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SYMPLECTIC HOMOLOGY OF COMPLEMENTS OF SMOOTH

DIVISORS

LUIS DIOGO AND SAMUEL T. LISI

Contents

1. Introduction 2

Part 1. Setup, almost complex structures and neck-stretching 42. Almost complex structures and neck-stretching 42.1. Symplectic hyperplane sections 42.2. Almost complex structures 6

Part 2. Moduli spaces and symplectic chain complex 113. The chain complex 114. Floer moduli spaces before and after splitting 144.1. Floer cascades before splitting 144.2. Floer cascades after splitting 175. Sketch of isomorphism of the two Floer complexes 206. Transversality for the Floer and holomorphic moduli spaces 256.1. Statements of transversality results 256.2. A Fredholm theory for Floer cascades 296.3. Transversality for chains of pearls in Σ 356.4. Transversality for spheres in X with order of contact constraints in Σ 416.5. Proof of Proposition 6.9 44

Part 3. Index calculations and ruling out configurations 557. Fredholm index for Floer cylinders in the symplectization 557.1. Conley–Zehnder indices 557.2. Index inequalities from monotonicity and transversality 598. Orientations 65

Part 4. Ansatz 699. From Floer cylinders to pseudoholomorphic curves 6910. From pseudoholomorphic curves to Floer cylinders 7410.1. Geometric interpretation 8311. Pseudoholomorphic spheres in Σ and meromorphic sections 8511.1. Back to orientations 8612. A formula for the symplectic homology differential 9612.1. Relation to Gromov–Witten invariants 98

Date: December 1, 2017.

1

2 LUIS DIOGO AND SAMUEL T. LISI

Part 5. Example: T˚S2 10113. The coefficients in the differential 10114. The group SH˚pT

˚S2q 103Appendix A. Morse–Bott Riemann–Roch 105References 110

1. Introduction

Symplectic homology is a version of Hamiltonian Floer homology for a class ofopen symplectic manifolds with contact boundary, including Liouville manifolds[FH94, Vit99]. Some of the applications of symplectic homology include specialcases of the Weinstein conjecture [Vit99], and a proof that there are infinitelymany distinct symplectic structures in R2n, for n ě 4 [McL09].

One reason why symplectic homology is so relevant is the fact that it relatesto other important objects of symplectic topology, including contact homology[BO09a], Rabinowitz Floer homology [CFO10] and the Fukaya category [Abo10,Gan12], as well as to string topology [AS12]. In this paper, we explore a rela-tion between symplectic homology and quantum cohomology. For other relationsbetween these two invariants, see Seidel [Sei16] as well as Borman and Sheridan[BS].

Symplectic homology has a wealth of algebraic structures, including a productand a Batallin–Vilkovisky operator, as observed by Seidel [Sei08]. This paper willnot address the computation of these algebraic structures, but we intend to explorethat direction in future work using the methods developed here.

Despite its great relevance, symplectic homology is often very hard to compute.Among the few known results is the fact that it vanishes for subcritical Weinsteinmanifolds (including pR2n, ωstdq) [Cie02] and for flexible Weinstein domains [?HERE].Symplectic homology of the cotangent bundle of a closed, orientable spin manifoldM is isomorphic as a ring to the Chas–Sullivan string topology ring of M [AS12,Abo15]. There is also a surgery formula for the symplectic homology of a Weinsteinmanifold [BEE12, BEE11], which has been used to compute new examples [EN15,EL15].

In this paper, we explore the problem of computing the symplectic homology ofa special class of symplectic manifolds, namely complements of certain symplectichypersurfaces Σ in closed symplectic manifolds X. An important source of examplescomes from smooth ample divisors in projective varieties. Allowing for Σ to be anormal crossings divisor would be a significant improvement in generality, but iscurrently beyond our reach. Important progress has been made in that direction,including [Pas13,Ngu14,GP16,McL].

In this paper, we work under the following assumptions:

‚ pX2n, ωq is a closed symplectic manifold;‚ X has a symplectic submanifold Σ2n´2 such that rΣs P H2n´2pX;Rq is

Poincare-dual to rKωs for some K ą 0 (in particular, rKωs admits a lift toH2pX;Zq);

‚ X is spherically monotone, so there is a constant τX ą 0 such that

xc1pTXq, Ay “ τXxrωs, Ay

SYMPLECTIC HOMOLOGY OF COMPLEMENTS OF SMOOTH DIVISORS 3

for any spherical homology class A (in the image of the Hurewicz mapπ2pXq Ñ H2pX;Zq);

‚ τX ´K ą 0.

Given a spherical homology class A in Σ, a simple use of the direct sum propertyfor Chern classes implies that xc1pTΣq, Ay “ pτX ´KqωpAq, hence pΣ, ωΣq is alsospherically monotone (where ωΣ is the restriction of ω to Σ), with

(1.1) τΣ “ τX ´K.

In particular, one can make sense of the (absolute) Gromov–Witten invariants ofpΣ, ωΣq, as in [MS04].

Our assumptions also imply that the tuple pX,Σ, ωq is strongly semi-positive, inthe sense of [TZ14, Definition 4.7(2)], hence its genus 0 relative Gromov–Witteninvariants can be defined without virtual techniques, by taking an almost complexstructure generic in a suitable sense [IP03,LR01,TZ14].

Under these assumptions on X and Σ, the complement XzΣ is the interior ofa Liouville domain, hence its symplectic homology can be defined. The goal ofthis paper is to compute the symplectic homology groups of such complements,in terms of counts of rigid pseudoholomorphic spheres in Σ and X. These countscan sometimes be expressed as absolute Gromov–Witten invariants of pΣ, ωΣq andrelative Gromov–Witten invariants of pX,Σ, ωq.

The strategy of the proof is inspired by [BO09a], which relates the symplectichomology of a symplectic manifold with contact boundary with the contact homol-ogy of the boundary. The idea consists of stretching the neck along the boundaryY of a tubular neighborhood of Σ in X, and keeping track of the degenerationsof Floer cascades that contribute to the differential. The limit configurations willinclude pieces in the symplectization of the contact manifold Y , and pieces in (thecompletion of) XzΣ. The former project to pseudohomolorphic spheres in Σ. Theseare related to Gromov–Witten invariants of pΣ, ωΣq. The curves in XzΣ are relatedto pseudoholomorphic spheres in X with tangency constraints in Σ. Under suitablehypotheses, they are counted by relative Gromov–Witten invariants of pX,Σ, ωq.The following is the simplest formulation of our main result. We call a Morsefunction perfect if the corresponding Morse differential vanishes.

Theorem 1.1. If Σ and XzΣ admit perfect Morse functions, then there is a chaincomplex computing the symplectic homology of XzΣ, whose differential is expressedin terms of Gromov–Witten invariants of pΣ, ωΣq, relative Gromov–Witten invari-ants of the pX,Σ, ωq and the Morse differentials in Y and in XzΣ.

For more general and precise statements, see Theorem 12.1 and Lemmas 12.4and 12.5.

An important step in the argument is the proof of a correspondence betweenmoduli spaces of solutions to Floer’s equation in RˆY and moduli spaces of punc-tured pseudoholomorphic spheres in R ˆ Y . This is done by showing that thedifference e (in a sense we make precise) between a Floer solution and a corre-sponding pseudoholomorphic curve solves a certain PDE, and then proving thatsolutions to this PDE exist and are unique (see Sections 9 and 10).

Remark 1.2. As mentioned earlier, this paper drew much inspiration from [BO09a]and its use of neck-stretching. We should point out some of the main differencesbetween the two papers. In [BO09a], the authors consider a more general situation,


in which the symplectic manifold is not necessarily obtained from the complementof a smooth symplectic divisor. For this reason, the contact boundaries Y theyconsider are not necessarily prequantization bundles over Σ, as in our case. The factthat our Y are so special has a number of advantages which we explore. Specifically,this enables us to obtain a bijection between moduli spaces of Floer trajectoriesand of pseudoholomorphic curves (rather than just a continuation map relatingsymplectic homology and a non-equivariant version of contact homology), whichis then used to compute the symplectic homology differential explicitly. Also, thespecial nature of Y is useful to achieve transversality of moduli spaces, adaptingstandard geometric arguments (particularly using the monotonicity condition andautomatic transversality) [MS04, Bou06, Dra04, Wen10]. For more on the relationbetween this work and [BO09a], see Remark 12.10.

Remark 1.3. The contact boundaries Y of the Liouville domains XzΣ are such thatthe Reeb flow is a free S1-action. As explained in [Sei08, (3.2)], there is a spectralsequence converging to symplectic homology, whose first page consists of homologygroups of Y and XzΣ. One can think of this paper as computing the differentialon that page of the spectral sequence (and there happen to be no higher orderdifferentials). This is also related to Remark 12.10.

This paper is organized as follows. Part 1 provides more details about our geo-metric setup and the almost complex structures that we use. Part 2 describes thechain complexes that compute symplectic homology, before and after stretching theneck. We use a Morse–Bott approach, so the differentials count cascades. This partincludes a discussion of transversality of moduli spaces and of evaluation maps, andthe relevant compactness results. In Part 3, we compute Conley–Zehnder indicesand use them to determine Fredholm indices of Floer cascades. We use mono-tonicity to show that only a limited collection of simple cascades can contribute tothe differential. In Part 4, we explain why counting Floer cylinders with cascadesin the differential is equivalent to counting punctured pseudohomolorphic curveswith cascades, and how the latter can be expressed in terms of counts of pseudo-holomorphic spheres in Σ and in X. We also relate these with Gromov–Witteninvariants. Finally, in Part 5, we illustrate this computation scheme by calculatingthe symplectic homology of T˚S2 (which, as we mentioned earlier, is isomorphic tothe well-known homology of the free loop space of S2, see for instance [CJY04]).

Part 1. Setup, almost complex structures and neck-stretching

2. Almost complex structures and neck-stretching

2.1. Symplectic hyperplane sections. We adopt some of our terminology from[BK13], with minor modifications.

Let pX,ωq be a closed symplectic manifold such that rKωs admits a lift toH2pX;Zq for some K ą 0. We refer to a closed codimension 2 symplectic subman-

ifold Σ2n´2 ιĂ X as a symplectic divisor. Write ωΣ “ ι˚ω.

Let π : E Ñ Σ be a complex line bundle over Σ such that c1pEq “ rKωΣs inH2pΣ;Rq. Choose a Hermitian metric on E. Then, there exists a connection 1-form Θ P Ω1pEzΣq such that dΘ “ ´Kπ˚ωΣ. The restriction of α :“ ´Θ to the


unit sphere bundle Y in E is a contact form.1 pY, αq is a prequantization bundleassociated to pΣ,KωΣq. If the coordinate ρ stands for the norm in E measuredwith respect to the Hermitian metric, and f : r0,`8q Ñ p0, 1s is a smooth functionwith fp0q “ 1, f 1 ă 0, then

(2.1) ωE :“ d

ˆ

fpρ2q

Kα

˙

“2ρf 1pρ2q

Kdρ^ α` fpρ2qπ˚ωΣ

is a symplectic form on Ez0 that extends smoothly to all of E. Its restriction to the

0-section is ωΣ. Following [BK13], we will explicitly use fpρ2q “ e´ρ2

. Observe thatif µ is a closed 1-form on Σ, we may take a different connection form by Θ ´ π˚µand change the contact form correspondingly by taking α` π˚µ|Y . This does notchange the Reeb vector field, nor does it change the symplectic structure of thenormal bundle.

Say that a symplectic tubular neighborhood of the symplectic divisor ΣιĂ X is

an extension of ι to a symplectomorphism ϕ : U Ñ X, where U is a disk neigh-borhood of the zero section on a complex line bundle E over Σ, equipped with aHermitian metric and a symplectic form ωE as above. It is well known that everysymplectic divisor has a symplectic tubular neighborhood (see [MS98]), where E isthe symplectic normal bundle to Σ in X.

Definition 2.1. The symplectic divisor Σ is a weak symplectic hyperplane sectionof X if rΣs is Poincare dual (over Q) to rKωs for some K ą 0.

The symplectic divisor Σ is a symplectic hyperplane section if furthermore, thereexists a a symplectic tubular neighborhood ϕ : U Ñ X of Σ such that pXzϕpUq, ωqis a Weinstein domain.

In the following, we will focus primarily on the case where the divisor Σ is aweak symplectic hyperplane section Σ. The stronger assumption is used only inProposition 7.6. Sources of examples of (weak) symplectic hyperplane sections arefor instance smooth ample divisors in smooth complex projective varietes [Bir01]and Donaldson divisors in symplectic manifolds with rational rωs [Gir02].

We now observe that the complement of the neighbourhood of the weak hyper-plane section is a Liouville domain.

Lemma 2.2. The restriction of ω to XzϕpUq is exact. There exists a 1-form λwith dλ “ Kω, and there exists a contact form α on Y induced by a connection1-form, so that so that λ|Y “ e´1 α.

Proof. First, we apply Poincare Duality in H2pX;Rq, H2pW ;Rq to conclude thatrω|W s “ 0 P H2pW ;Rq. Thus, Kω|W “ dλ0 for some 1-form λ0 on W .

Fix a contact form α0 on Y , coming from the above construction. Then, as

coming from Equation (2.1) and using the explicit function e´ρ2

, we obtain

ω|Y “e´1

Kdα0.

Thus, dλ0|Y “ e´1 dα0, and thus eλ0|Y “ α0 ` ν ` df , where ν is a closed 1-formon Y and f : Y Ñ R. From the Gysin sequence, however, we have

0 Ñ H1pΣ;Rq π˚ÝÝÑ H1pY ;Rq π˚

ÝÝÑ H0pΣ;Rq ^KωÝÝÝÑ H2pΣ;Rq.

1In [BK13], Θ is referred to as α∇.


The last map is an injection from R – H0pΣ;Rq to H2pΣ;Rq and so the closed1-form ν “ π˚µ` dg for some closed 1-form µ on Σ and some function g : Y Ñ R.

Extend now f, g to give smooth functions f, g : W Ñ R. It follows then thatλ “ λ0 ´ e´1 dpf ` gq is a primitive of Kω whose restriction to Y is e´1 α, whereα – α0 ` π˚µ, which is indeed a contact form induced by a connection 1-form onY .

Remark 2.3. Note that this implies that the vector field dual to λ points outwardsat the boundary, and is thus a Liouville vector field and λ is a Liouville form forKω. This fact was also pointed out to us by McLean, using a different argument.

Notice also that the lemma forces the choice of a connection 1-form.

We fix a complex line bundle E with Hermitian metric and connection form Θ asabove, such that, for the corresponding symplectic form ωE , there is a symplectictubular neighborhood ϕ : U 1 Ñ X for some neighborhood U 1 of the zero-sectionin E. Let U be an open neighbourhood of the zero section so that U Ă U 1. Thecomplement of ϕpUq therefore contains an open set, and we will allow our almostcomplex structures to be perturbed there.

2.2. Almost complex structures. We say that an almost complex structure JΣ

on Σ that is compatible with ωΣ is regular if somewhere injective JΣ-holomorphicspheres have surjective linearized Cauchy–Riemann operators. The set of regularalmost complex structures is co-meagre in the space of almost complex structurescompatible with ωΣ (see [MS04]).

Since the Hermitian connection on E defines a horizontal distribution, any almostcomplex structure JΣ on pΣ, ωΣq may be lifted to an almost complex structure JEby requiring that JE preserve the horizontal and vertical subspaces, is i on thevertical subspaces and is the lift of JΣ to the horizontal subspaces. In particular,if π : E Ñ Σ is the bundle projection map, we have dπ ¨ JE “ JΣ ¨ dπ.

Definition 2.4. Fix a symplectic tubular neighbourhood of Σ, ϕ : U Ñ X. Analmost complex structure JX on pX,ωq, compatible with ω, is called admissible if,in a neighbourhood of Σ, JX “ ϕ˚JE , where JE is the lift of an almost complexstructure JΣ on Σ. In particular, Σ has JX invariant tangent spaces (i.e. is JX -holomorphic).

In the following, we fix this tubular neighbourhood.In order to compute the symplectic homology of XzΣ, we will need to study

Floer trajectories in the completion W of XzΣ (which we construct below). Thiswill involve SFT-style splitting (neck-stretching) along a contact-type hypersurfacein W . We will construct a suitable family of almost complex structures on W toimplement this process.

Remark 2.5. The idea of this construction is very simply illustrated by an examplewhere this becomes trivial: let X “ S2 with area 1, and let Σ be a point. Itfollows that XzΣ is symplectomorphic to the open unit disk. Its completion involvesattaching r0,`8qˆS1 to this open disk to give C with its standard symplectic form.

However, X can be identified with CP1 and thus XzΣ is, holomorphically, thecomplex plane, which, as we argued, is the completion of the disk.

More generally, this same phenomenon occurs. First, XzΣ must be completedby attaching a cylindrical end. This completion is then biholomorphic (but notsymplectomorphic) to XzΣ. Our construction is to start with an admissible almost


complex structure JX in X, complete XzΣ and equip it with a family of almostcomplex structures that allow us to simultaneously define symplectic homology andstretch the neck. In the trivial example of X “ CP1, we stretch along a copy of S1.The resulting split almost complex manifold (as in [BEH`03]) will consist of twolevels: a lower level that is C and an upper level that is RˆS1. The lower level willthen be biholomorphic to XzΣ with JX , and the upper level will be biholomorphicto the normal bundle of Σ with the 0-section removed.

Let Y Ă E be the unit circle bundle (with respect to the Hermitian metric).This is then a contact-type hypersurface in E, with contact form α “ ´Θ. Recall

that KωE “ dpe´ρ2

αq on EzΣ. Denote by R the Reeb vector field of α. Recall alsothat, given a choice of almost complex structure JΣ on pΣ, ωΣq, there is an inducedalmost complex structure JE that is compatible with ωE .

We now observe that there is a symplectomorphism

ψ1 : pEzΣ,KωEq Ñ pp´8, 0q ˆ Y, dper αqq

v ÞÑ p´|v|2, v|v|q

Since Σ has a neighborhood in X modelled after a neighborhood of the zerosection in E, we can use ψ1 to glue a copy of

`

r0,8q ˆ Y, dper αq˘

to pXzΣ,Kωq.The resulting symplectic manifold is denoted by W and called the completion ofXzΣ. W is a Liouville manifold. Denote by λ a Liouville form on W that agreeswith er α on r0,8qˆ Y . Notice that the restriction of this λ to XzΣ may be takento be the 1-form λ given by Lemma 2.2.

Lemma 2.6. Let Σ be a weak symplectic hyperplane section of pX,ωq and let JX bean admissible almost complex structure on X. Fix a choice of Hermitian structureon NΣ and symplectic neighborhood of Σ.

Then, there exists a parametric family of almost complex structures JR on thecompletion W of XzΣ, which are cylindrical outside a compact set, so that pW,JRqconverge in the SFT sense to a split almost complex manifold whose bottom level isbiholomorphic to pXzΣ, JXq and whose upper level is Rˆ Y with an R translationand Reeb invariant JY .

Remark 2.7. Observe that J1 – ψ1˚JE becomes singular as r Ñ 0. Indeed, if R isthe Reeb vector field on Y for the contact form α, we obtain:

J1Br “ ´1

2rR.

Consider now the diffeomorphism (that is not symplectic!)

ψ2 : EzΣ Ñ Rˆ Yv ÞÑ p´ ln |v|, v|v|q

JY :“ pψ2q˚JE is a cylindrical almost complex structure on R ˆ Y , adapted tothe contact form α (where we re-interpret the Up1q-connection 1-form as a contactform). JY is both R–translation invariant and Reeb-flow invariant since these twoactions correspond to the real scaling and Up1q components of the C˚ action onthe hermitian line bundle E. Even though ψ2 is not a symplectomorphism, JY iscompatible with the symplectic form dper αq.

Having in mind to stretch the neck of W along t0u ˆ Y (in the sense of SFTcompactness [BEH`03]), we will define a family JR, for R ąą 0, of almost complex


x

gpxq

´ε2

´ε2

´ε4

´ε4

Figure 2.1. The function g

structures on W . Fix a small number ε ą 0 such that p´ε, εq ˆ Y Ă W . Weconstruct an almost complex structure on W which, on the symplectization piecep´ε,8q ˆ Y Ă W , interpolates between J1 and JY . Let g : p´8, 0q Ñ R be adiffeomorphism graphed Figure 2.1, such that

$

’

’

’

&

’

’

’

%

gpxq “ x in p´8,´ε2q

gp´ε4q “ ´ε4

g1 ą 0

g1pxq “ ´ 12x for r ě ´ε4

Let G : p´8,´ε4q ˆ Y Ñ p´8,´ε4q ˆ Y be the diffeomorphism Gpr, xq “pgprq, xq. Define an almost complex structure JW on W as follows:

JW :“

$

’

&

’

%

JX on W z`

r´ε2,8q ˆ Y˘

G˚J1 on r´ε2,´ε4q ˆ Y

JY on r´ε4,8q ˆ Y

Since g1pxq “ 1 if x ď ´ε2, we have

(2.2) JW Br “ ´1

2rg1prqR

on r´ε, 0q ˆ Y ,We can now construct a family of almost complex structures JR on W (the

positive real parameter R is not to be confused with the Reeb vector field). Foreach R ě ε4 take a diffeomorphism

fR : r´ε4, ε4s Ñ r´R,Rs


´ε2

´ε4

0

ε4

ε2JY

JWJX

pFRq˚JY

Figure 2.2. The almost complex structure JR in (2.3)

such that f 1R “ 1 near ˘ε4 and f 1R ą 0. Define the diffeomorphisms

FR : r´ε4, ε4s ˆ Y Ñ r´R,Rs ˆ Y

pr, xq ÞÑ pfRprq, xq

Let

(2.3) JR :“

$

’

&

’

%

JW on W z`

r´ε4,8q ˆ Y˘

pFRq˚JY on r´ε4, ε4s ˆ Y

JY on pε4,8q ˆ Y

See Figure 2.2. Note that if R “ ε4 and fRprq “ r, then Jε4 “ JW .

By SFT compactness [BEH`03], a sequence of finite energy JR-holomorphiccurves uR in W has a subsequence converging to a pseudoholomorphic buildingwith levels mapping to either pRˆ Y, JY q or pW,JW q.

To define symplectic homology of W , we will use an almost complex structureJR, for large enough R.

The following result compares JW -holomorphic planes with JX -holomorphicspheres, and will be useful to relate the symplectic homology differential with rel-ative Gromov–Witten invariants.

Lemma 2.8. There is a diffeomorphism ψ : W Ñ XzΣ such that ψ˚JW “ JX .This map defines a bijection between JW -holomorphic planes in W asymptotic toReeb orbits of multiplicity k in Y and JX-holomorphic spheres in X intersecting Σat precisely one point, with order of tangency k.

Proof. Define

ψ “

#

id on W z`

r´ε,8q ˆ Y˘

G on r´ε,8q ˆ Y


x

gpxq

´ε

´ε

´ε2

´ε2

´ε4


x

gpxq

´ε

´1ε

´ε2

´2ε

´ε4

´2


where Gpr, xq “ pϕ ˝ ψ´11 qpgprq, xq for the diffeomorphism g : r´ε,8q Ñ r´ε, 0q

graphed in Figure 2.3, such that

(2.4)

#

gpxq “ x near ´ε

g1pxq “ 1xg1pxq gpxq

Recall that gpxq :“ 1xg1pxq (graphed in Figure 2.4) satisfies

$

’

&

’

%

gpxq “ 1x near r´ε,´ε2q

gpxq ă 0 on r´ε,´ε4q

gpxq “ ´2 near ´ε4

hence we can extend g to r´ε,8q by setting it equal to ´2 on r´ε4,8q. Theordinary differential equation (2.4) has a solution g. Note that, since gpxq “ ´2near ´ε4, 2.4 becomes g1pxq “ ´2gpxq, hence gpxq “ C e´2x for some constantC ă 0 and for all x ě ´ε4. This explains why the right endpoint of the target ofg is 0.

We are left with showing that ψ˚JW “ JX . This is clear on W z`

r´ε,8q ˆ Y˘

.To try to minimize confusion, call the radial variable on r´ε,8q ˆ Y Ă W by r1

and call the radial variable on pϕ ˝ ψ´11 q

`

r´ε, 0q ˆ Y˘

Ă X by r2. Note that under


er “ ρ

hperq

bk

´Apγkq

Figure 3.1. An admissible Hamiltonian and the graphical proce-dure for computing the action of a periodic orbit

ψ we have r2 “ gpr1q. On r´ε,8q ˆ Y , we need to check that

pψ˚JW q

ˆ

pϕ ˝ ψ´11 q˚

B

Br2

˙

“ JX

ˆ

pϕ ˝ ψ´11 q˚

B

Br2

˙

“ pϕ˚JEq

ˆ

pϕ ˝ ψ´11 q˚

B

Br2

˙

“

“ ppψ1q˚JEqB

Br2“ J1

B

Br2“ ´

1

2r2R

This follows from the computation, in which we use 2.4 and 2.2

pψ˚JW q

ˆ

pϕ ˝ ψ´11 q˚

B

Br2

˙

“ JW

ˆ

pψ´1 ˝ ϕ ˝ ψ´11 q˚

B

Br2

˙

“ JW

ˆ

pg´1q˚B

Br2

˙

“

“ JW

ˆ

1

g1pr1q

B

Br1

˙

“r1g

1pr1q

gpr1qJW

B

Br1“

“r1g

1pr1q

gpr1q

´1

2r1g1pr1qR “ ´

1

2gpr1qR “ ´

1

2r2R

Part 2. Moduli spaces and symplectic chain complex

3. The chain complex

We will describe two chain complexes associated to W whose generators are(essentially) the same, but for which the differentials are a priori different. Ourmain theorem is then a comparison of a these differentials. We will now define thegroup underlying these chain complexes.

Definition 3.1. Consider a function h : p0,`8q Ñ R with the following properties:

(i) hpρq “ 0 for ρ ď 2;(ii) h1pρq ą 0 for ρ ą 2;

(iii) h1pρq Ñ `8 as ρÑ8;(iv) h2pρq ą 0 for ρ ą 2;(v) h2pρq ą C for some constant C ą 0 and all ρ ě ρ0, where h1pρ0q “ 1, and

h3pρq ě ´h2pρqρ .


From this, we obtain a Hamiltonian function H on either R ˆ Y or on W bysetting Hpr, yq “ hperq on R` ˆ Y and extending it by 0 elsewhere. We will referto such Hamiltonians as admissible. See Figure 3.1.

Condition (v) will give us control over asymptotic operators associated to theFloer equation.

Since ω “ dper αq on R` ˆ Y , the Hamiltonian vector field associated to H ish1perqR, where R is the Reeb vector field associated to α. Recall that pY, αq isa prequantization bundle over pΣ, ωΣq, and that the corresponding periodic Reeborbits correspond to covers of the S1-fibres of Y Ñ Σ. The periods of these orbitsare positive integers, giving the multiplicities of the covers. The 1-periodic orbitsof H are thus of two types:

(1) constant orbits: one for each point in

W :“ tw PW | pdHqw “ 0u “W z supppdHq;

(2) non-constant orbits: for each k P Z`, there is a Y -family of 1-periodic XH -orbits, contained in the level set Yk :“ tbku ˆ Y , for the unique bk ą log 2such that h1pebkq “ k. Each point in Yk is the starting point of one suchorbit.

Remark 3.2. It is important to observe that these Hamiltonians are Morse–Bottnon-degenerate except at the boundary B supppdHq. In Section 5, we prove thatthe moduli spaces of curves we consider will not interact with these degenerate,constant orbits.

Recall that a family of periodic Hamiltonian orbits for a time-dependent Hamil-tonian vector field is said to be Morse–Bott non-degenerate if the parametrized1-periodic orbits form a manifold, and the tangent space of the family of orbits ata point is given by the eigenspace of 1 for the corresponding Poincare return map.(Morse non-degeneracy requires these periodic orbits to be isolated, and the returnmap not to have 1 as an eigenvalue.)

By an abuse of notation, we will write πΣ : E Ñ Σ to denote the bundle projec-tion, but will also denote its restrictions as πΣ : Rˆ Y Ñ Σ, πΣ : Y Ñ Σ.

We also fix some auxiliary data, consisting of Morse functions and vector fields.Fix throughout a Morse function fΣ : Σ Ñ R and a gradient-like vector field ZΣ P

XpΣq, which means that 1c |dfΣ|

2 ď dfΣpZΣq ď c|dfΣ|2 for some constant c ą 0.

Denote the time-t flow of ZΣ by ϕtZΣ. Given p P CritpfΣq, its stable and unstable

manifolds (or ascending and descending manifolds, respectively) are

(3.1) W sΣppq :“

!

q P Σ| limtÑ8

ϕ´tZΣpqq “ p

)

,WuΣppq :“

"

q P Σ| limtÑ´8

ϕ´tZΣpqq “ p

*

.

Notice the sign of time in the flow. We further require that pfΣ, ZΣq be a Morse–Smale pair, i.e. that all stable and unstable manifolds of ZΣ intersect transversally.

The contact distribution ξ defines an Ehresmann connection on the circle bundleS1 ãÑ Y Ñ Σ. Denote the horizontal lift of ZΣ by π˚ΣZΣ P XpY q. We fix a Morsefunction fY : Y Ñ R and a gradient-like vector field ZY P XpY q such that pfY , ZY qis a Morse–Smale pair and the vector field ZY ´ π˚ΣZΣ is vertical (tangent to theS1-fibers). Under these assumptions, flow lines of ZY project under πΣ to flow linesof ZΣ.

Observe that critical points of fY must lie in the fibres above the critical pointsof fΣ (and these are zeros of ZY and ZΣ respectively). For notational simplicity,


we suppose that fY has two critical points in each fibre. In the following, given acritical point for fΣ, p P Σ, we denote the two critical points in the fibre above pby pp and qp, the fibrewise maximum and fibrewise minimum of fY , respectively.

We will denote by Mppq the Morse index of a critical point p P Σ of fΣ, and by

Mppq “ Mppq ` ippq the Morse index of the critical point p “ pp or p “ qp of fY .The fibrewise index has ipppq “ 1 and ipqpq “ 0.

Since H is admissible, we can identify W zW with plog 2,8q ˆ Y . Fix also aMorse function fW and a gradient-like vector field ZW on W , such that pfW , ZW qis a Morse–Smale pair and ZW restricted to r´ε4,8q ˆ Y is the constant vectorfield Br, where r is the coordinate function on the first factor and ε ą 0 is as inSection 2.2. Use the diffeomorphism in Lemma 2.8 to define a pair pfX , ZXq onXzΣ from the pair pfW , ZW q.

We now define the Morse–Bott symplectic chain complex of W and H. Recallthat for every k ą 0, each point in Yk :“ tbku ˆ Y Ă R` ˆ Y is the starting pointof a 1-periodic orbit of XH , which cover k times their underlying Reeb orbits. Foreach critical point p “ pp or p “ qp of fY , there is a generator corresponding to thepair pk, pq. We will denote this generator by pk. The complex is then given by:

(3.2) SC˚pW,Hq “

˜

à

ką0

à

pkPCritpfΣq

Zxqpk, ppky

¸

‘

˜

à

xPCritpfW q

Zxxy

¸

Recall that λ is a Liouville form on W .

Definition 3.3. The Hamiltonian action of a loop γ : S1 ÑW is

Apγq “ż

γ˚pλ´Hdtq.

In particular, for any constant orbit γ PW , Apγq “ 0 and for any orbit γk P Yk,we have

(3.3) Apγkq “ ebk h1pebkq ´ hpebkq ą 0,

where bk is as above. The action of γk is the negative of the y-intersept of thetangent line to the graph of h at ebk . See Figure 3.1. Note that the convexity of himplies that Apγkq is monotone increasing in k.

We will now define the gradings of the generators. These definitions will bejustified in Section 7.1.3. For a critical point rp of fY , and a multiplicity k, wedefine

(3.4) |rpk| “ Mprpq ` 1´ n` 2τX ´K

Kk P R,

where we recall that τX is the monotonicity constant of X and c1pNΣq “ rKωΣs.

Remark 3.4. If there is a spherical homology classA P H2pX;Zq such that xrωs, Ay ‰0, then τX and K are rationally dependent and we can say that |pk| P Q. If X issymplectically aspherical, the monotonicity condition is vacuously satisfied for anyτX ą K.

For a critical point x of fW , we define

(3.5) |x| “ n´Mpxq.


4. Floer moduli spaces before and after splitting

In this section, we describe the moduli spaces of cascades that contribute tothe differential in the Morse–Bott symplectic homology of W , before and afterstretching the neck.

We also define auxiliary moduli spaces of spherical “chains of pearls” in Σ andin X. (These are familiar objects, reminiscent of ones considered in the literaturefor Floer homology of compact symplectic manifolds [BC09,Oh96,PSS96].)

4.1. Floer cascades before splitting. Consider H : W Ñ R as defined in Section3, together with the auxiliary data of pfΣ, ZΣq, pfY , ZY q and pfW , ZW q. Fix analmost complex structure JR as in (2.2) for a large R ą 0.

Definition 4.1. A map v : Rˆ S1 ÑW is a Floer cylinder if

(4.1) Bsv ` JRpBtv ´XHpvqq “ 0.

Let Mpx´, x`q denote the space of all such Floer cylinders such that, for fixed1-periodic orbits x˘ of XH , limsÑ˘8 vps, .q “ x˘.

Let S´ and S` denote connected spaces of 1-periodic XH -orbits (either W orYk). Then we define

MpS´, S`q “ď

x´PS´,x`PS`

Mpx´, x`q.

Given a Floer cylinder v, we denote its asymptotic limits by vp`8q “ x` andvp´8q “ x´.

Note that Apvps0, .qq ď Apvps1, .qq if s0 ď s1.

Definition 4.2. The energy of a Floer cylinder v : Rˆ S1 ÑW is given by:

Epvq “

ż

RˆS1

v˚dλ´ v˚dH ^ dt “

ż

RˆS1

||vs||2JR ds^ dt,

where ||vs||2JR“ dλpvs, JRvsq.

Since the symplectic form is exact, a Floer cylinder v with vp˘8q “ x˘ has

Epvq “ Apx`q ´Apx´q “ż

x˚`pλ´ hperqdtq ´

ż

x˚´pλ´ hperqdtq.

A Floer cylinder is non-trivial if Epvq ą 0, or equivalently, if v is not of the formps, tq ÞÑ γptq for some 1-periodic XH -orbit γ.

Let us recall some convergence properties of finite energy Floer trajectories. Thisis a combination of statements of several theorems from the literature, with slightlydifferent hypotheses. In the Morse–Bott case, we refer to [BO09b,Bou02,HWZ96a].In the non-degenerate case, the relevant ideas have appeared in [RS01, HWZ96b],and of course the original work was done by Floer [Flo88].

Lemma 4.3. Suppose that v : R ˆ S1 Ñ W is a finite energy Floer cylindercontained in a compact subset of W . Then, for any sequence sk Ñ `8 [resp.,sk Ñ ´8] there is a subsequence we also denote pskq

8k“1 and a 1-periodic orbit γptq

of the Hamiltonian vector field so that vpsk, tq Ñ γptq in C8.

(1) If γ is a Morse–Bott non-degenerate orbit, then the limit γ does not de-pend on the initial choice of sequence, and furthermore, we have the muchstronger result that vps, tq converges to γ exponentially fast in s;


(2) If γ is a non-isolated degenerate orbit, the limit may depend on the initialsequence pskq

8k“1. Any two limits are connected by a family of periodic

orbits of the same action.

Furthermore, if the curve converges exponentially fast, at a negative [resp., pos-itive] puncture the rate of convergence is governed by the smallest positive [largestnegative] eigenvalue of the corresponding asymptotic operator [Sie08].

Remark 4.4. There are several constructions of examples where the limit is notunique in the degenerate case. In particular, Siefring has carefully constructedsuch examples in the case of pseudoholomorphic curves in symplectizations [Sie16].Similarly, a gradient trajectory of a smooth function that is not Morse–Bott canfail to converge to a single critical point, and may contain sequences converging todifferent critical points.

Remark 4.5. In the case of a Morse–Bott limit, a cylinder contained in a compactsubset of W has finite energy if and only if it converges to 1-periodic Hamiltonianorbits at its punctures, see [Sal99, Proposition 1.21; BEH`03, Proposition 5.13].

Yoel Groman pointed out to us that the assumption that v has finite energyalready implies that its image is contained in a compact subset of W , so the as-sumptions of Lemmas 4.3 and 4.6 can be simplified. This is because the actionfunctional for our Hamiltonians satisfies the Palais–Smale condition [Gro15].

The next result shows that there are no non-trivial Floer cylinders whose positiveasymptote x` is in W . Note that if such a cylinder has asymptotic limits at `8and ´8, then it is contained in a compact subset of W , by the maximum principle.

Lemma 4.6. Let v : Rˆ S1 ÑW be a solution of (4.1) that has finite energy andis contained in a compact subset of W . If there is a sequence s`k Ñ 8 such that

the C0-limit limkÑ8 vps`k , .q “ x` PW , then v is constant.

Proof. Take any sequence s´k Ñ ´8. Lemma 4.3 implies that there are subse-

quences (denoted also by s˘k ) such that vps˘k , .q converge in C1 (actually C8) to

1-periodic XH -orbits x˘, with x` PW . Then,

0 ď Epvq “ limkÑ8

Epv|rs´k ,s

`k sˆS

1q “ limkÑ8

`

Apvps`k , .qq ´Apvps´k , .qq˘

“

“ Apx`q ´Apx´q.

Since Apx`q “ 0 and Apγq ě 0 for every 1-periodic XH -orbit, we conclude thatEpvq “ 0. This implies the result.

Definition 4.7. Fix N ě 0. Let S0, S1, . . . , SN be a collection of connected spacesof orbits, which can either be one of the Yk or W . Let pfi, Ziq, i “ 0, . . . , N ` 1be the pair of Morse function and gradient-like vector field of fi “ fY , Zi “ ZY ifSi “ Yk for some k, and fi “ ´fW , Zi “ ´ZW if Si “W .

Let p be a critical point of fN and q a critical point of f0 (so p and q aregenerators of the chain complex (3.2)).

A Floer cylinder with 0 cascades (N “ 0), with positive end at p and negativeend at q, consists of a positive gradient trajectory ν : RÑ S0, such that νp´8q “ q,νp`8q “ p and 9ν “ Z0pνq.

A Floer cylinder with N cascades, N ě 1, with positive end at p and negativeend at q, consists of the following data:


q

p

v1

v2

v3

ν0

ν1

ν2

ν3

Figure 4.1. A Floer cylinder with 3 cascades, as in Definition 4.7,with positive end at p and negative end at q.

(i) N ´ 1 length parameters li ą 0, i “ 1, . . . , N ´ 1;(ii) Two half-infinite gradient trajectories, ν0 : p´8, 0s Ñ S0 and νN : r0,`8q Ñ

SN with ν0p´8q “ q, νN p`8q “ p and 9νi “ Zipνiq for i “ 0 or N ;(iii) N´1 gradient trajectories νi defined on intervals of length li, νi : r0, lis Ñ Si

for i “ 1, . . . , N ´ 1 such that 9νi “ Zipνiq;(iv) N non-trivial Floer cylinders vi : R ˆ S1 Ñ W , i “ 1, . . . , N , satisfying

equation (4.1) where (defining l0 “ 0)

vip`8, ¨q “ νip0q P Si, vip´8, ¨q “ νi´1pli´1q P Si´1.

In the case of a Floer cylinder with N ě 1 cascades, we refer to the non-trivialFloer cylinders vi as sublevels. See Figure 4.1 for a schematic illustration.

The energy of a Floer cylinder with N cascades is the sum of the energies ofeach of its N sublevels.

Lemma 4.8. Floer cylinders with cascades do not contain sublevels vi such thatlimsÑ`8 vips, .q PW . If limsÑ´8 vips, .q “ x´ PW , then i “ 1 and x´ R BW .

Proof. The case s Ñ `8 follows from Lemma 4.6. For the case s Ñ ´8, ob-serve that νi´1 is a negative flow line of ZW , which agrees with Br on r´ε4,8q ˆY (in particular on BW ). If i ą 0, then the sublevel vi´1 must be such thatlimsÑ`8 vi´1ps, .q P W , which as we saw is impossible. If i “ 1, then ν0p´8q P

CritpfW q, and all critical points of fW are “below BW”. Therefore, x´ R BW .

Remark 4.9. The points in BW , which were ruled out as asymptotic orbits ofsublevels by the previous Lemma, correspond to constant orbits of XH that are notof Morse–Bott type. If finite energy solutions of Floer’s equation were allowed toconverge to orbits that were not of Morse–Bott type, there would be no guarantee ofuniqueness of their asymptotic limits. In particular, one would not be able to claimexponential convergence to orbits, which is crucial to setup a Fredholm problem.

Lemma 4.10. Let S0, S1, . . . , SN be the spaces of orbits of a Floer cylinder withN cascades. Let kpSiq “ k if Si “ Yk and let kpSiq “ 0 if Si “ W . Then, kpSiq ismonotone increasing in i.


Proof. This follows immediately from the fact that 0 ă Epviq “ Apvip`8qq ´Apvip´8qq and from the monotonicity of A in the multiplicity k.

Remark 4.11. These lemmas have a number of important consequences. In partic-ular, a Floer cylinder with cascades between two critical points of fW must have 0cascades, which is to say that it consists of a flow line of ´ZW .

Furthermore, a Floer cylinder with cascades whose positive end is on a Yk, k ě 1,and whose negative end is on W must have the first sublevel connecting an orbitin a Yl to a (constant) orbit in the interior of W . This is the unique sub-level inthe cascade that converges to a point in W .

We are now able to define a differential on the chain complex (3.2). Givengenerators p, q, denote by

MH,N pq, p; JRq

the space of Floer cylinders with N cascades from q to p (i.e. with negative end atq and positive end at p).

We then define(4.2)

Bpre p “ÿ

|q|“|p|´1

# pMH,0pq, p; JRqRq q `ÿ

|q|“|p|´1

8ÿ

N“1

#`

MH,N pq, p; JRqRN˘

q.

Observe that if N “ 0, the Floer cylinder with 0 cascades is a flow line, and theR action is given by its standard reparametrization. If, instead, N ě 1, the RNaction is given by domain translation on the N sublevels.

We call Bpre the presplit Floer differential, to distinguish it from the split differ-ential we will define next.

Remark 4.12. To give a complete definition of Bpre, we should discuss the orien-tations of the moduli spaces involved and the associated signs. Instead, we willdefine another differential in the next section, and discuss in details the associatedorientation issues in Section 8

4.2. Floer cascades after splitting. Recall the definition of the almost complexstructures JR, JY and JW in Section 2.3. Given a sequence vRn of finite energy JRn -Floer cylinders in W , with Rn Ñ8, SFT compactness [BEH`03] implies that thereis a subsequence converging to an SFT building. Observe that the Hamiltonian His supported in plog 2,8qˆY , which is above the hypersurface t0uˆY along whichwe stretch the neck. This makes it possible to apply the usual SFT compactnessargument. A gluing argument gives that a transverse SFT-type building can beglued to obtain a “presplit” Floer cylinder, as in the previous section.

This Hamiltonian induces a function in R ˆ Y , vanishing in R´ ˆ Y , which wealso denote by H. The non-constant orbits again form manifolds Yk, where k ě 1.The critical points of the Morse function fW : W Ñ R are below the neck-stretchingregion. We get a different description of the differential on the chain complex (3.2)(defined above in terms of Floer cylinders with cascades) by counting split Floercylinders with cascades. We begin by defining split Floer cylinders whose limits arenon-constant orbits.

Definition 4.13. Let x˘ P Yk˘ be 1-periodic orbits of XH . A split Floer cylinder

from x´ to x` consists of a map v “ pb, vq : R ˆ S1zΓ Ñ R ˆ Y , where Γ “


tz1, . . . , zku Ă Rˆ S1 is a (possibly empty) finite subset, such that

(4.3) Bsv ` JY pBtv ´XHpvqq “ 0

and in addition

‚ limsÑ˘8 vps, .q “ x˘;‚ if Γ ‰ H, then, for conformal parametrizations ϕi : p´8, 0s ˆ S1 Ñ R ˆS1ztz1, . . . , zku of neighborhoods of the zi, limsÑ´8 vpϕips, .qq “ p´8, γiq,where the γi are periodic Reeb orbits in Y ;

‚ for each Reeb orbit γi above, there is a JW -holomorphic plane Ui : CÑWthat is asymptotic to p`8, γiq.

Define a split Floer cylinder with cascades between two non-constant generatorsof the chain complex (3.2) by repeating Definition 4.7, replacing all Floer cylinderswith split Floer cylinders.

See Figure 4.2 for a representation of a cascade built out of split Floer cylinders.We consider now split Floer cylinders between non-constant and constant orbits.

Definition 4.14. Let x` P Yk and x´ P W . A split Floer cylinder from x´ to x`is a pair pv1, v2q, where v1 “ pb, vq : Rˆ S1zΓ Ñ Rˆ Y , Γ “ tz1, . . . , zku Ă Rˆ S1

is a (possibly empty) finite subset, solving equation (4.3). In addition

‚ limsÑ`8 v1ps, .q “ x`;‚ limsÑ´8 v1ps, .q “ p´8, γq, for some Reeb orbit γ in Y ;‚ if Γ ‰ H, then, for conformal parametrizations ϕi : p´8, 0s ˆ S1 Ñ R ˆS1ztz1, . . . , zku of neighborhoods of the zi, limsÑ´8 vpϕips, .qq “ p´8, γiq,where the γi are periodic Reeb orbits in Y ;

‚ for each Reeb orbit γi above, there is a JW -holomorphic plane Ui : CÑWthat is asymptotic to p`8, γiq.

v2 : Rˆ S1 ÑW is JW -holomorphic and

‚ limsÑ`8 v2ps, .q “ p`8, γq;‚ limsÑ´8 v2ps, .q “ x´.

Define a split Floer cylinder with cascades between a non-constant generatorand a constant generator of the chain complex (3.2) by repeating Definition 4.7,replacing all Floer cylinders with split Floer cylinders.

Definition 4.15. We refer to the punctures Γ appearing in Definitions 4.13 and4.14 as augmentation punctures. The corresponding JW holomorphic planes, Ui : CÑW are referred to as augmentation planes.

This terminology is by analogy to linearized contact homology, where rigid suchplanes give an (algebraic) augmentation of the full contact homology differential.We make no such claim here.

For split Floer cylinders in Rˆ Y , we introduce a new form of energy, a hybridbetween the standard Floer energy and the Hofer energy used in symplectizations.Recall that the Hofer energy of a punctured pseudoholomorphic curve u in thesymplectization of Y with contact form α is given by:

supt

ż

u˚dpψαq |ψ : RÑ r0, 1s smooth and nondecreasingu.


In a symplectic manifold either compact or convex at infinity, the standard Floerenergy of a cylinder v : Rˆ S1 ÑW is given by

ż

v˚ω ´ v˚dH ^ dt.

In our situation, however, the target manifold is Rˆ Y , which has a concave end.We therefore need to combine these two types of energy.

Definition 4.16. Consider a Hamiltonian H : Rˆ Y Ñ R so that dH has supportin rR,8q ˆ Y , for some R P R.

Let ϑR be the set of all non-decreasing smooth functions ψ : R Ñ r0,8q suchthat ψprq “ er for r ě R.

The hybrid energy ER of v : R ˆ S1zΓ Ñ R ˆ Y solving Floer’s equation 4.3 isthen given by:

(4.4) ERpvq “ supψPϑ

ż

RˆS1

v˚ pdpψαq ´ dH ^ dtq .

Notice that this is equivalent to partitioning Rˆ S1zΓ “ S0 Y S1, so that S0 “

v´1prR,`8q ˆ Y q and S1 “ SzS0. Then, v has finite hybrid energy if and only ifv|S0

has finite Floer energy and v|S1has finite Hofer energy.

Equivalently, v has finite hybrid energy if and only if the punctures t˘8u Y Γcan be partitioned into ΓF and ΓC (with `8 P ΓF , Γ Ă ΓC and ´8 either inΓF or in ΓC), such that in a neighbourhood of each puncture in ΓF , the map v isasymptotic to a Hamiltonian trajectory and in a neighbourhood of each puncture inΓC , the map is proper and negatively asymptotic to an orbit cylinder in RˆY . Thisfollows from a variation on the arguments in [BEH`03, Proposition 5.13, Lemma5.15]. By a slight abuse of notation, we will use the following notation to denotethe Hamiltonian orbits to which such a punctured cylinder v is asymptotic:

vp`8, tq “ limsÑ8

vps, tq

vp´8, tq “ limsÑ´8

vps, tq if ´8 P ΓF .

Instead, if ´8 P ΓC , we will write vp´8, tq “ limsÑ´8 vps, tq for the Reeb orbitin Y that the curve converges to. Notice that since the cylinder is parametrized,the asymptotic limit is parametrized as well. Since there is an ambiguity of the S1

parametrization of the Reeb orbits to which v is asymptotic at punctures z P Γ, wewill avoid using the analogous notation at punctures in Γ.

Remark 4.17. Lemmas 4.8 and 4.10 and Remark 4.11 have natural analogues forsplit Floer cascades.

We can now define a new differential B on the chain complex (3.2). Given gen-erators p, q, denote by

MH,N pq, p; JY , JW q

the space of split Floer cylinders with N cascades from q to p (with negative endat q and positive end at p). We use again formula (4.2) to define the differential B,replacing the spaces MH,N pq, p; JRq of presplit Floer cylinders with cascades withthe spaces MH,N pq, p; JY , JW q of split Floer cylinders with cascades.

We call B the split Floer differential on (3.2). We will obtain later that this dif-ferential is of degree ´1 (corresponding to our homology conventions). See Remark6.10.


q

p

v1

v2

v3

ν0

ν1

ν2

ν3

U1 U2 U3

Figure 4.2. A split Floer cylinder with 3 cascades.

5. Sketch of isomorphism of the two Floer complexes

Proposition 5.1. For a generic choice of almost complex structures of the cylin-drical type JR, JW and JY described in Section 2.2, and for R ą 0 large enough,the presplit and split chain complexes are well-defined and are chain isomorphic.Furthermore, these complexes compute the symplectic homology of W .

We will provide a sketch of the proof of this proposition. The main steps of sucha proof are as follows:

‚ Show that these complexes are well-defined. For this, we need to obtaintransversality for generic almost complex structures in the (very restrictive)class we consider, which are both R and S1 (Reeb) invariant in the cylin-drical end. This problem is similar in both the presplit and split complexes,and we address it in detail in Section 6 in the more difficult setting of thesplit complex. (The additional difficulty in the split complex comes fromFloer cylinders with cascades that have punctures capped with planes inW . The planes can be made transverse by a perturbation of J supportedin W , where the symmetry condition on J is vacuous. Such a perturbationdoes not make the upper level(s) transverse. In the presplit case, such abuilding corresponds to a single cylinder with “toes” that dip in to W , andthus a perturbation in W suffices to obtain transversality. See Figure 5.1.)

‚ For these complexes to be well-defined, we also need to establish that thereare no curves counted either in B or in the proof of B2 “ 0 that are as-ymptotic to the degenerate constant orbits at BW . As we pointed out inRemark 4.9, these orbits fail to even be Morse–Bott, and thus represent abreak-down of the standard analytical theory. We saw in Lemma 4.8 whythese orbits don’t appear in B. A similar argument (applying Lemma 4.6)implies that they also don’t need to be considered when proving B2 “ 0.


q

W

Rˆ S1

W

q

p p

stretch

the neck

Figure 5.1. By perturbing J in W (where the symmetry con-dition is vacuous), the curve on the left can be made transverse.Such a perturbation only makes the plane in W transverse for thecurve on the right.

‚ Compactness and gluing arguments allow us to identify the presplit andsplit moduli spaces for sufficiently large stretching parameter. Notice thatby formula (2.3), each of the almost complex structures JR used in stretch-ing is biholomorphic to JW , though the support of H is moved towards`8 by the biholomorphism. Thus, a slight modification of the standardSFT compactness theorem works in our setting. It is important to pointout that after stretchig the neck, the component of a Floer cylinder that iscontained in Rˆ Y is connected, which is why the split Floer cascades wedescribed above contain information about all the presplit Floer cascades.This follows from an argument in [BO09a, Step 1 in proof of Proposition5]. By a gluing argument, for each subcomplex of the split complex givenby a bound in action, there exists a sufficiently large stretching parameterfor which we get an identification with the corresponding subcomplex ofthe presplit complex.

‚ Show that the presplit complex is quasi-isomorphic to a symplectic homol-ogy complex obtained from a non-degenerate Hamiltonian. This involvesconstructing a continuation map Φ connecting two chain complexes. Thedelicate part of the proof that this is a chain map stems again from thefailure of H to be Morse–Bott non-degenerate along the boundary BW . Weaddress this difficulty by means of the Abouzaid-Seidel Lemma 4.8 below.A usual action filtration argument gives that this is a chain isomorphism.

We now address the difficulty in the proof of the fact that a continuation mapbetween presplit symplectic homology and a Morse perturbed version is a chainmap. Recall that we defined a differential Bpre on SC˚pW,Hq, as in (3.2), whichwe called presplit differential. We could construct a non-degenerate HamiltonianH : S1 ˆW Ñ R such that Hpt, wq “ Hpwq outside a small neighbourhood of the

periodic orbits of XH . We require that H be C2-small and Morse on W z`

p´8, εqˆ

Y˘

for some ε ą 0. In addition, H is a small time-dependent perturbation of Hnear the non-constant periodic orbits of XH , using auxiliary Morse functions onthe manifolds of orbits in a manner similar to [BO09b, page 73]. Picking a generic


almost complex structure on W , we get a chain complex SC˚pW, Hq that computes

the symplectic homology of W . Denote its differential by B.We could construct a chain map

Φ: SC˚pW,Hq Ñ SC˚pW, Hq

as follows. Let H : Rˆ S1 ˆW Ñ R be such that

‚ Hps, t, wq “ Hpwq if s ą 1;

‚ Hps, t, wq “ Hpt, wq if s ă ´1;‚ BsH ď 0.

We will impose two more technical conditions on H, on Lemma 5.4. Picking adomain-dependent almost complex structure interpolating between the ones usedin the two chain complexes, we could define Φ via counts of index 0 Floer cylinderswith cascades for H (satisfying (5.1) below with β “ dt).

Showing that Φ is a chain map depends on being able to glue cascades contribut-ing to Φ with cylinders contributing to B, obtain 1-parameter families of cylindersconnecting generators of the two complexes, and then show that the other bound-ary component of this family contributes to either Φ ˝ Bpre or to B ˝ Φ (and doingthe same when gluing a contribution to Bpre with a contribution to Φ).

The presence of the degenerate constant orbits of XH along BW could, in princi-ple, be problematic for both gluing and compactness. These problems are overcomeusing a combination of the convergence of sequences in Lemma 4.3 and of the fol-lowing result originally due to Abouzaid and Seidel (see [AS10, Lemma 7.2] andalso [Rit13, Lemma 19.3]), and useful to prove a “no escape” lemma. We will statethe result in more generality than necessary. In the following lemma, for instance,the annulus S “ p´ε, 0s ˆ S1 should be thought of as a subset of the puncturedcylinder, and β will then be the restriction of the form dt to this annulus.

Lemma 5.2. Let Y be a co-oriented contact manifold with contact form α anddenote the corresponding Reeb vector field by R.

Let S be the annulus S “ p´ε, 0s ˆ S1 with complex structure i.Let J be an S-dependent family of almost complex structures on rr0, r0 ` δs ˆ Y

and let H : Sˆrr0, r0` δsˆY Ñ R be an S-dependent family of Hamiltonians withthe following locally radial behaviour:

(1) Jpz, r, yqBr “ R, Jpz, r, yq kerα “ kerα,(2) Hpz, r, yq “ hpz, erq, for some function hpz, ρq.

Let β be a closed 1-form on S.Suppose that v : S Ñ rr0, r0 ` δq ˆ Y satisfies the following equation:

(5.1) 0 “ 2pdv ´XH b βq0,1 “ dv ` Jpz, vqdv ˝ i´XH b β ´ Jpz, vqXH b β ˝ i,

and vpt0u ˆ S1q Ă tr0u ˆ Y .Then, taking λ “ er α,

ż

t0uˆS1

v˚λ´Hpz, vpzqqβ ď

ż

t0uˆS1

pλpXHq ´Hqβ “

“

ż

t0uˆS1

ˆ

er0Bh

Bρpz, er0q ´ hpz, er0q

˙

β

Proof. Applying λ to (5.1) and using λ ˝ J “ er dr and λpJXHq “ 0, we obtain

v˚λ` er drpdv ˝ iq “ λpXHqβ.


By hypothesis, we have that rpvp0, tqq ” r0. Furthermore, since the image of theannulus S is not below r “ r0, we have drpdv ˝ iqp B

Bt q ě 0 along t0u ˆ S1, hence

v˚λpB

Btq ď λpXHqβp

B

Btq.

The result now follows.

Following [AS10,Rit13], we introduce the following terminology:

Definition 5.3. Let pS, jq be a Riemann surface and let β be a 1-form on S sodβ ď 0. Let pW,ωq be a symplectic manifold, and let J be an S-dependent familyof almost complex structures on W , compatible with ω. Let H : S ˆW Ñ R beas S-dependent Hamiltonian. We write dH to indicate the (total) differential of Hand dWH to denote the S-dependent family of 1-forms on W given by dWH|zPS “dpHpz, ¨qq.

For a map v : S ÑW , we define its

‚ Topological energy: Etopopvq “ş

Sv˚pω ´ dpHβqq;

‚ Geometric energy: Egeompvq “ş

Sv˚ω ´ pv˚dWHq ^ β.

Note that for any solution to Floer’s equation (5.1), Egeompvq “ş

SBsv

2 ě 0is (pointwise) non-negative. We say that the triple pβ,H, Jq is monotone if thetopological energy dominates the geometric energy pointwise (in the sense that theintegrand in the geometric energy is equal to the integrand in the topological energytimes a function f ď 1). Observe that they are equal if H is z-independent andβ “ dt.

Lemma 5.4. Let pW,dλq be an exact symplectic manifold with cylindrical endppr0,`8q ˆ Y, dpe

r αqq.Let pβ,H, Jq be a monotone triple on the cylinder pRˆS1, iq with 1-form β “ dt

and with Hp`8,tq an admissible Hamiltonian as in Definition 3.1.Let γ` be a 1-periodic orbit of Hp`8,tq contained in the level tr1u ˆ Y for some

r0 ă r1 ď log 2 and let γ´ be a 1-periodic orbit of Hp´8,tq, contained entirely belowthe the level tr1u ˆ Y .

Suppose furthermore that H : R ˆ S1 ˆW Ñ R and J are locally radial (in thesense of Lemma 5.2) on pr1 ´ δ, r1 ` δq ˆ Y , with the additional condition thatHps, t, r, yq “ hps, erq for some function hps, ρq, with

erBh

Bρps, erq ´ hps, erq ě 0

Bs

´

erBh

Bρps, erq ´ hps, erq

¯

ď 0.

Then, there does not exist any v : RˆS1 ÑW solving Floer’s equation (5.1) forwhich there exists a sequence sk Ñ `8 along which vpsk, tq Ñ γ`ptq Ă tr1u ˆ Yand for which vps, tq Ñ γ´ptq, sÑ ´8.

Proof. Suppose that there was a v as in the statement. Consider a smooth functionr : W Ñ R that is equal to the first coordinate r on pr1 ´ δ2, r1 ` δ2q ˆ Y andconstant outside of pr1 ´ δ, r1 ` δq ˆ Y . We can assume that δ ą 0 is small enoughthat r ˝ γ´ptq ă r1 ´ δ2. Take a regular value r1 ´ c of r ˝ v : Rˆ S1 Ñ R in theinterval pr1 ´ δ2, r1q.

By hypothesis, there exists a k0 sufficiently large so that for each k ě k0,mintPS1 rpvpsk, tqq ą r1 ´ c2. Fix such a k.


Define the subdomain Sk of Rˆ S1 by

Sk “ tps, tq P p´8, skq ˆ S1 | rpvps, tqq ą r1 ´ cu.

Note that while this subdomain may be disconnected, its closure in R ˆ S1 iscompact, due the fact that vps, tq Ñ γ´ptq as sÑ ´8 and that r˝γ´ptq ă r1´δ2.Since k ě k0, this set is non-empty.

By the fact that r1´ c is a regular value of r ˝ v, we have that BSk is a collectionof smooth, embedded closed curves. Let Γ be the collection of embedded closedcurves in R ˆ S1 so that the boundary of Sk is BSk “ Γ Y tsku ˆ S1. Let Γ beoriented with the orientation induced as the boundary of Sk. As an aside, notethat the projection of Γ Ă Rˆ S1 to S1 has total degree ´1, since it separates thecylinder.

By construction, vpΓq Ă tr1ćuˆY , and by definition rpvpSkqq Ă pr1ć, r1`δs.Combining Stokes’s Theorem with Lemma 5.2 (and implicitly using biholomor-phisms between neighborhoods of the connected components of Γ in Sk and annuliof the form p´ε, 0s ˆ S1), we obtain

Etopopv|Skq “

ż

tskuˆS1

v˚λ´Hps, t, vps, tqqdt`

ż

Γ

v˚λ´Hps, t, vps, tqqdt

ď

ż

s“sk

v˚λ´Hpsk, t, vpsk, tqqdt`

ż

Γ

per1ćBh

Bρps, er1ćq ´ hper1ćqqdt

“

ż

s“sk

v˚λ´Hpsk, t, vpsk, tqqdt´

´

ż

s“sk

per1ćBh

Bρps, er1ćq ´ hper1ćqqdt`

`

ż

Sk

Bs`

er1ćBh

Bρps, er1ćq ´ hper1ćq

˘

ds^ dt

ď

ż

s“sk

v˚λ´Hpsk, t, vpsk, tqqdt.

As the parameter k Ñ `8, the right hand side converges to the action ofγ`, which is 0. Furthermore, since the Floer data is assumed to be monotone,Egeompv|Skq ď Etopopu|Skq and since the Sk are nested, Egeompv|Skq is monotonenon-decreasing in k. It follows that Egeompv|Skq “ 0 for all k. As Sk has non-emptyinterior, we conclude that Bsv “ 0, which contradicts the fact that vps, tq Ñ γ´ assÑ ´8.

We now explain how the previous result is useful in showing that Φ is a chainmap. We may arrange for H to satisfy the radial conditions of Lemma 5.4. Theonly way that Φ could fail to be a chain map is if there were a family of finiteenergy Floer cylinders for H with a subsequence converging to a Floer cylinderv : RˆS1 ÑW with the following property: there is a sequence sk Ñ8 such thatthe sequence of loops vpsk, .q converges to BW as in Lemma 4.3. But this situationis ruled out by Lemma 5.4.

Combining these compactness results with the standard constructions in sym-plectic homology, we obtain that our chain complex SC˚pW,Hq is chain homotopic

to SC˚pW, Hq, which we know computes symplectic homology.


6. Transversality for the Floer and holomorphic moduli spaces

In this section, we will build the transversality theory needed for the Floer cas-cades in Section 4.2. In the process, we will also discuss transversality for pseu-doholomorphic curves in X and in Σ, which will be necessary for the proof of ourmain result.

6.1. Statements of transversality results. Before stating the main result ofthis section, we will introduce some definitions.

Recall from Section 4.2 that the differential B counts elements in moduli spacesMH,N py, x; JY , JW q of split Floer cylinders with cascades (modulo automorphisms),where JY is a lift to Rˆ Y of an almost complex structure JΣ on Σ and JW is analmost complex structure in W that agrees with JY outside of a compact subset,as in Section 2.2, and y, x are critical points of fY together with a multiplicity, orare critical points of fW .

Definition 6.1. Denote the space of almost complex structures in Σ that arecompatible with ωΣ by JΣ.

Let JW denote the the space almost complex structures JW as described inSection 2.2, interpreted either as an almost complex structure on W or as an almostcomplex structure on X.

There is a surjective map P : JW Ñ JΣ, obtained by restricting JW to JY , andthen projecting to JΣ.

Let ϕpUq Ă X be the fixed open neighbourhood of Σ on which any JW P JW isstandard (where ϕ is the tubular neighbourhood of Σ of Definition 2.1).

(In particular, none of the spheres in X is permitted to be constant.)

Notice that P is an open map since the almost complex structures in JW have astandard form in the fixed open neighborhood of Σ in X, with respect to the diffeo-morphism in Lemma 2.8. Also notice that Lemma 2.8 allows us to simultaneouslyview JW P JW as an almost complex structure on W and as an almost complexstructure on X.

Lemma 6.2. Let v : Rˆ S1zΓ Ñ Rˆ Y be a finite hybrid energy Floer cylinder inRˆY (as in Definition 4.16), converging to Hamiltonian orbits in the manifolds Y`and Y´ at ˘8, and with finitely many punctures at Γ Ă RˆS1 converging to Reeborbits in t´8uˆY . Then, the projection πΣ˝v extends to a smooth JΣ-holomorphicsphere πΣ ˝ v : CP 1 Ñ Σ.

Proof. The projection πΣ ˝ v is JΣ-holomorphic on R ˆ S1, since H is admissible(as in Definition 3.1) The result now follows from Gromov’s removal of singularitiestheorem together with the finiteness of the energy of πΣ ˝ v.

Similarly, by using the biholomorphism constructed in Lemma 2.8, we obtainthe following.

Lemma 6.3. Let u : C Ñ W be a JW -holomorphic plane with finite Hofer en-ergy. Then, under the identification of W with XzΣ from Lemma 2.8, u admits anextension to a JX-holomorphic sphere u : CP 1 Ñ X.

In order to describe the projection to Σ of the levels of a split Floer cylinderwith N cascades that map to Rˆ Y , we introduce the following:


8

0

w1

8

0

w2

8

0

w3

8

0w4

z1z2

z3

q pZΣ

ZΣZΣ

ZΣ

ZΣ

Figure 6.1. A chain of 4 pearls from q to p with 3 marked points.

Definition 6.4. A chain of pearls from q to p, where p and q are critical points offΣ, consists of the following:

‚ N ě 0 parametrized JΣ-holomorphic spheres wi in Σ with two distinguishedmarked points at 0 and 8 and a possibly empty collection of additionalmarked points z1, . . . , zk on the union of the N domains (distinct from 0 or8 in each of the N spherical domains); the spheres are either non-constantor contain at least one additional marked point;

‚ if N “ 0, an infinite positive flow trajectory of ZΣ from q to p; if N ě 1, ahalf-infinite trajectory of ZΣ from wN p8q to p, a half-infinite trajectory ofZΣ from q to w1p0q and positive finite length trajectories of ZΣ connectingwip8q to wi`1p0q , i “ 1, . . . , N ´ 1.

See Figure 6.1. If such a chain of pearls is the projection to Σ of the componentsin R ˆ Y is a split Floer cylinder, then the additional marked points in the pseu-doholomorphic spheres correspond to punctures in the Floer cylinders, where theyconverge to cylinders over Reeb orbits that are capped by planes in W .

Notice that the geometric configuration of two spheres touching at a criticalpoint of fΣ admits an interpretation as a chain of pearls in Σ, since the criticalpoint is the image of any positive length flow line with that initial condition.

A chain of pearls with a sphere in X from q to p, where p is a critical point offΣ and q is a critical point of fX , consists of the following:

‚ N ě 1 parametrized JΣ-holomorphic spheres wi in Σ with two distinguishedmarked points at 0 and 8 and a possibly empty collection of additionalmarked points z1, . . . , zk on the union of the N domains (distinct from 0 or8);

‚ a parametrized non-constant JX -holomorphic sphere v in X;‚ a half-infinite trajectory of ZΣ from wN p8q to p, a half-infinite trajectory

of ´ZX from q to vp0q (recall that ZX is the push-forward of ZW by theinverse of the map from Lemma 2.8);

‚ positive length trajectories of ZΣ from wip8q to wi`1p0q for i “ 1, . . . , N´1;‚ the sphere in X touches the first sphere in Σ: w1p0q “ vp`8q;‚ the spheres w2, . . . , wN satisfy the stability condition that they are either

non-constant or contain at least one of the additional marked points (v isautomatically non-constant and w1 is allowed to be unstable).

An augmented chain of pearls [or an augmented chain of pearls with a sphere inX] from q to p is a chain of pearls [or chain of pearls with a sphere in X] togetherwith k JX -holomorphic spheres vi : CP1

Ñ X, i “ 1, . . . , k, with the followingadditional properties:

‚ for each z P CP1, vipzq P Σ if and only if z “ 8;


‚ if the puncture zi is in the domain of the holomorphic sphere wji : CP1Ñ Σ,

then wjipziq “ vip8q.

From the Lemmas above and the fact that the trajectories of ZY cover trajecto-ries of ZΣ, it follows that a Floer cylinder with N cascades projects to a chain ofpearls or a chain of pearls with a sphere in X. Additionally, if any of the sublevelshave augmentation planes, then those correspond to spheres in X passing throughΣ at the images of the corresponding marked points in the chain of pearls.

Observe that we allow the sphere w1 to be unstable in the definition of a chainof pearls in Σ with a sphere in X. The case in which w1 is a constant curvewithout marked points corresponds to the situation in which the correspondingFloer cascade contains a non-trivial Floer cylinder v1 contained in a single fibreof R ˆ Y Ñ Σ, and has the asymptotic limits v1p`8, tq on a Hamiltonian orbitand v1p´8, tq on a closed Reeb orbit in t´8u ˆ Y (see Figure 7.3 below, wherethis case is considered in detail). The Floer cylinder v1 in Rˆ Y is non-trivial andhence stable, whereas the corresponding sphere w1 in Σ is unstable. Since we donot quotient by automorphisms (yet), this does not pose a problem.

Definition 6.5. A chain of pearls in Σ is simple if each sphere is either simple (i.e.not multiply covered, [MS04, Section 2.5]) or is constant, and if the image of nosphere is contained in the image of another. If the chain of pearls has a sphere vin X, we require v to be somewhere injective.

An augmented chain of pearls is simple if the chain of pearls is simple and theaugmentation spheres in X are somewhere injective, none has image contained inthe open neighbourhood ϕpUq (as in Definition 6.1), and no sphere in X has imagecontained in the image of another sphere in X.

Remark 6.6. Recall that a chain of pearls with a sphere in X has a distinguishedsphere v in X for which vp0q is on the descending manifold of a critical point q of fW .By the construction of fW , this forces the image of v to intersect the complementof the tubular neighbourhood of Σ. As we will see in Section 7.2, Fredholm indexconsiderations related to monotonicity will force the augmentation planes/spheresto leave the tubular neighbourhood.

Remark 6.7. Notice that our condition on a simple chain of pearls is slightly dif-ferent than the condition imposed in [MS04, Section 6.1], with regard to constantspheres. For a chain of pearls to be simple by our definition, constant spheres maynot be contained in another sphere, constant or not. In [MS04], there is no suchcondition on constant spheres.

Definition 6.8. Given a finite hybrid energy Floer cylinder with N cascades, weobtain an augmented chain of pearls (possibly with a sphere in X) by the followingconstruction:

(1) cylinders in RˆY are projected to Σ: by Lemma 6.2 these form holomorphicspheres in Σ;

(2) planes in W are interpreted as spheres in X by Lemma 6.3;(3) flow lines of the gradient-like vector field ZY are projected to flow lines of

ZΣ.

We refer to this augmented chain of pearls in Σ (possibly with a sphere in X) asthe projection of the Floer cylinder with N cascades.


A finite hybrid energy Floer cylinder with N cascades is simple if the resultingchain of pearls is simple.

Given generators x, y of the chain complex (3.2), denote by

M˚H,N py, x; JY , JW q

the space of simple split Floer cylinders with N cascades from y to x. Recall thatif x or y is in in Rˆ Y , the corresponding generator is described by a critical pointrp of fY (which can be either qp or pp), together with a multiplicity k. If instead, xor y is in W , it corresponds to a critical point of fW .

Proposition 6.9. There exists a residual set J regW Ă JW of almost complex struc-

tures such that for each JW P J regW , M˚

H,N py, x; JY , JW q is a manifold.

If N “ 0 and thus x, y are generators in R ˆ Y , then x “ rpk, y “ rqk for thesame multiplicity k, and

dimR M˚H,0py, x; JY , JW q “ |rpk| ´ |rqk|.

If N ě 1, and q, p are generators in Rˆ Y , then x “ rpk` , y “ rqk´ and

dimR M˚H,N py, x; JY , JW q “ |x| ´ |y| `N ´ 1

Finally, if y PW and x P Rˆ Y , then x “ rpk and y “ q P CritpfW q and

dimR M˚H,N py, x; JY , JW q “ |x| ´ |y| `N.

Furthermore, the image P pJ regW q Ă JΣ (recall Definition 6.1) is residual and

consists of almost complex structures that are regular for simple pseudoholomorphicspheres in Σ.

The two different formulas involving N reflect the fact that N counts the numberof cylinders in Rˆ Y . In the case of a Floer cascade that descends to W , there aretherefore N ` 1 cylinders in the cascade.

Remark 6.10. These index formulas imply that the split symplectic homology dif-ferential is of degree ´1. Indeed, observe that the case N “ 0 corresponds to apure Morse configuration and doesn’t depend on any almost complex structure.We count rigid flow lines modulo the R-action, and thus require |x| ´ |y| “ 1. Forgenerators y, x P RˆY , we consider these N cylinders modulo the R-action on eachone, giving an RN -action. From this, a rigid configuration has |x|´|y|`N´1 “ N .For the case with y PW , we have N ` 1 cylinders in the Floer cascade, so we havea rigid configuration modulo the RN`1-action when N ` 1 “ |x| ´ |y| `N .

Note that when both x, y are generators in RˆY , the moduli space M˚H,N py, x; JY , JW q

will depend on JW only insofar as augmentation planes appear, otherwise it dependsonly on JY .

The split Floer differential B, introduced at the end of Section 4.2, was definedby counting elements in MH,N py, x; JY , JW q. We will see in Propositions 7.4 and7.5 that our standing assumptions imply that this is equivalent to counting simpleconfigurations in M˚

H,N py, x; JY , JW q.The rest of this section will be devoted to the proof of Proposition 6.9. It will

proceed in the following steps:

‚ Section 6.2 describes the Fredholm set-up for Floer cascades. In Section6.2.1 and in Appendix A, we discuss the necessary function spaces andlinear theory for the Morse–Bott problems. Then, Section 6.2.2 splits the


linearization of the Floer operator in such a way as to split the transversalityproblem into two problems. The first is a Cauchy–Riemann-type operatoracting on sections of a complex line bundle, and it is transverse for topo-logical reasons (automatic transversality). The second is a transversalityproblem for a Cauchy–Riemann operator in Σ.

‚ Section 6.3 adapts the transversality arguments from [MS04] in order toobtain transversality for chains of pearls in Σ.

‚ Section 6.4 shows transversality for the components of the cascades con-tained in W . This problem is translated into the equivalent problem ofobtaining transversality for spheres in X with order of contact conditionsat Σ, together with evaluation maps. The main technical point is an ex-tension of the transversality results from [CM07].

‚ Finally, Section 6.5 uses the splitting from Section 6.2.2 to lift the transver-sality results in Σ to obtain transversality for Floer cylinders with cascades,and to finish the proof of Proposition 6.9.

6.2. A Fredholm theory for Floer cascades.

6.2.1. Sobolev spaces for the Morse–Bott problem. The first step in the proof ofProposition 6.9 is to set-up the appropriate Fredholm theory. Recall that oneof the important technical difficulties is that the periodic Hamiltonian orbits andthe periodic Reeb orbits that we take as asymptotic boundary conditions for ourpunctured cylinders come in Morse–Bott families. Given a Floer solution v : R ˆS1zΓ: RˆY , we need to consider exponentially weighted Sobolev spaces of sectionsof the pull-back bundle v˚T pR ˆ Y q, so that the relevant linearized operators areFredholm. For δ ą 0, we denote by W 1,p,δpR ˆ S1zΓ, v˚T pR ˆ Y qq the space ofsections that decay exponentially like e´δ|s| near the punctures (also in cylindricalcoordinates near the punctures Γ). (In certain circumstances, we will find it usefulto consider sections with exponential growth, and denote these by taking δ ă 0.)

In order to consider a parametric family of punctured cylinders in which theasymptotic limits move in their Morse–Bott families, we let V be a collection ofvector spaces, associating to each puncture z P Γ Y t˘8u a vector subspace Vz ofthe tangent space to the corresponding Morse–Bott family of orbits. For δ ą 0, we

then consider the space of sections W 1,p,δV pR ˆ S1zΓ, v˚T pR ˆ Y qq that converge

exponentially at each puncture z to a vector in the corresponding vector space Vz.These vector spaces also admit a description as the kernels of asymptotic oper-

ators associated to the linearized Floer operator at v, and can also be described bythe kernel of a differential operator associated to the linearized Hamiltonian/Reebflow. (See, for instance, [Sie08, Section 2.1] and the discussion in the Appendix A.)

These function spaces and background on Cauchy–Riemann operators on Her-mitian vector bundles over punctured surfaces are discussed in greater detail inAppendix A.

Using this point of view, and the geometry of our problem, we will now proceed tosplit the linearized problem into two sub-problems, each of which can be addressedseparately.

Remark 6.11. In this paper, we will not always be careful to specify how small δ hasto be. It is worth pointing out that there is no value of δ that works for all modulispaces of (possibly perturbed) holomorphic curves. The reason is that we need |δ|to be smaller than the absolute value of all eigenvalues in the spectra of the relevant


linearized operators. As we can see in Table 2 in Appendix A, some of these as-ymptotic operators have smallest positive eigenvalue 1

2

`

´C `?C2 ` 16π2

˘

, which

becomes arbitrarily small as C Ñ 8. The relevant value for C here is h2pebkqebk ,which can become arbitrarily large as the multiplicity k Ñ 8. Since we are onlyever interested in curves connecting orbits of bounded multiplicities, we can alwayschoose δ sufficiently small.

6.2.2. The linearization at a Floer solution. We now adapt an observation first usedin [Dra04, Bou06], to show that the linearization of the Floer operator is uppertriangular with respect to the splitting of T pR ˆ Y q as pR ‘ RRq ‘ ξ. We thendescribe the non-zero blocks in this upper triangular presentation of the operator.The two diagonal terms are of special importance: one will be a Cauchy–Riemann-type operator acting on sections of a complex line bundle, and the other can beidentified with the linearization of the Cauchy–Riemann operator for spheres in Σ.

We now explain this construction in more detail, starting with some notation.Let v : R ˆ S1zΓ Ñ R ˆ Y be a Floer solution with punctures Γ, of finite hybridenergy, asymptotic to a closed Hamiltonian orbit at vp`8, tq, either asymptoticto a closed Hamiltonian orbit or with a negative end converging to a Reeb orbitat vp´8, tq, and with negative ends converging to Reeb orbits at the puncturesin Γ. Let w “ πΣ ˝ v : CP 1 Ñ Σ be the smooth extension of the projection of vto the divisor (as given by Lemma 6.2). The linearized projection dπΣ induces anisomorphism of complex vector bundles

v˚`

T pRˆ Y q˘

– pR‘ RRq ‘ w˚TΣ.

To see this, note that for each point p P Y , dπΣ induces a symplectic isomorphismpξp, dαq – pTπΣppqΣ,KωΣq. By the Reeb invariance of the almost complex structure

(and thus S1-invariance under rotation in the fibre), this then gives a complex vectorbundle isomorphism.

Let V associate to each puncture z P Γ Y t˘8u the tangent space to Y if thecorresponding limit of v is a closed Hamiltonian orbit and the tangent space toR ˆ Y if the corresponding limit of v is a closed Reeb orbit at ´8. As will beclearer shortly, this is associating to each puncture the entirety of the kernel of thecorresponding asymptotic operator. Let

(6.1) Dv : W 1,p,δV pv˚T pRˆ Y qq Ñ Lp,δpHom0,1

pT pRˆ S1q, v˚T pRˆ Y qqqbe the linearization of the nonlinear Floer operator at the solution v, for δ ą 0sufficiently small. The vector spaces V correspond to allowing the asymptoticlimits to move in their Morse–Bott families. We have then a linearized evaluationmap at the punctures with values in ‘zPt˘8uYΓVz.

LetDΣw : W 1,ppw˚TΣq Ñ LppHom0,1

pTCP 1, w˚TΣqq

be the linearized Cauchy–Riemann operator in Σ at the holomorphic sphere w.We also have the linearized Cauchy–Riemann operator 9DΣ

w at the holomorphiccylinder s` it ÞÑ wpe2πps`itqq “ πΣpvps, tqq. Then, pπΣ ˝ vq

˚TΣ “ w˚TΣ|RˆS1 is aHermitian vector bundle over Rˆ S1zΓ. Let VΣ be the kernels of the asymptotic

operators of 9DΣw at each of the punctures, ˘8 and Γ. (These are explicitly given

by VΣp´8q “ Twp0qΣ, VΣp`8q “ Twp8qΣ, VΣpzq “ TwpzqΣ for each marked pointz P Γ.) We consider this operator acting on the space of sections

9DΣw : W 1,p,δ

VΣpw˚TΣ|RˆS1q Ñ Lp,δpHom0,1

pRˆ S1, w˚TΣ|RˆS1qq.


The operator DΣw is Fredholm independently of the weight, but 9DΣ

w is only Fredholmwhen the weight δ is not an integer multiple of 2π. Furthermore, by combining[Wen16b, Proposition 3.15] with Lemma A.10, for 0 ă δ ă 2π, these operators havethe same Fredholm index and their kernels and cokernels are isomorphic by themap induced by restricting a section of w˚TΣ to the punctured cylinder.

Finally, define DCv by

(6.2)

DCv : W 1,p,δ

V0pRˆ S1zΓ,Cq Ñ Lp,δpHom0,1

pT pRˆ S1q,Cqq

pDCvF qpBsq “ Fs ` iFt `

ˆ

h2pebq eb 00 0

˙

F

where V0 associates the vector space iR to the punctures at which v converges toa closed Hamiltonian orbit (i.e. at `8 and possibly at ´8), and associates thevector space C at punctures at which v converges to a closed Reeb orbit, i.e. at Γand possibly at ´8. Notice that again these are chosen so that they precisely givethe kernels of the corresponding asymptotic operators of DC

v .

Lemma 6.12. The isomorphism v˚T pR ˆ Y q – pR ‘ RRq ‘ w˚TΣ induces adecomposition:

Dv “

ˆ

DCv M

0 9DΣw

˙

where M is a multiplication operator that evaluates on Bs to a fibrewise linear mapM : w˚TΣ Ñ R‘ RR. (In particular, M is compact.) Furthermore, if w “ πΣ ˝ vis non-constant, then M is pointwise surjective except at finitely many points.

Proof. In our setting, the nonlinear Floer operator takes the form:

dv ` JY pvqdv ˝ i´ h1perqRb dt` h1perqBr b ds “ 0.

Write v “ pb, vq : R ˆ S1zΓ Ñ R ˆ Y . If we apply dr to the previous equation,and use the fact that dr ˝ JY “ ´α, we get:

db´ v˚α ˝ i` h1pebqds “ 0

Denoting by πξ : TY Ñ ξ the projection along the Reeb vector field, we get

(6.3) πξdv ` JY pvqπξdv ˝ i “ 0.

Let g be the metric on Rˆ Y given by g “ dr2 ` α2 ` dαp¨, JY ¨q. This metric is

JY -invariant. Let r∇ be the Levi-Civita connection for g. Let ∇ be the Levi-Civitaconnection on TΣ for the metric ωΣp¨, JΣ¨q.

Then, it follows that the linearization Dv applied to a section ζ of v˚T pRˆ Y qsatisfies

(6.4)Dvζ pBsq “ r∇sζ ` JY pvqr∇tζ `

`

r∇ζJY pvq˘

Btv ´ r∇ζpJYXHqpvq

“ r∇sζ ` JY pvqr∇tζ ``

r∇ζJY pvq˘

Btv ` r∇ζph1perqBrqr“b.

Notice that r∇Br “ 0 since g is a product metric. We have then

r∇ζph1perqBrq|r“b “ h2pebq eb drpζqBr.

Observe also that for any vector field V in TΣ, there is a unique horizontal liftV to Y with the property αpV q “ 0. For any two vector fields V and W in TΣ,

since dαpV , W q “ KωΣpV,W q, we have the following

rV , W s “ ČrV,W s ´KωΣpV,W qR.


From this, it follows that the Levi-Civita connection r∇ satisfies the followingidentities:

r∇V W “ Č∇VW ´K

2ωΣpV,W qR

r∇RR “ 0

r∇RV “ ´1

2JY V .

A simple computation using the Reeb-flow invariance of JY and the torsion-freeproperty of the connection gives

r∇BrJY “ 0 “ r∇RJY .

We will now compute Dvζ pBsq, first when ζ “ ζ1Br ` ζ2R “ pζ1 ` iζ2qBr, andthen when ζ is a section of v˚ξ.

For the first computation, it suffices to notice the following two identities

DvBr pBsq “ h2pebq eb Br

DvR pBsq “ 0.

It follows then from the Leibniz rule that we have

Dvpζ1 ` iζ2q Br pBsq “ pζs ` iζtq ``

h2pebqebζ1˘

Br “ DCv pζ1 ` iζ2q BrpBsq.

Now consider the case when ζ is a section of v˚ξ, and is thus the lift ζ “ η of asection η of w˚TΣ. We compute

r∇sR “ r∇πξvsR “ ´1

2JY πξvs

r∇sζ “Č∇wsη ´K

2ωΣpws, ηqR´

1

2αpvsqJY ζ,

and similarly for r∇t. We then obtain the following covariant derivatives of JY ,where W is a section of v˚ξ:

pr∇ζJY qBr “ r∇ζR´ JY r∇ζBr “ ´1

2JY ζ

pr∇ζJY qR “ ´r∇ζBr ´ JY r∇ζR “ ´1

2ζ

pr∇ζJY qW “ r∇ζpJY W q ´ JY r∇ζW

“ Č∇ηJΣW ´K

2ωΣpη, JΣW qR´ JY

ˆ

Č∇ηW ´K

2ωΣpη,W qR

˙

“ Čp∇ηJΣqW ´K

2ωΣpη, JΣW qR´

K

2ωΣpη,W qBr.

It follows then

Dvζ pBsq “ r∇sζ ` JY r∇tζ ` pr∇ζJY qvt

“ Ą∇sη ´1

2αpvsqJY ζ ´

K

2ωΣpws, ηqR` JY Ą∇tη `

1

2αpvtqζ `

K

2ωΣpwt, ηqBr

´1

2btJY ζ ´

1

2αpvtqζ ` Čp∇ηJΣqwt ´

K

2ωΣpη, JΣwtqR´

K

2ωΣpη, wtqBr

“Ć9DΣwη `KωΣpwt, ηqBr ´KωΣpws, ηqR.

(Note that we use the fact that vs ` JY vt ` h1pebqBr “ 0 in the cancellations.)


Writing ζ “ pζa, ζbq under the isomorphism v˚T pR ˆ Y q – pR ‘ RRq ‘ w˚TΣ,we obtain the decomposition:

Dvpζa, ζbqpBsq “

ˆ

Daa Dab

Dba Dbb

˙ˆ

ζaζb

˙

pBsq

Our calculations now establish that Daa “ DCv and Dba “ 0, Dbb “ 9DΣ

w, andDabζpBsq “ KωΣpwt, πΣζqBr ´KωΣpws, πΣζqR. Observe that in particular, Dab isa pointwise linear map from v˚ξ|p to RBr ‘ RR. The map is surjective except atcritical points of the pseudoholomorphic map w, of which there are finitely many ifw is non-constant.

Remark 6.13. Notice that for each puncture z P t˘8u Y Γ, if γptq denotes thecorresponding asymptotic Hamiltonian or Reeb orbit, the previous result allows usto identify Vz with Tγp0qY at a Hamiltonian orbit and with R ˆ Tγp0qY at a Reeborbit.

Lemma 6.14. Let v : R ˆ S1zΓ Ñ R ˆ Y be a finite hybrid energy Floer cylinderwith punctures Γ.

Then, the operator DCv defined in Equation (6.2) is Fredholm for δ ą 0 sufficiently

small.The restriction

DCv |W 1,p,δ : W 1,p,δpRˆ S1zΓ,Cq Ñ Lp,δpHom0,1

pT pRˆ S1q,Cqq

has Fredholm index ´1´ 2#Γ and is injective.If v converges at ´8 to a closed Hamiltonian orbit,

DCv : W 1,p,δ

V0pRˆ S1zΓ,Cq Ñ Lp,δpHom0,1

pT pRˆ S1q,Cqq

has Fredholm index 1 and is surjective.If, instead, v converges at ´8 to a closed Reeb orbit in t´8uˆY , then DC

v hasFredholm index 2 and is surjective.

In either of these cases, the kernel of DCv contains the constant section i, which

can be identified with the Reeb vector field.

Proof. We will apply the punctured Riemann–Roch theorems A.6 and A.8.If the Floer solution v converges to a Hamiltonian orbit in tb˘uˆY as sÑ ˘8,

then the associated asymptotic operator associated to DCv at ˘8 is given by

A˘ “ ´id

dt´

ˆ

h2peb˘q eb˘ 00 0

˙

.

We recall that for δ ą 0, the space of functions W 1,p,δpR ˆ S1zΓ,Cq has expo-nentially fast decay to zero in cylindrical coordinates near the punctures in Γ andalso approaching ˘8ˆ S1.

By Theorem A.6, we need to compute the Conley–Zehnder indices of the appro-priately perturbed asymptotic operators. We do so using Corollary A.4. Start withthe case δ ą 0. The Conley–Zehnder index relevant at `8 is that of A`` δ, whichis 0. The Conley–Zehnder index relevant at ´8 is that of A´ ´ δ, which is 1. Ifwe consider instead δ ă 0, then the Conley–Zehnder index of A` ` δ is 1 and thatof A´ ´ δ is 0.


Associated to a Reeb puncture at ´8 or at P P Γ, we have the asymptoticoperator

íd

dt.

Writing v “ pb, vq : RˆS1zΓ Ñ RˆY , we have bÑ ´8 at both types of punctures.The same argument as above implies that if δ ą 0, these punctures have the Conley–Zehnder index of í ddt ´ δ, which is 1. If δ ă 0, we have instead that the Conley–

Zehnder index of í ddt ´ δ is ´1.The exponential weight, δ, must be smaller than the spectral gap for any of

these punctures. In view of Lemma A.3, we take |δ| ăĆ``

?C2``16π2

2 ď 2π,

where C` “ h2peb`q eb` . Notice that by Condition (v) of Definition 3.1, we haveh2peb`q eb` ě h2peb´q eb´ so this condition guarantees that |δ| is smaller than anyspectral gap.

Applying now the punctured Riemann–Roch theorem A.6, and using the factthat the Euler characteristic of the punctured cylinder is ´#Γ, if δ ą 0, we obtainthe Fredholm index

´#Γ` 0´ 1´#Γ “ ´1´ 2#Γ,

as claimed.The injectivity of DC

v restricted to W 1,p,δ follows from automatic transversality,applying [Wen10, Proposition 2.2]. Using the notation of Wendl [Wen10, Equation(2.5)], the fact that the curve has genus 0 and that the unique puncture with evenConley–Zehnder index is the positive puncture, we have

c1pE, l,AΓq “1

2p´1´#Γ´ 2` 1q “ ´1´

#Γ

2ă 0

as necessary.

Now, in order to consider the operator DCv : W 1,p,δ

V0Ñ Lp,δ, it will be convenient

to consider a related operator with the same formula, but on the much larger spaceof functions with exponential growth. By a slight abuse of notation, we will use thesame name:

DCv : W 1,p,´δ Ñ Lp,´δ

pDCvF qpBsq “ Fs ` iFt `

ˆ

h2pebq eb 00 0

˙

F

(See also Lemma A.10). Then, the kernel and cokernel of the operator acting onthe spaces of sections with with exponential growth will have kernel and cokernel

that can be identified with the kernel and cokernel of the operator acting on W 1,p,δV0

.Applying Theorem A.8, we compute the Fredholm index to be:

´#Γ`1´ p´#Γq ´

#

0 if vp´8q converges to a Hamiltonian orbit

´1 if vp´8q converges to a Reeb orbit

“ 1 or 2, depending on the negative end of v.

Furthermore, the fact that the curve has genus 0 and one puncture with evenConley–Zehnder index precisely if limsÑ´8 v is a Hamiltonian orbit implies that

c1pE, l,AΓq “

#

12 p1´ 2` 1q “ 0 ă 1 if vp´8q converges to a Hamiltonian orbit12 p2´ 2q “ 0 ă 2 if vp´8q converges to a Reeb orbit


Therefore, DCv satisfies the automatic transversality criterion and is thus surjective,

as wanted.It follows immediately from the expression for DC

v that the constant i is in thekernel. Recalling that C “ v˚pR ‘ RRq in the splitting given by Lemma 6.12, wethen may identify this with the Reeb vector field R.

To summarize then the results of this section, by Lemma 6.12, a punctured Floercylinder in RˆS1 is regular if the operators DC

v and 9DΣw are surjective. Surjectivity

of the latter is equivalent to surjectivity of DΣw. Lemma 6.14 gives the surjectivity

of DCv . It thus remains to study transversality for DΣ

w, specifically with respectto the evaluation maps that will allow us to define the moduli spaces of chains ofpearls in Σ. Additionally, we need to consider transversality for moduli spaces ofplanes in W asymptotic to Reeb orbits in Y , or equivalently, the moduli spaces ofspheres in X with an order of contact condition at Σ.

Remark 6.15. In Part ??, we will also consider punctured JY -holomorphic cylindersu : RˆS1zΓ Ñ RˆY . They satisfy the JY -holomorphic curve equation, instead ofits Hamiltonian-perturbed Floer counterpart (4.3). These are the types of curvescontributing to the Morse–Bott contact homology of Y [Bou02]. The methods ofthis section can be adapted to show that the moduli spaces of such JY -holomorphiccurves are also cut out transversely. The main difference is that the analogue ofthe vertical operator in (6.2) no longer has the term involving the Hamiltonian.Automatic transversality can again be used to adapt Lemma 6.14 to this situation.

6.3. Transversality for chains of pearls in Σ. In this section and the next, weshow that for generic almost complex structure (in a sense to be made precise), themoduli spaces of chains of pearls and moduli spaces of chains of pearls with spheresin X (possibly augmented as well) are transverse. We begin with the definition ofseveral moduli spaces that will be useful.

Definition 6.16. Let JΣ P JΣ be an almost complex structure compatible withωΣ. Given p, q P CritpfΣq and a finite collection A1, . . . , AN P H2pΣ;Zq, let

M˚k,ΣppA1, . . . , AN q; q, p; JΣq

denote the space of simple chains of pearls in Σ from p to q (see Definition 6.5),such that pwiq˚rCP 1s “ Ai, with k marked points.

Let

M˚k,ΣppA1, . . . , AN q; JΣq

denote the moduli space of N parametrized holomorphic spheres in Σ, representingthe classes Ai, i “ 1, . . . , N , with k marked points, also satisfying the simplicitycriterion of Definition 6.5, i.e. so each sphere is either somewhere injective or con-stant, each constant sphere has at least one augmentation marked point, and nosphere has image contained in the image of another.

For JW P JW , let JΣ “ P pJW q be the corresponding almost complex structurein JΣ and JX the corresponding almost complex structure on X. Define

M˚k,pX,ΣqppB;A1, . . . , AN q; q, p, JW q

to be the moduli space of simple chains of pearls in Σ with a sphere in X (asin Definitions 6.4 and 6.5), where q is a critical point of fX and p is a critical


point of fΣ, and representing the spherical homology classes rwis “ Ai P H2pΣ;Zq,i “ 1, . . . , N and rvs “ B P H2pX;Zqz0. In the following, we will write

l “ B ‚ Σ “ KωpBq

which is the order of contact of v with Σ.Let

M˚k,pX,ΣqppB;A1, . . . , AN q; JW q

denote the moduli space of N parametrized holomorphic spheres in Σ, representingthe classes Ai, and of a holomorphic sphere in X representing the class B with orderof contact l “ B ‚Σ “ KωpBq, also satisfying the simplicity criterion of Definition6.5, i.e. so each sphere in Σ is either somewhere injective or constant (if constant,it has at least one augmentation marked point), no image of a sphere is containedin the image of another and the image of the sphere in X is not contained in thetubular neighbourhood ϕpUq of Σ. Furthermore, the spheres in Σ have k markedpoints.

Let

M˚XppB1, B2, . . . , Bkq; JW q

denote the moduli space of k parametrized holomorphic spheres in X, where eachsphere is somewhere injective, no image of a sphere is contained in the image ofanother sphere, and so the image of each sphere is not contained in the tubularneighbourhood ϕpUq of Σ, and such that each sphere intersects Σ only at 8 P CP1

with order of contact Bi ‚ Σ.Finally, let

Mak,ΣppA1, . . . , AN q, pB1, . . . , Bkq; q, p; JΣ, JW q

denote the moduli space of simple augmented chains of pearls in Σ with k augmen-tation planes, and let

Mak,pX,ΣqppB;A1, . . . , AN q; pB1, . . . , Bkq; q, p; JΣ, JW q

denote the moduli space of simple augmented chains of pearls with a sphere in X.(See Definitions 6.4 and 6.5.)

In order to apply the Sard–Smale Theorem, we need to consider Banach spacesof almost complex structures, so we let J r

Σ,J rW be the space of Cr–regular almost

complex structures otherwise satisfying the conditions of being in JΣ, JW . Weimpose r ě 2 and in general will require r to be sufficiently large that the Sard–Smale theorem holds (this will depend on the Fredholm indices associated to thecollection of homology classes and will also depend on the order of contact to Σ forthe spheres in X).

For each of these moduli spaces, we also consider the corresponding universalmoduli spaces as we vary the almost complex structure. For instance, we denoteby Mk,ΣppA1, . . . , AN q,J r

Σq the moduli space of pairs ppwiqNi“1, JΣq with JΣ P J r

Σ

and pwiqNi“1 PMk,ΣppA1, . . . , AN q, JΣq.

The main goal of this section and of the next is to prove that these modulispaces of simple chains of pearls are transverse for generic J . This is analogous to[MS04, Theorem 6.2.6], and indeed, the transversality theorem of McDuff-Salamonwill be an ingredient of our proof. We will furthermore require an extension of theresults from [CM07] (see Section 6.4), and an additional technical transversalitypoint required to be able to consider the lifted problem in Rˆ Y .


Proposition 6.17. There is a residual set J regW Ă JW such that J reg

Σ :“ P pJ regW q

is a residual set in JΣ and such that for all JΣ P J regΣ and JW P J reg

W , p P CritpfΣq,q P CritpfΣq or q P CritpfW q, the moduli spaces M˚

k,ΣppA1, . . . , AN q; q, p; JΣq,

M˚k,pX,ΣqppB;A1, . . . , AN q; q, p, JW q, Ma

k,ΣppA1, . . . , AN q, pB1, . . . , Bkq; q, p; JΣq and

Mak,pX,ΣqppB;A1, . . . , AN q; pB1, . . . , Bkq; q, p; JΣ, JW q are manifolds. Their dimen-

sions are

dim M˚k,ΣppA1, . . . , AN q; q, p; JΣq “ Mppq `

Nÿ

i“1

2 xc1pTΣq, Aiy ´Mpqq `N ´ 1` 2k,

dim M˚k,pX,ΣqppB;A1, . . . , AN q; q, p, JW q

“ Mppq `Nÿ

i“1

2 xc1pTΣq, Aiy ` 2pxc1pTXq, By ´B ‚ Σq `Mpqq ´ 2pn´ 1q `N ´ 1` 2k

dim Mak,ΣppA1, . . . , AN q, pB1, . . . , Bkq; q, p; JΣ, JW q

“ Mppq `Nÿ

i“1

2 xc1pTΣq, Aiy ´Mpqq `N ´ 1` 4k `kÿ

i“1

p2 xc1pTXq, Biy ´ 2Bi ‚ Σq ,

dim Mak,pX,ΣqppB;A1, . . . , AN q; pB1, . . . , Bkq; q, p; JΣ, JW q

“ Mppq `Nÿ

i“1

2 xc1pTΣq, Aiy ` 2pxc1pTXq, By ´B ‚ Σq `Mpqq ´ 2pn´ 1q`

`N ´ 1` 4k `kÿ

i“1

p2 xc1pTXq, Biy ´ 2Bi ‚ Σq .

where Mpqq is the Morse index of the critical point q, for fΣ if q is a critical pointof fΣ and for fX if q is a critical point of fX , and Mppq is the Morse index of pfor fΣ.

Remark 6.18. Notice that we will eventually consider the augmentation planesmodulo their 4 parameter family of automorphisms, thus effectively reducing thedimensions of each of Ma

Σ and MapX,Σq by 4k.

Proposition 6.19 ([MS04, Proposition 6.2.7]). M˚k,ΣppA1, . . . , AN q;J r

Σq is a Ba-nach manifold.

We will also make use of the following proposition, which we prove in the nextsection.

Definition 6.20. There is a universal evaluation map

evΣ : M˚k,ΣppA1, . . . , AN q;JΣq Ñ Σ2N

pw1, . . . , wN q ÞÑ pw1p0q, w1p8q, w2p0q, w2p8q, . . . , wN p8qq.

Similarly, we have

evX,Σ : M˚k,pX,ΣqppB;A1, . . . , AN q;JW q Ñ X ˆ Σ2N`1

pv, w1, . . . , wN q ÞÑ pvp0q, vp8q, w1p0q, w1p8q, w2p0q, . . . , wN p8qq,

where v is the holomorphic sphere in X and the wi are the spheres in Σ.


We have an evaluation map coming from simple collections of spheres in X:

evaΣ : M˚XppB1, B2, . . . , Bkq;JW q Ñ Σk

pv1, . . . , vkq ÞÑ pv1p8q, v2p8q, . . . , vkp8qq.

For spheres in Σ, We also obtain evaluation maps at the augmentation punctures

evaΣ : M˚k,ΣppA1, . . . , AN q;JΣq Ñ Σk

and

evaΣ : M˚k,pX,ΣqppB;A1, . . . , AN q;JW q Ñ Σk.

We refer to these three maps denoted eva as augmentation evaluation maps.

Proposition 6.21. Let B0, . . . , Bk be spherical classes in H2pXq. Let

r ě maxBi ‚ Σ` 2.

The universal moduli space M˚XppB1, . . . , Bkq;J r

W q is a Banach manifold andthe evaluation maps

evaΣ : M˚XppB1, . . . , Bkq;J r

W q Ñ Σk : pf1, f2, . . . , fkq ÞÑ pf1p8q, . . . , fkp8qq

evX,Σ : M˚XppB0q;J r

W q Ñ X ˆ Σ : f ÞÑ pfp0q, fp8qq

are submersions.

Recall that we have chosen a Morse function fΣ : Σ Ñ R and a correspondinggradient-like vector field ZΣ, such that pfΣ, ZΣq is a Morse–Smale pair. The time-t flow of ZΣ is denoted by ϕtZΣ

and the stable (ascending) W sΣpqq and unstable

(descending) manifolds WuΣppq were defined in Equation (3.1).

Definition 6.22. The flow diagonal in Σˆ Σ associated to the pair pfΣ, ZΣq is

∆fΣ:“

!

px, yq P pΣzCritpfΣqq2| Dt ą 0 so x “ ϕtZΣ

pyq)

where CritpfΣq is the set of critical points of fΣ.

We will now establish transversality of the evaluation maps to appropriate prod-ucts of stable/unstable manifolds, critical points, diagonals and flow diagonals. By[MS04, Proposition 6.2.8], the key difficulty will be to deal with constant spheres.For this, we will need the following lemma about evaluation maps intersecting withthe flow diagonals, essentially exploiting the composition property of the linearizedflow of ϕtZΣ

, and interpreting a chain of pearls with a constant sphere in it as a chainof pearls without that constant sphere, but with an additional “stopping point” onone of the intermediate flow lines.

Lemma 6.23. Suppose M is a manifold so that the map

ev “ pev´, ev`q : MÑ Σˆ Σ

is transverse to the flow diagonal ∆fΣ.

Then

ev : Mˆ Σ Ñ Σ4

pm, zq ÞÑ pev´pmq, z, z, ev`pmqq

is transverse to ∆fΣ ˆ∆fΣ .


Proof. Suppose that evpm, zq P ∆fΣˆ ∆fΣ

. Let m´ “ ev´pmq, m` “ ev`pmq.

Then, there exist times t1, t2 ą 0 so that ϕt1ZΣpm´q “ z and ϕt2ZΣ

pzq “ m`. In

particular then, if t “ t1 ` t2, we have ϕtZΣpev´pmqq “ ev`pmq so evpmq P ∆fΣ .

If px, yq P ∆fΣ, there exists t ą 0 so that ϕtZΣ

pxq “ y. Its tangent space isexplicitly given by

Tpx,yq∆fΣ “ tpv, dϕtZΣpxqv ` cZΣpyqq | v P TxΣ, c P Ru Ă TxΣ‘ TyΣ.

Let P1 “ dϕt1ZΣpm´q : Tm´Σ Ñ TzΣ and P2 “ dϕt2ZΣ

pzq : TzΣ Ñ Tm`Σ. Then

P2 ˝ P1 “ dϕtZΣpm´q : Tm´ Ñ Tm` .

By hypothesis, we have

d evm ¨TmM` Tpm´,m`q∆fΣ “ Tm´Σ‘ Tm`Σ.

Thus, for every pw´, w`q P Tm´Σ ‘ Tm`Σ, there exist pu´, u`q P d evm ¨TmM,c P R, v P Tm´Σ so that

u´ ` v “ w´

u` ` P2P1v ` cZΣpm´q “ w`.

We need to show that for every pw1, w2, w3, w4q P Tpm´,z,z,m`qΣ4, there exist

pu´, u`q P d evm ¨TmM, c1, c2 P R, v1 P Tm´Σ, v2, v3 P TzΣ so that

u´ ` v1 “ w1

v3 ` P1v1 ` c1ZΣpzq “ w2

v3 ` v2 “ w3

u` ` P2v2 ` c2ZΣpm`q “ w4.

From the hypothesis, there exist pu´, u`q P d evm ¨TmM, c2 P R and a v P Tm´Σso that

u´ ` v “ w1 ´ P´11 w2

u` ` P2P1v ` c2ZΣpm´q “ w4 ´ P2w3.

Now, let v3 “ ´P1v P TzΣ, c1 “ 0, v2 “ w3 ´ v3 and v1 “ P´11 pw2 ´ v3q “

P´11 w2 ` v. We now verify:

u´ ` v1 “ u´ ` v ` P´11 w2 “ w1

v3 ` P1v1 “ v3 ` w2 ´ v3 “ w2

v3 ` v2 “ v3 ` w3 ´ v3 “ w3

u` ` P2v2 ` c2ZΣpm`q “ u` ` P2w3 ´ P2v3 ` c2ZΣpm`q

“ u` ` P2P1v ` P2w3 ` c2ZΣpm`q

“ w4 ´ P2w3 ` P2w3 “ w4,

as required.

Lemma 6.24. Let N ě 1, and let A1, . . . , AN be spherical homology classes in Σand let B be a spherical homology class in X.

Suppose that S Ă Σ2N´2 is obtained by taking the product of some number ofcopies of ∆fΣ Ă Σ2 and of the complementary number of copies of tpp, pq | p PCritpfΣqu Ă Σ2, in arbitrary order. Let ∆ Ă Σ2 denote the diagonal.

Then ifřNi“1Ai ‰ 0, the universal evaluation map



is transverse to the submanifold ptˆ S ˆ pt.If B ‰ 0, the universal evaluation map

evX,Σ : M˚k,pX,ΣqppB;A1, . . . , AN q;JW q Ñ X ˆ Σ2N`1

is transverse to the submanifold ptˆ∆ˆ S ˆ pt.

Proof. We consider the case of M˚k,Σ in detail, since the argument is essentially the

same for M˚k,pX,Σq, though notationally more cumbersome.

Suppose that ppv1, . . . , vN q, Jq PM˚k,ΣppA1, . . . , AN q;JΣq is in the pre-image of

ptˆSˆ pt. Write S “ S1ˆS2ˆ ¨ ¨ ¨ ˆSN´1, where each Si Ă Σ2 is either the flowdiagonal or the set of critical points.

Notice that the simplicity condition then requires that if some sphere vi is con-stant, 1 ă i ă N , we must have that Si´1 and Si are flow diagonals. If v1 isconstant, then S1 is a flow diagonal and if vN is constant, SN´1 is a flow diagonal.

We will proceed by induction on N . The case N “ 1 follows from [MS04,Proposition 3.4.2].

Now, for the inductive argument, we suppose the result holds for any S Ă

Σ2pN´1q´2 of the form specified, and for any k ě 0, for any collection of N ´ 1spherical classes, not all of which are zero.

Let now A1, . . . , AN be spherical homology classes, not all of which are zero.Then, at least one of A1, . . . , AN´1 or A2, . . . , AN is a collection of spheres satisfyingthe hypotheses. For simplicity of notation, let us assume that A1`¨ ¨ ¨`AN´1 ‰ 0.Let S0 “ S1ˆS2ˆ¨ ¨ ¨ˆSN´1. Let k “ k0`kN where kN is the number of markedpoints we consider on the last sphere. By the induction hypothesis, we have that

evΣ : M˚k0,Σ

ppA1, . . . , AN´1q;JΣq Ñ Σ2pN´1q

is transverse to ptˆ S0 ˆ pt. Denote this evaluation map by ev0.Notice that M˚

k,ΣppA1, . . . , AN q;JΣq ĂM˚k0,Σ

ppA1, . . . , AN´1q;JΣqˆM˚k1,Σ

pAN ;JΣq.

Let then evN : M˚k,ΣppA1, . . . , AN q;JΣq Ñ Σ2 be the evaluation at 0 and 8 in the

Nth sphere. We therefore have


given by evΣ “ pev0, evN q.If AN ‰ 0, the result follows again from [MS04, Proposition 6.2.8].If, instead, AN “ 0, we have from above that SN “ ∆fΣ

. Notice that theevaluation map of constant spheres on Σ has image on the diagonal in ΣˆΣ. Theresult now follows by applying Lemma 6.23.

The case with a sphere in X follows a nearly identical induction argument,though the base case consists of a single sphere in X. The required submersion toX ˆ Σ now follows from Proposition 6.21, and the induction proceeds as before.

Proposition 6.25. Let N ě 0. Suppose that S Ă Σ2N´2 is obtained by taking theproduct of some number of copies of ∆fΣ Ă Σ2 and of the complementary numberof copies of tpp, pq | p P CritpfΣqu Ă Σ2, in arbitrary order.

Let ∆ Ă ΣˆΣ denote the diagonal and let ∆k denote the diagonal Σk in ΣkˆΣk.Let p, q be critical points of fΣ and let x be a critical point of fW .


Then the universal evaluation maps together with augmentation evaluation maps

evΣˆ evaΣˆ evaΣ : M˚k,ΣppA1, . . . , AN q;JΣq ˆM˚

0 ppB1, . . . , Bkq;JW q Ñ Σ2N ˆ Σk ˆ Σk

evX,Σˆ evaΣˆ evaΣ : M˚k,pX,ΣqppB;A1, . . . , AN q;JW q ˆM˚

0 ppB1, . . . , Bkq;JW q Ñ X ˆ Σ2N`1 ˆ Σk ˆ Σk

are transverse to

W sΣpqq ˆ S ˆW

uΣppq ˆ∆k

WuXpqq ˆ∆ˆ S ˆWu

Σppq ˆ∆k.

Proof. Notice first that by Proposition 6.21 the augmentation evaluation map

eva : M˚0 ppB1, . . . , Bkq;JW q Ñ Σk

is a submersion. It suffices therefore to prove the result in the case k “ 0.The proposition follows immediately if at least one of the Ai, i “ 1, . . . , N is

non-zero, or if we are considering the case of a chain of pearls with a sphere in X,by applying Lemma 6.24.

The only case then that must be examined is of a chain of pearls entirely in Σwith all spheres constant. In this case, the moduli space M˚

0,Σpp0, 0, . . . , 0q,JW qcan be identified with

tpz1, . . . , zN q P ΣN | zi “ zj ùñ i “ ju ˆ JW .Observe that in the case N “ 1, transversality follows from the Morse–Smalecondition on the gradient-like vector field ZΣ. This gives that the intersection ofW s

Σpqq and WuΣppq is transverse, and hence that the diagonal in ΣˆΣ is transverse

to W sΣpqqˆW

uΣppq. The case of N ě 2 then follows from this together with Lemma

6.23.

Proposition 6.25 can be combined with standard Sard–Smale arguments, the factthat P : JW Ñ JΣ is an open and surjective map and Taubes’s method for passingto smooth almost complex structures (see for instance [MS04, Theorem 6.2.6]) togive the following proposition:

Proposition 6.26. There exist residual sets of almost complex structures J regW Ă

JW and J regΣ “ P pJ reg

W q, so that for fixed JW P J regW and JΣ “ P pJW q, the restric-

tion of the evaluation maps to M˚k,ΣppA1, . . . , AN q; JΣq, M˚

k,pX,ΣqppB;A1, . . . , AN q; JW q,

and M˚0 ppB1, . . . , Bkq; JW q are transverse to the submanifolds of Proposition 6.25.

The transversality statement of the main result of this section, Proposition 6.17now follows. The dimension formulas follow from usual index arguments, combiningRiemann–Roch with contributions from the constraints imposed by the evaluationmaps.

6.4. Transversality for spheres in X with order of contact constraints inΣ. We will now consider transversality for a chain of pearls with a sphere in X.We will extend the results from Section 6 in [CM07]. In that paper, Cieliebak andMohnke prove that the moduli space of simple curves not contained in Σ, with anorder of contact condition, can be made transverse by a perturbation of the almostcomplex structure away from Σ. We will extend this result to show that additionallythe evaluation map to Σ at the point of contact can be made transverse.


Recall that Σ is a symplectic divisor and NΣ is its symplectic normal bundleequipped with a Hermitian structure. We have fixed a symplectic neighbourhoodϕ : U 1 Ñ X where U,U 1 with U Ă U 1 Ă NΣ are open neighbourhoods of the zerosection, as in Definition 2.1. From Definition 6.1, we require that all JX P JW havethat JX is standard in the image ϕpUq Ă X of this neighbourhood.

Fix an almost complex structure J0 P JW . We may suppose that P pJ0q P JΣ isan almost complex structure in the residual set given by Proposition 6.17, thoughthis isn’t strictly speaking necessary.

Let V :“ XzϕpUq. Following Cieliebak-Mohnke [CM07], let J pV q be the set ofall almost complex structures compatible with ω that are equal to J0 on ϕpUq.

To define the order of contact, consider an almost complex structure JX P JWand a JX -holomorphic sphere f : CP 1 Ñ X with fp0q P Σ, an isolated intersection.Choose coordinates s`it “ z P C on the domain and local coordinates near fp0q P Σon the target, such that fp0q P Σ Ă X corresponds to 0 P Cn´1 “ Cn´1 ˆ t0u ĂCn´1 ˆ C. Write πC : Cn Ñ C for projection onto the last coordinate (which isto be thought of as normal to Σ). Assume also that JXp0q “ i. Then, f hasorder l tangency at 0 if the vector of all partial derivatives up to order l has trivialprojection to C. We define then the order of contact at an arbitrary point in CP1

by precomposing with a Mobius transformation.Define the space of simple pseudoholomorphic maps into X that have order of

tangency l at 8 to a point in Σ to be

M˚8,l,pX,ΣqpJW ; pΣ, lqq “ tpf, JXq PBm,p ˆ JW | BJXf “ 0,

fp8q P Σ, dlfp8q P Tfp8qΣ,

f simple, f´1pV q ‰ Hu

where dkξp0q is the vector of all partial derivatives of ξ of orders 1 through k of fat 0 and we require m ě l ` 2. Note that our notation differs somewhat from thenotation in [CM07].

In this section, we will prove Proposition 6.21, which was stated and used above:

Proposition 6.21. Let B0, . . . , Bk be spherical classes in H2pXq. Let

r ě maxBi ‚ Σ` 2.

The universal moduli space M˚XppB1, . . . , Bkq;J r

W q is a Banach manifold andthe evaluation maps

evaΣ : M˚XppB1, . . . , Bkq;J r

W q Ñ Σk : pf1, f2, . . . , fkq ÞÑ pf1p8q, . . . , fkp8qq

evX,Σ : M˚XppB0q;J r

W q Ñ X ˆ Σ : f ÞÑ pfp0q, fp8qq

are submersions.

Notice that this is a local question, so it suffices to show this for pf, JXq PM˚8,l,pX,ΣqpJ

mW ; pΣ, lqq specifically for JX P J pV q (and not for all JX P JW ).

The proposition will follow by a modification of the proof given in [CM07, Section6]. Instead of reproducing their proof, we indicate the necessary modifications. Inorder to be as consistent as possible with their notation, we consider the point ofcontact with Σ to be at 0.

Consider a JX -holomorphic map f : CP 1 Ñ X such that fp0q P Σ with orderof tangency l. In the notation of [CM07], we are interested in the case of only one


component Z “ Σ. We will obtain transversality of the evaluation map at 0 byvarying JX freely in V .

The linearized Cauchy–Riemann operator at f with respect to a torsion-freeconnection is

pDfξqpzq “ ∇sξpzq ` JXpfpzqq∇tξpzq ` p∇ξpzqJXpfpzqqq ftpzq.

Specializing to the standard Euclidean connection in R2n “ Cn (which preservesCn´1 along Cn´1), we get

pDfξqpzq “ ξspzq ` JXpfpzqqξtpzq Àpzqξpzq

where

Apzqξpzq “ pDξpzqJXpfpzqqq ftpzq

(see page 317 in [CM07]).We need the following adaptation of Corollary 6.2 in [CM07].

Lemma 6.27. Suppose pf, JXq P M˚8,l,pX,ΣqpJ

mW ; pΣ, lqq with JX P JmpV q. Let

ξ : pD, 0q Ñ pCn, 0q be such that Dfξ “ 0, with JX preserving Cn´1. Given 0 ă

k ď l, if ξp0q P Cn´1, dk´1ξp0q P Cn´1 and BkξBskp0q P Cn´1, then dkξp0q P Cn´1.

Proof. We need to show that Bkξ

BskíBtip0q P Cn´1 for all 0 ď i ď k. It will be con-

venient to use multi-index notation for partial derivatives, and denote the previousexpression by Dpkí,iqξp0q. The case i “ 0 is part of the hypotheses of the Lemma.For the induction step, note that Dfξ “ 0 combined with the product rule impliesthat

Dpkí,iqξpzq “ JXpfpzqq´

Dpkí`1,i´1qξpzq`ÿ

α,β

DαpJXpfpzqqDβξpzq`

ÿ

α1,β1

Dα1ApzqDβ1ξpzq¯

Here, α and β are multi-indices such that α “ pa1, a2q for 0 ď a1 ď k ´ i, 0 ď a2 ď

i ´ 1, α ‰ p0, 0q and α ` β “ pk ´ i, iq. Similarly, α1 and β1 are multi-indices suchthat α1 “ pa11, a

12q for 0 ď a11 ď k´ i, 0 ď a12 ď i´ 1 and α1`β1 “ pk´ i, i´ 1q. The

hypotheses of the Lemma and the induction hypothesis implie that the derivativesof ξ on the right hand side take values in Tfp0qΣ. The fact that JX and ∇ preserve

Cn´1 along Cn´1, and that dlfp0q P Tfp0qΣ, implies the induction step.

Proposition 6.28. For m´ 2p ą l, the universal evaluation map

evX,Σ : M˚8,l,pX,ΣqpJ

mW ; pΣ, lqq Ñ Σ

pf, JXq ÞÑ fp0q

is a submersion.

Proof. We show that for every 0 ď k ď l and pf, JXq P ĂM˚

z pJ pV q; pΣ, kqq

pd evX,Σqpf,JXq : Tpf,JXqĂM˚

z pJ pV q; pΣ, kqq Ñ Tfp0qΣ

pξ, Y q ÞÑ ξp0q

is surjective. By Lemma 6.5 in [CM07],

Tpf,JXqĂM˚

z pJ pV q; pΣ, kqq :“ tpξ, Y q P TfBm,pˆTJXJ pV q |Dfξ `1

2Y pfq ˝ df ˝ j “ 0,

ξp0q P Tfp0qΣ, dkξp0q P Tfp0qΣu


We argue by induction on k. The case k “ 0 is a special case of Proposition 3.4.2in [MS04]. We assume that the claim is true for k ´ 1 and prove it for k.

Take any v P Tfp0qΣ. By induction, there is pξ1, Y1q P Tpf,JXqĂM˚

z pJ pV q; pΣ, k ´1qq such that pd evX,Σqpf,JXqpξ1, Y1q “ v and dk´1ξ1p0q P Tfp0qΣ. Let now ξ PTfBm,p be given by

ξpzq “ ´zk

k!βpzqπC

ˆ

Bk

Bskξ1

˙

p0q

where β : C Ñ r0, 1s is a smooth function that is identically 1 near 0 and hascompact support contained in Czf´1pV q. Writing

pDfξqpzq “ ξspzq ` iξtpzq ` pJXpfpzqq ´ iqξtpzq `Apzqξpzq

we have pDf ξqp0q “ 0 and dk´1pDf ξqp0q “ 0 (this follows the fact that ξs` iξt ” 0

near 0). By Lemma 6.6 in [CM07], there is pξ, Y q P TfBm,p ˆ TJXJ pV q such that

ξp0q “ 0, dkpξqp0q “ 0 and

Df ξ `1

2Y pfq ˝ df ˝ j “ ´Df ξ

Let now ξ2 “ ξ1 ` ξ ` ξ and Y2 “ Y1 ` Y . We have

Dfξ2 `1

2Y2pfq ˝ df ˝ j “ 0

as well as ξ2p0q “ v, dk´1pξ2qp0q P Tfp0qΣ and πC

´

Bk

Bskξ2

¯

p0q “ 0. Lemma 6.27

implies that dkpξ2qp0q P Tfp0qΣ, which completes the proof.

Observe also that by combining this with standard arguments (see, for instance,[MS04, Proposition 3.4.2], which is also used in the proof of Proposition 6.17 above),we obtain the transversality for the evaluation at a point, taking values in X. Thisfinishes the proof of Proposition 6.21.

6.5. Proof of Proposition 6.9. We are now ready to complete the proof of Propo-sition 6.9. To this end, we will show that the transversality problem for a cascadereduces to the already solved transversality problem for chains of pearls. The twokey ingredients of this are the splitting of the linearized operator given by Lemma6.12 and a careful study of the flow-diagonal in Y ˆ Y .

We denote by JY the set of cylindrical, Reeb-flow invariant almost complexstructures on RˆY obtained as lifts of the almost complex structures in JΣ. (Notethat dπΣ : JY Ñ JΣ gives a bijection between these sets.) Let J reg

Y be the set ofalmost complex structures on RˆY that are lifts of the almost complex structuresin J reg

Σ . Recall from Section 2.2, if JW P JW is an almost complex structure on Wthat is of the type we consider, it induces an almost complex structure JΣ P JΣ.The restriction of JW to the cylindrical end of W , JY , is then a translation andReeb-flow invariant almost complex structure on Rˆ Y that has dπΣJY “ JΣdπΣ.

Recall that the biholomorphism ψ : W Ñ XzΣ given in Lemma 2.8 allows us toidentify holomorphic planes in W with holomorphic spheres in X. In the following,we will suppress the distinction when convenient.

Recall also that by the definition of an admissible Hamiltonian (Definition 3.1),for each non-negative integer m, there exists a unique bm so that h1pebmq “ m.Then Ym “ tbmuˆY Ă RˆY is the corresponding Morse–Bott family of 1-periodicHamiltonian orbits that wind m times around the fibre of Y Ñ Σ.


We now define the moduli spaces of Floer cylinders, from which we will extractthe moduli spaces of cascades by imposing the gradient flow-line conditions. First,we define the moduli spaces relevant for the differential connecting two generatorsin R ˆ Y . Then, we will define the moduli spaces relevant for the differentialconnecting to a critical point in W .

Definition 6.29. Let N ě 1, let A1, . . . , AN P H2pΣ;Zq be spherical homologyclasses. Let JY P JY .

Define ĂM˚

H,k,RˆY ;k´,k`ppA1, . . . , AN q; JY q to be a set of tuples of punctured

cylinders prw1, . . . , rwN q with the following properties:

(1) There is a partition of Γ “ Γ1Y ¨ ¨ ¨ YΓN of k augmentation marked pointswith

rwi : Rˆ S1zΓi Ñ Rˆ Yso that rwi is a finite hybrid energy punctured Floer cylinder. For eachzj P Γ, there is a positive integer multiplicity kpzjq. Let wi denote theprojection to Y .

(2) There is an increasing list of N ` 1 multiplicities from k´ to k`:

k´ “ k0 ă k1 ă k2 ă ¨ ¨ ¨ ă kN “ k`

(3) For each i, the cylinder rwi has multiplicities ki and ki´1 at ˘8: wip`8, ¨q PYki , wip´8, ¨q P Yki´1

.(4) The Floer cylinders rwi are simple in the sense that their projections to Σ

are either somewhere injective or constant, if constant, they have at leastone augmentation puncture, and their images are not contained one in theother.

(5) For each i, and for every puncture zj P Γi, the augmentation puncture has

a limit whose multiplicity is given by kpzjq; i.e. limρÑ´8 wipzj ` e2πpρ`iθqq

is a Reeb orbit of multiplicity kpzjq.(6) The projections of the Floer cylinders to Σ represent the homology classes

Ai, i “ 1, . . . , N ; i.e. pπΣpwiqqNi“1 PM˚

kppA1, . . . , AN q, JΣq.

Let B P H2pX;Zq be a spherical homology class, B ‰ 0. Let JW be an almostcomplex structure on W as given by Lemma 2.8, matching JY on the cylindricalend.

Definition 6.30. Define the moduli space

ĂM˚

H,k,W ;k` ppB;A1, . . . , AN q; JW q

to consist of tuples

pv, w1, . . . , wN q

with the properties

(1) The map v : R ˆ S1 Ñ W is a finite energy holomorphic cylinder withremovable singularity at ´8.

(2) There is a partition of Γ “ Γ1Y ¨ ¨ ¨ YΓN of k augmentation marked pointswith

rwi : Rˆ S1zΓi Ñ Rˆ Yso that each rwi is a finite hybrid energy punctured Floer cylinder. Foreach zj P Γ, there is a positive integer multiplicity kpzjq. Denote by wi theprojection of rwi to Y .


(3) There is an increasing list of N ` 1 multiplicities:

k0 ă k1 ă k2 ă ¨ ¨ ¨ ă kN “ k`

(4) For each i, and for every puncture zj P Γi, the augmentation puncture has

a limit whose multiplicity is given by kpzjq; i.e. limρÑ´8 wipzj ` e2πpρ`iθqq

is a Reeb orbit of multiplicity kpzjq.(5) The Floer cylinders wi are simple, in the strong sense that the projections

to Σ are somewhere injective or constant, and have images not containedone in the other. The cylinder v is somewhere injective in W .

(6) The projections of the Floer cylinders to Σ represent the homology classesAi, i “ 1, . . . , N ; i.e. πΣpwiqq

Ni“1 PM˚

kppB;A1, . . . , AN q, JW q.(7) After identifying v with a holomorphic sphere in X, v represents the ho-

mology class B P H2pX;Zq.(8) The cylinder rw1 has multiplicity k1 at `8 and w1p`8, ¨q P Yk1

. At ´8,rw1 converges to a Reeb orbit in t´8uˆY . This Reeb orbit has multiplicityk0.

(9) For each i ě 2, the cylinder rwi has multiplicities ki and ki´1 at ˘8:wip`8, ¨q P Yki , wip´8, ¨q P Yki´1

.(10) The cylinder v converges at `8 to a Reeb orbit of multiplicity k0.

Observe that these moduli spaces are non-empty only if for each i “ 1, . . . , N ,

KωpAiq “ ki ´ ki´1 ´ÿ

zPΓi

kpzq.

Furthermore, for ĂMH,k,W , we must also have

B ‚ Σ “ KωpBq “ k0.

Note also that these moduli spaces have a large number of connected components,where different components have different partitions of Γ or different intermediatemultiplicities.

Identifying holomorphic spheres in X with finite energy JW -planes in W , weconsider also the moduli space of holomorphic planes M˚

0 ppB1, . . . , Bkq; JW q as inDefinition 6.16.

The space ĂM˚

H,k,RˆY ppA1, . . . , AN q; JY q consists of N -tuples of somewhere in-

jective punctured Floer cylinders in RˆY . Similarly, ĂM˚

H,k,W consist of N -tuples ofpunctured Floer cylinders in RˆY together with a holomorphic plane in W (whichwe can therefore also interpret as a holomorphic sphere in X). The cylinders andthe eventual plane have asymptotics with matching multiplicities, but are otherwise

unconstrained. These two moduli spaces, ĂMH,k,RˆY and ĂMH,k,W fail to be simplesplit Floer cascades (as in Definitions 4.1, 4.3) in two ways: they are missing thegradient trajectory constraints on their asymptotic evaluation maps, and they aremissing their augmentation planes. In order to impose these conditions, we willneed to study these evaluation maps and establish their transversality.

Proposition 6.31. For JY P J regY , ĂM

˚

H,k,RˆY ppA1, . . . , AN q; JY q is a manifold ofdimension

Np2n´ 1q `Nÿ

i“i

2 xc1pTΣq, Aiy ` 2k


For JW P J regW , ĂM

˚

H,k,W ppB;A1, . . . , AN q; JW q is a manifold of dimension

Np2n´ 1q ` 2n` 1`Nÿ

i“i

2 xc1pTΣq, Aiy ` 2pxc1pTXq, By ´B ‚ Σq ` 2k.

Proof. Consider first the case of cylinders in Rˆ Y . Let

pw1, . . . , wN q P ĂM˚

H,k,RˆY ppA1, . . . , AN q; JY q .

Recall from Lemma 6.17 that for JΣ P J regΣ , we have transversality for DΣ

ui foreach sphere ui “ πΣpwiq.

Let δ ą 0 be sufficiently small. For each i “ 1, . . . , N , by Lemma 6.14, DCwi

is

surjective when considered onW 1,p,´δ (with exponential growth), and has Fredholm

index 1. The operator considered instead on the space W 1,p,δV , with V´8 “ V`8 “

iR and VP “ C for any puncture P on the domain of wi, has the same kernel andcokernel by Lemma A.10. Thus, the operator, acting on sections free to move inthe Morse–Bott family of orbits is surjective and has index 1.

Since the operator Dwi is upper triangular from Lemma 6.12, and its diagonalcomponents are both surjective, the operator is surjective. Since the Fredholmindex is the sum of these, each component wi contributes an index of 1` 2n´ 2`2 xc1pTΣq, Aiy ` 2ki “ 2n ´ 1 ` 2 xc1pTΣq, Aiy ` 2ki, where ki is the number ofpunctures.

We now consider the case of a collection

pv, w1, . . . , wN q P ĂM˚

H,k,W ppB;A1, . . . , AN q; JW q .

The same consideration as previously gives that w2, . . . , wN are transverse andeach contributes an index of 2n´ 1` 2 xc1pTΣq, Aiy ` 2ki, where ki is the numberof punctures. For the component w1, again applying Lemma 6.12 and applyingLemma 6.14 in the case where the ´8 end of the cylinder converges to a Reeborbit at t´8uˆY , we obtain that the vertical Fredholm operator is surjective andhas index 2. The linearized Floer operator at w1 is then surjective and has index2n ` 2 xc1pTΣq, A1y ` 2k1. By Lemma 2.8, the plane v can be identified with asphere in X with an order of contact l “ B ‚ Σ with Σ. Its Fredholm index is2n` 2pxc1pTXq, By ´ lq. The total Fredholm index is therefore

pN ´ 1qp2n´ 1q ` 2n` 2n`Nÿ

i“i

2 xc1pTΣq, Aiy ` 2pxc1pTXq, By ´B ‚ Σq ` 2k.

For both cases, the result now follows from the implicit function theorem.

It now suffices to prove the transversality of evaluation maps to the products ofstable/unstable manifolds and flow diagonals, and also transversality of the aug-mentation evaluation maps, in order to obtain the constraints coming from pseudo-gradient flow lines. Indeed, let prw1, . . . , rwN q be a collection of N cylinders inĂM˚

H,k,RˆY ;k´,k` ppA1, . . . , AN q; JY q. Write each of the rwi : R ˆ S1 Ñ R ˆ Y as a


pair rwi “ pbi, wiq. We then have asymptotic evaluation maps

revY : ĂM˚

H,k,RˆY ;k´,k` ppA1, . . . , AN q; JY q Ñ Y 2N

prw1, . . . , rwN q ÞÑ

ˆ

limsÑ´8

w1ps, 1q, limsÑ`8

w1ps, 1q, . . . , limsÑ´8

wN ps, 1q, limsÑ`8

wN ps, 1q

˙

If pv, rw1, . . . , rwN q P ĂM˚

H,k,W ppB;A1, . . . , AN q; JW q, we have

revW,Y : ĂM˚

H,k,W ;k` ppB;A1, . . . , AN q; JW q ÑW ˆ Y 2N`1

pv, rw1, . . . , rwN q ÞÑ

ˆ

vp0q, limrÑ`8

πY vpr ` i0q, limsÑ´8

w1ps, 1q, . . . , limsÑ`8

wN ps, 1q

˙

This map is C1 smooth, exploiting the asymptotic expansion of the Floer cylindernear its asymptotic limit, as described by [Sie08]. This is proved in [FS17].

We also have augmentation evaluation maps. For each puncture z0 P Γ, thereexists an index i P t1, . . . , Nu so that the augmentation puncture z0 is a puncturein the domain of wi. For this augmentation puncture, we have the asymptoticevaluation map wi ÞÑ limzÑz0 πΣpwipzqq P Σ. Combining all of these evaluationmaps over all punctures in Γ, we obtain

revaΣ : ĂM˚

H,k,RˆY ;k´,k` ppA1, . . . , AN q; JY q Ñ Σk

revaΣ : ĂM˚

H,k,W ;k` ppB;A1, . . . , AN q; JW q Ñ Σk.

Note that this map is smooth by [Wen16b, Proposition 3.15].Define the flow diagonal in Y ˆ Y to be

r∆fY :“

px, yq P pY zCritpfY qq2 : Dt ą 0 s.t. ϕtZY pxq “ y

(

where CritpfY q is the set of critical points of fY .Let p, q P Y be critical points of fY and let Wu

Y ppq, WsY pqq be the unstable/stable

manifolds of p, q with respect to the anti-gradient-like vector fields ´ZY .We may now describe the moduli space of simple split Floer cylinders from qk´ to

pk` as the unions of the fibre products of these moduli spaces under the asymptoticevaluation maps and augmentation evaluation maps. For notational convenience,we write

rev : ĂM˚

H,k,RˆY ;k´,k` ppA1, . . . , AN q; JY q ˆM˚0 ppB1, . . . , Bkq; JW q Ñ Y 2N ˆ Σk ˆ Σk

pw,vq ÞÑ p revY pwq, revaΣpwq, evaΣpvqq .

Write ∆Σk Ă Σk ˆ Σk to denote the diagonal Σk. Then, define

ĂM˚

Hpqk´ , pk` ; pA1, . . . , AN q, pB1, . . . , Bkq; JW q “ rev´1

ˆ

W sY pqq ˆ

´

∆fY

¯N´1

ˆWuY ppq ˆ∆Σk

˙

.

From this, we have

ĂM˚

H,N pqk´ , pk` ; JY , JW q “ď

pA1,...,AN q

ď

kě0

ď

pB1,...,Bkq

ĂM˚

Hpqk´ , pk` ; pA1, . . . , AN q, pB1, . . . , Bkq; JW q(6.5)


Similarly, if x PW is a critical point of fW , and letting WuW pxq be the descending

manifold of x in W for the gradient-like vector field ´ZW , we similarly define

rev : ĂM˚

H,k,W ;k` ppB;A1, . . . , AN q; JW q ˆM˚0 ppB1, . . . , Bkq; JW q ÑW ˆ Y 2N`1 ˆ Σk ˆ Σk

ppv, wq,vq ÞÑ p revW,Y pv,wq, revaΣpwq, evaΣpvqq .

Then, define

ĂM˚

Hpx, pk` ; pB;A1, . . . , AN q,pB1, . . . , Bkq; JW q “

rev´1

ˆ

WuW pxq ˆ

r∆ˆ

´

r∆fY

¯N´1

ˆWuY ppq ˆ∆Σk

˙

.

Finally, we obtain

ĂM˚

H,N px, pk` ; JW q “ď

pB;A1,...,AN q

ď

kě0

ď

pB1,...,Bkq

ĂM˚

Hpx, pk` ; pB;A1, . . . , AN q, pB1, . . . , Bkq; JW q(6.6)

In order to establish transversality for our moduli spaces, it then becomes nec-essary to show transversality of the evaluation maps to these products of descend-ing/ascending manifolds, diagonals and flow diagonals. Recall that the spaces ofalmost complex structures JY P J reg

Y , JW P JregW are given from Proposition 6.26.

Proposition 6.32. Let JW P JregW and let JY P J regY be the induced almost complex

structure on Rˆ Y .Let q, p denote critical points of fY , and let x be a critical point of fW in W .

Let k` and k´ be non-negative multiplicities, k` ą k´.Let A1, . . . , AN be spherical homology classes in Σ, let B,B1, . . . , Bk be spherical

homology classes in X, k ě 0.Let ∆ Ă Y ˆ Y and ∆Σk Ă Σk ˆ Σk be the diagonals.Then,

(1) the evaluation map

revYˆ revaΣˆevaΣ : : ĂM˚

H,k,RˆY ;k´,k` ppA1, . . . , AN q; JY qˆM˚0 ppB1, . . . , Bkq; JW q Ñ Y 2NˆΣkˆΣk

is transverse to the submanifold

W sY pqq ˆ

´

∆fY

¯N´1

ˆWuY ppq ˆ∆Σk

(2) the evaluation map

revW,Yˆ revaΣˆevaΣ : ĂM˚

H,k,W ;k` ppB;A1, . . . , AN q; JW qˆM˚0 ppB1, . . . , Bkq; JW q ÑWˆY 2N`1ˆΣkˆΣk

is transverse to the submanifold

WuW pxq ˆ ∆ˆ

´

∆fY

¯N´1

ˆWuY ppq ˆ∆Σk .

In order to prove this proposition, we will need a better description of the rela-tionship between the moduli spaces of spheres in Σ, and the moduli spaces of Floercylinders in Rˆ Y (or in W ).


Lemma 6.33. The maps

πMΣ : ĂM

˚

H,k,RˆY ;k´,k` ppA1, . . . , AN q; JY q ÑM˚k,ΣppA1, . . . , AN q; JΣq

πMΣ : ĂM

˚

H,k,W ;k` ppB;A1, . . . , AN q; JW q ÑM˚k,X,ΣppB;A1, . . . , AN q; JW q

induced by πΣ : R ˆ Y Ñ Σ are submersions. The fibres have a locally free pS1qN

torus action by constant rotation by the action of the Reeb vector field.

Proof. We will study the case of

πMΣ : ĂM

˚

H,k,RˆY ppA1, . . . , AN q; JY q ÑM˚k,ΣppA1, . . . , AN q; JΣq

in detail. The case with a sphere in X follows by the same argument with a smallnotational change. It also suffices to consider the case with N “ 1, since modulispaces with more spheres are open subsets of products of these.

Suppose πΣpwq “ w with w P ĂMH,k,RˆY ;k´,k`pA; JY q and w P M˚k,ΣpA; JΣq.

Recall the splitting given in Lemma 6.12, which splits the linearized Floer operatorat w as

Dw “

ˆ

DCw M

0 DΣw

˙

By definition, TwM˚k,ΣpAq “ kerDΣ

w and TwĂMH,k,RˆY ;k´,k`pA; JY q “ kerDw.

By Lemma 6.14, DCw is surjective. It follows then that any section ζ0 of w˚TΣ that

is in the kernel of DΣw can be lifted to a section pζ1, ζ0q of w˚TΣ – pR‘RRq‘w˚TΣ

that is in the kernel of Dw.Notice now that dπΣpζ1, ζ0q “ ζ0, establishing that the evaluation map is a

submersion.Also observe that S1 acts on the curve w by the Reeb flow. From the Reeb-

invariance of JY and of the admissible Hamiltonian, this is then clearly in the fibre.If A ‰ 0, this new solution is geometrically distinct from the previous solution,at least for small rotation. If A “ 0, w has image contained in a fibre R ˆ Y Ñ

Σ, and thus the Reeb rotation has the same image, but the parametrization isdifferent. In this case also, we have then a different curve, at least for small rotationparameter.

We now consider all Floer cylinders that project to the same unparametrizedsphere in Σ. This will also be useful elsewhere in the paper. Let v “ pb, vq : R ˆS1zΓ Ñ R ˆ Y be a finite hybrid energy solution to Floer’s equation (4.3). Thereis an action of S1 ˆ S1 on the space of such solutions, with each circle action byrotation on the domain and on the target, respectively:

pθ1, θ2q.v “ vpθ1,θ2q

where

vpθ1,θ2qps, tq “ pbps, t` θ1q, φθ2R ˝ vps, t` θ1qq.

Here, φθR denotes the Reeb flow for time θ. If we identify the simple Reeb orbitsunderlying x˘ptq “ limsÑ˘8 vps, tq with S1, then the following map keeps track ofthe asymptotic effect of the action:

rotv : S1 ˆ S1 Ñ S1 ˆ S1

pθ1, θ2q ÞÑ

ˆ

limsÑ´8

vps, θ1q ` θ2, limsÑ`8

vps, θ1q ` θ2

˙

.


If k˘ are the multiplicities of the periodic orbits x˘ptq, then, rotv is the linear maprepresented in matrix form as

(6.7) rotv “

ˆ

k´ 1k` 1

˙

.

Suppose now that w : R ˆ S1 Ñ W is a non-constant JW -holomorphic cylinderwith removable singularity at ´8 (such as v2 in Definition 4.14). There is anS1-action on such curves, by rotation on the domain:

θ.wps, tq “ wθps, tq “ wps, t` θq.

Identifying the simple Reeb orbit underlying γptq “ limsÑ8 πY pwps, tqq with S1,we have

rotv : S1 Ñ S1

θ ÞÑ limsÑ`8

wps, θq.

If the multiplicity of the periodic Reeb orbit γptq is k, then rotw is multiplicationby k.

Lemma 6.34. Let A :“ rws P H2pΣ;Zq, where w : CP 1 Ñ Σ is the continuousextension of πΣ ˝ v. Assume that either A ‰ 0 or Γ ‰ H. Then, k` ą k´.

Proof. Denote by w˚Y the pullback under w of the S1-bundle Y Ñ Σ. The mapv gives a section s of w˚Y , defined in the complement of Γ Y t0,8u. By [BT82,Theorem 11.16], the Euler number

ş

CP 1 epw˚Y q (where e is the Euler class) is the

sum of the local degrees of the section s at the points in ΓY t0,8u.Denote the multiplicities of the periodic XH -orbits x˘ptq “ limsÑ˘8 vps, tq by

k˘, respectively, and denote the multiplicities of the asymptotic Reeb orbits at thepunctures z1, . . . , zm P Γ by k1, . . . , km, respectively. The positive integers k˘ andki are the absolute values of the degrees of s at the respective points. Taking signsinto account, we get

ż

CP 1

epw˚Y q “ k` ´ k´ ´ k1 ´ . . .´ km.

Butż

CP 1

epw˚Y q “

ż

CP 1

w˚epY Ñ Σq “

ż

CP 1

w˚epNΣq

where NΣ is the normal bundle to Σ in Y . Now, epNΣq “ s˚ThpNΣq, wheres : Σ Ñ NΣ is the zero section and ThpNΣq is the Thom class of NΣ [BT82,Proposition 6.41]. If j : NΣ Ñ X is a tubular neighborhood, then j˚ThpNΣq “PDprΣsq “ rKωs P H2pX;Rq [BT82, Equation (6.23)]. So,

ż

CP 1

w˚epNΣq “

ż

CP 1

w˚s˚ThpNΣq “

ż

CP 1

w˚ι˚j˚ThpNΣq “

“

ż

CP 1

w˚ι˚Kω “ KωpAq ě 0

since K ą 0 and w is a JΣ-holomorphic sphere. We conclude that

k` ´ k´ ´ k1 ´ . . .´ km “ KωpAq ě 0.

If A ‰ 0, we get a strict inequality. If A “ 0, we get an equality, but the assumptionsof the Lemma imply that

řmi“1 ki ą 0. In either case, we get k` ą k´, as wanted.


Recall that the gradient-like vector field ZY has the property that dπΣZY “ ZΣ.Also recall that we may use the contact form α as a connection to lift vectorfields from Σ to vector fields on Y , tangent to ξ. If V is a vector field on Σ, we

write π˚ΣV :“ rV to be the vector field on Y uniquely determined by the conditions

αpV q “ 0, dπΣrV “ V . This extends as well to lifting vector fields on Σ ˆ Σ to

vector fields on Y ˆ Y .

Lemma 6.35. The flow diagonal in Y satisfies

πΣpr∆fY q Ă ∆fΣ Y tpp, pq | p P CritpfΣqu.

Let px, yq P r∆fY and x “ πΣpxq, y “ πΣpyq. Let t so that y “ ϕtZY pxq. Then, ifx “ y, we have x P CritpfΣq and(6.8)

Tpx,yq r∆fY “ tpaR` v, bR` π˚Σdϕ

tZΣdπΣvq P TY ‘ TY | a, b P R and αpvq “ 0u.

If x ‰ y, then px, yq P ∆fΣ . Then, there exists a positive g “ gpx, yq ą 0 so that

(6.9) Tpx,yq r∆fY “ RpR, gRq ‘H

where dπΣ|H : H Ñ T∆fΣ induces a linear isomorphism.

Proof. Observe first that if x “ πΣx, we have

πΣϕtZY pxq “ ϕtZΣ

pxq.

This gives dπΣ dϕtZYpxq “ dϕtZΣ

dπΣpxq. From this, it follows that dϕtZY pxqR is a

multiple of the Reeb vector field. Observe also that ϕtZΣand ϕtZY are both orienta-

tion preserving diffeomorphisms for all t. We therefore obtain that if y “ ϕtZΣpxq,

ϕtZY induces a diffeomorphism between the fibres π´1Σ pxq Ñ π´1

Σ pyq. Additionally,

we must have then that dϕtZY pxqR is a positive multiple of the Reeb vector field.

Let gpx, yq ą 0 such that dϕtZY pxqR “ gpx, yqR.

In general, if y “ ϕtZY pxq, we have

(6.10) Tpx,yq r∆fY “ tpv, dϕtZY pxqv ` cZY pyqq | v P TxY, c P Ru.

Consider first the case of x “ y. Then, both x and y are in the same fibre ofY Ñ Σ. By definition of the flow diagonal, there exists t ą 0 so that ϕtZY pxq “ y,and hence ZY is vertical, ZΣpxq “ 0. It follows x P CritfΣ

. From this, it now

follows that πΣpr∆fY q Ă ∆fΣY tpp, pq | p P CritpfΣqu.

We now consider the consequences of Equation (6.10) in this case of x “ y.Any v P TxY may be written as v0 ` aR where αpv0q “ 0. Furthermore, sincex “ y P CritpfΣq, and by definition, neither x nor y are critical points of fY , weobtain that ZY pyq is a non-zero multiple of the Reeb vector field. Equation (6.8)now follows from the fact that dπΣϕ

tZYpxq “ dϕtZΣ

pxqdπΣ.

We now consider when x ‰ y. Let H “ tpv, dϕtZY pxqv ` cZY pyq |αpvq “ 0u.Then,

dφΣpHq “ tpv, dϕtZΣpxqv ` cZΣq | v P TxΣu “ T∆fΣ .

By assumption, y is not a critical point of fΣ, so dπΣ induces an isomorphism. The

decomposition of T r∆fY now follows immediately from the definition of g and fromEquation (6.10).


Proof of Proposition 6.32. We consider first the case of

revY : ĂM˚

H,k,RˆY ;k´,k` ppA1, . . . , AN q; JY q Ñ Y 2N

Suppose that v “ pv1, . . . , vN q P ĂM˚

H,k,RˆY ;k´,k` ppA1, . . . , AN q; JY q. For each

i “ 1, . . . N , let yi´1 “ vip´8, 0q P Y and xi “ vip`8, 0q P Y , with

y0 PWsY pqq, xN PW

uY ppq

pxi, yiq P r∆fY for 1 ď i ď N ´ 1

Let vi “ πSigmapviq and xi “ πΣpxiq, yi “ πΣpyiq. Then, it follows that

y0 PWsΣpqq, xN PW

uΣppq

pxi, yiq P ∆fΣ Y tpp, pq | p P CritpfΣqu for 1 ď i ď N ´ 1.

Let S Ă Σ2N´2 be the appropriate product of a number of copies of ∆fΣ and oftpp, pq | p P CritpfΣqu. By Proposition 6.26, the evaluation map on MΣppA1, . . . , AN q; JY qis transverse to S.

Then,

TS Ă d evΣ

´

Ty0WsY pqq ˆ Tpx1,y1q

r∆fY ˆ ¨ ¨ ¨ ˆ TpxN´1,yN´1qr∆fY ˆ TxNW

uY ppq

¯

.

It suffices therefore to obtain transversality in the vertical direction. Notice thatby rotating by the action of the Reeb vector field on vi, we obtain that the imageof dev contains the subspace

tpa1R, a1R, a2R, a2R, . . . , aNR, aNRq | pa1, . . . , aN q P RNu Ă pTY q2N .

In the case of the chain of pearls in Σ, all of the spheres vi, i “ 1, . . . , N musteither be non-constant or have a non-trivial collection of augmentation punctures.Then, by Lemma 6.34, each punctured cylinder vi has different multiplicities k`i , k

´i

at ˘8, and thus the action of rotating the domain marker gives that the imageof devY pviq contains pk´R, k`Rq P Tyi´1Y ‘ TxiY . While this holds for each i “1, . . . , N , we only require such a vector for one cylinder. Then, by taking this in thecase of i “ 1, we see that the following N`1 vertical vectors in pRRq2N Ă TY 2N arein the image of the linearized evaluation map (the first two obtained by combiningthe two Reeb actions on v1, the remainder by the Reeb action on vi, i ě 2):

pR, 0, 0, . . . , 0q,

p0, R, 0, . . . , 0q,

p0, 0, R,R, 0, 0, . . . , 0q,

p0, 0, 0, 0, R,R, 0, . . . , 0q,

. . .

p0, 0, . . . , 0, R,Rq.

By Lemma 6.23, the tangent space Tpxi,yiqr∆fY contains at least the vertical

vector pR, giRq, where gi :“ gpxi, yiq ą 0, for each 1 ď i ď N ´ 1. In the vertical


direction, this then contains the following N ´ 1 vectors:

p0, R, g1R, 0, . . . , 0q,

p0, 0, 0, R, g2R, 0, . . . , 0q

. . .

p0, . . . , 0, R, gN´1R, 0q.

We now observe that this collection of 2N vectors spans pRRq2N . This now

establishes the result in the case of ĂM˚

H,k,RˆY ;k´,k` ppA1, . . . , AN q; JW q.We now consider the case of

revW,Y : ĂM˚

H,k,W ;k` ppB;A1, . . . , AN q; JW q ÑW ˆ Y 2N`1.

We will show this evaluation map is transverse to

S :“WuW pxq ˆ ∆ˆ

´

∆fY

¯N´1

ˆWuY ppq.

As before, it suffices to show transversality in a vertical direction, since, byProposition 6.26, the projections to X, Σ are transverse. Let S ĂW ˆΣˆΣ2N beof the form S “Wu

W pxqˆ∆ˆS1ˆWuΣppq, where S1 Ă Σ2N´2 is a product of some

number of ∆fΣand of tpp, pq | p P CritpfΣqu so that TS Ă TdπΣpSq. Proposition

6.26 gives transversality to S.Notice that the tangent space T S contains at least the following vertical vectors

(we put 0 in the first component since TW has no vertical direction):

p0, R,R, 0, 0, . . . , 0q

p0, 0, 0, R, g1R, 0, . . . , 0q

. . .

p0, . . . , 0, R, gN´1R, 0q.

Let pw, v1, . . . , vN q P ĂM˚

H,k,W ;k` . The plane w converges to a Reeb orbit ofmultiplicity l “ B ‚ Σ. Observe that domain rotation on the plane w then givesthat p0, lR, 0, . . . , 0q P TW ‘ TY ‘ TY 2N is in the image of d revW,Y .

As before, the Reeb rotation on each of the punctured cylinders v1, . . . , vN givesthat the following vertical vectors are in the image of d revW,Y :

p0, 0, R,R, 0, 0, . . . , 0, 0q

p0, 0, 0, 0, R,R, . . . , 0, 0q

p0, 0, 0, 0, . . . , 0, R,Rq.

We notice then that these vectors span 0‘ pRRq2N´1, so it follows that the evalu-

ation map is transverse to S.Finally, the transversality of the evaluation maps at augmentation punctures

comes from the fact that the augmentation evaluation maps

revaΣ ˆ evaΣ : ĂM˚

H,k,RˆY ;k´,k` ppA1, . . . , AN q; JY q ˆM˚0 ppB1, . . . , Bkq; JW q Ñ Σk ˆ Σk

revaΣ ˆ evaΣ : ĂM˚

H,k,W ;k` ppB;A1, . . . , AN q; JW q ˆM˚0 ppB1, . . . , Bkq; JW q Ñ Σk ˆ Σk

factor through the evaluation maps eva : MppA1, . . . , AN q; JY qq Ñ Σk and eva : MppB;A1, . . . , AN q; JW q.The required transversality for these maps is given by Proposition 6.26. Further-more, these evaluation maps are invariant under the domain rotation and Reeb


rotations used to obtain transversality in the vertical directions, so the transversal-ity follows immediately.

Part 3. Index calculations and ruling out configurations

7. Fredholm index for Floer cylinders in the symplectization

7.1. Conley–Zehnder indices. In this section, we revisit the gradings of thegenerators of the Floer complex, as given in Equations (3.4) and (3.5), and relatethis to the relevant Fredholm indices, as given by Proposition 6.9.

To this end, we compute the Conley–Zehnder indices of closed Reeb orbits in Yand of closed Hamiltonian orbits in Rˆ Y and in W .

We recall that the Conley–Zehnder index was originally defined for non-degenerateperiodic orbits, and thus we need a modification to take into account the Morse–Bott nature of our situation. Furthermore, the Conley–Zehnder index depends ona choice of trivialization, either of the contact structure ξ over the orbit in the caseof a Reeb orbit, or of the tangent bundle TW over the orbit in the Hamiltoniancase. We refer the reader to Appendix A on Riemann–Roch for punctured holomor-phic curves for a very brief summary of the relevant background. We also refer thereader to Abreu and Macarini’s exposition of Conley–Zehnder and Robbin–Salamonindices in [AM16, Section 3], and also to Gutt [Gut14].

Suppose that we have two different trivializations of TW over a periodic orbit.The transition from one to the other gives a loop of matrices in Upnq. The differencein corresponding Conley–Zehnder indices is given by twice the winding numberof the determinant of this loop. In particular, then, the Conley–Zehnder indexassociated to an orbit depends only on a trivialization of ΛnCTW .

In the situation we consider, with Y obtained as a principal Up1q-bundle overΣ, there are three natural trivializations of T pR ˆ Y q that are useful to consider.The first is the constant trivialization, which uses the identification T pR ˆ Y q –R‘RR‘ ξ and which is so that the linearized Reeb flow is the identity map on ξ.This has the advantage of corresponding to our construction of Floer solutions interms of holomorphic spheres in Σ and of sections of the corresponding pull-backbundle. The second trivialization is closely related to the first, and involves seeingeach Reeb orbit in Y as the boundary of the corresponding disk fibre in the normalbundle to Σ in X. This second version of the Conley–Zehnder index is most naturalfrom the perspective of closed spheres in X. The final trivialization corresponds tothe fractional SFT grading [EGH00] as used, for instance in [BO09b]. We use thispoint of view to define the grading of our complex. This fractional grading existsbecause our Reeb orbits are all torsion elements of H1pY ;Zq. We will also relateour grading to the gradings introduced by Seidel in the case c1pTW q P H2pW ;Zqvanishes and by McLean in the case c1pTW q P H2pW ;Zq is torsion [Sei08, Section(3a); McL16, Section (4.1)]

Recall that the 1-periodic Hamiltonian orbits occur at levels tru ˆ Y , where rverifies h1perq P Z.

7.1.1. Constant trivializations along a fibre. We recall that we denote the bundlemap πΣ : Y Ñ Σ. The contact structure ξ is a horizontal distribution, and the Reebvector field generates the S1 action on Y . Then, by definition, TY “ RR ‘ ξ –RR‘ π˚ΣTΣ.


While each cover of a fibre of Y is a closed Reeb orbit, our chain complexadditionally keeps track of the Morse function fΣ : Σ Ñ R. This Morse functionrepresents infinitesimal perturbation data to perturb our contact form to a nearbynon-degenerate one. (See [Bou02,BEH`03].) In our current calculation, we will notmake any perturbations, but use this Morse function to define our moduli spacesof cylinders with cascades.

Let p P Σ be a point. Then, for a fixed trivialization of TpΣ and any point

q P π´1Σ ppq Ă Y in the corresponding fibre, we obtain a trivialization of ξ|q – TpΣ.

Furthermore, this trivialization is invariant under the linearized Reeb flow. We willrefer to this trivialization as the constant trivialization over the orbit. In particular,the linearized Reeb flow, with respect to this trivialization, is the (constant) identitymap.

The collection of unparametrized periodic Reeb orbits in Y is parametrized bya point p in Σ and an integer k ą 0 indicating how many times the orbit coversthe fiber of πΣ : Y Ñ Σ over p. Let p P CritpfΣq and take k ą 0. We denote thecorresponding unparametrized Reeb orbit by pk. This has an associated Conley–Zehnder index, as in Definition A.9:

CZ0ξppkq “Mppq ` 1´ n,

where Mppq denotes the Morse index of p for the Morse function fΣ : Σ Ñ R. Thesuperscript 0 indicates that the index is calculated with respect to the constanttrivialization. Note that the contribution 1 ´ n comes from the last statement inCorollary A.4.

Hamiltonian orbits of period 1 in Rˆ Y are contained in level sets tru ˆ Y . Forfixed such r, the corresponding Hamiltonian orbits cover fibres of πΣ : Y Ñ Σ withmultiplicity given by h1perq “ k P Z`. For each critical point p of fΣ : Σ Ñ R, thereare two distinguished Hamiltonian orbits of multiplicity k. They correspond to thetwo critical points of the restriction of fY to the circle π´1

Σ ppq, where fY : Y Ñ Ris the fixed Morse function that lifts fΣ. These two critical points are denoted byby pp and qp and the corresponding Hamiltionian orbits are denoted by ppk and qpk.The latter are generators of the chain complex 3.2.

Given a periodic Reeb orbit, we extend the previously defined constant trivi-alization of ξ to a trivialization of T pR ˆ Y q over the corresponding Hamiltonianorbits, by noticing that T pR ˆ Y q “ R ‘ RR ‘ ξ is obtained by stabilizing ξ by atrivialized complex line bundle.

With respect to this trivialization, the linearized Hamiltonian flow is no longerthe identity map, but splits a non-trivial factor on R ‘ RR and the identity onξ. The non-trivial factor corresponds to the following linearized flow differentialoperator (asymptotic operator),

´id

dt´

ˆ

h2perq er 00 0

˙

.

In the language of Appendix A, the Conley–Zehnder index of the perturbation ofthis asymptoptic operator in R2 by δ ą 0 is 0, by Corollary A.4. The Conley–Zehnder indices of a Reeb orbit and of the corresponding Hamiltonian orbits in thistrivialization therefore agree.

Denote by rp either qp or pp and write the Morse index of rp for fY as Mprpq “Mppq ` iprpq (so ipppq “ 1 and ipqpq “ 0). Using again Definition A.9, we obtain that


with respect to the trivialization we fixed for T pRˆ Y q

CZ0Hprpkq “ CZ0

ξppkq ` iprpq “ Mprpq ` 1´ n.

Notice that with respect to this constant trivialization, the Conley–Zehnder in-dex does not depend on the covering multiplicity of the orbit.

Finally, for a constant orbit x P CritpfW q, the constant trivialization gives

CZ0Hpxq “ ´n` p2n´Mpxqq “ n´Mpxq

where Mpxq is the Morse index of x as a critical point of fW .

7.1.2. Orbits capped by the fibres of NΣ. We take now a closely related trivial-ization: consider Y as a submanifold of X, arising as the concave boundary of afixed r radius normal disk bundle NrΣ Ă NΣ Ñ Σ. Observe that we then haveTX|NrΣ – NΣ‘ ξ (where we extend ξ over Σ Ă NrΣ as TΣ). A simple Reeb orbitin Y bounds a fibre of this disk bundle NrΣ Ñ Σ. The constant trivialization ofthe contact structure ξ extends across this disk fibre. The restriction NΣ|Y is atrivial complex line bundle. The computation of CZ0 uses the trivialization of thisline bundle given by the radial (Liouville) vector field and Reeb vector fields on Y .The trivialization of NΣ|Y that extends to all of NrΣ, however, has winding 1 withrespect to that one.

From this we obtain, for an orbit with multiplicity k, and considering the as-ymptotic boundary condition given by the descending manifold of p P CritpfΣq, thefollowing indices for Reeb and Hamiltonian orbits, with respect to the trivializationcoming from the inclusion Y Ă X:

(7.1) CZXξ ppkq “ CZ0ξppkq “Mppq ` 1´ n

and

(7.2) CZXHprpkq “ CZ0Hprpkq ´ 2k “ Mprpq ` 1´ n´ 2k.

7.1.3. The grading. In order to construct the grading we defined in Equations (3.4)and (3.5), we consider a third trivialization of ξ and TW in the case of a Reeborbit/Hamiltonian orbit that is contractible in W . We will later consider moregeneral orbits.

As discussed above, in order to define a Conley–Zehnder index, it suffices totrivialize the line bundle

L– ΛnCTX

over the orbit. To compare two Conley–Zehnder indices coming from two differenttrivializations of L over an orbit, it suffices to understand the relative winding ofthese two trivializations.

Suppose that γl is a Reeb orbit of Y which is a l-fold cover of a simple orbit γ(i.e. of a fibre of Y Ñ Σ). Suppose that γl is contractible in W . Denote by 9B adisk in W whose boundary is γl. Note that γl is also the boundary of a l-fold coverof a fibre of the normal bundle to Σ, as described in the previous Section. Thiscover of a fibre can be concatenated with 9B to produce a spherical homology classB P HS

2 pXq such that B ‚Σ “ l ą 0. Note that any B P HS2 pXq such that B ‚Σ “ l

gives rise to a disk 9B bounding γl. The complex line bundle L| 9B is trivial, since 9Bis a disk. This induces a trivialization of L over γl, which can be identified with atrivialization of Lbl over γ.


The orbit γ bounds the fibre of the normal bundle to Σ, which induces a trivial-ization of L over γ. This is precisely the trivialization used in the previous section.The l-fold cover of the fibre of the normal bundle induces a trivialization over γl.Now, observe that the trivialization in the previous paragraph has a winding withrespect to this second given by xc1pLq, By, since this represents the obstruction to

extending the trivialization of L over 9B to all of B. Recall that c1pLq “ c1pTXq.Suppose now that γ is the Reeb orbit corresponding to p P CritpfΣq. By (7.2),

the Conley–Zehnder index with respect to the new trivialization is

(7.3) CZWH prplq “ Mprpq ` 1´ n´ 2l ` 2xc1pTXq, By

Using the fact that KωpBq “ l and the spherical monotonicity of X, we have:

(7.4) CZWH prplq “ Mprpq ` 1´ n` 2pτX ´KqωpBq.

Now, for any k ą 0, we define a fractional grading

(7.5)

|rpk| “ Mprpq ` 1´ n` 2pτX ´KqωpBqk

l

“ Mprpq ` 1´ n`2pτX ´KqωpBq

KωpBqk

“ Mprpq ` 1´ n` 2

ˆ

τX ´K

K

˙

k

As before, for any critical point x P CritpfW q, we have the grading

(7.6) |x| “ n´Mpxq

Finally, it will be convenient to introduce an index similar to the SFT gradingfor the Reeb orbits that arise as the limits of a Floer cylinder with punctures att´8uˆ Y . If γ is such a Reeb orbit, it is a k-fold cover a fibre of Y Ñ Σ for somek. We then define its index to be:

(7.7) |γ|0 “ n´ 3` 1´ n` 2

ˆ

τX ´K

K

˙

k “ ´2` 2

ˆ

τX ´K

K

˙

k

Observe that if X is aspherical and thus vacuously spherically monotone, we nolonger have capping disks 9B as above for any cover of a simple orbit γ. In that case,however, the differential will only consist of gradient trajectories. We therefore mayalways take the grading from Equations (3.4) and (3.5).

Notice that if Σ has a spherical homology class A with l :“ xc1pNΣq, Ay “KωpAq ‰ 0, then the l-fold cover of every Reeb orbit in Y is contractible in Y . Inthis case, using the corresponding cappings in Y , we obtain the fractional Conley–Zehnder index of [EGH00, Section 2.9.1].

Remark 7.1. Even though the idea of a fractional grading may seem unnatural atfirst, it can be thought of as a way of keeping track of some information aboutthe homotopy classes of the Hamiltonian orbits. Indeed, two generators of ourchain complex can only be homotopic Hamiltonian orbits (which is a necessarycondition for the existence of a Floer cylinder connecting them) if the difference oftheir degrees is an integer. Alternatively, we could have introduced an additionalparameter to keep track of the homotopy classes of Hamiltonian orbits, as done forinstance in [BO09b].


Finally, we compare our gradings with those described by Seidel [Sei08] andgeneralized by McLean [McL16] (he considers Reeb orbits, but there is an analogousconstruction for Hamiltonian orbits). As before, assume that there exists a sphericalclass B P H2pX;Zq such that l “ xc1pTXq, By ą 0. Suppose now that c1pTW q PH2pW ;Zq is torsion, so Nc1pTW q “ 0 for a suitable choice of N ą 0.

As explained earlier, let 9B be the disk associated to B that bounds γl. L “ΛnCpTW q is trivial over the disk 9B, and induces a unique trivialization of TW over9B. This trivialization defines the Conley–Zehnder of γl.

Now, let us consider the associated trivialization of pTW q‘N over 9B, which isalso unique up to homotopy. This trivialization induces a Conley–Zehnder indexon pγlq

‘N .Now, since Nc1pTW q “ 0, it follows that LbN is trivial. We fix a trivialization

of this line bundle, which induces a trivialization of pTW q‘N over the 2-skeletonof W . The Seidel–McLean grading convention is to use this global trivialization todefine the Conley–Zehnder index of pγlq

‘N .

Since pTW q‘N has a unique trivialization over 9B, up to homotopy, the twoConley–Zehnder indices we obtained for pγlq

‘N must agree. In our maximallyMorse–Bott setting, the linearized return Reeb map is the identity. Thus, theConley–Zehnder index of the cover of a fibre of Y is obtained as a constant termand a term linear in the covering multiplicity of the orbit. This linear term isthe Robbin–Salamon index [RS93,Gut14] of the degenerate Reeb orbit. From thislinearity, it follows then that our grading convention agrees for all covers with that ofSeidel and McLean. It is interesting to note that our approach to fractional gradingsuses the fact that in this Morse–Bott setting, the Conley–Zehnder indices are affinein the covering multiplicity and the linear part is given by the Robbin–Salamonindex. The Seidel-McLean approach primarily uses the direct sum property, thoughequality of our two gradings also relies upon linearity.

7.2. Index inequalities from monotonicity and transversality. We now in-vestigate the implications of our monotonicity assumptions (specifically that X isspherically monotone with xc1pTXq, Ay “ τXωpAq for spherical classes A and thatτX´K ą 0), specifically in combination with the transversality results from Section6.

First, we consider the Fredholm index contributions of a plane in W that couldappear as an augmentation plane to obtain some bounds on the possible indices.

Lemma 7.2. If v : CÑW is a JW holomorphic plane asymptotic to a given closedReeb orbit γ in Y , the Fredholm index for the deformations of v (as an unparame-terized curve) keeping γ fixed is non-negative. Furthermore, if v is multiply covered,this Fredholm index is at least 2.

Proof. First, we observe that the Fredholm index (as an unparametrized curve),keeping the asymptotic limit fixed at γ is given by:

Indpvq “ 2pxc1pTXq, By ´m´ 1q “ 2pτXωpBq ´KωpBq ´ 1q.

Since the plane is holomorphic, the induced spherical class B has ωpBq ą 0.By our monotonicity assumptions, we have

τXωpBq ´KωpBq “ pτX ´KqωpBq ą 0.


q

p

v

Figure 7.1. Option (1) in Proposition 7.4, call it Case 1

q

p

v

U

Figure 7.2. Option (2) in Proposition 7.4, call it Case 2

Finally, observe that τXωpBq “ xc1pTXq, By P Z and KωpBq “ B ‚ Σ P Z, so thisquantity is an integer, and is thus at least 1..

It therefore follows that Indpvq ě 0.Suppose now that v is a k-fold cover of an underlying simple holomorphic plane

v0, representing classes B “ kB0 respectively. Then,

Indpvq ` 2 “ 2pτX ´KqωpkB0q “ kpIndpv0q ` 2q.

Hence, Indpvq ě 2pk ´ 1q.

Lemma 7.3. Let u : CÑW be a holomorphic plane asymptotic to a multiplicity mReeb orbit in Y corresponding to the critical point p, with marker condition goingto rp “ pp or rp “ qp, and with 0 mapping to the descending manifold in W of thecritical point x. Denote by B the homology class B P H2pX;Zq obtained by cappingu with an m-fold cover of a fibre of E. The Fredholm index of u, as a parametrizedcurve, is given by

2 xc1pTXq, By ´ 2m`Mprpq `Mpxq ´ 2n` 1

Proof. First, applying the Conley–Zehnder index formula from Equation (7.1), to-gether with the standard index formula from SFT, we obtain that the index of anunparametrized plane satisfying the marked point conditions is:

n´ 3` 2 xc1pTXq, By ` p1´ nq `Mppq ´ 2m` 2´ 2n`Mpxq.

We then obtain an additional two from the parametrization, but the marker con-dition reduces the index by 1.

Proposition 7.4. Any Floer cascade in R ˆ Y that can appear in the differentialmust be one of the following configurations:


(0) An index 1 gradient trajectory in Y without any (non-constant) holomorphiccomponents and without any augmentation punctures.

(1) A smooth cylinder in RˆY without any augmentation punctures and a non-trivial projection to Σ. The positive puncture converges to a qp orbit and thenegative puncture converges to a pq orbit. The difference in multiplicites ofthe orbits is given by KωpAq, where A P H2pΣ;Zq is the homology classrepresented by the projection of the cylinder to Σ. See Figure 7.1.

(2) A cylinder with one augmentation puncture and whose projection to Σis trivial. The positive puncture converges to a qp orbit and the negativepuncture converges to a pq orbit. The augmentation plane has index 0. IfB P H2pX;Zq is the class represented by the augmentation plane, then thedifference in multiplicities is given by KωpBq. Furthermore, qp and pq arecritical points of fY contained in the same fibre of Y Ñ Σ. See Figure 7.2.

Proof. Consider a cascade with N levels and l augmentation planes appearing in thedifferential drpk` “ ¨ ¨ ¨ ` rqk´ ` ¨ ¨ ¨ . Let A1, . . . , AN P H2pΣq denote the homology

classes of the projections to Σ, let B1, . . . , Bk P H2pXq denote the homology classescorresponding to the augmentation planes. Let γi, i “ 1, . . . , k denote the limits at

the augmentation punctures, and let ki denote their multiplicites. Let A “řNi“1Ai.

We therefore have k` ´ k´řkj“1 kj “ KωpAq. We also have kj “ Bj ‚ Σ “

KωpBjq. Notice then that |γi|0 “ 2xc1pTXq, Bjy ´ 2Bj ‚ Σ´ 2.We therefore have

(7.8)

1 “ |rpk` | ´ |rqk´ |

“ iprpq `Mppq ´ iprqq ´Mpqq ` 2τX ´K

Kpk` ´ k´ ´ k0q ` 2

τX ´K

Kk0

“ iprpq `Mppq ´ iprqq ´Mpqq ` 2xc1pTΣq, Ay ` 2k `kÿ

j“1

|γj |0

By Lemma 7.2, we have that for each j “ 1, . . . , k, 2xc1pTXq, Bjy´2Bj‚Σ´2 ě 0,so for each j, |γj |0 ě 0.

Consider the chain of pearls in Σ obtained by projecting the upper level of thissplit Floer trajectory to Σ. By Proposition 6.17, if this is a simple chain of pearls,it has Fredholm index

IΣ :“Mppq ` 2xc1pTΣq, Ay ´Mpqq `N ´ 1` 2k.

If the chain of pearls is not simple, by monotonicity, we have that the index is atleast as large as the index of the underlying simple chain of pearls.

Now let N0 be the number of sub-levels that project to constant curves in Σand let N1 be the number of sub-levels that project to non-constant curves in Σ,N “ N0 `N1. Note that by the stability condition, each cylinder that projects toa constant curve in Σ must have at least one augmentation puncture, so N0 ď k.

By transversality for simple chains of pearls (Proposition 6.17), we obtain theinequality

IΣ ě 2N1 ` 2k

by considering the 2-dimensional automorphism group for the N1 non-constantspheres and by considering the 2l parameter family of moving augmentation markedpoints on the domains.


x

p

v1

v2

Figure 7.3. Configuration as in Proposition 7.5, call it Case 3

Combining with Equation (7.8), we obtain

1 “ iprpq ´ iprqq ` pIΣ Ń ` 1q `kÿ

j“1

|γj |0

1 ě piprpq ´ iprqq ` 1q ` 2N1 ` 2k Ń `kÿ

j“1

|γj |0

1 ě piprpq ´ iprqq ` 1q `N1 ` k ` pk Ń0q `

kÿ

j“1

|γj |0

Observe now that each term on the right-hand-side of the inequality is non-negative.In particular, there is at most one augmentation plane, and if there is one, it musthave |γ1|0 “ 2xc1pTXq, B1y ´ 2B1 ‚ Σ´ 2 “ 0.

The inequality now reduces to

1 ě piprpq ´ iprqq ` 1q `N1 ` k ` pk Ń0q.

Notice that N1 ` 2k Ń0 ě N .This inequality can be satisfied in one of the following ways:

0 N “ 0. Then, either iprpq “ iprqq or rp “ p and rq “ q. Since N “ 0, this is apure Morse differential term.

(1) N1 “ 1, N0 “ k “ 0 and rp “ qp, rq “ pq. This case corresponds to anon-constant sphere in Σ without any augmentation punctures.

(2) N1 “ 0, k “ 1, N0 “ 1, and rp “ qp, rq “ pq. In this case, the Floer cylinderhas one augmentation puncture, but projects to a constant in Σ.

We now consider the possible terms in the differential that connect non-constantHamiltonian trajectories in Rˆ Y to Morse critical points in X.

Proposition 7.5. Any Floer cascade appearing in the differential, connecting anon-constant Hamiltonian orbit p in R ˆ Y to a Morse critical point x in W ,consists of two levels. The upper level, in Rˆ Y , projects to a point in Σ and is acylinder asymptotic at `8 to a qp orbit and at ´8 to a Reeb orbit in t´8u ˆ Y .


The lower level is a holomorphic plane in W with marker condition going toqp and with 0 mapping to the descending manifold of the critical point x. As aparametrized curve, this has Fredholm index 1. See Figure 7.3.

Proof. Suppose such a cascade occurs in the differential, connecting the non-constantorbit rpk` to the critical point q in the filling W .

Let N be the number of cylinders in R ˆ Y that appear in the split Floercylinder. Let Ai P H2pΣq, i “ 1, . . . , N denote the spherical classes represented

by the projections of these cylinders to Σ. Let A “řNi“1Ai.

Let k be the number of augmentation planes, and let Bj P H2pXq, j “ 1, . . . , kbe the corresponding spherical homology classes in X. Let γj , j “ 1, . . . , k be thecorresponding Reeb orbits with multiplicities kj “ Bj ‚ Σ “ KωpBjq.

Let B P H2pXq be the spherical homology class in X represented by the lowerlevel v in W connecting to the critical point x. Let k´ “ B ‚Σ be the multiplicityof the orbit to which the plane v converges. As before, we have

k` ´ k´ ´kÿ

j“1

kj “ KωpAq.

We then have(7.9)1 “ |rpk` | ´ |x|

“ iprpq `Mppq ` 1´ 2n`Mpxq ` 2τX ´K

K

˜

k` ´ k´ ´kÿ

j“1

kj ` k´ `kÿ

j“1

kj

¸

“ iprpq `Mppq ` 1´ 2n`Mpxq ` 2xc1pTΣq, Ay ` 2xc1pTXq, By ´ 2B ‚ Σ` 2k `kÿ

j“1

|γj |0

Projecting to Σ, we obtain a chain of pearls with a sphere in X. Let N0 be thenumber of constant spheres in Σ and let N1 be the number of non-constant spheresin Σ, N “ N0 `N1. Notice that each non-constant sphere in Σ has a 2-parameterfamily of automorphisms, and each augmentation marked point can be moved in a2-parameter family. Furthermore, the holomorphic sphere v also has a 2-parameterfamily of automorphisms. By passing to a simple underlying chain of pearls asnecessary, and applying monotonicity and Proposition 6.17, we obtain

IX :“Mppq ` 2xc1pTΣq, Ay ` 2 pxc1pTXq, By ´B ‚ Σq `Mpxq ´ 2n` 1`N ` 2k

ě 2N1 ` 2k ` 2

We now combine the inequality with Equation (7.9):

1 “ iprpq ` IX Ń `kÿ

j“1

|γj |0

1 ě iprpq ` 2N1 ` 2k ` 2Ń0 Ń1 `

kÿ

j“1

|γj |0

0 ě iprpq `N1 ` k ` pk ` 1Ń0q `

kÿ

j“1

|γj |0.


Notice that we have N0 ď k ` 1 since the first sphere in the chain of pearls witha sphere in X is allowed to be constant without any marked points. This observa-tion together with Lemma 7.2 gives that each term on the right-hand-side of theinequality is non-negative. It follows therefore that each term must vanish: N0 “ 0,N1 “ 1, k “ 0 and rp “ qp. Notice that the Floer cylinder in Rˆ Y is contained in asingle fibre of RˆY Ñ Σ, so the marker condition coming from qp can be interpretedas a marker condition on the holomorphic plane v. Without the marker condition,v has Fredholm index 2, and thus with the marker constraint, it has index 1.

Case (2) in Proposition 7.4 allows for the existence of augmented configura-tions contributing to the symplectic homology differential. We will now adapt anargument originally due to Biran and Khanevsky [BK13] to show that, if Σ is asymplectic hyperplane section in the sense of Definition 2.1 (so that the comple-ment of a tubular neighborhood of Σ is a Weinstein domain) and has minimal Chernnumber at least 2, then there can only be rigid augmentation planes if the isotropicskeleton has codimension at most 2 (in particular, dimRX “ 2n ď 4).

Lemma 7.6. If W is a Weinstein domain with isotropic skeleton of real codimen-sion at least 3, then X is symplectically aspherical if and only if Σ is.

Furthermore, any symplectic sphere in X is in the image of the inclusion

ı˚π2pΣq Ñ π2pXq.

Proof. The trivial direction is that if there exists a spherical class A P π2pΣq withωpAq ą 0, then ı˚A P π2pXq and still has positive area.

We will now prove that any symplectic sphere in X is in the image of the in-clusion. Let C Ă W be the isotropic skeleton of W . Notice that by following theflow of the Liouville vector field on W , we obtain that W zC is symplectomorphicto a piece of the symplectization p´8, aqˆY . Thus, we have that XzC is a subsetof a symplectic disk bundle over Σ. We denote this bundle’s projection map byπ : XzC Ñ Σ.

Suppose A P π2pXq is a spherical class with ωpAq ą 0. By hypothesis, theskeleton C is of codimension at least 3. We may therefore perturb A in a neigh-bourhood of the skeleton so that it does not intersect the skeleton C. If ι : Σ Ñ Xand j : XzC Ñ X are the inclusion maps, then ωΣ “ ι˚ω and ι ˝ π is homotopic toj. This implies that ωXpAq “ ωΣpπ˚Aq, and the result follows.

Lemma 7.7. Suppose W is a Weinstein domain with isotropic skeleton of realcodimension at least 3 and Σ has minimal Chern number at least 2. Then, theredo not exist any augmentation planes.

Proof. Recall from Proposition 7.4 that an augmentation plane in the class B musthave index 0, so 0 “ 2pxc1pTXq, By ´ B ‚ Σ ´ 1q. Now xc1pTXq, By ´ B ‚ Σ “

pτX ´ KqωpBq ě 1. Thus, the augmentation plane can only exist if there is aspherical class B with pτX ´KqωpBq “ 1.

Now, by applying Lemma 7.6, we have B “ ı˚A, where A P π2pΣq is a sphericalclass in Σ. It now follows from equation (1.1) that 1 “ pτX ´KqωpAq “ τΣ ωpAq “xc1pTΣq, Ay.

The minimal Chern number of Σ is at least 2 by hypothesis, so the augmentationplane cannot exist.


Remark 7.8. The condition that the minimal Chern number of Σ is at least 2 isalso used in upcoming joint work of the first author with D. Tonkonog, R. Viannaand W. Wu, studying the effect of the Biran circle bundle construction on super-potentials associated to Lagrangians [DTVW].

8. Orientations

In order to orient our moduli spaces, we will take the point of view of coherentorientations, and follow the method of orienting Morse–Bott moduli spaces as donein [Bou02,BO09b]. Some authors [Zap15,Sch16] have taken the point of view ofcanonical orientations. We find it more straightforward to use coherent orientationsin our specific computational example, especially since there are very few choicesinvolved.

We briefly recall the general method for obtaining signs in Floer homology, asfirst introduced in [FH93] and since generalized. Consider all Cauchy–Riemannoperators on Hermitian vector bundles over punctured spheres, as described inAppendix A. Fix the asymptotic operators at each puncture, and suppose they arenon-degenerate. The space of all Cauchy–Riemann operators with these asymptoticoperators is then contractible and each such operator is Fredholm. Associated tothis space of operators is the determinant line bundle, whose fibre at an operatorD is given by

detD “ pΛmax kerDq bR pΛmax cokerDq˚.

An orientation corresponds to a nowhere vanishing continuous section of this de-terminant bundle over the space of Cauchy–Riemann operators (topologized in away compatible with the discrete topology on the space of asymptotic operators).

An orientation is coherent if it respects the gluing operation on these Cauchy–Riemann operators. Indeed, given two such operators D and D1 that have a match-ing asymptotic operator at a positive puncture for D and a negative puncture forD1, we may form a glued operator D#D1. This operator is not unique, but dependson a contractible family of choices, in particular on a gluing parameter. For suf-ficiently large gluing parameter, a section of detD and a section of detD1 inducea section of detD#D1, well-defined up to a positive multiple. (See, for instance,[BM04].) Thus, an orientation of D and an orientation of D1 induce an orientationof D#D1.

We may also arrange for the coherent orientation to have the following twoproperties:

‚ the orientation of the direct sum of two operators is the tensor product oftheir orientations,

‚ the orientation of a complex linear operator is its canonical orientation.

Because of the Morse-Bott degeneracy in our situation, we must consider Cauchy-Riemann operators acting on exponentially weighted spaces in order to obtain Fred-holm problems. Specifically, we consider a space of sections with exponential decaytogether with a (finite dimensional) space of infinitesimal movements in the corre-sponding Morse-Bott family of orbits. For a given degenerate asymptotic operator,we focus on two associated non-degenerate operators. The first corresponds to keep-ing the asymptotic orbit fixed. The second corresponds to letting the orbit to whichthe cylinder converges move in its Morse-Bott family. These are the δ-perturbedasymptotic operators of Definition A.5. Notice that the free/fixed asymptotic op-erator are perturbed by ˘δ differently depending on the sign of the puncture.


We now describe how to attach a sign to a Floer cylinder with cascades con-tributing to the differential. We will follow [BO09b] closely. By Propositions 7.4and 7.5, there are four types of contributions to the differential, referred to as Cases0 through 3. We will explain how to associate signs to each of these.

We begin with Case 1. Recall that such configurations consist of a Floer cylinderwithout augmentation punctures, together with two flow lines of ´ZY at the ends.Suppose that the Floer cylinder converges to orbits of multiplicity k˘ at ˘8. Such

a cylinder is an element of the space ĂM˚

H,1pYk´ , Yk` ;A; JY q, for some A P H2pΣ;Zq.In accordance with the prescription above for the non-degenerate case, we choose

capping operators for the relevant asymptotic operators. By Lemma 6.12, thelinearized operator associated to a Floer cylinder v is a compact perturbation of asplit operator DC

v ‘9DΣw, where w “ πΣ˝ v. There is also a corresponding splitting of

the asymptotic operators at the asymptotic limits. In particular, 9DΣw has complex

linear asymptotic operators, and thus is a compact perturbation of a complex linearCauchy–Riemann operator. Hence, its orientation is induced by the canonical one,and is independent of choice of trivialization or capping operator.

Remark 8.1. If it had been necessary to write down asymptotic operators associatedto 9DΣ

w, we would have trivialized the contact distribution ξ over each stable orunstable manifold of the vector field ZY (even though it is not possible to trivializeξ over all of Y , this can be done over the stable and unstable manifolds, since theyare contractible).

It now remains to orient the operator DCv . Write v “ pb, vq : RˆS1 Ñ RˆY and

b˘ “ limsÑ˘8 bps, tq. At ˘8, the δ-perturbed asymptotic operators (see DefinitionA.5) associated to v are

(8.1) A˘ :“ ´

ˆ

Jd

dt`

ˆ

h2peb˘q eb˘ 00 0

˙˙

˘

ˆ

δ 00 δ

˙

at ˘8, for a choice of δ ą 0 sufficiently small. Recall that these asymptotic op-erators are associated to converging to the descending manifold of the maximumof fY at `8, and to the ascending manifold of the minimum at ´8, or equiv-alently, allowing the asymptotic S1 marker to move in its Morse-Bott family atboth punctures. We will choose capping operators for the A˘, which determinesan orientation of DC

v by the coherent orientation scheme.By Lemma 6.14, DC

v has Fredholm index 1, is surjective and its kernel containsan element that can be identified with the Reeb vector field.

Lemma 8.2. There is a choice of capping operators for the asymptotic operatorsabove, such that DC

v is oriented in the Reeb direction.

Proof. Recall that for each bk ą 0 that satisfies h1pebkq “ k P Z`, we have a Y -parametric family of 1-periodic Hamiltonian orbits. We can associate to each ofthese orbits two operators as in (8.1). We will define capping operators

Φ˘k : W 1,ppC,Cq Ñ LppHom0,1pT pCq,Cqq

with these asymptotic operators.We first define two families of auxiliary Fredholm operators. For each k ą 0,

Ψk : W 1,ppRˆ S1,Cq Ñ LppHom0,1pT pRˆ S1q,Cqq


is an operator given by

ΨkpF qpBsq “ Fs ` iFt `

ˆ

apsq ´ δ 00 ´δ

˙

F

where a : R Ñ R is such that limsÑ´8 apsq “ h2peb1qeb1 and limsÑ`8 apsq “h2pebkqebk .

Ξk : W 1,ppRˆ S1,Cq Ñ LppHom0,1pT pRˆ S1q,Cqq

is an operator given by

ΞkpF qpBsq “ Fs ` iFt `

ˆ

h2pebkqebk ` δpsq 00 δpsq

˙

F

where δ : RÑ R is such that limsÑ´8 δpsq “ ´δ ă 0 and limsÑ`8 δpsq “ δ.The operators Ψk are isomorphisms (in particular, they are canonically oriented).

This follows from an argument analogous to the proof of Lemma 6.14. A versionof the same argument implies that the operators Ξk are Fredholm of index 1 andsurjective, and that their kernels contain elements that can be identified with theReeb vector field.

Now, pick any capping operator Φ´1 . Define Φ´k for k ą 1 by gluing Φ´1 #Ψk.

Define Φ`k for all k ą 0 by gluing Φ´k #Ξk. For these choices of capping operators,DCv are oriented in the direction of the Reeb flow, as wanted.

We have now oriented the operators DCv and 9DΣ

w. Since the linearized Floeroperator is a compact perturbation of their direct sum, we get induced orientations

on the spaces of Floer cylinders ĂM˚

H,1pYk´ , Yk` ;A; JY q.

Remark 8.3. Recall that in Symplectic Field Theory [EGH00], bad orbits arise whenthe Conley-Zehnder index of the even covers of a simple Reeb orbit have differentparity from the odd covers (in that case, the bad orbits are the even covers). Inour situation, all covers of orbits have the same parity (as in Equation (7.2)). Thissimplifies our discussion of orientations. See [BO09b] for a treatment of coherentorientations in a setting where bad orbits can exist.

Observe that cylinders with cascades in Case 1 that contribute to the differentialare elements of spaces M˚

H,1pq, p; JY , JW q, for p ‰ q P CritpfY q. These spaces areunions of fiber products

(8.2) W sfY pqq ˆev

ĂM˚

H,1pYk´ , Yk` ;A; JY q ˆev WufY ppq

defined with respect to the inclusion maps

W sfY pqq,W

ufY ppq Ñ Y

and the evaluation maps

ĄevY : ĂM˚

H,1pYk´ , Yk` ;A; JY q Ñ Y ˆ Y

for appropriate A P H2pΣ;Zq.Let us now recall the convention in [BO09b] for how to orient a fiber sum (which

agrees with [Joy12, Convention 7.2.(b)]).

Definition 8.4. Given linear maps between oriented vector spaces fi : Vi Ñ W ,i “ 1, 2, such that f1 ´ f2 : V1 ‘ V2 ÑW is surjective, the fiber sum orientation onV1 ‘fi V2 “ kerpf1 ´ f2q is such that


(1) f1´f2 induces an isomorphism pV1‘V2q kerpf1´f2q ÑW which changesorientations by p´1qdimV2. dimW ,

(2) where a quotient UV of oriented vector spaces is oriented in such a waythat the isomorphism V ‘ pUV q Ñ U (associated to a section of thequotient short exact sequence) preserves orientations.

One key property of this orientation convention for fiber products is that it isassociative. Pick now orientations for all unstable manifolds of pfΣ, ZΣq (which,since Y is oriented, induces orientations also on the stable manifolds). We use thefiber sum convention to orient 8.2.

Observe now that if M˚H,1pq, p; JY , JW q is one-dimensional, then its tangent

space at every point is generated by the infinitesimal translation in the s-directionon the domain of the Floer cylinder. This also induces an orientation on the spacesof Floer cylinders with cascades in Case 1. Comparing this orientation with theone defined above with the fiber sum rule, we get the sign of such a contributionto the differential.

Remark 8.5. In [BO09b], the authors describe the often studied process of adia-batic degeneration, in which one approximates a Morse–Bott Hamiltonian H bya small perturbation Hδ that is non-degenerate, and compares moduli spaces ofFloer cylinders with cascades for H with moduli spaces of Floer cylinders for Hδ.In Proposition 3.9, they compare the signs prescribed by a coherent orientationscheme to rigid floer cylinders with cascades for H to the signs prescribed to thecorresponding rigid cylinders for Hδ. A similar argument shows that the signs weobtain for cylinders with cascades in Case 1 agree with the signs we would getfor the corresponding rigid Floer cylinders we would get from perturbing H to anon-degenerate Hamiltonian.

We adapt the argument above to associate a sign to a Case 2 contribution to thedifferential. The analogue of (8.2) is now

(8.3) W sfY ppqqêv

`

pM0pB; JW qAutpCqqêvĂM˚

H,1pYk´ , Yk` ; 0;JY q˘

êvWufY pqqq.

The manifold pM0pB; JW qAutpCqqêvĂM˚

H,1pYk´ , Yk` ; 0;JY q is diffeomorphic via

gluing to the space of presplit Floer cylinders inW , denoted ĂM˚

H,1pYk´ , Yk` ;B; JW q,

for an almost complex structure JW close enough to being split. If we give M0pB; JW q

and ĂM˚

H,1pYk´ , Yk` ; 0;JY q their coherent orientations, then pM0pB; JW qAutpCqqêv

ĂM˚

H,1pYk´ , Yk` ; 0;JY q can be given the fiber product orientation. If we also give

ĂM˚

H,1pYk´ , Yk` ;B; JW q its coherent orientation, then the diffeomorphism abovepreserves orientations, by the properties of the fiber product orientation and coher-ent orientation.

We remark here that in Cases 2 and 3, we do not obtain an additional sign comingfrom the number of cylinders in the Floer cascade (in contrast to the situation in[BO09b, Proposition 3.9]) since the cylinders in question have matching asymptotics(whereas in [BO09b], there is a finite length gradient flow line).

Note that one might wonder if the fact that cascades in Case 2 have a holomor-phic curve and a Floer cylinder would cause an additional sign to appear in thedifferential, as in [BO09b, Proposition 3.9] with m “ 2. But that is not the case,because...


As for Case 3 Floer cylinders with cascades that contribute to the differential,they are elements of fiber products

(8.4) WufW pxq êv M˚

H,1pB; JW q êv M˚H,1pYk, 0; JY q êv W

ufY pqpq.

The fiber product M˚H,1pB; JW q êv M˚

H,1pYk, 0; JY q has an action of R1 ˆ R2,where the 1-dimensional real vector space R1 acts by s-translation on an elementof M˚

H,1pB; JW q and R2 acts by s-translation on an element of M˚H,1pYk, 0; JY q.

The fiber product can be oriented by using the coherent orientation on the spacesof cylinders, combined wih the fiber product rule for orientations. If JW is analmost complex structure close to split, then the moduli space of presplit cylindersM˚

H,1pB; JW q has an R-action, corresponding to s-translations on the domain. Themoduli space of presplit Floer cylinders has a coherent orientation. The quotients´

M˚H,1pB; JW q êv M˚

H,1pYk, 0; JY q¯

pR1ˆR2q and M˚H,1pB; JW qR are oriented

diffeomorphic.We might again be led to think that an analogue of [BO09b, Proposition 3.9]

with m “ 2 would cause a sign to appear in the differential, but that is not thecase...

Part 4. Ansatz

We now show that Floer cylinders in R ˆ Y are equivalent punctured JY -holomorphic curves. We will first show how to obtain a pseudoholomorphic curvefrom a Floer cylinder and then explain how to go in the other direction.

9. From Floer cylinders to pseudoholomorphic curves

We show that every split Floer cylinder as in Definition 4.13 can be obtainedfrom a JY -holomorphic curve and a solution to an auxiliary equation.

It is useful to first recall that for every integer T ě 1, there is a Y -family of Reeborbits of period T , and a bijective correspondence to the Y -family of 1-periodicHamiltonian orbits contained in tbu ˆ Y Ă R ˆ Y , where h1pebq “ T . A Reeborbit γ corresponds to the Hamiltonian orbit t ÞÑ pb, γpTtqq. We say that two Reeborbits of the same period are the same as unparametrized orbits if they differ by anadditive constant on the domain.

Proposition 9.1. Let Γ Ă R ˆ S1 be either the empty set or the singleton tP u.Suppose v “ pb, vq : R ˆ S1zΓ Ñ R ˆ Y is the upper level of a split Floer cylinder,as in Definition 4.13. Then, there exists a pair pu, eq consisting of a map u “pa, uq : RˆS1zΓ Ñ Rˆ Y and of a map e “ pe1, e2q : RˆS1zΓ Ñ RˆS1, with thefollowing properties:

(1) The map u is a finite energy JY -holomorphic curve.(2) If v converges to a Hamiltonian orbit at ˘8, then u converges to the cor-

responding parametrized Reeb orbit γ˘ at ˘8, i.e. vp˘8, tq “ up˘8, tq.(3) If Γ “ tP u, then u is asymptotic to the same unparameterized Reeb orbit

that v converges to at P , i.e. the asymptotic limit is the same orbit up to aphase shift.

(4) The map e : Rˆ S1zΓ Ñ Rˆ S1 satisfies the equation

(9.1) de1 ´ de2 ˝ i` h1pebqds “ 0.


(5) The original Floer solution is given by

pb, vq “ pa` e1, φe2R ˝ uq

where φtR : Y Ñ Y denotes Reeb flow for time t P S1.

The pair pu, eq is unique, up to replacing u “ pa, uq with pa ` c1, uq and replacinge “ pe1, e2q with pe1 ´ c1, e2q, for some constant c1 P R.

Suppose now that Γ “ H and that v is the upper level of a split Floer cylinder,as in Definition 4.14. Then, there exists a pair pu, eq as above, with the followingdifference:

(2’) If v converges to a Hamiltonian orbit at `8 and to a Reeb orbit at ´8,then u converges to the corresponding Reeb orbit γ` at `8 and to the sameReeb orbit γ´ at ´8.

The pair pu, eq is again unique up to an R-action.

In the following, we will construct the function e from Proposition 9.1 by study-ing the differential equation (9.1).

To this end, we need to construct function spaces of maps e : R ˆ S1zΓ Ñ

R ˆ S1. These will be subspaces of W 1,ploc pR ˆ S1zΓ,R ˆ S1q satisfying different

types of asymptotic conditions. The maps e we consider will induce trivial mapsof fundamental groups. It will be more convenient therefore to consider the lift tothe universal cover e : Rˆ S1zΓ Ñ C, and study Equation (9.1) on this space.

As we will see below, all the punctures have the same asymptotic operator,namely

(9.2) A :“ ´id

dt: W 1,ppS1,Cq Ñ LppS1,Cq.

Note that A is self-adjoint, as a partially defined operator L2pS1,Cq Ñ L2pS1,Cq.Lemma A.3 describes the spectrum of A. For the remainder of this section, wewill fix 0 ă δ ă 2π, which is smaller than the absolute value of every non-zeroeigenvalue of A. We will use this δ below when defining weighted Sobolev spaces.

We recall some notation first — this is discussed in more detail in Appendix A.For a fixed z P R ˆ S1, or z “ ˘8, we fix µz to be a bump function supported

near z and identically 1 near z. The space of functions W 1,p,δp0,CqpRˆ S

1,Cq consists

of complex-valued functions exponentially decaying to 0 at ´8 and exponentiallydecaying to an unspecified constant at `8. In the case of a punctured cylinder,

W 1,p,δp0,C;CqpRˆS

1ztP u,Cq denotes the space of functions exponentially decaying to 0

at ´8, decaying to a free constant at P and decaying to a free constant at `8.In the case with Γ “ tP u, we fix a parametrization

ϕP : p´8,´1s ˆ S1 Ñ Rˆ S1zΓ

pρ, ηq ÞÑ P ` e2πpρ`iηq

of a neighborhood of P , under the identification of R ˆ S1 with Cpz „ z ` iq.Recall that the measure on RˆS1zΓ and the norm on derivatives used for definingW 1,p and Lp comes from these cylindrical coordinates near P . In particular, wemake the following observation that will be used multiple times,

(9.3) ϕ˚P ds “ 2π e2πρ cosp2πθqdρ´ 2π e2πρ sinp2πθqdθ

In particular then, |ϕ˚P ds| “ C e2πρ.


We will consider solutions to equation (9.1) in three different spaces, one cor-responding to each of the three possible Floer configurations appearing in Propo-sitions 7.4 and 7.5. Cases 1 and 2 appear in Proposition 7.4, and are relevantfor split Floer solutions in Definition 4.13 (connecting non-constant Hamiltonianorbits), represented by Figures 7.1 and 7.2, respectively. Case 3 is the one fromProposition 7.5, and is relevant for split Floer solutions in Definition 4.14 (connect-ing non-constant Hamiltonian orbits to constant ones), as in Figure 7.3.

Case 1: Γ “ H and both asymptotic limits of v are Hamiltonian orbits.We say e P X 1 if

e` pT`s` i0qµ`8 ` pT´s` i0qµ´8 PW1,p,δpC;CqpRˆ S

1,Cq.

(In particular then, e is of degree 0. The lift e is well-defined up to anadditive constant.)

Case 2: Γ “ tP u and both asymptotic limits of v are Hamiltonian orbits. We say

e P X 2 if

e` pT`s` i0qµ`8 ` pT´s` i0qµ´8 PW1,p,δpC,C;CqpRˆ S

1ztP u,Cq

Case 3: Γ “ H and the `8 asymptotic limit of v is a Hamiltonian orbit, whereasat ´8, it converges to an orbit cylinder over a Reeb orbit in t´8u ˆ Y .

We say e P X 3 if

e` pT`s` i0qµ`8 PW1,pδpC;CqpRˆ S

1,Cq.

We then define X 1,X 2,X 3 to be the set of functions obtained as compositions offunctions in X 1, X 2, X 3 with the projection map CÑ Rˆ S1.

Remark 9.2. The summands pT˘s ` i0qµ˘8 keep track of the fact that the Floertrajectory v “ pb, vq and the pseudoholomorphic curve u “ pa, uq in Proposition9.1 have different asymptotic behavior at ˘8, and that e1 “ b´ a.

Observe that in all three cases, we impose δ-decay to unspecified constants at˘8 (and possibly P ).

Denote the spaces of solutions to (9.1) in X j by Cjpvq, where j P t1, 2, 3u, and

let Cjpvq be the corresponding space of maps to RˆS1. We can write Cjpvq as thepreimage of zero under the operator

X j Ñ Lp,δpT˚pRˆ S1zΓqq

e “ pe1, e2q ÞÑ de1 ´ de2 ˝ i` h1pebqds.

Observe that this map descends to a map X j Ñ Lp,δpT˚pRˆS1zΓqq. We then havethat Cjpvq is the preimage of zero under this quotient operator.

It will be useful to consider the corresponding linearized operators. Let e P Cjpvq.The corresponding linearized operator is

(9.4)Dj : TeX j Ñ Lp,δpT˚pRˆ S1zΓqq

E “ E1 ` iE2 ÞÑ dE1 ´ dE2 ˝ i

where in Cases (1) and (3), we have TeX j “ W 1,p,δpC;CqpR ˆ S1,Cq and and in Case

(2), we have TeX 2 “W 1,p,δpC,C;CqpRˆ S

1zΓ,Cq.


Evaluating (9.4) at Bs and Bt and rearranging terms, we get operators with valuesin Lp,δpRˆ S1zΓ,Cq given by

(9.5)BE

Bs` iBE

Bt.

By an abuse of notation, denote these operators also by Dj . Observe that theasymptotic operators to Dj at all punctures are given by Equation (9.2) above.

Lemma 9.3. Fix e P Cjpvq, for j P t1, 2, 3u. Then, the linearized operator Dj

has index 2, is surjective and its kernel includes the span of the generator of theS1-action on the target R ˆ S1. As a consequence, the space Cjpvq has virtualdimension 2.

Proof. The operator Dj in (9.4) is just the standard Cauchy–Riemann operator.This proof is perhaps more involved than necessary, but illustrates in a simpleexample the ideas used in the more technical setting of the next section. This proofis also similar to that of Lemma 6.14, the main difference being that we now applythe Morse–Bott Riemann–Roch Theorem A.8 instead of Theorem A.6.

We start with the Cases j “ 1 or 3. Fix e as in the statement. Recall thatthe asymptotic operator is A “ ´i ddt at all punctures. We need to compute theConley–Zehnder index of the perturbed asymptotic operator A` δ, which is ´1 byCorollary A.4. Therefore using the fact that all vector spaces in V are the kernelsof the corresponding asymptotic operators, which we identified with C, TheoremA.8 implies that

IndDj “ p´1` 2q ´ p´1q “ 2.

Following the notation of Wendl [Wen10, Equation (2.5)], we have

c1pE, l,AΓq “1

2p2´ 2q “ 0 ă 2 “ Ind pDj

and surjectivity and the statement about the kernel follow from [Wen10, Proposition2.2] together with Lemma A.10.

Case 2 is slightly different, since there is an additional puncture at P . Its as-ymptotic operator is again A. The Euler characteristic of the domain is now ´1.Therefore,

IndD2 “ ´1``

p´1` 2q ´ p´1q ´ p´1q˘

“ 2,

c1pE, l,AΓq “1

2p2´ 2q “ 0 ă 2 “ Ind pD2

and we can once more apply [Wen10, Proposition 2.2] together with Lemma A.10.

Lemma 9.4. Let e P Cj, j “ 1, 2, 3. Then, e admits a smooth extension RˆS1 Ñ

Rˆ S1 that satisfies Equation (9.1).

Proof. Let e P Cj be a solution. Then, in Cases (1) and (3), the result followsimmediately from elliptic regularity and the fact that h1pebq is a smooth functionon the cylinder.

For Case (2), observe that h1perq vanishes for r ăă 0. Therefore, h1pebq vanishesin a neighbourhood of the puncture P , and thus extends as a smooth function onthe cylinder. Furthermore, e is holomorphic in this same punctured neighbourhood,

and the requirement it be in W 1,p,δC pR´ ˆ S1q forces it to have a continuous and

thus smooth extension across the puncture.It follows therefore that e is smooth in Case (2) as well.


Lemma 9.5. Let e “ pe1, e2q : Rˆ S1 Ñ C be a lift of a solution e “ pe1, e2q P Cj.Then, for every t P S1

limsÑ´8

e2ps, tq “ limsÑ8

e2ps, tq.

Proof. By Lemma 9.4, it follows that e is smooth and satisfies the equation

de1 ´ de2 ˝ i` h1pebqds “ 0.

For each c ąą 1, let Sc :“ rć, cs ˆ S1 Ă Rˆ S1. Then,ż

BSc

e2 dt “

ż

Sc

de2 ^ dt “

ż

Sc

de2 ^ dt

“

ż

Sc

de2 ˝ i^ dt ˝ i

“

ż

Sc

pde1 ` h1pebqdsq ^ ds

“

ż

Sc

de1 ^ ds

“

ż

BSc

e1 ds “ 0

since ds|BSc “ 0. Hence, we obtainż

tcuˆS1

e2pc, tqdt “

ż

tćuˆS1

e2pć, tqdt.

The result now follows from taking a limit as cÑ `8.

Lemma 9.6. Equation (9.1) has a solution in X j for j P t1, 2, 3u, which is uniqueup to adding a constant in the target Rˆ S1.

Proof. We will instead prove the result for the lifts e P X j , solving the samedifferential equation. Notice that any solution e admits a Z family of lifts. Itsuffices then to show that the solutions e “ pe1, e2q are unique up to adding aconstant in C.

Write (9.1) as an inhomogeneous Cauchy–Riemann equation on Rˆ S1zΓ:

(9.6) de1 ´ de2 ˝ i “ ´h1pebqds.

We will consider Cases 1, 2 and 3. Let νj : R Ñ R be smooth functions suchthat

#

νjpsq “ T´s for s ăă 0 and νjpsq “ T`s for s ąą 0 j “ 1, 2

νjpsq “ 0 for s ăă 0 and νjpsq “ T`s for s ąą 0 j “ 3

Instead of solving equation (9.6), we will consider the equivalent problem offinding g “ pg1, g2q “ pe1 ` νj , e2q such that

g P

#

W 1,p,δpC;CqpRˆ S

1,Cq, in cases j “ 1, 3

W 1,p,δpC,C;CqpRˆ S

1zΓ,Cq, in case j “ 2

solves the equation

(9.7) dg1 ´ dg2 ˝ i “ ´h1pebqds` dνj .

Notice that g is taken in the function space TeXj .


LetK “ ´h1pebqds`dνj . In Cases (1) and (3), the fact thatK P Lp,δpT˚pRˆS1qq

follows immediately from the exponential rate of convergence of b at ˘8 to b˘, andT˘ “ h1peb˘q. In case (2), the fact K P Lp,δpT˚pRˆ S1ztP uq follows from this andalso from Equation 9.3 so that ν12psqds has exponential decay near P .

According to Lemma 9.3, the linearized operators (9.5) have index 2 and aresurjective. Their kernels have dimension 2 and consist of constant functions. Letnow g P TeX j , with TeX j defined as in (9.4), be such that

dg1 ´ dg2 ˝ i “ K.

Any two solutions differ then by an element of the kernel of this linear operator,and thus differ by a constant. From this, we obtain the same statement for e andhence for e.

Proof of Proposition 9.1. Using the fact that XH “ h1perqR, the Floer equation(4.3) satisfied by v “ pb, vq is equivalent to:

(9.8)

#

db´ v˚α ˝ i` h1pebqds “ 0

πdv ` Jπdv ˝ i “ 0

By Lemma 9.6, we have a solution e to (9.1). Now, a :“ b ´ e1 and u :“ φ´e2R ˝ vare such that

da´ u˚α ˝ i “ db´ de1 ´ v˚α ˝ i´ de2 ˝ i

“ db´ v˚α ˝ i` h1pebqds

“ 0,

πdu` Jπdu ˝ i “ πdv ` Jπdv ˝ i

“ 0.

So, u :“ pa, uq is JY -holomorphic. The asymptotic constraints on the spaces X i aresuch that u converges to Reeb orbits at ˘8 (and at P , if Γ “ tP u). The uniquenessstatement in Lemma 9.6 together with Lemma 9.5, imply that there is an R-familyof solutions e “ pe1, e2q P X j , such that

limsÑ˘8

e2ps, tq “ 0 P S1,

where distinct solutions differ by adding a constant to e1. This implies the existencestatements in Proposition 9.1, and uniqueness up to an R-action.

10. From pseudoholomorphic curves to Floer cylinders

Proposition 9.1 implies that every split Floer cylinder v : RˆS1zΓ Ñ RˆY hasan underlying JY -holomorphic curve u : R ˆ S1zΓ Ñ R ˆ Y . We will show thatevery JY -holomorphic curve u : RˆS1zΓ Ñ Rˆ Y underlies a split Floer cylinder.

Proposition 10.1. Let Γ Ă R ˆ S1 be either the empty set or the singleton tP u.Suppose u “ pa, uq : R ˆ S1zΓ Ñ R ˆ Y is a finite energy JY -holomorphic curve,asymptotic at P to a Reeb orbit at ´8, if Γ “ tP u. Then, there exists a pairpv, eq consisting of a map v “ pb, vq : R ˆ S1zΓ Ñ R ˆ Y and of a map e “

pe1, e2q : Rˆ S1 Ñ Rˆ S1, with the following properties:

(1) The map v solves the Floer equation (4.3).(2) If u converges to a Reeb orbit γ˘ at ˘8, then v converges to the Hamil-

tonian orbit corresponding to γ˘ at ˘8.


(3) If Γ “ tP u, then v is asymptotic to the same unparameterized Reeb orbitthat u converges to at P .

(4) The map e : Rˆ S1ztP u Ñ Rˆ S1 satisfies the equation

(10.1) de1 ´ de2 ˝ i` h1peaè1qds “ 0.

(5) The original JY -holomorphic curve is given by

pa, uq “ pb´ e1, φé2R ˝ vq.

The pair pv, eq is unique.In the case that Γ “ H and ups, tq “ pTs ` c, γpTtqq is a trivial cylinder, for

some Reeb orbit γ of period T , there also exists a pair pv, eq as above, except for

(2’) v converges to a Hamiltonian orbit whose underlying Reeb orbit is γ at `8,and v converges to γ at ´8.

The pair pv, eq is now unique only up to replacing bps, tq with bps, tq “ bps` s0, tq,for some constant s0 P R, and replacing e1 with e1 “ b´ a.

Equation (10.1) has a solution e “ pe1, e2q iff there is a solution f “ pf1, f2q “

pe1 ` a, e2q to the equation

(10.2)f “pf1, f2q : Rˆ S1zΓ Ñ Rˆ S1

df1 ´ df2 ˝ i` h1pef1qds´ da “ 0,

so we will study this equation instead. As we will see below equation (10.12), thelinearized operator associated to (10.2) has asymptotic operators of the form

AC “ íd

dt´

ˆ

C 00 0

˙

: W 1,ppS1,Cq Ñ LppS1,Cq

for some C ě 0. Lemma A.3 implies that the kernel of AC has real dimension 2 ifC “ 0 (given by constant functions c1 ` ic2 P C) and 1 if C ‰ 0 (given by constantfunctions ic2 P iR).

Recall that we write W 1,p,δp0,C;iRqpRˆ S

1ztP u,Cq to denote maps converging expo-

nentially fast to 0 at´8, to an arbitrary complex constant at P and to an imaginary

constant at `8, and write W 1,p,δp0,iRqpRˆS

1,Cq to denote maps converging exponen-

tially fast to 0 at ´8 and to an arbitrary imaginary constant at `8. Recall thatwe have chosen cylindrical coordinates near P , ϕ : p´8,´1s ˆ S1 Ñ R ˆ S1 byϕpρ, θq “ P ` e2πpρìθq, and also a bump function µP that is identically 1 near Pand supported in the image of ϕ.

Similarly to the previous section, we will study solutions to (10.2) in W 1,ploc pRˆ

S1zΓ,Rˆ S1q satisfying three types of asymptotic conditions.

Case 1: Γ “ H. Define the space Y1 by the condition f P Y1 if it admits a liftf PW 1,p

loc pRˆ S1,Cq with

f ´ pb` ` i0qµ`8 ´ pb´ ` i0qµ´8 PW1,p,δp0,iRqpRˆ S

1,Cq

Case 2: Γ “ tP u. Define the space Y2 by the condition that f P Y2 if it admits a

lift f PW 1,ploc pRˆ S1,Cq with

f ´ pb` ` i0qµ`8 ´ pb´ ` i0qµ´8 ´ µP a PW1,p,δp0,C;iRqpRˆ S

1ztP u,Cq.

Case 3: Γ “ H. Define the space Y3 by the condition that f admits a lift f with

f ´ pb` ` i0qµ`8 ´ pb´ ` i0qµ´8 PW1,p,δpC;iRqpRˆ S

1,Cq.


Remark 10.2. Similarly to the previous section, the last part of the Proposition,where u is assumed to be a trivial cylinder, is related to Case 3 (see Figure 7.3),and the first part of the Proposition is related to Cases 1 and 2 (see Figures 7.1and 7.2, respectively).

It is important to observe that the asymptotic conditions imposed in this sectionare different from the ones that were imposed in the previous section. This will havean effect on the indices of the relevant Fredholm problems. Just to give an example,in Case 1 we are considering functions that converge at `8 to an unspecifiedconstant in tb`u ˆ S

1 Ă Rˆ S1, and that converge at ´8 to pb´, 0q P Rˆ S1.

In this section, it will also be useful to consider auxiliary parametric versions ofEquation (10.2). We will deform the solutions to (10.2) to solutions of a PDE forwhich we can explicitly describe the solution space. The deformation will give anidentification of the solution spaces.

For Case 1, we will consider the map

(10.3)F : Y1 ˆ r0, 1s Ñ Lp,δpRˆ S1, pR2q˚q

pf , τq “ ppf1, f2q, τq ÞÑ df1 ´ df2 ˝ i` h1pef1qds´ τda´ p1´ τqdν

where ν : R Ñ R is a smooth function such that νpsq “ T´s for s ăă 0 andνpsq “ T`s for s ąą 0 (just like ν1 in the proof of Lemma 9.6). We will studysolutions to equation

(10.4) Fpf , τq “ 0,

which interpolates between (10.2) and the equation

(10.5) df1 ´ df2 ˝ i` h1pef1qds´ dν “ 0.

The latter has the advantage that dν is independent of t. Denote the space ofsolutions to (10.4) in Y1 for fixed τ by C1,τ puq.

In Case 2, it will be more convenient to study functions g “ pg1, g2q “ pf1 ´

µP a, f2q. We think of g as an element of the subspace Y2 ĂW 1,ploc pRˆS1zΓ,RˆS1q,

with g P Y2 if it has a lift g with

g ´ pb` ` i0qµ`8 ´ pb´ ` i0qµ´8 PW1,p,δp0,C;iRqpRˆ S

1ztP u,Cq.

We will consider the map(10.6)

G : Y2 ˆ r0, 1s Ñ Lp,δpRˆ S1zΓ, pR2q˚q

pg, τq “ ppg1, g2q, τq ÞÑ dg1 ´ dg2 ˝ i` h1peg1`τµP aqds´ τdpp1´ µP qaq ´ p1´ τqdν

where ν : RÑ R is as above.We will study solutions to

(10.7) Gpg, τq “ 0.

This equation interpolates between the equivalent of equation (10.2) for g:

dg1 ´ dg2 ˝ i` h1peg1`µP aqds´ dpp1´ µP qaq “ 0

and equation (10.5) applied to g:

dg1 ´ dg2 ˝ i` h1peg1qds´ dν “ 0,

The latter has again the advantage that dν is independent of t. Denote the space

of solutions to (10.7) in Y2 for fixed τ by C2,τ puq.


Also, denote the space of solutions to (10.2) in Y3 by C3puq. In Case 3, we willstudy solutions directly and do not require any deformation argument.

We need the following smoothness in order to apply the implicit function theo-rem, which we will need in the proof of Proposition 10.1.

Lemma 10.3. The nonlinear operators F and G are C1 in Y1ˆ r0, 1s Ñ Lp,δpRˆS1, pR2q˚q and Y2 ˆ r0, 1s Ñ Lp,δpRˆ S1ztP u, pR2q˚q, respectively.

Proof. Notice that Fpf , τq is a Floer-type differential operator with an inhomoge-neous term depending on τ . The claimed result holds for F since ´da` ν1psq ds isin Lp,δ, which follows from the fact that aps, tq converges exponentially fast to T˘sat ˘8.

In order to show the result for Gpg, τq, it suffices to show that the following mapis C1, since the remaining terms in G are C1 by standard arguments.

(10.8)H : W 1,p,δ

p0,C;iRqpRˆ S1zΓ,Cq Ñ Lp,δpT˚pRˆ S1zΓqq

pg1, g2, τq ÞÑ h1peg1`τµP aq ds.

First, we restrict our attention to a W 1,p,δp0,C;iRq-ball of radius R. We will show the map

is C1 on each such ball by showing its second derivative is bounded by a constantthat depends on R. Exhausting the total space by such balls will prove the result.

Notice that the W 1,p,δ bound implies a uniform C0 bound on g1. Also noticethat µPa is bounded above. Thus, g1 ` τµPa is bounded above (where the bounddepends on R). Hence, there exists a C (depending on R) so that h1peg1`τµP aq ď C,h2peg1`τµP aq eg1`τµP a ď C and h2peg1`τµP aq e2pg1`τµP aq`h3peg1`τµP aq eg1`τµP a ď

C.The weak derivative of H is given by

dHpg, τqpG, τ 1q “ Gh2peg1`τµP aq eg1`τµP a ds` τ 1h2peg1`τµP aq eg1`τµP a µPa ds

Notice that this expression is in W 1,p,δ away from P . Near P , in the cylindricalcoordinates pρ, θq, we note that this is bounded by Cp|G|`|τ 1|pT |ρ|`Dqq e2πρp|dρ|`|dθ|q for suitable constants C,D and where T is the period of the Reeb orbit towhich pa, uq converges at P . From this, we obtain that dH is uniformly boundedon each ball of radius R, showing H is Lipschitz on each ball of radius R, and henceH is continuous.

By a similar argument, we estimate the second derivative to show C1 smoothnessof H. The C1-smoothness of G then follows.

We now discuss the linearized operators associated to F and G, for fixed valuesof τ . In Case 1, writing a solution to (10.4) as f “ f1` if2, the linearization of theoperator Fτ is

(10.9)D1,τ : TfY1 Ñ Lp,δpRˆ S1, pR2q˚q

F “ F1 ` iF2 ÞÑ dF1 ´ dF2 ˝ i` h2pef1q ef1 F1ds.

where TfY1 “W 1,p,δV pRˆ S1,Cq with V´ “ 0 and V` “ iR.

In Case 2, writing g “ g1 ` ig2 and again fixing τ , the linearized operator is

(10.10)D2,τ : TgY2 Ñ Lp,δpRˆ S1zΓ, pR2q˚q

G “ G1 ` iG2 ÞÑ dG1 ´ dG2 ˝ i` h2peg1`τµaq eg1`τµaG1ds.

where TgY2 “W 1,p,δV pRˆ S1zΓ,Cq with V´ “ 0, V` “ iR and VP “ C.


In Case 3, the linearized operator

D3 : TfY3 Ñ Lp,δpRˆ S1, pR2q˚q,

where TfY3 “W 1,p,δV pRˆS1,Cq with V´ “ C and V` “ iR, is also given by (10.9)

Evaluating (10.9) and (10.10) at Bs and Bt and rearranging, we get operatorswith values in Lp,δpRˆ S1,Cq, given by

(10.11)BF

Bs` iBF

Bt` h2pef1q ef1 F1

and

(10.12)BG

Bs` iBG

Bt` h2peg1`τµaq eg1`τµaG1,

respectively.The asymptotic operators at ˘8 in Cases 1 and 2, and at `8 in Case 3, are

A˘ :“ íd

dt´

ˆ

h2peb˘q eb˘ 00 0

˙


In Case 2, there is also an asymptotic operator at the augmentation puncture:

AP :“ íd

dt: W 1,ppS1,Cq Ñ LppS1,Cq

In Case 3, we have the asymptotic operator at ´8 given by A´ “ íddt .

All these asymptotic operators are self-adjoint, as partially defined operatorsL2pS1,Cq Ñ L2pS1,Cq. Their eigenvalues can be computed using Lemma A.3.Note that they all have eigenvalue 0. For the remainder of this section, we will fix

0 ă δ ă 2π such that δ ă C˘ “ h2peb˘q eb˘ ą 0 and δ ă 12

´

Ć˘ `b

C2˘ ` 16π2

¯

(see Remark 6.11). So, δ is smaller than the absolute value of every non-zeroeigenvalue of A˘ and AP . This δ will be used in the definitions of the relevantweighted Sobolev spaces.

Lemma 10.4. Fix τ P r0, 1s and f P C1,τ puq. Then, the linearized operator D1,τ isFredholm of index 0 and is an isomorphism.

Fix τ P r0, 1s and g P C2,τ puq. Then, the linearized operator D2,τ is Fredholm ofindex 0 and is an isomorphism.

For fixed f P C3puq, D3 is Fredholm of index 2, is surjective and its kernelincludes the span of the generator of the S1-action on the target Rˆ S1.

Proof. The proof has the same structure as that of Lemma 9.3. We start with Case1. Fix τ and f as in the statement. Corollary A.4 implies that the Conley–Zehnderindices of the perturbed asymptotic operators A˘` δ are 0. Recall that the kernelsof the asymptotic operators A are identified with iR and that the vector spaces inV are V´ “ 0 and V` “ iR. Theorem A.8 now implies that

Ind pD1,τ “ p0` 1q ´ p0` 1q “ 0.

Following the notation of Wendl [Wen10, Equation (2.5)], we have

c1pE, l,AΓq “1

2p0´ 2q “ ´1 ă 0 “ Ind pD1,τ

and pD1,τ is an isomorphism by [Wen10, Proposition 2.2].In Case 3, recall that the asymptotic operator at ´8 is now A´ “ í

ddt . By

Corollary A.4, the Conley–Zehnder index of the perturbed asymptotic operator


A´ ` δ is ´1. The kernel of the asymptotic operator A´ can be identified with C,and the vector spaces in V are now V´ “ C and V` “ iR. By Theorem A.8,

Ind pD3 “ p0` 1q ´ p´1` 0q “ 2,

c1pE, l,AΓq “1

2p2´ 2q “ 0 ă 2 “ Ind pD3

and pD3 is again an isomorphism by [Wen10, Proposition 2.2].In Case 2, we have the additional puncture P , with asymptotic operator AP “

´i ddt , and the Euler charateristic of the domain is ´1. We now have V´ “ 0,V` “ iR and VP “ C, so Theorem A.8 implies

Ind pD2,τ “ ´1` p0` 1q ´ p0` 1q ´ p´1` 0q “ 0,

c1pE, l,AΓq “1

2p0´ 2q “ ´1 ă 0 “ Ind pD2,τ

and we can apply once more [Wen10, Proposition 2.2].

We now prove a result analogous to Lemma 10.5.

Lemma 10.5. Let τ P r0, 1s.

Suppose that f P Y1, g P Y2 or f P Y3 satisfy

Fpf , τq “ 0, Gpg, τq “ 0, or df1 ´ df2 ˝ i` h1pef1q ´ da “ 0,

respectively.Then, the asymptotic constants satisfy f2p`8, ¨q “ f2p´8, ¨q, g2p`8, ¨q “ g2p´8, ¨q

and f2p`8, ¨q “ f2p´8, ¨q.In particular, in Cases (1) and (2), these asymptotic constants are 0.

Proof. The proof is very similar to the proof of Lemma 9.5.Notice that if we have such a solution f , g, the resulting function is smooth on

the punctured cylinder, by standard elliptic regularity arguments.We will prove the result for functions that solve the equation

df1 ´ df2 ˝ i` h1pef1`xq ds´ τ dy ´ p1´ τq dν,

on Rˆ S1zΓ, where x, y satisfy the following:

Case 1: x “ 0, y “ a, Γ “ H;Case 2: x “ τµPa, y “ p1´ µP qa, Γ “ tP u;Case 3: τ “ 1, x “ 0, y “ a, Γ “ H.

We assume furthermore that f2 converges exponentially fast to a constant at ˘8,and that f1 converges exponentially fast to a constant at P in Case 2.

Notice in particular that if the cylinder has a puncture, y vanishes in a neigh-bourhood of the puncture.

For each c ąą 1, let Sc :“ r´c, cs ˆ S1zDP p1cq Ă R ˆ S1zΓ, where DP pεqdenotes a disk of radius ε centered at P .


We then obtain (noticing that dν “ ν1psq ds)ż

BSc

f2dt “

ż

Sc

df2 ^ dt “

ż

Sc

df2 ^ dt

“

ż

Sc

df2 ˝ i^ dt ˝ i

“

ż

Sc

pdf1 ` h1pef1`xqds´ τdy ´ p1´ τqdνq ^ ds

“

ż

BSc

pf1 ´ τyq ^ ds

This vanishes in Cases (1) and (3), and we are done. In Case (2), BSc also containsa small loop around P . In that case we continue to obtain:

“

ż

BDP p1cq

pf1 ´ τyq ds

“

ż

BDP p1cq

f1 ds.

Notice thatş

BDP p1cqf1 ds decays like 1

c2πδ`1 .

The result now follows by taking the limit as cÑ `8.

Lemma 10.6. There is a unique solution to (10.5) in Y1 and in Y2.If a “ Ts` c, then there is an 2-parameter family of solutions to (10.2) in Y3,

parametrized by Rˆ S1.

Proof. We begin with Case 1. Equation (10.5) can be rewritten in coordinates as

(10.13)

#

Bsf1 ´ Btf2 ` h1pef1q ´ ν1 “ 0

Btf1 ` Bsf2 “ 0

Let us start by considering solutions where f2 ” k is constant, and f1ps, tq “f1psq is independent of t. Then, we get the s-dependent ODE

df1

ds` h1pef1q ´ ν1 “ 0.

Write it asdf1

ds“ F pf1, sq

where F px, sq “ ´h1pexq ` ν1psq.

Claim. For every s0 P R and every c1 P R, there is a unique solution f1 : R Ñ Rsuch that f1ps0q “ c1.

To prove the claim, recall the assumptions that h1pρq ě 0 and h1pρq “ 0 forρ ď 2. These and the asymptotic behavior of ν imply that there are constants Cand C such that:

‚ F px, sq ď C for all px, sq;

‚ F px, sq ě C for x ď ln 2, uniformly in s.

The claim now follows from standard existence and uniqueness theory for solutionsto ODE’s.


s

f1psq

b`

b´

F ě 0

F ď 0

ν “ T´s

F ě 0

F ď 0

ν “ T`s

Figure 10.1. The function f1

The facts that for s ăă 0 we have that b´ is the only zero of the functionx ÞÑ F px, sq and that BxF px, sq ď 0 (a consequence of h2pxq ě 0) imply that there

is a unique solution f1 of the ODE such that limsÑ´8 f1psq “ b´, see Figure 10.1.

Now, f0ps, tq “ pf1psq, 0q is a solution to (10.13) in Y1. The fact that it has

the correct exponential convergence properties at ˘8 follow from f1 solving theODE above. We want to show that there are no other solutions to (10.13) in Y1.The proof will be an application of positivity of intersections of pseudoholomorphiccurves in dimension 4.

Suppose that there is a solution f1 “ pf11 , f

12 q of (10.13) in Y1 such that f1

2 isnot constant. By the definition of Y1 and by Lemma 10.5, limsÑ˘8 f

12 ps, tq “ 0.

Since f12 is not constant, there is ps0, t0q P R ˆ S1 such that f1

2 ps0, t0q ‰ 0. Letf2

1 : RÑ R be the unique solution to the ODE above, for which f21 ps0q “ f1

1 ps0, t0q.Then, f2ps, tq “ pf2

1 psq, f12 ps0, t0qq is another solution to (10.13) (although pf2

1 , 0qdoes not have the correct asymptotic behavior to belong to Y1).

Now, observe that (10.13) is the Floer equation on the Kahler manifold pR ˆS1, dx^ dy, iq, for the time-dependent Hamiltonian

H : Rˆ pRˆ S1q Ñ R

ps, x, yq ÞÑ

ż x

0

h1pezqdz ´ ν1psqx

By Gromov’s trick [Gro85], solutions to the Floer equation can be thought of aspseudoholomorphic curves Rˆ S1 Ñ M :“ pRˆ S1q ˆ pRˆ S1q, for some twistedalmost complex structure on M . The projection of these curves to the first cylinderfactor is the identity.

Since limsÑ´8 f12 “ limsÑ´8 f

02 , Lemma 10.5 implies that they have lifts f1

2

and f02 to R, with the same limits as sÑ ˘8. By assumption, f1 and f0 are both

in Y1, which implies that limsÑ˘8 f11 ps, tq “ limsÑ˘8 f

01 ps, tq “ b˘. Therefore,

there is a homotopy from f1 to f0 that is C0-small on neighborhoods of ˘8ˆ S1

in R ˆ S1. This gives a homotopy from the graph of f1 to the graph of f0 in M ,and the intersections with the graph of f2 during the homotopy will remain in a


compact region of M . So, we have an equality of signed intersection numbers

#pGraphpf1q XGraphpf2qq “ #pGraphpf0q XGraphpf2qq “ 0,

since the second components of f0 and f2 are different constants. But then positivityof intersections of pseudoholomorphic curves in dimension 4 [MS04, Exercise 2.6.1]implies that the graphs of f1 and f2 do not intersect, which, by the constructionof f2, is a contradiction. Therefore, there is no such f1 to start with, which provesthe first statement in the Lemma in Case 1.

Case 2 can be dealt with by the same argument, applied to g P Y2. We againapply Lemma 10.5. The argument for Case 1 above produces a solution g “

pg1psq, 0q P Y1. We need to argue that such a solution is indeed in Y2. This means

that pg ˝ ϕP q P W1,p,δC´ pp´8,´1s ˆ S1,R ˆ S1q. This will be a consequence of the

fact that g1 is C1, and of its asymptotics. Write P “ pPs, Ptq. Then,

pg ˝ ϕP qpρ, θq “ pg1pe2πρ cosp2πθq ` Psq, 0q

and for ρ ăă ´1 the Mean Value Theorem gives

|g1pe2πρ cosp2πθq ` Psq ´ g1pPsq| ď ||g

11||L8 e2πρ

where the sup norm is finite by the exponential convergence of g1 at ˘8. Sinceδ ă 2π, we get the desired exponential convergence at P .

Let us now consider the case when u “ pa, uq is such that a “ Ts` c, and lookin Y3 for solutions to (10.2). This is equivalent to the system of equations

(10.14)

#

Bsf1 ´ Btf2 ` h1pef1q ´ T “ 0

Btf1 ` Bsf2 “ 0

We again have solutions of the form fps, tq “ pf1psq, f2q with f2 constant. Thereis an R ˆ S1 family of such solutions, parametrized by limsÑ´8pf1psq ´ Ts, f2q P

R ˆ S1. The union of the graphs of these solutions foliates the region pR ˆ S1q ˆ

pp´8, b`qˆS1q Ă pRˆS1qˆpRˆS1q. We can apply Lemma 10.5 to equation (10.2),

since it is a special case of (10.4) when τ “ 1. The argument above using positivityof intersections can then be adapted to prove that there are no other solutions withthe same asymptotics. To get different solutions, we can replace pf1psq, f2q withpf1psq, f2q “ pf1ps` c1q, f2 ` c2q, for a constant pc1, c2q P Rˆ S1.

Proof of Proposition 10.1. Most of the claims made in the statement follow fromthe existence of a solution e to (10.1) with the appropriate asymptotics. Indeed,given such e “ pe1, e2q, one can construct v “ pb, vq by taking

pb, vq “ pa` e1, φe2R ˝ uq.

As we pointed out, the existence of such e is equivalent to the existence of a solutionf to (10.2). Let us consider different cases separately.

In Case 1, note that Lemmas 10.4 and 10.3 and the Implicit Function Theoremimply that solutions to (10.4) vary smoothly in τ . Lemma 10.6 guarantees theexistence and uniqueness of solutions to (10.5) in Y1 (which corresponds to τ “ 0).Therefore, we conclude that (10.2) (which corresponds to τ “ 1) also has a uniquesolution in Y1, which finishes the proof.

Case 2 works the same way, with solution g to (10.7) instead of solutions f to(10.4).


Case 3 corresponds to the final part of Proposition 10.1, when u is assumedto be a trivial cylinder. Lemma 10.6 implies that there is an R ˆ S1-family ofsolutions to (10.2) in Y3. By Lemma 10.5, there is an R-family of such solutionswith limsÑ˘8 f2ps, tq “ 0 P S1. This finishes the proof.

Remark 10.7. We now briefly compare the Fredholm indices of the Floer cylinders,the holomorphic cylinders in Rˆ Y and the cylinders satisfying equation (10.1).

Cases (1) and (2) are similar. We discuss Case (1) in detail. The Floer cylin-der from pqk´ to qpk` has Fredholm index Mppq ´ Mpqq ´ 1 ` 2 τX´KK pk` ´ k´q.Observe that if this projects to the class A P H2pΣ;Zq, we have k` ´ k´ “

KωpAq, and thus this index is Mppq ´Mpqq ´ 1` 2 xc1pTΣq, Ay. The correspond-ing holomorphic cylinder, without any marker constraints, has Fredholm indexMppq ´Mpqq ` 2 xc1pTΣq, Ay ` 2 and thus, with the two marker constraints, hasindex Mppq´Mpqq`2 xc1pTΣq, Ay. Equation (10.1) has Fredholm index 0. Noticethat a holomorphic cylinder in R ˆ Y comes in a R-parametric family obtainedby translation in R ˆ Y . Each translate gives rise to the same Floer solution, sowe must consider the quotient by this action. Therefore, the index of the Floerequation is 1 less than the sum of the index of the holomorphic curve equation withthat of (10.1).

Case (3) is different in that the split Floer cylinder goes from a critical pointx PW to qpk P RˆY . Notice that as a split Floer cylinder, this has N “ 2, and thuswill need to be quotiented by an R2 action, hence the Fredholm index must be 2.This will appear in the differential when the grading difference of x and of qpk is 1:1 “Mppq`1´2n`Mpxq`2 τX´KK k. The lower level of this is a plane in W , whichwe interpret as a sphere in X, representing the spherical homology class B. Wehave k “ B ‚ Σ “ KωpBq. The Fredholm index of the plane W with asymptoticmarker condition is given by Mppq ` Mpxq ` 1 ´ 2n ` 2 xc1pTXq, By ´ 2B ‚ Σ.Notice that this is the grading difference between qpk and x. The Fredholm indexof Equation (10.1) is 2, with no marker condition. The marker condition cuts itsdimension down by 1, giving us the index 2 required.

10.1. Geometric interpretation. Propositions 9.1 and 10.1 play a very impor-tant role in our description of the symplectic homology differential, so we willsummarize their geometric meaning.

Recall that by Propositions 7.4 and 7.5, there are four types of contributions tothe differential, referred to as Cases 0, 1, 2 and 3.

Case 0 configurations are gradient flow lines in Y , so they can be described bythe Morse differential in Y .

Case 1 configurations are elements of 1-dimensional fiber products

W sfY ppqq ˆev

ĂM˚

H,1pYk, Yl;A; JY q ˆev WufY pqpq

where q, p P CritpfΣq and A P H2pΣ;Zq. Recall that M˚H,1pYk, Yl;A; JY q is the

space of parametrized unpunctured Floer cylinders v, going from orbits of multi-plicity k (as s Ñ ´8) to orbits of multiplicity l (as s Ñ `8), where l ą k. Thefiber product above has a free action of R by translations of v in the s-direction, soit is a finite disjoint union of copies of R. Applying Proposition 9.1 to such a Floercylinder v, we get that v can be expressed in terms of a pseudoholomorphic cylinderu in Rˆ Y and a map e : Rˆ S1 Ñ Rˆ S1 satisfying equation (9.1). If we requirethat the asymptotic limits of u be the Reeb orbits underlying the asymptotic limits


of v, we can say that the pair pu, eq is unique up to an R-shift on the domains of uand e.

If we now start with a pseudoholomorphic curve u going from a Reeb orbit ofmultiplicity k to one of multiplicity l (like the one in the previous paragraph),Proposition 10.1 says that there is a Floer cylinder v and a solution e of (10.1),such that u can be expressed in terms of v and e. If we require that the Reeb orbitsunderlying the asymptotic limits of v be the asymptotic limits of u, then the pairpv, eq is unique. So, combining Propositions 9.1 and 10.1, we get a bijection betweenM˚

H,1pYk, Yl;A; JY q and the space of JY -holomorphic cylinders in RˆY , going fromReeb orbits of multiplicity k to Reeb orbits of multiplicity l. In Section 11, we willrelate such pseudoholomorphic cylinders with pseudoholomorphic spheres in Σ andmeromorphic sections of line bundles over CP 1.

Case 2 configurations are elements of 1-dimensional moduli spaces

W sfY pppq êv

´

M˚0 pB; JW q êv

ĂM˚

H,1pYk, Yl; 0;JY q¯

êv WufY pqpq

where p P CritpfΣq, B P H2pX;Zq. Recall that M˚0 pB; JW q is a space of JW -

holomorphic planes U in W (which can be identified with a space of JX -holomorphicspheres in X, by Lemma 2.8) and M˚

H,1pYk, Yl; 0;JY q is a space of Floer cylinders

v : Rˆ S1ztP u Ñ Rˆ Y projecting to a point in Σ, asymptotic at ´8 (resp. `8)to an orbit of multiplicity k (resp. l) and with a negative puncture at P P Rˆ S1

that is asymptotic to a Reeb orbit of multiplicity l ´ k. The stable and unstablemanifolds in the fiber product force v to project to p P Σ. Hence, U is asymptoticto a Reeb orbit of multiplicity l ´ k “ B ‚ Σ over p.

Applying Propositions 9.1 and 10.1 to v now implies that M˚H,1pYk, Yl; 0;JY q can

be identified with the space of JY -holomorphic cylinders in RˆY that are multiplecovers of a trivial cylinder, with negative punctures at ´8 and P asymptotic toorbits of multiplicity k and l´ k, respectively, and with a positive puncture at `8asymptotic to an orbit of multiplicity l.

Finally, Case 3 configurations are elements of fiber products

WufW pxq êv

`

M˚H,1pB; JW q êv M˚

H,1p0; JY , JW q˘

êv WufY pqpq

where p P CritpfΣq, x P CritpfW q and B P H2pX;Zq. Here, M˚H,1pB; JW q is a

space of JW -holomorphic planes U in W that are asymptotic to a Reeb orbit ofmultiplicity k “ B‚Σ (recall that H “ 0 in W after neck-stretching, so we can thinkof JW -holomorphic planes as Floer cylinders that asymptote to constants at ´8).M˚

H,1p0; JY , JW q is a space of Floer cylinders v in R ˆ Y that project to a pointin Σ, asymptote at `8 to a Hamiltonian orbit of multiplicity k, and asymptote at´8 to a Reeb orbit of multiplicity k. The unstable manifold at the right of thefiber product forces v to project to p in Σ. The orbit U asymptotes to must alsoproject to p in Σ.

The last parts of Propositions 9.1 and 10.1 applied to v now imply that M˚H,1p0; JY , JW q

can be identified with the space of trivial JY -holomorphic spheres in R ˆ Y , overReeb orbits of multiplicity k.


11. Pseudoholomorphic spheres in Σ and meromorphic sections

The previous sections explain that split Floer cylinders are in some sense equiv-alent to JY -holomorphic curves. We now want to describe JY -holomorphic curvesin a manner that is more suitable for computing the Floer differential.

Let u “ pa, uq : RˆS1zΓ Ñ RˆY be a JY -holomorphic curve, where Γ is eitherempty or the singleton tP u. Since JY is RˆS1-equivariant, it is a lift of an almostcomplex structure JΣ in Σ, and one can project u to obtain a JΣ-holomorphic mapw : R ˆ S1zΓ Ñ Σ. Since punctures of finite energy pseudoholomorphic curves inR ˆ Y are asymptotic to Reeb orbits in Y , and these are multiple covers of thefibres of Y Ñ Σ, w extends to a JΣ-holomorphic map CP 1 Ñ Σ [MS04, Theorem4.1.2]. We are using the standard identification of Rˆ S1 with C˚ Ă CP 1.

The symplectization R ˆ Y can be identified with the complement of the zerosection in a complex line bundle E Ñ Σ (the dual to the normal bundle to Σ inX). We have the following commutative diagram

w˚pEz0q Rˆ Y “ Ez0

CP 1 ΣRˆ S1zΓw

pa, uq

s

where s is a section of w˚E Ñ CP 1 with a zero at 0 (and at P , if Γ “ tP u) and apole at 8. The line bundle w˚E has an induced complex linear Cauchy–Riemannoperator, in the sense of [MS04, Section C.1]. Complex linearity is a consequenceof the Reeb-invariance of JY . Such a Cauchy–Riemann operator corresponds toa unique holomorphic structure on w˚E [Kob87, Proposition 1.3.7]. The sections is in the kernel of this operator and is therefore meromorphic. This proves thefollowing result.

Lemma 11.1. Every pseudoholomorphic curve u “ pa, uq : R ˆ S1zΓ Ñ R ˆ Ydefines a JΣ-holomorphic map w : CP 1 Ñ Σ and a meromorphic section of w˚E ÑCP 1, with zero at 0 (and at P , if Γ “ tP u) and pole at 8.

We can now reduce the question of counting punctured JY -holomorphic maps uto that of counting JΣ-holomorphic curves w : CP 1 Ñ Σ, together with meromor-phic sections s of w˚E, with prescribed zeros and poles. The count of maps w isrelated with the computation of genus zero Gromov–Witten numbers of Σ [MS04].We can also give a complete description of the relevant meromorphic sections s.Given a divisor of points D in CP 1 and a holomorphic line bundle L over CP 1,where both D and L have degree d, there is a C˚-family of meromorphic sectionsof L, such that the divisor associated with each section is D. One can justify thisfact by reducing it to the simplest case of trivial L: use a trivialization of L overC Ă CP 1 to identify meromorphic sections of L with meromorphic functions onCP 1 [Mir95, pp 342–345].

Remark 11.2. Ignoring transversality issues, the fact that meromorphic sections ofline bundles over CP 1 come in C˚-families implies that the contact homology dif-ferential (without point constraints) should vanish for prequantization bundles (see


[EGH00]), because the moduli spaces of non-constant pseudoholomorphic curveswith cylindrical ends have an S1-action without fixed points. This is not the casein our setting, though, because we impose marker conditions on our asymptoticlimits (as in non-equivariant contact homology, see Section 3.2 of [BO09a]).

Fix a JΣ-holomorphic map w : CP 1 Ñ Σ and let u “ pa, uq : Rˆ S1zΓ Ñ Rˆ Ybe a JY -holomorphic lift. Given pθ1, θ2q P S

1 ˆ S1, we can produce a new JY -holomorphic curve

pθ1, θ2q.u “ upθ1,θ2q

such that

upθ1,θ2qps, tq “ paps, t` θ1q, φθ2 ˝ ups, t` θ1qq.

Lemma 11.3. The number of such lifts that have prescribed asymptotic markersat ˘8 is k` ´ k´ “ ´xc1pE Ñ Σq, w˚rCP 1sy `

ř

PPΓ kP ą 0.

Proof. This result follows immediately from the observation that the discussion ofthe linear maps rotv before Lemma 6.34 can be adapted to also to JY -holomorphiccurves (instead of just Floer cylinders). The number of lifts we want is the absolutevalue of the degree of the appropriate linear map T 2 Ñ T 2, which is the absolutevalue of the determinant of the matrix in equation (6.7), i.e. k` ´ k´ ą 0 (wherethe inequality follows from Lemma 6.34). The alternative formula for k` ´ k´ inthe statement follows from the consideration in the proof of Lemma 6.34.

11.1. Back to orientations. We wish to compare the signs that appear in thedifferential to the signs involved in counts of pseudoholomorphic spheres (as inGromov–Witten theory). It is useful to begin with some general properties of thefiber sum orientation, introduced in Definition 8.4. The first result is Proposition7.5.(a) in [Joy12] (which has the same fiber sum orientation convention as Definition8.4). We include a proof here because the structure of its argument will be the modelfor other proofs in this section.

Lemma 11.4. Let fi : V1 ÑW be as in Definition 8.4. Consider the map r : V1 ‘

V2 Ñ V2 ‘ V1, such that rpv1, v2q “ pv2, v1q. Then, r induces an identification

ker`

pf1 ´ f2q : V1 ‘ V2 ÑW˘

Ñ ker`

pf2 ´ f1q : V2 ‘ V1 ÑW˘

,

which changes the fiber sum orientations by p´1qpdimV1`dimW qpdimV2`dimW q.

Proof. Call f :“ pf1´f2q : V1‘V2 ÑW and g :“ pf2´f1q : V2‘V1 ÑW . Denotingalso by f and g the appropriate isomorphisms in Part (1) of Definition 8.4, we havean isomorphism

g ˝ r ˝ f´1 : W ÑW,

which is ´ Id, hence it changes orientation by p´1qdimW .Denote also by r the induced map pV1 ‘ V2q ker f Ñ pV2 ‘ V1q ker g. A simple

calculation yields the following.

Claim. Let s12 : pV1 ‘ V2q ker f Ñ V1 ‘ V2 be a right-inverse for the projectionπ12 : V1‘V2 Ñ pV1‘V2q ker f . Then, s21 :“ r˝s12˝r

´1 : pV2‘V1q ker g Ñ V2‘V1

is a right-inverse for π21 : V2 ‘ V1 Ñ pV2 ‘ V1q ker g.


We can now justify the sign in the statement of the Lemma. By the Claim, thefollowing diagram commutes:

ker fà `

pV1 ‘ V2q ker f˘ i12`s12

> V1 ‘ V2

ker gà

`

pV2 ‘ V1q ker g˘

r‘r_

i21`s21> V2 ‘ V1

r

_

where i12 and i21 are inclusion maps.By Part (1) in Definition 8.4 and the argument above the Claim, r : pV1 ‘

V2q ker f Ñ pV2 ‘ V1q ker g changes orientations by p´1qpdimV1`dimV2`1q dimW .By Part (2) in Definition 8.4 and the fact that r : V1 ‘ V2 Ñ V2 ‘ V1 changes ori-entations by p´1qdimV1 dimV2 , we get the remaining contribution to the sign in theLemma.

Definition 11.5. Let f1, f2 be as in Definition 8.4. Denote by ∆ Ă W ‘ Wthe diagonal, and orient it as the image of the map w ÞÑ pw,wq. The preimageorientation on the fiber sum V1 f1

‘ f2V2 “ pf1, f2q

´1p∆q Ă V1‘V2 is so that, if H Ă

V1 ‘ V2 is a complementary subspace to pf1, f2q´1p∆q, the following isomorphisms

preserve orientations

(1) H ‘ pf1, f2q´1p∆q Ñ p´1qdimV2pdimV1`dimW qV1 ‘ V2;

(2) pf1, f2qpHq ‘∆ ÑW ‘W .

Remark 11.6. If g1 : X Ñ Y and g2 : Z Ñ Y are transverse maps of smooth orientedmanifolds, then the previous definition can be applied to maps of tangent spaces toorient the fiber product X g1

ˆ g2Z. In the special case when g2 is the inclusion of

a submanifold, this agrees with the preimage orientation defined in [GP74]. This isthe reason for the name used for this orientation convention, and for the complicatedsign in Part (1) above.

Lemma 11.7. The identification of ker f with the fiber sum orientation (as inDefinition 8.4) and pf1, f2q

´1p∆q with preimage orientation (as in Definition 11.5)changes orientations by p´1qpdimV1`dimW qpdimV2`dimW q.

Proof. For convenience, denote Parts (1) and (2) in Definition 11.5 by (P1) and(P2), respectively, and similarly denote by (F1) and (F2) the Parts in Definition8.4. Note that to write (F2) in this case, we need a section s : V1‘V2

ker f Ñ V1 ‘ V2.

We will take H “ spV1‘V2

ker f q in (P1).


The result follows from a sequence of oriented isomorphisms:

W ‘W ‘ ker f – p´1qpdimV1`dimV2´dimW q dimWW ‘ ker f ‘W –

pF1q– p´1qdimV1 dimW`dimWW ‘ ker f ‘

V1 ‘ V2

ker f–

pF2q– p´1qdimV1 dimW`dimWW ‘ V1 ‘ V2 –

pP1q– p´1qdimV1 dimW`dimV1 dimV2`dimV2 dimW`dimW .

.W ‘H ‘ pf1, f2q´1p∆q –

– p´1qdimV1 dimW`dimV1 dimV2`dimV2 dimW`dimW .

.∆‘ pf1, f2qpHq ‘ pf1, f2q´1p∆q –

– p´1qdimV1 dimW`dimV1 dimV2`dimV2 dimW .

.pf1, f2qpHq ‘∆‘ pf1, f2q´1p∆q –

pP2q– p´1qdimV1 dimW`dimV1 dimV2`dimV2 dimWW ‘W ‘ pf1, f2q

´1p∆q –ϕ– p´1qdimV1 dimW`dimV1 dimV2`dimV2 dimW`dimWW ‘W ‘ pf1, f2q

´1p∆q

where ϕ “

ˆ

1 11 ´1

˙

‘ id : W ‘ W ‘ pf1, f2q´1p∆q Ñ W ‘ W ‘ pf1, f2q

´1p∆q.

Note that ϕ changes orientations by p´1qdimW . The composition of the chain ofisomorphisms above is of the form

ˆ

2 ˚

0 1

˙

‘ id : W ‘W ‘ ker f ÑW ‘W ‘ pf1, f2q´1p∆q

so it preserves orientations iff the identity ker f Ñ pf1, f2q´1p∆q does. The result

now follows.

Remark 11.8. Since the signs appearing in Lemmas 11.4 and 11.7 are the same,the order-reversing map r in Lemma 11.4 induces an orientation-preserving isomor-phism between kerpf1 ´ f2q with the fiber sum orientation and pf2, f1q

´1p∆q withthe preimage orientation (note the order of the fi).

Let us now study the signs that appear in the split symplectic homology dif-ferential. We start with Case 1 configurations. Let p, q P CritpfΣq and consider aFloer cylinder with one cascade giving a contribution of pqk to the differential of qpl.As we saw in (8.2), these cascades form spaces

(11.1) Mppqk, qpl;Aq :“W sfY ppqq êv

ĂM˚

H,1pYk, Yl;A; JY q êv WufY pqpq

for suitable A P H2pΣ;Zq. Each such cascade projects to a chain of pearls in Σfrom q to p, with exactly one sphere, contained in

(11.2) Mpq, p;Aq :“W sfΣpqq êv M˚

2 pA; JΣq êv WufΣppq.

Note that Mpq, p;Aq can be identified with ev´1´

W sfΣpqq ˆWu

fΣppq

¯

, for

ev : M˚2 pA; JΣq Ñ Σˆ Σ

w ÞÑ pwp0q, wp8qq


Definition 11.9. The Gromov–Witten orientation on Mpq, p;Aq is the one induced

by the identification with ev´1´

W sfΣpqq ˆWu

fΣppq

¯

, where the latter is equipped

with the preimage orientation.

Remark 11.10. To understand the reason for this terminology, suppose that thesestable and unstable manifolds defined cycles in H˚pΣq. An element in Mpq, p;Aqwould then contribute to a two-point Gromov–Witten invariant of Σ, with insertionsgiven by the stable and unstable manifolds. The sign of this contribution wouldbe the the one prescribed by the Gromov–Witten orientation (see [MS04, Exercise7.1.2]).

Our goal is to compare

‚ the sign of the contribution of a Floer cylinder with cascades in (11.1) tothe symplectic homology differential

‚ to the sign induced on the projected chain of pearls in (11.2) by theGromov–Witten orientation.

This will be the content of Proposition 11.15. To compare the two signs, it will beuseful to also consider the fiber products

(11.3) Mpqqk, ppl;Aq :“W sfY pqqq êv

ĂM˚

H,1pYk, Yl;A; JY q êv WufY pppq

and

(11.4) Mpqqk, ppl;Aq :“ π´1Σ

`

W sfΣpqq

˘

êvĂM˚

H,1pYk, Yl;A; JY q êv π´1Σ

`

WufΣppq

˘

,

which contains Mpqqk, ppl;Aq as an open dense subset (which justifies the notation).

W sfYpqqq is an open subset of π´1

Σ

´

W sfΣpqq

¯

, which is an S1-bundle over W sfΣpqq.

Note also that W sfYppqq specifies a section of this bundle, which we use to trivialize

π´1Σ

´

W sfΣpqq

¯

– W sfΣpqq ˆ S1 (identifying W s

fYppqq with W s

fΣpqq ˆ t1u). We can

now write W sfYppqq “ π´1

Σ

´

W sfΣpqq

¯

qsˆ it1u, where the maps in the fiber product

are qs : π´1Σ

´

W sfΣpqq

¯

Ñ S1 and i : t1u ãÑ S1. Similarly, we can write WufYpqpq “

π´1Σ

´

WufΣppq

¯

puˆ it1u for the analogous map pu : π´1Σ

´

WufΣppq

¯

Ñ S1.

Note that Mppqk, qpl;Aq is a codimension 2 submanifold of Mpqqk, ppl;Aq. Givenan element px, u, yq PMppqk, qpl;Aq, there is a splitting

(11.5) Tpx,u,yqMpqqk, ppl;Aq –`

Tpx,u,yqMppqk, qpl;Aq˘

‘ Rdomain ‘ Rtarget

where Rdomain is the direction spanned by an infinitesimal rotation of the Floercylinder u on the domain (in the t-direction) and Rtarget is spanned by the infini-tesimal rotation of u on the target (in the Reeb direction).

Lemma 11.11. If we take the fiber product orientation on (11.1) and (11.4), thenthe splitting (11.5) preserves orientations....

Proof. Using the associativity of the fiber product orientation and Lemma 11.4 onthe third identity, we have the oriented diffeomorphisms (dropping A and JY from


the notation)

Mppqk, qplq “W sfY ppqq êv

ĂM˚

H,1pYk, Ylq êv WufY pqpq “

“`

π´1Σ

`

W sfΣpqq

˘

qsˆ it1u˘

êvĂM˚

H,1pYk, Ylq êv

`

π´1Σ

`

WufΣppq

˘

puˆ it1u˘

“

“ p´1qdimπ´1Σ pW

sfΣpqqq`1.

.t1u iˆ qs

`

π´1Σ

`

W sfΣpqq

˘

êvĂM˚

H,1pYk, Ylq êv π´1Σ

`

WufΣppq

˘ ˘

puˆ it1u “

“ p´1qindfΣ pqqt1u iˆ qsMpqqk, pplqpuˆ it1u

and we get an oriented isomorphism of vector spaces

Tpx,u,yqMppqk, qpl;Aq – p´1qindfΣ pqq0 di‘ dqs

`

Tpx,u,yqMpqqk, ppl;Aq˘

dpu‘ di 0

(where we wrote 0 instead of Tt1ut1u).

On the other hand, abbreviating R2 :“ Rdomain ‘ Rtarget and thinking of thedirect sum as a fiber product (of maps to the zero vector space), we get

0 di‘ dqs

` `


‘ R2˘

dpu‘ di 0 “

“ 0 di‘ dqs

`

Tpx,u,yqMppqk, qpl;Aq ‘ pR2dpu‘ di 0q

˘

“

“ p´1qdimMppqk,qpl;Aq0 di‘ dqs

`

pR2dpu‘ di 0q ‘ Tpx,u,yqMppqk, qpl;Aq

˘

“

“ ´`

0 di‘ dqs R2dpu‘ di 0

˘

‘ Tpx,u,yqMppqk, qpl;Aq “

“ Tpx,u,yqMppqk, qpl;Aq.

The first identity above uses the associativity of the fiber product, the second usesLemma 11.4, the third uses the fact that dimMppqk, qpl;Aq “ 1 (since we are studyingcontributions to the differential) and again the associativity property. The lastidentity follows from the fact that the fiber sum orientation on the zero dimensionalvector space 0 di‘ dqsR2

dpu‘ di 0 is negative, which is a simple calculation.The Lemma follows from combining the previous two computations.

In what follows, it is useful to work from a slightly more abstract point of view.Let πA : A Ñ A and πB : B Ñ B be S1-bundles. Suppose that there are S1-equivariant maps f1 : A Ñ Y and f2 : B Ñ Y , and denote their projections byf1 : A Ñ Σ and f2 : B Ñ Σ. Then, the fiber product A

f1ˆf2B is an S1-bundle

over A f1ˆ f2

B (with ‘diagonal’ S1-action on Af1ˆf2B).

Recall that the contact form on Y induces an Ehresmann connection on πΣ : Y ÑΣ. The S1-equivariance of f1 implies that there is a unique Ehresmann connectionon A inducing the top arrow in the commutative diagram

(11.6)

TaA‘ R > TaA

Tf1paqΣ‘ R

df1‘id

_

> Tf1paqY

df1

_

where a P A and a “ πApaq. Do the same for B.

These connections induce a connection on the S1-bundle Af1ˆf2B as follows.

Given pa, bq P Af1ˆf2B, we can take the horizontal subspaceHa dpπΣ˝f1q

‘dpπΣ˝f2q

Hb.


This can be identified with Tpa,bqpA f1ˆ f2

Bq, via the restriction of pπA, πBq to thefiber sum. This gives an identification

(11.7) Tpa,bqpA f1

ˆf2Bq –

`

Tpa,bqpA f1ˆ f2

Bq˘

‘ R,

where R stands for the vector space generated by the infinitesimal generator of theS1-action on A

f1ˆf2B. We rewrite (11.7) as an isomorphism

χ : kerpdf1 ´ df2qpa,bq ‘ RÑ kerpdf1 ´ df2qpa,bq.

Pick an orientation on A and orient A so that the splitting TaA – TaA ‘ Rinduced by the connection on A preserves orientations. Pick an orientation on Band orient B in an analogous manner.

Lemma 11.12. The identification (11.7) preserves orientations, if we use the fiberproduct orientation on both fiber products, and orient R in the direction of theinfinitesimal S1-action on A

f1ˆf2B.

Proof. The structure of the argument is parallel to that of Lemma 11.4.Fix pa, bq P A

f1ˆf2B, so that πApaq “ a and πBpbq “ b. Write df :“

pdf1 ´ df2q : TA ‘ TB Ñ TY and df :“ pdf1 ´ df2q : TA ‘ TB Ñ TΣ, where

we omited subscripts a, b, a, b to make the notation somewhat lighter. We will dothis throughout the proof. Denote by ξ : TΣ ‘ R Ñ TY the isomorphism inducedby the contact structure on Y . The commutative diagram

TA‘ TB

kerpdfq‘ R rdfs‘id

> TΣ‘ R

TA‘ TB

kerpdfq

ϕ_

rdfs> TY

ξ

_

defines the isomorphism ϕ.Denote by ψ the composition TA‘TB‘R‘RÑ TA‘R‘TB‘RÑ TA‘TB,

where the first isomorphism permutes the second and third factors and the secondisomorphism is induced by the connections on A and B.

Claim. Let s : TA‘TBkerpdfq Ñ TA ‘ TB be a right-inverse for the projection π : TA ‘

TB Ñ TA‘TBkerpdfq . Then, s :“ ψ ˝ ps ‘ p1, 0qq ˝ ϕ´1 : TA‘TB

kerpdfqÑ TA ‘ TB is a right-

inverse for π : TA‘ TB Ñ TA‘TBkerpdfq

.

The claim follows from the fact that the following diagram commutes, which isa simple calculation (using (11.6) and its analogue for B).

TA‘ TB ‘ R‘ Rπ‘

´

1 ´1¯

>TA‘ TB

kerpdfq‘ R

TA‘ TB

ψ_

π>TA‘ TB

kerpdfq

ϕ_


We can now justify the sign in the statement of the Lemma. The Claim can beused to show that the following diagram commutes:

`

kerpdfq ‘ R˘à

ˆ

TA‘ TB

kerpdfq‘ R

˙

i`α`s`β> TA‘ TB ‘ R‘ R

kerpdfqà TA‘ TB

kerpdfq

χ‘ϕ_

i`s> TA‘ TB

ψ_

where α‘ β : R‘ RÑ R2 is the linear map represented by

ˆ

1 11 0

˙

.

By Part (1) in Definition 8.4, ϕ changes orientations by p´1qdimB dim Σ`dim B dimY “

p´1qdimB`1. The map ψ changes orientations by p´1qdimB (the sign comes from

the permutation, since the orientations on A and B behave well with the connec-tions). Note that to get the map α‘ β (which has negative determinant), we needto permute R with TA‘TB

kerpdfq at the top of the diagram. The permutation picks up

no sign, since dim TA‘TBkerpdfq “ dim Σ is even. By Part (2) in Definition 8.4 and

the commutativity of the diagram, we conclude that the isomorphism χ changesorientations by p´1qdimB`1`dimB`1 “ 1, as wanted.

The space (11.4) is an S1-bundle over (11.2). With respect to the connectionsdiscussed above (see (11.6)), we have a decomposition

(11.8) Tpx,u,yqMpqqk, ppl;Aq –`

TpπΣpxq,πΣ˝u,πΣpyqqMpq, p;Aq˘

‘ R

Lemma 11.13. The isomorphism (11.8) preserves orientations, if the fiber prod-ucts on the left and right are given their fiber product orientations.

Proof. This follows from applying the Lemma 11.12 twice, to the S1-bundles π´1Σ

´

W sfΣpqq

¯

Ñ

W sfΣpqq, ĂM

˚

H,1pYk, Yl;A; JY q Ñ M˚2 pA; JΣq and π´1

Σ

´

WufΣppq

¯

Ñ WufΣppq. Note

that Lemma 8.2 implies that the coherent orientation on B “ ĂM˚

H,1pYk, Yl;A; JY qand the usual complex orientation on B “ M˚

2 pA; JΣq are such that the identifi-

cation TbB – TbB ‘ R is orientation-preserving, as was assumed in Lemma 11.12(see the paragraph above the Lemma).

Lemma 11.14. The fiber product orientation on (11.2) differs from the preimage

orientation on (11.2) by p´1qdimW sfΣpqq dimWu

fΣppq.

Proof. This follows from applying Lemma 11.7 twice. Writing (11.2) as

pW sfΣpqq êv M˚

2 pA; JΣqq êv WufΣppq

we apply Lemma 11.7 to W sfΣpqq ãÑ Σ and ev1 : M˚

2 pA; JΣq Ñ Σ to get a sign

p´1qpdimW sfΣpqq`dim ΣqpdimM˚

2 pA;JΣq`dim Σq“ 1

since M˚2 pA; JΣq and Σ are even dimensional. We then apply Lemma 11.7 to

ev2 : W sfΣpqq êv M˚

2 pA; JΣq Ñ Σ and WufΣppq ãÑ Σ to get the sign

p´1qdimpW sfΣpqqêvM˚

2 pA;JΣq`dim ΣqpdimWufΣppq`dim Σq

“ p´1qdimW sfΣpqq dimWu

fΣppq.


Combining these results, we get the following.

Proposition 11.15. The sign with which an element of (11.1) contributes to thesymplectic homology differential agrees with the sign with which the projection of thiselement (to an element in (11.2)) would contribute to a Gromov–Witten invariantin Σ.

Proof. Combining Lemmas 11.11 and 11.13, we can relate the fiber product orien-tations on (11.1) and (11.2):`


‘ Rdomain ‘ Rtarget – Tpx,u,yqMpqqk, ppl;Aq –

–`

TpπΣpxq,πΣ˝u,πΣpyqqMpq, p;Aq˘

‘ RThe factors Rtarget on the left and R on the right can be identified, yielding anorientation-preserving isomorphism

`


‘ Rdomain – TpπΣpxq,πΣ˝u,πΣpyqqMpq, p;Aq.

Observe that Mpq, p;Aq in (11.2) can be identified with

(11.9) M˚2 pA; JΣq êv

`

W sfΣpqq ˆWu

fΣppq

˘

.

The sign with which an element px, u, yq PMppqk, qpl;Aq contributes to the symplec-tic homology differential is the sign of the zero-dimensional vector space obtainedby taking the quotient of Tpx,u,yqMppqk, qpl;Aq by translation in the s-variable inthe domain of the Floer cylinder u. This oriented zero-dimensional vector spacecoincides with the quotient of

`


‘ Rdomain by the infinites-imal C˚-action on the domain of u. On the other hand, the sign with whichpπΣpxq, πΣ ˝ u, πΣpyqq P Mpq, p;Aq would contribute to a Gromov–Witten invari-ant is the sign of the zero-dimensional vector space obtained as the quotient ofTpπΣpxq,πΣ˝u,πΣpyqqMpq, p;Aq by the C˚-action, where Mpq, p;Aq is identified with(11.9) with the preimage orientation. The result now follows from

Claim. If we equip (11.2) with the fiber product orientation and (11.9) with thepreimage orientation, then the natural identification of the two spaces is orientation-preserving.

By Lemma 11.14, the fiber product and preimage orientations on (11.2) differ

by p´1qdimW sfΣpqq dimWu

fΣppq. But this is also the difference between the preimage

orientations on (11.2) and on (11.9). The Claim now follows.

Remark 11.16. The Claim in the previous proof is also a consequence of the fol-lowing argument. If we equip (11.2) and (11.9) with their fiber product orienta-tions, then their natural identification preserves orientations, by [Joy12, Proposition7.5.(c)]. On the other hand, the fiber product and preimage orientations on (11.9)are the same, by Lemma 11.7.

There is an analogous argument for Case 2 contributions to the symplectic ho-mology differential. Recall that they come from elements of fiber products

(11.10) W sfY pppq êv

`

M0pB; JW q êvĂM˚

H,1pYk´ , Yk` ; 0;JY q˘

êv WufY pqpq,

which are given the fiber product orientation. An element in this fiber producthas an underlying plane in M0pB; JW q, which can be thought of as a sphere inM0pB; JXq that is asymptotic to p P Σ. Denote by ev : M0pB; JXq Ñ Σ theevaluation map.


Definition 11.17. Given a generic point pt P Σ, the preimage orientation onev´1pptq is called its Gromov–Witten orientation.

Analogously to what was pointed out in Remark 11.10, this orientation deter-mines the sign with which an element in ev´1ppq contributes to a relative Gromov–Witten invariant of the pair pX,Σq, with a single insertion of the point p (chosento be generic) in Σ, with order of contact B ‚ Σ.

Proposition 11.18. The sign with which an element of (11.10) contributes to thesymplectic homology differential agrees with the sign with which the underlying JW -holomorphic plane in W contributes to a relative Gromov–Witten in pX,Σq, withone point constraint of order of contact k` ´ k´ “ B ‚ Σ at p P Σ.

Proof. The proof is analogous to that of Propostion 11.15. We have oriented iso-morphisms“

W sfY pppq êv

`


H,1pYk´ , Yk` ; 0;JY q˘‰

êv WufY pqpq

–“`


H,1pYk´ , Yk` ; 0;JY q˘

êv WsfY pppq

‰

êv WufY pqpq

–“

M0pB; JW q êv

`

ĂM˚

H,1pYk´ , Yk` ; 0;JY q êv WsfY pppq

˘‰

êv WufY pqpq

–M0pB; JW q êv

“`

ĂM˚

H,1pYk´ , Yk` ; 0;JY q êv WsfY pppq

˘

êv WufY pqpq

‰

– p´1q1`MppqM0pB; JW q êv

“

ĂM˚

H,1pYk´ , Yk` ; 0;JY q êv`

W sfY pppq ˆW

ufY pqpq

˘‰

– p´1qMppq`

∆Σ X pWsfΣppq ˆWu

fΣppqq

˘

M0pB; JW q êv R– pW s

fΣppq XWu

fΣppqq| ev´1ppq|R,

where we identified the space of Floer cylinders from pp to qp with R in the last twolines, with the orientation induced by s-translation on the domain (any two suchcylinders differ by translation by a constant s0 P R). The first isomorphism pre-

serves orientations, by Lemma 11.4 (dim`

M0pB; JW qêvĂM˚

H,1pYk´ , Yk` ; 0;JY q˘

`

dimY is even). The second and third isomorphisms are orientation-preserving, bythe associativity of the fiber product orientation. The sign in the fourth isomor-phism comes from [Joy12, Proposition 7.5.(c)]. Note that Lemma 11.7, the fiber

sum and preimage orientations on“

ĂM˚

H,1pYk´ , Yk` ; 0;JY qêv`

W sfYpppqˆWu

fYpqpq

˘‰

agree. In the penultimate line,`

∆Σ X pWsfΣppq ˆWu

fΣppqq

˘

stands for the sign of

the intersection, which is the same as the sign of p´1qMppqpW sfΣppq XWu

fΣppqq, by

Remark 11.6. In the last line, | ev´1ppq| is the absolute value of the signed count ofpoints in ev´1ppq.

Finally, Case 3 contributions to the symplectic homology differential come from

(11.11) WufW pxq êv M˚

H,1pB; JW q êv M˚H,1pYk, 0; JY q êv W

ufY pqpq

with the fiber product orientation. An element in the fiber product contains apseudoholomorphic plane in M˚

2 pB; JW q (recall that H “ 0 in W after stretchingthe neck), which can be thought of as a pseudoholomorphic sphere in M˚

2 pB; JW qwith one constraint at ψ

`

WufWpxq

˘

Ă X and one constraint at WufΣppq Ă Σ with

order of contact B ‚ Σ. Recall from Lemma 2.8 that the map ψ : W Ñ X is adiffeomorphism onto its image XzΣ. Denote by ev : M2pB; JXq Ñ X ˆ Σ theevaluation map.


Definition 11.19. The preimage orientation on ev´1´

ψ`

WufWpxq

˘

ˆWufΣppq

¯

is

called its Gromov–Witten orientation.

If WufWpxq Ă X and Wu

fΣppq Ă Σ represent homology classes, then the Gromov–

Witten orientation determines the signs with which elements in ev´1´

ψ`

WufWpxq

˘

ˆWufΣppq

¯

contribute to a relative Gromov–Witten invariant.

Proposition 11.20. The sign with which an element of (11.11) contributes to thesymplectic homology differential agrees with the sign with which the underlying JW -holomorphic plane in W contributes to a relative Gromov–Witten in pX,Σq, withone point constraint at Wu

fWpxq Ă X and one point constraint of order of contact

k “ B ‚ Σ at WufΣppq Ă Σ.

Proof. We have the oriented diffeomorphisms

WufW pxq êv M˚

H,1pB; JW q êv M˚H,1p0; JY , JW q êv W

ufY pqpq

– p´1qdimWufYpqpq`Wu

fW pxq êv M˚H,1pB; JW q êv W

ufY pqpq

˘

ˆ R2

– p´1qdimWufYpqpq`dimWu

fWpxq`M˚

H,1pB; JW q êv

`

WufW pxq ˆW

ufY pqpq

˘ ˘

ˆ R2

where we denoted by R2 the domain of the s-coordinate of a Floer cylinder inM˚

H,1p0; JY , JW q. We used the oriented diffeomorphism M˚H,1p0; JY , JW q – R2ˆY ,

the fact that W and M˚H,1pB; JW q are even dimensional and [Joy12, Proposi-

tion 7.5.(a)&(c)]. Now, if we identify M˚H,1pB; JW q with M˚

2 pB; JW q and denote

M˚2 pB; JW qC˚ by ĂM

˚

2 pB; JW q, then we also have the oriented diffeomorphisms`

M˚H,1pB; JW q êv

`

WufW pxq ˆW

ufY pqpq

˘ ˘

– p´1qdimWufWpxq`dimWu

fΣppq

´

ĂM˚

2 pB; JW q êv

`

WufW pxq ˆW

ufΣppq

˘

¯

ˆ R1

where R1 is the domain of the s-coordinate of a Floer cylinder in M˚H,1pB; JW q.

By Lemma 11.7, the fiber sum and preimage orientations agree on ĂM˚

2 pB; JW qêv´

WufWpxq ˆWu

fΣppq

¯

. Combining the two calculations, we get an oriented diffeo-

morphism between

WufW pxq êv M˚

H,1pB; JW q êv M˚H,1p0; JY , JW q êv W

ufY pqpq

with the fiber product orientation (which gives rigid contributions to the symplectichomology differential when we mod out by R1 ˆ R2) and

´

ĂM˚

2 pB; JW q êv

`

WufW pxq ˆW

ufΣppq

˘

¯

ˆ R1 ˆ R2

with the preimage orientation (which contributes to a relative Gromov–Witteninvariant when we mod out by R1 ˆR2, according to Definition 11.19). The resultnow follows.

We now define some curve counts that will be useful when we write a formulafor the symplectic homology differential.

Definition 11.21. Let q, p P CritpfΣq and A P H2pΣ;Zq. Define

nApq, pq :“ #MApq, pq,

whereMApq, pq :“Mpq, p;AqC˚.


Here, Mpq, p;Aq is given the Gromov–Witten orientation (see Definition 11.9).

This makes sense when dimMApp, qq “ 2 xc1pTΣq, Ay `Mppq ´Mpqq “ 0 (seeLemma ??).

Definition 11.22. Let B P H2pX;Zq and define

nB :“ #MB ,

where

MB :“ ev´1pptqAutpCP 1, 0q.

Here, ev´1pptq is as in Definition 11.17, and is given its Gromov–Witten orientation.2

This makes sense when dimMB “ 2 xc1pTXq, By´2pB ‚Σq´2 “ 0 (see Lemma??).

Definition 11.23. Let p P CritpfΣq, x P CritpfW q and B P H2pX;Zq. Define

nBpx, pq :“ #MBpx, pq

where

MBpx, pq :“ ev´1`

ψ`

WufW pxq

˘

ˆWufΣppq

˘

C˚.

The formula makes sense when dimMBpp, xq “ 2 xc1pTXq, By ´ 2pB ‚ Σq `Mppq `Mpxq ´ 2n “ 0 (see proof of Lemma 7.3).

Remark 11.24. The coefficients nApq, pq above can only be non-trivial if

xc1pTΣq, Ay “ pτX ´KqωpAq ď n.

by the dimension formula for MApq, pq. Similarly, the coefficients nB can only benon-trivial if

xc1pTXq, By ´B ‚ Σ “ p1´KτXq xc1pTXq, By “ pτX ´KqωpBq “ 1.

The coefficients nBpx, pq can only be non-zero if

xc1pTXq, By ´B ‚ Σ “ p1´KτXq xc1pTXq, By “ pτX ´KqωpBq ď n.

These inequalities are useful when computing the symplectic homology differential,as we will see in an example below.

12. A formula for the symplectic homology differential

Recall that the symplectic chain complex (3.2) is

SC˚pW,Hq “

˜

à

ką0

à

pkPCritpfΣq

Zxqpk, ppky

¸

‘

˜

à

xPCritpfW q

Zxxy

¸

as an abelian group, where fΣ : Σ Ñ R and fW : W Ñ R are Morse functions, withassociated Morse–Smale gradient-like vector fields ZΣ and ZW , respectively. Givenp P CritpfΣq, its stable and unstable manifolds (W s

Σppq and WuΣppq) were defined in

(3.1). We use the analogous conventions to define critical manifolds for fW and fY(recall that we also have a Morse–Smale pair pfY , ZY q in Y ).

2We thank Viktor Fromm for observing that the correct constraint is a generic point in Σ,rather than a submanifold of possibly higher dimension.


Recall that, given p P CritpfY q, its Morse differential is

BfY ppq “ÿ

qPCritpfY qindY ppq´indpqq“1

#Mpq, pq q

where the coefficients are signed counts of elements in

Mpq, pq “`

W sY pqq XW

uY ppq

˘

R.We are now ready to write the differential on SC˚pW,Hq.

Theorem 12.1. The differential on (3.2), denoted B, is as follows. Given p ‰ q PCritpfΣq, the coefficient of pql in Bqpk is

xBpqpkq, pqly “ pk ´ lqÿ

APH2pΣ;Zqδk´l,xc1pNΣq,Ay nApq, pq ` δk,l xBfY pqpkq, pqly

where 2xc1pTΣq, Ay`Mppq´Mpqq “ 0, BfY is the Morse differential in Y and theδ˚,˚ are Kronecker deltas.

If p P CritpfΣq, then

xBpqpkq, pply “ pk ´ lqÿ

BPH2pX;Zqδk´l,pB‚Σq nB

where p1 ´KτXq xc1pTXq, By ´ 1 “ 0. If W is Weinstein, this term is trivial ifn ě 3 and the minimal Chern number of spheres in Σ is at least 2.

Given p P CritpfΣq and x P CritpfW q,

xBpqpkq, xy “ kÿ

BPH2pX;Zqδk,pB‚Σq nBpx, pq

where 2p1´KτXq xc1pTXq, By `Mppq `Mpxq ´ 2n “ 0.Given x P CritpfW q,

Bpxq “ B´fW pxq,

where B´fW is the Morse differential in W with respect to the function ´fW .

Proof. The split Floer differential on (3.2) was introduced at the end of Section 4.2.Propositions 7.4 and 7.5 describe those split Floer cylinders with cascades that cancontribute to the differential.

The two types of contributions in xBpqpkq, pqly are (1) and (0), respectively, inProposition 7.4. The former are given by Floer cylinders in Rˆ Y and correspondto Case 1 in Sections 9 and 10. According to Propositions 9.1 and 10.1, counts ofFloer cylinders in R ˆ Y are equivalent to counts of pseudoholomorphic cylindersin RˆY . By Lemma 11.1 and the discussion that follows, the latter are equivalentto counts of rigid pseudoholomorphic spheres in Σ (given by the nApq, pq), withadditional factors k ´ l in accordance with Lemma 11.3.

The contributions in xBpqpkq, pply correspond to (2) in Proposition 7.4, namely splitFloer cylinders with one augmentation. They correspond to Case 2 in Sections 9and 10. We can appeal again to Propositions 9.1 and 10.1, to relate puncturedFloer cylinders in R ˆ Y to counts of punctured pseudoholomorphic cylinders inRˆ Y . The latter are branched covers of trivial cylinders in this case, and projectto constant spheres in Σ. The number of lifts of these constant spheres is k ´ l,again by Lemma 11.3 (in the notation of that Lemma, k´ ` k´P “ k` in this case,since the spheres in Σ are constant). The counts of augmentation planes in W ,capping the punctures in these cylinders, are given by the numbers nB , by Lemma


2.8. The comment about W Weinstein follows from Lemma 7.7 (and, by Lemma12.5 below, can be interpreted as saying that a certain relative Gromov–Witteninvariant of the pair pX,Σq vanishes).

The contributions in xBpqpkq, xy are the ones in Proposition 7.5, and consist ofa split Floer cylinder in R ˆ Y with one end asymptotic to a Reeb orbit in Y ,matching the asymptotics of a pseudohomorphic plane in W , which is in turnconnected to a critical point of fW by a flow line of ZW . They correspond to Case3 in Sections 9 and 10. We can appeal to Propositions 9.1 and 10.1, to relatethe Floer cylinders in Rˆ Y to trivial pseudoholomorphic cylinders in Rˆ Y . Thecomponent in W is a pseudoholomorphic cylinder v2 : RˆS1 ÑW with a removablesingularity at ´8. At `8, the cylinder converges to a Reeb orbit of multiplicityk. Therefore, precomposing v2 with constant rotations on the domain (as in thediscussion following Lemma 6.34), we get k many maps with the correct asymptoticmarker condition at `8. By Lemma 2.8, such v2 correspond to pseudoholomorphicspheres in X with appropriate tangency condition to Σ. The contribution of theseconfigurations to the differential will then be k nBpx, pq, as wanted.

The only contributions to the differential of x P CritpfW q are from the Morsedifferential, because W is exact and hence has no non-constant pseudoholomorphicspheres.

Remark 12.2. For every p P CritpfΣq and k ą 0, we have BfY pppkq “ 0. There aretwo flow lines of ´ZY from ppk, which cancel one another.

Remark 12.3. At least if there were no contributions xBpqpkq, xy, it would be imme-diate from Theorem 12.1 that B2 “ 0.

12.1. Relation to Gromov–Witten invariants. All the coefficients contributingto the differential in Theorem 12.1 are either combinatorial or topological, exceptfor the nApq, pq, the nB and the nBpx, pq. These are harder to determine, but cansometimes be related to Gromov–Witten invariants, absolute or relative.

We denote by GWΣ0,k,ApC1, . . . , Ckq the genus 0 Gromov–Witten invariant of

pΣ, ωΣq, counting pseudohomolorphic spheres in Σ in class A P H2pΣ;Zq withk marked points constrained to go through representatives of the classes Ci PH˚pΣ;Zq. To see a definition of these invariants when pΣ, ωΣq is monotone, see

[MS04]. We also denote denote by GWpX,Σq0,k,ps1,...,slq,B

pC1, . . . , Ck;D1, . . . , Dlq the

genus 0 relative Gromov–Witten invariant of pX,Σ, ωq, counting pseudohomolor-phic spheres in X class B P H2pX;Zq with k marked points constrained to gothrough representatives of the classes Ci P H˚pX;Zq, and l additional markedpoints constrained to be tangent to Σ with order of contact sj and go throughrepresentatives of the classes Dj P H˚pΣ;Zq. For details on how to define theseinvariants, see [IP03, LR01, TZ14] (for an algebro-geometric approach, see [Li01]).Note that our assumptions that pX,ωq is monotone with monotonicity constant τX ,that Σ is Poincare-dual to Kω and that τX ´K ą 0 imply that the tuple pX,Σ, ωqis strongly semi-positive, as in [TZ14, Definition 4.7].

Recall that a Morse function is perfect if the corresponding Morse differentialvanishes. Furthermore, it is lacunary if it does not have critical points of consecutiveMorse index.


Lemma 12.4. If the Morse function fΣ is perfect, then W sΣppq and Wu

Σppq representclasses in H˚pΣ;Zq and

nApq, pq “ GWΣ0,2,A

`

rW sΣpqqs, rW

uΣppqs

˘

.

If the Morse function fW is also perfect, then ψ`

WuW pxq

˘

represents a class inthe image of the map H˚pXzΣ;Zq Ñ H˚pX;Zq and

nBpx, pq “ GWX,Σ0,1,pB‚Σq,B

`

rψpWuW pxqqs; rW

uΣppqs

˘

.

If fΣ is lacunary, then

xBfY pqpkq, pqky “@

c1pNΣq,`

W sΣpqq XW

uΣppq

˘D

where NΣ is the normal bundle to Σ in X.

Proof. For the first statement, the fact that the stable and unstable submanifoldsof fΣ define homology classes in Σ implies that the nApq, pq are the Gromov–Witteninvariants in the statement, and not merely numbers defined at the (Morse) chainlevel. The orientation conventions in Definition 11.21 are precisely the ones usedin Gromov–Witten theory (recall Remark 11.10). The second statement is provenalong similar lines. Note that the Wu

W pxq define homology classes in W , but thatthe W s

W pxq (which we do not consider) would define relative classes in H˚pW,Y ;Zqinstead. On a technical note, an admissible JX as in Definition 2.4 satisfies thevanishing normal Nijenhuis tensor condition [TZ14, (4.6)], and is hence suitable fordefining relative Gromov–Witten invariants.

We now discuss the last statement. For index reasons, the coefficients xBfY pqpkq, pqkycan only be non-zero if indfΣppq ´ indfΣpqq “ 2. Since fΣ is lacunary, W s

Σpqq XWu

Σppq, together with p and q, form a 2-sphere S. The restriction of Y Ñ Σ to S isan S1-bundle E over S, and pfY , ZY q restricts to a Morse–Smale pair pfE , ZEq onE. By the description of the Gysin sequence in Morse homology [Oan03, (3.26)],xBfE pqpq, pqy is the Euler class of E Ñ S integrated over S, which is exactly what wewanted to show.

The following result does not need any hypotheses on the auxiliary Morse func-tions.

Lemma 12.5. The count of augmentation planes

nB “ GWX,Σ0,0,pB‚Σq,BpH; rptsq

is the genus 0 relative Gromov–Witten invariant of pX,Σq, in class B P H2pX,Zq,with no point constraints in X and order of contact B ‚Σ to a fixed (generic) pointin Σ.

The proof is as in the second case of Lemma 12.4.

Remark 12.6. When fΣ is not a perfect Morse function, the stable and unstablemanifolds of its critical points may not represent classes in H˚pΣ;Zq. In such cases,the nApq, pq can still be thought of as chain level Gromov–Witten numbers of Σ.These are not invariants, and may thus be harder to compute. Similarly, if fW isnot perfect, the nBpx, pq are only chain level relative Gromov–Witten numbers.

We will see below an example in which we compute Gromov–Witten invariantsof pCP 1ˆCP 1,∆, ωFS‘ωFSq using the integral J , even though it is not ‘cylindricalnear Σ’, as we require of admissible JX in Definition 2.4. That is not a problem,though, since the numbers we are computing are invariants.


Remark 12.7. Computing Gromov–Witten invariants can be a very hard problem,both in the absolute and relative case, but it has been done for a large class ofimportant examples, especially using tools from algebraic geometry. For absoluteinvariants, see for instance [Bea95, Zin14]. Gathmann wrote a computer programcalled GROWI to compute absolute and relative Gromov–Witten invariants of hyper-surfaces in projective spaces [Gat].

Relative invariants are often harder to compute than absolute ones. Nevertheless,under certain assumptions, a relative invariant where all intersections with Σ aretransverse (i.e. have order of tangency 1) can be related to an absolute invariantwhere one forgets Σ (in the spirit of the divisor axiom for absolute Gromov–Witteninvariants) [McD08,HR13,TZ16].

It is also worth pointing out that the relative invariants of a tuple pX,Σ, ωq canin principle be obtained from the absolute invariants of X and of Σ (incluing morecomplicated insertions like ψ-classes, in general) [MP06]. Nevertheless, it can becomputationally quite challenging to use this fact to compute relative invariants ofpX,Σ, ωq.

Remark 12.8. Since explicit computations of Gromov–Witten invariants are ofteneasier in algebraic geometry than in symplectic geometry, we implicitly assume thatthe two definitions agree in geometric settings where they both make sense. Formore details about the relation between symplectic and algebraic absolute Gromov–Witten invariants, see [Sie99,LT97].

Remark 12.9. Since the computation of relative Gromov–Witten invariants can bemore difficult than that of absolute invariants, it is worth pointing out that thesymplectic homology differential does not have contributions nA when n ě 3, theminimal Chern number of Σ is at least 2 and W is Weinstein, as we saw in Theorem12.1. Also, if one were only interested in the positive (or large-action) symplectichomology of W [BO09a], then one would ignore the contributions xBpqpkq, xy, andhence would not need to determine the coefficients nBpx, pq.

Remark 12.10. The following was pointed out to us by D. Pomerleano. We alsothank R. Leclercq and M. Sandon for helpful conversations about their related workin progress. Theorem 12.1 may seem unsatisfactory since it relates the differentialB, which is a chain level object depending on the model we use to compute sym-plectic homology, to counts of pseudoholomorphic spheres which, at least underthe assumption of Lemma 12.4, are given by Gromov–Witten invariants, which arehomological. The chain complex SC˚ in (3.2) has a quotient complex, referred to in[BO09a] as SC`˚ , obtained by modding out generators coming from critical pointsof fW . We can write SC`˚ as a complex of the form

(12.1) C˚pΣqrtst‘`

C˚pΣqrtst˘

r1s

where C˚pΣq denotes the Morse complex of fΣ and tk encodes generators associatedto Reeb orbits of multiplicity k. The variable t has degree 2 τX´KK , by (3.4) (whichshould also force us to take a global shift of C˚pΣq by 1 ´ n; we don’t do this toavoid making the notation even heavier). The first summand contains the orbitsdecorated withq and the second summand, with the degree shift up by 1, containsthe orbits decorated withp. Theorem 12.1 implies that the differential in (12.1) isgiven by a matrix

B` “

ˆ

dΣ ∆0 dΣ

˙


where dΣ is the Morse differential in Σ and ∆: C˚pΣqrtst Ñ C˚pΣqrtst is a chainmap counting: flow lines in Σ connecting critical points of index difference 2, pseu-dohomolorphic spheres in Σ (the terms involving nApp, qq) and in X (the termsinvolving nB). This means that SC`˚ is a cone complex on ∆. There is an inducedlong exact sequence

. . .Ñ H˚pΣqrtstr1s Ñ SH`˚ pW,Hq Ñ H˚pΣqrtstr∆sÑ H˚pΣqrtstr1s Ñ . . .

with connecting homomorphism r∆s (we are not being careful keeping track of thedegrees ˚ in the sequence). This can be thought of as the long exact sequence in[BO09a], in the special case when the Liouville manifold is the completion of XzΣ.It is also related to the spectral sequence in [Sei08, (3.2)].

We can think of this exact sequence as a more invariant consequence of Theorem12.1. Note that if we were interested in computing SH`˚ pW q with field coefficients,then the exact sequence suggests that it should be enough to compute Gromov–Witten invariants, instead of chain level counts of pseudoholomorphic spheres, evenif one does not start with a perfect Morse function in Σ. Observe that SC˚pW q isalso a cone complex on a chain map SC`˚ pW q Ñ C˚pW q, counting pseudoholomor-phic spheres in X with tangency conditions in Σ (the terms involving nBpp, xq).The same argument as above should enable us to compute SH˚pW q with the addi-tional information of relative Gromov–Witten invariants of the pair pX,Σq, insteadof the non-invariant counts nBpp, xq, at least over field coefficients.

Part 5. Example: T˚S2

We now illustrate the results in this paper with the computation of the symplectichomology of the completion W of XzΣ, where pX,Σ, ωq “ pCP 1 ˆ CP 1,∆, ωFS ‘ωFSq, where ∆ is the diagonal and ω restricts to the Fubini-Study form of area 1in each factor. The manifold W is symplectomorphic to T˚S2 (see Exercise 6.20in [MS98]), so its symplectic homology is isomorphic to the homology of the basedloop space of S2. We will see that our computation recovers this result. X and Σare both monotone, with τX “ 2 and τΣ “ 1. Also, rΣs is Poincare-dual to rωs, soK “ 1. The S1-bundle Y Ñ Σ is RP 3.

13. The coefficients in the differential

The manifold Σ “ CP 1 admits a lacunary Morse function fΣ : Σ Ñ R, with onezero of index 0, denoted by m (for minimum), and one zero of index 2, denoted byM (for maximum). Lemma 12.4 implies that all the numbers nApq, pq (giving theCase 1 contributions to the differential) are absolute Gromov–Witten invariants ofΣ “ CP 1. Definition 11.21 and Remark 11.24 indicate that the only potentiallynon-trivial invariant is

nLpM,mq “ GWCP 1

L,2 prpts, rptsq “ 1

where L P H2pCP 1;Zq is the class of a line. This invariant counts the number oflines in CP 1 through two generic points, which is 1 (the integral complex structureis regular in CP 1).

The manifold W “ T˚S2 also admits a lacunary Morse function fW : W Ñ Rgrowing at infinity, with one zero of index 0, denoted by e, and one zero of index 2,denoted by c. The manifold Wupcq represents the zero section in T˚S2. We orientit as L1´L2, where L1, L2 P H2pCP 1ˆCP 1;Zq are homology classes representing


the factors CP 1 ˆ tptu and tptu ˆCP 1, respectively. Definition 11.21 and Remark11.24 again indicate that the only potentially non-trivial coefficients in Cases 2 and3 are

nLi , nLipe,Mq, nLipc,mq, n2Lipe,mq and nL1`L2pe,mq.

Lemmas 12.4 and 12.5 imply that these numbers are relative Gromov–Witteninvariants of pCP 1 ˆ CP 1,∆, ωq. We will now compute these invariants.

Proposition 13.1. (1) nLi “ GWCP 1ˆCP 1,∆

Li,0,p1qpH; ptq “ 1,

for i “ 1, 2 (the point constraint is in ∆, not in CP 1 ˆ CP 1);

(2) nLipe,Mq “ GWCP 1ˆCP 1,∆

Li,1,p1qprpts; r∆sq “ 1;

(3) nLipc,mq “ GWCP 1ˆCP 1,∆

Li,1,p1qprS2s; rptsq “ p´1qi

where S2 Ă T˚S2 is the zero section, oriented so as to represent the ho-mology class L1 ´ L2;

(4) n2Lipe,mq “ GWCP 1ˆCP 1,∆

2Li,1,p2qprpts; rptsq “ 0;

(5) nL1`L2pe,mq “ GWCP 1

ˆCP 1,∆L1`L2,1,p2q

prpts; rptsq “ 1.

We need an auxiliary result.

Lemma 13.2. The product complex structure on X “ CP 1 ˆ CP 1 is regular forall immersed holomorphic spheres U : CP 1 Ñ CP 1 ˆCP 1 such that

ş

CP 1 U˚ω ě 1.

In particular, this is true when U˚rCP 1s P tL1, L2, L1 ` L2u.

Proof. By [MS04, Lemma 3.3.3], we just need to make sure that xc1pTXq, U˚rCP 1sy ě

1, but xc1pTXq, U˚rCP 1sy “ 2ş

CP 1 U˚ω ě 2.

For the proof of cases (1) through (3) of Proposition 13.1, one could appealto the relation between relative and absolute Gromov–Witten invariants, when allthe tangencies to Σ are of order 1, as mentioned in Remark 12.7. This is notnecessary, though, since Lemma 13.2 allows us to do the computations explicitlyusing the integral complex structure in CP 1 ˆ CP 1. This satisfies the vanishingnormal Nijenhuis tensor condition [TZ14, (4.6)], and can thus be used for computingrelative Gromov–Witten invariants.

Proof of Proposition 13.1. (1) Given i “ 1, 2,

GWCP 1ˆCP 1,∆

Li,0,p1qpH; rptsq “ 1

is the fact that if one fixes any point p P ∆, then there is exactly oneholomorphic sphere in class Li that goes through p.

(2) Given i “ 1, 2,

GWCP 1ˆCP 1,∆

Li,1,p1qprpts; r∆sq “ 1

expresses the fact that, if we fix any point p P CP 1 ˆ CP 1z∆, there is aunique holomorphic sphere in class Li that goes through p and intersects∆.

(3) Given i, j “ 1, 2,

GWCP 1ˆCP 1,∆

Li,1,p1qpLj ; rptsq “ 1´ δi,j .

This means, for instance, that there is a unique vertical sphere in CP 1ˆCP 1

intersecting a horizontal sphere and a generic point p P CP 1 ˆ CP 1, but


there is no horizontal sphere intersecting another horizontal sphere and ageneric p. This is because two different horizontal spheres do not intersect.Since S2 “ L1 ´ L2,

GWCP 1ˆCP 1,∆

Li,1,p1qprS2s; rptsq “ GWCP 1

ˆCP 1,∆Li,1,p1q

pL1; rptsq´GWCP 1ˆCP 1,∆

Li,1,p1qpL2; rptsq “ p´1qi

(4) Given i “ 1, 2,

GWCP 1ˆCP 1,∆

2Li,1,p2qprpts; rptsq “ 0

encodes the fact that holomorphic curves in classes 2Li P H2pCP 1ˆCP 1;Zqare covers of either vertical or horizontal spheres, and therefore cannot gothrough one generic point in CP 1 ˆ CP 1 and another generic point in ∆.

(5) To prove that

GWCP 1ˆCP 1,∆

L1`L2,1,p2qppt; ptq “ 1

we show that, up to domain automorphism, there is a unique holomorphicmap U : CP 1 Ñ CP 1 ˆ CP 1, such that U˚rCP 1s “ L1 ` L2 P H2pCP 1 ˆ

CP 1;Zq, Up8q “ p8, 1q, Up0q “ p0, 0q, and such that U intersects thediagonal ∆ Ă CP 1 ˆ CP 1 in a non-transverse way. Since

Up8q “ U`

r1; 0s˘

“ p8, 1q “`

r1; 0s, r1; 1s˘

,

we can write in homogeneous coordinates

U`

rz; 1s˘

“`

raz ` b; 1s, rz ` c; z ` ds˘

which we will abbreviate as

Upzq “

ˆ

az ` b,z ` c

z ` d

˙

.

Since Up0q “ p0, 0q, we get b “ c “ 0, and so Upzq “`

az, zpz ` dq˘

. Now,for the tangency condition, note that

U 1pzq “

ˆ

a,d

pz ` dq2

˙

and so U 1p0q “`

a, 1d˘

. So U is tangent to the diagonal at p0, 0q preciselywhen a “ 1d. Therefore, the space of maps U can be identified with thespace of a P C˚. Taking a quotient by the group C˚ of automorphisms ofthe domain pCP 1, t0,8uq, we get the uniqueness of U .

14. The group SH˚pT˚S2q

We can now compute the symplectic homology chain complex for T˚S2. Usingthe functions fΣ and fW introduced above, we have

SC˚pT˚S2q “ Z xe, cy ‘

à

ką0

ZA

qmk, pmk,|Mk,xMk

E

We can use Theorem 12.1, Lemma 12.4 and Proposition 13.1 to compute the dif-ferential. Given k ě 2,

B qmk`1 “ 2nLpM,mqxMk´1 ``

nL1` nL2

˘

pmk “

“ 2ˆ 1 xMk´1 ``

1` 1˘

pmk “ 2 xMk´1 ` 2 pmk


and

B|Mk “`

nL1 ` nL2

˘

xMk´1 ` xc1pRP 3 Ñ CP 1q,CP 1y pmk “ 2 xMk´1 ` 2 pmk

The remaining differentials are

B qm2 “`

nL1` nL2

˘

pm1 ` 2nL1`L2pe,mq e “ 2 pm1 ` 2 e,

B|M1 “`

nL1pe,Mq ` nL2pe,Mq˘

e ` xc1pRP 3 Ñ CP 1q,CP 1y pm1 “

“ 2 e` 2 pm1

and

B qm1 “`

nL1pc,mq ` nL2pc,mq˘

c “ p´1` 1q c “ 0.

Summing up, the non-trivial contributions to the differential are

$

’

’

&

’

’

%

B qmk`1 “ 2 pmk ` 2 xMk´1

B|Mk “ 2pmk ` 2 xMk´1

B qm2 “ 2 pm1 ` 2 e

B|M1 “ 2 pm1 ` 2 e

for k ě 2. We can conclude the following.

Proposition 14.1.

SH˚pT˚S2;Zq “ Z

A

c, qm1, e,|Mk ´ qmk`1,xMk

E

‘ Z2A

e` pm1,xMk ` pmk`1

E

where we take all k ě 1.

We can compare these results with earlier computations of the symplectic ho-mology of T˚S2. By Viterbo’s Theorem, it is isomorphic as a ring to the homologyof the free loop space of S2 [AS12]. As a graded abelian group, this was computedby McCleary [McC90]. The a ring, it was computed in [CJY04] to be

(14.1) H˚pLS2;Zq – pΛpbq b Zra, vsqpa2, ab, 2avq

for some a P H0pLS2;Zq, b P H1pLS

2;Zq and v P H4pLS2;Zq. The following tables

show how our computations match these, at least as graded abelian groups. Forthe free part, we get

SHdpT˚S2q c ´qm1 e |Mk ´ qmk`1

xMk

HdpLS2q a b 1 bvk vk

d 0 1 2 2k ` 1 2k ` 2

for k ě 1. For the Z2-torsion part:

SHdpT˚S2q e` pm1

xMk ` pmk`1

HdpLS2q av avk`1

d 2 2k ` 2

for k ě 1. One can also adapt the arguments presented above to describe thering structure on symplectic homology and show that it matches 14.1, but that isbeyond the scope of this paper.


Appendix A. Morse–Bott Riemann–Roch

In this section, we collect some facts about Cauchy–Riemann-type operators onHermitian vector bundles over punctured Riemann surfaces. These facts are well-established in parts of the literature, but we collect them here for the convenienceof the reader. The main reference for these results is [Sch95]. Additional referencesinclude [HWZ99; Sch95; Wen10; ACH05, Sections 2.1–2.3; BM04].

We begin by introducing some Sobolev spaces of sections of appropriate bundles.Let Γ Ă R ˆ S1 be a finite set of punctures and denote R ˆ S1zΓ by 9S. WriteΓ` “ t`8u and Γ´ “ t´8u Y Γ. Consider, for each puncture z P Γ, exponentialcylindrical polar coordinates of the form p´8,´1s ˆ S1 Ñ R ˆ S1zΓ: ρ ` iη ÞÑz0` ε e2πpρ`iηq. Choose ε ą 0 sufficiently small that the image of these embeddingsfor any two different punctures are disjoint.

Let E Ñ 9S be a (complex) rank n Hermitian vector bundle over 9S together witha preferred set of trivializations in a small neighbourhood of ΓY t˘8u. While the

bundle E over 9S is trivial if there is at least one puncture, this is no longer the caseonce we specify these preferred trivializations near Γ Y t˘8u. We therefore asso-ciate a first Chern number to this bundle relative to the asymptotic trivializations.There are several equivalent definitions. One approach is to consider the complexdeterminant bundle ΛnE. The trivialization of E at infinity gives a trivialization ofthis determinant bundle at infinity, and we can now count zeros of a generic sectionof ΛnE that is constant (with respect to the prescribed trivializations) near thepunctures. We denote this Chern number by c1pEq, but emphasize that it dependson the choice of these trivializations near the punctures.

Since we cannot specify where an augmentation puncture appears when westretch the neck on a Floer cylinder, we should have the punctures in Γ free tomove on the domain R ˆ S1. This creates a problem when we try to linearize theFloer operator in a family of domains where the positions of the punctures are notfixed. We will instead consider a 2#Γ parameter family of almost complex struc-tures on RˆS1, but fix the location of the punctures, as follows. We specify a fixedcollection Γ of punctures on R ˆ S1 and, for any other collection of augmentationpunctures, choose an isotopy with compact support from the new punctures to thefixed ones. We take the push-forward of the standard complex structure in Rˆ S1

by the final map of the isotopy, to produce a family of complex structures on RˆS1,which can be assumed standard near Γ and outside of a compact set.

For each z P Γ, let βz : 9S Ñ r0,`8q be a function supported in a small neighbour-hood of z, with βzpρ, ηq “ ´ρ near the puncture (where pρ, ηq are cylindrical polarcoordinates near z, as above). Similarly, let β` : R ˆ S1 Ñ r0,`8q be supportedin a region where s is sufficiently large and β`ps, tq “ s for s large enough. Letβ´ : Rˆ S1 Ñ r0,`8q have support near ´8, and β´ps, tq “ ´s for s sufficientlysmall.

In many situations, it will be convenient to consider the function

(A.1) β :“ÿ

zPΓ

βz ` β´ ` β`.

Given a vector of weights δ : Γ Y t˘8u Ñ R, we define W 1,p,δp 9S,Eq to be

the space of sections u of E for which u eř

δzβz`δ´β´`δ`β` P W 1,pp 9S,Eq (with

respect to a measure on 9S that equals ds dt for |s| sufficiently large, and dρ dη incylindrical polar coordinates near each puncture in Γ). Note that these sections


decay exponentially fast at the punctures where δ ą 0. We can similarly defineLp,δp 9S,Eq. While these definitions involve making various choices, the resultingmetrics are strongly equivalent.

We will say that a differential operator D : ΓpEq Ñ Λ0,1T˚ 9S b E is a Cauchy–Riemann operator if it is a real linear Cauchy–Riemann operator [MS04, DefinitionC.1.5] such that, near ˘8 , it takes the form:

(A.2) pDσqB

Bs“B

Bsσ ` Jps, tq

B

Btσ `Aps, tqσ

where Jps, tq is a smooth function on R˘ ˆ S1 with values in almost complexstructures on Cn compatible with the standard symplectic form, and Aps, tq takesvalues in real matrices on R2n – Cn. We further impose that these functionsconverge as sÑ ˘8, Jps, tq Ñ Jzptq and Aps, tq Ñ Azptq, where Azptq is a loop ofself-adjoint matrices. We impose the same conditions near punctures z P Γ, usingthe local coordinates pρ, ηq instead of ps, tq in (A.2).

Associated to such a Cauchy–Riemann operator D, we obtain asymptotic oper-ators at each puncture in ΓY t˘8u by Az :“ ´Jzptq

ddt ´Azptq. This is a densely

defined unbounded self-adjoint operator on L2pS1,R2nq. Let σpAzq Ă R denoteits spectrum. This will consist of a discrete set of eigenvalues. If an asymptoticoperator Az does not have 0 in its spectrum, we say the asymptotic operator isnon-degenerate. If all the asymptotic operators are non-degenerate, we say D itselfis non-degenerate.

Note that we obtain a path of symplectic matrices associated to the asymptoticoperator Az by finding the fundamental matrix Φ to the ODE d

dtx “ JzptqAzptqx.The asymptotic operator is non-degenerate if and only if the time-1 flow of the ODEdoes not have 1 in the spectrum. We will consider a description of the Conley–Zehnder index in terms of properties of the asymptotic operator itself [HWZ95,Lemmas 3.4, 3.5, 3.6, 3.9].

Remark A.1. In this paper, we also want to associate a Conley–Zehnder index toa periodic orbit of a Hamiltonian vector field, given a trivialization of the tangentbundle along the orbit. In order to do so, we take the linearized flow map, whichdefines a path Φ: r0, 1s Ñ Spp2nq with respect to the fixed trivialization. If we fixa path of almost complex structures, this path of symplectic matrices satisfies anODE as in the previous paragraph, which in turn specifes an asymptotic operator.The Conley–Zehnder index of the Hamiltonian orbit is by definition the Conley-Zehnder index of this asymptotic operator.

Proposition A.2. Suppose Az is non-degenerate and E is a rank 1 vector bundle.Then, each eigenfunction u : S1 Ñ C of Az has a winding number that dependsonly on its eigenvalue λ, wpλq. The function w : σpAzq Ñ Z is non-decreasing inλ and is surjective. If λ˘ are eigenvalues so that λ´ ă 0 ă λ` and there are noeigenvalues in the interval pλ´, λ`q, then

CZpAzq “ wpλ´q ` wpλ`q.

This formulation will be the most useful for our calculations. Furthermore, inthe case of a higher rank bundle, we use the axiomatic description, see for instance[HWZ95, Theorem 3.1] to observe that CZpAzq is invariant under deformations forwhich 0 is never in the spectrum, and that if the operator can be decomposed asthe direct sum of operators, then the CZ-index is additive.


The following statement is useful at several points in the paper. It can often becombined with Proposition A.2 to compute Conley–Zehnder indices of interest.

Lemma A.3. Given a constant C ě 0, the spectrum σpACq of the operator

AC :“ íd

dt´

ˆ

C 00 0

˙


is the set"

1

2

´

Ć á

C2 ` 16π2k2¯

| k P Z*

Y

"

1

2

´

Ć à

C2 ` 16π2k2¯

| k P Z*

.

If λ is an eigenvalue associated to k P Z, then the winding number of the corre-sponding eigenvector is |k| if λ ě 0 and ´|k| if λ ď 0. If C “ 0, then all eigenvalueshave multiplicity 2 (see Table 1). If C ą 0, then the same is true except for theeigenvalues Ć and 0, corresponding to k “ 0 above, both of which have multiplicity1 (see Table 2).

In particular, the σpA0q “ 2πZ and the winding number of 2πk is k.

Proof. An eigenvector v : S1 Ñ C of AC with eigenvalue λ solves the equation

´

ˆ

0 ´11 0

˙

9v ´

ˆ

C 00 0

˙

v “ λv ô 9v “

ˆ

0 ´λC ` λ 0

˙

v.

Computing the eigenvalues of the matrix on the right, and requiring that they beof the form 2πik, k P Z (since vpt` 1q “ vptq), yields the result.

eigenvalues . . . ´4π ´2π 0 2π 4π . . .multiplicities . . . 2 2 2 2 2 . . .

winding numbers . . . ´2 ´1 0 1 2 . . .Table 1. Eigenvalues of A0

eigenvalues . . . 12

`

Ć ´?C2 ` 16π2

˘

Ć 0 12

`

Ć `?C2 ` 16π2

˘

. . .multiplicities . . . 2 1 1 2 . . .winding #s . . . ´1 0 0 1 . . .

Table 2. Eigenvalues of AC in increasing order, if C ą 0

Corollary A.4. Take C ě 0 and δ ą 0 such that r´δ, δs X σpACq “ t0u. Then

CZpAC ` δq “

#

0 if C ą 0

´1 if C “ 0and CZpAC ´ δq “ 1.

For any n ě 0, taking

íd

dt: W 1,ppS1,Cnq Ñ LppS1,Cnq,

we have

CZ

ˆ

íd

dt˘ δ

˙

“ ¯n.

Proof. The case n “ 1 follows from Proposition A.2 and Lemma A.3. The case ofgeneral n uses the additivity of CZ under direct sums.


Definition A.5. A key observation for our computations of Fredholm indices (asnoted, for instance, in [HWZ99]) is that a Cauchy–Riemann operator

D : W 1,p,δp 9S,Eq Ñ Lp,δp 9S,Λ0,1T˚ 9S b Eq

with asymptotic operators Az is conjugate to the Cauchy–Riemann operator

Dδ : W 1,pp 9S,Eq Ñ Lpp 9S,Λ0,1T˚ 9S b Eq

Dδ “ eř

δzβzpsqD e´ř

δzβzpsq .

This has asymptotic operators Aδz “ Az ˘ δz (where the sign is positive at positive

punctures and negative at negative punctures). We refer to these as the δ-perturbedasymptotic operators.

This observation about the conjugation of the weighted operator to the non-degenerate case, combined with Riemann–Roch for punctured domains (see forinstance, [Sch95, Theorem 3.3.11; HWZ99, Theorem 2.8; Wen16a, Theorem 5.4])gives the following.

Theorem A.6. Let δ : Γ Ñ R such that ¯δz R σpAzq. Then, the Cauchy–Riemannoperator

D : W 1,p,δp 9S,Eq Ñ Lp,δp 9S,Λ0,1T˚ 9S b Eq

with asymptotic operators Az, z P Γ is Fredholm and its index is given by

IndpD, δq “ nχ 9S ` 2c1pEq `ÿ

zPΓ`

CZpAz ` δzq ´ÿ

zPΓ´

CZpAz ´ δzq.

Now, a useful fact for us is a description of how the Conley–Zehnder indexchanges as a weight crosses the spectrum of the operator:

Lemma A.7. Suppose that r´δ,`δs X σpAzq “ t0u. Then,

CZpAz ´ δq ´ CZpAz ` δq “ dimpker Azq

For a proof, see for instance [Wen05, Proposition 4.5.22].

To obtain a result that is useful for our moduli spaces of cascades asymptotic toMorse–Bott families of orbits, we consider the following modification of our functionspaces.

To each puncture, we associate a subspace of the kernel of the correspondingasymptotic operator, which we denote by Vz, z P Γ, V´, V` and write V for thiscollection. Then, for each puncture z P Γ and also ˘8, we associate a smooth bumpfunction µz, µ˘, supported near and identically 1 even nearer to its puncture. Wethen define

W 1,p,δV p 9S,Eq “tu PW 1,p

loc p9S,Eq | D cz P Vz, z P Γ, c´ P V´, c` P V`

such that u´ÿ

czµz ´ c´µ´ ´ c`µ` PW1,p,δp 9S,Equ.

(A.3)

We remark that we are using the asymptotic cylindrical coordinates near Γ andthe asymptotic trivialization of E in order to define the local sections czµz.

In this paper, we are primarily concerned with Cauchy-Riemann operators de-fined on 9S “ R ˆ S1 and on 9S “ R ˆ S1ztP u (a cylinder with one additionalnegative puncture). In the case of Rˆ S1, we will write V “ pV´;V`q, and in thecase of R ˆ S1ztP u, we will write V “ pV´, VP ;V`q. (The negative punctures areenumerated first, and separated from the positive puncture by a semicolon.)


Observe that since the vector spaces Vz are in the kernel of the correspondingasymptotic operators, for any choice of V and any vector of weights δ, we have thatthe Cauchy–Riemann-type operator D can be extended to

D : W 1,p,δV p 9S,Eq Ñ Lp,δp 9S,Λ0,1T˚ 9S b Eq.

Let dimVz denote the dimension of the vector space Vz and let codimVz “dim pker AzVzq. Combining Theorem A.6 with Lemma A.7, we have:

Theorem A.8. Let δ ą 0 be sufficiently small that for z P Γ`, r´δ, 0qXσpAzq “ H

and such that for z P Γ´, p0, δs X σpAzq “ H.For each z P Γ, fix the subspace Vz Ă ker Az.Then,

D : W 1,p,δV p 9S,Eq Ñ Lp,δp 9S,Λ0,1T˚ 9S b Eq.

is Fredholm, and its Fredholm index is given by

IndpDq “ nχ 9S`2c1pEq`ÿ

zPΓ`

`

CZpAz`δq`dimpVzq˘

´ÿ

zPΓ´

`

CZpAz`δq`codimpVzq˘

.

In applications where there are Morse–Bott manifolds of orbits, we will some-times use Vz to be the tangent space to the descending manifold of a critical pointpz on the manifold of orbits at a positive puncture, and Vz will be the tangent spaceto ascending manifold of a critical point pz at a negative puncture. In either case,the contribution to IndpDq of dimVz or of codimVz will be the Morse index of theappropriate critical point. This motivates the following definition.

Definition A.9. Let δ ą 0 be sufficiently small. If pz is a critical point of anauxiliary Morse function on the manifold of orbits associated to z, then the Conley–Zehnder index of the pair pAz, pzq is

CZpAz, pzq “ CZpAz ` δq `Mppzq,

where Mppzq is the Morse index of pz.

In this case, we can write the Fredholm index as

IndpDq “ nχ 9S ` 2c1pEq `ÿ

zPΓ`

CZpAz, pzq ´ÿ

zPΓ´

CZpAz, pzq.

We conclude with a lemma that is particularly useful when applying the auto-matic transversality result [Wen10, Proposition 4.22]. The lemma states that theFredholm index of an operator with a small negative weight at a puncture is thesame as that of the corresponding operator with a small positive weight at thatpuncture, if the puncture is decorated with the kernel of the corresponding asymp-totic operator. The former indices are used in [Wen10, Proposition 4.22], whereasthe latter can be computed using Theorem A.8.

We first learned this result from Wendl [Wen05]. We give a proof of this formu-lation since it is slightly stronger than what we found in the literature (and is stillnot as strong as can be proved.)

Lemma A.10. Let D be a Cauchy–Riemann-type operator. Fix a puncture z0 P

ΓY t˘8u.Let δ and δ1 be vectors of sufficiently small weights so that the differential operator

induces a Fredholm operator on W 1,p,δ and on W 1,p,δ1 , and for which there is a


z0 P Γ Y t˘8u with δz0 ą 0 and δ1z0 ă 0 and so that for each z P Γ Y t˘8u withz ‰ z0, the weights δz “ δ1z.

Let V be the trivial vector space at each puncture other than z0 and let Vz0 bethe kernel of the asymptotic operator at z0.

Then, the induced operators

Dδ : W 1,p,δV p 9S,Eq Ñ Lp,δp 9S,Λ0,1T˚ 9S b Eq

Dδ1 : W1,p,δ1p 9S,Eq Ñ Lp,δ

1

p 9S,Λ0,1T˚ 9S b Eq

have the same Fredholm index and their kernels and cokernels are isomorphic.

Proof. The main idea of the lemma is contained in [Wen05, Proposition 4.5.22],which contains a proof of the equality of Fredholm indices. See also the very closelyrelated [Wen16b, Proposition 3.15].

Note that W 1,p,δV p 9S,Eq is a subspace of W 1,p,δ1p 9S,Eq, and thus the kernel of Dδ

is contained in the kernel of Dδ1 .Now, by a linear version of the analysis done in [HWZ99, Sie08], any element

of the kernel of Dδ1 converges exponentially fast at z0 to an eigenfunction of theasymptotic operator, with exponential rate governed by the eigenvalue (in this case0). Therefore, any element of the kernel of Dδ1 must converge exponentially fast toan element of the kernel of the asymptotic operator at z0. Hence, the kernel of Dδ1

is contained in the kernel of Dδ.We conclude that the kernels of the two operators may be identified. Since their

Fredholm indices are the same, their cokernels are also isomorphic.

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