A Polyfold Proof of the Arnold Conjecture

We give a detailed proof of the homological Arnold conjecture for nondegenerate periodic Hamiltonians on general closed symplectic manifolds $M$ via a direct Piunikhin-Salamon-Schwarz morphism. Our constructions are based on a coherent polyfold description for moduli spaces of pseudoholomorphic curves in a family of symplectic manifolds degenerating from $\mathbb{C}\mathbb{P}^1\times M$ to $\mathbb{C}^+ \times M$ and $\mathbb{C}^-\times M$, as developed by Fish-Hofer-Wysocki-Zehnder as part of the Symplectic Field Theory package. To make the paper self-contained we include all polyfold assumptions, describe the coherent perturbation iteration in detail, and prove an abstract regularization theorem for moduli spaces with evaluation maps relative to a countable collection of submanifolds. The 2011 sketch of this proof was joint work with Peter Albers, Joel Fish.


Introduction
Let (M, ω) be a closed symplectic manifold and H : S 1 × M → R a periodic Hamiltonian function. It induces a time-dependent Hamiltonian vector field X H : S 1 × M → TM given by ω(X H (t, x), ·) = dH(t, ·). We denote the set of contractible periodic orbits by (1) P(H) := γ : S 1 → M γ(t) = X H (t, γ(t)) and γ is contractible and note that periodic orbits can be identified with the fixed points of the time 1 flow φ 1 H : M → M of X H . We call this Hamiltonian system nondegenerate if φ 1 H × id M is transverse to the diagonal and hence cuts out the fixed points transversely. In particular, this guarantees a finite set of periodic orbits. Arnold [A] conjectured in the 1960s that the minimal number of critical points of a Morse function on M is also a lower bound for the number of periodic orbits of a nondegenerate Hamiltonian system as above. In this strict form, the Arnold conjecture has been confirmed for Riemann surfaces [E] and tori [CZ]. A weaker form is accessible by Floer theory, introduced by Floer [F2, F3] in the 1980s. It constructs a chain complex generated by P(H) that can be compared with the Morse complex generated by the critical points of a Morse function. When Floer homology is well-defined, it is usually independent of the Hamiltonian, and on a compact symplectic manifold can in fact be identified with Morse homology, which is also independent of the Morse function and computes the singular homology. Using this approach, the following nondegenerate homological form of the Arnold conjecture was first proven by Floer [F1, F4] in the absence of pseudoholomorphic spheres. Floer's proof was later extended to general closed symplectic manifolds [HS, O, FO, LT], and in the presence of pseudoholomorphic spheres of negative Chern number requires abstract regularizations of the moduli spaces of Floer trajectories since perturbations of the geometric structures may not yield regular moduli spaces; see e.g. [MW]. Further generalizations and alternative proofs have been published in the meantime, using a variety of regularization methods. The purpose of this note is to provide a general and maximally accessible proof of Theorem 1.1 -using an abstract perturbation scheme provided by the polyfold theory of Hofer-Wysocki-Zehnder [HWZ], following an approach by Piunikhin-Salamon-Schwarz [PSS] based on [Sc2], and building on polyfold descriptions of Gromov-Witten moduli spaces [HWZ1] as well as their degenerations in Symplectic Field Theory [EGH, FH].
To be more precise, let CF = ⊕ γ∈P(H) Λ γ be the Floer chain group of the Hamiltonian H with coefficients in the Novikov field Λ (see §2). Let (CM, d) be the Morse complex with coefficients in Λ associated to a Morse function f : M → R and a suitable metric on M (see §3). We obtain a proof of Theorem 1.1 as consequence of Lemma 4.9, Definition 5.7, Lemma 6.4, 6.5, 6.6, summarized here.
(ii) ι is a Λ-module isomorphism. (iii) h is a chain homotopy between SSP • P SS and ι, that is ι − SSP • P SS = d • h + h • d.
Here we view the Floer chain group CF as a vector space over Λ -not as a chain complex, and in particular do not consider a Floer differential. Thus we are neither constructing a Floer homology for H, nor identifying it with the Morse homology of f . However, the algebraic structures in Theorem 1.2 suffice to deduce the homological Arnold conjecture for the Hamiltonian H as follows.
Proof of Theorem 1.1. Denote the sum of the Betti numbers k := dim M i=0 dim H i (M ; Q). Let (CM Q , d Q ) be the Morse complex over Q as defined in §3. Then by the isomorphism of singular and Morse homology there exist c 1 , . . . , c k ∈ CM Q that are cycles, d Q c i = 0, and linearly independent in the Morse homology over Q. Since the Morse differential d : CM → CM is given by Λ-linear extension of d Q from CM Q ⊂ CM the chains c 1 , . . . , c k ∈ CM are also cycles dc i = d Q c i = 0 and linearly independent in the Morse homology over Λ. By Theorem 1.2 (i),(ii), ι induces an isomorphism Hι : HM → HM on homology. This in particular implies that [ι(c 1 )], . . . , [ι(c k )] ∈ HM are also linearly independent in homology, that is for any λ 1 , . . . , λ k ∈ Λ we have We now show that P SS(c 1 ), . . . , P SS(c k ) ∈ CF are Λ-linearly independent, proving #P(H) ≥ k since the elements of P(H) generate CF by definition. This proves the theorem.
Let λ 1 , . . . , λ k ∈ Λ be a tuple such that k i=0 λ i · P SS(c i ) = 0. Then we deduce from Theorem 1.2 (iii) that k i=0 λ i · ι(c i ) = k i=0 λ i · SSP P SS(c i ) + dh(c i ) + h(dc i ) = SSP k i=0 λ i · P SS(c i ) + k i=0 λ i · dh(c i ) = d k i=0 λ i · h(c i ) , which implies λ 1 = . . . = λ k = 0 by (2). This algebraically minimalistic approach of deducing the homological Arnold conjecture from the existence of maps P SS and SSP whose composition is chain homotopic to an isomorphism on the Morse complex was developed in 2011 discussions of the second author, Peter Albers, and Joel Fish with Mohammed Abouzaid and Thomas Kragh. These were prompted by our observation that any rigorous proof of "Floer homology equals Morse homology" seemed to require equivariant transversality; see Remark 1.3. Since equivariant transversality is generally obstructed -even for simple equivariant sections of finite rank bundles -we were looking for a proof that requires the least amount of specific geometric insights or new abstract tools. We ultimately expect the [PSS]-approach to yield an isomorphism between Floer and Morse homology, as well as generalizations of spectral invariants [Sc3] to all closed symplectic manifolds, but the purpose of the present manuscript is to exemplify the use of existing polyfold results to obtain detailed, rigorous, and transparent proofs.
For that purpose we include Appendix A, which summarizes all necessary polyfold notions and abstract tools. Here we moreover establish in Theorem A.9 a relative perturbation result that should be of independent interest: It allows to bring moduli spaces with an evaluation map into general position to a countable collection of submanifolds. Besides these 10 pages of background, there are a few more technical complications due to the current lack of polyfold publications: Since the polyfold description of Floer-theoretic moduli spaces -while evident to experts -is not published apart from an outline in [W2], we reformulate all moduli spaces in our application into SFT moduli spaces, using the fact that graphs of perturbed pseudoholomorphic maps are J-curves for an appropriate J. Since the polyfold description of SFT moduli spaces [FH,FH1] is also not completely published yet, we summarize the anticipated results in Assumptions 4.3, 5.5, 6.3. Finally, we give a detailed account of the iterative construction of coherent perturbations in the proofs of Lemma 6.4 and 6.6. To strike a balance between supplying technical details that are not easily available in the literature and maximal accessibility, we have clearly labeled all such technical work. Readers willing to view polyfold theory as a black box can save 20 pages by skipping these parts.
For accessibility we begin with reviews of the pertinent facts on the Novikov field, §2, and Morse trajectories, §3. The proof of Theorem 1.2 then proceeds by constructing the P SS and SSP maps in §4 from curves in C ± × M , constructing the isomorphism ι and chain homotopy h in §5 from curves in CP 1 × M and its degeneration into C − × M and C + × M , and proving their algebraic relations in §6 by constructing coherent perturbations. Remark 1.3. (i) There are essentially two approaches to the general Arnold conjecture as stated in Theorem 1.1. The first -developed by [F4] and used verbatim in [HS, O, FO, LT] -is to establish the independence of Floer homology from the Hamiltonian function, and to identify the Floer complex for a C 2 -small S 1 -invariant Hamiltonian H : M → R with the Morse complex for H. This requires S 1 -equivariant transversality to argue that isolated Floer trajectories must be S 1 -invariant, hence Morse trajectories. A conceptually transparent construction of equivariant and transverse perturbations -under transversality assumptions at the fixed point set which are met in this setting -can be found in [Z], assuming a polyfold description of Floer trajectories.
(ii) The second approach to Theorem 1.1 by [PSS] is to construct a direct isomorphism between the Floer homology of the given Hamiltonian and the Morse homology for some unrelated Morse function. Two chain maps P SS : CM → CF , SSP : CF → CM between the Morse and Floer complexes are constructed from moduli spaces of once punctured perturbed holomorphic spheres with one marking evaluating to the unstable resp. stable manifold of a Morse critical point, and with the given Hamiltonian perturbation of the Cauchy-Riemann operator on a cylindrical neighbourhood of the puncture. Then gluing and degeneration arguments are used to argue that both P SS • SSP and SSP • P SS are chain homotopic to the identity, and hence SSP is the inverse of P SS on homology. However, sphere bubbling can obstruct these arguments: In the first chain homotopy it creates an ambiguity in the choice of nodal gluing when the intermediate Morse trajectory shrinks to zero length. (We expect to be able to avoid this by arguing that "index 1 solutions generically avoid codimension 2 strata" -another classical fact in differential geometry that should generalize to polyfold theory.) The second chain homotopy is as claimed in Theorem 1.2 (iii) but with ι = id, which requires arguing that the only isolated holomorphic spheres with two marked points evaluating to an unstable and stable manifold are constant. This again requires S 1 -equivariant transversality (which we expect to be able to achieve with the techniques in [Z]).
(iii) Theorem 1.2 is proven by following the [PSS]-approach as above but avoiding the use of new polyfold technology such as equivariant or strata-avoiding perturbations. In particular, ι is the map that results from counting holomorphic spheres that intersect an unstable and stable manifold; its invertibility is deduced from an "upper triangular" argument.
(iv) The techniques in this paper -combining existing perturbation technology with the polyfold descriptions of SFT moduli spaces -would also allow to define the Floer differential, prove d 2 = 0, establish independence of Floer homology from the Hamiltonian (and other geometric data), and prove that P SS and SSP are chain maps. Then the chain homotopy between SSP • P SS and the isomorphism ι implies that P SS is injective and SSP surjective on homology. However, proving that P SS and SSP are isomorphisms on homology, or directly identifying the Floer complex of a small S 1 -invariant Hamiltonian with its Morse complex, requires the techniques discussed in (ii).
Moreover, a proof of independence of Floer homology from the choice of abstract perturbation would require a study of the algebraic consequences of self-gluing Floer trajectories in expected dimension −1 during a homotopy of perturbations, as developed in the A ∞ -setting in [LW].
We thank Peter Albers and Joel Fish for helping develop the outline of this project -and Edi Zehnder for asking the initial question. The project was further supported by various discussions with Mohammed Abouzaid, Helmut Hofer, Thomas Kragh, Kris Wysocki, and Zhengyi Zhou. Crucial financial support was provided by NSF grants DMS-1442345 and DMS-1708916.

The Novikov field
We use the following Novikov field Λ associated to the symplectic manifold (M, ω). Let H 2 (M ) denote integral homology and consider the map ω : H 2 (M ) → R given by the pairing ω(A) := ω, A for A ∈ H 2 (M ). The image of this pairing is a finitely generated additive subgroup of the real numbers denoted Γ := im ω = ω(H 2 (M )) ⊂ R.
The Novikov field Λ is the set of formal sums where T is a formal variable, with rational coefficients λ r ∈ Q which satisfy the finiteness condition ∀c ∈ R #{r ∈ Γ | λ r = 0, r ≤ c} < ∞.
The multiplication is given by λ · µ = r∈Γ λ r T r · s∈Γ µ s T s := t∈Γ r+s=t λ r µ s T t . This defines a field Λ by [HS,Thm.4.1] and the discussion preceding the theorem in [HS,§4], the key being that Γ is a finitely generated subgroup of R.
We will moreover make use of the following generalization of the invertibility of triangular matrices with nonzero diagonal entries.
Proof. Since Λ is a field, invertibility of M is equivalent to det(M ) = 0. Write det(M ) = r∈Γ µ r T r ∈ Λ for some µ r ∈ Q. It suffices to show that µ 0 = 0. We proceed by induction on the size of the matrix M . In the ℓ = 1 base case, when M is a 1 × 1 matrix M = [λ 11 ], we have det(M ) = λ 11 = r∈Γ µ r T r with µ r = λ 11 r so µ 0 = λ 11 0 = 0. Now suppose that M is size ℓ × ℓ for some ℓ > 1 and inductively assume that, for any size (ℓ − 1) × (ℓ − 1) matrix N satisfying the hypotheses of the Lemma, we have det(N ) = r∈Γ µ N r T r with µ N 0 = 0. For 1 ≤ j ≤ ℓ, let C 1j denote the matrix obtained by deleting the first row and j-th column of M . Then N := C 11 is an (ℓ − 1) × (ℓ − 1) matrix that satisfies the hypotheses of the Lemma, and the cofactor expansion of the determinant yields det(M ) = λ 11 det(N ) + ℓ j=2 (−1) 1+j λ 1j det(C 1j ). By hypothesis, all entries of M are of the form λ ij = r≥0 λ ij r T r . Since the determinants det(N ) and det(C 1j ) are polynomials of those entries, they are of the same form -with zero coefficients for T r with r < 0. Since we moreover have λ 1j 0 = 0 for j ≥ 2 by hypothesis, it follows that the constant term (i.e. the coefficient on T 0 ) of λ 1j det(C 1j ) is 0. Hence the constant term of det(M ) = µ r T r is µ 0 = λ 11 0 · µ N 0 , where µ N 0 = 0 by induction and λ 11 0 = 0 by hypothesis. This implies det(M ) = µ 0 + . . . = 0 and thus finishes the proof.

The Morse complex and half-infinite Morse trajectories
This section reviews the construction of the Morse complex as well as the compactified spaces of half-infinite Morse trajectories which will appear in all our moduli spaces.
3.1. Euclidean Morse-Smale pairs. The Morse complex can be constructed for any Morse-Smale pair of function and metric on a closed smooth manifold M (and more general spaces). However, we will also work with half-infinite Morse trajectories, and to obtain natural manifold with boundary and corner structures on these, we will restrict ourselves to the following special setting.
Definition 3.1. A Euclidean Morse-Smale pair on a closed manifold M is a pair (f, g) consisting of a smooth function f ∈ C ∞ (M, R) and a Riemannian metric g on M satisfying a normal form and transversality condition as follows.
(i) For every critical point p ∈ Crit(f ) of index |p| ∈ N 0 there exists a local chart φ to a neighbourhood of 0 ∈ R n such that (ii) For every pair of critical points p, q ∈ Crit(f ) the intersection of unstable and stable manifolds is transverse, W − p ⋔ W + q . Remark 3.2. One can check that given any Morse function f and metric g (e.g. one that is Euclidean in Morse normal coordinates around the critical points), any generic perturbation of g on annuli around the critical points yields a Euclidean Morse-Smale pair, see e.g. [BH,Prp.2] or [Sc1,Prp.2.24]. Hence Euclidean Morse-Smale pairs exist on every closed manifold, and for any given Morse function.
3.2. The Morse complex. For distinct critical points p − = p + ∈ Crit(f ) the space of unbroken Morse trajectories (which are necessarily nonconstant) is . It is canonically identified with the intersection of unstable and stable manifold modulo the Raction given by the flow of −∇f , or their intersection with a level set for any regular value c ∈ (f (p + ), f (p − )). Both formulations equip it with a canonical smooth structure of dimension |p − | − |p + | − 1, see e.g. [Sc1,§2.4.1]. Moreover, any choice of orientation of the unstable manifolds W − p for all p ∈ Crit(f ) induces orientations on the trajectory spaces M(p − , p + ) by e.g. [Wb,§3.4]. Then the Morse chain complex of (f, g) is obtained by counting (with signs induced by the orientations) the zero dimensional spaces of unbroken trajectories, It computes the singular homology of M ; see e.g. [Sc1,§4.3]. More precisely, the Morse complex is graded The PSS and SSP morphisms will be constructed on the Morse complex with coefficients in the Novikov field Λ from Section 2, (5) CM = CM Λ := CM Q ⊗ Λ = p∈Crit(f ) Λ p , with differential d = d Λ the Λ-linear extension of d Q (defined as above on generators). This complex is naturally graded with differential of degree 1, 3.3. Compactified spaces of Morse trajectories. Our construction of moduli spaces will also make use of the following spaces of half-infinite unbroken Morse trajectories for p ± ∈ Crit(f ) These will be equipped with smooth structures of dimension dim M(M, which identify the trajectory spaces with the unstable and stable manifolds M(M, p + ) ∼ = W + p+ resp. M(p − , M ) ∼ = W − p− . Note that these spaces contain constant trajectories at a critical point, {τ ≡ p + } ∈ M(M, p + ) and {τ ≡ p − } ∈ M(p − , M ). To compactify these trajectory spaces in a manner compatible with Morse theory, we cannot simply take the closure of the unstable or stable manifold W ± p± ⊂ M , but must add broken trajectories involving the bi-infinite Morse trajectories.
The bi-infinite trajectories from (3) which appear in such a compactification are always nonconstant, i.e. between distinct critical points p − = p + . So, unlike constant half-infinite length trajectories, our constructions will not involve constant bi-infinite trajectories, and we simplify subsequent notation by setting M(p, p) := ∅ for all p ∈ Crit(f ). With that we first introduce spaces of k-fold broken half-or bi-infinite Morse trajectories for k ∈ N 0 and p ± ∈ Crit(f ), Now the compactifications of the spaces of half-or bi-infinite Morse trajectories are given by with topology given by the Hausdorff distance between the images of the broken or unbroken trajectories. Compactness of these spaces is proven analogously to the bi-infinite Morse trajectory spaces in e.g. [BH,Prp.3], using [W1,Lemma 3.5]. Moreover, [W1,Lemma 3.3] shows that the evaluation maps extend continuously to ev : Smooth structures on these spaces are obtained by the following variation of a folk theorem, which is proven in [W1], using techniques similar to those of [BH] for the bi-infinite trajectory spaces. For reference, we recall the definition of a manifold with (boundary and) corners and its strata.
Definition 3.4. A smooth manifold with corners of dimension n ∈ N 0 is a second countable Hausdorff space M together with a maximal atlas of charts φ ι : homeomorphisms between open sets such that ∪ ι U ι = M ) whose transition maps are smooth.
For k = 0, . . . , n the k-th boundary stratum ∂ k M is the set of all x ∈ M such that for some (and hence every) chart the point φ ι (x) ∈ [0, ∞) n has k components equal to 0. such that T p M = T p W − ⊕ T p W + induces the orientation on M given by the symplectic form. This also induces orientations on M(p − , p + ) = W − p− ∩ W + p+ /R that are coherent (by e.g. [Wb,§3.4]) in the sense that the top strata of the oriented boundaries of the compactified Morse trajectory spaces are products ∂ 1 M(·, ·) = q∈Crit(f ) o(·, q, ·)M(·, q) × M(q, ·) with universal signs o(·, q, ·) = ±1. We compute the relevant cases: Here we identify N p+ W + p+ ∼ = T p+ W − p+ , and the outer normal direction is represented by ∇f , so that the sign arises from (ii) For computational purposes in §6.3 we moreover determine the fiber products of the compactified Morse trajectory spaces of critical points p − , p + ∈ Crit(f ) with the same Morse index To verify this recall that the compactifications M(p − , M ) and M(M, p + ) are constructed in (7) via broken flow lines involving bi-infinite Morse trajectories in M(p i , p i+1 ), which are (defined to be) nonempty only for only for |p 1 | > |p + |, and thus the image of the evaluation maps are contained in unions of unstable/stable manifolds

The PSS and SSP maps
In this section we construct the PSS and SSP morphisms in Theorem 1.2 between Morse and Floer complexes. As in the introduction, we fix a closed symplectic manifold (M, ω) and a smooth function H : S 1 ×M → R. This induces a time-dependent Hamiltonian vector field X H : S 1 → Γ(TM ), which we assume to be nondegenerate. Thus it has a finite set of contractible periodic orbits, denoted by P(H) as in (1). We moreover pick a Morse function f : M → R and denote its -again finite -set of critical points by Crit(f ). Then we will work with the Floer and Morse complexes over the Novikov field from Section 2, and construct the Λ-linear maps P SS : CM → CF , SSP : CF → CM from moduli spaces which we introduce in §4.1. We provide these moduli spaces with a compactification and polyfold description in §4.2, and in §4.3 rigorously construct the PSS/SSP map by using polyfold perturbations to obtain well defined (but still choice dependent) counts of compactified-and-perturbed moduli spaces.
4.1. The Piunikhin-Salamon-Schwarz moduli spaces. To construct the moduli spaces, we need to make further choices as follows.
The vector-field-valued 1-form Y H encodes the Floer equation on both the positive cylindrical end {z ∈ C | |z| ≥ e} ∼ = [1, ∞) × S 1 and the negative end {|z| ≥ e} ∼ = (−∞, −1] × S 1 (where β ≡ 1) as follows: The reparametrization v(s, t) := u(e ±(s+it) ) of a map u : C → M satisfies the Floer Then for u : C ± → M with lim R→∞ u(Re ±it ) = γ(t), denote by u#u γ : CP 1 → M the continuous map given by gluing u to u ± γ (where the ± denotes the orientation of D 2 ). By abuse of language, we will call A := [u#u γ ] = (u#u γ ) * [CP 1 ] ∈ H 2 (M ) the homology class represented by u. Moreover, we denote by u γ : D 2 → D 2 × M the graph of u γ . Then the graph u : makes sense for other maps v : C → C × M with the same asymptotic behaviour, and we say v represents A. In fact, we will suppress the notation A and label spaces with A -as this specifies the topological type of v.
Given such choices, the (choice-dependent) morphisms P SS : CM → CF and SSP : CF → CM will be constructed from the following moduli spaces for critical points p ∈ Crit(f ), periodic orbits γ ∈ P(H), and A ∈ H 2 (M ) Each of these moduli spaces can be described as the zero set of a Fredholm section Here the Banach manifolds B ± are given by a weighted Sobolev closure of the set of smooth maps u : C ± → M representing the homology class A with point constraint u(0) ∈ W ∓ p and satisfying a decay condition lim R→∞ u(Re ±it ) = γ(t), but not necessarily satisfying the perturbed Cauchy-Riemann equation where CZ(γ) is the Conley-Zehnder index with respect to a trivialization of u * γ TM as in e.g. [Sc2], c 1 (A) is the first Chern class of (T M, J) paired with A, and |p| is the Morse index of p ∈ Crit(f ).
If the moduli spaces were compact oriented manifolds, then we could define P SS (and analogously SSP ) by a signed count of the index 0 solutions, where the sum is over γ ∈ P(H) and A ∈ H 2 (M ) with I(p, γ; A) = 0. In many cases -if sphere bubbles of negative Chern number can be excluded -this compactness and regularity can be achieved by a geometric perturbation of the equation, e.g. in the choice of almost complex structure. In general, obtaining well defined "counts" of the moduli spaces requires an abstract regularization scheme. We will use polyfold theory to replace "#M(p, γ; A)" by a count of 0-dimensional perturbed moduli spaces. In the presence of sphere bubbles with nontrivial isotropy, the perturbations will be multi-valued, yielding rational counts.
Remark 4.1. Compactness, or rather Gromov-compactifications, of the moduli spaces M(p, γ; A) and M(γ, p; A) will result from energy estimates [MS,Remark 8.1.7] S 1 |β ′ (r)|(max x∈M H(θ, x) − min x∈M H(θ, x)) dθ dr since β ′ has compact support in [1, e]. Since moreover P(H) is a finite set, we obtain the above estimate with a finite constant K := R H β + max γ∈P(H) D 2 u * γ ω. Thus the energy of the perturbed pseudoholomorphic maps in each of our moduli spaces will be bounded since we fix [u#u γ ] = A. Now SFT-compactness [BEHWZ] asserts that for any C > 0 the set of solutions of bounded energy {u : C → M | ∂ J u = Y H (u), lim R→∞ u(Re ±it ) = γ(t), E(u) ≤ C} is compact up to breaking and bubbling. This compactness will be stated rigorously in polyfold terms in Assumption 5.5 (ii).

4.2.
Polyfold description of moduli spaces. We will obtain a polyfold description for the moduli spaces in §4.1 by a fiber product construction motivated by the natural identifications This couples the half-infinite Morse trajectory spaces from §3.3 with a space of perturbed pseudoholomorphic maps via the evaluation maps (8) and More precisely, the general approach to obtaining counts or more general invariants from moduli spaces such as (11) is to replace them by compact manifolds -or more general 'regularizations' which still carry 'virtual fundamental classes'). Polyfold theory offers a universal regularization approach after requiring a compactification M(. The Morse trajectory spaces are compactified and given a smooth structure in Theorem 3.3. The Gromov compactification and perturbation theory for (12) will be achieved by identifying theses spaces with moduli spaces that appear in Symplectic Field Theory (SFT) as introduced in [EGH], compactified in [BEHWZ,CM1], and given a polyfold description in [FH1]. Here we identify u : C → M with the map to its graph u : C → C × M, z → (z, u(z)) as in [MS,§8.1] to obtain a homeomorphism (in appropriate topologies) M ± (γ; A) ∼ = M ± SFT ( γ; A)/Aut(C ± ) to an SFT moduli space for the symplectic cobordism 1 C ± × M between ∅ and S 1 × M . Here S 1 × M is equipped with the stable Hamiltonian structure (±dt, ω + dH t ∧ dt) whose Reeb field ±∂ t + X Ht has simply covered Reeb orbits 2 given by the graphsγ : t → (±t, γ(t)) of the 1-periodic orbits γ ∈ P(H). Moreover, Aut(C ± ) is the action of biholomorphisms φ : C → C by reparametrization v → v • φ on the SFT space for an almost complex structure J ± H on C ± × M induced by J, X H , and j = ±i on C ± , More precisely, the asymptotic requirement is d C×M v(Re ±i(t+t0) ), γ R (t) → 0 for some t 0 ∈ S 1 as R → ∞ for the graphs γ R (t) = (Re ±it , γ(t)) of the orbit γ parametrized by S 1 ∼ = {|z| = R} ⊂ C ± . To express the evaluation (13) in SFT terms note that a holomorphic map in the given homology class intersects the holomorphic submanifold {0} × M in a unique point 3 , so we can fix the point 0 ∈ C ± in the domain where this intersection occurs and rewrite the moduli space M ± (γ; with a slicing condition and quotient by the biholomorphisms which fix 0 ∈ C ± . Thus we rewrite (11) into the fiber products over using evaluation maps on the SFT moduli space with one marked point 1 For definitions of these notions see [CM1,§2]. For C × M the positive symplectization end is R + × S 1 × M → C × M, (r, θ, x) → (e r+iθ , x). After reversing orientation on C there is an analogous negative end R − × S 1 × M ֒→ C − × M .
2 Here we have implicitly chosen asymptotic markers that fix a parametrization of each Reeb orbit. 3 For solutions in M ± SFT ( γ; A) this follows from pr C ± • v : C ± → C ± being an entire function with a pole of order 1 at infinity (prescribed by the asymptotics). For J ± H -holomorphic curves in the compactification, it follows from positivity of intersections, see e.g. [CM2,Prop.7.1]. Now we will obtain a polyfold description of the PSS/SSP moduli spaces (14) by the slicing construction of [Fi1] applied to polyfold descriptions of the SFT-moduli spaces M ± SFT ( γ; A) (compactified as space of pseudoholomorphic buildings with one marked point). This result is outlined in [FH], but to enable a self-contained proof of our results, we formulate it as assumption, where we use as target factor for a simplified evaluation map, as explained in the following remark.
We do not need a precise description of this compactification (beyond the fact that it exists and is cut out by a sc-Fredholm section), but it affects the formulation of the evaluation maps [v, z 0 ] → v(z 0 ) for a marked point z 0 ∈ Σ that v might map to a cylinder factor R×S 1 ×M ⊂ Σ×M . We will simplify the resulting sc ∞ evaluation with varying target -being developed in [FH1] -to a continuous evaluation map ev ± : M ± SFT (γ; A) → C ± into the compactified target C ± . For that purpose we topologize C ± ∼ = {|z| ≤ 1} as a disk via a diffeomorphism C ± → {|z| < 1}, re iθ → f (r)e iθ induced by a diffeomorphism f : [0, ∞) → [0, 1) that is the identity near 0, and its extension to a homeomorphism C ± → {|z| ≤ 1} via S 1 = R / 2πZ → {|z| = 1}, θ → e ±iθ . Then for any marked point z 0 ∈ R × S 1 on a cylinder we project the evaluation v(z 0 ) ∈ R × S 1 × M to S 1 × M = ∂ C ± × M by forgetting the R-factor. The resulting simplified evaluation map will be unchanged and thus still sc ∞ when restricted to the open subset (ev ± ) −1 (C ± × M ) of the ambient polyfold -as stated in (v) below. This open subset inherits a scale-smooth structure, and still contains some broken curves -just not those on which the marked point leaves the main component. This suffices for our purposes since the fiber product construction uses the evaluation map only in an open set of curves [v, z 0  (iv) The intersection of the zero set with the dense subset σ −1 (v) The sections σ SFT have tame sc-Fredholm representatives in the sense of [Fi1,Def.5.4], and the evaluation maps ev ± restrict on the open subsets B ±,C SFT (γ; which are σ SFT -compatibly submersive in the sense of Definition A.4. Finally, this open subset contains the interior, ∂ 0 B ± SFT (γ; A) ⊂ B ±,C SFT (γ; A). Remark 4.4. The properties (iii),(iv) in this assumption are stated only to give readers an intuitive sense of what spaces we are working with. For the specific application in this paper it would be sufficient to assume the existence of sc-Fredholm sections and submersive maps, along with the further sections, maps, and boundary stratifications stated in Assumptions 5.5 and 6.3. Property (v), which is used to construct fiber products in Lemma 4.5, should follow similarly to the explanation given in [Fi1,Ex.5.1] for the Gromov-Witten polyfolds [HWZ1].
We also expect the existence of a direct polyfold description of the moduli space (12) in terms of a collection of sc-Fredholm sections σ : B ± (γ; A) → E ± (γ; A) with the same indices, and submersive sc ∞ maps ev ± : B ± (γ; A) → M with the following simplified properties.
(iii') The smooth maps u : C → M which equal u(Re ±it ) = γ(t) for sufficiently large R > 1 and represent the class A form a dense subset of B ± (γ; A) that is contained in the interior. On this subset, the section is σ(u) = ∂ J u − Y H (u), and the evaluation is ev ± (u) = u(0).
(iv') The intersection of M ± (γ; A) := σ −1 (0) with the dense subset from (iii') is naturally identified with the moduli space M ± (γ; A) in (12). While such a construction should follow from the same construction principles as in [FH], there is presently no writeup beyond [W2], which proves the Fredholm property in a model case. Alternatively, one could abstractly obtain this construction from restricting the setup in Assumption 4.3 to subsets consisting of maps of the form v(z) = (z, u(z)). Thus there is no harm in using (iii') and (iv') as intuitive guide for our work with the abstract setup.
Given one or another polyfold description of the naturally identified moduli spaces (12) or (15) and corresponding evaluation maps, we will now extend the identifications (11) or (14) to a fiber product construction of polyfolds which will contain these PSS/SSP moduli spaces. For p ∈ Crit(f ), γ ∈ P(H), and A ∈ H 2 (M ) we define the topological spaces . We will use [Fi1] to equip these spaces with natural polyfold structures and show that the pullbacks of the sections σ SFT by the projections to B ± SFT (γ; A) yield sc-Fredholm sections whose zero sets are compactifications of the PSS/SSP moduli spaces. This will require a shift in levels which is of technical nature as each m-level B m ⊂ B contains the dense "smooth level" B ∞ ⊂ B m , which itself contains the moduli space . Moreover, pullback of the sc-Fredholm sections of strong polyfold bundles σ ± SFT : resp. I(γ, p; A) given in (9). Their zero sets contain 5 the moduli spaces from §4.1, −1 (0) is compact, and given any p ∈ Crit(f ), γ ∈ P(H), and C ∈ R, there are only finitely many A ∈ H 2 (M ) with ω(A) ≤ C and nonempty zero set σ ± Proof. We will follow [Fi1,Cor.7.3] to construct the PSS polyfold, bundle, and sc-Fredholm section σ + p,γ;A in detail, and note that the construction of the SSP section σ − γ,p;A is analogous. Consider an ep-groupoid representative X = (X, X) of the polyfold B + SFT (γ; A) with source and target maps denoted s, t : X → X together with a strong bundle P : W → X over the M -polyfold X and a structure map µ : X s × P W → X such that the pair (P, µ) is a strong bundle over X representing the polyfold bundle E + SFT (γ; A) → B + SFT (γ; A). In addition, consider a sc-Fredholm 4 Here we can only make statements about the smooth level because we do not know what points of other levels are included in the fiber products. This is sufficient for applications as the zero set of any sc-Fredholm section (and its admissible perturbations) is contained in the smooth level. 5 As in Remark 4.4, this identification is stated for intuition and will ultimately not be used in our proofs.
section functor S SFT : X → W of (P, µ) that represents σ + SFT . The ep-groupoid X and the bundle (P, µ) are tame, since they represent a tame polyfold and a tame bundle, respectively. Moreover, S SFT is a tame sc-Fredholm section in the sense of [Fi1,Def.5.4

] by Assumption 4.3(v).
We view the Morse moduli space M(p, M ) as the object space of an ep-groupoid with morphism space another copy of M(p, M ) and with unit map a diffeomorphism; that is, the only morphisms are the identity morphisms. The unique rank-0 bundle over M(p, M ) is a strong bundle in the epgroupoid sense, and the zero section of this bundle is a tame sc-Fredholm section functor. Next, note thatB within the open subset X ev := (ev + ) −1 (C × M ) ⊂ X and the corresponding full ep-subgroupoid X ev of X , which represent the open subset B +,C SFT (γ, A) ⊂ |X|, and by Assumption 4.3(v) the restricted evaluation ev + : X ev → C × M is sc ∞ and S SFT -compatibly submersive (see Definition A.4). Denote by ev 0 : M(p, M ) → C × M, τ → (0, ev(τ )) the product of the trivial map to 0 ∈ C and the Morse evaluation map. We claim that the product map ev 0 × ev + : Fi1,Lem.7.1] for a discussion of the sc-Fredholm property on Cartesian products.) Next, note that M(p, x) ev0 × ev + X ev ∞ represents the smooth level of the fiber product topological spaceB + (p, γ; A). So [Fi1,Cor.7

.3] yields an open neighbourhood
1 is a tame ep-groupoid and the pullbacks of (P, µ) and S SFT to X ′ are a tame bundle and tame sc-Fredholm section. Here we used the fact that the smooth level M(p, x) ∞ = M(p, x) of any finite dimensional manifold is the manifold itself; see Remark A.3.

Construction of the morphisms.
To construct the Λ-linear maps PSS and SSP in Theorem 1.2 with relatively compact notation we index all moduli spaces from §4.1 by the two sets To simplify notation we then denote I := I − ∪ I + and drop the superscripts from the polyfolds B(α) = B ± (α). Since Lemma 4.5 provides each moduli space M(α) for α ∈ I with a compactification and polyfold description M(α) ⊂ σ −1 α (0), we can apply [HWZ,Theorems 18.2,18.3,18.8] to obtain admissible regularizations of the moduli spaces, and counts of the 0-dimensional perturbed solution spaces [HWZ,§15.4], in the following sense. Here we denote by Q + := Q ∩ [0, ∞) the groupoid with only identity morphisms.
(v) For every α ∈ I with Fredholm index I(α) = 1 and κ α : W α → Q + as in (ii) the boundary of the perturbed zero set is given by its intersection with the first boundary stratum of the polyfold, The statements in (iv) and (v) of Corollary 4.6 require orientations of the sections σ α for α ∈ I. By the fiber product construction in Lemma 4.5 they do indeed inherit orientations from the orientations of the Morse trajectory spaces in Remark 3.5, the orientations of σ ± SFT given in Assumption 4.3, and an orientation convention for fiber products.
In practice, we will construct the perturbations κ in Corollary 4.6 by pullback of perturbations λ = (λ ± γ,A ) γ∈P,A∈H2(M) of the oriented SFT-sections σ ± SFT . Thus it suffices to specify the orientations of the regularized zero sets, which is implicit in their identification with transverse fiber products of oriented spaces over the oriented manifold M ,

Orientations of the boundary restrictions in (v) are then induced by the orientations of
is an exterior normal vector at z ∈ ∂Z κ (α). (ii) Note that the counts in part (iv) of this Corollary may well depend on the choice of the multi-valued perturbations κ α -unless the ambient polyfold has no boundary, ∂B(α) = ∅. Indeed, although the moduli space M(α) is expected to have dimension 0, it may not be cut out transversely from the ambient polyfold B(α), and moreover it may not be compact. Assumption 4.3 provides an inclusion in a compact set M(α) ⊂ σ −1 α (0), and the perturbation theory for sc-Fredholm sections of strong bundles then associates to σ −1 α (0) a perturbed zero set Z κ (α) ⊂ B(α) with weight function κ α •S α : Z κ (α) → Q∩(0, ∞). This process generally adds points on the boundary σ −1 α (0) M(α) ⊂ B(α) ∂ 0 B(α), which may or may not persist under variations of the perturbation κ α .
The following construction of morphisms will depend on the choices of perturbations and orientation convention (see the previous remark) as well as geometric data fixed in §4.1, and possibly the choice of polyfold construction in Assumption 4.3 and ep-groupoid representation in Remark A.2.
The algebraic properties in Theorem 1.2 will be achieved in §6 -for any given choice of geometric data -by particular choices of ep-groupoids and perturbations κ ± , and an overall sign adjustment.
Definition 4.8. Given collections κ ± = (κ ± α ) α∈I ± of admissible sc + -multisections in general position as in Corollary 4.6, we define the maps P SS κ + : CM → CF and SSP κ − : CF → CM to be the Λ-linear extension of Lemma 4.9. The maps P SS κ + : CM → CF and SSP κ − : CF → CM in Definition 4.8 are well defined, i.e. the coefficients take values in the Novikov field Λ defined in §2.
Proof. To prove that P SS κ + is well defined we need to check finiteness of the following set for any p ∈ Crit(f ), γ ∈ P(H), and c ∈ R, Here ω : H 2 (M ) → R is given by pairing with the symplectic form on M , and recall from Lemma 4.5 that there are only finitely many homology classes A ∈ H 2 (M ) with ω(A) ≤ c and σ −1 α (0) = ∅. On the other hand, the perturbations κ + were chosen in Corollary 4.6 (iii),(iv) so that #Z κ + (. . . ; A) = 0 whenever σ −1 α (0) = ∅. Thus there are in fact only finitely many A ∈ H 2 (M ) with ω(A) ≤ c and #Z κ + (. . . ; A) = 0, which proves the required finiteness. The proof for SSP κ − is analogous.

The chain homotopy maps
In this section we construct Λ-linear maps ι : CM → CM and h : CM → CM on the Morse complex over the Novikov field Λ given in (5), which appear in Theorem 1.2. For that purpose we again fix a choice of geometric data as in §4.1 to construct moduli spaces in §5.1 and §5.2. We equip these with polyfold descriptions in §5.3, and define the maps ι, h for admissible regular choices of perturbations in Definitions 5.7. To obtain the algebraic properties claimed in Theorem 1.2 (i)-(iii) we will then construct particular "coherent" choices of perturbations in §6.

5.1.
Moduli spaces for the isomorphism ι. We will construct ι : CM → CM from the following moduli spaces for critical points p − , p + ∈ Crit(f ), A ∈ H 2 (M ), using the almost complex structure J and the unstable/stable manifolds (see §3.3) of the Morse-Smale pair (f, g) chosen in §4.1, Note that a cylinder acts on this moduli space by reparametrization with biholomorphisms of CP 1 that fix the two points [1 : 0], [0 : 1]. However, we do not quotient out this symmetry so describe these moduli spaces as the zero set of a Fredholm section over a Sobolev closure of the set of smooth maps u : As in §4.2 we will obtain a compactification and polyfold description of this moduli space by identifying it with a fiber product of Morse trajectory spaces and a space of pseudoholomorphic curves, in this case the space of parametrized J-holomorphic spheres with evaluation maps for z 0 ∈ CP 1 , With this we can describe the moduli space (17) as fiber product with the half-infinite Morse trajectory spaces from §3.3, using z + 0 := [1 : 0] and z − 0 : Note here that we are not working with a Gromov-Witten moduli space, as we do not quotient by Aut(CP 1 ). This is due to the chain homotopy in Theorem 1.2 (iii), which will result from identifying a compactification of M(A) with a boundary of the neck-stretching moduli space M SFT (A) in (26) that appears in Symplectic Field Theory [EGH]. For that purpose we identify a solution u : CP 1 → M with the map to its graph u : CP 1 → CP 1 × M, z → (z, u(z)) as in [MS,§8.1]. This yields is a bijection (and homeomorphism in appropriate topologies) . This uses the evaluation maps from a Gromov-Witten moduli space with two marked points, The polyfold setup in [HWZ1, Theorems 1.7,1.10,1.11] for Gromov-Witten moduli spaces now provides a strong polyfold bundle E GW (A) → B GW (A), and oriented sc-Fredholm section σ GW : which implicitly carries the two marked points z ± 0 ∈ CP 1 . Nodal curves in M GW (A) then explicitly come with the data of two marked points on their domain. On the dense subset the section is given by The setup in [HWZ1, Theorem 1.8] moreover provides sc ∞ evaluation maps ev ± : B GW (A) → CP 1 × M at the marked points, which on the dense subset are given by ev ± ([v]) = v(z ± 0 ). Thus we have given each factor in the fiber product (20) a compactification 6 that is either a manifold with corners given by the compactified Morse trajectory spaces in Theorem 3.3, or the compact zero set M GW (A) = σ −1 GW (0) of a sc-Fredholm section. In §5.3 we will combine the polyfold description of the Gromov-compactification of (21) with an abstract construction of fiber products in polyfold theory [Fi1] to obtain compactifications and polyfold descriptions of the moduli spaces. Then the construction of ι : CM → CM proceeds as in §4.3. The algebraic properties of ι in Theorem 1.2 (i) and (ii) will follow from the boundary stratifications of the Morse trajectory spaces M(p − , M ) and M(M, p + ) since the ambient polyfold B GW (A) has no boundary. However, this requires specific "coherent" choices of perturbations in §6.
Remark 5.1. Gromov-compactifications of the moduli spaces M ι (p − , p + ; A) will result from the energy identity [MS,Lemma 2.2.1] for solutions of ∂ J u = 0, ). This fixes the energy of solutions on each solution space M(A), and Gromov compactness asserts which is compact and cut out transversely.
Translated to graphs in CP 1 × M with two marked points, this means M GW (0) ≃ CP 1 × CP 1 × M by adding two marked points in the domain. That is, (z − , z + , x) ∈ CP 1 × CP 1 × M corresponds to the (equivalence class of) graphs u x : z → (z, x) with two marked points z − , z + ∈ CP 1 . For z − = z + this tuple can be reparametrized to the fixed marked points z − 0 , z + 0 ∈ CP 1 and then represents an Aut(CP 1 , z − 0 , z + 0 )-orbit. For z − = z + the tuple (z − , z + , x) corresponds to a stable map in M GW (0), given by the graph u x with a node at z − = z + attached to a constant sphere with two distinct marked points. This will be stated in polyfold terms in Assumption 5.5 (ii). 6 The term 'compactification' applied to spaces of pseudoholomorphic curves is always to be understood as Gromovcompactification, as M GW (A) ⊂ M GW (A) may not be dense.

5.2.
Moduli spaces for the chain homotopy h. To construct the moduli spaces from which we will obtain h : CM → CM , we again use the almost complex structure J and Morse-Smale pair (f, g) chosen in §4.1. In addition, we fixed an anti-holomorphic vector-field-valued 1-form Y H ∈ Ω 0,1 (C, Γ(TM )) that arises from the fixed Hamiltonian function H : S 1 × M → R and a choice of smooth cutoff function β : [0, ∞) → [0, 1] with β| [0,1] ≡ 0, β ′ ≥ 0, and β| [e,∞) ≡ 1. Gluing this 1-form to another copy of Y H over C − with neck length R > 0 in exponential coordinates yields the anti-holomorphic vector-field-valued Here ∞). Now perturbing the Cauchy-Riemann operator on CP 1 by Y R H yields the following moduli spaces for critical points and we will construct h from their union It is constructed so that it has the following properties: (17) from which ι will be constructed.
Towards a compactification and polyfold description of these moduli spaces we again -as in §4.2, §5.1, [MS, §8.1] -identify a solution u : CP 1 → M with the map to its graph. Moreover, we again fix marked points z + 0 = [1 : 0], z − 0 = [0 : 1] to implement evaluation maps to express the conditions u(z ∓ 0 ) ∈ W ± p± . This yields a homeomorphism (in appropriate topologies) between the moduli space (23) and the fiber product over CP 1 × M with the half-infinite Morse trajectory spaces from §3.3, (20) this replaces the Gromov-Witten moduli space in (21) with a family of moduli spaces for almost complex structures Here, again, we implicitly include the two marked points z ± 0 ∈ CP 1 . Then, for R → ∞, the degeneration of the PDE ∂ J R H v = 0 is the "neck stretching" 7 considered more generally in Symplectic Field Theory [EGH]. The evaluation maps from (21) directly generalize to . Now, as in §5.1, each factor in the fiber product (25) has natural compactifications -either the compactified Morse trajectory spaces from Theorem 3.3, or the compact zero set M SFT (A) = σ −1 SFT (0) of a sc-Fredholm section that we will introduce in §5.3. Combined with the construction of fiber products in polyfold theory [Fi1] this will yield compactifications and polyfold descriptions of the moduli spaces (23), and the construction of h : CM → CM then again proceeds as in §4.3. Establishing the algebraic properties in Theorem 1.2 relating h with ι and SSP • P SS will moreover require an in-depth discussion of the boundary stratification of the polyfold domains B SFT (A) of these sections, and "coherent" choices of perturbations in §6.
Remark 5.3. Gromov-compactifications of the moduli spaces M(p − , p + ; A) will result from energy estimates [MS,Remark 8 . This proves (28), and thus establishes energy bounds on the perturbed pseudoholomorphic maps in each of our moduli spaces, where we fix [u] = A. Now SFT-compactness [BEHWZ] asserts that for any C > 0 the set of solutions of bounded energy R∈[0,∞) ≤ C} is compact up to breaking and bubbling. This compactness will be stated rigorously in polyfold terms in Assumption 5.5 (ii). 5.3. Construction of the morphisms. In this section we construct the Λ-linear maps ι : CM → CM and h : CM → CM analogously to §4.3 by first obtaining compactifications and polyfold descriptions for the moduli spaces in §5.1 and §5.2 as in §4.2. This construction is motivated by the fiber product descriptions of the moduli spaces in (20), (25), which couple Morse trajectory spaces from §3.3 with moduli spaces of pseudoholomorphic curves in CP 1 × M via evaluation maps (21), (27). Polyfold descriptions of these moduli spaces and their properties are stated in the following Assumption 5.5 for reference, with proofs in [HWZ1] resp. outlined in [FH]. Here we formulate the evaluation map in the context of neck stretching, as explained in the following remark, using a splitting of the sphere as topological space with smooth structures on the complement of the equator Remark 5.4. (i) Recall from §5.1 that we denote by B GW (A) a Gromov-Witten polyfold of curves in class [CP 1 ] + A ∈ H 2 (CP 1 × M ) with 2 marked points. These are determined by A ∈ H 2 (M ) as we model graphs of maps CP 1 → M , but should not be confused with a polyfold of curves in M . In particular, B GW (A) never contains constant maps and hence is well defined for A = 0. The properties of the Gromov-Witten moduli spaces for ω(A) ≤ 0 are spelled out abstractly in (ii) below; for the geometric meaning see Remark 5.1. 7 Strictly speaking, R ∈ [0, 2] parametrizes a family of Gromov-Witten moduli spaces for varying almost complex structure. At R = 2, the manifold S 1 × M with its stable Hamiltonian structure (see §4.2) embeds as a stable hypersurface in CP 1 × M . Then R ∈ [2, ∞) parametrizes the SFT neck-stretching.
(ii) The SFT polyfolds B SFT (A) will similarly describe curves in class [CP 1 ] + A in a neck stretching family of targets (CP 1 R × M ) R∈[0,∞] as in [BEHWZ,§3.4], given by Here we identify the boundaries of the closed unit disks D ± = {z ∈ C | |z| ≤ 1} with the boundary components of the necks E R via To describe convergence and evaluation maps we also embed each CP 1 For R = 0 this is to be understood as    (A), and on which the section is given by On these dense subsets, ev ± ([v]) resp. ev ± (R, [v]) is the evaluation as in (27).
(v) The intersection of the zero sets with the dense subsets σ −1  (A), this map coincides with the Gromov-Witten evaluations ev + × ev − viewed as maps Here the properties (iv),(v) are stated to give an intuitive sense of what spaces we are working with. The polyfold description σ GW : B GW (A) → E GW (A) is developed for the homology classes [CP 1 ] + A ∈ H 2 (CP 1 × M ) in [HWZ1], and assumption (v) should follow similarly to the explanation given in [Fi1,Ex.5.1] for the Gromov-Witten polyfolds [HWZ1]. Given any such polyfold descriptions of the moduli spaces of pseudoholomorphic curves, we now extend the fiber product descriptions of the moduli spaces M (ι) (p − , p + ; .1 and §5.2 to obtain ambient polyfolds which contain compactifications of the moduli spaces. Towards this we define for each p − , p + ∈ Crit(f ) and A ∈ H 2 (M ) the topological spaces , where the last equality stems from the identification at the end of Remark 5.4 (ii). Then the abstract fiber product constructions in [Fi1] will be used as in Lemma 4.5 to obtain the following polyfold description for compactifications of the moduli spaces in §5.1 and §5.2.
Lemma 5.6. Given any p − , p + ∈ Crit(f ) and A ∈ H 2 (M ), there exist open subsets B ι (p − , p + ; A) ⊂ B ι (p − , p + ; A) 1 and B(p − , p + ; A) ⊂B(p − , p + ; A) 1 which contain the smooth levelsB (ι) (p − , p + ; A) ∞ of the fiber products and inherit natural polyfold structures with smooth level of the interior , and a scale-smooth inclusion  (18). Further, these are related via the inclusion φ ι by natural orientation preserving identification σ ι . The zero sets of these sc-Fredholm sections contain 8 the moduli spaces from §5.1 and §5.2, −1 (0) is compact, and given any p ± ∈ Crit(f ) and C ∈ R, there are only finitely many A ∈ H 2 (M ) with ω(A) ≤ C and nonempty zero set σ is a sc ∞ inclusion by Assumption 5.5 (iii). Apart from further relations involving φ ι , the proof is directly analogous to the fiber product construction in Lemma 4.5, using Assumption 5.5 -in particular the sc ∞ and σ SFT -compatibly submersive evaluation map (29) ) using Assumption 5.5 (iii). Finally, the index of the induced section σ (p−,p+; A) , and similarly of σ ι (p−,p+;A) , is computed by [Fi1,Cor.7 Given this compactification and polyfold description of the moduli spaces M(α) ⊂ σ −1 α (0) and M ι (α) ⊂ σ ι α −1 (0) for all tuples in the indexing set we can again apply [HWZ,Theorems 18.2,18.3,18.8] to the sc-Fredholm sections σ α and σ ι α and obtain Corollary 4.6 verbatim for these collections of moduli spaces. In §6 we will moreover make use of the fact that σ ι α = φ * ι σ α arises from restriction of σ α , so admissible perturbations of σ α pull back to admissible perturbations of σ ι α . For now, we choose perturbations independently and thus as in Definition 4.8 obtain perturbation-dependent, and not yet algebraically related, Λ-linear maps.
Remark 5.8. The determination in Corollary 4.6 of #Z κ (p − , p + ; A), #Z κ ι (p − , p + ; A) ∈ Q that is used in Definition 5.7 requires an orientation of the sections σ (p−,p+;A) and σ ι (p−,p+;A) . As in Remark 4.7 this is determined via the fiber product construction in Lemma 5.6 from the orientations of the Morse trajectory spaces in Remark 3.5 (i) and the orientations of σ GW , σ SFT given in Assumption 5.5. In practice, we will construct the perturbations κ, κ ι by pullback of perturbations λ = (λ A ) A∈H2(M) of the SFT-sections σ SFT and their restriction λ ι to {0} × B GW (A) ⊂ ∂B SFT (A). So we can specify the orientations of the regularized zero sets by expressing them as transverse fiber products of oriented spaces over CP 1 × M or C ± × M , (τ )).

Algebraic relations via coherent perturbations
In this section we prove parts (i)-(iii) of Theorem 1.2, that is the algebraic properties which relate the maps P SS : CM → CF , SSP : CF → CM constructed in §4, and the maps ι : CM → CM , h : CM → CM constructed in §5. More precisely, we will make so-called "coherent" choices of perturbations in §6.2, §6.3, and §6.4 which guarantee that (i) ι is a chain map, (ii) ι is a Λ-module isomorphism, and (iii) h is a chain homotopy between the composition SSP • P SS and ι.
6.1. Coherent polyfold descriptions of moduli spaces. The general approach to obtaining not just counts as discussed in §4.2 but well-defined algebraic structures from moduli spaces of pseudoholomorphic curves is to replace them by compact manifolds with boundary and corners (or generalizations thereof which still carry 'relative virtual fundamental classes') in such a manner that their boundary strata are given by Cartesian products of each other. In the context of polyfold theory, this requires a description of the compactified moduli spaces M(α) = σ −1 α (0) as zero sets of a "coherent collection" of sc-Fredholm sections σ α : B(α) → E(α) α∈I of strong polyfold bundles. Here "coherence" indicates a well organized identification of the boundaries ∂B(α) with unions of Cartesian products of other polyfolds in the collection I, which is compatible with the bundles and sections. A general axiomatic description of such coherent structures is being developed in [Fi2].
As a first example, the moduli spaces M ι (p − , p + ; A) in §5.1 which yield the map ι : CM → CM are given polyfold descriptions σ ι (p−,p+;A) : B ι (p − , p + ; A) → φ * ι E(p − , p + ; A) in Lemma 5.6 that arise as fiber products with polyfolds B GW (A) without boundary. Thus their coherence properties stated below follow from properties of the fiber product in [Fi1] and the boundary stratification of the Morse trajectory spaces in Theorem 3.3. We state this result to illustrate the notion of coherence. The full technical statement -on the level of ep-groupoids and including compatibility with bundles and sections -can be found in the second bullet point of Lemma 6.4. Lemma 6.1. For any p ± ∈ Crit(f ) and A ∈ H 2 (M ) the smooth level of the first boundary stratum of the fiber product B ι (p − , p + ; A) in Lemma 5.6 is naturally identified with Proof. By the fiber product construction [Fi1,Cor.7.3] of B ι (p − , p + ; A) in Lemma 5.6, the degener- Here we also used the identification of the interior smooth level in Lemma 5.6.
Next, the polyfold description in Lemma 5.6 for the moduli spaces M(p − , p + ; A) in §5.2, which yield the map h : CM → CM , are obtained as fiber products of the Morse trajectory spaces with polyfold descriptions σ SFT : B SFT (A) → E SFT (A) of SFT moduli spaces given in [FH,FH1]. We will state as assumption only those parts of their coherence properties that are relevant to our argument in §6. (ii) The union of the images l γ,A± B +,C SFT (γ; When restricted to the interiors, this yields a disjoint cover of the top boundary stratum, The restriction of ev ± : B SFT (A) → CP 1 ∞ to each boundary face im l γ,A± ⊂ ∂B SFT (A) takes values in C ± ⊂ CP 1 ∞ , and its pullback under l γ,A± coincides with ev ± : B ± SFT (γ; A ± ) → C ± × M . Moreover, pullback of the restricted sc ∞ evaluations ev + ×ev − : . This identification reverses the orientation of sections.
This identification preserves the orientation of sections. 6.2. Coherent perturbations for chain map identity. In this section we prove Theorem 1.2 (i), that is we construct ι κ ι in Definition 5.7 as a chain map on the Morse complex (5) with differential d : CM → CM given by (4). This requires the following construction of the perturbations κ ι that is coherent in the sense that it is compatible with the boundary identifications of the polyfolds B ι (p − , p + ; A) in Lemma 6.1. Here we will indicate smooth levels by adding ∞ as superscriptdenoting e.g. X ι,∞ p−,p+;A as the smooth level of an ep-groupoid representing B ι (p − , p + ; A) ∞ .
• Each κ ι α : W ι α → Q + for α ∈ I is an admissible sc + -multisection of a strong bundle P α : W ι α → X ι α that is in general position to a sc-Fredholm section functor S ι α : X ι α → W ι α which represents σ ι α | Vα on an open neighbourhood V α ⊂ B ι (α) of the zero set σ −1 α (0). • The identification of top boundary strata in Lemma 6.1 holds for the representing ep-groupoids, and the oriented section functors S ι α : X ι α → W ι α are compatible with these identifications in the sense that the restriction of S ι p−,p+;A to any face F ∞ (p−,q−),α ′ : and natural map pr * F : pr * F W ι α ′ → W ι α ′ . For any such choice of κ ι = (κ ι α ) α∈I , the resulting map ι κ ι : CM → CM in Definition 5.7 satisfies ι κ ι • d + d • ι κ ι = 0. By setting ι p := (−1) |p| ι κ ι p we then obtain a chain map ι : Proof. We will first assume the claimed coherence and discuss the algebraic consequences up to signs, then construct the coherent data, and finally use this construction to compute the orientations.
Construction of chain map: Assuming ι κ ι • d + d • ι κ ι = 0, recall that d decreases the degree on the Morse complex (6) by 1. Thus ι : C * M → C * M defined as above satisfies for any q ∈ Crit(f ) By Λ-linearity this proves ι • d = d • ι on C * M .
This finishes the identification of the boundary ∂Z κ ι (α). Now Corollary 4.6 (v) asserts that the sum of weights over this boundary is zero -when counted with signs that are induced by the orientation of Z κ ι (α). So in order to prove the identity (30) we need to compare the boundary orientation of ∂Z κ ι (α) with the orientations on the faces. We will compute the relevant signs in (31) below, after first making coherent choices of representatives S ι α : X ι α → W ι α of the oriented sections σ ι α , and constructing coherent sc + -multisections κ ι α : W ι α → Q + for α ∈ I. Coherent ep-groupoids, sections, and perturbations: Recall that the fiber product construction in Lemma 5.6 defines each bundle W ι α = pr * α W GW

→ X GW
A under a projection of ep-groupoids -with abbreviated notation Then the identification of the top boundary stratum proceeds exactly as the proof of Lemma 6.1. Coherence of the bundles and sections follows from coherence of the projections pr α : X ι α → X GW A in the sense that pr α | F ∞ = pr α ′ • pr F for all smooth levels of faces F ⊃ F ∞ ⊂ ∂ 1 X ι α and their projections pr F :  (8), which determine ev ± 0 (τ ) = (z ± 0 , ev(τ )). Thus we obtain transverse fiber products

Construction of admissible
M(M, p + ) for every α ∈ I. This translates into the pullbacks κ ι α = λ GW A • pr * α being in general position to the pullback sections S ι α for α ∈ I. Moreover, κ ι α is admissible with respect to a pullback of (N A , U A ), so the perturbed zero set is a compact weighted branched orbifold for each α = (p − , p + ; A), This finishes the construction of coherent perturbations.

Computation of orientations:
To prove the identity (30) it remains to compute the effect of the orientations in Remark 5.8 on the algebraic identity in Corollary 4.6 (v) that arises from the boundary ∂Z κ ι (α) of the 1-dimensional weighted branched orbifolds arising from regularization of the moduli spaces with index I ι (α) = I ι (p − , p + ; A) = 1. Here Z λ GW A is of even dimension and has no boundary since the Gromov-Witten polyfolds in Assumption 5.5 have no boundary, and the index of σ GW is even. For the Morse trajectory spaces, the boundary strata are determined in Theorem 3.3, with relevant orientations computed in Remark 3.5. Thus for I ι (α) = |p − | − |p + | + 2c 1 (A) = 1 we can compute orientations -at the level of well defined finite dimensional tangent spaces at a solution; in whose neighbourhood the evaluation maps are guaranteed to be scale-smooth - Here the signs in the first equality arise from the ambient Cartesian product in the second equality we used Remark 3.5; and in the final equality we use |p − | + |p + | + 1 ≡ I ι (α) = 1 ≡ 0 modulo 2. This finishes the computation of the oriented boundaries ∂Z κ ι (α) for I ι (α) = 1 that proves (30) and thus yields a chain map.
6.3. Admissible perturbations for isomorphism property. In this section we prove Theorem 1.2 (ii), i.e. construct ι = (−1) * ι κ ι : C * M → C * M in Definition 5.7 and Lemma 6.4 as a Λ-module isomorphism on the chain complex CM = CM Λ over the Novikov field as in (5). This requires a construction of the perturbations κ ι that preserves the properties of the zero sets in Remark 5.1 for nonpositive symplectic area ω(A) ≤ 0.
Lemma 6.5. The coherent collection of sc + -multisections κ ι in Lemma 6.4 can be chosen such Proof. The sc + -multisections κ ι in Lemma 6.4 are obtained from choices of sc + -multisections (κ A : We will first consider α = (p − , p + ; A) ∈ I for nontrivial homology classes A ∈ H 2 (M ) {0} with nonpositive symplectic area ω(A) ≤ 0. Recall from Remark 5.1 that these moduli spaces are empty |S −1 A (0)| = ∅, so as in Corollary 4.6 we can choose empty neighbourhoods ∅ = |U A | ⊂ |X GW A | to control compactness. Then the perturbed zero set Z(κ A ) = |{x ∈ X A | κ A (S A (x)) > 0}| ⊂ |U A | is forced to be empty, i.e. κ A • S A ≡ 0. This is an allowed choice in Lemma 6.4 since evaluation maps from an empty set are trivially transverse to any submanifold. This choice induces for any p ± ∈ Crit(f ) in α = (p − , p + ; A) an induced sc + -multisection κ ι α = κ A • pr * α : W ι α → Q + . Its perturbed zero set is Next we consider A = 0 ∈ H 2 (M ) and recall from Remark 5.1 and Assumption 5.5 (ii) that the Gromov-Witten moduli space M GW (0) = Z(κ 0 ) is already compact and transversely cut out. Thus the trivial sc + -multisection κ 0 : W 0 → Q + , given by κ 0 (0 x ) = 1 on zero vectors 0 x ∈ (W 0 ) x and κ 0 | (W0)x {0x} ≡ 0, is an admissible sc + -multisection in general position to S 0 : X GW 0 → W 0 . Recall moreover that the evaluation maps on the unperturbed zero set are In the CP 1 -factors this is submersive so transverse to the fixed points (z − 0 , z + 0 ) ∈ CP 1 × CP 1 . In the M -factors this is the diagonal map, which is transverse to the unstable and stable manifolds W − p− × W + p+ ⊂ M × M for any pair p − , p + ∈ Crit(f ) by the Morse-Smale condition on the metric on M chosen in §3. Thus the trivial multisection κ 0 is in fact an allowed choice in Lemma 6.4. Now with this choice, the tuples (p − , p + ; 0) ∈ I for which we need to compute are those with 0 = I ι (p − , p + ; 0) = 2c 1 (0) + |p − | − |p + |, i.e. |p − | = |p + |. These are the fiber products identified in Remark 3.5 (ii) as either empty or a one point set, Thus we have counts #Z κ ι (p − , p + ; 0) = 0 for p − = p + and #Z κ ι (p, p; 0) = 0 for each p ∈ Crit(f ).
(i) Each κ ··· α : W ··· α → Q + for α ∈ I + ⊔I − ⊔I ι ⊔I is an admissible sc + -multisection of a strong bundle P ··· α : W ··· α → X ··· α that is in general position to a sc-Fredholm section functor S ··· α : The tuple κ ι = (κ ι α ) α∈I ι satisfies the conclusions of Lemma 6.4 and 6.5. (ii) The smooth level of the first boundary stratum of X p−,p+,A for every (p − , p + , A) ∈ I is naturally identified -on the level of object spaces, and compatible with morphisms -with and the oriented section functors S ··· α are compatible with these identifications in the sense that the restriction of S p−,p+,A to any of these faces F ∞ ⊂ ∂ 1 X ∞ p−,p+,A is given by pullback S p−,p+,A | F ∞ = pr * F S F of another sc-Fredholm section of a strong bundle over an ep-groupoid S F : X F → W F given by S q,p+,A , S p−,q,A , S ι p−,p+,A , resp.
via the projection pr F : F → X F given by the natural maps where d is the Morse differential from §3. By setting ι p := (−1) |p| ι κ ι p as in Lemma 6.4, P SS p := (−1) |p| P SS κ + p , SSP := SSP κ − , and h := h κ we then obtain a chain homotopy between ι and Proof. This proof is similar to Lemma 6.4, with more complicated combinatorics of the boundary faces due to the boundary of B SFT described in Assumption 6.3, and presented in different order: We will first make the coherent constructions and then deduce the algebraic consequences.
Coherent ep-groupoids and sections: To construct coherent representatives S ··· α : X ··· α → W ··· α for α ∈ I + ⊔ I − ⊔ I ι ⊔ I as claimed in (ii) recall that the fiber product construction in Lemma 5.6 defines each bundle W α = pr * α W SFT 12 These disks should not be confused with the closed disks D ± in the construction of CP 1 R , as e.g. D + ⊂ C + ∼ = (D + ⊔ [−R, 0) × S 1 )/ ∼ R is a precompact subset of the first hemisphere in CP 1 R ∼ = C + ∪ S 1 ∪ C − for any R ≥ 0.
Next, restriction to the boundary faces given in Assumption 6.3 (i) induces representatives S GW A : resp. S ± γ,A± : X ± γ,A± → W ± γ,A± of the sections σ GW : B GW (A) → E GW (A) resp. σ SFT : B ± SFT (γ; A ± ) → E ± SFT (γ; A ± ) from Assumption 4.3 resp. 5.5. Moreover, the boundary of the open subset (ev . Then the evaluation maps restrict to sc ∞ functors ev ± : X GW A → D ± r × M resp. ev ± : X ± γ,A → D ± r × M , which yield -again independent of r > 0 -the fiber product construction of B ± (α) in Lemma 4.5, and of B ι (α) in Lemma 5.6. Now the identification of the top boundary strata ∂ 1 X ∞ p−,p+,A will proceed similar to the proof of Lemma 6.1 with B GW (A) replaced by B +,− SFT (A), apart from the fact that the SFT polyfold has boundary. This boundary is identified in Assumption 6.3 (ii) as By the fiber product construction [Fi1,Cor.7 . We obtain an identification that throughout is to be interpreted on the smooth level (as fiber product constructions drop some nonesmooth points) Here we also used the identification of evaluation maps in Assumption 6.3 (iii)(a). Then compatibility in (ii) of the oriented section functors S ··· α with the identification of these (smooth levels of) faces F ∞ ⊂ ∂ 1 X ∞ p−,p+,A follows from compatibility of pr p−,p+,A : X p−,p+,A → X SFT A with the projections pr ± α : X ± α → X ± γ,A± for α ∈ I ± used in Lemma 4.5 and pr ι α : X ι α → X GW A used in Lemma 5.6. More precisely, S p−,p+,A | F ∞ = pr * F S F follows from compatibility of the sections in Assumption 6.3 (iii) and Construction of coherent perturbations: Next, we construct admissible sc + -multisections κ ··· α : W ··· α → Q + for α ∈ I + ∪ I − ∪ I ι ∪ I as claimed in (i), i.e. in general position to the respective sections S ··· α : X ··· α →: W ··· α , while also coherent as claimed in (iii). The existence of such coherent transverse perturbations will ultimately be guaranteed by an abstract perturbation theorem for coherent systems of sc-Fredholm sections, as developed in the case of trivial isotropy in [Fi2]. Since we have to deal with nontrivial isotropy and the SFT perturbation package [FH1] has only been outlined in [FH], we give a detailed construction of the perturbations for our purposes. We proceed as in Lemma 6.4 and construct them all as pullbacks κ ··· α := λ ··· A • (pr ··· α ) * of a collection of sc +multisections on the SFT resp. Gromov-Witten polyfold bundles -without Morse trajectories - For this to induce a coherent collection of sc + -multisections as required in (iii), , it suffices to pick λ compatible with respect to the faces of the SFT neck stretching polyfolds X SFT A in (33). More precisely, using the natural identifications of bundles from Assumption 6.3 (iii), we will construct λ coherent in the sense that -for some choice of r > 0 in the construction of So to finish this proof it remains to choose the sc + -multisections λ so that each induced sc +multisection in the induced coherent collection for (κ ··· α ) α∈I + ∪I − ∪I ι ∪I is admissible and in general position, while also satisfying the coherence requirements (34), (35) and the requirements on κ ι in the proofs of Lemma 6.4 and 6.5. The construction of coherent perturbations for the SFT polyfolds outlined in [FH]

⊂ X SFT
A satisfies the auxiliary norm bounds N (λ ∂ A ) ≤ 1 2 and support requirements that guarantee compactness for extensions λ SFT A of λ ∂ A with N (λ SFT A ) ≤ 1 and appropriate support requirements. Moreover, we may choose each of the extensions λ SFT A using Theorem A.9 to ensure -as in Lemma 6.4 -that the induced multisections κ ··· α are in general position as well. The latter will automatically be admissible with respect to pullback of the pair controlling compactness. In more detail (but without specifying the auxiliary norm bounds) the inductive construction of perturbations in [FH1] -simplified to the subset of SFT moduli spaces considered here -proceeds as follows:

Construction of λ GW
A and κ ι : Since the Gromov-Witten ep-groupoids X GW A are boundaryless by Assumption 5.5 (iii), the sc + -multisections λ GW A can be chosen independently of all other multisections. So we construct λ GW A as in the proofs of Lemma 6.4 and 6.5, to ensure that the conclusions in these lemmas hold, as required by (i). This prescribes (34) on the boundary face X GW A ⊂ ∂ X SFT A . Moreover, recall that λ GW A is obtained by applying Theorem A.9 to the sc-Fredholm section functors S GW A , the sc ∞ submersion ev + × ev − : X GW A → CP 1 × M × CP 1 × M , and the collection of Cartesian products of stable and unstable manifolds {z As in the proof of Lemma 6.4 this ensures that the pullbacks κ ι = (κ ι α = λ GW A • (pr ι α ) * ) α∈I ι are in general position. Moreover, these pullbacks are admissible w.r.t. the pairs controlling compactness on W ι α → X ι α that result by pullback from the coherent compactness controlling pair on W GW A → X GW A , which is constructed in a preliminary step by [FH,Lecture 13].
Coherence for λ ± γ,A : The next step is to construct sc + -multisections λ ± γ,A : W ± γ,A → Q + over the SFT ep-groupoids X ± γ,A of planes with limit orbit γ ∈ P(H) from Assumption 4.3, which then induce the perturbations κ ± for the P SS/SSP moduli spaces. These constructions are independent of the choice of λ GW A since the corresponding boundary faces of X SFT A do not intersect by Assumption 6.3 (ii). However, to enable the subsequent construction of λ SFT A as extension of the boundary values prescribed in (34) and (35), we need to make sure that each sc + -multisection (λ + γ, and coincides with the other sc + -multisections (λ + Then this yields a well defined sc + -multisection on F γ, To describe these intersections we note that [FH] constructs the ep-groupoids X ± γ,A± with coherent boundaries -involving ep-groupoids (X Fl γ − ,γ + ,B ) γ ± ∈P(H),B∈H2(M) which contain the moduli spaces of Floer trajectories between periodic orbits γ ± , as well as further ep-groupoids for Floer trajectories carrying a marked point. We will avoid dealing with the latter by specifying values r < ∞ when pulling back perturbations from the ep-groupoids X ± γ,A ⊂ X ± γ,A given by , as this will prevent the appearance of marked Floer trajectories even in the closure. For any fixed value 0 < r ≤ ∞, the j-th boundary stratum is given by j Floer trajectories breaking off, Generally, the boundary of the Floer ep-groupoids is given by broken trajectories, and this yields a disjoint cover of ∂ R=∞ X SFT in which the embeddings l γ,A±,B coincide with each of the embeddings l γ j ,A j ± for 0 ≤ j ≤ k and Now on these subsets we require coherence λ + as this is equivalent to (35) being well defined on im l γ,A±,B = k j=0 F γ j ,A j ± . This will be achieved by constructing the sc + -multisections (λ ± γ,A± ) to have product structure on the boundary -where the bundles P γ,A : for a collection of sc + -multisections λ Fl γ − ,γ + ,B : W Fl γ − ,γ + ,B → Q over the Floer ep-groupoids X Fl γ − ,γ + ,B . While this guarantees coherence on each overlap of embeddings im l γ,A±, Construction of λ Fl γ − ,γ + ,B : To achieve the coherence in (37), [FH] first constructs the sc +multisections (λ Fl β ) β∈I Fl by iteration over the maximal degeneracy k β := max{k ∈ N 0 | (S Fl β ) −1 (0) ∩ ∂ k X Fl β = ∅} of unperturbed solutions (which is finite by Gromov compactness): For k β = −∞ the section S Fl β has no zeros so is already transverse, so that λ Fl β can be chosen as the trivial perturbation. (The trivial multivalued section functor λ : W → Q + is given by λ(0) = 1 and λ(w = 0) = 0.) For k β = 0 the section S Fl β has all zeros in the interior, so that λ Fl β can be chosen admissible and trivial on the boundary -by applying Corollary 4.6 (i) with a neighbourhood of the unperturbed zero set in the interior, . .·λ Fl γ k−1 ,γ k ,B k on all boundary faces that contain unperturbed solutions in their closure. Indeed, existence of a solution in X Fl γ 0 ,γ 1 ,B1 ×. . .×X Fl γ k−1 ,γ k ,B k implies k γ i−1 ,γ i ,Bi ≥ 0 for i = 1, . . . , k, and the Cartesian product of solutions of maximal degeneracy yields 1+k γ 0 ,γ 1 ,B1 +. . .+k γ k−1 ,γ k ,B k ≤ k β . Thus these prescriptions are made for 0 ≤ k γ i−1 ,γ i ,Bi ≤ k β − 1, and on boundary faces with no solutions in their closure we prescribe the trivial perturbation throughout. This yields a well defined sc + -multisection λ Fl β | P −1 β (∂X Fl β ) by coherence in the prior iteration steps, so that λ Fl β can be constructed by applying the extension result [HWZ,Thm.15.5] which provides general position and admissibility with respect to a pair controlling compactness that extends the pair which was chosen on the boundary in prior iteration steps.
Construction of λ ± γ,A and κ ± : With the Floer perturbations in place, [FH] next constructs the collections of sc + -multisections (λ ± γ,A ) γ∈P(H),A∈H2(M) to satisfy (37) by iteration over degeneracy k γ,A := max{k ∈ N 0 | (S ± γ,A ) −1 (0) ∩ ∂ k X ± γ,A = ∅}. For k γ,A = −∞ one takes λ ± γ,A to be trivial. For k γ,A = 0 one applies Theorem A.9 to the sc-Fredholm section functor S ± γ,A : X ± γ,A → W ± γ,A , the map ev ± : X ± γ,A → C ± × M , and the collection of stable resp. unstable manifolds {0} × W ± p for all critical points p ∈ Crit(f ). These satisfy the assumptions as the zero set |(S ± γ,A ) −1 (0)| is compact and the preimages (ev ± ) −1 ({0} × W ± p ) lie within the open subset X ± γ,A ⊂ X ± γ,A on which ev ± restricts to a sc ∞ submersion ev ± : X ± γ,A → C ± × M . We can moreover prescribe λ ± γ,A | P −1 γ,A (∂ X ± γ,A ) to be trivial, since in the absence of solutions the trivial perturbation is in general position. Then Theorem A.9 provides λ ± γ,A that is supported in the interior and transverse to each submanifold {0} × W ± p in the sense that these submanifolds are transverse to the evaluation from the perturbed zero set Now suppose that admissible λ ± γ ′ ,A ′ in general position have been constructed for k γ ′ ,A ′ ≤ k ∈ N 0 , and satisfy both the transversality in (38) and the coherence condition (37) over the ep-groupoids |X ± γ ′ ,A ′ | = (ev ± ) −1 (D ± r k × M ) with r k := 2 + 2 −k . Then for k γ,A = k + 1 we will construct λ ± γ,A to satisfy (37) over (ev ± ) −1 (D ± r k+1 ×M ) by first noting that the previous iteration -and requiring triviality on boundary faces without solutions -determines a well defined sc + -multisection λ ± γ,A | P −1 . This is well defined at (x ± , x, x ′ ) ∈ ∂ 0 X ± γ ′ ,A± × X Fl γ ′ ,γ ′′ ,B ′ × X Fl γ ′′ ,γ,B−B ′ , which appears both as (x ± , (x, x ′ )) ∈ ∂ 0 X ± γ ′ ,A± × ∂X Fl γ ′ ,γ,B and ((x ± , x), x ′ ) ∈ ∂X ± γ ′′ ,A±+B ′ × X Fl γ ′′ ,γ,B−B ′ , by the coherence of the Floer multisections and the prior iteration: For vectors in the respective fibers (w ± , w, w ′ ) ∈ However, this defines an admissible sc + -multisection in general position only over the open subset of the boundary ∂X ± γ,A = (ev ± ) −1 (D ± r k × M ) ∩ ∂ X ± γ,A . We multiply the given data by a scale-smooth cutoff function -guaranteed by the existence of partitions of unity for the open cover | Remark A.6 -to obtain an → Q + which coincides with the prescribed datathus in general position and with evaluation transverse to each {0} × W ± p -over the closed subset Then λ ± γ,A : W ± γ,A → Q + is constructed with these given boundary values using Theorem A.9 to achieve not just general position but also transversality as in (38). By admissibility of the prior iteration and coherence of the pairs controlling compactness, λ ± γ,A can moreover be chosen admissible.
As required in the coherence discussion, this determines right hand sides of (35) which agree on overlaps of different immersions l γ,A± (X + γ,A+ × X − γ,A− ) for r = 2. Thus it constructs a well defined sc + -multisection on ∂ R=∞ X SFT that is admissible and has evaluations transverse to the submanifolds {0} × W − p− × {0} × W + p+ for all pairs p − , p + ∈ Crit(f ). Moreover, for α ∈ I ± we obtain a pair controlling compactness by pullback of the coherent pairs constructed in [FH] on the bundles W ± γ,A . Then the pullback multisections κ ± = (κ ± α = λ ± γ,A • (pr ι α ) * ) α∈I ± are sc + , admissible w.r.t. the pullback pair, and in general position by the arguments in the proof of Lemma 6.4.

Construction of λ SFT
A and κ: The above constructions determine the right hand sides in the coherence requirements λ SFT A | P −1 A the image of the immersion l γ,A± on the ep-groupoids representing |X ± γ,A± | = (ev ± ) −1 (D ± r × M ) ⊂ B ± SFT (γ; A). By admissibility in the prior steps and existence of scale-smooth partitions of unity (see Remark A.6) these induce for every A ∈ H 2 (M ) an admissible sc + -multisection λ ∂ A : W ± A | ∂ XA → Q + which coincides with the prescribed data over X GW A ⊔ γ,A± F γ,A± (1) ⊂ ∂ X SFT A . Thus on this closed subset we have general position and transversality of the evaluation map for any pair of critical points p − , p + ∈ Crit(f ). Then the admissible sc + -multisection λ SFT A : W SFT A → Q + is constructed with these given boundary values -and auxiliary norm and support prescribed by the coherent pairs controlling compactness -using Theorem A.9 to achieve general position on all of X SFT A and extend transversality of the evaluation ev (A). As in the proof of Lemma 6.4, the transversality of the evaluation maps implies that the pullbacks κ = (κ α = λ A • (pr ι α ) * ) α∈I are in general position. They are also admissible with respect to the pullback of pairs controlling compactness. This finishes the construction of the sc + -multisections claimed in (i) with the boundary restrictions required in (iii).
Computation of orientations: To prove the identity (40) it remains to compute the effect of the orientations in Remark 5.8 on the algebraic identity in Corollary 4.6 (v) that arises from the boundary ∂Z κ (α) of the 1-dimensional weighted branched orbifolds arising from regularization of the moduli spaces with index I(α) = I(p − , p + ; A) = 1. Here Z λ SFT A is of odd dimension with oriented boundary determined by the orientation relations in Assumption 6.3 (iii)(b) and (c) as Moreover, the index of σ SFT is I(α) = |p − | − |p + | + 2c 1 (A) + 1 = 1, so we compute orientations in close analogy to (31) -while also giving an alternative identification of the boundary components - This computation should be understood in a neighbourhood of a solution, so in particular with scale-smooth evaluation maps to C ± × M . Based on this, Corollary 4.6 (v) implies -as claimed -

Appendix A. Summary of Polyfold Theory
This section gives an overview of the main notions of polyfold theory that are used in this paper. The following language is used to describe settings with trivial isotropy. 13 While ρ is generally not classically differentiable, it is required to be scale-smooth (sc ∞ ) with respect to a scale structure on E, which is indicated by E.
13 Trivial isotropy would be guaranteed in our settings by an almost complex structure J for which there are no nonconstant J-holomorphic spheres.
(i') An M-polyfold, as defined in [HWZ,Def.2.8], is a paracompact Hausdorff space X together with an atlas of charts φ ι : homeomorphisms between open sets U ι ⊂ X and sc-retracts O ι such that ∪ ι U ι = X), whose transition maps are sc-smooth. For k ∈ N 0 the k-th boundary stratum ∂ k X is the set of all x ∈ X of degeneracy index d(x) = k given 14 by the number of components equal to 0 for the point in a chart φ ι (x) ∈ [0, ∞) sι × E ι . In particular, ∂ 0 X is the interior of X.
(ii) A strong bundle over an M-polyfold X, as defined in [HWZ,Def.2.26], is a sc-smooth surjection P : W → X with linear structures on each fiber W x = P −1 (x) for x ∈ X, and an equivalence class of compatible strong bundle charts, which in particular encode a sc-smooth subbundle W ⊃ W 1 → X whose fiber inclusions W 1 x ֒→ W x are compact and dense. (iii) The notion of sc-Fredholm for a scale smooth section S : X → W of a strong bundle in [HWZ,Def.3.8] encodes elliptic regularity and a nonlinear contraction property [HWZ,Def.3.6,3.7]. The latter is a stronger condition than the classical notion of linearizations being Fredholm operators, and is crucial to ensure an implicit function theorem; see [FWZ].
A more detailed survey of these trivial isotropy notions can be found in [FFGW]. Then the generalization to nontrivial isotropy is directly analogous to the notion of smooth sections of orbibundles, in which orbifolds are realizations ofétale proper groupoids [Mo].
Remark A.2. A sc-Fredholm section σ : B → E of a strong polyfold bundle as introduced in [HWZ,Def.16.16,16.40] is a map between topological spaces together with an equivalence class of sc-Fredholm section functors s : X → W of strong bundles W over ep-groupoids X , whose realization |s| : |X | → |W| together with homeomorphisms |X | := Obj X /Mor X ∼ = B and |W| ∼ = E induces σ.
To summarize these notions we use conventions of [HWZ] in denoting object and morphism spaces as Obj X = X and Mor X = X. These will be equipped with M-polyfold structures, so that the k-th boundary stratum of a polyfold B ∼ = |X | is given as ∂ k B ∼ = ∂ k X/X ⊂ |X | for all k ∈ N 0 .
(i) An ep-groupoid as in [HWZ,Def.7.3] is a groupoid X = (X, X) equipped with M-polyfold structures on the object and morphism sets such that all structure maps are local sc-diffeomorphisms and every x ∈ X has a neighbourhood V (x) such that t : s −1 cl X (V (x)) → X is proper. As in [HWZ,§7.4] we require that the realization |X | is paracompact and thus metrizable. (ii) A strong bundle as in [HWZ,Def.8.4] over the ep-groupoid X is a pair (P, µ) of a strong bundle P : W → X and a strong bundle map µ : X s × P W → W so that P lifts to a functor P : W → X from an ep-groupoid W = (W, W) induced by (P, µ). Then P restricts to a functor W 1 → X on the full subcategory whose object space is the sc-smooth subbundle W 1 ⊂ W . (iii) A sc-Fredholm section functor of the strong bundle P : W → X as in [HWZ,Def.8.7] is a functor S : X → W that is sc-smooth on object and morphism spaces, satisfies P • S = id X , and such that S : X → W is sc-Fredholm on the M-polyfold X.
Now a polyfold description of a compact moduli space M is a sc-Fredholm section σ : B → E of a strong polyfold bundle with zero set σ −1 (0) ∼ = M. The polyfold descriptions used in this paper are obtained as fiber products of existing polyfolds and sc-Fredholm sections over them. This requires a technical shift in levels described in the following remark, and a notion of submersion below.
. Then B m := Xm / Mor X is well defined since morphisms of X -locally represented by scale-diffeomorphisms -preserve the levels on Obj X = X.
The restriction σ| Bm of a sc-Fredholm section σ : B → E is again sc-Fredholm with values in E m , and the choice of such a shift in levels is irrelevant for applications since the zero set σ −1 (0) ⊂ B ∞ -as well as the perturbed zero set for any admissible perturbation -is always contained in the so-called "smooth part" that is densely contained in each level B ∞ ⊂ B m .
14 The degeneracy index d(x) ∈ N 0 in [HWZ,Def.2.13,Thm.2.3] is a priori independent of the choice of chart φι only for points in a dense subset X∞ ⊂ X specified in Remark A.3. With that d(x) := max{lim sup d(x i ) | X∞ ∋ x i → x} is well defined for all x ∈ X and can also be computed in any fixed chart. Definition A.4. [Fi1,Def.5.8] A sc ∞ functor f : X → M from an ep-groupoid X = (X, X) to a finite dimensional manifold M is a submersion if for all x ∈ X ∞ the tangent map D x f : x X is the reduced tangent space [HWZ,Def.2.15]. Consider in addition a sc-Fredholm section functor S : X → W. Then the sc ∞ functor f : X → M is S-compatibly submersive if for all x ∈ X ∞ there exists a sc-complement L ⊂ T R x X of ker(D x f ) ∩ T R x X and a tame sc-Fredholm chart for S at x [Fi1,Def.5.4] in which the change of coordinates ψ : O → [0, ∞) s × R k−s × W that puts S in basic germ form -which by tameness has the form ψ(v, e) = (v, ψ(e)) for (v, e) ∈ O ⊂ [0, ∞) s × E and a linear sc-isomorphism ψ -moreover x X satisfying the above condition. The purpose of giving a moduli space a polyfold description is to utilize the perturbation theory for sc-Fredholm sections over polyfolds, which allows to "regularize" the moduli space by associating to it a well defined cobordism class of weighted branched orbifolds. (For a technical statement see Corollary 4.6 and the references therein.) Since the ambient space |X | is almost never locally compact, this requires "admissible perturbations" of the section to preserve compactness of the zero set. This admissibility is determined by the following data introduced in [HWZ,Def.12.2,15.4].
Definition A.5. A saturated open subset U ⊂ X of an ep-groupoid X = (X, X) is an open subset U ⊂ X with π −1 (π(U)) = U, where π : X → |X | = X / X is the projection to the realization.
A pair controlling compactness for a sc-Fredholm section S : X → W of a strong bundle P : W → X consists of an auxiliary norm N : [HWZ,Def.12.2]) and a saturated open subset U ⊂ X that contains the zero set S −1 (0) ⊂ U, such that {x ∈ U | N (S(x)) ≤ 1} ⊂ |X | has compact closure.
Given such a pair, a section s : X → W is (N, U)-admissible if N (s(x)) ≤ 1 and supp s ⊂ U.
The construction of perturbations moreover requires scale-smooth partitions of unity, which will be guaranteed by the following standing assumptions. Remark A.6. Throughout this paper we assume that the realizations |X | of ep-groupoids are paracompact, and the Banach spaces E in all M-polyfold charts are Hilbert spaces. This guarantees the existence of scale-smooth partitions of unity by [HWZ,§5.5,§7.5.2]. In order to guarantee the same on every level B m as discussed in Remark A.3, we moreover assume that each scale structure E = (E m ) m∈N0 consists of Hilbert spaces E m . These assumptions hold in applications, such as the ones cited [HWZ1,FH]. Then paracompactness and thus existence of scale-smooth partitions of unity on every level is guaranteed by [HWZ,Prop.7.12].
When discussing coherence of perturbations of a system of sc-Fredholm sections, the boundaries are described in terms of Cartesian products of polyfolds, bundles, and sections. So we will make use of Cartesian products of multivalued perturbations as follows, to obtain multisections over the boundary as summarized in the subsequent remark. Lemma A.7. Let S 1 : X 1 → W 1 and S 2 : X 2 → W 2 be sc-Fredholm section of strong bundles P i : W i → X i over ep-groupoids. Then the Cartesian product X 1 × X 2 is naturally an ep-groupoid and (S 1 × S 2 ) : X 1 × X 2 → W 1 × W 2 is a sc-Fredholm section of the strong bundle P 1 × P 2 .
Proof. A detailed treatment of sc-Fredholmness of the product section S 1 × S 2 can be found in [Fi1,Lemma 7.2]. The remaining statements follow easily from the definitions in [HWZ] (as do the statements in the first paragraph).
Recall in particular from [HWZ,Def.13.4] that a sc + -multisection on a strong bundle P : W → X is a functor λ : W → Q + that is locally of the form λ(w) = {j | w=pj (P (w))} q j , represented by sc + -sections p 1 , . . . , p k : V → P −1 (V) (i.e. sc ∞ sections of W 1 ; see [HWZ,Def.2.27]) and weights q 1 , . . . , q k ∈ Q ∩ [0, ∞) with j q j = 1. Then for local sections p i j and weights q i j representing λ i for i = 1, 2, the multisection λ 1 · λ 2 is locally represented by the sections (p 1 j , p 2 j ′ ) with weights q 1 j q 2 j ′ , and all admissibility and general position arguments are made at the level of these local sections.
Remark A.8. Let P : W → X be a strong bundle over a tame ep-groupoid X = (X, X). Then for every x ∈ X ∞ there is a chart φ : U x → O from a locally uniformizing 15 neighbourhood U x ⊂ X of x to a sc-retract O ⊂ [0, ∞) n × E, with φ(x) = 0 lying in the intersection of the n local faces F k := φ −1 ({(v, e) ∈ [0, ∞) n × E | v k = 0}) which cover the boundary ∂X ∩ U x = n k=1 F k . Now a sc + -multisection over the boundary is a functor λ ∂ : P −1 (∂X ) → Q + whose restriction λ ∂ | P −1 (F k ) to each local face is a sc + -multisection of the strong bundle P −1 (F k ) → F k . In the presence of a sc-Fredholm section S : X → W, such a sc + -multisection is in general position over the boundary if for each intersection of faces F K := k∈K F k ⊂ ∂X the restriction of the perturbed multi-section λ ∂ • S| FK : P −1 (F K ) → Q + has surjective linearizations at all solutions. If, moreover, (N, U) is a pair controlling compactness, then λ ∂ is (N, U)-admissible if each restriction λ ∂ | P −1 (F k ) is admissible w.r.t. the pair (N | P −1 (F k ) , U ∩ F k ).
In our applications, as described in Assumption 6.3, the local faces F k are images of open subsets of global face immersions l F : F → ∂X , where each F is a Cartesian product of two polyfolds, and the restriction to the interior l F | ∂0F is an embedding into the top boundary stratum ∂ 1 X . The bundles over each face are naturally identified with the pullbacks l * F W, and then the pushforwards of sc +multisections λ F : l * F W → Q + form a sc + -multisection over the boundary λ ∂ : P −1 ( im λ F ) → Q + if they agree on overlaps and self-intersections of the immersions l F , at the boundary ∂F of the faces. In this setting, general position of λ ∂ is equivalent to general position of the multisections λ F .
The following perturbation theorem allows us to refine the construction of coherent perturbations in [FH] for the SFT moduli spaces such that moreover the evaluation maps from the perturbed solution sets are transverse to the unstable and stable manifolds in the symplectic manifold. This is a generalization of the polyfold perturbation theorem over ep-groupoids and the extension of transverse perturbations from the boundary [HWZ,Theorems 15.4,15.5] (with norm bound given by h ≡ 1 for simplicity). Another version of this -with the submanifolds representing cycles whose Gromov-Witten invariants are then obtained as counts -also appears in [Sch]. We are working under the assumptions made in this section -e.g. paracompactness -without further mention. The limitation to finitely many submanifolds in the extension result seems to be of technical nature; we expect that joint work of the first author with Dusa McDuff -on coherent finite dimensional reductions of polyfold Fredholm sections -will establish the result for countably many submanifolds. Theorem A.9. Suppose S : X → W is a sc-Fredholm section functor of a strong bundle P : W → X over a tame ep-groupoid X with compact solution set |S −1 (0)| ⊂ |X |, and let (N, U) be a pair controlling compactness. Moreover, let e : X → M be a sc 0 -map to a finite dimensional manifold M which has a sc ∞ submersive restriction e| V : V → M on a saturated open set V ⊂ X .
Then, for any countable collection of smooth submanifolds (C i ⊂ M ) i∈I with e −1 ∪ i∈I (C i ) ⊂ V, there exists an (N, U)-admissible sc + -multisection λ : W → Q + so that (S, λ) is in general position (see [HWZ,Definition 15.6]) and the restriction e| Z λ : Z λ → M to the perturbed zero set Z λ = |{x ∈ X | λ(S(x)) > 0}| is in general position 16 to the submanifolds C i for all i ∈ I. 15 A neighbourhood Ux ⊂ X forms a local uniformizer as in [HWZ,Def.7.9] if the morphisms between points in Ux are given by a local action of the isotropy group Gx. 16 General position to C i requires transversality to C i of each restriction e| Z λ ∩F K to the perturbed solution set within an intersection of local faces F K = k∈K F k as defined in Remark A.8, including for F ∅ := Z λ .
Moreover, suppose I is finite and λ ∂ : P −1 (∂X ) → Q + for some 0 < α < 1 is an ( 1 α N, U)admissible structurable sc + -multisection in general position over the boundary such that the restriction e| Z ∂ : Z ∂ → M to the perturbed zero set in the boundary Z ∂ := |{x ∈ ∂X | λ ∂ (S(x)) > 0}| is in general position 17 to the submanifolds C i for all i ∈ I. Then λ above can be chosen with λ| P −1 (∂X ) = λ ∂ .
Proof. Our proof follows the perturbation procedure of [HWZ,Theorem 15.4], which proves the special case when there is no condition on a map e : X → M , i.e. when M = {pt} and C i = {pt}. To obtain the desired transversality of e to the submanifolds C i ⊂ M we will go through the proof and indicate adjustments in three steps: A local stabilization construction, which adds a finite dimensional parameter space to cover the cokernels near a point x ∈ S −1 (0); a local-toglobal argument which combines the local constructions into a global stabilization which covers the cokernels near S −1 (0); and a global Sard argument which shows that regular values yield transverse perturbations. Within these arguments we need to consider restrictions to any intersection of faces to ensure general position to the boundary, use submersivity of e to achieve transversality to the C i , and work with multisections due to isotropy. The statement with prescribed boundary values λ ∂ generalizes the extension result [HWZ,Theorem 15.5], which hinges on the fact that general position over the boundary persists in an open neighbourhood -something that is generally guaranteed only for finitely many transversality conditions; see the end of this proof. The first step in any construction of perturbations is the existence of local stabilizations which cover the cokernels, as follows.
It is constructed in [HWZ] to be structurable in the sense of [HWZ,Def.13.17], in general position in the sense that the linearization T (S x ,Λ x ) (0, x) : T 0 R l × T R x X → W x is surjective 18 and admissible in the sense that the domain support ofΛ x is contained in U and the auxiliary norm is bounded linearly, N (Λ)(t, y) ≤ c x |t| for some constant c x . In case x ∈ V ∩ S −1 (0) we refine this construction to require surjectivity of the restrictions where K x := ker(D x e| T R x X ) ⊂ T R x X is the kernel of the linearization D x e : T R x X → T e(x) M restricted to the reduced tangent space. For that purpose note that e is sc ∞ near x by assumption, so has a well defined linearization, and since its codomain is finite dimensional, its kernel has finite codimension. Moreover im D x S ⊂ W x has finite codimension by the sc-Fredholm property of S, and the reduced tangent space T R x X ⊂ T x X has finite codimension by the definition of M-polyfolds with corners. Thus we can find finitely many vectors w 1 , . . . , w l ∈ W x which together with D x S(K x ) span W x . These vectors are extended to sc + -sections of the form p j (t, y) = t j w j (y), multiplied with sc ∞ cutoff functions of sufficiently small support, and pulled back by local isotropy actions to construct the functorΛ x as in [HWZ,Thm.15.4]. We claim that this yields the following local properties with respect to the sc ∞ functorẽ x : R l × V → M, (t, y) → e(y).
Note that the auxiliary norm N on W pulls back to an auxiliary normÑ onW, and compactness ofS is controlled in the sense that for any compact subset K ⊂ Rl we have compactness of (47) {(t, x) ∈ K × U |Ñ (S(t, x)) ≤ 1} = K × {(x ∈ U | N (S(x)) ≤ 1} ⊂ Rl × |X |.