A polyfold proof of the Arnold conjecture

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 P(H ) := γ : S 1 → M γ (t) = X H (t, γ (t)) and γ is contractible (1) and note that periodic orbits can be identified with the fixed points of the time 2π flow φ 2π H : M → M of X H . (Here we choose the convention S 1 = R/2π Z, i.e. period 2π , for ease of notation later on.) We call this Hamiltonian system nondegenerate if

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 [1] 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 [7] and tori [6]. A weaker form is accessible by Floer theory, introduced by Floer [17,18] 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 [16,19] in the absence of pseudoholomorphic spheres.

Theorem 1.1 Let (M, ω) be a closed symplectic manifold and H : S 1 × M → R a nondegenerate periodic Hamiltonian function. Then

Floer's proof was later extended to general closed symplectic manifolds [20,21,25,29], 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. [27]. 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 [22], following an approach by Piunikhin-Salamon-Schwarz [30] based on [32], and building on polyfold descriptions of Gromov-Witten moduli spaces [23] as well as their degenerations in Symplectic Field Theory [8,15]. Remark 1. 2 Since the polyfold descriptions of SFT moduli spaces [12][13][14][15] are not completely published, we formulate them as Assumptions 4.3, 5.5, 6.3. While these descriptions of four kinds of moduli spaces and their relations involve a lot of structures (bundles, sections, evaluation maps, and compatible immersions from Cartesian products to boundaries), they will be familiar from classical descriptions of moduli spaces of pseudoholomorphic curves. Our assumptions in polyfold theoretic terms formalize the well known fact that the moduli spaces have local descriptions in terms of Fredholm sections and gluing theorems, which polyfold theory interprets as global smooth structure within an appropriately generalized differential geometry. Indeed, transition maps between the natural infinite dimensional local models fail to be classically differentiable for only two reasons which polyfold theory resolves as explained in e.g. [9, §2] and [23, §2.1]: Actions of reparameterization groups satisfy the new notion of scale-smoothness for maps between Banach spaces. Neighbourhoods of maps with broken or nodal domains are given local polyfold models as the image of a retraction (modulo a finite group action in the case of isotropy), which becomes scale-smooth after adjusting the smooth structure near nodal curves in Deligne-Mumford spaces. With this understood, there is little doubt in the existence of polyfold descriptions for moduli spaces. The much more audacious claim of polyfold theory is the existence of an abstract perturbation scheme for moduli spaces that are described as zero set of a scale-smooth section over a polyfold. However, this claim is fully substantiated in [22]. So the goal of this paper is to demonstrate the use of this abstract perturbation scheme once polyfold descriptions for the basic building blocks of moduli spaces are given.

We moreover chose this structure to give an example of how rigorous and transparent proofs can be written at a time when parts of their foundation are unpublished or in question.

To describe our proof, let C F = ⊕ γ ∈P(H ) γ be the Floer chain group of the Hamiltonian H with coefficients in the Novikov field (see Sect. 2). Let (C M, d) be the Morse complex with coefficients in associated to a Morse function f : M → R and a suitable metric on M (see Sect. 3). Then we will prove the following in Lemma 4.9, Definition 5.8, and Lemmas 6.4, 6.5, 6.6.

(ii) ι is a -module isomorphism. (iii) h is a chain homotopy between SS P • P SS and ι, that is ι -SS P

Here we view the Floer chain group C F as a vector space over -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 1.3 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 (C M Q , d Q ) be the Morse complex over Q as defined in Sect. 3. Then by the isomorphism of singular and Morse homology there exist c 1 , . . . , c k ∈ C M Q that are cycles, d Q c i = 0, and linearly independent in the Morse homology over Q. Since the Morse differential d : C M → C M is given by -linear extension of d Q from C M Q ⊂ C M the chains c 1 , . . . , c k ∈ C M are also cycles dc i = d Q c i = 0 and linearly independent in the Morse homology over . By Theorem 1.3 (i),(ii), ι induces an isomorphism H ι : H M → H M on homology. This in particular implies that [ι(c 1 )], . . . , [ι(c k )] ∈ H M are also linearly independent in homology, that is for any λ 1 , . . . , λ k ∈ we have

We now show that P SS(c 1 ), . . . , P SS(c k ) ∈ C F are -linearly independent, proving #P(H ) ≥ k since the elements of P(H ) generate C F by definition. This proves the theorem.

Let λ 1 , . . . , λ k ∈ be a tuple such that

Then we deduce from Theorem 1.3 (iii) that

This algebraically minimalistic approach of deducing the homological Arnold conjecture from the existence of maps P SS and SS P 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 proofs of "Floer homology equals Morse homology" require equivariant transversality which is generally obstructed-even for equivariant sections of finite rank bundles. Thus our goal was a proof using the least amount of geometric insights or new abstract tools. Beyond this we expect the [30]-approach to yield an isomorphism between Floer and Morse homology, and spectral invariants [33] on all closed symplectic manifolds, using refinements of polyfold theory described in Remark 1.4.

To maximize accessibility we begin with reviews of the pertinent facts on the Novikov field, Sect. 2, and Morse trajectories, Sect. 3. The Proof of Theorem 1.3 then proceeds by constructing the P SS and SS P maps in Sect. 4 from curves in C ± × M, constructing the isomorphism ι and chain homotopy h in Sect. 5 from curves in CP 1 × M and its degeneration into C -× M and C + × M, and proving their algebraic relations in Sect. 6 by constructing coherent perturbations. We give a detailed account of these iterative constructions in the Proofs of Lemma 6.4 and 6.6. While these results should be contained in [15], neck-stretching is not addressed in [12], and it seemed timely to give the proof in a case whose structure is vastly simplified by the absence of trivial cylinders compared with [12, §3.5]. To strike a balance between technical details 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 readers new to polyfold theory we provide in Appendix A a summary of all notions and facts that are necessary for the present application. Here we moreover establish in Theorem A.9 a relative perturbation result that should be of independent interest: It allows one to bring moduli spaces with an evaluation map into general position to a countable collection of submanifolds. We combine this result with [10] to construct polyfold descriptions of the [30] moduli spaces as fiber products of SFT moduli spaces with the Morse trajectory spaces constructed in [38].

Remark 1.4 (i)

There are essentially two approaches to the general Arnold conjecture as stated in Theorem 1.1. The first-developed by [19] and used verbatim in [20,21,25,29]-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 perturbationsunder transversality assumptions at the fixed point set which are met in this settingcan be found in [39], assuming a polyfold description of Floer trajectories. (ii) The second approach to Theorem 1.1 by [30] 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 : C M → C F, SS P : C F → C M 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 • SS P and SS P • P SS are chain homotopic to the identity, and hence SS P 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 1.3 (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 1equivariant transversality (which we expect to be able to achieve with the techniques in [39]). (iii) Theorem 1.3 is proven by following the [30]-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 one 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 SS P are chain maps. Then the chain homotopy between SS P • P SS and the isomorphism ι implies that P SS is injective and SS P surjective on homology. However, proving that P SS and SS P 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 [24].

The Novikov field

We use the following Novikov field 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

The Novikov field is the set of formal sums

where T is a formal variable, with rational coefficients λ r ∈ Q which satisfy the finiteness condition

The multiplication is given by

This defines a field by [21,Thm.4.1] and the discussion preceding the theorem in [21, §4], the key being that 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. Lemma 2.1 Let M = (λ i j ) 1≤i, j≤ ∈ × be a square matrix with entries λ i j ∈ in the Novikov field. Suppose that λ i j = r ∈ ,r ≥0 λ i j r T r with λ i j 0 = 0 for i = j and λ ii 0 = 0. Then M is invertible.

Proof Since is a field, invertibility of M is equivalent to det(M) = 0. Write det(M) = r ∈ μ r T r ∈ 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 = 1 base case, when M is a 1 × 1 matrix M = [λ 11 ], we have det(M) = λ 11 = r ∈ μ r T r with μ r = λ 11 r so μ 0 = λ 11 0 = 0. Now suppose that M is size × for some > 1 and inductively assume that, for any size ( -1) × ( -1) matrix N satisfying the hypotheses of the lemma, we have det(N ) = r ∈ μ N r T r with μ N 0 = 0. For 1 ≤ j ≤ , let C 1 j denote the matrix obtained by deleting the first row and j-th column of M. Then N := C 11 is an ( -1) × ( -1) matrix that satisfies the hypotheses of the lemma, and the cofactor expansion of the determinant yields det(M) = λ 11 det(N ) + j=2 (-1) 1+ j λ 1 j det(C 1 j ).

By hypothesis, all entries of M are of the form λ i j = r ≥0 λ i j r T r . Since the determinants det(N ) and det(C 1 j ) 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 1 j 0 = 0 for j ≥ 2 by hypothesis, it follows that the constant term (i.e. the coefficient on T 0 ) of λ 1 j det(C 1 j ) 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.

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

which identify the trajectory spaces with the unstable and stable manifolds M(M, 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 biinfinite Morse trajectory spaces in e.g. [3,Prp.3], using [38,Lemma 3.5]. Moreover, [38,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 [38], using techniques similar to those of [3] for the bi-infinite trajectory spaces.

Moreover, the evaluation maps (8) are smooth.

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

. 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.

that are coherent (by e.g. [36, §3.4]) in the sense that the top strata of the oriented boundaries of the compactified Morse trajectory spaces are products

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

Similarly, for

(ii) For computational purposes in Sect. 6.3 we determine the fiber products of the compactified Morse trajectory spaces of critical points p -, p

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

and thus the image of the evaluation maps are contained in unions of unstable/stable manifolds

Since the intersections W - q -∩ W + q + are transverse by the Morse-Smale condition, they can be nonempty only for

The Piunikhin-Salamon-Schwarz moduli spaces

To construct the moduli spaces, we need to make further choices as follows.

• Let J be an ω-compatible almost complex structure on M.

Then the Cauchy-Riemann operator on maps u : → M parametrized by a Riemann surface with complex structure j is

• Let g be a metric on M such that ( f , g) is a Euclidean Morse-Smale pair as in Definition 3.1. It exists by Remark 3.2.

Then we define the anti-holomorphic vector-field-valued 1-form

In the notation of [26, §8.1], we have Y H = -(X H β ) 0,1 given by the antiholomorphic part of the 1-form with values in Hamiltonian vector fields X H β which arises from the 1-form with values in smooth functions

We denote the oriented complex plane by C + := (C, i) = C, and denote its reversed complex structure and orientation by C -:= (C, -i). 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

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 : C M → C F and SS P : C F → C M 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 C Z(γ ) is the Conley-Zehnder index with respect to a trivialization of u * γ TM as in e.g. [32], 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 SS P) 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 [26,Remark 8.1.7] for solutions of

Here the curvature

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 [2] asserts that for any C > 0 the set of solutions of bounded energy {u :

≤ C} is compact up to breaking and bubbling. This compactness will be stated rigorously in polyfold terms in Assumption 5.5 (ii).

Polyfold description of moduli spaces

We will obtain a polyfold description for the moduli spaces in Sect. 4.1 by a fiber product construction motivated by the natural identifications

This couples the half-infinite Morse trajectory spaces from Sect. 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(. . .) ⊂ M(. . .) of the moduli space and a description of the compact moduli space M(. . .) = σ -1 (0) as zero set of a sc-Fredholm section σ : B(. . .) → E(. . .) of a strong polyfold bundle. For an introduction to the language [22] used here see Appendix §A. 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 [8], compactified in [2,4], and given a polyfold description in [15]. Here we identify u : C → M with the map to its graph u : C → C × M, z → (z, u(z)) as in [26, §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+d H t ∧dt) whose Reeb field ±∂ t + X H t has simply covered Reeb orbits 2 given by the graphs γ : t → (±t, γ (t)) of the periodic 1 For definitions of these notions see [4, §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. 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+t 0 ) ), 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

Now we will obtain a polyfold description of the PSS/SSP moduli spaces ( 14) by the slicing construction of [10] 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 [12], 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.

Remark 4.2 Note that the compactified moduli space

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 ∈ that v might map to a cylinder factor R×S 1 ×M ⊂ ×M. We will simplify the resulting sc ∞ 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. [5,Prop.7.1]. evaluation with varying target-being developed in [15]-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 / 2π Z → {|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 (iii) 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

In Assumption 4.3, Remark 4.4, and Lemma 4.5 we introduce some of the polyfolds under construction in [15] and their expected properties. To describe these objects we introduce a significant amount of notation. A summary of the types of curves in each polyfold and subsets thereof is displayed in Table 1 for the reader's convenience.

) is evaluation as in (15). The intersection of σ -1 SFT (0) with this dense subset is contained in the moduli space M ± SFT (γ ; A) from (15). The full moduli space M ± SFT (γ ; A) is obtained by enlarging B ± dense (γ ; A) to include equivalence classes with sup t∈S 1 d C×M v(Re ±it ), (Re ±it , γ (t)) → 0 as R → ∞. However, only classes with specific exponential decay of this quantity and related derivatives are contained in B ± SFT (γ ; A). (ii) The sc-smooth structure, sc-Fredholm property, and compactness is stated in [12,Thm.3.4]. The proof of polyfold and bundle structure outlined in [12, §7-11] extends the construction of Gromov-Witten polyfolds in [23] by local models for punctures and neck-stretching from [14, §3], using the implanting method in [13, §3, §5]. These constructions automatically satisfy the tameness assumed in (iii). The nonlinear Fredholm property needs to be proven globally-in close analogy to [23]. The Fredholm index stated in (i) is computed in a local chart, where the linearized section coincides with a restriction of the classical linearized Cauchy-Riemann operator to a local slice to the reparametrization action. The compactness properties follow from SFT-compactness of the moduli spaces [2] since the topology on the polyfolds given in [12, §3.4] generalizes the notion of SFT-convergence. Orientations are constructed in [12, §15]. Sc-smoothness of the evaluation maps is proven analogously to [23,Thm.1.8], and their submersion property in (iii), which is used to construct fiber products in Lemma 4.5, is proven as in [10,Ex.5.1]. (iii) 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.

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

and the evaluation is ev ± (u) = u(0). The intersection of σ -1 (0) with this dense subset is contained in the moduli space M ± (γ ; A) from (12). The full moduli space M ± (γ ; A) is obtained by enlarging the dense subset to include maps with sup t∈S 1 d M u(Re ±it ), γ (t) → 0 as R → ∞. However, only maps with specific exponential decay of this quantity and related derivatives are contained in B ± (γ ; A). While such a construction should follow from the same construction principles as in [12], there is presently no writeup beyond [37], 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 would be no harm in using this property as intuitive guide for following 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 [10] 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 : (9). Their zero sets contain 5 the moduli spaces from Sect. 4.1,

Finally, each zero set σ ± (...;A) -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 ± (...;A)

Proof We will follow [10,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 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 [10,Def.5.4] by Assumption 4.3(iii).

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 ep-groupoid sense, and the zero section of this bundle is a tame sc-Fredholm section functor. Next, note that

SFT (γ , A) ⊂ |X |, and by Assumption 4.3(iii) 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 + :

v X ev be a sc-complement of the kernel of the linearization of ev + at some v ∈ X ev ∞ that satisfies the conditions for S SFT -compatible submersivity in Definition A.4 w.r.t. a coordinate change ψ ev on a chart of X ev . Then the subspace {0} × L ⊂ T R τ M( p, M) × T R v X ev satisfies the conditions for S SFTcompatible transversality of ev 0 × ev + with at (τ , v) w.r.t. the product change of coordinates id × ψ ev in a product chart on the Cartesian product M( p, M) × X ev . (See [10, Lem.7.1, 7.2] for a discussion of the sc-Fredholm property on Cartesian products.)

Next, note that M( p, x) ev 0 × ev + X ev ∞ represents the smooth level of the fiber product topological space B+ ( p, γ ; A). So [10,Cor.7.3]

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.

The tame ep-groupoid X yields the claimed polyfold B + ( p, γ ; A) := |X |, and similarly the pullbacks of (P, μ) and S SFT through the projection X → X 1 define the claimed bundle and sc-Fredholm section σ + ( p,γ ;A) : The index formula in [10,Cor.7.3]

Finally, the zero set σ + ( p,γ ;A) -1 (0) is the fiber product of the zero sets as claimed, as these are contained in the smooth level, and the restriction to ev -1 ({0} × M) already restricts considerations to the domain X ev from which the fiber product polyfold is constructed. Moreover, σ + ( p,γ ;A) -1 (0) is compact as in [10,Cor.7.3], since both M( p, M) and σ + SFT -1 (0) are compact and both ev 0 and ev + are continuous. The final statement then follows from Assumption 4.3(ii).

Moduli spaces for the isomorphism Ã

We will construct ι : C M → C M 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 Sect. 3.3) of the Morse-Smale pair ( f , g) chosen in Sect. 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 :

This determines the Fredholm index as

As in Sect. 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 Sect. 3.3, using z

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 1.3 (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 [8]. For that purpose we identify a solution u : CP 1 → M with the map to its graph u :

as in [26, §8.1]. This yields is a bijection (and homeomorphism in appropriate topologies)

between the Cauchy-Riemann solution space for M and the Gromov-Witten moduli space for

To transfer the evaluation maps at z + 0 = [1 : 0] and z - 0 = [0 : 1] we keep track of these as (unique) marked points mapping to {z ± 0 } × M and thus replace (19) by a fiber product over

(20) This uses the evaluation maps from a Gromov-Witten moduli space with two marked points,

where Aut(CP 1 , z - 0 , z + 0 ) denotes the set of biholomorphisms φ :

The polyfold setup in [23,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 :

Here a dense subset of the base polyfold

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 σ

The setup in [23,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

). 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 Sect. 5.3 we will combine the polyfold description of the Gromov-compactification of ( 21) with an abstract construction of fiber products in polyfold theory [10] to obtain compactifications and polyfold descriptions of the moduli spaces. Then the construction of ι : C M → C M proceeds as in Sect. 4.3. The algebraic properties of ι in Theorem 1.3 (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 Sect. 6. Remark 5.1 Gromov-compactifications of the moduli spaces M ι ( p -, p + ; A) will result from the energy identity [26, 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 that {u :

Another consequence of ( 22) is that for ω(A) ≤ 0 we have no solutions M(A) = ∅ except for A = 0 ∈ H 2 (M) when the solution space is the space of constant maps

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).

Moduli spaces for the chain homotopy h

To construct the moduli spaces from which we will obtain h : C M → C M, we again use the almost complex structure J and Morse-Smale pair ( f , g) chosen in Sect. 4.1. In addition, we fixed an anti-holomorphic vector-field-valued 1-form Y H ∈ 0,1 (C, (TM)) that arises from the fixed Hamiltonian function H :

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-

and we will construct h from their union

It is constructed so that it has the following properties:

(i) For R = 0 we have Y 0 H ≡ 0 so that the moduli space M 0 ( p -, p + ; A) = M ι ( p -, p + ; A) is the same moduli space (17) from which ι will be constructed. (ii) The restriction of any solution u

The moduli space M( p -, p + ; A) is the zero set of a Fredholm section over a Banach manifold [0, ∞) × B, where B is the same Sobolev closure as in Sect. 5.1 of the set of smooth maps u :

Restricted to {0} × B this is the Fredholm section that cuts out M ι ( p -, p + ; A) in (17) with J 0 H = J . This determines the Fredholm index as

Towards a compactification and polyfold description of these moduli spaces we again-as in Sects. 4.2, 5.1, [26, §8.1]-identify a solution u : CP 1 → M with the map to its graph. Moreover, we again fix marked points

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 Sect. 3.3,

Compared with (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"foot_3 considered more generally in Symplectic Field Theory [8]. The evaluation maps from (21) directly generalize to

Now, as in Sect. 5.1, each factor in the fiber product (25) has natural compactificationseither 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 Sect. 5.3. Combined with the construction of fiber products in polyfold theory [10] this will yield compactifications and polyfold descriptions of the moduli spaces (23), and the construction of h : C M → C M then again proceeds as in Sect. 4.3. Establishing the algebraic properties in Theorem 1.3 relating h with ι and SS P • 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 Sect. 6.

Construction of the morphisms

In this section we construct the -linear maps ι : C M → C M and h : C M → C M analogously to Sect. 4.3 by first obtaining compactifications and polyfold descriptions for the moduli spaces in Sects. 5.1 and 5.2 as in Sect. 4.2. This construction is motivated by the fiber product descriptions of the moduli spaces in ( 20), (25), which couple Morse trajectory spaces from Sect. 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 [23] resp. outlined in [12]. A summary of the types of curves in each polyfold and subsets thereof is displayed in Table 2. 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

using the topologies and smooth structures on

Here we identify the boundaries of the closed unit disks

where we denote 0

To describe convergence and evaluation maps we also embed

A polyfold with dense subset B dense (A), which contains the Gromov-compactification M GW (A) of the moduli space in (21) Assumption 5.5

For R = 0 this is to be understood as

which are used in the construction of the PSS and SSP moduli spaces in Sect. 4.2. Moreover, this allows us to extend the evaluation maps from (27) to continuous maps ev ± :

and for any marked point with evaluation into a cylinder R × S 1 × M we project the result to

-will allow us to compare the evaluation at R = ∞ with the product of the evaluations ev ± :

SFT (0) is compact, and given any C ∈ R there are only finitely many

the evaluation maps ev ± :

On this domain intersected with {0} × B GW (A) ⊂ ∂ 1 B SFT (A), this map coincides with the Gromov-Witten evaluations ev + × ev -viewed as maps

where we identify C + C -= CP 1 S 1 and restrict to the domain

Remark 5.6 (i) While not needed for our proof, we state the following properties for intuition:

The Aut(CP 1 , z - 0 , z + 0 )-orbits of smooth maps v : CP 1 → CP 1 × M which represent the class [CP 1 ]+ A form a dense subset B dense (A) ⊂ B GW (A). On this subset, the section is given by σ

is a dense subset that intersects the boundary ∂B SFT (A) exactly in {0} × B dense (A), and on which the section is given by σ

On these dense subsets, ev ± ([v]) resp. ev ± (R, [v]) is the evaluation as in (27).

The intersection of the zero sets with the dense subsets

are naturally identified with the Gromov-Witten moduli space (21) and SFT moduli space in (26). (ii) The polyfold description σ GW : [23], with the submersion property shown in [10, Ex.5.1]. The properties for ω(A) ≤ 0 in Assumption 5.5 (ii) follow from the fact that nonconstant pseudoholomorphic curves have positive symplectic area, and linear Cauchy-Riemann operators on trivial bundles (arising from linearization at constant maps) are surjective. The construction of σ SFT starts by recognizing that the family of almost complex manifolds in Remark 5.4 (ii) for R < ∞ is equivalent to a degeneration of the almost complex structure on CP 1 ×M along the equator S 1 ⊂ CP 1 . This can be described by an R-dependent bundle and section over [0, ∞) × B GW (A). The construction for R → ∞ then proceeds analogous to [12, §3], with buildings consisting of a top and bottom floor curve in C ± × M and intermediate floors given by curves in R × S 1 × M. Thus Assumption 5.5 (iii) and the compatibility with B GW (A) in (iv) hold by construction. The polyfold and bundle structure are again obtained as in [12] by extending the constructions in [23] with local models for punctures and neck-stretching from [14, §3], using the implanting method in [13, §3, §5]. The remaining properties are proven as outlined in Remark 4.4 (ii).

Given any such polyfold descriptions of the moduli spaces of pseudoholomorphic curves, we now extend the fiber product descriptions of the moduli spaces

in Sects. 5.1 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 [10] will be used as in Lemma 4.5 to obtain the following polyfold description for compactifications of the moduli spaces in Sects.

and a scale-smooth inclusion

Moreover (18). Further, these are related via the inclusion φ ι by natural orientation preserving identification σ ι ( p -, p + ;A) ∼ = φ * ι σ ( p -, p + ;A) . The zero sets of these sc-Fredholm sections containfoot_4 the moduli spaces from Sects. 5.1 and 5.2,

Finally, each zero set σ (ι) ( p -, p + ;A) -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 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)

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 [22,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 Sect. 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, -linear maps. Definition 5.8 Given admissible sc + -multisections κ = (κ ( p -, p + ;A) ) p ± ∈Crit( f ),A∈H 2 in general position to (σ ( p -, p + ;A) ) and κ ι = (κ ι ( p -, p + ;A) ) p ± ∈Crit( f ),A∈H 2 in general position to (σ ι ( p -, p + ;A) ) as in Corollary 4.6, we define the maps h κ : C M → C M and ι κ ι : C M → C M to be the -linear extensions of

The proof that the coefficients of these maps lie in the Novikov field is verbatim the same as Lemma 4.9, based on the compactness properties in Lemma 5.7.

Coherent polyfold descriptions of moduli spaces

The general approach to obtaining not just counts as discussed in Sect. 4.2 but welldefined 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.

As a first example, the moduli spaces M ι ( p -, p + ; A) in Sect. 5.1 which yield the map ι :

7 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 [10] 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 5.7 is naturally identified with

Proof By the fiber product construction [10,Cor.7.3]

The polyfold B GW (A) and its open subset B +,- GW (A) are boundaryless by Assumption 5.5 (iii), which means

or the other way around. These two cases are disjoint but analogous, so it remains to show that the first case consists of points in the union q∈Crit( f ) M( p -, q) × ∂ 0 B ι (q, p + ; A) ∞ . For that purpose recall the identification

) in Theorem 3.3, which is compatible with the evaluation ev : M( p -, q) × M(q, M) → M, (τ 1 , τ 2 ) → ev(τ 2 ) by construction, and thus

Here we also used the identification of the interior smooth level in Lemma 5.7.

Next, the polyfold description in Lemma 5.7 for the moduli spaces M( p -, p + ; A) in Sect. 5.2, which yield the map h : C M → C M, 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 [15]. We will state as assumption only those parts of their coherence properties that are relevant to our argument in Sect. 6.4 for the chain homotopy ι-SS P

Here the contributions to d•h +h •d will arise from boundary strata of the Morse trajectory spaces, whereas ι -SS P • P SS arises from the following identification of the boundary of the polyfold B +,- SFT (A), which is given as open subset of B SFT (A) in Assumption 5.5 (iv).foot_5 Remark 6. 2 In the following we will use the word "face" loosely for Cartesian products of polyfolds such as

and their immersions into the boundary of another polyfold such as ∂B +,- SFT (A). We also refer to the image of the immersion F → ∂B +,- SFT (A) as a face of B +,- SFT (A). Compared with the formal definition of faces in [22,Definitions 2.21,11.1,16.13], ours are disjoint unions of faces. They describe the interaction between the moduli spaces -roughly speaking:

(i) The R → ∞ boundary parts of B SFT (A) in which the marked points separate are covered by immersions of products of the PSS and SSP polyfolds. This structure arises from generalizing the SFT compactification in [2] to buildings of not necessarily holomorphic maps. The parts of the boundary described here are given by buildings whose top and bottom floors are given by maps to C ± × M and intermediate floors given by maps to R × S 1 × M. The immersions then arise from stacking a building in B + SFT (γ ; A + ) (with top floor in C × M) on top of a building in B - SFT (γ ; A -) (with bottom floor in C -× M). Here a lack of injectivity arises at buildings with middle floors in R × S 1 × M from ambiguity in splitting such building into two parts. (ii) The immersions restrict to a disjoint cover of the top boundary stratum of B +,- SFT (A) by embeddings. This restriction is given by the buildings with a single floorguaranteeing injectivity by avoiding the ambiguous middle floors in R × S 1 × M. (iii) The immersions are compatible-simply by construction-with the evaluation maps, bundles, and sections for the boundary components at R = ∞, and the boundary B GW (A) at R = 0.

Assumption 6.3

The collection of oriented sc-Fredholm sections of strong polyfold bundles σ ± SFT :

B SFT (A) → E SFT (A) for γ ∈ P(H ) and A ∈ H 2 (M) together with the evaluation maps ev ± :

∞ × M, and their sc ∞ restrictions on open subsets, ev ± : B ±,C SFT (γ ; A) → C ± × M, ev ± : B +,- GW/SFT (A) → C ± ×M from Assumptions 4.3, 5.5 has the following coherence properties.

(i) For each γ ∈ P(H ) and A -,

whose restriction to the interior

SFT (γ ; A -) is an embedding into the boundary of the open subset B +,- SFT (A) ⊂ B SFT (A). They map into the limit set lim R→∞ {R} × B GW (A) from Assumption 5.5(iii), so cover most of the boundary 10

(ii) The union of the images l γ,

When restricted to the interiors, this yields a disjoint cover of the top boundary stratum,

(iii) The immersions l γ,A ± are compatible with the evaluation maps, bundles, and sections-as required for the construction [15] of coherent perturbations for SFT, that is:

(a) The boundary restriction of the evaluation maps ev ± | {0}×B GW (A)⊂∂B SFT (A) coincides with the Gromov-Witten evaluation maps ev ± : B GW (A) → CP 1 ∞ , and the same holds for their sc ∞ restriction ev + × ev -| {0}×B +,-

The extra boundary faces of B SFT (A) arise from both marked points mapping to the same component in the R → ∞ neck stretching limit. These will not be relevant to our construction of coherent perturbations.

C + S 1 C -. The restriction of ev ± :

∞ , 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. (c) The restriction of σ SFT to each face

This identification preserves the orientation of sections.

Proof of identity:

To prove ι κ ι • d + d • ι κ ι = 0 note that both ι κ ι and d are -linear, so the claimed identity is equivalent to the collection of identities (ι κ ι • d) p -+ (d • ι κ ι ) p -= 0 for all generators p -∈ Crit( f ). That is we wish to verify q, p + ,A

Here, by the index formula (18), both sides can be written as sums over p + ∈ Crit( f ) and A ∈ H 2 (M) for which I ι ( p -, p + ; A) = 1. Then it suffices to prove for any such pair α = ( p -, p + ; A) with

This identity will follow by applying Corollary 4.6 (v) to the sc + -multisection κ α : W ι α → Q + . Its perturbed zero set is a weighted branched 1-dimensional orbifold Z κ ι (α), whose boundary is given by the intersection with the smooth level 11 of the top boundary stratum

By coherence (and with orientations discussed below) this boundary is

Here the first summand of the third identification on the level of object spaces,

follows if we assume coherence of sections and multisections on the faces

x)) = κ α (S ι q, p + ;A (x)) = κ q, p + ;A (S ι q, p + ;A (x)).

The second summand is identified similarly by assuming coherence on the faces F α ,(q -, p + ) ⊂ ∂ 1 X ι α . Finally, the fourth identification in ∂ Z κ ι (α) for α = ( p -, p + ; A) with I ι (α) = 1 follows from index and regularity considerations as follows. Corollary 4.6 (iii),(iv) guarantees that the perturbed solution spaces Z κ ι (α ) are nonempty only for Fredholm index I ι (α ) ≥ 0, and for I ι (α ) = 0 are contained in the interior, Z κ ι (α ) ⊂ ∂ 0 B(α ). The Morse trajectory spaces M( p -, q) resp. M(q, p + ) are nonempty only for | p -|-|q| ≥ 1 resp. |q|-| p + | ≥ 1, so the perturbed solution spaces in the Cartesian products have Fredholm index (18)

and analogously I ι ( p -, q; A) = I ι ( p -, p + ; A) + |p + | -|q| ≤ 0. By the above regularity of the perturbed solution spaces this implies that the unions on the left hand side of the fourth identification are over |q| = |p -| -1 resp. |q| = |p + | + 1 as in (30), and for these critical points we have the inclusions Z κ ι (q, p + ; A) ⊂ ∂ 0 B(q, p + ; A) and Z κ ι ( p -, q; A) ⊂ ∂ 0 B( p -, q; A) that verify the equality. 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 5.7 defines each bundle W ι α = pr * α W GW A for α = ( p -, p + ; A) ∈ I as the pullback of a strong bundle W GW A → X GW A under a projection of ep-groupoids-with abbreviated notation ev ±

Moreover, the section S ι α = S GW A • pr α is induced by the section S GW A :

Then the identification of the top boundary stratum proceeds exactly as the Proof of Lemma 6.1.

Admissible perturbations for isomorphism property

In this section we prove Theorem 1.3 (ii), i.e. construct ι = (-1) * ι κ ι : C * M → C * M in Definition 5.8 and Lemma 6.4 as a -module isomorphism on the chain complex C M = C M 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 that #Z κ ι ( p -, p + ; A) = 0 for A ∈ H 2 (M) {0} with ω(A) ≤ 0, or for A = 0 and p -= p + , and

Proof The sc + -multisections κ ι in Lemma 6.4 are obtained from choices of sc +multisections (κ A :

A (0)|, and such that moreover the evaluation maps restricted to the perturbed zero sets, ev -× ev + :

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

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

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 | (W 0 ) x {0 x } ≡ 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 Sect. 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

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 ).

Finally, we will use these computations of #Z κ ( p -, p + ; A) for ω(A) ≤ 0 to prove that the resulting map ι := (-1) * ι κ ι : C M → C M is a -module isomorphism. For that purpose we choose an arbitrary total order of the critical points Crit( f ) = {p 1 , . . . , p } and for i, j ∈ {1, . . . , } denote the coefficients of ι( p j ) = i=1 λ i j p i by λ i j ∈ . We claim that the ( × )-matrix with entries λ i j = r ∈ λ i j r T r satisfies the conditions of Lemma 2.1. To check this recall that we have by construction in Definition 5.8 and change of signs in Lemma 6.4

For r < 0 we obtain λ i j r = 0 since each coefficient #Z κ ι ( p j , p i ; A) = 0 vanishes for ω(A) = r < 0. For r = 0 and i = j we also have λ i j 0 = 0 since # Z κ ι ( p j , p i ; A) = 0 also holds for ω(A) = 0 and p j = p i . Finally, for r = 0 and i = j we use #Z κ ι ( p j , p i ; A) = 0 for A = 0 with ω(A) = 0 to compute λ ii 0 = #Z κ ι ( p j , p i ; 0) = 0. This confirms that Lemma 2.1 applies, and thus ι ∼ = (λ i j ) 1≤i, j≤ is invertible. This finishes the proof.

Coherent perturbations for chain homotopy

In this section we prove Theorem 1.3 (iii) by constructing h κ : C M → C M in Definition 5.8 as a chain homotopy between SS P κ + • P SS κ -and ι κ ι from Definitions 4.8, 5.8, with appropriate sign adjustments as in Lemma 6.4. This requires a coherent construction of perturbations κ, κ ι , κ -, κ + over the indexing sets

Here we will use notation from Lemma A.7 for Cartesian products of multisections. Lemma 6.6 There is a choice of κ + = (κ + α ) α∈I + , κ -= (κ - α ) α∈I -, κ ι = (κ ι α ) α∈I , κ = (κ α ) α∈I in Definitions 4.8, 5.8 that is coherent in the following sense.

. The tuple κ ι = (κ ι α ) α∈I ι satisfies the conclusions of Lemmas 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

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

For any such choice of κ ι = (κ ι α ) α∈I , the resulting maps P SS κ + , SS P κ -, ι κ ι , h κ in Definitions 4.8, 5.8 satisfy (-1) | p| ι κ ι p = (-1) | p| SS P κ -P SS κ + p + h κ (d p ) + d(h κ p ), where d is the Morse differential from Sect. 3. By setting ι p := (-1) | p| ι κ ι p as in Lemma 6.4, PSS p := (-1) | p| P SS κ + p , SS P := SS P κ -, and h := h κ we then obtain a chain homotopy between ι and SS P • P SS, that is ι

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.

Moreover, the boundary of the open subset (ev

Then the evaluation maps restrict to sc ∞ functors ev ± :

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 5.7. 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 [10,Cor.7.3] Then the subsequent identifications result by comparing the resulting expressions with the interiors in Lemmas 4.5, 5.7. We obtain an identification that throughout is to be interpreted on the smooth level (as fiber product constructions drop some non-smooth points) 12 These disks should not be confused with the closed disks D ± in the construction of CP 1 R , as e.g.

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

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 5.7. 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

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. Since the SFT perturbation package [12, §14] has not yet been described for neck stretching, we give a detailed construction of the perturbations for our purposes. We proceed as in Lemma 6.4 and construct them all as pullbacks

* 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

-we have

where l γ,A ± is the map defined in Assumption 6.3(i). 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 analogously to [12, §14] proceeds by first choosing coherent compactness controlling data, i.e. pairs (N , U) of auxiliary norms on all the bundles and saturated neighbourhoods of the compact zero sets in all the ep-groupoids X ± γ,A , X GW A , X SFT A (c.f. Definition A.5), which are compatible with the immersions to boundary faces in (33). Then it constructs the perturbations λ GW A as in Lemma 6.5 and also λ ± γ,A ± to be in general position, admissible w.r.t. the coherent data (2N , U), and coherent in the sense that continuous extension of ( 34)-( 35) induces a well defined multisection

Here coherence of the perturbations on the intersection of faces (see Remark 6.2) is required to guarantee existence of scale-smooth extensions of λ ∂ A to multisections λ SFT A :

for the two split-

coincide under this identification. Generally, the boundary of the Floer ep-groupoids is given by broken trajectories, and this yields a disjoint cover of

in which the embeddings l γ ,A ± ,B coincide with each of the embeddings l γ j ,A j ± for 0 ≤ j ≤ k and A j

Now on these subsets we require coherence λ +

for all 0 ≤ j = j ≤ k, 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 :

are restricted to various faces of ∂X ± γ,A -

for a collection of sc

we are now faced with the challenge of satisfying the coherence conditions in (37). These conditions uniquely determine the boundary restrictions λ

via the identification of the boundaries with Cartesian products of interiors in (36). Thus (36) on Cartesian products involving boundary strata poses coherence conditions on the choice of λ Fl β for β ∈ I Fl := P(H

Construction of. λ Fl γ -,γ + ,B : To achieve the coherence in (37), [15] first constructs the sc + -multisections (λ Fl β ) β∈I Fl by iteration over the maximal degeneracy

unperturbed solutions (which is finite by Gromov compactness): We first consider classes β with k β = -∞. For these, 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.) Next, we consider β with k β = 0. For these, the section S Fl β has all zeros in the interior, so that λ Fl β can be chosen admissible and trivial on the boundaryby applying Corollary 4.6 (i) with a neighbourhood of the unperturbed zero set in the interior,

Once the iteration has constructed λ Fl β for all β with k β ≤ n for some n ∈ N 0 , we proceed to consider β = (γ -, γ + , B) ∈ I Fl with k β = n + 1. For these, the restriction λ Fl β | P -1

on all boundary faces that contain unperturbed solutions in their closure. Indeed, existence of a solution in

and the Cartesian product of solutions of maximal degeneracy yields 1

and on boundary faces with no solutions in their closure we prescribe the trivial perturbation throughout.

This yields a well defined sc

by coherence in the prior iteration steps, so that λ Fl β can be constructed by applying the extension result [22, 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, [15] next constructs the collections of sc + -multisections

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 ± :

) 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 ev

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

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

where

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 ) ∈

Moreover, ev

However, this defines an admissible sc + -multisection in general position only over the open subset of the boundary

We multiply the given data by a scale-smooth cutoff functionguaranteed by the existence of partitions of unity for the open cover

→ Q + which coincides with the prescribed data-thus in general position and with evaluation transverse to each

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

A 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 as in [12, §13] 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 (34), as well as

A in (35), where we denote by F γ,A ± (r )

A the image of the immersion l γ,A ± on the ep-groupoids representing

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

Thus on this closed subset we have general position and transversality of the evaluation map

to {0} × W - p -× {0} × W + p + 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

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).

Proof of identity: By -linearity of all maps involved, it suffices to fix two generators p -, p + ∈ Crit( f ) of C M and check that ι κ ι p -and

Here the sums on the right hand side are over counts of pairs of moduli spaces of index 0. From Sect. 9), ( 18), (24) this is moreover equivalent to

So all sums can be rewritten with the index condition I ( p -, p + ; A) = 1 for A = A -+ A + ∈ H 2 (M), and since the symplectic area is additive ω(A -) + ω(A + ) = ω(A -+ A + ), it suffices to show the following identity for each α = ( p -, p + ; A) ∈ I with I ( p -, p + ; A) = 1,

This identity will follow from Corollary 4.6 (v) applied to the weighted branched 1-dimensional orbifold Z κ (α) that arises from an admissible sc + -multisection κ α :

is given by the intersection with the top boundary stratum ∂ 1 B(α) ∩ V α = |∂ 1 X α |, and will be determined here-with orientations computed in (41) below.

Here the second identity uses coherence of the ep-groupoid as in (32). The third identity follows from coherence of sections S ••• α and sc + multisections κ ••• α stated in (ii), (iii), and the fact from Corollary 4.6 (iv) that perturbed zero sets

α | are contained in the interior of the polyfolds when the Fredholm index is 0. For the second summand we moreover use Lemma A.7 which ensures that each restriction κ α | P -1 α (F ) to a face F = ∂ 0 X + p -,γ ;A + ×∂ 0 X - γ, p + ;A -⊂ ∂ 1 X p -, p + ;A , given by κ + p -,γ ;A + •κ - γ, p + ;A -, is in general position to the section S + p -,γ ;A + × S - γ, p + ;A -. Then its perturbed zero set

A -| as the complement of the pairs of points (x + , x -) with 0 = κ p -, p + ;A (S p -, p + ;A (x + , x -))

Since a product in Q + = Q∩[0, ∞) is nonzero exactly when both factors are nonzero, this identifies the objects of the perturbed zero set of κ p -, p + ;A with the product of perturbed zero objects for κ ± ,

And the realization of this set is precisely

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- 14 by the number of components equal to 0 for the point in a chart φ ι (x) ∈ [0, ∞) s ι × E ι . In particular, ∂ 0 X is the interior of X . (ii) A strong bundle over an M-polyfold X , as defined in [22,Def.2.26], is a 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 [22,Def.3.8] encodes elliptic regularity and a nonlinear contraction property [22,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 [11].

A more detailed survey of these trivial isotropy notions can be found in [9]. Then the generalization to nontrivial isotropy is directly analogous to the notion of smooth sections of orbi-bundles, in which orbifolds are realizations of étale proper groupoids [28].

Remark A.2 A sc-Fredholm section σ : B → E of a strong polyfold bundle as introduced in [22,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 [22] 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

(i) An ep-groupoid as in [22,Def.7.3] is a groupoid X = (X , X) equipped with Mpolyfold 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 [22, §7.4] we require that the realization |X | is paracompact and thus metrizable. (ii) A strong bundle as in [22,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 [22,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.

Remark A. 3 Polyfolds carry a level structure B ∞ ⊂ . . . ⊂ B 1 ⊂ B 0 = B as follows: For any M-polyfold X , in particular the object space of the ep-groupoid representing

/ Mor X is well defined since morphisms of X -locally represented by scale-diffeomorphisms-preserve the levels on Obj X = X .

The restriction σ | B m 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 .

For a finite dimensional manifold or orbifold M-such as the Morse trajectory spaces in Sect. 3.3-viewed as polyfold, the level structure is trivial

M is surjective, where T R x X is the reduced tangent space [22,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 [10,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 satisfies ψ(L) ⊂ {0} k-s × W, where the chart identifies L

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 [22, 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 : W [1] → [0, ∞) (see [22,Def.12.2]) and a saturated open subset U ⊂ X that contains the zero set

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 E in all M-polyfold charts are Hilbert spaces. This guarantees the existence of scale-smooth partitions of unity by [22, §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∈N 0 consists of Hilbert spaces E m . These assumptions hold in applications, such as the ones cited [12,23]. Then paracompactness and thus existence of scale-smooth partitions of unity on every level is guaranteed by [22,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 .

Moreover, if λ i : W i → Q + are sc + -multisections for i = 1, 2, then there is a well defined sc

admissible for some fixed pair controlling compactness as in Definition

Proof A detailed treatment of sc-Fredholmness of the product section S 1 × S 2 can be found in [10,Lemma 7.2]. The remaining statements follow easily from the definitions in [22] (as do the statements in the first paragraph).

Recall in particular from [22,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= p j (P(w))} q j , represented by sc + -sections p 1 , . . . , p k : V → P -1 (V) (i.e. sc ∞ sections of W 1 ; see [22,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.

In particular, the (N i , U i )-admissibility can be phrased as the existence of local representations by sections with N i ( p i j (x)) ≤ 1 and Z (S i , p

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 15neighbourhood

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

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 | ∂ 0 F 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 [12] 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 [22,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 [34,35]. 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|

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 [22,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 .

Moreover, suppose I is finite and λ ∂ : 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 [22,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-to-global 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 [22,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.

multisection in general position over the boundary such that the restriction e| Z

Local stabilization constructions: For every zero x ∈ S -1 (0) of the unperturbed sc-Fredholm section we construct a finite dimensional parameter space R l for l = l x ∈ N 0 and sc + -multisection

such that x 0 is the trivial multisection, i.e. x 0 (0) = 1, x 0 (w) = 0 for w ∈ W x {0}. This multisection ˜ x is viewed as local perturbation near (0, x) of a sc-Fredholm 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.

It is constructed in [22] to be structurable in the sense of [22,Def.13.17], in general position in the sense that the linearization T ( Sx , ˜ 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 of ˜ x is contained in U and the auxiliary norm is bounded linearly, N ( )(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

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 functor ˜ x as in [22,Thm.15.4]. We claim that this yields the following local properties with respect to the sc ∞ functor ẽx : R l × V → M, (t, y) → e(y).

Local stabilization properties: There exists x > 0 and a locally uniformizing neighborhood Q(x) ⊂ X of x whose closure is contained in U, such that 18 This is shorthand for Sx + p j having surjective linearization for every section p j in a local representation of ˜ x with Sx (0, x) = 0 = p j (0, x), and restricted to the reduced tangent space T R x X . 19 As before, this is shorthand for surjectivity on each reduced tangent space ker D (t,y) (

S -1 (0)∩U x we have (0, y) ∈ supp x so that the reduced linearizations T R ( Sx , ˜ x ) (0, y)

and the restriction to their kernel D (0,y) ẽx | N x 0,y are surjective. These properties persist for y ∈ S -1 (0) with |y| ∈ |Q(x)|.

The structure of supp x and surjectivity of linearizations T

For each x = x i we constructed above a family of sc + -multisections

t∈R lx i . These are summed up, using [22,Def.13.11], to a sc + -multisection

Here each t : W → Q + for t ∈ R l is a structurable sc +multisection by [22,Prop.13.3]. We view the multisection ˜ as global perturbation of a sc-Fredholm section functor S of a bundle P, S : R l × X → R l × W =: W P : R l × W → R l × X (t, y) → (t, S(y)) (t, w) → (t, P(w)), and claim that e : X → M induces a submersion on its perturbed solution set in the following sense. Global stabilization properties: There exists 0 > 0 such that for every 0 < < 0 satisfies ( ẽ| supp ˜ ) -1 (C) ⊂ R l × V , and its restriction to supp ˜ ∩ (R l × V ) is classically smooth and submersive as in Definition A.4. Note that the auxiliary norm N on W pulls back to an auxiliary norm Ñ on W, and compactness of S is controlled in the sense that for any compact subset K ⊂ R l we have compactness of

(47) Next, the restriction of ˜ to each R l x i × X → R l × X is the local perturbation ˜ x i of Sx i , since we identify R l x i ∼ = {(t 1 , . . . , t k ) ∈ R l | t j = 0 ∀ j = i} and each x j 0 is trivial. In particular, 0 is the trivial multisection, with N ( 0 ) = 0. Moreover, we have an estimate N ( t ) ≤ c|t| that results from the linear estimates on each x i t . Now for 0 ≤ 1 c we can deduce compactness of the stabilized solution set as closed subset of (47), Z := (t, x) ∈ R l × X |t| ≤ 0 , t (S(x)) > 0 .

(48)

The next step is to argue that (48) is smooth in a neighbourhood of Z ∩ ({0}

X is an open neighbourhood of S -1 (0). So for any x ∈ Q we can use the local properties of some ˜ x i with |x| ∈ |Q(x i )| to deduce surjectivity of T R ( S, ˜ ) (0, x). Then the local implicit function theorems [22,Thms 15.2,15.3,Rmk.15.2] yield an open neighbourhood U (0, x) = {|t| < x } × U (x) ⊂ R l × X of (0, x) for some 0 < x < 0 , and hence a saturated neighbourhood Ũ (0, x) := {|t| < x } × π -1 (|U (x)|) ⊂ R l × X such that ˜ | Ũ (0,x) = ˜ • S| Ũ (0,x) is a tame branched ep + -subgroupoid of Ũ (0, x). As a consequence, the orbit space of the support supp ˜ | Ũ (0,x) is a weighted branched orbifold with boundary and corners.

For x ∈ S -1 (0) e -1 (C) we can moreover choose U (x) ∩ e -1 (C) = ∅, since |e -1 (C)| ⊂ |X | is closed. For x ∈ S -1 (0) ∩ e -1 (C) ⊂ V the covering (44) guarantees |x| ∈ |Q(x i )| for some 1 ≤ i ≤ r with Q(x i ) ⊂ V and we choose U (x) ⊂ Q(x i ). This guarantees that the restriction of ẽ : R l × X → M, (t, y) → e(y) to Ũ (0, x) is sc ∞ , and surjectivity of D (0,x) ẽx i | ker T R ( Sx i , ˜ x i ) (0,x) implies surjectivity of D (0,x) ẽ| N 0,x : N 0,x → T e(x) M on N 0,x := ker T R ( S, ˜ ) (0, x). Here N 0,x represents the reduced tangent space at |(0, x)| to the weighted branched orbifold | supp ˜ | Ũ (0,x) |. Now ẽ| supp ˜ ∩ Ũ (0,x) : supp ˜ | Ũ (0,x) → M is classically smooth since it is a restriction of an sc ∞ map to finite dimensions, and we have shown it to be submersive at (0, x). Hence, by openness of submersivity along each corner stratum, and local compactness of supp ˜ | Ũ (0,x) ⊂ Z it follows that Ũ (0, x) ⊂ R l × V can be chosen sufficiently small to ensure that ẽ| supp ˜ ∩ Ũ (0,x) is submersive as in Definition A. Moreover, from the properties of ẽ| supp ˜ ∩ Ũ (0,x i ) for i = 1, . . . , r we know that the restriction of ẽ to supp ˜ ∩ (R l × V ) for V := π -1 r i=1 U (x i ) ⊂ V is classically smooth and submersive. Here we have e -1 (C) ∩ A ⊂ V since U (x i ) for i > r was chosen disjoint from e -1 (C), and hence we have • U (y 0 ) ⊂ X admits the natural action of the isotropy group G y 0 ; see [22,Thm.7 ⊂ Ũ (t 0 , y 0 ), which intersect any intersection of local faces in a manifold with boundary and corners.

• On each branch M t 0 ,y 0 j , the reduced linearizations T R ( S, ˜ ) (t, y) are surjective for all (t, y) ∈ M t 0 ,y 0 j , and the restriction of ẽ| supp ˜ is a submersion M t 0 ,y 0 j ∩(R l ×V ) → M in general position to the boundary in the sense of Definition A.4. That is, D (t,y) ẽ| N t,y : N t,y → T e(y) M is surjective on N t,y := ker T R ( S, ˜ ) (t, y) for all (t, y) ∈ M t 0 ,y 0 j

∩ (R l × V ).

There is a countable cover supp ˜ ⊂ β∈Z Ũ (t β , y β ) indexed by (t β , y β ) β∈Z ⊂ supp ˜ , since R l × X -and hence its subspace supp ˜ -is second-countable, and every open cover of a second-countable space has a countable subcover. Moreover, for any given β ∈ Z there are finitely many choices FK := {|tt 0 | < δ} × k∈K F y β k ⊂ Ũ (t β , y β ) of intersections of finitely many local faces K ⊂ {1, . . . , d X (y β )}, with F∅ := Ũ (t β , y β ). Finally, for each β ∈ Z and intersection of faces FK , there are finitely many smooth manifolds FK ∩ M t β ,y β j indexed by j ∈ J β . For each of these countably many choices, Sard's theorem asserts that FK ∩ M t β ,y β j → R l , (t, y) → t has an open and dense subset R β K , j ⊂ R l of regular values. Then, since R l is a Baire space, the set of common regular values R 0 := β∈Z K , j R β K , j ⊂ R l is still dense. For any t 0 ∈ R 0 , the sc + -multisection t 0 : W → Q + is in general position by the usual linear algebra for each restriction of the linearized operators to intersections of faces: Consider (t 0 , x 0 ) ∈ FK ∩ M t β ,y β j ⊂ supp ˜ and a local section S + p j in the representation of ˜ = ˜ • S with M t β ,y β j ⊂ (S + p j ) -1 (0). The surjective differential along this intersection of faces can be written as D t 0 ,x 0 (S + p j )| FK = D ⊕ L, where L is a bounded operator (arising from differentiating p j in the direction of R l ) and D is the reduced linearization-on the intersection of faces F K := k∈K F y β k ⊂ U (y β ) ⊂ X -of the section S + p j (t 0 , •) that is a part of the representation of t 0 • S. Then regularity of t 0 implies surjectivity of the projection : ker(D ⊕ L) → R l , which in turn is equivalent to surjectivity of D; see e.g. [26,Lemma A.3.6].

Moreover, each t for |t| < is (N , U)-admissible, thus any sufficiently small regular t 0 ∈ R 0 yields an admissible sc + -multisection λ := t 0 in general position as in [22,Thm.15.4]. To prove our theorem, we have to moreover choose t 0 ∈ R 0 so that the restriction e| Z λ : Z λ → M to the solution set Z λ = | supp λ • S| is in general position to C i ⊂ M for all i ∈ I . For that purpose we consider the countably many projections → M is smooth and submersive in a neighborhood of ẽ-1 (C i ). In particular, it is transverse to C i so that there is a natural smooth structure on ẽ-1 (C i ) ∩ FK ∩ M t β ,y β j

. Thus we can apply the Sard theorem to each (50) to find open and dense subsets T i,β K , j ⊂ R l of regular values, and a dense set of common regular values T 0 := β∈Z K , j R β K , j ∩ i T i,β K , j ⊂ R l . Note that T 0 ⊂ R 0 , so sufficiently small t 0 ∈ T 0 yield admissible sc + -multisections λ := t 0 in general position. Moreover, general position of e| Z λ : Z λ → M to C i at x ∈ Z λ ∩ e -1 (C i ) means that the linearizations of e| F K ∩Z λ map onto T e(x) M / T e(x) C i for each intersection of local faces F K ⊂ U (y β ) ⊂ X that contains x. Here the tangent spaces of F K ∩ Z λ at x are given by those of FK ∩ M t β ,y β j ∩ ({t 0 } × X ) for each branch with (t 0 , x) ∈ M t β ,y β j ⊂ supp ˜ , so we need to ensure surjectivity of D (t 0 ,x) ẽ : ker → T e(x) M / T e(x) C i on the kernel of the projection : T (t 0 ,x) FK ∩ M t β ,y β j → R l .

Here D (t 0 ,x) ẽ : T (t 0 ,x) FK ∩ M t β ,y β j → T e(x) M is surjective (since ẽ| supp ˜ is submersive), and regularity t 0 ∈ T i,β K , j means that we have (D (t 0 ,x) ẽ) -1 (T e(x) C i ) = R l , so for any Y ∈ T e(x) M we find (T , X ) ∈ T (t 0 ,x) FK ∩ M t β ,y β j with D (t 0 ,x) ẽ(T , X ) = Y and (T , X ) ∈ (D (t 0 ,x) ẽ) -1 (T e(x) C i ), so that (0, X -X ) ∈ ker proves the required surjectivity D (t 0 ,x) ẽ(0, X -X ) = Y -D (t 0 ,x) ẽ(T , X ) = [Y ] ∈ T e(x) M / T e(x) C i . Thus this choice of sufficiently small t 0 ∈ T 0 also guarantees general position of e| Z λ to each of the countably many submanifolds C i , which finishes the proof of the theorem when no boundary values are prescribed.

Regular extension:

To prove the last paragraph of the theorem we consider a given ( 1 α N , U)-admissible structurable sc + -multisection λ ∂ : P -1 (∂X ) → Q + that is in general position over the boundary, and with e| Z ∂ : Z ∂ = supp λ ∂ • S| ∂X → M in general position to finitely many submanifolds C i . Then we will adjust the above construction of λ : W → Q + to also satisfy λ| P -1 (∂X ) = λ ∂ , by following the proof of the transversal extension theorem over ep-groupoids [22,Thm.15.5].

Since λ ∂ is supported in U ∩ ∂X with N (λ ∂ )(x) < α for all x ∈ ∂X we can find a continuous functor h : X → [0, 1) supported in U with N (λ ∂ )(x) < h(x) < for the stabilized multisection ˜ : R l × W → Q + , (t, w) → (w). For x ∈ S ∩ ∂X we need no stabilization by a R l factor (i.e. take l = 0) due to the general position of λ ∂ at x. However, we only obtain general position to the C i , rather than submersivity in the following claim.

Local properties relative to boundary: For each x ∈ S there exists l x ∈ N 0 -with l x = 0 for x ∈ S ∩ ∂X -and a locally uniformizing neighborhood Q(x) ⊂ X of x whose closure is contained in U, such that for some x > 0 we have a tame ep +subgroupoid x : {t ∈ R l | |t| < x } × Q(x) → Q + , (t, y) → ⊕ x t (S(y)) with surjective reduced linearizations, and thus a weighted branched orbifold | supp x |. Moreover, if x ∈ S ∩ V then ẽx induces a smooth map | supp x | → M, which is in general position to C i for each i ∈ I .

The structure of x is established in [22,Thm.15.5.], and the general position to each C i for x ∈ ∂ 0 X follows from submersivity. To achieve general position to the C i for x ∈ ∂X , recall that C = ∪ i∈I (C i ) ⊂ M is closed, so for x / ∈ e -1 (C) we can choose Q(x) disjoint from e -1 (C) so that general position to the C i ⊂ C is automatic. For x ∈ e -1 (C) ⊂ V we have e : supp x ∩ ∂X = supp λ ∂ • S| ∂X → M in general position to each C i by assumption on λ ∂ . Moreover, we choose Q(x) ⊂ V so that e : Q(x) ∩ supp x → M is smooth, and thus general position to each C i extends to a neighbourhood Q i ⊂ X of x. Then Q := i∈I Q i is a neighbourhood of x since I is finite, and we can replace Q(x) by a uniformizing neighbourhood in Q to achieve general position to all C i . |Q(x i )| with neighbourhoods of interior points x i ∈ S ∩ ∂ 0 X whose associated multisections x i are supported in the interior, dom-supp x i ⊂ R l x ∩ ∂ 0 X . Then we define ˜ : R l × W → Q + by ˜ (t, w) := t (w) := ⊕ x 1 t 1 ⊕ • • • ⊕ x k t k (w). This multisection is constructed so that 0 = and t | P -1 (∂X ) = λ ∂ for any t ∈ R l . Moreover, the estimate N ( t ) ≤ N ( ) + c|t| ≤ 1+α 2 + c|t| allows us to guarantee admissibility N ( t ) ≤ 1 by choosing |t| ≤ 1-α 2c . Then compactness of Z in (48) follows as above, and its smoothness is established using a covering |S -1 (0)| ⊂ k i=-k ∂ |U (x i )| where |U (x i )| for i ≤ 0 arise from x i ∈ S ∩ ∂X and cover a neighbourhood of |∂X |. Moreover, U (x i ) ⊂ R l × Q(x i ) can be chosen as in the prior proof of the local properties such that ẽ| supp : U (x i ) → M is in general position to C i for each i ∈ I . This establishes the following.

Global stabilization properties with fixed boundary values: There exists 0 > 0 such that ˜ := ˜ • S : {|t| < } × X → Q + is a tame ep + -subgroupoid with surjective reduced linearizations for every 0 < < 0 . In particular, | supp ˜ | is a weighted branched orbifold. Moreover, there is a neighbourhood V ⊂ X of S -1 (0) ∩ e -1 (C) such that ẽ| supp ˜ : supp ˜ → M satisfies ( ẽ| supp ˜ ) -1 (C) ⊂ R l × V , and its restriction to supp ˜ ∩ (R l × V ) is classically smooth and in general position to each C i .