Morse-Bott Split Symplectic Homology

We introduce a chain complex associated to a Liouville domain $(\overline{W}, d\lambda)$ whose boundary $Y$ admits a Boothby--Wang contact form (i.e. is a prequantization space). The differential counts cascades of Floer solutions in the completion $W$ of $\overline{W}$, in the spirit of Morse--Bott homology (as in work of Bourgeois, Frauenfelder arXiv:math/0309373 and Bourgeois-Oancea arXiv:0704.1039). The homology of this complex is the symplectic homology of the completion $W$. We identify a class of simple cascades and show that their moduli spaces are cut out transversely for generic choice of auxiliary data. If $X$ is obtained by collapsing the boundary along Reeb orbits and $\Sigma$ is the quotient of $Y$ by the $S^1$-action induced by the Reeb flow, we also establish transversality for certain moduli spaces of holomorphic spheres in $X$ and in $\Sigma$. Finally, under monotonicity assumptions on $X$ and $\Sigma$, we show that for generic data, the differential in our chain complex counts elements of moduli spaces that are transverse. Furthermore, by some index estimates, we show that very few combinatorial types of cascades can appear in the differential.


Introduction and statement of main results
In this paper we define Morse-Bott split symplectic homology theory for Liouville manifolds W of finite-type whose boundary Y " BW is a prequantization space. This is inspired by the construction of Bourgeois and Oancea for positive symplectic homology, SH` [BO09a]. While the chain complex we define should compute symplectic homology for general Liouville manifolds, we focus on the case when the boundary of the Liouville manifold is a prequantization space. Under this geometric condition, our main result is that we obtain transversality for all the relevant moduli spaces and thus have a well-defined theory. This is obtained by means of generic choice of the geometric data, as opposed to using abstract perturbations. This is an important preliminary step in the computation of this chain complex in [DL18].
In the following, we consider a 2n-dimensional Liouville domain pW , dλq with BW " Y . Denoting by α " λ| Y the contact form on the boundary induced by the Liouville form λ, we require that α is a Boothby-Wang contact form, i.e. its Reeb vector field induces a free S 1 action. Such a contact manifold is also called a prequantization space. We let Σ 2n´2 denote the quotient by the Reeb vector field, and we note that dα descends to a symplectic form ω Σ on the quotient. It follows that there exists a closed symplectic manifold pX, ωq for which Σ is a codimension 2 symplectic submanifold, Poincaré dual to a multiple of ω, with the property that pXzΣ, ωq is symplectomorphic to pW zY, dλq. See Proposition 2.1 for a more detailed description of this, or see [Gir18,Proposition 5].
We let pW, dλq be the completion of W , obtained by attaching a cylindrical end R`ˆY , and take a Hamiltonian H : RˆY Ñ R with a growth condition (see Definition 3.1 for details). This Hamiltonian will have Morse-Bott families of 1periodic orbits, and we then use the formalism of cascades (similar in style to the Morse-Bott theories of [Bou02,Fra04,HN17]) in order to construct a Morse-Bott Floer homology associated to H, but we also consider configurations that interact between RˆY and W , generalizing from the situation considered in [BO09a].
In Section 3, we introduce the chain complex for split symplectic homology and describe the moduli spaces that are relevant for the differential. The chain complex will be generated by critical points of auxiliary Morse functions associated to our Morse-Bott manifolds of orbits. The differential will be obtained from moduli spaces of cascades of Floer cylinders together with asymptotic boundary conditions given in terms of the auxiliary Morse functions.
We then prove three main results. The first is a transversality theorem for "simple cascades". These are elements of the relevant moduli spaces that are also somewhere injective when projected to Σ. This result holds without any monotonicity assumptions. A more precise formulation is provided in Proposition 5.9: Theorem 1.1. Simple Floer cascades in W and RˆY are transverse for a comeagre set of Reeb-invariant cylindrical almost complex structures on W .
Theorem 1.1 builds on a transversality theorem for moduli spaces of spheres in X and in Σ, which may be of independent interest. This construction is somewhat analogous to the strings of pearls that Biran and Cornea study in the Lagrangian case [BC09]. A more precise formulation is given in Proposition 5.26: Theorem 1.2. For generic almost complex structure, moduli spaces of somewhere injective spheres in Σ and of somewhere injective spheres in X with order-of-contact constraints at Σ, connected by gradient trajectories, are transverse assuming no two spheres have the same image.
The third result builds on these two to describe the moduli spaces that are relevant in the case in which X and Σ are assumed to be spherically monotone (with a resulting constraint on the degree of the normal bundle to Σ). In particular, the differential is computed from only four types of simple cascades. Theorem 1.3. Assume that pX, ωq is spherically monotone with monotonicity constant τ X and assume that Σ Ă X is Poincaré dual to Kω with τ X ą K ą 0.
Then, the split Floer homology of W is well-defined, and does not depend on the choice of Hamiltonian or of Reeb-invariant almost complex structure (in a comeagre set of such almost complex structures).
The only moduli spaces of split Floer cascades that count in the differential are the following: (0) Morse trajectories in Y or in W ; (1) Floer cylinders in RˆY , projecting to non-trivial classes in Σ, with asymptotics constrained by descending/ascending manifolds of critical points in Y ; (2) holomorphic planes in W that converge to a generic Reeb orbit in Y ; (3) holomorphic planes in W constrained to have a marked point in the descending manifold of a Morse function in W and whose asymptotic limit is constrained by the auxiliary Morse function on the manifold of orbits in Y .
We remark that Cases (2) and (3) are non-trivial cascades, but their components in RˆY lie in fibres of RˆY Ñ Σ. This is formulated more precisely in Propositions 6.2 and 6.3. See Figures 6.1, 6.2 and 6.3 for a depiction of Cases (1)-(3).
U of the 0-section in N Σ and a symplectic embedding ϕ : U Ñ X. By shrinking U as necessary, we may arrange that ϕ extends to an embedding of U.
After choosing a Hermitian connection on N Σ, we may lift any almost complex structure J Σ on Σ to an almost complex structure J on N Σ. We refer to any almost complex structure obtained in this way as a bundle almost complex structure on N Σ. Define the open set V " XzϕpUq. We will later perturb our almost complex structures in V.
Then, there exists a diffeomorphism ψ : W Ñ XzΣ with the following properties: (i) if J W is an almost complex structure on W , cylindrical on W zW and Reebinvariant, then ψ˚J W extends to an almost complex structure on X that restricts on ϕpUq to the push-forward by ϕ of a bundle almost complex structure; (ii) if J X in an almost complex structure on X that restricts to ϕpUq as the pushforward by ϕ of a bundle almost complex structure, then ψ˚J X is a cylindrical almost complex structure on W that is Reeb-invariant; (iii) if J X is the extension to X of ψ˚J W , then J Σ :" J X | Σ is also given by restricting J W to a parallel copy of tcuˆY for some c ą 0, and taking the quotient by the Reeb S 1 action.
Note that the diffeomorphism ψ is not symplectic. Furthermore, it can be arranged so that ψpW zBW q " V.
In Section 6.1, we impose additional conditions of monotonicity. These will also be relevant for the grading given in Definition 3.4.
Definition 2.3. The pair pX, Σq of a symplectic manifold pX 2n , ωq and a codimension-2 symplectic submanifold Σ 2n´2 Ă X is a monotone pair if (i) X is spherically monotone: there exists a constant τ X ą 0 so that for each spherical homology class A P H S 2 pXq, ωpAq " τ X xc 1 pT Xq, Ay; (ii) rΣs P H 2n´2 pX; Qq is Poincaré dual to rKωs for some K ą 0 with τ X ą K. In this case, we write ω Σ " ω| Σ .
Observe that Condition (ii) is automatically verified if rΣs is Poincaré dual to rKωs and is non-trivially spherically monotone. In that case, τ X´K is then the monotonicity constant of pΣ, ω Σ q, which we denote by τ Σ .
We will let N Σ denote the symplectic normal bundle to Σ. This can be equipped with the structure of a Hermitian complex line bundle over Σ.
We then obtain the following useful characterization of J W -holomorphic planes in W (see also [HK99]): Lemma 2.4. Under the diffeomorphism of Proposition 2.2, finite energy J Wholomorphic planes in W correspond to J X -holomorphic spheres in X with a single intersection with Σ. The order of contact gives the multiplicity of the Reeb orbit to which the plane converges.
Proof. Under the map ψ, a finite energy J W -plane in W gives a finite area J X -plane in XzΣ. The singularity at 8 is removable by Gromov's removal of singularities theorem, and thus the plane admits an extension to a J X -holomorphic sphere. The order of contact follows from considering the winding around Σ of a loop near the puncture.  Since ω " dpe r αq on R`ˆY , the Hamiltonian vector field associated to H is h 1 pe r qR, where R is the Reeb vector field associated to α. The fibres of Y Ñ Σ are periodic Reeb orbits. Their minimal periods are T 0 :" ş π´1 Σ ppq α, for p P Σ. The 1-periodic orbits of H are thus of two types: (1) constant orbits: one for each point in W and at each point in p´8, log 2sŶ Ă RˆY ; (2) non-constant orbits: for each k P Z`, there is a Y -family of 1-periodic X Horbits, contained in the level set Y k :" tb k uˆY , for the unique b k ą log 2 such that h 1 pe b k q " kT 0 . Each point in Y k is the starting point of one such orbit.
Remark 3.2. Notice that these Hamiltonians are Morse-Bott non-degenerate except at tlog 2uˆY . These orbits will not play a role because they can never arise as boundaries of relevant moduli spaces.
Recall that a family of periodic Hamiltonian orbits for a time-dependent Hamiltonian vector field is said to be Morse-Bott non-degenerate if the connected components of the space of parametrized 1-periodic orbits form manifolds, and the tangent space of the family of orbits at a point is given by the eigenspace of 1 for the corresponding Poincaré return map. (Morse non-degeneracy requires the return map not to have 1 as an eigenvalue and hence such periodic orbits must be isolated.) By an abuse of notation, we will both write π Σ : Y Ñ Σ to denote the quotient map that collapses the Reeb fibres, π Σ : RˆY Ñ Σ to denote the composition of this projection with the projection to Y .
We also fix some auxiliary data, consisting of Morse functions and vector fields. Fix throughout a Morse function f Σ : Σ Ñ R and a gradient-like vector field Z Σ P XpΣq, which means that 1 c |df Σ | 2 ď df Σ pZ Σ q ď c|df Σ | 2 for some constant c ą 0. Denote the time-t flow of Z Σ by ϕ t ZΣ . Given p P Critpf Σ q, its stable and unstable manifolds (or ascending and descending manifolds, respectively) are (3.1) W s Σ ppq :" Notice the sign of time in the flow, so that these are the stable/unstable manifolds for the flow of the negative gradient. We further require that pf Σ , Z Σ q be a Morse-Smale pair, i.e. that all stable and unstable manifolds of Z Σ intersect transversally. The contact distribution ξ defines an Ehresmann connection on the circle bundle S 1 ãÑ Y Ñ Σ. Denote the horizontal lift of Z Σ by πΣZ Σ P XpY q. We fix a Morse function f Y : Y Ñ R and a gradient-like vector field Z Y P XpY q such that pf Y , Z Y q is a Morse-Smale pair and the vector field Z Y´πΣ Z Σ is vertical (tangent to the S 1 -fibers). Under these assumptions, flow lines of Z Y project under π Σ to flow lines of Z Σ .
Observe that critical points of f Y must lie in the fibres above the critical points of f Σ (and these are zeros of Z Y and Z Σ respectively). For notational simplicity, we suppose that f Y has two critical points in each fibre. In the following, given a critical point for f Σ , p P Σ, we denote the two critical points in the fibre above p by p p and q p, the fibrewise maximum and fibrewise minimum of f Y , respectively. We will denote by M ppq the Morse index of a critical point p P Σ of f Σ , and bỹ M ppq " M ppq`ippq the Morse index of the critical pointp " p p orp " q p of f Y . The fibrewise index has ipp pq " 1 and ipq pq " 0.
Fix also a Morse function f W and a gradient-like vector field Z W on W , such that pf W , Z W q is a Morse-Smale pair and Z W restricted to r0, 8qˆY is the constant vector field B r , where r is the coordinate function on the first factor. We denote also by pf W , Z W q the Morse-Smale pair that is induced on XzΣ by the diffeomorphism in Lemma 2.4. Denote by M pxq the Morse index of x P Critpf W q with respect to f W .
We now define the Morse-Bott symplectic chain complex of W and H. Recall that for every k ą 0, each point in Y k :" tb k uˆY Ă R`ˆY is the starting point of a 1-periodic orbit of X H , which covers k times its underlying Reeb orbit.
For each critical pointp " p p orp " q p of f Y , there is a generator corresponding to the pair pk,pq. We will denote this generator byp k . The complex is then given by: In particular, Apγq " 0 for any constant orbit γ, and for any orbit γ k P Y k we have where b k is the unique solution to the equation h 1 pe b k q " kT 0 , as above. The action of γ k is the negative of the y-intercept of the tangent line to the graph of h at e b k . See Figure 3.1. The convexity of h implies that Apγ k q is monotone increasing in k.
3.1. Gradings. We will now define the gradings of the generators. For this, we will assume that pX, Σq is a monotone pair as in Definition 2.3.
Definition 3.4. For a critical point r p of f Y , and a multiplicity k, we define where we recall that τ X is the monotonicity constant of X and c 1 pN Σq " rKω Σ s. For a critical point x of f W , we define (3.5) |x| " n´M pxq.
Finally, for convenience, we introduce an index similar to the SFT grading for the Reeb orbits to which a split Floer cylinder converges at augmentation punctures. If γ is such a Reeb orbit, it is a k-fold cover of a fibre of Y Ñ Σ for some k. We then define its index to be: The justifications of these gradings comes from analyzing the Conley-Zehnder indices of the 1-periodic Hamiltonian orbits. These are defined for Morse nondegenerate Hamiltonian/Reeb orbits, and depend on a choice of trivialization of T W or of T pRˆY q over the orbit. See Definition 5.19 for the Morse-Bott analogue, and also [AM16, Section 3; Gut14]. The first key observation is that the Conley-Zehnder index of an orbit only depends on the trivialization of the complex line bundle L :" Λ n C T W over the orbit. For a constant orbit, we may take a constant trivialization, and applying Definition 5.19, we obtain the Conley-Zehnder index of the constant orbit to bé n`p2n´M pxqq " n´M pxq. A non-constant orbit γ in RˆY projects to a point in Σ. From this, we may take a "constant" trivialization of γ˚ξ by taking the horizontal lift of a constant trivialization of T π pγqΣ. Then, by considering the linearized Hamiltonian flow in the vertical direction, we obtain the Conley-Zehnder index of the corresponding generator r p k to beM pr pq`1´n.
Notice also that Y may be capped off by the normal disk bundle over Σ, and each orbit bounds a disk fibre. The trivialization induced by the fibre differs from the constant trivialization only in a k-fold winding in the RB r ' RR direction. The resulting Conley-Zehnder index of r p k for this trivialization induced by the disk fibre is thenM pr pq`1´n´2k. Now, suppose that γ k is the k-fold cover a simple Reeb orbit γ, and suppose it is contractible in W . Denote by 9 B a disk in W whose boundary is γ k . Note that γ k is also the boundary of a k-fold cover of a fibre of the normal bundle to Σ in X This cover of a fibre can be concatenated with 9 B to produce a spherical homology class B P H S 2 pXq such that B ‚ Σ " k ą 0. Conversely, note that any B P H S 2 pXq such that B ‚ Σ " k gives rise to a disk 9 B bounding γ k . The complex line bundle L| 9 B is trivial, since 9 B is a disk. This induces a trivialization of L over γ k , which can be identified with a trivialization of L bk over γ.
The orbit γ bounds the fibre of the normal bundle to Σ, which induces a trivialization of L over γ. This is precisely the trivialization considered above. The k-fold cover of the fibre of the normal bundle induces a trivialization over γ k . The relative windings of these two trivializations is given by xc 1 pLq, By, since this represents the obstruction to extending the trivialization of L over 9 B to all of B. Recall that c 1 pLq " c 1 pT Xq.
Putting this together, we obtain that the Conley-Zehnder index of the orbit with respect to the trivialization induced by the disk 9 B is given by (3.7) CZ W H pr p k q "M pr pq`1´n´2k`2xc 1 pT Xq, By Finally, we obtain the grading from Equation (3.4) by using the spherical monotonicity of X and the fact that k " B ‚ rΣs " KωpBq.

K˙k
This formula holds when k P ωpπ 2 pXqq, and we extend it as a fractional grading for all k P Z. (This corresponds to the fractional SFT grading from [EGH00, Section 2.9.1].) Finally, we compare our gradings with those described by Seidel [Sei08] and generalized by McLean [McL16] (he considers Reeb orbits, but there is an analogous construction for Hamiltonian orbits) in the case that c 1 pT W q P H 2 pW ; Zq is torsion, so N c 1 pT W q " 0 for a suitable choice of N ą 0. Note that in our setting, this holds if X is monotone (and not just spherically monotone) and Σ is Poincaré dual to a multiple of ω.
As before, we consider the case of the non-constant 1-periodic Hamiltonian orbit that is contractible in W , γ k that bounds a disk 9 B, and hence so there exists a spherical class B P H S 2 pX; Zq such that k " xc 1 pT Xq, By ą 0. L " Λ n C pT W q is trivial over the disk 9 B, and there is a trivialization of T W over 9 B that is unique up to homotopy. This trivialization defines the Conley-Zehnder index of γ k , CZ W H pγ k q. Let us consider the associated trivialization of pT W q 'N over 9 B, which is also unique up to homotopy. This trivialization induces a Conley-Zehnder index on pγ k q 'N , which is, by additivity, N CZ W H pγ k q. Now, since N c 1 pT W q " 0, it follows that L bN is trivial. We fix a trivialization of this line bundle, which induces a trivialization of pT W q 'N over the 2-skeleton of W . The Seidel-McLean grading convention consists of using this global trivialization to define the Conley-Zehnder index of pγ k q 'N .
Since pT W q 'N has a unique trivialization over 9 B, up to homotopy, the two Conley-Zehnder indices we obtained for pγ k q 'N must agree, and hence agree for γ k .
In our maximally Morse-Bott setting, the linearized return Reeb map is the identity. Therefore, if T W is trivialized over an orbit γ and this trivialization is used to obtain a trivialization over a higher cover γ k , the Conley-Zehnder CZpγ k q is affine in k, where the linear term is the Robbin-Salamon index [RS93,Gut14] of the Morse-Bott degenerate Reeb orbit γ.
We will now argue that both the Seidel-McLean Conley-Zehnder index and our grading are affine in the covering multiplicity k. Our grading is affine by construction. Suppose now that γ is a Hamiltonian orbit that projects to a simply covered Reeb orbit. Then, the fact that c 1 pT W q is torsion again gives a trivialization of L bN and hence chooses a preferred homotopy class of trivializations of γ˚pT W 'N q. By the fact that the linearized return map (restricted to the contact structure) is the identity, it follows that the Conley-Zehnder index of pγ k q 'N is affine in the covering multiplicity k. It remains affine after dividing by N , and thus the Seidel-McLean grading is also affine in k.
As the two gradings agree on all contractible orbits and are both affine in the covering multiplicity, our grading conventions agree.
The index of the Reeb orbit comes from similar considerations for the Conley-Zehnder index of the Reeb orbit, together with the n´3 shift coming from the grading of SFT. In particular, the Fredholm index for an unparametrized holomorphic plane in W asymptotic to the closed Reeb orbit γ 0 (free to move in its Morse-Bott family) will be given by |γ 0 |.
Remark 3.5. Even though the idea of a fractional grading may seem unnatural at first, it can be thought of as a way of keeping track of some information about the homotopy classes of the Hamiltonian orbits. Indeed, two generators of our chain complex can only be homotopic Hamiltonian orbits (which is a necessary condition for the existence of a Floer cylinder connecting them) if the difference of their degrees is an integer. Alternatively, we could have introduced an additional parameter to keep track of the homotopy classes of Hamiltonian orbits, as done for instance in [BO09b].

Split Symplectic Homology moduli spaces
In this section, we describe the moduli spaces of cascades that contribute to the differential in the Morse-Bott split symplectic homology of W .
We also define auxiliary moduli spaces of spherical "chains of pearls" in Σ and in X. (These are familiar objects, reminiscent of ones considered in the literature for Floer homology of compact symplectic manifolds [BC09,Oh96,PSS96].) 4.1. Split Floer cascades. We now identify the moduli spaces of split Floer cylinders with cascades we use to define the differential on the chain complex (3.2).
First, we define the basic building blocks: split Floer cylinders. We consider two types of basic split Floer cylinders: one connecting two non-constant 1-periodic Hamiltonian orbits and one that connects a non-constant 1-periodic orbit to a constant one (in W ).
Notice that we may identify a 1-periodic orbit of H with its starting point, and in this way, we have an identification between Y k and the set of (parametrized) 1-periodic orbits of H that have covering multiplicity k over the underlying simple periodic orbit.
Definition 4.1. Let x˘P Y k˘b e 1-periodic orbits of X H in RˆY . A split Floer cylinder from x´to x`consists of a mapṽ " pb, vq : RˆS 1 zΓ Ñ RˆY , where Γ " tz 1 , . . . , z k u Ă RˆS 1 is a (possibly empty) finite subset, together with equivalence classes rU i s of J W -holomorphic planes U i : C Ñ W for each z i P Γ, such that ‚ṽ satisfies Floer's equation (4.1) B sṽ`JY pB tṽ´XH pṽqq " 0; ‚ lim sÑ˘8ṽ ps, .q " x˘; ‚ if Γ ‰ H, then, for conformal parametrizations ϕ i : p´8, 0sˆS 1 Ñ RŜ 1 ztz 1 , . . . , z k u of neighborhoods of the z i , lim sÑ´8ṽ pϕ i ps, .qq " p´8, γ i p.qq, where the γ i are periodic Reeb orbits in Y ; ‚ for each Reeb orbit γ i above, U i : C Ñ W is asymptotic to p`8, γ i q. We consider U i up to the action of AutpCq.
Callṽ the upper level of the split Floer cylinder.
For the upper levels of split Floer cylinders in RˆY , we introduce a new form of energy, a hybrid between the standard Floer energy and the Hofer energy used in symplectizations. Recall that the Hofer energy of a punctured pseudoholomorphic curveũ in the symplectization of Y with contact form α is given by: supt żũ˚d pψαq | ψ : R Ñ r0, 1s smooth and nondecreasingu.
In a symplectic manifold either compact or convex at infinity, the standard Floer energy of a cylinderṽ : RˆS 1 Ñ W is given by żṽ˚ω´ṽ˚d H^dt.
In our situation, however, the target manifold is RˆY , which has a concave end. We therefore need to combine these two types of energy.
Definition 4.3. Consider a Hamiltonian H : RˆY Ñ R so that dH has support in rR, 8qˆY , for some R P R. Let ϑ R be the set of all non-decreasing smooth functions ψ : R Ñ r0, 8q such that ψprq " e r for r ě R.
The hybrid energy E R ofṽ : RˆS 1 zΓ Ñ RˆY solving Floer's equation (4.1) is then given by: Notice that this is equivalent to partitioning RˆS 1 zΓ " S 0 Y S 1 , so that S 0 " v´1prR,`8qˆY q and S 1 " SzS 0 . Then,ṽ has finite hybrid energy if and only if v| S0 has finite Floer energy andṽ| S1 has finite Hofer energy.
Equivalently,ṽ has finite hybrid energy if and only if the punctures t˘8uYΓ can be partitioned into Γ F and Γ C (with`8 P Γ F , Γ Ă Γ C and´8 either in Γ F or in Γ C ), such that in a neighbourhood of each puncture in Γ F , the mapṽ is asymptotic to a Hamiltonian trajectory and in a neighbourhood of each puncture in Γ C , the map is proper and negatively asymptotic to an orbit cylinder in RˆY . This follows from a variation on the arguments in [BEH`03, Proposition 5.13, Lemma 5.15]. We will use the following notation to denote the Hamiltonian orbits to which such a punctured cylinderṽ is asymptotic: Instead, if´8 P Γ C , we will write vp´8, tq " lim sÑ´8 vps, tq for the Reeb orbit in 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 S 1 parametrization of the Reeb orbits to which v is asymptotic at punctures z P Γ, we will avoid using the analogous notation at punctures in Γ. We now define a split Floer cylinder with cascades between two generators of the chain complex (3.2).
Definition 4.4. Fix N ě 0. Let S 0 , S 1 , . . . , S N be a collection of connected spaces of orbits, with S 0 " Y k0 or S 0 " W , and S i " Y ki for 1 ď i ď N . Let pf i , Z i q, i " 0, . . . , N`1 be the pair of Morse function and gradient-like vector field of Let x be a critical point of f 0 and y a critical point of f N (so x and y are generators of the chain complex (3.2)).
A Floer cylinder with 0 cascades (N " 0), from x to y, consists of a positive gradient trajectory ν : R Ñ S 0 , such that νp´8q " x, νp`8q " y and 9 ν " Z 0 pνq. A Floer cylinder with N cascades, N ě 1, from x to y, consists of the following data: (i) N´1 length parameters l i ą 0, i " 1, . . . , N´1; (ii) two half-infinite gradient trajectories, ν 0 : p´8, 0s Ñ S 0 and ν N : r0,`8q Ñ S N with ν 0 p´8q " x, ν N p`8q " y and 9 ν i " Z i pν i q for i " 0 or N ; (iii) N´1 gradient trajectories ν i defined on intervals of length l i , ν i : r0, l i s Ñ S i for i " 1, . . . , N´1 such that 9 ν i " Z i pν i q; (iv) N non-trivial split Floer cylinders from ν i´1 pl i´1 q P S i´1 to ν i p0q P S i , where we take l 0 " 0. In the case of a Floer cylinder with N ě 1 cascades, we refer to the non-trivial Floer cylindersṽ i as sublevels. Notice that if S 0 (and thus all S i ) is of the form Y k , all the split Floer cylinders are as in Definition 4.1. If S 0 " W , then the bottom-most level is a split Floer cylinder as in Definition 4.2.
See Figure 4.1 for a schematic illustration.
Definition 4.5. We refer to the punctures Γ appearing in Definitions 4.1 and 4.2 as augmentation punctures. The corresponding J W holomorphic planes, U i : C Ñ W are referred to as augmentation planes. This terminology is by analogy to linearized contact homology, where rigid planes of this type give an (algebraic) augmentation of the full contact homology differential. We make no such claim about algebraic properties here.
Remark 4.6. Notice that the hybrid energy of each sublevel must be non-negative. Since we require that the sublevels are non-trivial, it follows that any such cascade with collections of orbits S i " Y ki , i " 1, . . . , N and S 0 " Y k0 , or, if S 0 " W , with k 0 " 0, we must have that the sequence of multiplicities is monotone increasing By a standard SFT-type compactness argument, the Floer-Gromov-Hofer compactification of a moduli spaces of split Floer cylinders with cascades will consist of several possible configurations. The length parameters can go to 0 or to 8 (in the latter case, corresponding to a Morse-type breaking of the gradient trajectory). The split Floer cylinders can break at Hamiltonian orbits, thus increasing the number of cascades but with a length parameter of 0. They can also split off a holomorphic building with levels in RˆY and in W . We will see that this latter degeneration can't occur in low dimensional moduli spaces, at least under our monotonicity assumptions. In this way, we will see that the N and N`1 dimensional moduli spaces of split Floer cylinders with N cascades have boundary consisting of concatenations of split Floer cylinders with cascades and of the boundaries of such moduli spaces of split Floer cascades with more sublevels. Notice also that for energy reasons, these Floer cylinders with cascades will not break at constant Hamiltonian trajectories in p´8, log 2sˆY .
We now define the split Floer differential B on the chain complex (3.2). Given generators x, y, denote by M H,N px, y; J W q the space of split Floer cylinders with N cascades from x to y (with negative end at x and positive end at y).
For N ě 1, this moduli space M H,N px, y; J W q has an R N action by domain automorphisms corresponding to R-translation of the domain cylinders RˆS 1 . When N " 0, this moduli space is of gradient trajectories, and also admits an R reparametrization action.
We now define (4.3) We call B the split Floer differential on (3.2).

Transversality for the Floer and holomorphic moduli spaces
In this section, we will build the transversality theory needed for the Floer cascades that appear in the split Floer differential as in Equation (4.3). In the process, we will also discuss transversality for pseudoholomorphic curves in X and in Σ, which will be necessary for the proof of our main result.  5.1. Statements of transversality results. Before stating the main result of this section, we will introduce some definitions allowing us to relate transversality for split Floer cylinders with cascades to transversality problems for spheres in Σ and in X with various constraints.
Lemma 5.1. Letṽ : RˆS 1 zΓ Ñ RˆY be a finite hybrid energy Floer cylinder in RˆY (as in Definition 4.3), converging to a Hamiltonian orbit in the manifolds Y`at`8 and converging at´8 either to a Hamiltonian orbit in the manifold Yó r to a Reeb orbit at t´8uˆY , and with finitely many punctures at Γ Ă RˆS 1 converging to Reeb orbits in t´8uˆY . Then, the projection π Σ˝ṽ extends to a smooth J Σ -holomorphic sphere π Σ˝ṽ : CP 1 Ñ Σ.
Proof. The projection π Σ˝ṽ is J Σ -holomorphic on RˆS 1 , since H is admissible (as in Definition 3.1). The result now follows from Gromov's removal of singularities theorem together with the finiteness of the energy of π Σ˝ṽ .
In order to describe the projection to Σ of the levels of a split Floer cylinder with N cascades that map to RˆY , we introduce the following: Definition 5.2. A chain of pearls from q to p, where p and q are critical points of f Σ , consists of the following: ‚ N ě 0 parametrized J Σ -holomorphic spheres w i in Σ with two distinguished marked points at 0 and 8 and a possibly empty collection of additional marked points z 1 , . . . , z k on the union of the N domains (distinct from 0 or 8 in each of the N spherical domains); the spheres are either non-constant or contain at least one additional marked point; ‚ if N " 0, an infinite positive flow trajectory of Z Σ from q to p; if N ě 1, a half-infinite trajectory of Z Σ from q to w 1 p0q, a half-infinite trajectory of Z Σ from w N p8q to p; ‚ if N ě 1, positive length parameters l i , i " 1, . . . , N´1, so that ϕ li ZΣ pw i p8qq " w i`1 p0q, i " 1, . . . , N´1.
See Figure 5.1. If such a chain of pearls is the projection to Σ of the components in RˆY is a split Floer cylinder, then the additional marked points in the pseudoholomorphic spheres correspond to augmentation punctures in the Floer cylinders, where they converge to cylinders over Reeb orbits that are capped by planes in W .
Notice that the geometric configuration of two spheres touching at a critical point of f Σ admits an interpretation as a chain of pearls in Σ, since the critical point is the image of any positive length flow line with that initial condition.
Definition 5.3. A chain of pearls with a sphere in X from x to p, where x is a critical point of f W and p is a critical point of f Σ , consists of the following: ‚ N ě 1 parametrized J Σ -holomorphic spheres w i in Σ with two distinguished marked points at 0 and 8 and a possibly empty collection of additional marked points z 1 , . . . , z k on the union of the N domains (distinct from 0 or 8); ‚ a parametrized non-constant J X -holomorphic sphere v in X; ‚ a half-infinite trajectory of Z Σ from w N p8q to p, a half-infinite trajectory of´Z X from x to vp0q (where Z X is the push-forward of Z W by the inverse of the map from Lemma 2.4); ‚ positive length trajectories of Z Σ from w i p8q to w i`1 p0q for i " 1, . . . , N´1; ‚ the sphere in X touches the first sphere in Σ: w 1 p0q " vp`8q; ‚ the spheres w 2 , . . . , w N satisfy the stability condition that they are either non-constant or contain at least one of the additional marked points (v is automatically non-constant and w 1 is allowed to be constant and unstable).
Definition 5.4. An augmented chain of pearls [or an augmented chain of pearls with a sphere in X] is a chain of pearls [or chain of pearls with a sphere in X] together with k equivalence classes rU i s of J X -holomorphic spheres U i : CP 1 Ñ X, i " 1, . . . , k, with the following additional properties: ‚ for each z P CP 1 , U i pzq P Σ if and only if z " 8; ‚ if the puncture z i is in the domain of the holomorphic sphere w ji : CP 1 Ñ Σ, then w ji pz i q " U i p8q; ‚ each U i is considered up to the action of AutpCP 1 , 8q " AutpCq, that is, as an unparametrized sphere.
From Lemma 5.1 and Lemma 2.4 and the fact that the trajectories of Z Y cover trajectories of Z Σ , it follows that a Floer cylinder with N cascades projects to a chain of pearls or a chain of pearls with a sphere in X. Additionally, again by Lemma 2.4, if any of the sublevels have augmentation planes, then those correspond to spheres in X passing through Σ at the images of the corresponding marked points in the chain of pearls.
Observe that we allow the sphere w 1 to be unstable in the definition of a chain of pearls in Σ with a sphere in X. The case in which w 1 is a constant curve without marked points corresponds to the situation in which the corresponding Floer cascade contains a non-trivial Floer cylinderṽ 1 contained in a single fibre of RˆY Ñ Σ, and has the asymptotic limitsṽ 1 p`8, tq on a Hamiltonian orbit and v 1 p´8, tq on a closed Reeb orbit in t´8uˆY . The Floer cylinderṽ 1 in RˆY is non-trivial and hence stable, whereas the corresponding sphere w 1 in Σ is unstable. Since we do not quotient by automorphisms (yet), this does not pose a problem. (see Figure 6.3 and Proposition 6.3 below, where this situation is analysed.) Definition 5.5. 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 no sphere is contained in the image of another. If the chain of pearls has a sphere v in X, we require v to be somewhere injective (but the first sphere in Σ is allowed to be constant, with image contained in the image of v).
An augmented chain of pearls is simple if the chain of pearls is simple and the augmentation spheres in X are somewhere injective, none has image contained in the fixed open neighbourhood ϕpUq and no sphere in X has image contained in the image of another sphere in X.
Remark 5.6. Recall that a chain of pearls with a sphere in X has a distinguished sphere v in X for which vp0q is on the descending manifold of a critical point x of f W . By the construction of f W , this forces the image of v to intersect the complement of the tubular neighbourhood of Σ. As we revisit in Remark 6.7, Fredholm index considerations related to monotonicity will force the augmentation planes/spheres to leave the tubular neighbourhood.
Remark 5.7. Notice that our condition on a simple chain of pearls is slightly different than the condition imposed in [MS04, Section 6.1], with regard to constant spheres. For a chain of pearls to be simple by our definition, constant spheres may not be contained in another sphere, constant or not. In [MS04], there is no such condition on constant spheres.
Definition 5.8. Given a finite hybrid energy Floer cylinder with N cascades, we obtain an augmented chain of pearls (possibly with a sphere in X) by the following construction: (1) cylinders in RˆY are projected to Σ: by Lemma 5.1 these form holomorphic spheres in Σ; (2) planes in W are interpreted as spheres in X by Lemma 2.4; (3) flow lines of the gradient-like vector field Z Y are projected to flow lines of Z Σ .
We refer to this augmented chain of pearls in Σ (possibly with a sphere in X) as the projection of the Floer cylinder with N cascades. A finite hybrid energy Floer cylinder with N cascades is simple if the projected chain of pearls is simple.
Given generators x, y of the chain complex (3.2), denote by MH ,N px, y; J W q the space of simple split Floer cylinders with N cascades from x to y. Recall that if x or y is in RˆY , the corresponding generator is described by a critical point r p of f Y (which can be either q p or p p), together with a multiplicity k. If instead, x or y is in W , it corresponds to a critical point of f W .
Proposition 5.9. There exists a residual set J reg W Ă J W of almost complex structures such that for each J W P J reg W , MH ,N px, y; J W q is a manifold. If N " 0, and thus x, y are generators in RˆY , then x " r q k , y " r p k for the same multiplicity k, and If N ě 1, and x, y are generators in RˆY , then x " r q k´, y " r p k`a nd dim R MH ,N pr q k´, r p k`; J W q " |r p k`|´| r q k´|`N´1 Finally, if x P W and y P RˆY , then x P Critpf W q, y " r p k and dim R MH ,N px, r p k ; J W q " |r p k |´|x|`N.
Furthermore, the image P pJ reg W q Ă J Σ (recall Definition 2.6) is residual and consists of almost complex structures that are regular for simple pseudoholomorphic spheres in Σ.
The two different formulas involving N reflect the fact that N counts the number of cylinders in RˆY . In the case of a Floer cascade that descends to W , there are therefore N`1 cylinders in the cascade.
Remark 5.10. These index formulas justify that the moduli spaces are rigid (modulo their R, R N and R N`1 actions) when the index difference is 1, which then justifies the definition of the differential given in Equation (4.3). Indeed, observe that the case N " 0 corresponds to a pure Morse configuration and doesn't depend on any almost complex structure. We count rigid flow lines modulo the R action, and thus require |y|´|x| " 1. For generators x, y in RˆY , we consider these N cylinders modulo the R action on each one, giving an R N action. From this, a rigid configuration has |y|´|x|`N´1 " N . For the case with x P W , we have N`1 cylinders in the Floer cascade, so we have a rigid configuration modulo the R N`1 action when N`1 " |y|´|x|`N .
Note that when both x, y are generators in RˆY , the moduli space MH ,N px, y; J W q will depend on J W only insofar as augmentation planes appear, otherwise it depends only on J Y . The split Floer differential B, introduced in Equation (4.3) was defined by counting elements in M H,N px, y; J W q. We will see in Propositions 6.2 and 6.3 that our monotonicity assumptions imply that this is equivalent to counting simple configurations in MH ,N px, y; J W q.
The rest of this section will be devoted to the proof of Proposition 5.9. It will proceed in the following steps: ‚ Section 5.2 describes the Fredholm set-up for Floer cascades. In Section 5.2.1, we discuss the necessary function spaces and linear theory for the Morse-Bott problems. Then, Section 5.2.2 splits the linearization of the Floer operator in such a way as to split the transversality problem into two problems. The first is a Cauchy-Riemann-type operator acting on sections of a complex line bundle, and it is transverse for topological reasons (automatic transversality). The second is a transversality problem for a Cauchy-Riemann operator in Σ. ‚ Section 5.3 adapts the transversality arguments from [MS04] in order to obtain transversality for chains of pearls in Σ. ‚ Section 5.4 shows transversality for the components of the cascades contained in W . This problem is translated into the equivalent problem of obtaining transversality for spheres in X with order of contact conditions at Σ, together with evaluation maps. The main technical point is an extension of the transversality results from [CM07]. ‚ Finally, Section 5.5 uses the splitting from Section 5.2.2 to lift the transversality results in Σ to obtain transversality for Floer cylinders with cascades, and to finish the proof of Proposition 5.9.

A Fredholm theory for Morse-Bott asymptotics.
In this section, we collect some facts about Cauchy-Riemann-type operators on Hermitian vector bundles over punctured Riemann surfaces, specifically in the context of degenerate asymptotic operators. These facts can mostly be found in the literature, but not in a unified way. The main reference for these results is [Sch95]. Additional references include [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 RˆS 1 be a finite set of punctures and denote RˆS 1 zΓ by 9 S. Write Γ`" t`8u and Γ´" t´8u Y Γ. Consider, for each puncture z P Γ, exponential cylindrical polar coordinates of the form p´8,´1sˆS 1 Ñ RˆS 1 zΓ : ρ`iη Þ Ñ z 0` e 2πpρ`iηq . Choose ą 0 sufficiently small these are embeddings and that the image of these embeddings for any two different punctures are disjoint.
Let E Ñ 9 S be a (complex) rank n Hermitian vector bundle over 9 S together with a preferred set of trivializations in a small neighbourhood of Γ Y t˘8u. While the bundle E over 9 S is trivial if there is at least one puncture, this is no longer the case once we specify these preferred trivializations near Γ Y t˘8u. We therefore associate a first Chern number to this bundle relative to the asymptotic trivializations. There are several equivalent definitions. One approach is to consider the complex determinant bundle Λ n C E. The trivialization of E at infinity gives a trivialization of this determinant bundle at infinity, and we can now count zeros of a generic section of Λ n C E that is constant (with respect to the prescribed trivializations) near the punctures. We denote this Chern number by c 1 pEq, but emphasize that it depends on the choice of these trivializations near the punctures.
Since we cannot specify where an augmentation puncture appears when we stretch the neck on a Floer cylinder, we should have the punctures in Γ free to move on the domain RˆS 1 . This creates a problem when we try to linearize the Floer operator in a family of domains where the positions of the punctures are not fixed. We will instead consider a 2#Γ parameter family of almost complex structures on RˆS 1 , but fix the location of the punctures. Specify a fixed collection Γ of punctures on RˆS 1 and, for any other collection of augmentation punctures, choose an isotopy with compact support from the new punctures to the fixed ones. We take the push-forward of the standard complex structure in RˆS 1 by the final map of the isotopy, to produce a family of complex structures on RˆS 1 , which can be assumed standard near Γ and outside of a compact set.
For each z P Γ, let β z : 9 S Ñ r0,`8q be a function supported in a small neighbourhood of z, with β z pρ, ηq "´ρ near the puncture (where pρ, ηq are cylindrical polar coordinates near z, as above). Similarly, let β`: RˆS 1 Ñ r0,`8q be supported in a region where s is sufficiently large and β`ps, tq " s for s large enough. Let β´: RˆS 1 Ñ r0,`8q have support near´8, and β´ps, tq "´s for s sufficiently small.
In many situations, it will be convenient to consider the function Finally, on the punctured cylinder 9 S, we take the measure induced by an area form on 9 S that has the form ds^dt for |s| large and that has the form dρ^dη in the cylinder polar coordinates near each puncture in Γ. Notice that pairing this with the domain complex structure induces a metric on 9 S for which the vector field B η , defined near a puncture in Γ by the exponential cylindrical polar coordinates, has norm comparable to 1.
Given a vector of weights δ : Γ Y t˘8u Ñ R, we define W 1,p,δ p 9 S, Eq to be the space of sections u of E for which u e ř δzβz`δ´β´`δ`β`P W 1,p p 9 S, Eq (with respect to the measure and metric described above). Note that these sections decay exponentially fast at the punctures where δ ą 0 and are allowed to have exponential growth at punctures where δ ă 0. We can similarly define L p,δ p 9 S, Eq. While these definitions involve making various choices, the resulting metrics are strongly equivalent. In practice, we'll typically take p ą 2 to obtain continuity of the sections. By a similar construction, we may define W m,p,δ as well.
We will say that a differential operator D : ΓpEq Ñ Λ 0,1 T˚9 S b E is a Cauchy-Riemann operator if it is a real linear Cauchy-Riemann operator [MS04, Definition C.1.5] such that, near˘8 , it takes the form: where Jps, tq is a smooth function on R˘ˆS 1 with values in almost complex structures on C n compatible with the standard symplectic form, and Aps, tq takes values in real matrices on R 2n -C n . We further impose that these functions converge as s Ñ˘8, Jps, tq Ñ J z ptq and Aps, tq Ñ A z ptq, where A z ptq is a loop of self-adjoint matrices. We impose the same conditions near punctures z P Γ, using the local coordinates pρ, ηq instead of ps, tq in (5.2). Associated to such a Cauchy-Riemann operator D, we obtain asymptotic operators at each puncture in Γ Y t˘8u by A z :"´J z ptq d dt´A z ptq. This is a densely defined unbounded self-adjoint operator on L 2 pS 1 , R 2n q. Let σpA z q Ă R denote its spectrum. This will consist of a discrete set of eigenvalues. If an asymptotic operator A z does not have 0 in its spectrum, we say the asymptotic operator is non-degenerate. If all the asymptotic operators are non-degenerate, we say D itself is non-degenerate.
Note that we obtain a path of symplectic matrices associated to the asymptotic operator A z by finding the fundamental matrix Φ to the ODE d dt x " J z ptqA z ptqx. The asymptotic operator is non-degenerate if and only if the time-1 flow of the ODE does 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 5.11. An asymptotic operator induces a path of symplectic matrices, and this identification (understood correctly) is a homotopy equivalence. This will allow us to associate a Conley-Zehnder index to a periodic orbit of a Hamiltonian vector field, given a trivialization of the tangent bundle along the orbit. In order to do so, we take the linearized flow map, which defines a path Φ : r0, 1s Ñ Spp2nq with respect to the fixed trivialization. If we fix a path of almost complex structures, this path of symplectic matrices satisfies an ODE 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. This is homotopic to the asymptotic operator coming from the linearized Floer operator.
Proposition 5.12. Suppose A z is non-degenerate and E is a rank 1 vector bundle. Then, each eigenfunction u : S 1 Ñ C of A z has a winding number that depends only on its eigenvalue λ, wpλq. The function w : σpA z q Ñ Z is non-decreasing in λ and is surjective. If λ˘are eigenvalues so that λ´ă 0 ă λ`and there are no eigenvalues in the interval pλ´, λ`q, then CZpA z q " wpλ´q`wpλ`q.
This formulation will be the most useful for our calculations. Furthermore, in the case of a higher rank bundle, we use the axiomatic description, see for instance [HWZ95,Theorem 3.1] to observe that CZpA z q is invariant under deformations for which 0 is never in the spectrum, and that if the operator can be decomposed as the direct sum of operators, then the Conley-Zehnder index is additive.
The following computation is useful at several points in the paper. It can often be combined with Proposition 5.12 to compute Conley-Zehnder indices of interest.
Lemma 5.13. Given a constant C ě 0, the spectrum σpA C q of the operator If λ is an eigenvalue associated to k P Z, then the winding number of the corresponding eigenvector is |k| if λ ě 0 and´|k| if λ ď 0. If C " 0, then all eigenvalues have multiplicity 2 (see Table 1). If C ą 0, then the same is true except for the eigenvalues´C and 0, corresponding to k " 0 above, both of which have multiplicity 1 (see Table 2). In particular, the σpA 0 q " 2πZ and the winding number of 2πk is k.
Proof. An eigenvector v : Computing the eigenvalues of the matrix on the right, and requiring that they be of the form 2πik, k P Z (since vpt`1q " vptq), yields the result. Corollary 5.14. Take C ě 0 and δ ą 0 such that r´δ, δs X σpA C q " t0u. Then and CZpA C´δ q " 1.
For any n ě 0, takinǵ Proof. The case n " 1 follows from Proposition 5.12 and Lemma 5.13. The case of general n uses the additivity of CZ under direct sums.
Definition 5.15. A key observation for our computations of Fredholm indices (as noted, for instance, in [HWZ99]) is that a Cauchy-Riemann operator with asymptotic operators A z is conjugate to the Cauchy-Riemann operator This has asymptotic operators A δ z " A z˘δz (where the sign is positive at positive punctures and negative at negative punctures). We refer to these as the δ-perturbed asymptotic operators.
This observation about the conjugation of the weighted operator to the nondegenerate case, combined with Riemann-Roch for punctured domains (see for instance, [Sch95, Theorem 3.3.11; HWZ99, Theorem 2.8; Wen16a, Theorem 5.4]) gives the following.
Theorem 5.16. Let δ : Γ Ñ R such that¯δ z R σpA z q. Then, the Cauchy-Riemann operator D : W 1,p,δ p 9 S, Eq Ñ L p,δ p 9 S, Λ 0,1 T˚9 S b Eq with asymptotic operators A z , z P Γ is Fredholm and its index is given by Now, a useful fact for us is a description of how the Conley-Zehnder index changes as a weight crosses the spectrum of the operator: Lemma 5.17. Suppose that r´δ,`δs X σpA z q " t0u. Then, To obtain a result that is useful for our moduli spaces of cascades asymptotic to Morse-Bott families of orbits, we consider the following modification of our function spaces.
To each puncture, we associate a subspace of the kernel of the corresponding asymptotic operator, which we denote by V z , z P Γ, V´, V`and write V for this collection. Then, for each puncture z P Γ and also˘8, we associate a smooth bump function µ z , µ˘, supported near and identically 1 even nearer to its puncture. We then define uch that u´ÿ c z µ z´c´µ´´c`µ`P W 1,p,δ p 9 S, Equ.
We remark that we are using the asymptotic cylindrical coordinates near Γ and the asymptotic trivialization of E in order to define the local sections c z µ z .
In this paper, we are primarily concerned with Cauchy-Riemann operators defined on 9 S " RˆS 1 and on 9 S " RˆS 1 ztP u (a cylinder with one additional negative puncture). In the case of RˆS 1 , we will write V " pV´; V`q, and in the case of RˆS 1 ztP u, we will write V " pV´, V P ; V`q. (The negative punctures are enumerated first, and separated from the positive puncture by a semicolon.) Observe that since the vector spaces V z are in the kernel of the corresponding asymptotic operators, for any choice of V and any vector of weights δ, we have that the Cauchy-Riemann-type operator D can be extended to Let dim V z denote the dimension of the vector space V z and let codim V z " dim pker A z {V z q. Combining Theorem 5.16 with Lemma 5.17, we have: Theorem 5.18. Let δ ą 0 be sufficiently small that for z P Γ`, r´δ, 0qXσpA z q " H and such that for z P Γ´, p0, δs X σpA z q " H.
For each z P Γ, fix the subspace V z Ă ker A z . Then, D : W 1,p,δ V p 9 S, Eq Ñ L p,δ p 9 S, Λ 0,1 T˚9 S b Eq. is Fredholm, and its Fredholm index is given by In applications where there are Morse-Bott manifolds of orbits, we will typically take V z to be the tangent space to the descending manifold of a critical point p z on the manifold of orbits at a positive puncture, and V z will be the tangent space to ascending manifold of a critical point p z at a negative puncture. In either case, the contribution to IndpDq of dim V z or of codim V z will be the Morse index of the appropriate critical point. This motivates the following definition.
Definition 5.19. Let δ ą 0 be sufficiently small. If p z is a critical point of an auxiliary Morse function on the manifold of orbits associated to z, then the Conley-Zehnder index of the pair pA z , p z q is In this case, we can write the Fredholm index as We conclude with a lemma that is particularly useful when applying the automatic transversality result [Wen10,Proposition 4.22]. The lemma states that the Fredholm index of an operator with a small negative weight at a puncture is the same as that of the corresponding operator with a small positive weight at that puncture, if the puncture is decorated with the kernel of the corresponding asymptotic operator. The former indices are used in [Wen10,Proposition 4.22], whereas the latter can be computed using Theorem 5.18. Additionally, the latter arises naturally in the linearization of the non-linear problem.
We first learned this result from Wendl [Wen05]. We give a proof of this formulation since it is slightly stronger than what we have found in the literature (and is still not as strong as can be proved.) Lemma 5.20. Let D be a Cauchy-Riemann-type operator. Fix a puncture z 0 P Γ Y t˘8u.
Let δ and δ 1 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,δ 1 , and for which there is a z 0 P Γ Y t˘8u with δ z0 ą 0 and δ 1 z0 ă 0 and so that for each z P Γ Y t˘8u with z ‰ z 0 , the weights δ z " δ 1 z . Let V be the trivial vector space at each puncture other than z 0 and let V z0 be the kernel of the asymptotic operator at z 0 .
Then, the induced operators Note that W 1,p,δ V p 9 S, Eq is a subspace of W 1,p,δ 1 p 9 S, Eq, and thus the kernel of D δ is contained in the kernel of D δ 1 . Now, by a linear version of the analysis done in [HWZ99,Sie08], any element of the kernel of D δ 1 converges exponentially fast at z 0 to an eigenfunction of the asymptotic operator, with exponential rate governed by the eigenvalue (in this case 0). Therefore, any element of the kernel of D δ 1 must converge exponentially fast to an element of the kernel of the asymptotic operator at z 0 . Hence, the kernel of D δ 1 is contained in the kernel of D δ .
We conclude that the kernels of the two operators may be identified. Since their Fredholm indices are the same, their cokernels are also isomorphic. 5.2.2. The linearization at a Floer solution. The first step in the proof of Proposition 5.9 is to set up the appropriate Fredholm problem. Given a Floer solutioñ v : RˆS 1 zΓ : RˆY , we consider exponentially weighted Sobolev spaces of sections of the pull-back bundleṽ˚T pRˆY q since the asymptotic limits are (Morse-Bott) degenerate. For δ ą 0, we denote by W 1,p,δ pRˆS 1 zΓ, v˚T pRˆY qq the space of sections that decay exponentially like e´δ |s| near the punctures (also in cylindrical coordinates near the punctures Γ), as in the previous section.
We similarly define W m,p,δ sections with exponential decay/growth. The following results will not depend on m except in the case of jet conditions considered in Section 5.4, where m will need to be sufficiently large that the order of contact condition can be defined.
In order to consider a parametric family of punctured cylinders in which the asymptotic limits move in their Morse-Bott families, we let V be a collection of vector spaces, associating to each puncture z P Γ Y t˘8u a vector subspace V z of the tangent space to the corresponding Morse-Bott family of orbits. For δ ą 0, we then consider the space of sections W 1,p,δ V pRˆS 1 zΓ, v˚T pRˆY qq that converge exponentially at each puncture z to a vector in the corresponding vector space V z .
Remark 5.21. In this paper, we will not always be careful to specify how small δ has to be. It is worth pointing out that there is no value of δ that works for all moduli spaces. 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. Lemma 5.13 computes the spectrum of a number of these relevant asymptotic operators, and as we see in Table 2, the smallest positive eigenvalue 1 2`´C`? C 2`1 6π 2˘b ecomes arbitrarily small as C Ñ 8. As will become clear from Lemma 5.22 and Equation (5.5), the relevant value for C here is h 2 pe b k qe b k , which can become arbitrarily large as the multiplicity k Ñ 8. Since the relevant moduli spaces in the differential involve connecting orbits of bounded multiplicities, for any given moduli space, we may choose δ sufficiently small.
We now adapt an observation first used in [Dra04,Bou06], to show that the linearization of the Floer operator is upper triangular with respect to the splitting of T pRˆY q as pR ' RRq ' ξ. We then describe 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 be identified with the linearization of the Cauchy-Riemann operator for spheres in Σ.
We now explain this construction in more detail. Letṽ : RˆS 1 zΓ Ñ RˆY be a Floer solution with punctures Γ. The Hamiltonian need not be admissible, but needs to be radial (i.e. depending only on r, the symplectization variable). The almost complex structure J Y is assumed to be admissible. We consider three possible cases for the asymptotics of such a curve.
In the first case,ṽ is asymptotic to a closed Hamiltonian orbit atṽp`8, tq, to a closed Hamiltonian orbit atṽp´8, tq, and with negative ends converging to Reeb orbits at the punctures in Γ. The second case hasṽ asymptotic to a closed Hamiltonian orbit atṽp`8, tq, but with negative ends converging to Reeb orbits in t´8uˆY at t´8u Y Γ. These two cases correspond to an upper level of a split Floer cylinder as in Definitions 4.1 and 4.2, respectively.
The third case we consider is most directly applicable to studying holomorphic curves in RˆY :ṽ has a positive cylindrical end at`8 converging to a Reeb orbit in t`8uˆY , and has negative cylindrical ends at the punctures t´8u Y Γ. For such a curve, we may assume that H is identically 0, and thus this example includes J Y -holomorphic curves. This is of independent interest, and is useful in [DL18]. Part of this was sketched in [EGH00, Section 2.9.2].
Let w " π Σ˝ṽ : CP 1 Ñ Σ be the smooth extension of the projection ofṽ to the divisor (as given by Lemma 2.4). The linearized projection dπ Σ induces an isomorphism of complex vector bundles v˚`T pRˆY q˘-pR ' RRq ' w˚T Σ.
To see this, note that for each point p P Y , dπ Σ induces a symplectic isomorphism pξ p , dαq -pT πΣppq Σ, Kω Σ q. By the Reeb invariance of the almost complex structure (and thus S 1 -invariance under rotation in the fibre), this then gives a complex vector bundle isomorphism.
Let V associate to each puncture z P Γ Y t˘8u the tangent space to Y if the corresponding limit ofṽ is a closed Hamiltonian orbit and the tangent space to RˆY if the corresponding limit ofṽ is a closed Reeb orbit. As will be clearer shortly, this is associating to each puncture the entirety of the kernel of the corresponding asymptotic operator. Let D Σ w at the holomorphic cylinder s`it Þ Ñ wpe 2πps`itq q " π Σ pṽps, tqq. Then, pπ Σ˝ṽ q˚T Σ " w˚T Σ| RˆS 1 zΓ is a Hermitian vector bundle over RˆS 1 zΓ. Let V Σ be the kernels of the asymptotic operators of 9 D Σ w at each of the punctures,˘8 and Γ. (These are explicitly given by V Σ p´8q " T wp0q Σ, V Σ p`8q " T wp8q Σ, V Σ pzq " T wpzq Σ for each marked point z P Γ.) We consider this operator acting on the space of sections 9 D Σ w : W 1,p,δ V Σ pw˚T Σ| RˆS 1 zΓ q Ñ L p,δ pHom 0,1 pT pRˆS 1 zΓq, w˚T Σ| RˆS 1 zΓ qq.
The operator D Σ w is Fredholm independently of the weight, but 9 D Σ w is only Fredholm when the weight δ is not an integer multiple of 2π. Furthermore, by combining [Wen16b, Proposition 3.15] with Lemma 5.20, for 0 ă δ ă 2π, these operators have the same Fredholm index and their kernels and cokernels are isomorphic by the map induced by restricting a section of w˚T Σ to the punctured cylinder.
Finally, define D C v by (5.5) where V 0 associates the vector space iR to the punctures at whichṽ converges to a closed Hamiltonian orbit and associates the vector space C at punctures at which v converges to a closed Reeb orbit. Notice that again these are chosen so that they precisely give the kernels of the corresponding asymptotic operators of D C v . Lemma 5.22. The isomorphismṽ˚T pRˆY q -pR ' RRq ' w˚T Σ induces a decomposition: Dṽ "ˆD Furthermore, if w " π Σ˝ṽ is non-constant, then M is pointwise surjective except at finitely many points.
Proof. In our setting, the nonlinear Floer operator takes the form: dṽ`J Y pṽqdṽ˝i´h 1 pe r qR b dt`h 1 pe r qB r b ds " 0.
Let g be the metric on RˆY given by g " dr 2`α2`d αp¨, J Y¨q . This metric is J Y -invariant. Let r ∇ be the Levi-Civita connection for g. Let ∇ be the Levi-Civita connection on T Σ for the metric ω Σ p¨, J Σ¨q .
Then, it follows that the linearization Dṽ applied to a section ζ ofṽ˚T pRˆY q satisfies Notice that r ∇B r " 0 since g is a product metric. We have then Observe also that for any vector field V in T Σ, there is a unique horizontal lift V to Y with the property αpṼ q " 0. For any two vector fields V and W in T Σ, since dαpṼ ,W q " Kω Σ pV, W q, we have the following rṼ ,W s " Č rV, W s´Kω Σ pV, W qR.
From this, it follows that the Levi-Civita connection r ∇ satisfies the following identities: A simple computation using the Reeb-flow invariance of J Y and the torsion-free property of the connection gives We will now compute Dṽζ pBsq, first when ζ " ζ 1 B r`ζ2 R " pζ 1`i ζ 2 qB r , and then when ζ is a section ofṽ˚ξ.
For the first computation, it suffices to notice the following two identities DṽR pB s q " 0.
It follows then from the Leibniz rule that we have Now consider the case when ζ is a section ofṽ˚ξ, and is thus the lift ζ "η of a section η of w˚T Σ. We compute and similarly for r ∇ t . We then obtain the following covariant derivatives of J Y , whereW is a section ofṽ˚ξ: It follows then (Note that we use the fact thatṽ s`JYṽt`h 1 pe b qB r " 0 in the cancellations.) Writing ζ " pζ a , ζ b q under the isomorphismṽ˚T pRˆY q -pR ' RRq ' w˚T Σ, we obtain the decomposition: Our calculations now establish that D aa " D C v and D ba " 0, D bb " 9 D Σ w , and D ab ζpB s q " Kω Σ pw t , π Σ ζqB r´K ω Σ pw s , π Σ ζqR. Observe that in particular, D ab is a pointwise linear map fromṽ˚ξ| p to RB r ' RR. The map is surjective except at critical points of the pseudoholomorphic map w, of which there are finitely many if w is non-constant. The decay claim follows since w converges to a point, and thus its derivatives decay exponentially fast.
Remark 5.23. Notice that for each puncture z P t˘8u Y Γ, if γptq denotes the corresponding asymptotic Hamiltonian or Reeb orbit, the previous result allows us to identify V z with T γp0q Y at a Hamiltonian orbit and with RˆT γp0q Y at a Reeb orbit.
Then, the operator D C v defined in Equation (5.5) is Fredholm for δ ą 0 sufficiently small.
If, instead,ṽ converges at`8 to a closed Hamiltonian orbit, and at´8 to a closed Reeb orbit in t´8uˆY , then D C v has Fredholm index 2 and is surjective. Finally, ifṽ converges at˘8 to closed Reeb orbits in t˘8uˆY , then D C v has Fredholm index 2 and is surjective.
In all three cases, the kernel of D C v contains the constant section i, which can be identified with the Reeb vector field.
Proof. We will apply the punctured Riemann-Roch Theorems 5.16 and 5.18. For this, we need to compute the Conley-Zehnder indices of the appropriately perturbed asymptotic operators. We will first identify the (Morse-Bott degenerate) asymptotic operators at each of the punctures, and then apply Corollary 5.14 to obtain the Conley-Zehnder indices of the˘δ-perturbed operators.
Recall from Remark 5.21 that we have |δ| ą 0 smaller than the spectral gap for any of these punctures.
In order to consider the operator D C v : W 1,p,δ V0 Ñ L p,δ , it will be convenient to consider a related operator with the same formula, but on the much larger space of functions with exponential growth. By a slight abuse of notation, we will use the same name: Then, the kernel and cokernel of the operator acting on the spaces of sections with exponential growth can be identified with the kernel and cokernel of the operator acting on W 1,p,δ V0 , by Lemma 5.20. First, consider the case whenṽ converges to a closed Hamiltonian orbit in tb˘uŶ as s Ñ˘8. Then, the asymptotic operator associated to D C v at˘8 is given by In the case of δ-exponential decay, the relevant asymptotic operators are given by A``δ at the positive puncture`8 and by A´´δ at the negative puncture´8.
In the case of δ-exponential growth, the relevant asymptotic operators are A`´δ and A´`δ, respectively. For the case of exponential decay, Corollary 5.14 then gives the Conley-Zehnder index of 0 for A``δ and of 1 for A´´δ.
In the case of exponential growth, Corollary 5.14 gives instead that the Conley-Zehnder index of A`´δ is 1 and that of A´`δ is 0.
Associated to a Reeb puncture at˘8 or at P P Γ, we have the asymptotic operator´i d dt .
Writingṽ " pb, vq : RˆS 1 zΓ Ñ RˆY , we have b Ñ´8 at both types of negative punctures and b Ñ`8 at the positive puncture. As above, in the case of exponential decay, the relevant asymptotic operators are´i d dt`δ at a positive puncture and´i d dt´δ at a negative puncture. Again, by Corollary 5.14, we obtain a Conley-Zehnder index of´1 at`8 and a Conley-Zehnder indices of 1 at a negative puncture (´8 or P P Γ).
If, instead, we consider exponential growth, we obtain Conley-Zehnder indices of`1 at positive punctures and´1 at negative punctures.
Applying now the punctured Riemann-Roch theorem 5.16, and using the fact that the Euler characteristic of the punctured cylinder is´#Γ, we obtain that the Fredholm index of as necessary to apply [Wen10, Proposition 2.2]. Now, applying Theorem 5.18, we compute that the Fredholm index of D C v : W 1,p,δ V0 pRˆS 1 zΓ, Cq Ñ L p,δ pHom 0,1 pT pRˆS 1 zΓq, Cqq is given bý #Γ`1´p´#Γq´# 0 ifṽp´8q converges to a Hamiltonian orbit 1 ifṽp´8q converges to a Reeb orbit " 1 or 2, depending on the negative end ofṽ.
Furthermore, the fact that the curve has genus 0 and one puncture with even Conley-Zehnder index precisely if lim sÑ´8ṽ is a Hamiltonian orbit implies that c 1 pE, l, A Γ q " # 1 2 p1´2`1q " 0 ifṽp´8q converges to a Hamiltonian orbit 1 2 p2´2q " 0 ifṽp´8q converges to a Reeb orbit In either case, the adjusted Chern number is less than the Fredholm index. Therefore, D C v satisfies the automatic transversality criterion and is thus surjective, as wanted.
It follows immediately from the expression for D C v that the constant i is in the kernel. Recalling that C "ṽ˚pR ' RRq in the splitting given by Lemma 5.22, we then may identify this constant with the Reeb vector field R.
To summarize the results of this section, by Lemma 5.22, a punctured Floer cylinder in RˆS 1 is regular if the operators D C v and 9 D Σ w are surjective. Surjectivity of the latter is equivalent to surjectivity of D Σ w . Lemma 5.24 gives the surjectivity of D C v . It thus remains to study transversality for D Σ w , specifically with respect to the evaluation maps that will allow us to define the moduli spaces of chains of pearls in Σ (see Section 5.3). Additionally, we need to consider transversality for moduli spaces of planes in W asymptotic to Reeb orbits in Y , or equivalently, the moduli spaces of spheres in X with an order of contact condition at Σ (see Section 5.4).

5.3.
Transversality for chains of pearls in Σ. In this section and the next, we show that for generic almost complex structure (in a sense to be made precise), the moduli spaces of chains of pearls and moduli spaces of chains of pearls with spheres in X (possibly augmented as well) are transverse. We begin with the definition of several moduli spaces that will be useful.
Definition 5.25. Let J Σ P J Σ be an almost complex structure compatible with ω Σ . Given p, q P Critpf Σ q and a finite collection A 1 , . . . , A N P H 2 pΣ; Zq, let Mk ,Σ ppA 1 , . . . , A N q; q, p; J Σ q denote the space of simple chains of pearls in Σ from q to p (see Definition 5.5), such that pw i q˚rCP 1 s " A i , with k marked points.
Let Mk ,Σ ppA 1 , . . . , A N q; J Σ q denote the moduli space of N parametrized J Σ -holomorphic spheres in Σ, representing the classes A i , i " 1, . . . , N , with k marked points, also satisfying the simplicity criterion of Definition 5.5, i.e. so each sphere is either somewhere injective or constant, each constant sphere has at least one augmentation marked point, and no sphere has image contained in the image of another.
For J W P J W , let J Σ " P pJ W q be the corresponding almost complex structure in J Σ and J X the corresponding almost complex structure on X. Define Mk ,pX,Σq ppB; A 1 , . . . , A N q; x, p, J W q to be the moduli space of simple chains of pearls in Σ with a sphere in X (as in Definitions 5.3 and 5.5), where x is a critical point of f W and p is a critical point of f Σ , and representing the spherical homology classes rw i s " A i P H 2 pΣ; Zq, i " 1, . . . , N and rvs " B P H 2 pX; Zqz0. In the following, we will write l " B ‚ Σ " KωpBq which is the order of contact of v with Σ.
Let Mk ,pX,Σq ppB; A 1 , . . . , A N q; J W q denote the moduli space of N parametrized J Σ -holomorphic spheres in Σ, representing the classes A i , and of a J X -holomorphic sphere in X representing the class B with order of contact l " B ‚ Σ " KωpBq, also satisfying the simplicity criterion of Definition 5.5, 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 in Σ is contained in the image of another and the image of the sphere in X is not contained in the tubular neighbourhood ϕpUq of Σ. Furthermore, the spheres in Σ have k marked points. Let MX ppB 1 , B 2 , . . . , B k q; J W q denote the moduli space of k unparametrized J X -holomorphic spheres in X, where each sphere is somewhere injective, no image of a sphere is contained in the image of another sphere, and so the image of each sphere is not contained in the tubular neighbourhood ϕpUq of Σ, and such that each sphere intersects Σ only at 8 P CP 1 with order of contact B i ‚ Σ. We can think of an unparametrized sphere as an equivalence class of parametrized spheres, modulo the action of AutpCP 1 , 8q " AutpCq on the domain. Finally, let M a k,Σ ppA 1 , . . . , A N q, pB 1 , . . . , B k q; q, p; J W q denote the moduli space of simple augmented chains of pearls in Σ with k unparamentrized augmentation planes, and let In order to apply the Sard-Smale Theorem, we need to consider Banach spaces of almost complex structures, so we let J r Σ , J r W be the space of C r -regular almost complex structures otherwise satisfying the conditions of being in J Σ , J W . We impose 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 the collection of homology classes and will also depend on the order of contact to Σ for the spheres in X).
For each of these moduli spaces, we also consider the corresponding universal moduli spaces as we vary the almost complex structure. For instance, we denote by Mk ,Σ ppA 1 , . . . , A N q, J r Σ q the moduli space of pairs ppw i q N i"1 , J Σ q with J Σ P J r Σ and pw i q N i"1 P Mk ,Σ ppA 1 , . . . , A N q, J Σ q. The main goal of this section and of the next is to prove that these moduli spaces of simple chains of pearls are transverse for generic almost complex structures. This is analogous to [MS04, Theorem 6.2.6], and indeed, the transversality theorem of McDuff-Salamon will be a key ingredient of our proof. Their Theorem 6.2.6 is about transversality of the universal evaluation map to a specific submanifold ∆ E of the target, whereas our work in this section establishes transversality to some other submanifolds. We will furthermore require an extension of the results from [CM07] (see Section 5.4), and an additional technical transversality point needed to be able to consider the lifted problem in RˆY .
Proposition 5.26. There is a residual set J reg W Ă J W such that J reg Σ :" P pJ reg W q is a residual set in J Σ and such that for all J Σ P J reg Σ and J W P J reg W , p P Critpf Σ q, q P Critpf Σ q and x P Critpf W q, the moduli spaces Mk ,Σ ppA 1 , . . . , A N q; q, p; J Σ q, Mk ,pX,Σq ppB; A 1 , . . . , A N q; x, p, J W q, M a k,Σ ppA 1 , . . . , A N q, pB 1 , . . . , B k q; q, p; J W q and M a k,pX,Σq ppB; A 1 , . . . , A N q; pB 1 , . . . , B k q; x, p; J W q are manifolds. Their dimensions are dim Mk ,Σ ppA 1 , . . . , A N q; q, p; J Σ q " M ppq`N where M ppq and M pqq are the Morse indices of p, q P Critpf Σ q and M pxq is the Morse index of x P Critpf W q.
We will also make use of the following definition and proposition, the latter of which we prove in the next section.
For spheres in Σ, we also obtain evaluation maps at the augmentation punctures ev a Σ : Mk ,Σ ppA 1 , . . . , A N q; J r Σ q Ñ Σ k and ev a Σ : Mk ,pX,Σq ppB; A 1 , . . . , A N q; J r W q Ñ Σ k . We refer to these three maps denoted ev a as augmentation evaluation maps.
Recall that we have chosen a Morse function f Σ : Σ Ñ R and a corresponding gradient-like vector field Z Σ , such that pf Σ , Z Σ q is a Morse-Smale pair. The timet flow of Z Σ is denoted by ϕ t ZΣ and the stable (ascending) W s Σ pqq and unstable (descending) manifolds W u Σ ppq were defined in Equation (3.1). (Note that these are the stable/unstable manifolds for the negative gradient flow.) Definition 5.30. The flow diagonal in ΣˆΣ associated to the pair pf Σ , Z Σ q is where Critpf Σ q is the set of critical points of f Σ .
We will now establish transversality of the evaluation maps to appropriate products 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 with the flow diagonals. Proof. Suppose F pm 0 , m 1 q " px, p, pq P ∆ fΣˆt pu. Then, there exists t so that φ t ZΣ pxq " φ t ZΣ pf 0 pm 0 qq " f 1 pm 1 q " p. Notice that For notational simplicity, we write Φ " dφ´t ZΣ ppq. It follows then that dF pm 0 , m 1 q¨T pM 0ˆM1 q`pE ' 0q using the surjectivity of df 0 , df 1 . This then establishes the result, since E Ă T px,pq ∆ fΣ .
From this, we now obtain the following: Proof. We apply the previous Lemma, using f 0 " ev`and f 1 " e. Then,êvpm, nq " pev´pmq, F pm, nqq. The transversality to Aˆ∆ fΣˆp t follows by the transversality of F to ∆ fΣˆp t together with the transversality of ev´to A.
Lemma 5.33. Let N ě 1, and let A 1 , . . . , A N be spherical homology classes in Σ and let B be a spherical homology class in X.
Suppose that S Ă Σ 2N´2 is obtained by taking the product of some number of copies of ∆ fΣ Ă Σ 2 and of the complementary number of copies of tpp, pq | p P Critpf Σ qu Ă Σ 2 , in arbitrary order. Let ∆ Ă Σ 2 denote the diagonal.
Then if ř N i"1 A i ‰ 0, the universal evaluation map ev Σ : Mk ,Σ ppA 1 , . . . , A N q; J r Σ q Ñ Σ 2N is transverse to the submanifold txuˆSˆtyu for all x, y P Σ.
If B ‰ 0, the universal evaluation map ev X,Σ : Mk ,pX,Σq ppB; A 1 , . . . , A N q; J r W q Ñ XˆΣ 2N`1 is transverse to the submanifold txuˆ∆ˆSˆtyu for any x P X, y P Σ.
Proof. We consider the case of Mk ,Σ in detail, since the argument is essentially the same for Mk ,pX,Σq , though notationally more cumbersome.
Suppose that ppv 1 , . . . , v N q, Jq P Mk ,Σ ppA 1 , . . . , A N q; J r Σ q is in the pre-image of txuˆSˆtyu. Write S " S 1ˆS2ˆ¨¨¨ˆSN´1 , where each S i Ă Σ 2 is either the flow diagonal or the set of critical points.
Notice that the simplicity condition then requires that if some sphere v i is constant, 1 ă i ă N , we must have that S i´1 and S i are flow diagonals. If v 1 is constant, then S 1 is a flow diagonal and if v N is constant, S N´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 Σ 2pN´1q´2 of the form specified, and for any k ě 0, for any collection of N´1 spherical classes, not all of which are zero.
Let now A 1 , . . . , A N be spherical homology classes, not all of which are zero. Then, at least one of A 1 , . . . , A N´1 or A 2 , . . . , A N is a collection of spheres satisfying the hypotheses. For simplicity of notation, let us assume that A 1`¨¨¨`AN´1 ‰ 0. Let S 0 " S 1ˆS2ˆ¨¨¨ˆSN´2 . Let k " k 0`kN where k N is the number of marked points we consider on the last sphere. By the induction hypothesis, we have that the evaluation map Mk 0,Σ ppA 1 , . . . , A N´1 q; J r Σ q Ñ Σ 2pN´1q is transverse to ptˆS 0ˆp t. Denote this map by ev 0 . Notice that Mk ,Σ ppA 1 , . . . , A N q; J r Σ q Ă Mk 0,Σ ppA 1 , . . . , A N´1 q; J r Σ qˆMk N ,Σ pA N ; J r Σ q. Let then ev N : Mk ,Σ ppA 1 , . . . , A N q; J r Σ q Ñ Σ 2 be the evaluation at 0 and 8 in the N -th sphere. We therefore have ev Σ : Mk ,Σ ppA 1 , . . . , A N q; J r Σ q Ñ Σ 2N given by ev Σ " pev 0 , ev N q.
If A N ‰ 0, the result follows again from [MS04, Proposition 3.4.2]. If, instead, A N " 0, we have from above that S N´1 " ∆ fΣ . Notice that the evaluation map of constant spheres on Σ has image on the diagonal in ΣˆΣ. The result now follows by applying Lemma 5.32.
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 to XˆΣ now follows from Proposition 5.29, and the induction proceeds as before.
Proposition 5.34. Let N ě 0. Suppose that S Ă Σ 2N´2 is obtained by taking the product of some number of copies of ∆ fΣ Ă Σ 2 and of the complementary number of copies of tpp, pq | p P Critpf Σ qu Ă Σ 2 , in arbitrary order.
Let ∆ Ă ΣˆΣ denote the diagonal and let ∆ k denote the diagonal Σ k in Σ kˆΣk . Let p, q be critical points of f Σ and let x be a critical point of f W . Then the universal evaluation maps together with augmentation evaluation maps Σˆe v a Σ : Mk ,pX,Σq ppB; A 1 , . . . , A N q; J r W qˆMX ppB 1 , . . . , B k q; J r W q Ñ XˆΣ 2N`1ˆΣkˆΣk are transverse to, respectively, Proof. We will consider only the first case, the second being analogous. Notice first that by Proposition 5.29 the augmentation evaluation map ev a Σ : MX ppB 1 , . . . , B k q; J r W q Ñ Σ k is a submersion. It suffices therefore to prove that The proposition follows immediately if at least one of the A i , 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 5.33.
The only case then that must be examined is that of a chain of pearls entirely in Σ with all spheres constant. In this case, the evaluation map from the moduli space Mk ,Σ pp0, 0, . . . , 0q, J r W q factors through the evaluation map tpz 1 , . . . , z N q P Σ N | z i " z j ùñ i " juˆJ r W Ñ Σ 2N . Transversality follows from the Morse-Smale condition on the gradient-like vector field Z Σ . This gives that the intersection of W s Σ pqq and W u Σ ppq is transverse, and hence that the diagonal in ΣˆΣ is transverse to W s Σ pqqˆW u Σ ppq, which is what we need when N " 1. The case of N ě 2 is similar, using the description of the tangent space to the flow diagonal at px, yq P ∆ fΣ , such that ϕ t ZΣ pxq " y for some t ą 0, as Proposition 5.34 can be combined with standard Sard-Smale arguments, the fact that P : J r W Ñ J r Σ is an open and surjective map and Taubes's method for passing to smooth almost complex structures (see for instance [MS04, Theorem 6.2.6]) to give the following proposition: Proposition 5.35. There exist residual sets of almost complex structures J reg W Ă J W and J reg Σ " P pJ reg W q, so that for fixed J W P J reg W and J Σ " P pJ W q, the restrictions of the evaluation maps ev Σˆe v a Σˆe v a Σ and ev X,Σˆe v a Σˆe v a Σ to Mk ,Σ ppA 1 , . . . , A N q; J Σ qˆMX ppB 1 , . . . , B k q; J W q and Mk ,pX,Σq ppB; A 1 , . . . , A N q; J W qˆMX ppB 1 , . . . , B k q; J W q, respectively, are transverse to the submanifolds of Proposition 5.34.
The transversality statement of the main result of this section, Proposition 5.26 now follows. The dimension formulas follow from usual index arguments, combining Riemann-Roch with contributions from the constraints imposed by the evaluation maps.
5.4. Transversality for spheres in X with order of contact constraints in Σ. 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 and Mohnke prove that the moduli space of simple curves not contained in Σ, with a condition on the order of contact with Σ, can be made transverse by a perturbation of the almost complex structure away from Σ. We will extend this result to show that additionally the evaluation map to Σ at the point of contact can be made transverse. This can be useful, for instance, to define relative Gromov-Witten invariants with constraints on homology classes in Σ.
Recall that Σ is a symplectic divisor and N Σ is its symplectic normal bundle equipped with a Hermitian structure. We have fixed a symplectic neighbourhood ϕ : U Ñ X where ϕ : U Ñ X is an embedding. From Definition 2.6, we require that all J X P J W have that J X is standard in the image ϕpUq Ă X of this neighbourhood (recall that we identify XzΣ with W using Lemma 2.4, as usual).
Fix an almost complex structure J 0 P J W . We may suppose that P pJ 0 q P J Σ is an almost complex structure in the residual set J reg Σ given by Proposition 5.26, though this isn't strictly speaking necessary.
Let V :" XzϕpUq. Following Cieliebak-Mohnke [CM07], let J pVq be the set of all almost complex structures on X compatible with ω that are equal to J 0 on ϕpUq. Similarly, we will let J r pVq be the compatible almost complex structures of C r regularity.
To define the order of contact, consider an almost complex structure J X P J W and a J X -holomorphic sphere f : CP 1 Ñ X with f p0q P Σ, an isolated intersection. Choose coordinates s`it " z P C on the domain and local coordinates near f p0q P Σ on the target, such that f p0q P Σ Ă X corresponds to 0 P C n´1 " C n´1ˆt 0u Ă C n´1ˆC . Write π C : C n Ñ C for projection onto the last coordinate (which is to be thought of as normal to Σ). Assume also that J X p0q " i. Then, f has contact of order l at 0 if the vector of all partial derivatives of orders 1 through l (denoted by d l f p0q) has trivial projection to C. We can write this condition as d l f p0q P T f p0q Σ. We define then the order of contact at an arbitrary point in CP 1 by precomposing with a Möbius transformation. (This is well-defined, by [CM07,Lemma 6.4].) Define the space of simple pseudoholomorphic maps into X that have order of contact l at 8 to a point in Σ to be where we require m ě l`2. Note that our notation differs somewhat from the notation in [CM07].
In this section, we need to have a higher regularity on our Sobolev spaces to make sense of the order of contact condition. For the remaining moduli spaces, for simplicity of notation, we have taken m " 1, where this is not a problem. Notice that by elliptic regularity, the moduli spaces themselves are manifolds of smooth maps, and are independent of the choice of m. This only affects the classes of deformations we consider in setting up the Fredholm theory.
Notice that it suffices to prove this when considering only pairs pf, J X q P MX ppB 0 q; J r W q with the additional condition that J X P J r pVq.
We also observe that if l " B 0 ‚Σ, we have that MX ppB 0 q; J W q Ă M8 ,k,pX,Σq pJ W q for each k ď l. Furthermore, MX ppB 0 q; J W q is a connected component of M8 ,l,pX,Σq pJ W q. This observation will enable us to obtain the result by inducting on k.
The proposition will follow by a modification of the proof given in [CM07, Section 6]. Instead of reproducing their proof, we indicate the necessary modifications. In order to be as consistent as possible with their notation, we consider the point of contact with Σ to be at 0.
Consider a J X -holomorphic map f : CP 1 Ñ X such that f p0q P Σ with order of contact 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 by varying J X freely in the complement of our chosen neighbourhood of the divisor, V " XzϕpUq.
The linearized Cauchy-Riemann operator at f with respect to a torsion-free connection is pD f ξqpzq " ∇ s ξpzq`J X pf pzqq∇ t ξpzq`p∇ ξpzq J X pf pzqqq f t pzq.
At a coordinate chart around z " 0, we can specialize to the standard Euclidean connection in R 2n " C n (which preserves C n´1 along C n´1 ), we get Apzqξpzq " pD ξpzq J X pf pzqqq f t pzq (see also page 317 in [CM07]).
We need the following adaptation of Corollary 6.2 in [CM07].
Lemma 5.36. Suppose pf, J X q P M8 ,l,pX,Σq pJ r W q with J X P J r pVq, r ě m. After choosing local coordinates, suppose f p0q P Σ and in coordinates around f p0q, Σ is mapped to C n´1 and is thus preserved by J X .
Denote the unit disk by D 2 and let ξ : pD 2 , 0q Ñ pC n , 0q be such that D f ξ " 0.
Proof. We need to show that B k ξ Bs k´i Bt i p0q P C n´1 for all 0 ď i ď k. It will be convenient to use multi-index notation for partial derivatives, and denote the previous expression by D pk´i,iq ξp0q. The case i " 0 is part of the hypotheses of the Lemma. For the induction step, note that D f ξ " 0 combined with the product rule implies that ApzqD β 1 ξpzqH ere, α and β are multi-indices such that α " pa 1 , a 2 q for 0 ď a 1 ď k´i, 0 ď a 2 ď i´1, α ‰ p0, 0q and α`β " pk´i, iq. Similarly, α 1 and β 1 are multi-indices such that α 1 " pa 1 1 , a 1 2 q for 0 ď a 1 1 ď k´i, 0 ď a 1 2 ď i´1 and α 1`β1 " pk´i, i´1q. The hypotheses of the Lemma and the induction hypothesis implie that the derivatives of ξ on the right hand side take values in T f p0q Σ. The fact that J X and ∇ preserve C n´1 along C n´1 , and that d l f p0q P T f p0q Σ, implies the induction step.
We now prove the key property of the linearized evaluation map: Proposition 5.37. For m´2{p ą l, r ě m, the universal evaluation map ev X,Σ : M8 ,l,pX,Σq pJ r W q Ñ Σ pf, J X q Þ Ñ f p0q is a submersion.
Proof. We show that for every 0 ď k ď l, and pf, J X q P M8 ,k,pX,Σq pJ r pVqq, is surjective. By Lemma 6.5 in [CM07], We argue by induction on k. The case k " 0 is a special case of Proposition 3.4.2 in [MS04]. We assume that the claim is true for k´1 and prove it for k.
Observe now that by combining this with standard arguments (see, for instance, [MS04, Proposition 3.4.2], which is also used in the proof of Proposition 5.26 above), we obtain the transversality for the evaluation at a point, taking values in X. This finishes the proof of Proposition 5.29. 5.5. Proof of Proposition 5.9. We are now ready to complete the proof of Proposition 5.9. To this end, we will show that the transversality problem for a cascade reduces to the already solved transversality problem for chains of pearls. The two key ingredients of this are the splitting of the linearized operator given by Lemma 5.22 and a careful study of the flow-diagonal in YˆY .
Recall from Definition 2.6 that J Y denotes the space of cylindrical, Reebinvariant almost complex structures on RˆY . These are obtained as lifts of the almost complex structures in J Σ . Let J reg Y be the set of almost complex structures on RˆY that are lifts of the almost complex structures in J reg Σ . Recall from Definition 2.6 and from Proposition 2.2, if J W P J W is an almost complex structure on W that is of the type we consider, it induces an almost complex structure P pJ W q " J Σ P J Σ . The restriction of J W to the cylindrical end of W , J Y , is then a translation and Reeb-flow invariant almost complex structure on RˆY that has dπ Σ J Y " J Σ dπ Σ .
Recall that the biholomorphism ψ : W Ñ XzΣ given in Lemma 2.4 allows us to identify 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 b m so that h 1 pe bm q " m. Then Y m " tb m uˆY Ă RˆY is the corresponding Morse-Bott family of 1-periodic Hamiltonian orbits that wind m times around the fibre of Y Ñ Σ.
We now define moduli spaces of Floer cylinders, from which we will extract the moduli spaces of cascades by imposing the gradient flow-line conditions. First, we define the moduli spaces relevant for the differential connecting two generators in RˆY . Then, we will define the moduli spaces relevant for the differential connecting to a critical point in W .
Define MH ,k,RˆY ;k´,k`p pA 1 , . . . , A N q; J Y q to be a set of tuples of punctured cylinders pṽ 1 , . . . ,ṽ N q with the following properties: (1) There is a partition of Γ " Γ 1 Y¨¨¨Y Γ N of k augmentation marked points withṽ i : RˆS 1 zΓ i Ñ RˆY so thatṽ i is a finite hybrid energy punctured Floer cylinder. For each z j P Γ, there is a positive integer multiplicity kpz j q. Let v i denote the projection to Y .
(2) There is an increasing list of N`1 multiplicities from k´to k`: 3) For each i, the cylinderṽ i has multiplicities k i and k i´1 at˘8:ṽ i p`8,¨q P Y ki ,ṽ i p´8,¨q P Y ki´1 . (4) The Floer cylindersṽ i are simple in the sense that their projections to Σ are either somewhere injective or constant, if constant, they have at least one augmentation puncture, and their images are not contained one in the other. (5) For each i, and for every puncture z j P Γ i , the augmentation puncture has a limit whose multiplicity is given by kpz j q; i.e. lim ρÑ´8 v i pz j`e 2πpρ`iθq q is a Reeb orbit of multiplicity kpz j q. (6) The projections of the Floer cylinders to Σ represent the homology classes A i , i " 1, . . . , N ; i.e. pπ Σ pṽ i qq N i"1 P Mk ppA 1 , . . . , A N q, J Σ q. Let B P H 2 pX; Zq be a spherical homology class, B ‰ 0. Let J W be an almost complex structure on W as given by Lemma 2.4, matching J Y on the cylindrical end.
Definition 5.39. Define the moduli space MH ,k,W ;k`p pB; A 1 , . . . , A N q; J W q to consist of tuples pṽ 0 ,ṽ 1 , . . . ,ṽ N q with the properties (1) The mapṽ 0 : RˆS 1 Ñ W is a finite energy holomorphic cylinder with removable singularity at´8. (2) There is a partition of Γ " Γ 1 Y¨¨¨Y Γ N of k augmentation marked points withṽ i : RˆS 1 zΓ i Ñ RˆY, i ě 1, so that eachṽ i is a finite hybrid energy punctured Floer cylinder. For each z j P Γ, there is a positive integer multiplicity kpz j q. Denote by v i the projection ofṽ i to Y .
(3) There is an increasing list of N`1 multiplicities: k 0 ă k 1 ă k 2 ă¨¨¨ă k N " k( 4) For each i ě 1, and for every puncture z j P Γ i , the augmentation puncture has a limit whose multiplicity is given by kpz j q; i.e. lim ρÑ´8 v i pz jè 2πpρ`iθq q is a Reeb orbit of multiplicity kpz j q. (5) The Floer cylindersṽ i for i ě 1 are simple, in the strong sense that the projections to Σ are somewhere injective or constant, and have images not contained one in the other. The cylinderṽ 0 is somewhere injective in W . (6) The projections of the Floer cylinders to Σ represent the homology classes A i , i " 1, . . . , N ; i.e. π Σ pṽ i qq N i"1 P Mk ppB; A 1 , . . . , A N q, J W q. (7) After identifyingṽ 0 with a holomorphic sphere in X,ṽ 0 represents the homology class B P H 2 pX; Zq. (8) The cylinderṽ 1 has multiplicity k 1 at`8 andṽ 1 p`8,¨q P Y k1 . At´8,ṽ 1 converges to a Reeb orbit in t´8uˆY . This Reeb orbit has multiplicity k 0 . (9) For each i ě 2, the cylinderṽ i has multiplicities k i and k i´1 at˘8: v i p`8,¨q P Y ki ,ṽ i p´8,¨q P Y ki´1 . (10) The cylinderṽ 0 converges at`8 to a Reeb orbit of multiplicity k 0 .
Observe that these moduli spaces are non-empty only if for each i " 1, . . . , N , Furthermore, for MH ,k,W , we must also have Note also that these moduli spaces have a large number of connected components, where different components have different partitions of Γ or different intermediate multiplicities.
Identifying holomorphic spheres in X with finite energy J W -planes in W , we consider also the moduli space of holomorphic planes MX ppB 1 , . . . , B k q; J W q as in Definition 5.25.
The space MH ,k,RˆY ppA 1 , . . . , A N q; J Y q consists of N -tuples of somewhere injective punctured Floer cylinders in RˆY . Similarly, MH ,k,W consist of N -tuples of punctured Floer cylinders in RˆY together with a holomorphic plane in W (which we can therefore also interpret as a holomorphic sphere in X). The cylinders and the eventual plane have asymptotics with matching multiplicities, but are otherwise unconstrained. These two moduli spaces, MH ,k,RˆY and MH ,k,W fail to be simple split Floer cascades (as in Definition 5.8) in two ways: they are missing the gradient trajectory constraints on their asymptotic evaluation maps, and they are missing their augmentation planes. In order to impose these conditions, we will need to study these evaluation maps and establish their transversality.
Recall from Proposition 5.26 that for J Σ P J reg Σ , we have transversality for D Σ wi for each sphere w i " π Σ pv i q.
Let δ ą 0 be sufficiently small. For each i " 1, . . . , N , by Lemma 5.24, D C vi is surjective when considered on W 1,p,´δ (with exponential growth), and has Fredholm index 1. The operator considered instead on the space W 1,p,δ V , with V´8 " V`8 " iR and V P " C for any puncture P on the domain ofṽ i , has the same kernel and cokernel by Lemma 5.20. Thus, the operator, acting on sections free to move in the Morse-Bott family of orbits, is surjective and has index 1.
Since the operator Dṽ i is upper triangular from Lemma 5.22, and its diagonal components are both surjective, the operator is surjective. Since the Fredholm index is the sum of these, each componentṽ i contributes an index of 1`2n´22 xc 1 pT Σq, A i y`2k i " 2n´1`2 xc 1 pT Σq, A i y`2k i , where k i is the number of punctures.
The same consideration as previously gives thatṽ 2 , . . . ,ṽ N are transverse and each contributes an index of 2n´1`2 xc 1 pT Σq, A i y`2k i , where k i is the number of punctures. For the componentṽ 1 , again applying Lemma 5.22 and applying Lemma 5.24 in the case where the´8 end of the cylinder converges to a Reeb orbit at t´8uˆY , we obtain that the vertical Fredholm operator is surjective and has index 2. The linearized Floer operator atṽ 1 is then surjective and has index 2n`2 xc 1 pT Σq, A 1 y`2k 1 . By Lemma 2.4, the planeṽ 0 can be identified with a sphere in X with an order of contact l " B ‚ Σ with Σ. Its Fredholm index is 2n`2pxc 1 pT Xq, By´lq. The total Fredholm index is therefore 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 of stable/unstable manifolds and flow diagonals, and also transversality of the augmentation evaluation maps, in order to obtain the constraints coming from pseudo-gradient flow lines. Indeed, let pṽ 1 , . . . ,ṽ N q be a collection of N cylinders in MH ,k,RˆY ;k´,k`p pA 1 , . . . , A N q; J Y q. Write each of theṽ i : RˆS 1 Ñ RˆY as a pairṽ i " pb i , v i q. We then have asymptotic evaluation maps (5.8) r ev Y : MH ,k,RˆY ;k´,k`p pA 1 , . . . , A N q; J Y q Ñ Y 2N pṽ 1 , . . . ,ṽ N q Þ Ñˆlim If pṽ 0 ,ṽ 1 , . . . ,ṽ N q P MH ,k,W ppB; A 1 , . . . , A N q; J W q, we have (5.9) r ev W,Y : MH ,k,W ;k`p pB; A 1 , . . . , A N q; J W q Ñ WˆY 2N`1 pṽ 0 ,ṽ 1 , . . . ,ṽ N q Þ Ñˆṽ 0 p0q, lim hese maps are C 1 smooth, which follows from exploiting the asymptotic expansion of a Floer cylinder near its asymptotic limit, as described by [Sie08]. Details for this are given in [FS17].
We also have augmentation evaluation maps. For each puncture z 0 P Γ, there exists an index i P t1, . . . , N u so that the augmentation puncture z 0 is a puncture in the domain of v i . For this augmentation puncture, we have the asymptotic evaluation map v i Þ Ñ lim zÑz0 π Σ pv i pzqq P Σ. Combining all of these evaluation maps over all punctures in Γ, we obtain r ev a Σ : MH ,k,RˆY ;k´,k`p pA 1 , . . . , Note that these maps are C 1 smooth, either by [FS17] or by combining [Wen16b, Proposition 3.15] with the smoothness for the evaluation map for closed spheres. Define the flow diagonal in YˆY to be where Critpf Y q is the set of critical points of f Y . Letp,q P Y be critical points of f Y and let W u Y ppq, W s Y pqq be the unstable/stable manifolds ofp,q, as in (3.1).
We may now describe the moduli space of simple split Floer cylinders fromq k´t õ p k`a s the unions of the fibre products of these moduli spaces under the asymptotic evaluation maps and augmentation evaluation maps. For notational convenience, we write (5.10) r ev : MH ,k,RˆY ;k´,k`p pA 1 , . . . , A N q; J Y qˆMX ppB 1 , . . . , B k q; J W q Ñ Y 2NˆΣkˆΣk pṽ, vq Þ Ñ p r ev Y pṽq, r ev a Σ pṽq, ev a Σ pvqq .
Similarly, if x P W is a critical point of f W , and letting W u W pxq be the descending manifold of x in W for the gradient-like vector field´Z W , we define r ev : MH ,k,W ;k`p pB; A 1 , . . . , A N q; J W qˆMX ppB 1 , . . . , B k q; J W q Ñ WˆY 2N`1ˆΣkˆΣk ppṽ 0 ,ṽq, vq Þ Ñ p r ev W,Y pṽ 0 ,ṽq, r ev a Σ pṽq, ev a Σ pvqq . Then, define MH px,p k`; pB; A 1 , . . . , A N q,pB 1 , . . . , B k q; J W q " Finally, we obtain In order to establish transversality for our moduli spaces, it then becomes necessary to show transversality of the evaluation maps to these products of descending/ascending manifolds, diagonals and flow diagonals. Recall the space of almost complex structures J reg W given in Proposition 5.35. We denoted by J reg Y the space of cylindrical almost complex structures on RˆY obtained from restrictions of elements in J reg W . The following result will provide the final step in the proof of Proposition 5.9.
Proposition 5.41. Let J W P J reg W and let J Y P J reg Y be the induced almost complex structure on RˆY .
Letq,p denote critical points of f Y , and let x be a critical point of f W in W . Let k`and k´be non-negative multiplicities, k`ą k´.
Let∆ Ă YˆY and ∆ Σ k Ă Σ kˆΣk be the diagonals. Then, (1) the evaluation map r ev Yˆr ev a Σˆe v a Σ : MH ,k,RˆY ;k´,k`p pA 1 , . . . , A N q; J Y qˆMX ppB 1 , . . . , B k q; J W q Ñ Y 2NˆΣkˆΣk is transverse to the submanifold (2) the evaluation map r ev W,Yˆr ev a Σˆe v a Σ : MH ,k,W ;k`p pB; A 1 , . . . , A N q; J W qˆMX ppB 1 , . . . , B k q; J W q Ñ WˆY 2N`1ˆΣkˆΣk is transverse to the submanifold In order to prove this proposition, we will need a better description of the relationship between the moduli spaces of spheres in Σ, and the moduli spaces of Floer cylinders in RˆY (or in W ).
Proof. We will study the case of π M Σ : MH ,k,RˆY ppA 1 , . . . , A N q; J Y q Ñ Mk ,Σ ppA 1 , . . . , A N q; J Σ q in detail. The case with a sphere in X follows by the same argument with a small notational change. It also suffices to consider the case with N " 1, since moduli spaces with more spheres are open subsets of products of these.
Suppose π Σ pṽq " w withṽ P MH ,k,RˆY ;k´,k`p A; J Y q and w P Mk ,Σ pA; J Σ q. Recall the splitting of the linearized Floer operator atṽ, given in Lemma 5.22 as w and TṽMH ,k,RˆY ;k´,k`p A; J Y q " ker Dṽ. By Lemma 5.24, D C v is surjective. It follows then that any section ζ 0 of w˚T Σ that is in the kernel of D Σ w can be lifted to a section pζ 1 , ζ 0 q ofṽ˚T Σ -pR'RRq'w˚T Σ that is in the kernel of Dṽ.
Also observe that S 1 acts on the curveṽ by the Reeb flow. By the Reeb invariance of J Y and of the admissible Hamiltonian H, the rotated curve is in the same fibre of π M Σ . Furthermore, for small rotation parameter, the curve will be distinct (as a parametrized curve) fromṽ.
The next result justifies why it was reasonable to assume k`ą k´in Proposition 5.41. The fact that k`‰ k´will also be used below.
Proof. Denote by w˚Y the pullback under w of the S 1 -bundle Y Ñ Σ. The map v 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 Γ Y t0, 8u.
Denote the multiplicities of the periodic X H -orbits x˘ptq " lim sÑ˘8 vps, tq by k˘, respectively, and denote the multiplicities of the asymptotic Reeb orbits at the punctures z 1 , . . . , z m P Γ by k 1 , . . . , k m , respectively. The positive integers k˘and k i are the absolute values of the degrees of s at the respective points. Taking signs into account, we get ż CP 1 epw˚Y q " k`´k´´k 1´. . .´k m .
We will show that this quantity is non-negative. We have ż w˚ι˚Kω " KωpAq ě 0 since K ą 0 and w is a J Σ -holomorphic sphere. We conclude that k`´k´´k 1´. . .´k m " KωpAq ě 0.
If A ‰ 0, we get a strict inequality. If A " 0, we get an equality, but the assumptions of the Lemma imply that ř m i"1 k i ą 0. In either case, we get k`ą k´, as wanted.
Recall that the gradient-like vector field Z Y has the property that dπ Σ Z Y " Z Σ . Also recall that we may use the contact form α as a connection to lift vector fields from Σ to vector fields on Y , tangent to ξ. If V is a vector field on Σ, we write πΣV :" r V to be the vector field on Y uniquely determined by the conditions αpV q " 0, dπ Σ r V " V . This extends as well to lifting vector fields on ΣˆΣ to vector fields on YˆY .
Lemma 5.44. The flow diagonal in Y satisfies Let px,ỹq P r ∆ f Y and x " π Σ pxq, y " π Σ pỹq. Let t ą 0 be so thatỹ " ϕ t Z Y pxq. Then, if x " y, we have x P Critpf Σ q and (5.13) T px,ỹq r ∆ f Y " tpaR`v, bR`πΣdϕ t ZΣ dπ Σ vq P T Y ' T Y | a, b P R and αpvq " 0u. If x ‰ y, then px, yq P ∆ fΣ . Then, there exists a positive g " gpx,ỹq ą 0 so that (5.14) where the subspace H is such that dπ Σ | H : H Ñ T ∆ fΣ induces a linear isomorphism.
Proof. Observe first that if x " π Σx , we have From this, it follows that dϕ t Z Y pxqR is a multiple of the Reeb vector field. Observe also that ϕ t ZΣ and ϕ t Z Y are both orientation-preserving diffeomorphisms for all t. We therefore obtain that if y " ϕ t ZΣ pxq, ϕ t Z Y induces a diffeomorphism between the fibres π´1 Σ pxq Ñ π´1 Σ pyq. Additionally, we must have then that dϕ t Z Y pxqR is a positive multiple of the Reeb vector field. Let gpx,ỹq ą 0 such that dϕ t Z Y pxqR " gpx,ỹqR. In general, ifỹ " ϕ t Z Y pxq, we have (5.15) Consider first the case of x " y. Then, bothx andỹ are in the same fibre of Y Ñ Σ. By definition of the flow diagonal, there exists t ą 0 so that ϕ t Z Y pxq "ỹ, and hence Z Y is vertical, Z Σ pxq " 0. It follows that x P Crit fΣ . From this, it now follows that π Σ p r ∆ f Y q Ă ∆ fΣ Y tpp, pq | p P Critpf Σ qu. We now consider the consequences of Equation (5.15) in this case of x " y.
Any v P T x Y may be written as v 0`a R where αpv 0 q " 0. Furthermore, since x " y P Critpf Σ q, and by definition, neitherx norỹ are critical points of f Y , we obtain that Z Y pỹq is a non-zero multiple of the Reeb vector field. Equation (5.13) now follows from the fact that dπ Σ ϕ t Z Y pxq " dϕ t ZΣ pxqdπ Σ . We now consider when x ‰ y. Let H " tpv, dϕ t Z Y pxqv`cZ Y pỹq | αpvq " 0u. Then, By assumption, y is not a critical point of f Σ , so dπ Σ induces an isomorphism. The decomposition of T r ∆ f Y now follows immediately from the definition of g and from Equation (5.15).
Proof of Proposition 5.41. We consider first the case of r ev Y : MH ,k,RˆY ;k´,k`p pA 1 , . . . , Let w i " π Σ pṽ i q and x i " π Σ px i q, y i " π Σ pỹ i q. Then, it follows that Let S Ă Σ 2N´2 be the appropriate product of a number of copies of ∆ fΣ and of tpp, pq | p P Critpf Σ qu. By Proposition 5.35, the evaluation map on M Σ ppA 1 , . . . , A N q; J Y q is transverse to S.
Then, by the previous Lemma, It suffices therefore to obtain transversality in the vertical direction. Notice that by rotating by the action of the Reeb vector field onṽ i , we obtain that the image of dẽv contains the subspace tpa 1 R, a 1 R, a 2 R, a 2 R, . . . , a N R, a N Rq | pa 1 , . . . , a N q P R N u Ă pT Y q 2N .
In the case of the chain of pearls in Σ, each of the spheres w i , i " 1, . . . , N must either be non-constant or have a non-trivial collection of augmentation punctures. Then, by Lemma 5.43, each punctured cylinderṽ i has different multiplicities kì , kí at˘8, and thus the action of rotating the domain marker gives that the image of dẽv Y pṽ i q contains pk´R, k`Rq P Tỹ i Y ' Tx i Y . While this holds for each i " 1, . . . , N , we only require such a vector for one cylinder. Then, by taking this in the case of i " 1, we see that the following N`1 vertical vectors in pRRq 2N Ă T Y 2N are in the image of the linearized evaluation map (the first two obtained by combining the two Reeb actions onṽ 1 , the remainder by the Reeb action onṽ i , 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.
We now observe that this collection of 2N vectors spans pRRq 2N . This establishes thatẽv Y defined on MH ,k,RˆY ;k´,k`p pA 1 , . . . , We now consider the case of r ev W,Y : MH ,k,W ;k`p pB; A 1 , . . . , A N q; J W q Ñ WˆY 2N`1 .
We will show this evaluation map is transverse tõ As before, it suffices to show transversality in a vertical direction, since, by Proposition 5.35, the projections to X, Σ are transverse. More precisely, let S Ă WˆΣˆΣ 2N be of the form S " W u W pxqˆ∆ˆS 1ˆW u Σ ppq, where S 1 Ă Σ 2N´2 is a product of some number of ∆ fΣ and of tpp, pq | p P Critpf Σ qu so that T S Ă T dπ Σ pSq. Proposition 5.35 gives transversality of r ev W,Y to S. Notice that the tangent space TS contains at least the following vertical vectors (we put 0 in the first component since T W has no vertical direction): p0, R, R, 0, 0, . . . , 0q p0, 0, 0, R, g 1 R, 0, . . . , 0q . . . p0, . . . , 0, R, g N´1 R, 0q.
Let pṽ 1 ,ṽ 1 , . . . ,ṽ N q P MH ,k,W ;k`. The planeṽ 1 converges to a Reeb orbit of multiplicity l " B ‚ Σ. Observe that domain rotation on the planeṽ 1 then gives that p0, lR, 0, . . . , 0q P T W ' T Y ' T Y 2N is in the image of d r ev W,Y .
The required transversality for these maps is given by Proposition 5.35. Furthermore, these evaluation maps are invariant under the domain and Reeb rotations used to obtain transversality for r ev Y and for r ev W,Y in the vertical directions, so the transversality follows immediately.

Monotonicity and the differential
The results of the previous section show that the moduli spaces of Floer cascades that project to simple chains of pearls are transverse.
We now impose monotonicity conditions on pX, ωq and on pΣ, ω Σ q in order to show that these moduli spaces are sufficient for the purposes of defining the split Floer differential. We suppose pX, ωq is spherically monotone, so there exists a constant τ X ą 0 with xc 1 pT Xq, Ay " τ X ωpAq for every spherical homology class A. Also, if we let K ą 0 such that A ‚ Σ " KωpAq, then we require τ X´K ą 0. Observe that Σ must be spherically monotone if it has a spherical homology class, with monotonicity constant τ Σ " τ X´K .
6.1. Index inequalities from monotonicity and transversality. First, we consider the Fredholm index contributions of a plane in W that could appear as an augmentation plane, to obtain some bounds on the possible indices.
Lemma 6.1. If v : C Ñ W is a J W holomorphic plane asymptotic to a given closed Reeb orbit γ in Y , the Fredholm index for the deformations of v (as an unparameterized curve) keeping γ fixed is |γ| 0 and it is non-negative. Furthermore, if v is multiply covered, this Fredholm index is at least 2.
Proof. The fact that the Fredholm index Indpvq in the statement is given by |γ| 0 as in (3.6) can be seen using Theorem 5.18. On the other hand, thinking of v as giving a J X -holomorphic sphere in homology class B P H 2 pX; Zq, with an order of contact B ‚ Σ with Σ, we see that Indpvq " 2pxc 1 pT Xq, By´B ‚ Σ´1q " 2pτ X ωpBq´KωpBq´1q. Since the plane is holomorphic, the class B has ωpBq ą 0. By our monotonicity assumptions, we have τ X ωpBq´KωpBq " pτ X´K q ωpBq ą 0.
Finally, since τ X ωpBq " xc 1 pT Xq, By P Z and KωpBq " B ‚ Σ P Z, we have that τ X ωpBq´KωpBq ą 0 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 v 0 , representing classes B " kB 0 and B 0 , respectively. Then, Indpvq`2 " 2pτ X´K qωpkB 0 q " kpIndpv 0 q`2q.
Proposition 6.2. Any Floer cascade appearing in the differential, connecting two periodic orbits in RˆY , must be one of the following configurations: (0) An index 1 gradient trajectory in Y without any (non-constant) holomorphic components 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 an orbit q p k`a nd the negative puncture converges to an orbit p q k´. The difference in multiplicites of the orbits is given by k`´k´" KωpAq, where A P H 2 pΣ; Zq is the homology class represented by the projection of the cylinder to Σ. See Figure 6.1.
(2) A cylinder with one augmentation puncture and whose projection to Σ is trivial. The positive puncture converges to an orbit q p k`a nd the negative puncture converges to an orbit p q k´. The augmentation plane has index 0. If B P H 2 pX; Zq is the class represented by the augmentation plane, then the difference in multiplicities is given by k`´k´" KωpBq. Furthermore, q p and p q are critical points of f Y contained in the same fibre of Y Ñ Σ, which we can write as q " p. See Figure 6.2.
Proof. Consider a cascade with N levels and k augmentation planes appearing in the differential dr p k`"¨¨¨`r q k´`¨¨¨. Let A 1 , . . . , A N P H 2 pΣq denote the homology classes of the projections to Σ, let B 1 , . . . , B k P H 2 pXq denote the homology classes corresponding to the augmentation planes. Let γ i , i " 1, . . . , k denote the limits at the augmentation punctures, and let k i denote their multiplicites. Let A " ř N i"1 A i . We therefore have k`´k´´ř k j"1 k j " KωpAq " K xc1pT Σq,Ay τ X´K . We also have k j " B j ‚ Σ " KωpB j q. Notice then that |γ i | 0 " 2xc 1 pT Xq, B j y´2B j ‚ Σ´2.
We therefore have (6.1) 1 " |r p k`|´| r q k´| " ipr pq`M ppq´ipr qq´M pqq`2 τ X´K K pk`´k´´k By Lemma 6.1, we have that for each j " 1, . . . , k, |γ j | 0 ě 0. Consider the chain of pearls in Σ obtained by projecting the upper level of this split Floer trajectory to Σ. By Proposition 5.26, if this is a simple chain of pearls, it has Fredholm index I Σ :" M ppq`2xc 1 pT Σq, Ay´M pqq`N´1`2k.
If the chain of pearls is not simple, by monotonicity, we have that the index is at least as large as the index of the underlying simple chain of pearls. Now let N 0 be the number of sub-levels that project to constant curves in Σ and let N 1 be the number of sub-levels that project to non-constant curves in Σ, N " N 0`N1 . Note that by the stability condition, each cylinder that projects to a constant curve in Σ must have at least one augmentation puncture, so N 0 ď k.
By transversality for simple chains of pearls (Proposition 5.26), we obtain the inequality I Σ ě 2N 1`2 k by considering the 2-dimensional automorphism group for the N 1 non-constant spheres and by considering the 2k-parameter family of moving augmentation marked points on the domains. Combining with Equation (6.1), we obtain 1 " ipr pq´ipr qq`pI Σ´N`1 q`k ÿ j"1 |γ j | 0 1 ě pipr pq´ipr qq`1q`2N 1`2 k´N`k 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 (k ď 1) and if there is one, it must have |γ 1 | 0 " 2xc 1 pT Xq, B 1 y´2B 1 ‚ Σ´2 " 0 (so the augmentation plane cannot be multiply covered, by Lemma 6.1). We can further write 1 ě pipr pq´ipr qq`1q`N 1`k`p k´N 0 q.
Notice that N 1`2 k´N 0 ě N . This inequality can be satisfied in one of the following ways: (0) N " 0. Then, either ipr pq " ipr qq or r p "p and r q "q. Since N " 0, this is a pure Morse differential term. (1) N 1 " 1, N 0 " k " 0 and r p " q p, r q " p q. This case corresponds to a non-constant sphere in Σ without any augmentation punctures.
(2) N 1 " 0, k " 1, N 0 " 1, and r p " q p, r q " p q. In this case, the Floer cylinder has one augmentation puncture, and projects to a constant in Σ, so q " p P Σ.
We now consider the possible terms in the differential that connect non-constant Hamiltonian trajectories in RˆY to Morse critical points in X. Proposition 6.3. Any Floer cascade appearing in the differential, connecting a non-constant Hamiltonian orbitp k`i n RˆY to a Morse critical point x in W , consists of two levels. The upper level, in RˆY , projects to a point in Σ and is a cylinder asymptotic at`8 to an orbit q p k`a nd at´8 to a Reeb orbit γ in t´8uˆY . This γ is the parametrized Reeb orbit associated to q p k`. The lower level is a holomorphic plane in W converging to the parametrized orbit γ at 8 and with 0 mapping to the descending manifold of the critical point x. As a parametrized curve, this has Fredholm index 1. See Figure 6.3.
Proof. Suppose such a cascade occurs in the differential, connecting the non-constant orbit r p k`t o the critical point x in the filling W .
Let N be the number of cylinders in RˆY that appear in the split Floer cylinder. Let A i P H 2 pΣq, i " 1, . . . , N , denote the spherical classes represented by the projections of these cylinders to Σ. Let A " Let k be the number of augmentation planes, and let B j P H 2 pXq, j " 1, . . . , k be the corresponding spherical homology classes in X. Let γ j , j " 1, . . . , k, be the corresponding Reeb orbits with multiplicities k j " B j ‚ Σ " KωpB j q.
Let B P H 2 pXq be the spherical homology class in X represented by the lower level v 0 in W , connecting to the critical point x. Let k´" B ‚ Σ be the multiplicity of the orbit to which the plane v converges. As before, we have We then have (6.2) 1 " |r p k`|´| x| Projecting to Σ, we obtain a chain of pearls with a sphere in X. Let N 0 be the number of constant spheres in Σ and let N 1 be the number of non-constant spheres in Σ, N " N 0`N1 . Notice that each non-constant sphere in Σ has a 2-parameter family of automorphisms, and each augmentation marked point can be moved in a 2-parameter family. Furthermore, the holomorphic sphere v 0 also has a 2-parameter family of automorphisms. By passing to a simple underlying chain of pearls as necessary, and applying monotonicity and Proposition 5.26 (to Mk ,pX,Σq ppB; A 1 , . . . , A N q; x, p, J W q), we obtain I X :" M ppq`2xc 1 pT Σq, Ay`2 pxc 1 pT Xq, By´B ‚ Σq`M pxq´2n`1`N`2k ě 2N 1`2 k`2 We now combine the inequality with Equation (6.2): Notice that we have N 0 ď k`1 since the first sphere in the chain of pearls with a sphere in X is allowed to be constant without any marked points. This observation together with Lemma 6.1 gives that each term on the right-hand-side of the inequality is non-negative. It follows therefore that each term must vanish: N 1 " 0, N 0 " 1, k " 0 and r p " q p. Notice that the Floer cylinder in RˆY is contained in a single fibre of RˆY Ñ Σ, so the marker condition coming from q p can be interpreted as a marker condition on the holomorphic plane v 0 (via the parametrized Reeb orbit γ in the statement). Without the marker condition, v 0 has Fredholm index 2, and thus with the marker constraint, it has index 1.
Remark 6.4. Similar analysis applied to continuation maps gives that our construction doesn't depend on the choices of almost complex structure J Y , J W or of the auxiliary Morse functions and pseudogradient vector fields. Index considerations immediately give that d 2 " 0.
Case (2) in Proposition 6.2 allows for the existence of augmented configurations contributing to the symplectic homology differential. We will now adapt an argument originally due to Biran and Khanevsky [BK13] to show that if W is a Weinstein domain (or equivalently, if W is a Weinstein manifold of finite-type), and Σ has minimal Chern number at least 2, then there can only be rigid augmentation planes if the isotropic skeleton has codimension at most 2 (in particular, dim R X " 2n ď 4).
Lemma 6.5. If W is a Weinstein domain with isotropic skeleton of real codimension 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˚: π 2 pΣq Ñ π 2 pXq.
Proof. The trivial direction is that if there exists a spherical class A P π 2 pΣq with ωpAq ą 0, then ı˚A P π 2 pXq and still has positive area.
We will now prove that any symplectic sphere in X is in the image of the inclusion. Let C Ă W be the isotropic skeleton of W . Notice that by following the flow of the Liouville vector field on W , we obtain that W zC is symplectomorphic to a piece of the symplectization p´8, aqˆY . Thus, we have that XzC is an open subset of a symplectic disk bundle over Σ (the normal bundle to Σ in X). We denote this bundle's projection map by π : XzC Ñ Σ.
Suppose A P π 2 pXq is a spherical class with ωpAq ą 0. By hypothesis, the skeleton C is of codimension at least 3. We may therefore perturb A in a neighbourhood of the skeleton so that it does not intersect the skeleton C. If ι : Σ Ñ X and j : XzC Ñ X are the inclusion maps, then ω Σ " ι˚ω and ι˝π is homotopic to j. This implies that ω X pAq " ω Σ pπ˚Aq, and the result follows.
Lemma 6.6. Suppose W is a Weinstein domain with isotropic skeleton of real codimension at least 3 and Σ has minimal Chern number at least 2. Then, there do not exist any augmentation planes.
Proof. Recall from Proposition 6.2 that an augmentation plane in the class B must have index 0, so 0 " 2pxc 1 pT Xq, By´B ‚ Σ´1q. Now, xc 1 pT Xq, By´B ‚ Σ " pτ X´K q ωpBq ě 1. Thus, the augmentation plane can only exist if there is a spherical class B with pτ X´K q ωpBq " 1.
By applying Lemma 6.5, we have B " ı˚A, where A P π 2 pΣq is a spherical class in Σ. Now observe that xc 1 pT Σq, Ay`xc 1 pN Σq, Ay " xc 1 pT Xq, Ay, so we have xc 1 pT Σq, Ay " pτ X´K qω Σ pAq. Hence, 1 " pτ X´K q ωpAq " xc 1 pT Σq, Ay. This contradicts the assumption that the minimal Chern number of Σ is at least 2, so the augmentation plane cannot exist.
Remark 6.7. Observe that this lemma applies more generally: if Σ has minimal Chern number at least 2, then an augmentation plane cannot represent a spherical class in the image of ı˚: π 2 pΣq Ñ π 2 pXq.
Additionally, we have that an augmentation plane cannot have image entirely contained in ϕpUq. Indeed, any holomorphic sphere contained in ϕpUq will have index too high to be an augmentation plane: the J X -holomorphic sphere with image in ϕpUq automatically comes in a 2-parameter family (corresponding to the C˚action on the normal bundle to Σ). To make this argument more precise, we use our index computations. Suppose a sphere in ϕpUq is an augmentation plane. It then represents a class ı˚A with A P H 2 pΣq. By the same index argument as in Lemma 6.6, 1 " xc 1 pT Σq, Ay. Since the image is assumed to be in ϕpUq, the projection of the curve to Σ is J Σ -holomorphic. The index of this projection is given by´4`2xc 1 pT Σq, Ay "´2. This must be non-negative, however, since the projection is J Σ -holomorphic, and represents an indecomposable homology class. This contradiction then rules this possibility out.
[FH93, Section 3] and [BM04].) Thus, an orientation of D and an orientation of D 1 induce an orientation of D#D 1 .
We may also arrange for the coherent orientation to have the following two properties: ‚ the orientation of the direct sum of two operators is the tensor product of their 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 Fredholm problems. Specifically, we consider a space of sections with exponential decay together with a (finite dimensional) space of infinitesimal movements in the corresponding Morse-Bott family of orbits. For a given Morse-Bott degenerate asymptotic operator, we focus on two associated non-degenerate operators. The first corresponds to keeping the asymptotic orbit fixed. The second corresponds to letting the orbit to which the cylinder converges move in its Morse-Bott family. These are the δ-perturbed asymptotic operators of Definition 5.15. Notice that the free/fixed asymptotic operators are perturbed by˘δ differently depending on the sign of the puncture.
After these preliminary observations, let us now describe how to attach a sign to a Floer cylinder with cascades contributing to the differential. We will follow [BO09b] closely. By Propositions 6.2 and 6.3, there are four types of contributions to the differential, referred to as Cases 0 through 3. We will explain how to determine the signs in each case.
In Case 0, we have gradient flow lines of Morse-Smale pairs pf, Zq on manifolds of orbits, which can either be pf Y , Z Y q on Y or p´f W ,´Z W q on W (see Definition 4.4). Let us stipulate the orientation conventions for Morse homology that we will use. The Morse complex of a Morse-Smale pair pf, Zq on a manifold S is generated by critical points of f and the differential B f is such that, given p P Critpf q, (7.1) B f ppq " ÿ qPCritpf q ind f ppq´ind f pqq"1 #``W s S pqq X W u S ppq˘{R˘q.
In this formula, we use the notation of (3.1) for critical manifolds of Z. Note that they intersect transversely, by the Morse-Smale assumption. We need to make sense of the signed count in the formula. We will be interested in the cases where S is Σ, Y or W , all of which are oriented (by their chosen symplectic, contact and symplectic forms, respectively). If we fix an orientation on a critical manifold at a critical point p, then we get an induced orientation on the other critical manifold, by imposing that the splitting (7.2) T p W s S ppq ' T p W u S ppq -T p S preserves orientations. Pick orientations on all unstable manifolds of Σ and W . For all p P Critpf Σ q, we will assume that the orientations on critical submanifolds of Σ and Y are such that the restrictions of π Σ : Y Ñ Σ to (7.3) W u Y pq pq Ñ W u Σ ppq and W s Y pp pq Ñ W s Σ ppq are orientation-preserving diffeomorphisms.
If γ : R Ñ S is a rigid flow line from q to p (critical points of consecutive indices), then it induces a diffeomorphism onto its image γpRq Ă W s S pqq X W u S ppq. The source R has its usual orientation, corresponding to increasing values of time in the orbit γ, and the image W s S pqq X W u S ppq is a transverse intersection, and can be oriented in such a way that the splitting (7.4) T W u S ppq -T pW s S pqq X W u S ppqq ' T W u S pqq is orientation-preserving (see [Hut02,Equation (2)] for a similar convention). The flow line γ contributes to #``W s S pqq X W u S ppq˘{R˘in (7.1) positively iff it preserves orientations. The Case 0 contribution of a flow line to the symplectic homology differential is the same as its contribution to the Morse differential.
Let us now consider Case 1, which is more interesting. Recall that such configurations consist of a Floer cylinder without augmentation punctures, together with two flow lines of Z Y at the ends. Suppose that the Floer cylinder converges to orbits of multiplicities k˘at˘8. Such a cylinder is an element of the space MH ,0,RˆY ;k´,k`p A; J Y q, for some A P H 2 pΣ; Zq.
To orient these spaces of Floer cylinders, we begin by choosing capping operators for the relevant asymptotic operators. By Lemma 5.22, the linearized operator associated to a Floer cylinderṽ is a compact perturbation of a split operator D C v ' 9 D Σ w , where w " π Σ˝ṽ . There is also a corresponding splitting of the asymptotic operators at the asymptotic limits. In particular, 9 D Σ w has complex linear asymptotic operators, and thus is a compact perturbation of a complex linear Cauchy-Riemann operator. Hence, its orientation is induced by the canonical one, and is independent of choice of trivialization or capping operator.
Remark 7.1. If it had been necessary to write down asymptotic operators associated to 9 D Σ w , we would have trivialized the contact distribution ξ over each stable and unstable manifold of the vector field Z Y (this is possible since the critical manifolds are contractible, but there is no global trivialization of ξ over Y ).
We are left with the task of orienting the operator D C v . We will orient the operator with its asymptotic ends free to move in the S 1 families of orbits. Writẽ v " pb, vq : RˆS 1 Ñ RˆY and b˘" lim sÑ˘8 bps, tq. At˘8, the δ-perturbed asymptotic operators (see Definition 5.15) associated toṽ are (7.5) A˘:"´ˆJ d dt`ˆh 2 pe b˘q e b˘˘δ 0 0˘δ˙ȧ t˘8, for a choice of δ ą 0 sufficiently small (see Remark 5.21).
Recall that these asymptotic operators are associated to converging to the descending manifold of the maximum of f Y at`8, and to the ascending manifold of the minimum at´8, or equivalently, allowing the asymptotic S 1 marker to move in its Morse-Bott family at both punctures. We will choose capping operators for the A˘, which determines an orientation of D C v by the coherent orientation scheme. By Lemma 5.24, D C v has Fredholm index 1, is surjective and its kernel contains an element that can be identified with the Reeb vector field.
Lemma 7.2. There is a choice of capping operators for the asymptotic operators above, such that D C v is oriented in the Reeb direction.
Proof. Recall that for each b k ą 0 satisfying h 1 pe b k q " k P Z`, we have a Yparametric family of 1-periodic Hamiltonian orbits. We can associate to each of these orbits two operators, as in (7.5). We will define capping operators Φk : W 1,p pC, Cq Ñ L p pHom 0,1 pT pCq, Cqq with these asymptotic operators. We first define two families of auxiliary Fredholm operators. For each k ą 0, Ψ k : W 1,p pRˆS 1 , Cq Ñ L p pHom 0,1 pT pRˆS 1 q, Cqq is an operator given by Ψ k pF qpB s q " F s`i F t`ˆa psq´δ 0 0´δ˙F where the function a : R Ñ R is such that lim sÑ´8 apsq " h 2 pe b1 qe b1 and lim sÑ`8 apsq " h 2 pe b k qe b k . Let now Ξ k : W 1,p pRˆS 1 , Cq Ñ L p pHom 0,1 pT pRˆS 1 q, Cqq be an operator given by Ξ k pF qpB s q " F s`i F t`ˆh 2 pe b k qe b k`δ psq 0 0 δpsq˙F where δ : R Ñ R is such that lim sÑ´8 δpsq "´δ ă 0 and lim sÑ`8 δpsq " δ.
The operators Ψ k are isomorphisms (in particular, they are canonically oriented). This follows from an argument analogous to the proof of Lemma 5.24. A version of the same argument implies that the operators Ξ k are Fredholm of index 1 and surjective, and that their kernels contain elements that can be identified with the Reeb vector field. Now, pick any capping operator Φ1 . Define Φḱ for k ą 1 by gluing Φ1 #Ψ k . Define Φk for all k ą 0 by gluing Φḱ #Ξ k . For these choices of capping operators, D C v are oriented in the direction of the Reeb flow, as wanted.
We have now oriented the operators D C v and 9 D Σ w . Since the linearized Floer operator is a compact perturbation of their direct sum, we get induced orientations on the spaces of Floer cylinders MH ,0,RˆY ;k´,k`p A; J Y q.
Remark 7.3. Recall that in Symplectic Field Theory [EGH00], bad orbits arise when the parity of the Conley-Zehnder indices of the even covers of a simple Reeb orbit is different from that of the odd covers (in such cases, the bad orbits are the even covers). In our situation, if we take the constant trivialization (i.e. the one that comes from a horizontal lift of a trivialization of T π pγqΣ), the Conley-Zehnder index does not depend on covering multiplicity. (See Section 3.1.) This simplifies our discussion of orientations. See [BO09b] for a treatment of coherent orientations in a setting where bad orbits can exist.
Cylinders with cascades that contribute to the differential in Case 1 are elements of spaces MH ,1 pp q k´, q p k`; J W q, for p ‰ q P Critpf Σ q (recall the notation in (5.11)). These spaces are unions of fiber products Definition 7.4. Given linear maps between oriented vector spaces f i : V i Ñ W , i " 1, 2, such that f 1´f2 : V 1 ' V 2 Ñ W is surjective, the fiber sum orientation on V 1 ' fi V 2 " kerpf 1´f2 q is such that (1) f 1´f2 induces an isomorphism pV 1 ' V 2 q{ kerpf 1´f2 q Ñ W which changes orientations by p´1q dim V2. dim W , (2) where a quotient U {V of oriented vector spaces is oriented in such a way that the isomorphism V ' pU {V q Ñ U (associated to a section of the quotient short exact sequence) preserves orientations.
One key property of this orientation convention for fiber sums is that it is associative (this property specifies the orientation convention almost uniquely, as explained in [Joy12, Remark 7.6.iii] and [RB09]; this was pointed out to us by Maksim Maydanskiy). Recall the discussion of Case 0 above, which included a specification of orientations on all critical submanifolds of Y . We use the fiber sum convention to orient the fiber product (7.6).
Observe now that if MH ,1 pp q k´, q p k`; J W q is one-dimensional, then its tangent space at every point is generated by the infinitesimal translation in the s-direction on the domain of the Floer cylinder. This induces an orientation on MH ,1 pp q k´, q p k`; J W q. Comparing this orientation with the one defined above with the fiber sum rule, we get the sign of such a contribution to the split symplectic homology differential.
We adapt the argument above to associate a sign to a contribution to the differential in Case 2. Such cascades are elements of spaces MH ,1 pp p k´, q p k`; J W q, for p P Critpf Σ q. The analogue of (7.6) is now (using the notation of (5.10)): (7.7) W s Y pp pqˆe v`MX pB; J W qˆẽ v MH ,1,RˆY ;k´,k`p 0; J Y q˘ˆe v W u Y pq pq. Notice that in this case we also need capping operators for augmentation punctures. The asymptotic operators at such punctures are which are complex linear. We may therefore take (canonically oriented) complex linear capping operators at these punctures. We can now orient the fiber product (7.7) using the fiber sum convention. The sign of such a contribution to the differential is obtained by comparing this orientation with the one induced by s-translation on the domain of the punctured Floer cylinder in MH ,1,RˆY ;k´,k`p 0; J Y q.
Finally, Case 3 Floer cylinders with cascades that contribute to the differential are elements of MH ,1 px, q p k`; J W q, which are unions of fiber products W u W pxqˆe v´MH ,0,W ;k`p pB; 0q; J W qˆe v r ∆¯ˆe v W u Y pq pq. The relevant evaluation maps are given by factors ofẽv W,Y in (5.9). It will be useful to write an alternative description of this fiber product. Recall that MH ,0,W ;k`p pB, 0q; J W q is a space of pairs pṽ 0 ,ṽ 1 q with the following properties. The simple J W -holomorphic cylinderṽ 0 : RˆS 1 Ñ W has a removable singularity at´8, defining a pseudoholomorphic sphere in X in class B P H 2 pX; Zq, with order of contact k`" B ‚ Σ at 8. Denote the space of such cylinders by MH pB; J W q. The Floer cylinderṽ 1 : RˆS 1 Ñ RˆY converges at`8 to a Hamiltonian orbit of multiplicity k`and at´8 to a Reeb orbit of the same multiplicity in t´8uˆY . It projects to a constant in Σ. Denote the space of such cylinders by MH ,k`p 0; J Y q. We have evaluation maps pev 1 , ev 1 q : MH pB; J W q Ñ WˆY and pev 2 , ev 2 q : MH ,k`p 0; J Y q Ñ YˆY and can write MH ,0,W ;k`p pB, 0q; J W q " MH pB; J W qˆMH ,k`p 0; J Y q and MH ,0,W ;k`p pB, 0q; J W qˆe v r ∆ " MH pB; J W q ev 1ˆe v 2 MH ,k`p 0; J Y q.
We now rewrite the Case 3 contributions to the differential as (7.8) W u W pxqˆe v 1´MH pB; J W q ev 1ˆe v 2 MH ,k`p 0; J Y q¯ˆe v 2 W u Y pq pq, which is oriented using coherent orientations on the spaces of cylinders and the fiber sum orientation convention.
The space MH pB; J W q ev 1ˆe v 2 MH ,k`p 0; J Y q has an action of R 1ˆR2 , where the 1-dimensional real vector space R 1 acts by s-translation on the domain of the cylinder in W and R 2 acts by s-translation on the domain of the cylinder in RˆY . The sign of a Case 3 contribution to the differential is obtained by comparing the coherent/fiber product orientation on (7.8) with the usual orientation on R 1ˆR2 , corresponding to s-translation on the domain ofṽ 0 followed by s-translation on the domain ofṽ 1 .