbb{N}}$ for $i=1,\ldots,N$ such that exactly $k_{i}$ boundary components of each page lie in $\Sigma_{i}\times S^{1}$. In the following, we will assume the spinal open book decomposition satisfies the hypotheses of Theorem 1, so, in particular, we may assume that the spinal open book decomposition is symmetric.

Remark 3*.*

A result for planar spinal open books is also stated in [Kaloti:Stein], but its proof is framed in terms of Dehn twist factorizations, thus it needs to assume that the fillings of $(M,\xi)$ are all characterized in terms of Lefschetz fibrations. The latter is not true for all spinal open books with planar pages, but only for a special class, satisfying a technical condition known as Lefschetz-amenability (see [LisiVanhornWendl1]*§1.1). More generally, a spinal open book may have the property that its monodromy permutes boundary components of the pages, in which case [LisiVanhornWendl2] produces on any filling a foliation by $J$-holomorphic curves that can include finitely many so-called exotic fibers, i.e. singularities that are different from Lefschetz singular fibers. The classification problem in these cases requires something more than an understanding of positive factorizations in the mapping class group. The following example exhibits a class of contact manifolds to which our theorem applies, but whose fillings cannot generally be understood in terms of Lefschetz fibrations with fixed boundary.

Example 4**.**

Suppose $B$ is a closed and connected (but not necessarily orientable) surface, and $\Gamma\subset B$ is a nonempty multicurve such that $B\setminus\Gamma$ is orientable. We say that $\Gamma$ inverts orientations if for every sufficiently small open subset ${\mathcal{U}}\subset B$ that is divided by $\Gamma$ into two components ${\mathcal{U}}_{+}$ and ${\mathcal{U}}_{-}$, ${\mathcal{U}}$ can be given an orientation that matches that of $B\setminus\Gamma$ on ${\mathcal{U}}_{+}$ and is the opposite on ${\mathcal{U}}_{-}$. Under this condition, a slight generalization of a well-known construction of Lutz [Lutz:77] (see [LisiVanhornWendl1]§1.4) assigns to any $S^{1}$-bundle $M\to B$ with oriented total space a canonical isotopy class of contact structures $\xi_{\Gamma}$, which are tangent to the fibers over $\Gamma$ and positively transverse to the fibers in $B\setminus\Gamma$ (with respect to their orientations induced by the orientations of $M$ and $B\setminus\Gamma$). As shown in [LisiVanhornWendl1]§1.4, $(M,\xi_{\Gamma})$ admits a supporting spinal open book with a family of annular pages corresponding to each component of $\Gamma$, where the monodromy exchanges boundary components of the annulus for each component of $\Gamma$ whose normal bundle is nontrivial. It is not hard to cook up concrete examples of this phenomenon where $B$ is the Klein bottle, in which case $(M,\xi_{\Gamma})$ can also be described as a contact parabolic torus bundle (see [LisiVanhornWendl2]). Theorem 1 applies to all contact manifolds of this type, and thus gives bounds on the geography of their minimal symplectic fillings.

Before proving Theorem 1 in detail, here is a sketch. We construct the symplectic manifold $X$ as a spine removal cobordism that caps every boundary component of the pages (see Figure 1). Planarity implies that the rest of $\partial X$ is then a disjoint union of symplectic $S^{2}$-fibrations over $S^{1}$, so for a generic choice of compatible almost complex structure $J$ making these $S^{2}$-fibers complex, the usual holomorphic curve methods (as in [McDuff:rationalRuled]) produce a symplectic Lefschetz fibration

[TABLE]

over some compact oriented surface $\Sigma_{0}$ with boundary, where the regular fibers are $J$-holomorphic spheres and the singular fibers each consist of a pair of transversely intersecting $J$-holomorphic exceptional spheres. The co-cores of the spine removal handles

[TABLE]

define multisections of this Lefschetz fibration with degrees determined by the original spinal open book, and they are also symplectic submanifolds, so they can be arranged to be $J$-holomorphic. Now since $\partial\Sigma_{0}\neq\emptyset$, blowing down an exceptional sphere in each singular fiber produces a trivial fibration

[TABLE]

implying that the topology of $Z$ is $(\Sigma_{0}\times S^{2})\#m\overline{{\mathbb{C}}P}^{2}$, where $m$ is the number of singular fibers. In order to obtain bounds on both $\Sigma_{0}$ and $m$, we observe first that each $\Pi|_{C_{i}}:C_{i}\to\Sigma_{0}$ is a branched cover, so that the Riemann-Hurwitz formula implies a lower bound on $\chi(\Sigma_{0})$. Finally, we will find a bound on $m$ by blowing down $m$ exceptional spheres and considering the resulting multisections

[TABLE]

Since the original filling was assumed to be minimal, each exceptional sphere in $Z$ must intersect at least one of the $C_{i}$’s, thus each blow-down operation makes some positive contribution to the relative first Chern numbers of the normal bundles of $\widecheck{C}_{i}$. But since the topology of $\Sigma_{0}\times S^{2}$ is quite simple, we will also be able to show that these Chern numbers depend only on the framed link $\coprod_{i=1}^{N}\partial\widecheck{C}_{i}\subset\partial\Sigma_{0}\times S^{2}$, which (up to some finite ambiguity due to choices of trivialization) again depends only on the original spinal open book.

The motivating idea in this argument is that the topology of the unknown filling $W$ can be encoded in the arrangement of positively intersecting multisections $\widecheck{C}_{1}\cup\ldots\cup\widecheck{C}_{N}\subset\Sigma_{0}\times S^{2}$, whose pattern of intersections depends on the intersections of the co-cores $C_{i}\subset Z$ with the singular fibers. Crucially, if we make the right choices in blowing down $Z$ to construct $\widecheck{Z}$, then the relative homology classes of the $\widecheck{C}_{i}$ are determined by their restrictions to the boundary, and are thus independent of the choice of filling. Figures 3 and 3 show an example of how two slightly different arrangements with the same boundary can correspond to distinct minimal symplectic fillings of the same contact manifold.

Note that while it is convenient in our argument to assume the co-cores $C_{i}\subset X$ are $J$-holomorphic, it is not essential—we use this assumption mainly in order to ensure that their intersections are positive and that they have well-defined blow-downs $\widecheck{C}_{i}$, but we do not need any Fredholm or compactness theory for these curves. That is fortunate, because in most cases, they live in moduli spaces of negative virtual dimension and would thus disappear under any nontrivial deformation of $J$. There is one exception: if the original spinal open book is an ordinary supporting open book in the sense of Giroux [Giroux:ICM], then each co-core $C_{i}$ is a disk that can be completed to a $J$-holomorphic plane having index [math] and an unobstructed deformation theory. This provides a reason to expect strictly more rigidity in the presence of planar open books, suggesting in particular the following conjecture:

Conjecture**.**

Every planar contact $3$-manifold has at most finitely many distinct deformation classes of minimal symplectic fillings.

Acknowledgements

This paper emerged as a side project from our long-running collaboration with Jeremy Van Horn-Morris, and we are deeply indebted to him for many enlightening discussions. The second author would also like to thank Tom Mark, who originally suggested the idea of using the symplectic co-cores of spine removal handles to extract information about symplectic fillings.

2. The proof

2.1. Spine removal

Assume for the rest of this paper that $(M,\xi)$ satisfies the hypotheses of Theorem 1. In particular, $\xi$ is supported by the symmetric, planar spinal open book decomposition

[TABLE]

Fix a framing of the spinal open book, which then identifies each component of the spine with $\Sigma_{i}\times S^{1}$. Since the spinal open book decomposition is symmetric, there are integers $k_{i}$ that give the incidence of any page with the spine component $\Sigma_{i}\times S^{1}$.

Denote by

[TABLE]

the cobordism obtained by performing spine removal surgery on every component of $M_{\Sigma}$, attaching $\Sigma_{i}\times{\mathbb{D}}^{2}$ along its boundary in the obvious way to each spinal component $\Sigma_{i}\times S^{1}\subset\{1\}\times M$ and then smoothing the corners (see Figure 1). According to [LisiVanhornWendl1], this cobordism carries a symplectic structure $\omega_{X}$ such that we can write

[TABLE]

where $\partial_{-}X=M$ is concave with induced contact structure $\xi$, and $\partial_{+}X$ is a disjoint union of symplectic fibrations over $S^{1}$, one for each connected component of $M_{P}$, whose fibers are closed surfaces obtained by capping off all boundary components of the pages with disks. By the assumption that the spinal open book is planar, these are spheres. Since $\operatorname{Diff}_{+}(S^{2})\simeq\operatorname{SO}(3)$ is connected, these fibrations are all trivial and are thus diffeomorphic to $S^{1}\times S^{2}$, and each of the fibers has exactly $k_{i}$ transverse and positive intersections with the symplectic co-cores

[TABLE]

for $i=1,\ldots,N$. The boundaries

[TABLE]

form a link transverse to the sphere fibers, which comes with a natural framing dependent on our original choice of framing for $\boldsymbol{\pi}$.

2.2. Lefschetz fibration and multisections

Now suppose $(W,\omega_{W})$ is a minimal strong filling of $(M,\xi)$. After a rescaling and deformation of $\omega_{W}$ near $\partial W$, we glue symplectically to form an enlarged symplectic manifold

[TABLE]

with boundary $\partial Z=\partial_{+}X$. Choose an $\omega$-compatible almost complex structure $J$ on $Z$ that makes both the co-cores $C_{1},\ldots,C_{N}$ and the $S^{2}$-fibers on $\partial_{+}X$ into $J$-holomorphic curves and is generic everywhere else. Then by standard arguments as in [McDuff:rationalRuled, Wendl:rationalRuled], the compactified moduli space of $J$-holomorphic spheres homotopic to the fibers on $\partial_{+}X$ forms the fibers of a Lefschetz fibration

[TABLE]

where $\Sigma_{0}$ is a compact oriented surface (homeomorphic to the compactified moduli space) whose number of boundary components is the number of connected components of $M_{P}$. The singular fibers of $\Pi:Z\to\Sigma_{0}$ each have two irreducible components, both exceptional spheres which intersect each other once transversely. Let $[F]\in H_{2}(Z)$ denote the homology class of the fibers and $[C_{i}]\in H_{2}(Z,\partial C_{i})$ the relative homology class of the co-core $C_{i}\subset X\subset Z$ for each $i=1,\ldots,N$; the intersection products $[F]\cdot[C_{i}]\in{\mathbb{Z}}$ are then well defined and satisfy

[TABLE]

By positivity of intersections, it follows that $C_{i}$ intersects each fiber at most $k_{i}$ times, with equality for the fibers to which it is transverse. A standard genericity argument (cf. [CieliebakMohnke:transversality]*Prop. 9.1(b)) implies:

Lemma 5**.**

The following holds for generic choices of $J$ satisfying the conditions described above. All singular fibers of $\Pi:Z\to\Sigma_{0}$ are transverse to each co-core $C_{i}$, with intersections occurring only at regular points, while the collection of all regular fibers that intersect one of the co-cores nontransversely is finite. Moreover, every such fiber $F\subset Z$ has at most one non-transverse intersection with any of the co-cores, and it has local intersection index $2$. ∎

We will need a slightly more precise picture of these curves in certain regions.

Lemma 6**.**

Suppose $E_{1},\ldots,E_{m}\subset Z$ is a collection of pairwise disjoint exceptional spheres that are each irreducible components of singular fibers for $\Pi$. Then after smooth and $C^{0}$-small deformations of $J$, the Lefschetz fibration, and the co-cores $C_{1},\ldots,C_{N}$ such that the co-cores and fibers remain $J$-holomorphic, the following can be assumed without loss of generality:

1. (1)

For each $i=1,\ldots,m$, $J$ is integrable on a neighborhood of $E_{i}$, which is biholomorphically equivalent to a neighborhood of the zero-section in the tautological line bundle over $\mathbb{CP}^{1}$. 2. (2)

For each $i=1,\ldots,N$ and each point $z_{0}\in\Sigma_{0}$ such that the fiber over $z_{0}$ has a tangential intersection with $C_{i}$, we can identify a neighborhood ${\mathcal{U}}\subset\Sigma_{0}$ of $z_{0}$ with ${\mathbb{D}}^{2}$ and identify $\Pi^{-1}({\mathcal{U}})$ with ${\mathbb{D}}^{2}\times S^{2}$ where $S^{2}:={\mathbb{C}}\cup\{\infty\}$, such that on this neighborhood, $J=i\oplus i$, $\Pi(z,w)=z$, and $(C_{1}\cup\ldots\cup C_{N})\cap\Pi^{-1}({\mathcal{U}})$ is the disjoint union of one surface of the form

[TABLE]

with a finite collection of other surfaces of the form

[TABLE]

Proof.

Two preliminary remarks: first, the $J$-holomorphic fibers of the Lefschetz fibration will deform smoothly under any generic deformation of $J$. Indeed, deformations are unobstructed for all $J$ since the curves satisfy the automatic transversality criterion of [HoferLizanSikorav], and for index reasons, the only danger of new bubbling under these deformations would come from index $-1$ curves when $J$ becomes nongeneric, but closed holomorphic curves can only have even index (cf. [Wendl:rationalRuled]*Remark 2.20). Second, the deformation of $J$ does not need to be generic everywhere, as it suffices to have genericity only in some open region that intersects all of the curves we are concerned about.

Now, denote by $\widetilde{{\mathbb{C}}}^{2}\to\mathbb{CP}^{1}$ the tautological line bundle, which can be obtained by performing a complex blowup on ${\mathbb{C}}^{2}$ at the origin, so let $i$ denote its standard complex structure. Since the co-cores $C_{i}$ intersect each of the exceptional spheres $E_{j}$ transversely, we can identify the complex normal bundle of $E_{j}$ with a $J$-invariant subbundle of $TZ|_{E_{j}}$ that is tangent to each $C_{i}$, and feeding this into the tubular neighborhood theorem then identifies a neighborhood of $E_{j}$ with a neighborhood of the zero-section $\mathbb{CP}^{1}\subset\widetilde{{\mathbb{C}}}^{2}$ such that $J$ matches $i$ along $E_{j}=\mathbb{CP}^{1}$ and each $C_{i}$ is tangent to fibers of $\widetilde{{\mathbb{C}}}^{2}\to\mathbb{CP}^{1}$ at the zero-section. We can then make a $C^{1}$-small deformation of each $C_{i}$ near $E_{j}$ so that it matches fibers precisely in some smaller neighborhood of $E_{j}$, and a corresponding $C^{0}$-small deformation of $J$ to make it match $i$ in this smaller neighborhood; since the change in $C_{i}$ was $C^{1}$-small, we can assume it is still symplectic and thus adjust $J$ correspondingly away from $E_{j}$ to make sure that $C_{i}$ is $J$-holomorphic. Notice that $E_{j}$ remains holomorphic throughout this deforation, which we can also assume is supported in a tubular neighborhood of $E_{j}$; then since $[E_{j}]\cdot[E_{j}]=-1$, positivity of intersections implies that $E_{j}$ is always the only closed holomorphic curve that is contained fully in this neighborhood, so that all other fibers (or irreducible components of fibers) of $\Pi$ pass through regions in which $J$ can still be assumed generic, hence they deform smoothly as indicated in the previous paragraph.

The changes near $E_{j}$ can be assumed to have no effect on any fiber $F:=\Pi^{-1}(z_{0})$ that intersects some co-core $C_{i}$ non-transversely, since $F$ is a regular fiber. To understand the neighborhood of $F$, identify its (necessarily trivial) complex normal bundle with a $J$-invariant subbundle of $TZ|_{F}$ that is tangent to every co-core $C_{i}$ except at the unique point where one of them is tangent to $F$, and use this to identify the neighborhood of $F$ with ${\mathbb{D}}^{2}\times S^{2}$ such that $F=\{0\}\times S^{2}$ and $J|_{F}=i\oplus i$. One can also arrange this so that the surfaces $\{z\}\times S^{2}$ are all fibers of $\Pi$, and after suitably reparametrizing $S^{2}$, we can assume all transverse intersections of $C_{1}\cup\ldots\cup C_{N}$ with $F$ occur at points $(0,b)\in{\mathbb{D}}^{2}\times S^{2}$ with $b\neq\infty$, while the tangential intersection occurs at $(0,0)$. In light of our choice of normal subbundle, the surfaces $C_{j}$ are tangent to surfaces of the form ${\mathbb{D}}^{2}\times\{\operatorname{const}\}$ whenever they intersect $F$ transversely. We can now modify $J$ near $F$ without changing it in directions tangent to the fibers such that, after a $C^{0}$-small deformation, $J=i\oplus i$ near $F$; after a similar $C^{1}$-small adjustment to the co-cores $C_{j}$ near their tranverse intersections with $F$, these will also have the form ${\mathbb{D}}^{2}\times\{b\}$ near those intersections, and $J$ can still be adjusted outside this small neighborhood to make the modified $C_{j}$ holomorphic. Near the tangential intersection of $C_{i}$ and $F$, Taylor’s theorem and the assumption that $C_{i}$ intersects $F$ with local index $2$ implies that $C_{i}$ can be parametrized by an embedded holomorphic curve of the form

[TABLE]

for some $a\in{\mathbb{C}}\setminus\{0\}$ and a smooth remainder function $R(z)\in{\mathbb{C}}$ satisfying $\lim_{z\to 0}\frac{R(z)}{|z|^{2}}=0$. Choosing $\epsilon>0$ small and a smooth cutoff function $\beta:[0,\infty)\to[0,1]$ with $\beta(s)=1$ for $s\geq 1$ and $\beta(s)=0$ near $s=0$, we modify $u$ to an embedding of the form

[TABLE]

and modify $C_{i}$ near $(0,0)$ by defining it as the image of $u_{\epsilon}$. Modifying $J$ to match $i\oplus i$ near $F$ now makes $u_{\epsilon}$ $J$-holomorphic near the intersection, and for $\epsilon>0$ sufficiently small it still traces out a symplectic submanifold $C^{1}$-close to the original $C_{i}$, thus $J$ can be modified further outside the neighborhood of $(0,0)$ to make the new $C_{i}$ everywhere $J$-holomorphic. The latter modification can be assumed to take place in a small neighborhood that contains no closed holomorphic curves, thus we can assume all fibers passing through that neighborhood also pass through regions where $J$ is generic, and they therefore survive the deformation. ∎

Corollary 7**.**

For each $i=1,\ldots,N$, the map

[TABLE]

is a branched cover of degree $k_{i}$. Moreover, its branch points are all simple (i.e. of order $2$), and for any $i,j=1,\ldots,N$, any two distinct branch points of $\varphi_{i}:C_{i}\to\Sigma_{0}$ and $\varphi_{j}:C_{j}\to\Sigma_{0}$ have distinct images in $\Sigma_{0}$, all of which are regular values of $\Pi:Z\to\Sigma_{0}$. ∎

We shall refer to the surfaces $C_{i}\subset Z$ as multisections of $\Pi:Z\to\Sigma_{0}$ with degree $k_{i}$. The multisection $C_{i}$ is an honest section if and only if $k_{i}=1$. Let $b_{i}$ be the number of branch points of the branched covering $\varphi_{i}\colon C_{i}\to\Sigma_{0}$ from Corollary 7. Then, the Riemann-Hurwitz formula gives

[TABLE]

since each branch point is simple. We then have

[TABLE]

Recall that the number of boundary components of $\Sigma_{0}$ is given by the number of connected components of the paper $M_{P}$. This therefore gives an upper bound on the genus of $\Sigma_{0}$ that is determined by the spinal open book $\boldsymbol{\pi}$ and hence is independent of the choice of filling $W$. The topological type of $\Sigma_{0}$ thus belongs to a finite list of possibilities determined by $(M,\xi)$ and $\boldsymbol{\pi}$.

Let $m\geq 0$ denote the number of singular fibers in the Lefschetz fibration $\Pi:Z\to\Sigma_{0}$. Blowing down one exceptional sphere in each of these fibers then gives a smooth $S^{2}$-fibration over $\Sigma_{0}$, which is necessarily trivial since $\partial\Sigma_{0}\neq\emptyset$, hence

[TABLE]

The remaining task is thus to establish an a priori upper bound on the number of singular fibers $m$.

Lemma 8**.**

Every symplectic exceptional sphere in $Z$ intersects at least one of the co-cores $C_{1},\ldots,C_{N}$.

Proof.

If not, then one can remove tubular neighborhoods of the co-cores and find a symplectic filling that is symplectic deformation equivalent to $(W,\omega)$ but contains an exceptional sphere, contradicting the assumption that $(W,\omega)$ is minimal. ∎

The argument for bounding $m$ is slightly more straightforward if we can assume one of the numbers $k_{i}$ is $1$, so let us consider the case $k_{1}=1$ before tackling the general situation.

2.3. The case $k_{1}=1$

Under this assumption, $C_{1}$ is a section of $\Pi:Z\to\Sigma_{0}$, so in particular, every singular fiber consists of exactly one exceptional sphere that intersects $C_{1}$ once and one that is disjoint from $C_{1}$. We shall denote the spheres of the latter type by

[TABLE]

and note that they are in one-to-one correspondence with the singular fibers of $\Pi:Z\to\Sigma_{0}$. Applying Lemma 6 to deform $J$ so that it becomes integrable near $E_{1}\cup\ldots\cup E_{m}$, define $\widecheck{Z}$ to be the symplectic manifold obtained from $Z$ by performing a blow-down operation on each $E_{1},\ldots,E_{m}$. This symplectic manifold inherits a compatible almost complex structure $\widecheck{J}$ such that the blowdown map

[TABLE]

defined as the identity outside of $E_{1}\cup\ldots\cup E_{m}$ but collapsing each $E_{i}$ to a point, is pseudoholomorphic. Since all singular fibers of $\Pi:Z\to\Sigma_{0}$ are in the interior, the boundary of $\widecheck{Z}$ is still $\partial Z$ and thus contains the same framed link $\coprod_{i=1}^{N}\partial C_{i}\subset\widecheck{Z}$. The Lefschetz fibration $\Pi:Z\to\Sigma_{0}$ now becomes a smooth symplectic $S^{2}$-fibration

[TABLE]

with $\widecheck{J}$-holomorphic fibers, and the co-cores $C_{i}$ for $i=1,\ldots,N$ project through $\beta:Z\to\widecheck{Z}$ to immersed (but possibly non-injective) $\widecheck{J}$-holomorphic curves

[TABLE]

with boundary $\partial\widecheck{C}_{i}=\partial C_{i}\subset\partial\widecheck{Z}$. These curves are immersed since each $C_{i}$ is transverse to every singular fiber by Lemma 5, but they have transverse self-intersections whenever $C_{i}$ intersects one of the $E_{j}$ more than once. They are again multisections of $\widecheck{\Pi}:\widecheck{Z}\to\Sigma_{0}$ with degree $k_{i}$. Notice that since $C_{i}$ and $C_{j}$ might both intersect some $E_{k}$, $\widecheck{C}_{i}$ and $\widecheck{C}_{j}$ might intersect. The co-cores $C_{i}$ and $C_{j}$ intersect $E_{k}$ in different points, however, so the intersection of $\widecheck{C}_{i}$ and $\widecheck{C}_{j}$ will be transverse and positive.

If $\tau$ denotes the framing of $\partial C_{i}\subset\partial Z$ we fixed previously, then by construction, $\tau$ extends to a global trivialization of the complex normal bundle $N_{C_{i}}\subset TZ|_{C_{i}}$ of $C_{i}$, so that its relative first Chern number is $c_{1}^{\tau}(N_{C_{i}})=0$. After blowing down, each of these Chern numbers will in general be larger, as can be measured by counting the intersections of $C_{i}$ with a small perturbation of itself: a positive transverse intersection in the blowdown is forced wherever $C_{i}$ and its perturbation pass through one of the $E_{j}$, proving

[TABLE]

Since we chose $E_{1},\ldots,E_{m}$ to be disjoint from $C_{1}$, we have $c_{1}^{\tau}(N_{\widecheck{C}_{1}})=0$. Note that $\widecheck{C}_{1}$ also does not intersect any of $\widecheck{C}_{2},\ldots,\widecheck{C}_{N}$.

We claim that for each $i=1,\ldots,N$, the left hand side of (2.2) satisfies a bound that is independent of the choice of minimal filling for $(M,\xi)$. To see this, note first that we can write

[TABLE]

where $c_{1}^{\tau}(\widecheck{C}_{i})$ is an abbreviation for the relative first Chern number of the complex vector bundle $T\widecheck{Z}$ pulled back along the immersion $\widecheck{C}_{i}\looparrowright\widecheck{Z}$, relative to the obvious trivialization induced by $\tau$ at the boundary. In particular, $\chi(C_{i})$ is determined by the spinal open book, and $c_{1}^{\tau}(\widecheck{C}_{i})$ depends only on the relative homology class $[\widecheck{C}_{i}]\in H_{2}(\widecheck{Z},\partial C_{i})$.

To understand the latter, observe first that there are finitely many homotopy classes of trivializations of the oriented $S^{2}$-bundle

[TABLE]

corresponding to choices of elements in $\pi_{1}(\operatorname{Diff}_{+}(S^{2}))\cong\pi_{1}(\operatorname{SO}(3))\cong{\mathbb{Z}}_{2}$ for each component of $\partial\Sigma_{0}$. Each of these trivializations identifies the framed link $\coprod_{i=1}^{N}\partial\widecheck{C}_{i}\subset\partial\widecheck{Z}=\partial Z$ with some isotopy class of framed links in $\partial\Sigma_{0}\times S^{2}$ transverse to the $S^{2}$-fibers, producing a finite list of such framed links that depends only on the original framed spinal open book and not on the filling $W$.

Note that $\Sigma_{0}$ is a surface with boundary and retracts to its $1$-skeleton. The fibration $\widecheck{\Pi}:\widecheck{Z}\to\Sigma_{0}$ is thus globally trivializable. After choosing such a trivialization, we may identify

[TABLE]

where the isotopy class of the framed link $\coprod_{i=1}^{N}\partial\widecheck{C}_{i}\subset\partial\Sigma_{0}\times S^{2}$ must belong to the aforementioned finite list, which is independent of the filling.

We now claim that the isotopy class of this framed link uniquely determines the relative homology classes $[\widecheck{C}_{i}]$ for $i=1,\ldots,N$. Indeed, let $[\widecheck{C}_{i}]\cdot_{\tau}[\widecheck{C}_{i}]\in{\mathbb{Z}}$ denote the relative self-intersection number of $[\widecheck{C}_{i}]$, defined by counting the signed intersections of $\widecheck{C}_{i}$ with a small generic perturbation of itself that is pushed in the direction of $\tau$ at the boundary (cf. [Hutchings:index, Siefring:intersection]). Then

[TABLE]

by construction. If $C^{\prime}\looparrowright\widecheck{Z}$ is then another properly immersed surface with $\partial C^{\prime}=\partial\widecheck{C}_{1}$, the relative homology classes of $C^{\prime}$ and $\widecheck{C}_{1}$ differ by some absolute class in $H_{2}(\Sigma_{0}\times S^{2})\cong{\mathbb{Z}}$, which is generated by the fiber class $[F]$, thus $[C^{\prime}]=[\widecheck{C}_{1}]+\ell[F]$ for some $\ell\in{\mathbb{Z}}$, and we have

[TABLE]

since $[\widecheck{C}_{1}]\cdot[F]=k_{1}=1$ and $[F]\cdot[F]=0$. It follows that $C^{\prime}$ can only have relative self-intersection zero if it is homologous to $\widecheck{C}_{1}$. Thus $[\widecheck{C}_{1}]\in H_{2}(\Sigma_{0}\times S^{2},\partial C_{1})$ is uniquely determined. For $\widecheck{C}_{i}$ with $i=2,\ldots,N$, the fact that

[TABLE]

and $[\widecheck{C}_{1}]\cdot[F]\neq 0$ similarly implies that $[\widecheck{C}_{i}]\in H_{2}(\Sigma_{0}\times S^{2},\partial C_{i})$ is uniquely determined.

From this, it follows that for $i=1,\ldots,N$, there exist integers $m_{i}\in{\mathbb{Z}}$ depending only on the framed spinal open book $\boldsymbol{\pi}$ such that

[TABLE]

At the same time, Lemma 8 implies

[TABLE]

for each $j=1,\ldots,m$, thus

[TABLE]

and this bound depends only on $\boldsymbol{\pi}$ with its chosen framing. Since $Z\cong(\Sigma_{0}\times S^{2})\#m\overline{{\mathbb{C}}P}^{2}$, this concludes the proof for the case $k_{1}=1$.

2.4. The general case

The above strategy fails if none of the $k_{i}=[C_{i}]\cdot[F]$ equal $1$ since it could then happen that for every singular fiber of $\Pi:Z\to\Sigma_{0}$, both exceptional spheres intersect every co-core $C_{i}$. We can deal with this by replacing $Z$ by a branched cover, defined essentially as the pullback of the Lefschetz fibration $\Pi:Z\to\Sigma_{0}$ via the map $\varphi_{1}:C_{1}\to\Sigma_{0}$.

Let us begin with a construction at the boundary that requires no knowledge of the filling. Recall that $\partial Z$ and $\partial\Sigma_{0}$ depend only on the spinal open book decomposition $\boldsymbol{\pi}$, and do not depend on the filling (though $Z$ and $\Sigma_{0}$ do). Since $\partial C_{1}$ is transverse to the fibers of $\partial Z\stackrel{{\scriptstyle\Pi}}{{\to}}\partial\Sigma_{0}$, the map $\partial C_{1}\stackrel{{\scriptstyle\varphi_{1}}}{{\to}}\partial\Sigma_{0}$ is a smooth $k_{1}$-fold covering map and there exists a smooth closed $3$-manifold111The notation $\partial Z^{\prime}$ is chosen because it will turn out to be the boundary of something constructed further below, but the definition of $\partial Z^{\prime}$ itself does not require this knowledge.

[TABLE]

which admits a smooth oriented $S^{2}$-fibration

[TABLE]

and a $k_{1}$-fold covering map

[TABLE]

such that $\varphi_{1}\circ\Pi^{\prime}=\Pi\circ\Phi$. The framed link $\coprod_{i=1}^{N}\partial C_{i}\subset\partial Z$ then gives rise to a new framed link consisting of the disjoint union of

[TABLE]

These links are similarly transverse to the $S^{2}$-fibers, and $\partial C_{1}^{\prime}$ contains a distinguished component given by the tautological section

[TABLE]

which is a lift of $\partial C_{1}\subset\partial Z$ to the cover. All of this depends only on the framed link $\coprod_{i=1}^{N}\partial C_{i}$ in $\partial Z$, and thus on the original framed spinal open book, but not on the filling $W$.

Next we extend the covering map $\partial Z^{\prime}\to\partial Z$ to a branched cover $Z^{\prime}\to Z$. The construction is completely analogous to that of the previous paragraph, except that it does depend on the filling: recall first that by Corollary 7, the branch points of $\varphi_{1}:C_{1}\to\Sigma_{0}$ occur in regular fibers, implying that $\varphi_{1}$ is transverse to $\Pi$ and the set

[TABLE]

is therefore a smooth compact $4$-manifold with boundary $\partial Z^{\prime}$, while

[TABLE]

is a Lefschetz fibration and

[TABLE]

is a $k_{1}$-fold branched cover satisfying $\varphi_{1}\circ\Pi^{\prime}=\Pi\circ\Phi$. The regular fibers of $\Pi^{\prime}:Z^{\prime}\to C_{1}$ are again spheres, and $\Phi$ is locally $2$-to-$1$ near a branching locus that consists of a finite collection of regular fibers corresponding to the branch points of $\varphi_{1}$. Since singular fibers are disjoint from the branching locus, $\Pi^{\prime}:Z^{\prime}\to C_{1}$ has exactly $k_{1}m$ singular fibers.

For each $i=1,\ldots,N$, let

[TABLE]

For $i=2,\ldots,N$, $C^{\prime}_{i}\subset Z^{\prime}$ is a submanifold since (by Corollary 7) the branch points of $\varphi_{i}:C_{i}\to\Sigma_{0}$ occur outside the branching locus of $\Phi$, hence $\Phi$ is transverse to $C_{i}$. The situation is slightly different for $i=1$ since $\Phi$ is not transverse to $C_{1}$, and $\Phi^{-1}(C_{1})$ is the set of all pairs $(z,x)\in C_{1}\times C_{1}$ such that $z$ and $x$ belong to the same fiber of $\Pi$. This contains the tautological section of $\Pi^{\prime}:Z^{\prime}\to C_{1}$ defined by

[TABLE]

Lemma 9**.**

We have $C_{1}^{\prime}=\sigma_{1}\cup C_{1}^{\prime\prime}$, where $C_{1}^{\prime\prime}$ is a (possibly disconnected) smooth submanifold of $Z^{\prime}$ that has finitely many intersections with $\sigma_{1}$, all transverse and positive, occurring at each of the points $(z,z)\in\sigma_{1}$ for the branch points $z\in C_{1}$ of $\varphi_{1}:C_{1}\to\Sigma_{0}$. Moreover, the map

[TABLE]

is a branched cover of degree $k_{1}-1$.

Proof.

As a set, we define $C_{1}^{\prime\prime}$ to be the closure of $C_{1}^{\prime}\setminus\sigma_{1}$:

[TABLE]

Recall that $Z^{\prime}$ was defined as a smooth submanifold of $C_{1}\times Z$, and $C_{1}^{\prime}$ is the intersection of this with the submanifold $C_{1}\times C_{1}\subset C_{1}\times Z$. We claim that for any $(z,x)\in Z^{\prime}\cap(C_{1}\times C_{1})$ with the property that $z$ and $x$ are not both branch points of $\varphi_{1}:C_{1}\to\Sigma_{0}$, this intersection of submanifolds in $C_{1}\times Z$ is transverse. This is equivalent to the claim that the tangent space

[TABLE]

contains elements of the form $(X,Y)$ for arbitrary vectors $Y$ spanning some subspace of $T_{x}Z$ transverse to $T_{x}C_{1}$. If $z$ is a regular point of $\varphi_{1}$, then $T_{z}\varphi_{1}:T_{z}C_{1}\to T_{\varphi_{1}(x)}\Sigma_{0}$ is an isomorphism, so for any $Y\in T_{x}Z$ there is a unique $X\in T_{z}C_{1}$ satisfying $T\varphi_{1}(X)=T\Pi(Y)$, which gives $(X,Y)\in T_{(z,x)}Z^{\prime}$. The alternative is that $T_{z}\varphi_{1}$ vanishes but $x$ is a regular point of $\varphi_{1}$, so $T_{(z,x)}Z^{\prime}=T_{z}C_{1}\oplus\ker T_{x}\Pi$, where $\ker T_{x}\Pi$ is the tangent space to the fiber at $x$. The regularity of $\varphi_{1}$ at $x$ then implies that $\ker T_{x}\Pi$ is transverse to $T_{x}C_{1}$ and thus proves the claim.

Observe that by Corollary 7, any $(z,x)\in Z^{\prime}\cap(C_{1}\times C_{1})$ with $z\neq x$ has the property that $z$ and $x$ cannot both be branch points of $\varphi_{1}$, since $\varphi_{1}(z)=\varphi_{1}(x)$. We have thus proved that the only possible singularities of $C_{1}^{\prime}$ are at points $(z,z)\in\sigma_{1}$ such that $z$ is a branch point of $\varphi_{1}$.

We now analyze a neighborhood of any such point, using the local model from Lemma 6. Locally, we can assume $\Sigma_{0}={\mathbb{D}}^{2}\subset{\mathbb{C}}$, $Z={\mathbb{D}}^{2}\times S^{2}$, $\Pi(\zeta,w)=\zeta$ and $C_{1}\subset Z$ is parametrized by an embedding of the form

[TABLE]

for some $a\in{\mathbb{C}}\setminus\{0\}$. Since there are no singular fibers in this local picture, $\Pi^{\prime}:Z^{\prime}\to C_{1}$ is actually just the pullback of $\Pi:Z\to\Sigma_{0}$ through $\varphi_{1}:C_{1}\to\Sigma_{0}$, and the parametrization for $C_{1}$ above then provides a smooth trivialization of $\Pi^{\prime}$ identifying $((\zeta^{2},a\zeta),(\zeta^{2},x))\in C_{1}\times Z$ with $(\zeta,x)\in{\mathbb{D}}^{2}\times S^{2}$. Under this identification, we can write

[TABLE]

with $\Phi(\zeta,x)=(\zeta^{2},x)$ and $\Pi^{\prime}(\zeta,x)=(\zeta^{2},a\zeta)$, and the tautological section is now parametrized by

[TABLE]

We then have

[TABLE]

which can be written as the union of two transversely and positively intersecting submanifolds

[TABLE]

where the first is the tautological section $\sigma_{1}$. It thus follows that $C_{1}^{\prime\prime}$ is a smooth submanifold of $Z^{\prime}$.

Finally, the coordinate description above implies that the map $\varphi_{1}^{\prime}:C_{1}^{\prime\prime}\to C_{1}$ is a local diffeomorphism near $C_{1}^{\prime\prime}\cap\sigma_{1}$, and it is clearly also a local diffeomorphism near all points of the form $(z,x)\in C_{1}^{\prime\prime}$ such that $z$ and $x$ are regular points of $\varphi_{1}$. If on the other hand $(z,x)\in C_{1}^{\prime\prime}$ where $x$ is a branch point of $\varphi_{1}$ and $z$ is not, then we can choose coordinates as above so that locally $\Sigma_{0}={\mathbb{D}}^{2}$ and $Z={\mathbb{D}}^{2}\times S^{2}$, with $\Pi(\zeta,w)=\zeta$ and a neighborhood of $x$ in $C_{1}$ parametrized by a map in the form of (2.3) with $x=(0,0)$, while a neighborhood of $z$ in $C_{1}$ is a straightforward section

[TABLE]

with $z=(0,b)$ for some constant $b\in{\mathbb{C}}\setminus\{0\}$. The neighborhood of $(z,x)$ in $C_{1}^{\prime}$ is thus parametrized by the embedding

[TABLE]

which satisfies $\varphi_{1}^{\prime}(h(z))=(\zeta^{2},b)$, showing that $(z,x)$ is a simple branch point of $\varphi_{1}^{\prime}$. The degree of $\varphi_{1}^{\prime}$ can be deduced by counting $\varphi_{1}^{-1}(z)$ for a generic point $z\in C_{1}$: it is the set of all pairs $(z,x)$ such that $x\in C_{1}$ as in the same fiber as $z$ but is not equal to it, so outside of the finitely many fibers that are not transverse to $C_{1}$, the number of points in this set will be exactly $k_{1}-1$, and they are all regular points by the discussion above. ∎

The next lemma follows from Lemma 8:

Lemma 10**.**

For each singular fiber of $\Pi^{\prime}:Z^{\prime}\to C_{1}$, each of the two irreducible components intersects $C^{\prime}_{1}\cup\ldots\cup C^{\prime}_{N}$, and the intersections are transverse and positive.∎

Lemma 11**.**

Each of the surfaces $C_{1}^{\prime\prime},C_{2}^{\prime},\ldots,C_{N}^{\prime}\subset Z^{\prime}$ has topological type belonging to a finite list of possibilities that depend on the spinal open book $\boldsymbol{\pi}$ and its chosen framing, but not on the filling $W$.

Proof.

For $i=2,\ldots,N$, $C^{\prime}_{i}$ is a (possibly disconnected) multisection of $\Pi^{\prime}:Z^{\prime}\to C_{1}$ with degree $k_{i}$, and

[TABLE]

is a $k_{1}$-fold branched cover with exactly $k_{i}$ simple branch points in every component of the branching locus of $\Phi$. Then, using again the notation introduced following Corollary 7, where $b_{i}$ is the number of branch points of $\varphi_{i}=\Pi|_{C_{i}}\colon C_{i}\to\Sigma_{0}$, this gives that $\psi_{i}\colon C^{\prime}_{i}\to C_{i}$, is a $k_{1}$-fold branched cover with $k_{i}b_{1}$ simple branch points. Riemann-Hurwitz then implies

[TABLE]

Recall from Corollary 7 and the subsequent discussion that Riemann-Hurwitz also gives us

[TABLE]

It follows then

[TABLE]

Recall that $\chi(\Sigma_{0})\leq 1$ since $\Sigma_{0}$ is connected (and has boundary). This then gives a lower bound

[TABLE]

which only depends on the spinal open book decomposition. We also immediately obtain the upper bound $\chi(C^{\prime}_{i})\leq k_{1}\chi(C_{i})$, which also depends only on the spinal open book decomposition.

We now consider $C^{\prime}_{1}=\sigma_{1}\cup C^{\prime\prime}_{1}$. Observe that the number of intersections of $\sigma_{1}$ with $C^{\prime\prime}_{1}$ is precisely $b_{1}$, the number of branch points of $\varphi_{1}$. Applying the Riemann-Hurwitz formula and again using $\chi(\Sigma_{0})\leq 1$, we obtain the following bound that depends only on $\boldsymbol{\pi}$:

[TABLE]

The topologies of $\sigma_{1}$ and $C^{\prime\prime}_{1}$ satisfy similar bounds in terms of $\boldsymbol{\pi}$: $\sigma_{1}$ is diffeomorphic to $C_{1}$ since it is a section, and the topology of $C^{\prime\prime}_{1}$ can be bounded via the fact that $\varphi_{1}^{\prime}:C^{\prime\prime}_{1}\to C_{1}$ is a $(k_{1}-1)$-fold branched cover with $k_{1}-2$ simple branch points in each connected component of the branching locus of $\Phi$, i.e. it has $(k_{1}-2)b_{1}$ branch points. Again by Riemann-Hurwitz,

[TABLE]

Hence,

[TABLE]

This is bounded below by $(2k_{1}-3)\chi(C_{1})-k_{1}(k_{1}-2)$ since $\chi(\Sigma_{0})\leq 1$, and above by $(2k_{1}-3)\chi(C_{1})-(k_{1}-2)\chi(C_{1})=(k_{1}-1)\chi(C_{1})$ due to (2.1). ∎

We now proceed by adapting the blow-down argument that worked in the case $k_{1}=1$, but with $Z$ replaced by $Z^{\prime}$ and the tautological section $\sigma_{1}$ playing the role previously played by $C_{1}$. Let $\tau$ denote the framing of the link $\coprod_{i=1}^{N}\partial C_{i}^{\prime}\subset\partial Z^{\prime}$ defined by lifting the framing of $\coprod_{i=1}^{N}\partial C_{i}\subset\partial Z$. In light of the bounds we have already established on the topology of $\Sigma_{0}$, the next lemma implies both lower and upper bounds on the value of $[\sigma_{1}]\cdot_{\tau}[\sigma_{1}]$.

Lemma 12**.**

$[\sigma_{1}]\cdot_{\tau}[\sigma_{1}]=\chi(C_{1})-k_{1}\chi(\Sigma_{0})$.

Proof.

We construct a perturbed section $\sigma_{1}^{\epsilon}\subset Z^{\prime}$ as follows. Each branch point of $\varphi_{1}:C_{1}\to\Sigma_{0}$ corresponds to a point at which $C_{1}$ touches a fiber tangentially, and the Riemann-Hurwitz formula dictates that the number of such points is $-\chi(C_{1})+k_{1}\chi(\Sigma_{0})$. Now consider a coordinate neighborhood of such a point as in Lemma 6: the local model near this intersection is

[TABLE]

for some $a\in{\mathbb{C}}\setminus\{0\}$. For $\epsilon>0$ small, define the perturbed surface

[TABLE]

which has a single transverse and positive intersection with $C_{1}$ in this neighborhood. Now extend $C_{1}^{\epsilon}$ globally to a small perturbation of $C_{1}$ that is pushed in the direction of the framing $\tau$ at the boundary, and assume it is generic so that all other intersections between $C_{1}$ and $C_{1}^{\epsilon}$ are transverse and occur in fibers that are transverse to $C_{1}$. Since $[C_{1}]\cdot_{\tau}[C_{1}]=0$, the positive intersections we see in the coordinate neighborhoods above must be canceled by further intersections, the total signed count of which is therefore $\chi(C_{1})-k_{1}\chi(\Sigma_{0})$. But $C_{1}^{\epsilon}$ can also be lifted to a section $\sigma_{1}^{\epsilon}$ of $\Pi^{\prime}:Z^{\prime}\to C_{1}$ that will be disjoint from $\sigma_{1}$ near the branching locus, while each of the other intersections of $C_{1}$ with $C_{1}^{\epsilon}$ in $Z$ lifts to an intersection of $\sigma_{1}$ with $\sigma_{1}^{\epsilon}$ in $Z^{\prime}$, giving

[TABLE]

∎

The rest of the argument is completely analogous to the $k_{1}=1$ case, so a sketch should now suffice. Recall that the goal is to establish a bound on the number $m\geq 0$ of singular fibers in $\Pi:Z\to\Sigma_{0}$. Using the local model in Lemma 6 to understand the branching locus of $\Phi:Z^{\prime}\to Z$, there is a well-defined almost complex structure $J^{\prime}:=\Phi^{*}J$ on $Z^{\prime}$ for which the multisections $\sigma_{1},C_{1}^{\prime\prime},C_{2}^{\prime},\ldots,C_{N}^{\prime}$ are all $J^{\prime}$-holomorphic. Each of the singular fibers of $\Pi^{\prime}:Z^{\prime}\to C_{1}$ contains a unique exceptional sphere disjoint from $\sigma_{1}$, so denote these by $E_{1},\ldots,E_{k_{1}m}$ and blow them down to produce a smooth $S^{2}$-fibration

[TABLE]

with $\widecheck{J}^{\prime}$-holomorphic fibers for some almost complex structure such that the blowdown map $(Z^{\prime},J^{\prime})\to(\widecheck{Z}^{\prime},\widecheck{J}^{\prime})$ is pseudoholomorphic. Composing the latter with our multisections in $Z^{\prime}$ gives rise to immersed $\widecheck{J}^{\prime}$-holomorphic multisections

[TABLE]

Trivializing the fibration $\widecheck{\Pi}^{\prime}:\widecheck{Z}^{\prime}\to C_{1}$ then identifies $\widecheck{Z}^{\prime}$ with $C_{1}\times S^{2}$ so that the framed link $\coprod_{i=1}^{N}\partial\widecheck{C}_{i}^{\prime}\subset\partial\widecheck{Z}^{\prime}$ belongs to one of a finite collection of isotopy classes of framed links in $\partial C_{1}\times S^{2}$ that are independent of the choice of filling. Using the established bounds on $[\widecheck{\sigma}_{1}]\cdot_{\tau}[\widecheck{\sigma}_{1}]=[\sigma_{1}]\cdot_{\tau}[\sigma_{1}]$ and the fact that $[\widecheck{\sigma}_{1}]\cdot[F]=1$ for the fiber class $[F]\in H_{2}(C_{1}\times S^{2})$, this framed link determines $[\widecheck{\sigma}_{1}]\in H_{2}(C_{1}\times S^{2},\partial\widecheck{\sigma}_{1})$ up to a finite ambiguity. Similarly, $\sigma_{1}$ has exactly $-\chi(C_{1})+k_{1}\chi(\Sigma_{0})$ positive and transverse intersections with $C_{1}^{\prime\prime}$, which does not change after blowing down, nor does the fact that $\sigma_{1}$ and $C_{i}^{\prime}$ are disjoint for all $i=2,\ldots,N$, hence these relations also determine the relative homology classes $[\widecheck{C}_{1}^{\prime\prime}]$ and $[\widecheck{C}_{i}^{\prime}]$ for $i=2,\ldots,N$ up to finite ambiguity. These relative homology classes together with the respective Euler characteristics determine the relative first Chern numbers of the normal bundles of these surfaces, which are then used to bound the numbers

[TABLE]

By Lemma 10, we also have

[TABLE]

for each $j=1,\ldots,k_{1}m$, and thus conclude

[TABLE]

giving an upper bound on $m$ that depends only on the framed spinal open book $\boldsymbol{\pi}$. This concludes the proof in the general case.