Morse–Bott split symplectic homology
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Abstract
We associate a chain complex to a Liouville domain \((\overline{W}, \mathrm{d}\lambda )\) whose boundary Y admits a Boothby–Wang contact form (i.e. is a prequantization space). The differential counts Floer cylinders with cascades in the completion W of \(\overline{W}\), in the spirit of Morse–Bott homology (Bourgeois in A Morse–Bott approach to contact homology, Ph.D. Thesis. ProQuest LLC, Stanford University, Ann Arbor 2002; Frauenfelder in Int Math Res Notices 42:2179–2269, 2004; Bourgeois and Oancea in Duke Math J 146(1), 71–174, 2009). The homology of this complex is the symplectic homology of W (Diogo and Lisi in J Topol 12:966–1029, 2019). Let X be obtained from \(\overline{W}\) by collapsing the boundary Y along Reeb orbits, giving a codimension two symplectic submanifold \(\Sigma \). Under monotonicity assumptions on X and \(\Sigma \), we show that for generic data, the differential in our chain complex counts elements of moduli spaces of cascades that are transverse. Furthermore, by some index estimates, we show that very few combinatorial types of cascades can appear in the differential.
Mathematics Subject Classification
53D40 53D421 Introduction and statement of main results
In this paper, we define Morse–Bott split symplectic homology theory for Liouville manifolds W of finitetype whose boundary \(Y = \partial W\) is a prequantization space. This is inspired by the construction of Bourgeois and Oancea for positive symplectic homology, \(SH^+\) [8]. Our main result is that we obtain transversality for all the relevant moduli spaces and thus have a welldefined 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 [13]. Furthermore, in [13], we justify that the homology of this complex is indeed the symplectic homology of W.
The main purpose of this definition of a split version of symplectic homology is to enable computations in certain examples. For instance, we expect to be able to compute symplectic homology for completions W of \(X{\setminus } \Sigma \), where both X and \(\Sigma \) are smooth projective complete intersections. Many of these examples fit in our framework and additionally enough is known about their Gromov–Witten invariants for the computation to be possible. In [13, Part 4], we illustrate our results by computing the wellknown symplectic homology of \(T^*S^2\).
In the following, we consider a 2ndimensional Liouville domain \((\overline{W}, \mathrm{d}\lambda )\) with \(\partial \overline{W} = Y\). Denoting by \(\alpha = \lambda _Y\) the contact form on the boundary induced by the Liouville form \(\lambda \), we require that \(\alpha \) 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 \(\Sigma ^{2n2}\) denote the quotient by the Reeb vector field, and we note that \(\mathrm{d}\alpha \) descends to a symplectic form \(\omega _\Sigma \) on the quotient. It follows that there exists a closed symplectic manifold \((X, \omega )\) for which \(\Sigma \) is a codimension 2 symplectic submanifold, Poincaré dual to a multiple of \(\omega \), with the property that \((X {\setminus } \Sigma , \omega )\) is symplectomorphic to \((\overline{W} {\setminus } Y, \mathrm{d}\lambda )\). See Proposition 2.1 for a more detailed description of this, or see [21, Proposition 5]. We can think of X as the symplectic cut of \({\overline{W}}\) along Y.
We let \((W, \mathrm{d}\lambda )\) be the completion of \(\overline{W}\), obtained by attaching a cylindrical end \(\mathbb {R}^+ \times Y\), and take a Hamiltonian \(H :\mathbb {R}\times Y \rightarrow \mathbb {R}\) 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 [5, 19, 27]) to construct a Morse–Bott Floer homology associated with H. We also consider configurations that interact between \(\mathbb {R}\times Y\) and W, as in [8].
In Sect. 3, we introduce the chain complex for split symplectic homology, and in Sect. 4 we describe the moduli spaces that are relevant to the differential. The chain complex will be generated by critical points of auxiliary Morse functions associated with our Morse–Bott manifolds of orbits. The differential will be obtained from moduli spaces of Floer cylinders with cascades 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 \(\Sigma \). This result holds without any monotonicity assumptions. A more precise formulation is provided in Proposition 5.9.
Theorem 1.1
Simple Floer cylinders with cascades in W and \(\mathbb {R}\times Y\) are transverse for a comeagre set of compatible, Reebinvariant and cylindrical almost complex structures on \((W,\mathrm{d}\lambda )\).
Theorem 1.1 builds on a transversality theorem for moduli spaces of spheres in X and in \(\Sigma \), 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 [3]. Our result builds essentially on results from [29]. A more precise formulation is given in Proposition 5.26.
Theorem 1.2
For a generic compatible almost complex structure, moduli spaces of somewhere injective spheres in \(\Sigma \) and of somewhere injective spheres in X with order of contact constraints at \(\Sigma \), 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 under suitable monotonicity assumptions on X and \(\Sigma \). In particular, the differential is computed from only four types of simple cascades. See Propositions 6.2 and 6.3 for more details.
Theorem 1.3
Assume that \((X, \omega )\) is spherically monotone with monotonicity constant \(\tau _X\) and assume that \(\Sigma \subset X\) is Poincaré dual to \(K \omega \) with \(\tau _X> K > 0\).
Then the split Floer homology of W is well defined, and does not depend on the choice of Hamiltonian or of compatible, Reebinvariant and cylindrical almost complex structure on W (in a comeagre set of such almost complex structures).
 (0)
Morse trajectories in Y or in W;
 (1)
Floer cylinders in \(\mathbb {R}\times Y\), projecting to nontrivial spheres in \(\Sigma \), and 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 nontrivial cascades, but their components in \(\mathbb {R}\times Y\) lie in fibres of \(\mathbb {R}\times Y \rightarrow \Sigma \). This is formulated more precisely in Propositions 6.2 and 6.3. See Figs. 4, 5 and 6 for a depiction of Cases (1)–(3).
The paper concludes with a discussion of orientations of the moduli spaces of cascades contributing to the differential, in Sect. 7.
2 Setup
We now provide details of the classes of Liouville manifolds for which we prove transversality in split Floer homology.
We begin by summarizing some constructions from [13], specifically
Proposition 2.1
[13, Lemma 2.2]. Let \((\overline{W}, \mathrm{d}\lambda )\) be a Liouville domain with boundary \(Y = \partial \overline{W}\). Assume that \(\alpha :=\lambda _Y\) has a Reeb vector field generating a free \(S^1\) action.
 (i)
\((X{\setminus } \Sigma , \omega )\) is symplectomorphic to \((\overline{W} {\setminus } \partial {{\overline{W}}}, \mathrm{d}\lambda )\);
 (ii)
\([\Sigma ] \in H_{2n2}(X; \mathbb {Q})\) is Poincaré dual to \([K \omega ] \in H^2(X; \mathbb {Q})\) for some \(K > 0\);
 (iii)
if \(N\Sigma \) denotes the symplectic normal bundle to \(\Sigma \) in X, equipped with a Hermitian structure (and hence symplectic structure), there exist a neighbourhood \(\mathcal {U}\) of the 0section in \(N\Sigma \) and a symplectic embedding \(\varphi :\mathcal {U}\rightarrow X\). By shrinking \(\mathcal {U}\) as necessary, we may arrange that \(\varphi \) extends to an embedding of \({\overline{\mathcal {U}}}\).\(\square \)
Definition 2.2
Let \(X, \omega , \Sigma , \omega _\Sigma \) and \(N\Sigma \) be as in Proposition 2.1.
Fix a Hermitian line bundle structure on the symplectic normal bundle \(\pi :N\Sigma \rightarrow \Sigma \). A Hermitian connection on \(N\Sigma \) can be encoded in terms of a connection 1form \(\Theta \in \Omega ^1(N\Sigma {\setminus } \Sigma )\) with the property that \(\mathrm{d}\Theta = K \pi ^*\mathrm{d}\omega _\Sigma \). Fix such a Hermitian connection 1form \(\Theta \).
Since \(N\Sigma \) is a Hermitian line bundle, we have the action of U(1) on the bundle by rotation in the \(\mathbb {C}\)fibres. The infinitesimal generator of this action is a vector field on which \(\Theta \) evaluates to 1.
Let \(\rho :N\Sigma \rightarrow [0, +\infty )\) denote the norm in \(N\Sigma \) measured with respect to the Hermitian metric. Then, for each \(x \in N\Sigma {\setminus } \Sigma \), let \(\xi _x = \left( \ker \Theta _x \cap \ker \mathrm{d}\rho \right) \subset T_x N\Sigma \). We may then extend the distribution \(\xi _x\) smoothly to the 0section by defining \(\xi _x = T_x \Sigma \) if \(x \in \Sigma \).
Notice that this gives a splitting \(T_x N\Sigma = \ker \mathrm{d}\pi \oplus \xi _x\) and \(\mathrm{d}\pi _{\xi } :\xi _x \rightarrow T_{\pi (x)} \Sigma \) is a symplectic isomorphism.
Any almost complex structure \(J_\Sigma \) on \(\Sigma \) can then be lifted to an almost complex structure on \(N\Sigma \) by taking it to be the linearization of the bundle complex structure on \(\ker \mathrm{d}\pi \) and to be the pullback of \(J_\Sigma \) to \(\xi _x\) by the isomorphism \(\mathrm{d}\pi :\xi _x \rightarrow T_{\pi (x)} \Sigma \).
We refer to any almost complex structure obtained in this way as a bundle almost complex structure on \(N\Sigma \).
Define the open set \(\mathcal {V}= X {\setminus } \overline{ \varphi (\mathcal {U})}\). We will later perturb our almost complex structures in \(\mathcal {V}\).
Proposition 2.3
[13, Lemma 2.6]. Let \(\overline{W}\), Y, \(\lambda \), \(\alpha \), X, \(\Sigma \) and \(\varphi :\mathcal {U}\rightarrow X\) be as in Proposition 2.1.
 (i)
if \(J_X\) in a compatible almost complex structure on X that restricts to \(\varphi (\mathcal {U})\) as the pushforward by \(\varphi \) of a bundle almost complex structure, then \(\psi ^*J_X\) is a compatible almost complex structure on W that is cylindrical and Reebinvariant on \(W {\setminus } \overline{W}\);
 (ii)
if \(J_W\) is a compatible almost complex structure on W, cylindrical and Reeb invariant on \(W {\setminus } \overline{W}\), then the pushforward \(\psi _* J_W\) extends to an almost complex structure \(J_X\) on X and \(J_\Sigma :=J_X_\Sigma \) is also given by restricting \(J_W\) to a parallel copy of \(\{ c \} \times Y\) for some \(c > 0\), and taking the quotient by the Reeb \(S^1\) action.
Note that the diffeomorphism \(\psi \) is not symplectic.
In Sect. 6.1, we impose additional conditions of monotonicity. These will also be relevant to the grading given in Definition 3.4.
Definition 2.4
 (i)
X is spherically monotone: there exists a constant \(\tau _X > 0\) so that for each spherical homology class \(A \in H_2^S(X)\), \(\omega (A) = \tau _X \langle c_1(TX), A \rangle \);
 (ii)
\([\Sigma ] \in H_{2n2}(X; \mathbb {Q})\) is Poincaré dual to \([K\omega ]\) for some \(K > 0\) with \(\tau _X > K\).
Observe that Condition (ii) implies that \((\Sigma , \omega _\Sigma )\) is spherically monotone, monotonicity constant \(\tau _\Sigma = \tau _X  K\).
We then obtain the following useful characterization of \(J_W\)holomorphic planes in W (see also [23]):
Lemma 2.5
[13, Lemma 2.6]. Under the diffeomorphism of Proposition 2.3, finite energy \(J_W\)holomorphic planes in W correspond to \(J_X\)holomorphic spheres in X with a single intersection with \(\Sigma \). The order of contact gives the multiplicity of the Reeb orbit to which the plane converges.
Proof
Under the map \(\psi \), a finite energy \(J_W\)plane in W gives a \(J_X\)plane in \(X {\setminus } \Sigma \). By the fact that \(\psi \) is radial, the restriction of \(\psi ^*\lambda \) to the cylindrical end \([r_0, +\infty ) \times Y\) takes the form \(g(r)\alpha \) for some function g with \(g' > 0\) and \(\lim _{r\rightarrow \infty } g(r) = 1\). It follows then that the integral of \(\psi ^*\mathrm{d}\lambda \) over the plane is dominated by its Hofer energy as given in [10, Section 5.3]. Hence, the image of the plane under the map is a plane in \(X {\setminus } \Sigma \) with finite \(\omega \)area.
It follows then that the singularity at \(\infty \) 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 \(\Sigma \) of a loop near the puncture. \(\square \)
We now formalize the class of almost complex structures that will be relevant to this paper.
Definition 2.6
An admissible almost complex structure \(J_X\) on X is compatible with \(\omega \) and its restriction to \(\varphi (\mathcal {U})\) is the pushforward by \(\varphi \) of a bundle almost complex structure on \(N\Sigma \).
An almost complex structure \(J_W\) on \((W, \mathrm{d}\lambda )\) is admissible if \(J_W = \psi ^*J_X\) for an admissible \(J_X\). In particular, such almost complex structures are cylindrical and Reebinvariant on \(W {\setminus } \overline{W}\).
A compatible almost complex structure \(J_Y\) on the symplectization \(\mathbb {R}\times Y\) is admissible if \(J_Y\) is cylindrical and Reebinvariant.
In the following, we will identify W with \(X {\setminus } \Sigma \) by means of the diffeomorphism \(\psi \) and identify the corresponding almost complex structures. By an abuse of notation, we will both write \(\pi _\Sigma :Y \rightarrow \Sigma \) to denote the quotient map that collapses the Reeb fibres, and \(\pi _\Sigma :\mathbb {R}\times Y \rightarrow \Sigma \) to denote the composition of this projection with the projection to Y.
Definition 2.7
Denote the space of almost complex structures in \(\Sigma \) that are compatible with \(\omega _\Sigma \) by \({\mathcal {J}}_\Sigma \).
Let \({\mathcal {J}}_Y\) denote the space of admissible almost complex structures on \(\mathbb {R}\times Y\). Then the projection \(\mathrm{d}\pi _\Sigma \) induces a diffeomorphism between \({\mathcal {J}}_Y\) and \({\mathcal {J}}_\Sigma \).
Let \({\mathcal {J}}_W\) denote the space of admissible almost complex structures on W. By Proposition 2.3, for any \(J_W \in {\mathcal {J}}_W\), we obtain an almost complex structure \(J_\Sigma \).
Denote this map by \(P :{\mathcal {J}}_W \rightarrow {\mathcal {J}}_\Sigma \). This map is surjective and open by Proposition 2.3. For any given \(J_\Sigma \in {\mathcal {J}}_\Sigma \), \(P^{1}(J_\Sigma )\) consist of almost complex structures on W that differ in \(\overline{W}\), or equivalently, can be identified (using \(\psi \)) with almost complex structures on X that differ in \(\mathcal {V}= X {\setminus } \varphi (\mathcal {U})\).
3 The chain complex
We will now describe the chain complex for the split symplectic homology associated with W.
Definition 3.1
 (i)
\(h(\rho ) = 0\) for \(\rho \leqslant 2\);
 (ii)
\(h'(\rho ) > 0\) for \(\rho > 2\);
 (iii)
\(h'(\rho ) \rightarrow +\infty \) as \(\rho \rightarrow \infty \);
 (iv)
\(h''(\rho ) > 0\) for \(\rho > 2\).
 (1)
constant orbits: one for each point in W and at each point in \((\infty , \log 2] \times Y \subset \mathbb {R}\times Y\);
 (2)
nonconstant orbits: for each \(k\in \mathbb {Z}_+\), there is a Yfamily of 1periodic \(X_H\)orbits, contained in the level set \(Y_k :=\{b_k\}\times Y\), for the unique \(b_k>\log 2\) such that \(h'({\text {e}}^{b_k}) = kT_0\). Each point in \(Y_k\) is the starting point of one such orbit.
Remark 3.2
Notice that these Hamiltonians are Morse–Bott nondegenerate except at \(\{\log 2\} \times Y\). These orbits will not play a role because they can never arise as boundaries of relevant moduli spaces (see [13, Lemma 4.8]).
Recall that a family of periodic Hamiltonian orbits for a timedependent Hamiltonian vector field is said to be Morse–Bott nondegenerate if the connected components of the space of parametrized 1periodic 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 nondegeneracy requires the return map not to have 1 as an eigenvalue and hence such periodic orbits must be isolated.)
The contact distribution \(\xi \) defines an Ehresmann connection on the circle bundle \(S^1 \hookrightarrow Y\rightarrow \Sigma \). Denote the horizontal lift of \(Z_\Sigma \) by \(\pi _\Sigma ^*Z_\Sigma \in \mathfrak X(Y)\). We fix a Morse function \(f_Y :Y \rightarrow \mathbb {R}\) and a gradientlike vector field \(Z_Y\in {\mathfrak {X}}(Y)\) such that \((f_Y,Z_Y)\) is a Morse–Smale pair and the vector field \(Z_Y\pi _\Sigma ^*Z_\Sigma \) is vertical (tangent to the \(S^1\)fibres). Under these assumptions, flow lines of \(Z_Y\) project under \(\pi _\Sigma \) to flow lines of \(Z_\Sigma \).
Observe that critical points of \(f_Y\) must lie in the fibres above the critical points of \(f_\Sigma \) (and these are zeros of \(Z_Y\) and \(Z_\Sigma \), respectively). For notational simplicity, we suppose that \(f_Y\) has two critical points in each fibre. In the following, given a critical point for \(f_\Sigma \), \(p \in \Sigma \), we denote the two critical points in the fibre above p by \({\widehat{p}}\) and \(\check{p}\), the fibrewise maximum and fibrewise minimum of \(f_Y\), respectively.
We will denote by M(p) the Morse index of a critical point \(p \in \Sigma \) of \(f_\Sigma \), and by \({\tilde{M}} ({\tilde{p}}) = M(p) + i({\tilde{p}})\) the Morse index of the critical point \({\tilde{p}} = {\widehat{p}}\) or \(\tilde{p} = \check{p}\) of \(f_Y\). The fibrewise index has \(i({\widehat{p}}) = 1\) and \(i(\check{p}) = 0\).
Fix also a Morse function \(f_W\) and a gradientlike vector field \(Z_W\) on W, such that \((f_W,Z_W)\) is a Morse–Smale pair and \(Z_W\) restricted to \([0,\infty )\times Y\) is the constant vector field \(\partial _r\), where r is the coordinate function on the first factor. We denote also by \((f_W,Z_W)\) the Morse–Smale pair that is induced on \(X{\setminus }\Sigma \) by the diffeomorphism in Lemma 2.5. Denote by M(x) the Morse index of \(x\in {{\,\mathrm{Crit}\,}}(f_W)\) with respect to \(f_W\).
We now define the Morse–Bott symplectic chain complex of W and H. Recall that for every \(k>0\), each point in \(Y_k :=\{ b_k \} \times Y \subset \mathbb {R}^+ \times Y\) is the starting point of a 1periodic orbit of \(X_H\), which covers k times its underlying Reeb orbit.
Definition 3.3
3.1 Gradings
We will now define the gradings of the generators. For this, we will assume that \((X, \Sigma , \omega )\) is a monotone triple as in Definition 2.4.
Definition 3.4
The justifications of these gradings comes from analyzing the Conley–Zehnder indices of the 1periodic Hamiltonian orbits. These are defined for Morse nondegenerate Hamiltonian/Reeb orbits, using a trivialization of TW or of \(T(\mathbb {R}\times Y)\) over the orbit. See Definition 5.19 for the Morse–Bott analogue, and also [2, Section 3]; [22]. The first key observation is that the Conley–Zehnder index of an orbit only depends on the trivialization of the complex line bundle \(L :=\Lambda ^{n}_\mathbb {C}TW\) 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 be \(n+ (2nM(x)) = n  M(x)\).
Notice also that Y may be capped off by the normal disk bundle over \(\Sigma \), and each orbit bounds a disk fibre. The trivialization induced by the fibre differs from the constant trivialization only in a kfold winding in the \(\mathbb {R}\partial _r \oplus \mathbb {R}R\) direction. The resulting Conley–Zehnder index of \(\widetilde{p}_k\) for this trivialization induced by the disk fibre is then \({\tilde{M}}( {\widetilde{p}}) +1  n  2k\). We refer to this trivialization as the normal bundle trivialization.
Now, suppose that \(\gamma _k\) is the kfold cover a simple Reeb orbit \(\gamma \), and suppose it is contractible in W. Denote by \(\dot{B}\) a disk in W whose boundary is \(\gamma _k\). As we pointed out, \(\gamma _k\) is also the boundary of a kfold cover of a fibre of the normal bundle to \(\Sigma \) in X. This cover of a fibre can be concatenated with \(\dot{B}\) to produce a spherical homology class \(B\in H_2^S(X)\) such that the intersection \(B\bullet \Sigma = k>0\). Conversely, note that any \(B\in H_2^S(X)\) such that \(B\bullet \Sigma = k\) gives rise to a disk \(\dot{B}\) bounding \(\gamma _k\). The complex line bundle \(L_{\dot{B}}\) is trivial since \(\dot{B}\) is a disk. This induces a trivialization of L over \(\gamma _k\), which can be identified with a trivialization of \(L^{\otimes k}\) over \(\gamma \). We refer to this as the trivialization induced by \(\dot{B}\).
The relative winding of the trivialization of L over \(\gamma _k\) induced by \(\dot{B}\) and the normal bundle trivialization considered above is given by \(\langle c_1(L), B \rangle \) since this represents the obstruction to extending the trivialization of L over \(\dot{B}\) to all of B. Recall that \(c_1(L) = c_1(TX)\).
This formula holds when \(k \in \omega ( \pi _2(X) )\), and we extend it as a fractional grading for all \(k\in \mathbb {Z}\). (This corresponds to the fractional SFT grading from [16, Section 2.9.1].)
Finally, we compare our gradings with those described by Seidel [38] and generalized by McLean [30] (the latter considers Reeb orbits, but there is an analogous construction for Hamiltonian orbits) in the case that \(c_1(TW) \in H_2(W;\mathbb {Z})\) is torsion, so \(Nc_1(TW) = 0\) for a suitable choice of \(N>0\). Note that in our setting, this holds if X is monotone (and not just spherically monotone) and \(\Sigma \) is Poincaré dual to a multiple of \(\omega \).
The index (3.6) of the Reeb orbit was introduced for convenience in writing formulas for expected dimensions of moduli spaces. It comes from similar considerations for the Conley–Zehnder index of the Reeb orbit, together with the \(n3\) 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 \(\gamma _0\) (free to move in its Morse–Bott family) will be given by \(\gamma _0\).
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, there can only be a Floer cylinder connecting two Hamiltonian orbits if the difference of their degrees is an integer. Hence, one could write the symplectic homology as a direct sum indexed by the fractional parts of the degrees. Alternatively, one could also decompose it as a direct sum over homotopy classes of Hamiltonian orbits, as done, for instance, in [9].
4 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 \(\Sigma \) and in X. (These are familiar objects, reminiscent of ones considered in the literature for Floer homology of compact symplectic manifolds [3, 32, 33].)
4.1 Split Floer cylinders with 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 nonconstant 1periodic Hamiltonian orbits and one that connects a nonconstant 1periodic orbit to a constant one (in W).
Notice that we may identify a 1periodic orbit of H with its starting point, and in this way, we have an identification between \(Y_{k}\) and the set of (parametrized) 1periodic orbits of H that have covering multiplicity k over the underlying simple periodic orbit.
Definition 4.1
 \({\tilde{v}}\) satisfies Floer’s equation$$\begin{aligned} \partial _s {\tilde{v}} + J_Y(\partial _t {\tilde{v}}  X_H({\tilde{v}})) = 0; \end{aligned}$$(4.1)

\(\lim _{s \rightarrow \pm \infty } {\tilde{v}}(s,.) = x_{\pm }\);

if \(\Gamma \ne \varnothing \), then, for conformal parametrizations \(\varphi _i: (\infty ,0] \times S^1 \rightarrow \mathbb {R}\times S^1{\setminus }\{z_1, \ldots , z_k\}\) of neighbourhoods of the \(z_i\), \(\lim _{s\rightarrow \infty } {\tilde{v}}(\varphi _i(s,.)) = (\infty ,\gamma _i(.))\), where the \(\gamma _i\) are periodic Reeb orbits in Y;

for each Reeb orbit \(\gamma _i\) above, \(U_i: \mathbb {C}\rightarrow W\) is asymptotic to \((+\infty ,\gamma _i)\). We consider \(U_i\) up to the action of \({{\,\mathrm{Aut}\,}}(\mathbb {C})\).
See Figs. 4 and 5 for an illustration (ignore the horizontal segments in the figures, which represent gradient flow lines).
Definition 4.2

\({\tilde{v}}_1\) solves Eq. (4.1);

\({\tilde{v}}_0\) is \(J_W\)holomorphic;

\(\lim _{s \rightarrow +\infty } {\tilde{v}}_1(s,.) = x_+\);

\(\lim _{s \rightarrow \infty } {\tilde{v}}_1(s,.) = (\infty ,\gamma (\cdot ))\), for some Reeb orbit \(\gamma \) in Y;

\(\lim _{s \rightarrow +\infty } {\tilde{v}}_0(s,.) = (+\infty ,\gamma (\cdot ))\), where \(\gamma \) is the same Reeb orbit;

\(\lim _{s \rightarrow \infty } {\tilde{v}}_0(s,.) = x_\);

if \(\Gamma \ne \varnothing \), then, for conformal parametrizations \(\varphi _i: (\infty ,0] \times S^1 \rightarrow \mathbb {R}\times S^1{\setminus }\{z_1, \ldots , z_k\}\) of neighbourhoods of the \(z_i\), \(\lim _{s\rightarrow \infty } {\tilde{v}}(\varphi _i(s,.)) = (\infty , \gamma _i(.))\), where the \(\gamma _i\) are periodic Reeb orbits in Y;

for each Reeb orbit \(\gamma _i\) above, \(U_i: \mathbb {C}\rightarrow W\) is asymptotic to \((+\infty ,\gamma _i)\). We consider \(U_i\) up to the action of \({{\,\mathrm{Aut}\,}}(\mathbb {C})\).
See Fig. 6 for an illustration.
Definition 4.3
Consider a Hamiltonian \(H :\mathbb {R}\times Y \rightarrow \mathbb {R}\) so that \(\mathrm{d}H\) has support in \([R, \infty ) \times Y\), for some \(R \in \mathbb {R}\).
Let \(\vartheta _R\) be the set of all nondecreasing smooth functions \(\psi :\mathbb {R}\rightarrow [0, \infty )\) such that \(\psi (r) = {\text {e}}^r\) for \(r \geqslant R\).
Notice that this is equivalent to partitioning \(\mathbb {R}\times S^1 {\setminus } \Gamma = S_0 \cup S_1\), so that \(S_0 = {\tilde{v}}^{1} ( [R, +\infty ) \times Y )\) and \(S_1 = S {\setminus } S_0\). Then \({\tilde{v}}\) has finite hybrid energy if and only if \({\tilde{v}}_{S_0}\) has finite Floer energy and \({\tilde{v}}_{S_1}\) has finite Hofer energy.
We now define a split Floer cylinder with cascades between two generators of the chain complex (3.2).
Definition 4.4
Fix \(N \geqslant 0\). Let \(S_0, S_1, \dots , S_{N}\) be a collection of connected spaces of orbits, with \(S_0 = Y_{k_0}\) or \(S_0 = W\), and \(S_i = Y_{k_i}\) for \(1 \leqslant i \leqslant N\). Let \((f_i, Z_i)\), \(i=0, \dots , N\) be the pair of Morse function and gradientlike vector field of \(f_i = f_Y\), \(Z_i = Z_Y\) if \(S_i = Y_{k_i}\) and \(f_i = f_W\), \(Z_i=Z_W\) if \(S_i = W\).
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 \(\nu :\mathbb {R}\rightarrow S_0\), such that \(\nu (\infty ) = x\), \(\nu (+\infty ) = y\) and \({\dot{\nu }} = Z_0(\nu )\).
 (i)
\(N1\) length parameters \(l_i > 0, i=1, \dots , N1\);
 (ii)
two halfinfinite gradient trajectories, \(\nu _0 :(\infty , 0] \rightarrow S_0\) and \(\nu _{N} : [0, +\infty ) \rightarrow S_{N}\) with \(\nu _0(\infty ) = x\), \(\nu _N(+\infty ) = y\) and \({\dot{\nu }}_i = Z_i(\nu _i)\) for \(i=0\) or N;
 (iii)
\(N1\) gradient trajectories \(\nu _i\) defined on intervals of length \(l_i\), \(\nu _i :[0, l_i] \rightarrow S_i\) for \(i=1, \dots , N1\) such that \({\dot{\nu }}_i = Z_i(\nu _i)\);
 (iv)
N nontrivial split Floer cylinders from \(\nu _{i1}(l_{i1}) \in S_{i1}\) to \(\nu _{i}(0) \in S_i\), where we take \(l_0 = 0\).
Definition 4.5
We refer to the punctures \(\Gamma \) appearing in Definitions 4.1 and 4.2 as augmentation punctures. The corresponding \(J_W\) holomorphic planes, \(U_i :\mathbb {C}\rightarrow W\) 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.
Remark 4.6
Notice that the hybrid energy of each sublevel must be nonnegative. Since we require that the sublevels are nontrivial, it follows that any such cascade with collections of orbits \(S_i = Y_{k_i}\), \(i=1, \dots , N\) and \(S_0 = Y_{k_0}\), or, if \(S_0 = W\), with \(k_0 = 0\), we must have that the sequence of multiplicities is monotone increasing \(k_0< k_1< \dots < k_N\).
By a standard SFTtype compactness argument, the Floer–Gromov–Hofer compactification of a moduli space of split Floer cylinders with cascades will consist of several possible configurations. The length parameters can go to 0 or to \(\infty \) (in the latter case, corresponding to a Morsetype 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 \(\mathbb {R}\times Y\) and in W. We will see that this latter degeneration cannot occur in lowdimensional moduli spaces, at least under our monotonicity assumptions. For energy reasons, these Floer cylinders with cascades will not break at constant Hamiltonian trajectories in \((\infty , \log 2] \times Y\) (see [13, Lemma 4.8]).
For \(N \geqslant 1\), this moduli space \(\mathcal {M}_{H,N}(x,y; J_W)\) has an \(\mathbb {R}^{N}\) action by domain automorphisms corresponding to \(\mathbb {R}\)translation of the domain cylinders \(\mathbb {R}\times S^1\). When \(N = 0\), this moduli space is of gradient trajectories, and also admits an \(\mathbb {R}\) reparametrization action.
When \(x = y 1\), these moduli spaces will be rigid modulo these actions. See Remark 5.10.
5 Transversality for the Floer and holomorphic moduli spaces
In this section, we will build the transversality theory needed for the Floer cylinders with cascades that appear in the split Floer differential as in Eq. (4.3). In the process, we will also discuss transversality for pseudoholomorphic curves in X and in \(\Sigma \), 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 \(\Sigma \) and in X with various constraints.
Lemma 5.1
Let \({\tilde{v}} :\mathbb {R}\times S^1 {\setminus } \Gamma \rightarrow \mathbb {R}\times Y\) be a finite hybrid energy Floer cylinder in \(\mathbb {R}\times Y\) (as in Definition 4.3), converging to a Hamiltonian orbit in the manifolds \(Y_+\) at \(+\infty \) and converging at \(\infty \) either to a Hamiltonian orbit in the manifold \(Y_\) or to a Reeb orbit at \(\{\infty \} \times Y\), and with finitely many punctures at \(\Gamma \subset \mathbb {R}\times S^1\) converging to Reeb orbits in \(\{\infty \} \times Y\). Then the projection \(\pi _\Sigma \circ {\tilde{v}}\) extends to a smooth \(J_\Sigma \)holomorphic sphere \(\pi _\Sigma \circ {\tilde{v}} :{\mathbb {C}}{\mathbb {P}}^1 \rightarrow \Sigma \).
Proof
The projection \(\pi _\Sigma \circ {\tilde{v}}\) is \(J_\Sigma \)holomorphic on \(\mathbb {R}\times S^1\) 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 \(\pi _\Sigma \circ {\tilde{v}}\). \(\square \)
To describe the projection to \(\Sigma \) of the levels of a split Floer cylinder with N cascades that map to \(\mathbb {R}\times Y\), we introduce the following:
Definition 5.2

\(N \geqslant 0\) parametrized \(J_\Sigma \)holomorphic spheres \(w_i\) in \(\Sigma \) with two distinguished marked points at 0 and \(\infty \) and a possibly empty collection of additional marked points \(z_1, \dots , z_k\) on the union of the N domains (distinct from 0 or \(\infty \) in each of the N spherical domains); the spheres are either nonconstant or contain at least one additional marked point;

if \(N=0\), an infinite positive flow trajectory of \(Z_\Sigma \) from q to p; if \(N\ge 1\), a halfinfinite trajectory of \(Z_\Sigma \) from q to \(w_1(0)\), a halfinfinite trajectory of \(Z_\Sigma \) from \(w_N(\infty )\) to p;

if \(N \geqslant 1\), positive length parameters \(l_i, i=1, \dots , N1\), so that \(\varphi ^{l_i}_{Z_\Sigma }( w_i(\infty ) ) = w_{i+1}(0)\), \(i=1, \dots , N1\).
See Fig. 3. If such a chain of pearls is the projection to \(\Sigma \) of the components in \(\mathbb {R}\times Y\) 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 \(f_\Sigma \) admits an interpretation as a chain of pearls in \(\Sigma \) since the critical point is the image of any positive length flow line with that initial condition.
Definition 5.3

\(N \geqslant 1\) parametrized \(J_\Sigma \)holomorphic spheres \(w_i\) in \(\Sigma \) with two distinguished marked points at 0 and \(\infty \) and a possibly empty collection of additional marked points \(z_1, \dots , z_k\) on the union of the N domains (distinct from 0 or \(\infty \));

a parametrized nonconstant \(J_X\)holomorphic sphere v in X;

a halfinfinite trajectory of \(Z_\Sigma \) from \(w_N(\infty )\) to p, a halfinfinite trajectory of \(Z_X\) from x to v(0) (where \(Z_X\) is the pushforward of \(Z_W\) by the inverse of the map from Lemma 2.5);

positive length trajectories of \(Z_\Sigma \) from \(w_{i}(\infty )\) to \(w_{i+1}(0)\) for \(i=1, \dots , N1\);

the sphere in X touches the first sphere in \(\Sigma \): \(w_1(0) = v(+\infty )\);

the spheres \(w_2, \dots , w_N\) satisfy the stability condition that they are either nonconstant or contain at least one of the additional marked points (v is automatically nonconstant and \(w_1\) is allowed to be constant and unstable).
Definition 5.4

for each \(z \in {\mathbb {C}}{\mathbb {P}}^1\), \(U_i(z) \in \Sigma \) if and only if \(z = \infty \);

if the puncture \(z_i\) is in the domain of the holomorphic sphere \(w_{j_i} :{\mathbb {C}}{\mathbb {P}}^1 \rightarrow \Sigma \), then \(w_{j_i}(z_i) = U_i(\infty )\);

each \(U_i\) is considered up to the action of \({{\,\mathrm{Aut}\,}}({\mathbb {C}}{\mathbb {P}}^1,\infty ) = {{\,\mathrm{Aut}\,}}(\mathbb {C})\), that is, as an unparametrized sphere.
From Lemmas 5.1 and 2.5 and the fact that the trajectories of \(Z_Y\) cover trajectories of \(Z_\Sigma \), 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 2.5, if any of the sublevels have augmentation planes, then those correspond to spheres in X passing through \(\Sigma \) 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 \(\Sigma \) 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 nontrivial Floer cylinder \({\tilde{v}}_1\) contained in a single fibre of \(\mathbb {R}\times Y \rightarrow \Sigma \), and has the asymptotic limits \({\tilde{v}}_1(+\infty , t)\) on a Hamiltonian orbit and \({\tilde{v}}_1(\infty , t)\) on a closed Reeb orbit in \(\{ \infty \} \times Y\). The Floer cylinder \({\tilde{v}}_1\) in \(\mathbb {R}\times Y\) is nontrivial and hence stable, whereas the corresponding sphere \(w_1\) in \(\Sigma \) is unstable. Since we do not quotient by automorphisms (yet), this does not pose a problem. (See Fig. 6 and Proposition 6.3 below, where this situation is analysed.)
Definition 5.5
A chain of pearls in \(\Sigma \) is simple if each sphere is either simple (i.e. not multiply covered, [29, 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 \(\Sigma \) 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 \(\varphi (\mathcal {U})\) 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 v(0) 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 \(\Sigma \). 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 [29, 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 [29], there is no such condition on constant spheres.
Definition 5.8
A finite hybrid energy Floer cylinder with N cascades is simple if the projected chain of pearls is simple.
Proposition 5.9
There exists a residual set \({\mathcal {J}}^{reg}_W \subset {\mathcal {J}}_W\) of almost complex structures such that for each \(J_W \in {\mathcal {J}}^{reg}_W\), \(\mathcal {M}_{H,N}^*(x,y;J_W)\) is a manifold.
The two different formulas involving N reflect the fact that N counts the number of cylinders in \(\mathbb {R}\times Y\). 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 \(\mathbb {R}\), \(\mathbb {R}^{N}\) and \(\mathbb {R}^{N+1}\) actions) when the index difference is 1, which then justifies the definition of the differential given in Eq. (4.3). Indeed, observe that the case \(N=0\) corresponds to a pure Morse configuration and does not depend on any almost complex structure. We count rigid flow lines modulo the \(\mathbb {R}\) action, and thus require \(y  x = 1\). For generators x, y in \(\mathbb {R}\times Y\), we consider these N cylinders modulo the \(\mathbb {R}\) action on each one, giving an \(\mathbb {R}^N\) action. From this, a rigid configuration has \(yx+N1 = N\). For the case with \(x \in W\), we have \(N+1\) cylinders in the Floer cascade, so we have a rigid configuration modulo the \(\mathbb {R}^{N+1}\) action when \(N+1 = yx+N\).
The split Floer differential \(\partial \), introduced in Eq. (4.3) was defined by counting elements in \(\mathcal {M}_{H,N}(x,y;J_W)\). We will see in Propositions 6.2 and 6.3 that our monotonicity assumptions imply that this is equivalent to counting simple configurations in \(\mathcal {M}_{H,N}^*(x,y;J_W)\).

Section 5.2 describes the Fredholm setup for Floer cylinders with cascades. On a first reading, it can be skipped and used as a reference. In Sect. 5.2.1, we discuss the necessary function spaces and linear theory for the Morse–Bott problems. Then Sect. 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 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 \(\Sigma \).

Section 5.3 adapts the transversality arguments from [29] to obtain transversality for chains of pearls in \(\Sigma \).

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 \(\Sigma \), together with evaluation maps. The main technical point is an extension of the transversality results from [12].

Finally, Sect. 5.5 uses the splitting from Sect. 5.2.2 to lift the transversality results in \(\Sigma \) to obtain transversality for Floer cylinders with cascades, and to finish the proof of Proposition 5.9.
5.2 A Fredholm theory for Floer cylinders with cascades
5.2.1 A Fredholm theory for Morse–Bott asymptotics
In this section, we collect some facts about Cauchy–Riemann 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 [37]. Additional references include [1, Sections 2.1–2.3], [7, 25, 37, 41].
We begin by introducing some Sobolev spaces of sections of appropriate bundles. Let \(\Gamma \subset \mathbb {R}\times S^1\) be a finite set of punctures and denote \(\mathbb {R}\times S^1 {\setminus } \Gamma \) by \(\dot{S}\). Write \(\Gamma _+ = \{+\infty \}\) and \(\Gamma _ = \{\infty \} \cup \Gamma \). Consider, for each puncture \(z\in \Gamma \), exponential cylindrical polar coordinates of the form \((\infty , 1] \times S^1 \rightarrow \mathbb {R}\times S^1 {\setminus } \Gamma :\rho + i \eta \mapsto z_0 + \epsilon {\text {e}}^{2\pi (\rho + i \eta )}\). Choose \(\epsilon > 0\) sufficiently small; these are embeddings and that the image of these embeddings for any two different punctures are disjoint.
Let \(E \rightarrow \dot{S}\) be a (complex) rank n Hermitian vector bundle over \(\dot{S}\) together with a preferred set of trivializations in a small neighbourhood of \(\Gamma \cup \{\pm \infty \}\). While the bundle E over \(\dot{S}\) is trivial if there is at least one puncture, this is no longer the case once we specify these preferred trivializations near \(\Gamma \cup \{\pm \infty \}\). 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 \(\Lambda _\mathbb {C}^n E\). 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 \(\Lambda _\mathbb {C}^n E\) that is constant (with respect to the prescribed trivializations) near the punctures. We denote this Chern number by \(c_1(E)\), 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 \(\Gamma \) free to move on the domain \(\mathbb {R}\times S^1\). 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 \(2\#\Gamma \) parameter family of almost complex structures on \(\mathbb {R}\times S^1\), but fix the location of the punctures. Specify a fixed collection \(\Gamma \) of punctures on \(\mathbb {R}\times S^1\) 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 pushforward of the standard complex structure in \(\mathbb {R}\times S^1\) by the final map of the isotopy, to produce a family of complex structures on \(\mathbb {R}\times S^1\), which can be assumed standard near \(\Gamma \) and outside of a compact set.
For each \(z \in \Gamma \), let \(\beta _{z} :\dot{S} \rightarrow [0, +\infty )\) be a function supported in a small neighbourhood of z, with \(\beta _{z}(\rho , \eta ) = \rho \) near the puncture (where \((\rho , \eta )\) are cylindrical polar coordinates near z, as above). Similarly, let \(\beta _+ :\mathbb {R}\times S^1 \rightarrow [0, +\infty )\) be supported in a region where s is sufficiently large and \(\beta _+(s,t) = s\) for s large enough. Let \(\beta _ :\mathbb {R}\times S^1 \rightarrow [0, +\infty )\) have support near \(\infty \), and \(\beta _(s,t) = s\) for s sufficiently small.
Note that we obtain a path of symplectic matrices associated with the asymptotic operator \(\mathbf {A}_z\) by finding the fundamental matrix \(\Phi \) to the ODE \(\frac{\mathrm{d}}{\mathrm{d}t} x = J_z(t) A_z(t) x\). The asymptotic operator is nondegenerate if and only if the time1 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 [24, 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. To do so, we take the linearized flow map, which defines a path \(\Phi :[0,1] \rightarrow {{\,\mathrm{Sp}\,}}(2n)\) 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 specifies 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 \(\mathbf {A}_z\) is nondegenerate and E is a rank 1 vector bundle.
If \(u :S^1 \rightarrow \mathbb {C}\) is an eigenfunction of \(\mathbf {A}_z\) corresponding to the eigenvalue \(\lambda \), it must be nowhere vanishing. The winding number of u is then the degree of the map \(\frac{u}{u} :S^1 \rightarrow S^1\). Any two eigenfunctions corresponding to the same eigenvalue \(\lambda \) have the same winding number. This is then referred to as the winding number of the eigenvalue, and is denoted by \(w(\lambda )\).
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, [24, Theorem 3.1] to observe that \({\text {CZ}}( \mathbf {A}_z )\) 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
In particular, the \(\sigma (\mathbf {A}_0) = 2\pi \mathbb {Z}\) and the winding number of \(2\pi k\) is k.
Proof
Eigenvalues of \(\mathbf {A}_0\)
Eigenvalues of \(\mathbf {A}_C\) in increasing order, if \(C>0\)
Corollary 5.14
Proof
The case \(n=1\) follows from Proposition 5.12 and Lemma 5.13. The case of general n uses the additivity of \({\text {CZ}}\) under direct sums. \(\square \)
Definition 5.15
Notice that the operator \(D^\mathbf {\delta }\) depends on the choice of cutoff functions \(\beta _z, z \in \Gamma \cup \{ \pm \infty \}\). A different choice of cutoff function will give an operator that differs only by a compact operator. This is thus of secondary importance for what we discuss here.
Note that with the sign conventions that we have chosen, a positive weight \(\delta _z > 0\) always corresponds to the constraint of exponential decay at the puncture. A negative weight \(\delta _z < 0\) always corresponds to allowing exponential growth.
Theorem 5.16
Let \(\mathbf {\delta } :\Gamma \cup \{ \pm \infty \} \rightarrow \mathbb {R}\) such that \(\delta _z \notin \sigma ( \mathbf {A}_z )\) for positive punctures \(z \in \Gamma _+\) and such that \(+\delta _z \notin \sigma ( \mathbf {A}_z )\) for negative punctures \(z \in \Gamma _\).
This observation about the conjugation of the weighted operator to the nondegenerate case, combined with Riemann–Roch for punctured domains (see, for instance, [25, Theorem 2.8], [37, Theorem 3.3.11], [42, Theorem 5.4]) gives the following.
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
For a proof (using the spectral flow idea of [35]), see, for instance, [40, Proposition 4.5.22].
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.
In this paper, we are primarily concerned with Cauchy–Riemann operators defined on \(\dot{S} = \mathbb {R}\times S^1\) and on \(\dot{S} = \mathbb {R}\times S^1 {\setminus } \{ P \}\) (a cylinder with one additional negative puncture). In the case of \(\mathbb {R}\times S^1\), we will write \({\mathbf {V}} = (V_; V_+)\), and in the case of \(\mathbb {R}\times S^1 {\setminus } \{ P \}\), we will write \({\mathbf {V}} = (V_, V_P; V_+)\). (The negative punctures are enumerated first, and separated from the positive puncture by a semicolon.)
Theorem 5.18
Let \(\delta > 0\) be sufficiently small that, for each puncture \(z\in \Gamma \cup \{ \pm \infty \}\), \([\delta , \delta ] \cap \sigma ( \mathbf {A}_z) \subset \{0\}\). For each \(z \in \Gamma \), fix a subspace \(V_z \subset \ker \mathbf {A}_z\). Denote the collection of all \(V_z\) by \({\mathbf {V}}\).
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 \({\text {Ind}}(D)\) of \(\dim V_z\) or of \({\text {codim}}V_z\) will be the Morse index of the appropriate critical point. This motivates the following definition.
Definition 5.19
We first learned this result from Wendl [40]. 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 operator. Fix a puncture \(z_0 \in \Gamma \cup \{ \pm \infty \}\).
Let \(\mathbf {\delta }\) and \(\mathbf {\delta '}\) be vectors of sufficiently small weights so that the differential operator induces a Fredholm operator on \(W^{1,p,\mathbf {\delta }}\) and on \(W^{1,p,\mathbf {\delta '}}\), and \(\delta _{z_0} > 0 \) and \(\delta _{z_0}' < 0\), the interval \([\delta _{z_0}', \delta _{z_0}] \cap \sigma (\mathbf {A}_{z_0}) = \{ 0 \}\), and for each \(z \in \Gamma \cup \{ \pm \infty \}\) with \(z \ne z_0\), the weights \(\delta _z = \delta '_z\).
Let \(\mathbf {V}\) be the trivial vector space at each puncture other than \(z_0\) and let \(V_{z_0}\) be the kernel of the asymptotic operator \(\mathbf {A}_{z_0}\) at \(z_0\).
Proof
The main idea of the lemma is contained in [40, Proposition 4.5.22], which contains a proof of the equality of Fredholm indices. See also the very closely related [43, Proposition 3.15].
Note that \(W^{1,p, \mathbf \delta }_{\mathbf {V}}( \dot{S}, E) \) is a subspace of \(W^{1,p, \mathbf \delta '}(\dot{S}, E)\), and thus the kernel of \(D_\delta \) is contained in the kernel of \(D_{\delta '}\).
Now, by a linear version of the analysis done in [25, 39], any element of the kernel of \(D_{\delta '}\) 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 \(D_{\delta '}\) must converge exponentially fast to an element of the kernel of the asymptotic operator at \(z_0\). Hence, the kernel of \(D_{\delta '}\) is contained in the kernel of \(D_{\delta }\).
We conclude that the kernels of the two operators may be identified. Since their Fredholm indices are the same, their cokernels are also isomorphic.
\(\square \)
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 solution \({\tilde{v}} :\mathbb {R}\times S^1 {\setminus } \Gamma \rightarrow \mathbb {R}\times Y\), we consider exponentially weighted Sobolev spaces of sections of the pullback bundle \({\tilde{v}}^*T( \mathbb {R}\times Y)\) since the asymptotic limits are (Morse–Bott) degenerate. For \(\delta > 0\), we denote by \(W^{1,p,\delta }(\mathbb {R}\times S^1 {\setminus } \Gamma , v^*T(\mathbb {R}\times Y))\) the space of sections that decay exponentially like \({\text {e}}^{\delta s}\) near the punctures (also in cylindrical coordinates near the punctures \(\Gamma \)), as in the previous section.
We similarly define \(W^{m,p,\delta }\) sections with exponential decay/growth. The following results will not depend on m except in the case of jet conditions considered in Sect. 5.4, where m will need to be sufficiently large that the order of contact condition can be defined.
To consider a parametric family of punctured cylinders in which the asymptotic limits move in their Morse–Bott families, we let \({\mathbf {V}}\) be a collection of vector spaces, associating with each puncture \(z \in \Gamma \cup \{ \pm \infty \}\) a vector subspace \(V_z\) of the tangent space to the corresponding Morse–Bott family of orbits. For \(\delta >0\), we then consider the space of sections \(W^{1,p,\delta }_{{\mathbf {V}}}( \mathbb {R}\times S^1 {\setminus } \Gamma , v^*T(\mathbb {R}\times Y))\) 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 \(\delta \) has to be. It is worth pointing out that there is no value of \(\delta \) that works for all moduli spaces. The reason is that we need \(\delta \) 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 \(\frac{1}{2}\left( C + \sqrt{C^2 + 16 \pi ^2} \right) \) becomes arbitrarily small as \(C \rightarrow \infty \). As will become clear from Lemma 5.22 and Eq. (5.5), the relevant value for C here is \(h''(e^{b_k}) e^{b_k}\), which can become arbitrarily large as the multiplicity \(k \rightarrow \infty \). Since the relevant moduli spaces in the differential involve connecting orbits of bounded multiplicities, for any given moduli space, we may choose \(\delta \) sufficiently small.
We now adapt an observation first used in [6, 15] to show that the linearization of the Floer operator is upper triangular with respect to the splitting of \(T(\mathbb {R}\times Y)\) as \((\mathbb {R}\oplus \mathbb {R}R) \oplus \xi \). We then describe the nonzero blocks in this upper triangular presentation of the operator. The two diagonal terms are of special importance: one will be a Cauchy–Riemann 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 \(\Sigma \).
We now explain this construction in more detail. Let \({\tilde{v}} :\mathbb {R}\times S^1 {\setminus } \Gamma \rightarrow \mathbb {R}\times Y\) be a Floer solution with punctures \(\Gamma \). 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, \({\tilde{v}}\) is asymptotic to a closed Hamiltonian orbit at \({\tilde{v}}(+\infty , t)\), to a closed Hamiltonian orbit at \({\tilde{v}}(\infty , t)\), and with negative ends converging to Reeb orbits at the punctures in \(\Gamma \). The second case has \({\tilde{v}}\) asymptotic to a closed Hamiltonian orbit at \({\tilde{v}}(+\infty , t)\), but with negative ends converging to Reeb orbits in \(\{\infty \} \times Y\) at \(\{ \infty \} \cup \Gamma \). 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 \(\mathbb {R}\times Y\): \({\tilde{v}}\) has a positive cylindrical end at \(+\infty \) converging to a Reeb orbit in \(\{+\infty \} \times Y\), and has negative cylindrical ends at the punctures \(\{ \infty \} \cup \Gamma \). 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 [13]. Part of this was sketched in [16, Section 2.9.2].
Let \(\mathbf {V}\) associate with each puncture \(z \in \Gamma \cup \{ \pm \infty \}\) the tangent space to Y if the corresponding limit of \({\tilde{v}}\) is a closed Hamiltonian orbit and the tangent space to \(\mathbb {R}\times Y\) if the corresponding limit of \({\tilde{v}}\) is a closed Reeb orbit. As will be clearer shortly, this is associating with each puncture the entirety of the kernel of the corresponding asymptotic operator.
Lemma 5.22
Proof
Remark 5.23
Notice that for each puncture \(z \in \{ \pm \infty \} \cup \Gamma \), if \(\gamma (t)\) denotes the corresponding asymptotic Hamiltonian or Reeb orbit, the previous result allows us to identify \(V_z\) with \(T_{\gamma (0)} Y\) at a Hamiltonian orbit and with \(\mathbb {R}\times T_{\gamma (0)} Y\) at a Reeb orbit.
Lemma 5.24
Let \({\tilde{v}} :\mathbb {R}\times S^1 {\setminus } \Gamma \rightarrow \mathbb {R}\times Y\) be a finite hybrid energy Floer cylinder with punctures \(\Gamma \).
Then the operator \(D^\mathbb {C}_{{\tilde{v}}}\) defined in Eq. (5.5) is Fredholm for \(\delta > 0\) sufficiently small.
If, instead, \({\tilde{v}}\) converges at \(+\infty \) to a closed Hamiltonian orbit, and at \(\infty \) to a closed Reeb orbit in \(\{\infty \} \times Y\), then \(D^\mathbb {C}_{{\tilde{v}}}\) has Fredholm index 2 and is surjective.
Finally, if \({\tilde{v}}\) converges at \(\pm \infty \) to closed Reeb orbits in \(\{ \pm \infty \} \times Y\), then \(D^\mathbb {C}_{{\tilde{v}}}\) has Fredholm index 2 and is surjective.
In all three cases, the kernel of \(D^\mathbb {C}_{{\tilde{v}}}\) 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 \(\pm \delta \)perturbed operators.
Recall from Remark 5.21 that we have \(\delta  > 0\) smaller than the spectral gap for any of these punctures.
For the case of exponential decay, Corollary 5.14 then gives the Conley–Zehnder index of 0 for \(\mathbf {A}_+ + \delta \) and of 1 for \(\mathbf {A}_  \delta \).
In the case of exponential growth, Corollary 5.14 gives instead that the Conley–Zehnder index of \(\mathbf {A}_+  \delta \) is 1 and that of \(\mathbf {A}_ + \delta \) is 0.
As above, in the case of exponential decay, the relevant asymptotic operators are \(i\frac{\mathrm{d}}{\mathrm{d}t} +\delta \) at a positive puncture and \(i\frac{\mathrm{d}}{\mathrm{d}t} \delta \) at a negative puncture. Again, by Corollary 5.14, we obtain a Conley–Zehnder index of \(1\) at \(+\infty \) and a Conley–Zehnder indices of 1 at a negative puncture (\(\infty \) or \(P \in \Gamma \)).
If, instead, we consider exponential growth, we obtain Conley–Zehnder indices of \(+1\) at positive punctures and \(1\) at negative punctures.
It follows immediately from the expression for \(D^\mathbb {C}_{{\tilde{v}}}\) that the constant i is in the kernel. Recalling that \(\mathbb {C}= {\tilde{v}}^*(\mathbb {R}\oplus {\mathbb {R}}R)\) in the splitting given by Lemma 5.22, we then may identify this constant with the Reeb vector field R. \(\square \)
To summarize the results of this section, by Lemma 5.22, a punctured Floer cylinder in \(\mathbb {R}\times S^1\) is regular if the operators \(D^\mathbb {C}_{{\tilde{v}}}\) and \(\dot{D}^\Sigma _w\) are surjective. Surjectivity of the latter is equivalent to surjectivity of \(D^\Sigma _w\). Lemma 5.24 gives the surjectivity of \(D^\mathbb {C}_{\tilde{v}}\). It thus remains to study transversality for \(D^\Sigma _w\), specifically with respect to the evaluation maps that will allow us to define the moduli spaces of chains of pearls in \(\Sigma \) (see Sect. 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 \(\Sigma \) (see Sect. 5.4).
5.3 Transversality for chains of pearls in \(\Sigma \)
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
To apply the Sard–Smale Theorem, we need to consider Banach spaces of almost complex structures, so we let \({\mathcal {J}}^r_\Sigma , {\mathcal {J}}^r_W\) be the space of \(C^r\)regular almost complex structures otherwise satisfying the conditions of being in \({\mathcal {J}}_\Sigma \), \({\mathcal {J}}_W\). We impose \(r \geqslant 2\) and in general will require r to be sufficiently large that the Sard–Smale theorem holds (this will depend on the Fredholm indices associated with the collection of homology classes and will also depend on the order of contact to \(\Sigma \) 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 \(\mathcal {M}_{k, \Sigma }^*((A_1, \dots , A_N), {\mathcal {J}}^r_\Sigma )\) the moduli space of pairs \(((w_i)_{i=1}^N, J_\Sigma )\) with \(J_\Sigma \in {\mathcal {J}}^r_\Sigma \) and \((w_i)_{i=1}^N \in \mathcal {M}_{k, \Sigma }^*( (A_1, \dots , A_N), J_\Sigma )\).
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 [29, 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 \(\Delta ^E\) 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 [12] (see Sect. 5.4), and an additional technical transversality point needed to be able to consider the lifted problem in \(\mathbb {R}\times Y\).
Proposition 5.26
Proposition 5.27
[29, Proposition 6.2.7] \(\mathcal {M}_{k, \Sigma }^*((A_1, \dots , A_N); {\mathcal {J}}^r_\Sigma )\) is a Banach manifold.
We will also make use of the following definition and proposition, the latter of which we prove in the next section.
Definition 5.28
Proposition 5.29
Recall that we have chosen a Morse function \(f_\Sigma :\Sigma \rightarrow \mathbb {R}\) and a corresponding gradientlike vector field \(Z_\Sigma \), such that \((f_\Sigma ,Z_\Sigma )\) is a Morse–Smale pair. The timet flow of \(Z_\Sigma \) is denoted by \(\varphi ^t_{Z_\Sigma }\) and the stable (ascending) \(W^s_\Sigma (q)\) and unstable (descending) manifolds \(W^u_\Sigma (p)\) were defined in Eq. (3.1). (Note that these are the stable/unstable manifolds for the negative gradient flow.)
Definition 5.30
We will now establish transversality of the evaluation maps to appropriate products of stable/unstable manifolds, critical points, diagonals and flow diagonals. By [29, 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.
Lemma 5.31
Suppose \(f_0 :\mathcal {M}_0 \rightarrow \Sigma \) and \(f_1 :\mathcal {M}_1 \rightarrow \Sigma \) are submersions.
Proof
Suppose \(F(m_0, m_1) = (x, p, p) \in \Delta _{f_\Sigma } \times \{ p \}\). Then there exists t so that \( \phi ^t_{Z_\Sigma } ( x ) = \phi ^t_{Z_\Sigma } ( f_0(m_0) ) = f_1(m_1) = p\).
From this, we now obtain the following:
Lemma 5.32
Proof
We apply the previous lemma, using \(f_0 = {\text {ev}}_+\) and \(f_1 = e\). Then \({\hat{{\text {ev}}}}(m, n) = ({\text {ev}}_(m), F(m, n))\). The transversality to \(A \times \Delta _{f_\Sigma } \times \text {pt}\) follows by the transversality of F to \(\Delta _{f_\Sigma } \times \text {pt}\) together with the transversality of \({\text {ev}}_\) to A. \(\square \)
Lemma 5.33
Let \(N \geqslant 1\), and let \(A_1, \dots , A_N\) be spherical homology classes in \(\Sigma \) and let B be a spherical homology class in X.
Suppose that \(S \subset \Sigma ^{2N2}\) is obtained by taking the product of some number of copies of \(\Delta _{f_\Sigma } \subset \Sigma ^2\) and of the complementary number of copies of \(\{ (p, p) \,  \, p \in {{\,\mathrm{Crit}\,}}(f_\Sigma ) \} \subset \Sigma ^2\), in arbitrary order. Let \(\Delta \subset \Sigma ^2\) denote the diagonal.
Proof
We consider the case of \(\mathcal {M}_{k, \Sigma }^*\) in detail since the argument is essentially the same for \(\mathcal {M}_{k, (X, \Sigma )}^*\), though notationally more cumbersome.
Suppose that \(((v_1, \dots , v_N), J) \in \mathcal {M}_{k, \Sigma }^*((A_1, \dots , A_N); {\mathcal {J}}^r_\Sigma )\) is in the preimage of \(\{x\} \times S \times \{y\}\). Write \(S = S_1 \times S_2 \times \dots \times S_{N1}\), where each \(S_i \subset \Sigma ^2\) 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, \(1< i < N\), we must have that \(S_{i1}\) and \(S_{i}\) are flow diagonals. If \(v_1\) is constant, then \(S_1\) is a flow diagonal and if \(v_N\) is constant, \(S_{N1}\) is a flow diagonal.
We will proceed by induction on N. The case \(N =1\) follows from [29, Proposition 3.4.2].
Now, for the inductive argument, we suppose the result holds for any \(S \subset \Sigma ^{2(N1)2}\) of the form specified, and for any \(k \geqslant 0\), for any collection of \(N1\) spherical classes, not all of which are zero.
If \(A_N \ne 0\), the result follows again from [29, Proposition 3.4.2].
If, instead, \(A_N = 0\), we have from above that \(S_{N1} = \Delta _{f_\Sigma }\). Notice that the evaluation map of constant spheres on \(\Sigma \) has image on the diagonal in \(\Sigma \times \Sigma \). 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 \(X \times \Sigma \) now follows from Proposition 5.29, and the induction proceeds as before. \(\square \)
Proposition 5.34
Let \(N \geqslant 0\). Suppose that \(S \subset \Sigma ^{2N2}\) is obtained by taking the product of some number of copies of \(\Delta _{f_\Sigma } \subset \Sigma ^2\) and of the complementary number of copies of \(\{ (p, p) \,  \, p \in {{\,\mathrm{Crit}\,}}(f_\Sigma ) \} \subset \Sigma ^2\), in arbitrary order.
Let \(\Delta \subset \Sigma \times \Sigma \) denote the diagonal and let \(\Delta _k\) denote the diagonal \(\Sigma ^k\) in \(\Sigma ^k \times \Sigma ^k\).
Let p, q be critical points of \(f_\Sigma \) and let x be a critical point of \(f_W\).
Proof
The proposition follows immediately if at least one of the \(A_i, i=1, \dots , N\) is nonzero, or if we are considering the case of a chain of pearls with a sphere in X, by applying Lemma 5.33.
Proposition 5.34 can be combined with standard Sard–Smale arguments, the fact that \(P:{\mathcal {J}}^r_W \rightarrow {\mathcal {J}}^r_\Sigma \) is an open and surjective map and Taubes’s method for passing to smooth almost complex structures (see for instance [29, Theorem 6.2.6]) to give the following proposition:
Proposition 5.35
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 \(\Sigma \)
We will now consider transversality for a chain of pearls with a sphere in X. We will extend the results from Section 6 in [12]. In that paper, Cieliebak and Mohnke prove that the moduli space of simple curves not contained in \(\Sigma \), with a condition on the order of contact with \(\Sigma \), can be made transverse by a perturbation of the almost complex structure away from \(\Sigma \). We will extend this result to show that additionally the evaluation map to \(\Sigma \) 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 \(\Sigma \).
Recall that \(\Sigma \) is a symplectic divisor and \(N\Sigma \) is its symplectic normal bundle equipped with a Hermitian structure. Keeping in mind the discussion in Sect. 2 (in particular the identification of \(X{\setminus } \Sigma \) with W in Lemma 2.5), we will by an abuse of notation identify an almost complex struture on W with the corresponding almost complex structure on X. We have fixed a symplectic neighbourhood \(\varphi :\mathcal {U}\rightarrow X\) where \(\varphi :\overline{\mathcal {U}} \rightarrow X\) is an embedding. From Definition 2.7, we require that all \(J_X \in {\mathcal {J}}_W\) have that \(J_X\) is standard in the image \(\varphi (\mathcal {U}) \subset X\) of this neighbourhood.
Fix an almost complex structure \(J_0 \in {\mathcal {J}}_W\). We may suppose that \(P(J_0) \in {\mathcal {J}}_\Sigma \) is an almost complex structure in the residual set \({\mathcal {J}}_\Sigma ^{reg}\) given by Proposition 5.26, though this is not strictly speaking necessary.
Let \(\mathcal {V}:=X {\setminus } \varphi (\overline{\mathcal {U}})\). Following Cieliebak–Mohnke [12], let \({\mathcal {J}}(\mathcal {V})\) be the set of all almost complex structures on X compatible with \(\omega \) that are equal to \(J_0\) on \(\varphi (\mathcal {U})\). Similarly, we will let \({\mathcal {J}}^r(\mathcal {V})\) be the compatible almost complex structures of \(C^r\) regularity.
To define the order of contact, consider an almost complex structure \(J_X \in {\mathcal {J}}_W\) and a \(J_X\)holomorphic sphere \(f :{\mathbb {C}}{\mathbb {P}}^1 \rightarrow X\) with \(f(0) \in \Sigma \), an isolated intersection. Choose coordinates \(s+it = z\in \mathbb {C}\) on the domain and local coordinates near \(f(0)\in \Sigma \) on the target, such that \(f(0)\in \Sigma \subset X\) corresponds to \(0\in \mathbb {C}^{n1} = \mathbb {C}^{n1}\times \{0\} \subset \mathbb {C}^{n1} \times \mathbb {C}\). Write \(\pi _\mathbb {C}:\mathbb {C}^n\rightarrow \mathbb {C}\) for projection onto the last coordinate (which is to be thought of as normal to \(\Sigma \)). Assume also that \(J_X(0) = i\). Then f has contact of order l at 0 if the vector of all partial derivatives of orders 1 through l (denoted by \(d^l f(0)\)) has trivial projection to \(\mathbb {C}\). We can write this condition as \(d^l f(0) \in T_{f(0)}\Sigma \). We define then the order of contact at an arbitrary point in \({\mathbb {C}}{\mathbb {P}}^1\) by precomposing with a Möbius transformation. (This is well defined, by [12, Lemma 6.4].)
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.
In this section, we will prove Proposition 5.29, which was stated and used above:
Notice that it suffices to prove this when considering only pairs \((f, J_X) \in \mathcal {M}^*_X( (B_0); {\mathcal {J}}^r_W)\) with the additional condition that \(J_X \in {\mathcal {J}}^r(\mathcal {V})\).
We also observe that if \(l = B_0 \bullet \Sigma \), we have that \(\mathcal {M}^*_X((B_0); {\mathcal {J}}_W) \subset \mathcal {M}^*_{\infty , k, (X, \Sigma )}({\mathcal {J}}_W)\) for each \(k \leqslant l\). Furthermore, \(\mathcal {M}^*_X((B_0); {\mathcal {J}}_W)\) is a connected component of \(\mathcal {M}^*_{\infty , l, (X, \Sigma )}({\mathcal {J}}_W)\). 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 [12, Section 6]. Instead of reproducing their proof, we indicate the necessary modifications. To be as consistent as possible with their notation, we consider the point of contact with \(\Sigma \) to be at 0.
Consider a \(J_X\)holomorphic map \(f :{\mathbb {C}}{\mathbb {P}}^1\rightarrow X\) such that \(f(0)\in \Sigma \) with order of contact l. In the notation of [12], we are interested in the case of only one component \(Z=\Sigma \). 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, \(\mathcal {V}= X {\setminus } \varphi ({\overline{\mathcal {U}}})\).
We need the following adaptation of Corollary 6.2 in [12].
Lemma 5.36
Suppose \((f, J_X) \in \mathcal {M}^*_{\infty , l, (X, \Sigma )}({\mathcal {J}}^r_W)\) with \(J_X \in {\mathcal {J}}^r(\mathcal {V})\), \(r \geqslant m\).
After choosing local coordinates, suppose \(f(0) \in \Sigma \) and in coordinates around f(0), \(\Sigma \) is mapped to \(\mathbb {C}^{n1}\) and is thus preserved by \(J_X\).
Denote the unit disk by \(D^2\) and let \(\xi :(D^2,0) \rightarrow (\mathbb {C}^n,0)\) be such that \(D_f\xi = 0\). Given \(0<k\le l\), if \(\xi (0)\in \mathbb {C}^{n1}\), \(d^{k1}\xi (0) \in \mathbb {C}^{n1}\) and \(\frac{\partial ^k \xi }{ \partial s^k}(0) \in \mathbb {C}^{n1}\), then \(d^{k}\xi (0)\in \mathbb {C}^{n1}\).
Proof
We now prove the key property of the linearized evaluation map:
Proposition 5.37
Proof
Observe now that by combining this with standard arguments (see, for instance, [29, 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 \(Y \times Y\).
Recall from Definition 2.7 that \({\mathcal {J}}_Y\) denotes the space of compatible, cylindrical, Reeb–invariant almost complex structures on \(\mathbb {R}\times Y\). These are obtained as lifts of the almost complex structures in \({\mathcal {J}}_\Sigma \). Let \({\mathcal {J}}_Y^{reg}\) be the set of almost complex structures on \(\mathbb {R}\times Y\) that are lifts of the almost complex structures in \({\mathcal {J}}_\Sigma ^{reg}\) (see Proposition 5.35).
Recall from Definition 2.7 and from Proposition 2.3, if \(J_W \in {\mathcal {J}}_W\) is an almost complex structure on W that is of the type we consider, it induces an almost complex structure \(P(J_W) = J_\Sigma \in {\mathcal {J}}_\Sigma \). The restriction of \(J_W\) to the cylindrical end of W, \(J_Y\), is then a translation and Reebflow invariant almost complex structure on \(\mathbb {R}\times Y\) that has \(\mathrm{d}\pi _\Sigma J_Y = J_\Sigma \mathrm{d}\pi _\Sigma \).
Recall that the biholomorphism \(\psi :W \rightarrow X {\setminus } \Sigma \) given in Lemma 2.5 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 nonnegative integer m, there exists a unique \(b_m\) so that \(h'({\text {e}}^{b_m}) = m\). Then \(Y_m = \{ b_m \} \times Y \subset \mathbb {R}\times Y\) is the corresponding Morse–Bott family of 1periodic Hamiltonian orbits that wind m times around the fibre of \(Y \rightarrow \Sigma \).
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 to the differential connecting two generators in \(\mathbb {R}\times Y\). Then, we will define the moduli spaces relevant to the differential connecting to a critical point in W.
Definition 5.38
Let \(N \geqslant 1\), let \(A_1, \dots , A_N \in H_2(\Sigma ; \mathbb {Z})\) be spherical homology classes. Let \(J_Y \in {\mathcal {J}}_Y\).
 (1)There is a partition of \(\Gamma = \Gamma _1 \cup \dots \cup \Gamma _N\) of k augmentation marked points withso that \({\tilde{v}}_i\) is a finite hybrid energy punctured Floer cylinder. For each \(z_j \in \Gamma \), there is a positive integer multiplicity \(k(z_j)\). Let \(v_i\) denote the projection to Y.$$\begin{aligned} {\tilde{v}}_i :\mathbb {R}\times S^1 {\setminus } \Gamma _i \rightarrow \mathbb {R}\times Y \end{aligned}$$
 (2)There is an increasing list of \(N+1\) multiplicities from \(k_\) to \(k_+\):such that, for each i, the cylinder \({\tilde{v}}_i\) has multiplicities \(k_{i}\) and \(k_{i1}\) at \(\pm \infty \), i.e. \({\tilde{v}}_i(+\infty , \cdot ) \in Y_{k_i}, {\tilde{v}}_i(\infty , \cdot ) \in Y_{k_{i 1}}\).$$\begin{aligned} k_ = k_0< k_1< k_2< \dots < k_N = k_+ \end{aligned}$$
 (3)
The Floer cylinders \({\tilde{v}}_i\) are simple in the sense that their projections to \(\Sigma \) 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.
 (4)
For each i, and for every puncture \(z_j \in \Gamma _i\), the augmentation puncture has a limit whose multiplicity is given by \(k(z_j)\); i.e. \(\lim _{\rho \rightarrow \infty } v_i(z_j + {\text {e}}^{2\pi (\rho + i\theta )})\) is a Reeb orbit of multiplicity \(k(z_j)\).
 (5)
The projections of the Floer cylinders to \(\Sigma \) represent the homology classes \(A_i, i=1, \dots , N\); i.e. \((\pi _\Sigma ( {\tilde{v}}_i ) )_{i=1}^N \in \mathcal {M}^*_k( (A_1, \dots , A_N), J_\Sigma )\).
Let \(B \in H_2(X; \mathbb {Z})\) be a spherical homology class, \(B \ne 0\). Let \(J_W\) be an almost complex structure on W as given by Lemma 2.5, matching \(J_Y\) on the cylindrical end.
Definition 5.39
 (1)
The map \({\tilde{v}}_0 :\mathbb {R}\times S^1 \rightarrow W\) is a finite energy holomorphic cylinder with removable singularity at \(\infty \).
 (2)There is a partition of \(\Gamma = \Gamma _1 \cup \dots \cup \Gamma _N\) of k augmentation marked points withso that each \({\tilde{v}}_i\) is a finite hybrid energy punctured Floer cylinder. For each \(z_j \in \Gamma \), there is a positive integer multiplicity \(k(z_j)\). Denote by \(v_i\) the projection of \({\tilde{v}}_i\) to Y.$$\begin{aligned} {\tilde{v}}_i :\mathbb {R}\times S^1 {\setminus } \Gamma _i \rightarrow \mathbb {R}\times Y, \qquad i\ge 1, \end{aligned}$$
 (3)There is an increasing list of \(N+1\) multiplicities:$$\begin{aligned} k_0< k_1< k_2< \dots < k_N = k_+. \end{aligned}$$
 (4)
For each \(i\ge 1\), and for every puncture \(z_j \in \Gamma _i\), the augmentation puncture has a limit whose multiplicity is given by \(k(z_j)\), i.e. \(\lim _{\rho \rightarrow \infty } v_i(z_j + {\text {e}}^{2\pi (\rho + i\theta )})\) is a Reeb orbit of multiplicity \(k(z_j)\).
 (5)
The Floer cylinders \({\tilde{v}}_i\) for \(i\ge 1\) are simple, in the strong sense that the projections to \(\Sigma \) are somewhere injective or constant, and have images not contained one in the other. The cylinder \({\tilde{v}}_0\) is somewhere injective in W.
 (6)
The projections of the Floer cylinders to \(\Sigma \) represent the homology classes \(A_i, i=1, \dots , N\); i.e. \(\pi _\Sigma ( {\tilde{v}}_i ) )_{i=1}^N \in \mathcal {M}^*_k( (B; A_1, \dots , A_N), J_W)\).
 (7)
After identifying \({\tilde{v}}_0\) with a holomorphic sphere in X, \({\tilde{v}}_0\) represents the homology class \(B \in H_2(X; \mathbb {Z})\).
 (8)
The cylinder \({\tilde{v}}_1\) has multiplicity \(k_1\) at \(+\infty \) and \(\tilde{v}_1(+\infty , \cdot ) \in Y_{k_1}\). At \(\infty \), \({\tilde{v}}_1\) converges to a Reeb orbit in \(\{ \infty \} \times Y\). This Reeb orbit has multiplicity \(k_0\).
 (9)
For each \(i \geqslant 2\), the cylinder \({\tilde{v}}_i\) has multiplicities \(k_{i}\) and \(k_{i1}\) at \(\pm \infty \): \({\tilde{v}}_i(+\infty , \cdot ) \in Y_{k_i}, {\tilde{v}}_i(\infty , \cdot ) \in Y_{k_{i 1}}\).
 (10)
The cylinder \({\tilde{v}}_0\) converges at \(+\infty \) to a Reeb orbit of multiplicity \(k_0\).
Identifying holomorphic spheres in X with finite energy \(J_W\)planes in W, we consider also the moduli space of holomorphic planes \(\mathcal {M}^*_X( (B_1, \dots , B_k); J_W)\) as in Definition 5.25.
The space \({\mathcal {M}}_{H, k, \mathbb {R}\times Y}^*\left( (A_1,\ldots ,A_N);J_Y\right) \) consists of Ntuples of somewhere injective punctured Floer cylinders in \(\mathbb {R}\times Y\). Similarly, \({\mathcal {M}}_{H, k, W}^*\) consist of Ntuples of punctured Floer cylinders in \(\mathbb {R}\times Y\) 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, \({\mathcal {M}}_{H, k, \mathbb {R}\times Y}^*\) and \({\mathcal {M}}_{H, k, W}^*\) fail to be simple split Floer cylinders with 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. To impose these conditions, we will need to study these evaluation maps and establish their transversality.
Proposition 5.40
Proof
Let \(\delta > 0\) be sufficiently small. For each \(i=1, \dots , N\), by Lemma 5.24, \(D_{{\tilde{v}}_i}^\mathbb {C}\) is surjective when considered on \(W^{1,p,\delta }\) (with exponential growth), and has Fredholm index 1. The operator considered instead on the space \(W^{1,p,\delta }_{{\mathbf {V}}}\), with \(V_{\infty } = V_{+\infty } = i\mathbb {R}\) and \(V_P=\mathbb {C}\) for any puncture P on the domain of \({\tilde{v}}_i\), 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 \(D_{{\tilde{v}}_i}\) 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 component \({\tilde{v}}_i\) contributes an index of \(1 + 2n2 + 2 \, \langle c_1(T\Sigma ), A_i \rangle + 2k_i = 2n1 + 2 \, \langle c_1(T\Sigma ), A_i \rangle + 2k_i\), where \(k_i\) is the number of punctures.
Let \({\tilde{p}}, {\tilde{q}} \in Y\) be critical points of \(f_Y\) and let \(W_Y^u({\tilde{p}})\), \(W_Y^s({\tilde{q}})\) be the unstable/stable manifolds of \({\tilde{p}}, {\tilde{q}}\), as in (3.1).
Proposition 5.41
Let \(J_W \in {\mathcal {J}}_W^{reg}\) and let \(J_Y \in {\mathcal {J}}_Y^{reg}\) be the induced almost complex structure on \(\mathbb {R}\times Y\).
Let \({\tilde{q}}, {\tilde{p}}\) denote critical points of \(f_Y\), and let x be a critical point of \(f_W\) in W. Let \(k_+\) and \(k_\) be nonnegative multiplicities, \(k_+ > k_\).
Let \(A_1, \dots , A_N\) be spherical homology classes in \(\Sigma \), let \(B, B_1, \dots , B_k\) be spherical homology classes in X, \(k \geqslant 0\).
Let \({\tilde{\Delta }} \subset Y \times Y\) and \(\Delta _{\Sigma ^k} \subset \Sigma ^k \times \Sigma ^k\) be the diagonals.
 (1)the evaluation mapis transverse to the submanifold$$\begin{aligned}&{\widetilde{{\text {ev}}}}_{Y} \times {\widetilde{{\text {ev}}}}_\Sigma ^a \times {\text {ev}}^a_\Sigma :{\mathcal {M}}_{H, k, \mathbb {R}\times Y; k_, k_+}^*\left( (A_1,\ldots ,A_N);J_Y\right) \\&\quad \times \mathcal {M}^*_X( (B_1, \dots , B_k); J_W) \rightarrow Y^{2N} \times \Sigma ^k \times \Sigma ^k \end{aligned}$$$$\begin{aligned} W_Y^s({\tilde{q}}) \times \left( {\tilde{\Delta }}_{f_Y} \right) ^{N1} \times W^u_{Y}({\tilde{p}}) \times \Delta _{\Sigma ^k}; \end{aligned}$$
 (2)the evaluation mapis transverse to the submanifold$$\begin{aligned}&{\widetilde{{\text {ev}}}}_{W,Y} \times {\widetilde{{\text {ev}}}}_\Sigma ^a \times {\text {ev}}^a_\Sigma :{\mathcal {M}}_{H, k, W; k_+}^*\left( (B; A_1,\ldots ,A_N);J_W\right) \\&\quad \times \mathcal {M}^*_X( (B_1, \dots , B_k); J_W) \rightarrow W \times Y^{2N+1} \times \Sigma ^{k} \times \Sigma ^k \end{aligned}$$$$\begin{aligned} W_W^u(x) \times {\tilde{\Delta }} \times \left( {\tilde{\Delta }}_{f_Y} \right) ^{N1} \times W^u_{Y}({\tilde{p}}) \times \Delta _{\Sigma ^k}. \end{aligned}$$
To prove this proposition, we will need a better description of the relationship between the moduli spaces of spheres in \(\Sigma \), and the moduli spaces of Floer cylinders in \(\mathbb {R}\times Y\) (or in W).
Lemma 5.42
Proof
Notice now that \(\mathrm{d}\pi _\Sigma (\zeta _1, \zeta _0) = \zeta _0\), establishing that the evaluation map is a submersion.
Also observe that \(S^1\) acts on the curve \({\tilde{v}}\) 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 \(\pi ^\mathcal {M}_\Sigma \). Furthermore, for small rotation parameter, the curve will be distinct (as a parametrized curve) from \({\tilde{v}}\). \(\square \)
The next result justifies why it was reasonable to assume \(k_+ > k_\) in Proposition 5.41. The fact that \(k_+\ne k_\) will also be used below.
Lemma 5.43
Let \(A :=[w] \in H_2(\Sigma ;\mathbb {Z})\), where \(w:{\mathbb {C}}{\mathbb {P}}^1 \rightarrow \Sigma \) is the continuous extension of \(\pi _\Sigma \circ {\tilde{v}}\). Assume that either \(A\ne 0\) or \(\Gamma \ne \varnothing \). Then \(k_+ > k_\).
Proof
Denote by \(w^*Y\) the pullback under w of the \(S^1\)bundle \(Y\rightarrow \Sigma \). The map \({\tilde{v}}\) gives a section s of \(w^*Y\), defined in the complement of \(\Gamma \cup \{0,\infty \}\). By [11, Theorem 11.16], the Euler number \(\int _{{\mathbb {C}}{\mathbb {P}}^1}e(w^*Y)\) (where e is the Euler class) is the sum of the local degrees of the section s at the points in \(\Gamma \cup \{0,\infty \}\).
Recall that the gradientlike vector field \(Z_Y\) has the property that \(\mathrm{d}\pi _\Sigma Z_Y = Z_\Sigma \). Also recall that we may use the contact form \(\alpha \) as a connection to lift vector fields from \(\Sigma \) to vector fields on Y, tangent to \(\xi \). If V is a vector field on \(\Sigma \), we write \(\pi _\Sigma ^*V :={\widetilde{V}} \) to be the vector field on Y uniquely determined by the conditions \(\alpha (V) = 0\), \(\mathrm{d}\pi _\Sigma {\widetilde{V}} = V\). This extends as well to lifting vector fields on \(\Sigma \times \Sigma \) to vector fields on \(Y \times Y\).
Lemma 5.44
Proof
We now consider the consequences of Eq. (5.15) in this case of \(x=y\). Any \(v \in T_xY\) may be written as \(v_0 + aR\) where \(\alpha (v_0) = 0\). Furthermore, since \(x=y \in {{\,\mathrm{Crit}\,}}(f_\Sigma )\), and by definition, neither \({\tilde{x}}\) nor \({\tilde{y}}\) are critical points of \(f_Y\), we obtain that \(Z_Y({\tilde{y}})\) is a nonzero multiple of the Reeb vector field. Equation (5.13) now follows from the fact that \(\mathrm{d}\pi _\Sigma \varphi ^t_{Z_Y}({\tilde{x}}) = \mathrm{d}\varphi ^t_{Z_\Sigma }(x) \mathrm{d}\pi _\Sigma \).
Proof of Proposition 5.41
6 Monotonicity and the differential
The results of the previous section show that the moduli spaces of Floer cylinders with cascades that project to simple chains of pearls are transverse.
We will now assume that \((X,\Sigma ,\omega )\) is a monotone triple, as in Definition 2.4, to show that these moduli spaces are sufficient for the purposes of defining the split Floer differential. Recall that this yields that \((X, \omega )\) is spherically monotone, with \(c_1(TX) = \tau _X [\omega ]\) on spherical homology classes for some \(\tau _X>0\), and \(A \bullet \Sigma = K\omega (A)\) for some fixed \(K>0\) and every spherical homology class A in X. It is further assumed that \(\tau _\Sigma :=\tau _X  K > 0\), which implies that \((\Sigma ,\omega _\Sigma = \omega _\Sigma )\) is spherically monotone with monotonicity constant \(\tau _\Sigma \).
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 \(v :\mathbb {C}\rightarrow W\) is a \(J_W\) holomorphic plane asymptotic to a given closed Reeb orbit \(\gamma \) in Y, the Fredholm index for the deformations of v (as an unparameterized curve) keeping \(\gamma \) fixed is \(\gamma _0\) and it is nonnegative. Furthermore, if v is multiply covered, this Fredholm index is at least 2.
Proof
It, therefore, follows that \({\text {Ind}}(v) \geqslant 0\).
Proposition 6.2
 (0)
An index 1 gradient trajectory in Y without any (nonconstant) holomorphic components and without any augmentation punctures.
 (1)
A smooth cylinder in \(\mathbb {R}\times Y\) without any augmentation punctures and a nontrivial projection to \(\Sigma \). The positive puncture converges to an orbit \(\check{p}_{k_+}\) and the negative puncture converges to an orbit \({\widehat{q}}_{k_}\). The difference in multiplicities of the orbits is given by \(k_+k_= K \omega (A)\), where \(A \in H_2(\Sigma ;\mathbb {Z})\) is the homology class represented by the projection of the cylinder to \(\Sigma \). See Fig. 4.
 (2)
A cylinder with one augmentation puncture and whose projection to \(\Sigma \) is trivial. The positive puncture converges to an orbit \(\check{p}_{k_+}\) and the negative puncture converges to an orbit \({\widehat{q}}_{k_}\). The augmentation plane has index 0. If \(B \in H_2(X;\mathbb {Z})\) is the class represented by the augmentation plane, then the difference in multiplicities is given by \(k_+k_=K \omega (B)\). Furthermore, \(\check{p}\) and \({\widehat{q}}\) are critical points of \(f_Y\) contained in the same fibre of \(Y \rightarrow \Sigma \), which we can write as \(q=p\). See Fig. 5.
Proof
We, therefore, have \(k_+  k_  \sum _{j=1}^k k_j = K \omega (A) = K \frac{\langle c_1(T\Sigma ), A \rangle }{\tau _X  K}\). We also have \(k_j = B_j \bullet \Sigma = K\omega (B_j)\). Notice then that \(\gamma _i_0 = 2 \langle c_1(TX), B_j \rangle  2 B_j \bullet \Sigma 2\).
Now let \(N_0\) be the number of sublevels that project to constant curves in \(\Sigma \) and let \(N_1\) be the number of sublevels that project to nonconstant curves in \(\Sigma \), \(N = N_0 + N_1\). Note that by the stability condition, each cylinder that projects to a constant curve in \(\Sigma \) must have at least one augmentation puncture, so \(N_0 \leqslant k\).
 (0)
\(N=0\). Then, either \(i({\widetilde{p}}) = i({\widetilde{q}})\) or \({\widetilde{p}} = {\check{p}}\) and \({\widetilde{q}} = {\hat{q}}\). Since \(N =0\), this is a pure Morse differential term.
 (1)
\(N_1 = 1\), \(N_0 = k = 0\) and \({\widetilde{p}} = \check{p}\), \({\widetilde{q}} = {\widehat{q}}\). This case corresponds to a nonconstant sphere in \(\Sigma \) without any augmentation punctures.
 (2)
\(N_1 = 0\), \(k=1\), \(N_0 = 1\), and \({\widetilde{p}} = \check{p}\), \({\widetilde{q}} = {\widehat{q}}\). In this case, the Floer cylinder has one augmentation puncture, and projects to a constant in \(\Sigma \), so \(q=p \in \Sigma \). \(\square \)
We now consider the possible terms in the differential that connect nonconstant Hamiltonian trajectories in \(\mathbb {R}\times Y\) to Morse critical points in X.
Proposition 6.3
Any Floer cascade appearing in the differential, connecting a nonconstant Hamiltonian orbit \({\tilde{p}}_{k_+}\) in \( \mathbb {R}\times Y\) to a Morse critical point x in W, consists of two levels. The upper level, in \( \mathbb {R}\times Y \), projects to a point in \(\Sigma \) and is a cylinder asymptotic at \(+\infty \) to an orbit \(\check{p}_{k_+}\) and at \(\infty \) to a Reeb orbit \(\gamma \) in \(\{ \infty \} \times Y\). This \(\gamma \) is the parametrized Reeb orbit associated with \(\check{p}_{k_+}\).
The lower level is a holomorphic plane in W converging to the parametrized orbit \(\gamma \) at \(\infty \) and with 0 mapping to the descending manifold of the critical point x. As a parametrized curve, this has Fredholm index 1. See Fig. 6.
Proof
Suppose such a cascade occurs in the differential, connecting the nonconstant orbit \({\widetilde{p}}_{k_+}\) to the critical point x in the filling W.
Let N be the number of cylinders in \(\mathbb {R}\times Y\) that appear in the split Floer cylinder. Let \(A_i \in H_2(\Sigma ), i=1, \dots , N\), denote the spherical classes represented by the projections of these cylinders to \(\Sigma \). Let \(A = \sum _{i=1}^N A_i\).
Let k be the number of augmentation planes, and let \(B_j \in H_2(X), j=1, \dots , k\) be the corresponding spherical homology classes in X. Let \(\gamma _j, j=1, \dots , k\), be the corresponding Reeb orbits with multiplicities \(k_j = B_j \bullet \Sigma = K \omega (B_j)\).
Remark 6.4
Similar analysis applied to continuation maps gives that our construction does not depend on the choices of almost complex structure \(J_Y\), \(J_W\) or of the auxiliary Morse functions and gradientlike vector fields.
In general, \(\partial ^2 = 0\) is obtained through analyzing gluing and considering the boundary of 1dimensional moduli spaces. In our situation, if additionally \(f_\Sigma \) and \(f_W\) are assumed to be lacunary (i.e. have no critical points of consecutive indices), all contributions to the differential of an orbit \(\check{p}\) are either of the form \({\widehat{q}}\) or constant orbits. This automatically gives that \(\partial ^2 = 0\) for split symplectic homology.
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 [4] to show that if \({\overline{W}}\) is a Weinstein domain (or equivalently, if W is a Weinstein manifold of finitetype), and \(\Sigma \) 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, \(\dim _\mathbb {R}X = 2n\le 4\)).
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 \(\Sigma \) is.
Proof
The trivial direction is that if there exists a spherical class \(A \in \pi _2(\Sigma )\) with \(\omega (A) > 0\), then \(\imath _* A \in \pi _2( X)\) and still has positive area.
We will now prove that any symplectic sphere in X is in the image of the inclusion. Let \(C \subset W\) be the isotropic skeleton of W. Notice that by following the flow of the Liouville vector field on W, we obtain that \(W {\setminus } C\) is symplectomorphic to a piece of the symplectization \((\infty , a) \times Y\). Thus, we have that \(X {\setminus } C\) is an open subset of a symplectic disk bundle over \(\Sigma \) (the normal bundle to \(\Sigma \) in X). We denote this bundle’s projection map by \(\pi :X {\setminus } C \rightarrow \Sigma \).
Suppose \(A \in \pi _2(X)\) is a spherical class with \(\omega (A) > 0\). 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 \(\iota :\Sigma \rightarrow X\) and \(j:X{\setminus } C \rightarrow X\) are the inclusion maps, then \(\omega _\Sigma = \iota ^*\omega \) and \(\iota \circ \pi \) is homotopic to j. This implies that \(\omega _X(A) = \omega _\Sigma ( \pi _* A)\), and the result follows. \(\square \)
Lemma 6.6
Suppose W is a Weinstein domain with isotropic skeleton of real codimension at least 3 and \(\Sigma \) 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 \(0 = 2(\langle c_1(TX),B\rangle  B\bullet \Sigma  1)\). Now, \(\langle c_1(TX),B\rangle  B\bullet \Sigma = (\tau _X  K)\,\omega (B) \geqslant 1\). Thus, the augmentation plane can only exist if there is a spherical class B with \((\tau _X  K ) \,\omega (B) = 1\).
By applying Lemma 6.5, we have \(B = \imath _*A\), where \(A \in \pi _2(\Sigma )\) is a spherical class in \(\Sigma \).
Now observe that \(\langle c_1(T\Sigma ), A \rangle + \langle c_1(N\Sigma ), A \rangle = \langle c_1(TX), A \rangle \), so we have \(\langle c_1(T\Sigma ), A \rangle = (\tau _X  K) \omega _\Sigma (A).\) Hence, \(1 = (\tau _XK)\,\omega (A) = \langle c_1(T\Sigma ),A\rangle \). This contradicts the assumption that the minimal Chern number of \(\Sigma \) is at least 2, so the augmentation plane cannot exist. \(\square \)
Remark 6.7
Observe that this lemma applies more generally: if \(\Sigma \) has minimal Chern number at least 2, then an augmentation plane cannot represent a spherical class in the image of \(\imath _* :\pi _2(\Sigma ) \rightarrow \pi _2(X)\).
Additionally, we have that an augmentation plane cannot have image entirely contained in \(\varphi (\overline{\mathcal {U}})\). Indeed, any holomorphic sphere contained in \(\varphi (\overline{\mathcal {U}})\) will have index too high to be an augmentation plane: the \(J_X\)holomorphic sphere with image in \(\varphi (\overline{\mathcal {U}})\) automatically comes in a twoparameter family (corresponding to the \(\mathbb {C}^*\) action on the normal bundle to \(\Sigma \)). To make this argument more precise, we use our index computations. Suppose a sphere in \(\varphi (\overline{\mathcal {U}})\) is an augmentation plane. It then represents a class \(\imath _*A\) with \(A \in H_2(\Sigma )\). By the same index argument as in Lemma 6.6, \(1 = \langle c_1(T\Sigma ), A \rangle \). Since the image is assumed to be in \(\varphi (\overline{\mathcal {U}})\), the projection of the curve to \(\Sigma \) is \(J_\Sigma \)holomorphic. The index of this projection is given by \(4 + 2 \langle c_1(T\Sigma ), A \rangle = 2\). This must be nonnegative; however, since the projection is \(J_\Sigma \)holomorphic, and represents an indecomposable homology class. This contradiction then rules this possibility out.
Remark 6.8
The dichotomy between \(\Sigma \) with minimal Chern number equal to 1 and bigger than 1 is also explored in upcoming joint work of the first named author with D. Tonkonog, R. Vianna and W. Wu, studying the effect of the Biran circle bundle construction on superpotentials of monotone Lagrangian submanifolds [14].
7 Orientations
To orient our moduli spaces, we will take the point of view of coherent orientations, which is implemented in the Morse–Bott setting in [5, 9]. Some authors [36, 44] have used the alternative approach of canonical orientations. We find it more straightforward to use coherent orientations in our computations, especially since there are very few choices involved. Notice also that if one has a canonical orientation scheme, it is possible to extract a coherent orientation from this by making choices of preferred orientations of certain capping operators.
The geometry of our specific situation allows us to avoid some of the technical difficulties present in the general Morse–Bott situation. In particular, we have two key features that make our analysis more straightforward. First of all, we do not have any “bad” orbits appearing in our setting (recall from Sect. 3.1 that, if we take a “constant” trivialization, the Conley–Zehnder index does not depend on covering multiplicity). For another, the manifolds of orbits are all orientable, and are even oriented quite naturally by the symplectic/contact structures that exist on them.
We now recall the general method for obtaining signs in Floer homology, as first introduced in [18] and since generalized. First of all, over the space of all Fredholm Cauchy–Riemann operators, there is a determinant bundle. A choice of a section of this bundle then induces an orientation on moduli spaces of holomorphic curves. This (together with some additional choices in the Morse–Bott situation) allows us to orient all moduli spaces that occur in Floer homology. On the other hand, configurations that are counted in the differential have a natural \(\mathbb {R}\)action on them by reparametrization, which also induces an orientation on these moduli spaces. The sign of such a term in the differential is positive if they agree and negative if they disagree.
7.1 Orienting the moduli spaces of curves
We now explain the first part of this method: how to orient the moduli spaces of Floer punctured cylinders, but without considering their constraints coming from evaluation maps. We begin by sketching the situation for the nondegenerate case and then discuss the modifications needed for the Morse–Bott situation.

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.
An orientation of \(D^\delta :W^{1,p}(\dot{S}, E) \rightarrow L^{p}(\dot{S}, \Lambda ^{0,1}T^*\dot{S}\otimes E)\) induces an orientation of \(D :W^{1,p,\delta }(\dot{S}, E) \rightarrow L^{p,\delta }(\dot{S}, \Lambda ^{0,1}T^*\dot{S}\otimes E)\) by this conjugation.
From [7, 18, 16, Section 1.8], a coherent orientation of the determinant bundle over nondegenerate Cauchy–Riemann operators exists and may be specified by choosing a preferred section of the determinant bundle over certain capping operators. These are operators whose domain is the once punctured sphere \(\mathbb {C}\) (where the puncture is positive), with a trivial Hermitian vector bundle over them and specified asymptotic operator. To achieve the two properties listed above, it suffices to enforce them on these capping operators since the linear gluing operation described in [18] respects direct sums and complex linearity.
We now describe how we orient capping operators for the relevant asymptotic operators. By Lemma 5.22, the linearized operator associated with a Floer cylinder \({\tilde{v}}\) is a compact perturbation of a split operator \(D^\mathbb {C}_{{\tilde{v}}} \oplus \dot{D}^\Sigma _w\), where \(w = \pi _\Sigma \circ {\tilde{v}}\). There is also a corresponding splitting of the asymptotic operators at the asymptotic limits. In particular, \(\dot{D}^\Sigma _{ w}\) 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 of capping operator (which may always be taken to be complex linear).
We now choose capping operators for the \(\mathbf {A}_\pm \), which determines an orientation of \(D^\mathbb {C}_{{\tilde{v}}}\) by the coherent orientation scheme.
Lemma 7.1
Proof
The operators \(\Psi _k\) 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 \(\Xi _k\) are Fredholm of index 1 and surjective, and that their kernels contain elements that can be identified with the Reeb vector field.
Now, pick an arbitrary capping operator \(\Phi _1^\). Define \(\Phi _k^\) for \(k>1\) by gluing \(\Phi _1^ \# \Psi _k\). Define \(\Phi _k^+\) for all \(k>0\) by gluing \(\Phi _k^ \# \Xi _k\). For these choices of capping operators, \(D^\mathbb {C}_{{\tilde{v}}}\) are oriented in the direction of the Reeb flow, as wanted. \(\square \)
Lemma 7.2
Let \(\mathbf {A}\) be a degenerate asymptotic operator, let V be its kernel and let \(\delta > 0\) be chosen small enough that \([\delta , \delta ] \cap \sigma (\mathbf {A}) = \{ 0 \}\).
Then, the kernel of \(\frac{\partial }{\partial s}  \mathbf {A} :W^{1,p,\delta }(\mathbb {R}\times S^1, \mathbb {C}^n) \rightarrow L^{p,\delta }(\mathbb {R}\times S^1, \mathbb {C}^n)\) consists of constant maps with values in V, and its cokernel is trivial.
In particular, an orientation of this operator corresponds to an orientation of V.
Proof
The proof follows from expanding \(L^2(S^1, \mathbb {C}^n)\) in a Hilbert basis given by eigenvectors of the asymptotic operator A seen as an elliptic selfadjoint unbounded operator on \(L^2(S^1, \mathbb {C}^n)\). Then, the kernel of \(\frac{\partial }{\partial s}  \mathbf {A}\) is spanned by solutions of the form \({\text {e}}^{\lambda s}v(t)\), where v(t) is an eigenfunction for the eigenvalue \(\lambda \). Since we require exponential growth of rate \(\delta \), this forces \(\delta< \lambda < \delta \). The result for the kernel now follows since 0 is the only such eigenvalue.
The statement about the cokernel now follows from similar analysis of the adjoint operator. Indeed, the adjoint is \(\frac{\partial }{\partial s}  \mathbf {A}\), and so nonzero elements of its kernel will take a similar form, but with \(\lambda < \delta \) and \(\lambda > \delta \), showing no such element exists. \(\square \)
We thank Chris Wendl and Richard Siefring for suggesting this argument. See also [31, Theorem 10.4.19].
Let \(D^0_\delta :W^{1,p, \mathbf \delta }(\dot{S}, E) \rightarrow L^{p, \mathbf \delta }(\dot{S}, \Lambda ^{0,1} T^*S \otimes E)\) be the restriction of this operator to the subspace where all sections decay to 0 at punctures for which \(\delta _z > 0\).
Definition 7.3
Suppose \(\delta \), \(\mathbf {V}\), \(D_\delta \) and \(D_\delta ^0\) as above. Given orientations of \(V_{\pm \infty }\) and assuming that \(V_z\) is a complex linear vector space for each \(z \in \Gamma \), an orientation of \(D^0_\delta \) induces an orientation on \(D_\delta \). We define then the orientation on \(D_\delta \) and hence on D to be such that this map is orientation preserving.
Finally, we verify that this orientation of \(D_\delta \) is consistent with the one induced from Lemma 5.20.
Lemma 7.4
Let D be a Cauchy–Riemann operator on the punctured cylinder \(\dot{S} = \mathbb {R}\times S^1 {\setminus } \Gamma \), and let \(z_0 \in \{ \pm \infty \} \cup \Gamma \), \(\mathbf {\delta }\), \(\mathbf {\delta '}\) and \(\mathbf {V}\) as in Lemma 5.20. Suppose furthermore that the asymptotic operator at each puncture in \(\Gamma \) is complex linear. For each puncture \(z \in \Gamma \cup \{ \pm \infty \}\), let \(V_z = \ker \mathbf {A}_z\) be oriented by Lemma 7.2.
Proof
For notational simplicity, we will consider the case \(z_0 = +\infty \). The other cases will be similar, aside from a reordering of the terms.
Let \(D_+ = \frac{\partial }{\partial s}  \mathbf {A}_{+\infty } :W^{1,p,\delta }(\mathbb {R}\times S^1, \mathbb {C}^n) \rightarrow L^{p,\delta }(\mathbb {R}\times S^1, \mathbb {C}^n)\).
Let \(D^0_\delta :W^{1,p, \mathbf \delta }( \dot{S}, E) \rightarrow L^{p, \mathbf {\delta }}(\dot{S}, \Lambda ^{0,1} T^*\dot{S} \otimes E)\) be the restriction of \(D_\delta \) to the space of sections decaying to 0 at each puncture with \(\delta _z > 0\).
We remark here that our asymptotic operators are either complex linear or have a kernel that is naturally identified with the Reeb vector field or with the tangent space to the contact manifold Y. The Reeb vector field and the contact form on Y induce orientations on these asymptotic operators, which are compatible with the choices of orientations of capping operators in Lemma 7.1.
We have now oriented the operators \(D^\mathbb {C}_{{\tilde{v}}}\) and \(\dot{D}^\Sigma _w\) acting on spaces of sections free to move in their Morse–Bott families. Since the linearized Floer operator is a compact perturbation of their direct sum, we get induced orientations on the transverse moduli spaces of Floer cylinders with punctures.
7.2 Orientations with constraints
We have now explained how to orient all of the moduli spaces of punctured cylinders with ends free to move in the corresponding Morse–Bott families of orbits. This is not yet sufficient to orient our moduli spaces of cascades. The additional ingredient necessary is to orient moduli spaces of holomorphic curves with constraints on their asymptotic evaluation maps, with the asymptotic evaluation map constrained to lie in stable/unstable manifolds of critical points of the auxiliary Morse functions, or, in the case of multilevel cascades, constrained to lie in [flow] diagonals in manifolds of orbits.
Let us begin by stating the convention in [9] for how to orient a fibre sum (which agrees with [28, Convention 7.2.(b)]).
Definition 7.5
 (1)
\(f_1  f_2\) induces an isomorphism \((V_1 \oplus V_2) / \ker (f_1  f_2) \rightarrow W\) which changes orientations by \((1)^{\dim V_2 . \dim W}\),
 (2)
where a quotient U / V of oriented vector spaces is oriented in such a way that the isomorphism \(V \oplus (U/V) \rightarrow U\) (associated with a section of the quotient short exact sequence) preserves orientations.
One key property of this orientation convention for fibre sums is that it is associative (this property specifies the orientation convention almost uniquely, as explained in [28, Remark 7.6.iii] and [34]; this was pointed out to us by Maksim Maydanskiy).
To orient our constrained moduli spaces, we follow the point of view in the literature [3, 5, 9, 20, 36]. Specifically, we begin by orienting moduli spaces of unconstrained Floer cylinders as in the previous section, by the chosen coherent orientation of Cauchy–Riemann operators with free asymptotics. We also fix orientations on all stable and unstable manifolds of the relevant manifolds of orbits (see the next section for more details), as well as on the relevant diagonals and flow diagonals. Then, we orient the moduli spaces of Floer cylinders with cascades by the rule that the asymptotic constraints are obtained as fibre products over descending and ascending manifolds of the Morse functions in the manifolds of orbits \(Y_k\) and W and as fibre products over flow diagonals and diagonals in \(Y_k \times Y_k\) and in \(Y \times Y\). The fibre products are oriented using Definition 7.5. For this scheme to induce a differential, we then need the orientations of the various boundary components of these moduli spaces of cascades to be consistent.
Observe that in a general Morse–Bott situation, there are additional orientation difficulties that are not present in our problem. Specifically, [5, 9, 36] have to deal with parametric families of asymptotic operators that move in the space of asymptotic operators of fixed degeneracy. In our problem, the asymptotic operators of \(D^\mathbb {C}_{{\tilde{v}}}\) are constant on each Morse–Bott family of orbits, dramatically simplifying the problem to consider.
Lemma 7.6
The coherent orientation of the Cauchy–Riemann operator that corresponds to gluing \({\tilde{v}}_\) and \({\tilde{v}}_+\) is the opposite of the orientation induced as the boundary of the fibre sum orientation of \({\tilde{v}}_\pm \) over the flow diagonal.
Sketch of proof
Similarly, by taking the fibre product over the diagonal, we see this as oriented with the opposite orientation of the boundary of the flow diagonal, which is oriented as \([0, \infty ) \times \Delta \). This allows us to conclude that our orientation scheme is coherent with respect to the additional breakings that appear in the Morse–Bott setting. \(\square \)
7.3 A calculation of signs
Having now explained the general framework of our orientations, let us now give an explicit description of the signs associated with a Floer cylinder with cascades contributing to the differential. 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.
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 \(k_\pm \) at \(\pm \infty \). Such a cylinder is an element of the space \({\mathcal {M}}^*_{H,0,\mathbb {R}\times Y;k_,k_+}(A;J_Y)\), for some \(A\in H_2(\Sigma ;\mathbb {Z})\).
Observe now that if \({\mathcal {M}}^*_{H,1}({\widehat{q}}_{k_},\check{p}_{k_+};J_W)\) is onedimensional, then its tangent space at every point is generated by the infinitesimal translation in the sdirection on the domain of the Floer cylinder. This induces an orientation on \({\mathcal {M}}^*_{H,1}({\widehat{q}}_{k_},\check{p}_{k_+};J_W)\). Comparing this orientation with the one defined above with the fibre sum rule, we get the sign of such a contribution to the split symplectic homology differential.
The space \(\mathcal {M}^*_{H}(B;J_W) {}_{{\text {ev}}^1_+\,}\times {}_{{\text {ev}}^2_}{} \mathcal {M}^*_{H,k_+}(0;J_Y)\) has an action of \(\mathbb {R}_1 \times \mathbb {R}_2\), where the onedimensional real vector space \(\mathbb {R}_1\) acts by stranslation on the domain of the cylinder in W and \(\mathbb {R}_2\) acts by stranslation on the domain of the cylinder in \(\mathbb {R}\times Y\). The sign of a Case 3 contribution to the differential is obtained by comparing the coherent/fibre product orientation on (7.10) with the usual orientation on \(\mathbb {R}_1 \times \mathbb {R}_2\), corresponding to stranslation on the domain of \({\tilde{v}}_0\) followed by stranslation on the domain of \({\tilde{v}}_1\).
Footnotes
 1.
Technically, to define this requires a trivialization of \(\xi \) along this orbit. In our setting the asymptotic operator is complex linear in the \(\xi \) direction, so this choice does not matter.
Notes
Acknowledgements
We would like to thank Yasha Eliashberg, Paul Biran and Dusa McDuff for many helpful conversations, guidance and ideas. We also thank JeanYves Welschinger and Felix Schmäschke for helping us to understand coherent orientations better. We also thank Frédéric Bourgeois, Joel Fish, Richard Siefring and Chris Wendl for helpful suggestions. Finally, we thank the referee for their careful reading of our paper and for many helpful suggestions for improvement. S.L. was partially supported by the ERC Starting Grant of Frédéric Bourgeois StG239781ContactMath and also by Vincent Colin’s ERC Grant geodycon. L.D. thanks Stanford University, ETH Zürich, Columbia University and Uppsala University for excellent working conditions. L.D. was partially supported by the Knut and Alice Wallenberg Foundation.
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