Exact, graded, immersed Lagrangians and Floer theory
aa r X i v : . [ m a t h . S G ] O c t Exact, graded, immersed Lagrangians and Floertheory
Garrett Alston and Erkao Bao
Abstract
We develop Lagrangian Floer Theory for exact, graded, immersed La-grangians with clean self-intersection using Seidel’s setup [21]. A positivityassumption on the index of the self intersection points is imposed to ruleout certain (but not all) disc bubbles. This allows the Lagrangians to beincluded in the exact Fukaya category. We also study quasi-isomorphismof Lagrangians under certain exact deformations which are not Hamilto-nian.
Contents Floer cohomology 26 ι ; H , J ) is a chain complex . . . . . . . . . . . . . . . . . . . . 276.2 Invariance of HF( ι ) . . . . . . . . . . . . . . . . . . . . . . . . . . 286.3 Euler characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . 33 A ∞ -structure 34 A ∞ -structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407.5 Units, A ∞ categories, and invariance . . . . . . . . . . . . . . . . 42 A ∞ structure . . . . . . . . . . . . . . . . 528.3 Case when R = L : Equivalence with local systems . . . . . . . . 52 Exact, embedded Lagrangians are involved in many interesting classical ques-tions in symplectic geometry. They also constitute the class of objects in theexact Fukaya category of a symplectic manifold. In general, exact Lagrangiansare rather rigid objects, and existence of even one has strong implications. Onthe other hand, immersed exact Lagrangians satisfy an h-principle and alwaysexist in abundance. The goal of this paper is two-fold: one, to show that exact,graded, immersed Lagrangians with clean self-intersection, and which satisfya certain positivity condition, can be admitted as objects of the exact Fukayacategory; and two, to examine some of the invariance properties of these objectsin the Fukaya category.Immersed Floer theory was first developed by Akaho [2], and Akaho andJoyce [3]. In the context of curves on Riemann surfaces it also appears in workof Seidel [22] and Abouzaid [1]. Immersed Floer theory also plays a central rolein Sheridan’s work on homological mirror symmetry in [23] and [24].The results of this paper fall somewhere between [2] and [3]: Our theory ismore general than [2], but less general than [3]. In [2], a topological condition isimposed that rules out the existence of all holomorphic discs; in this paper, weimpose a positivity condition that only rules out some discs. As a consequence,the Floer differential can have a non-topological part. The A ∞ algebras howeverhave m = 0. What is gained over [3] is that we work over Z instead of theNovikov ring, and we employ the less abstract and more computable perturba-tion scheme of Seidel [21]. One could work over Z if one wanted to deal with2rientation issues. The positivity condition rules out bad degenerations of holo-morphic curves that would need a more advanced perturbation scheme to dealwith. Our theory allows immersions with clean, double-point self-intersection;and also transverse self-intersections without the double point restriction. Moregenerally, the theory works for arbitrary immersions with clean self-intersectionas long as there exists a totally geodesic metric on the image of the immer-sion. Also, inspired by [3] and [24], we prove invariance under certain exactdeformations which are not Hamiltonian.We would like to also mention our debt to the ideas and work of Fukaya,Oh, Ohta and Ono [12]. We start with (
M, ω, σ, J M , Ω M ) where M is a compact manifold with boundary, ω = dσ is an exact symplectic form, J M is an ω -compatible almost complexstructure, and Ω M is a nowhere-vanishing section of Λ top C ( T ∗ M, J M ). Actually,Ω M only needs to be well-defined up to a ± sign. Following Section (7a) of[21], the following convexity conditions are imposed: the Liouville vector field X σ defined by ω ( X σ , · ) = σ points strictly outwards along ∂M , and any J M -holomorphic curve touching ∂M is completely contained in ∂M . In particular,( ∂M, σ ) is a contact manifold. If J M is compatible with the contact structurenear the boundary then it automatically satisfies the convexity condition. Inthis case, one can allow M to be noncompact by attaching a conical end tothe boundary and taking J M to be cylindrical on the end. Up to homotopyequivalence, the Floer cohomology we will construct does not depend on J M .Also, Ω M is used for grading puposes and so the constructions only depend onthe homotopy equivalence class of Ω ⊗ M .Let L be a closed connected manifold and ι : L → M a Lagrangian immer-sion, and let L = ι ( L ) ⊂ M . For simplicity, we primarily focus on the Floertheory of the single Lagrangian immersion ι , but note that with some obviousmodifications the theory can be applied to any number of Lagrangian immer-sions. Assume that ι is exact in the sense that there exists f L : L → R suchthat df L = ι ∗ σ . Also, assume that L is graded in the sense that there exists θ L : L → R such that e πiθ L = Det M ◦ Dι . Here, Det M is the (squared)phase function, defined by Ω M in the following way: for X , . . . , X n a basis ofa Lagrangian plane in T M ,Det M ( X ∧ · · · ∧ X n ) = Ω M ( X ∧ · · · ∧ X n ) | Ω M ( X ∧ · · · ∧ X n ) | ∈ S . Let R = { ( p, q ) ∈ L × L | ι ( p ) = ι ( q ) , p = q } , R = ι ( R ) ⊂ L. (1)(Abusing notation, ι also denotes the obvious map R → M .) We call R the setof branch jump types and R the branch points or self-intersection points of L .3or the time being we assume that ι has only transverse double points; later onwe will allow ι to have clean self-intersection. It is interesting to note that L ∐ R = L × L L = { ( x, y ) ∈ L × L | ι ( x ) = ι ( y ) } . (2)In the clean intersection case, this serves to define a topology on R . An impor-tant concept that is a consequence of exactness is the notion of energy: Definition 1.1.
The energy E ( p, q ) of ( p, q ) ∈ R is E ( p, q ) = − f L ( p ) + f L ( q ).In addition to graded and exact, the main condition we impose on the La-grangian is a positivity condition . This condition is needed to control disc bub-bling. Condition 1.2.
If ( p, q ) ∈ R and E ( p, q ) > p, q ) ≥
3. Here, Ind( p, q )is a Maslov type index and is defined in Definition 2.5.From Definition 2.5 and Definition 1.1 it can be seen that E ( p, q ) = − E ( q, p )and Ind( q, p ) = dim L − Ind( p, q ). In Section 1.5 we discuss the possibility ofweakening the positivity condition.Now we explain our results. First, we need to choose some additional data.For a time dependent Hamiltonian function H = { H t } ≤ t ≤ with H t : M → R , denote the associated time dependent Hamiltonian vector field by X H andthe Hamiltonian flow by φ H t . (Our convention on Hamiltonian vector fields is dH t ( · ) = ω ( X H t , · ).) Definition 1.3.
Floer data H , J for ι : L → L consists of • a time dependent Hamiltonian H = { H t } ≤ t ≤ such that H t = 0 nearthe boundary of M , and • a time dependent ω -compatible almost complex structure J = { J t } ≤ t ≤ such that J t = J M near the boundary of M .Let Γ H denote the set of Hamiltonian chords γ : [0 , → M that start andstop on L . Assume that • (cid:2) ( φ H ) − ( L ) ∪ φ H ( L ) (cid:3) ∩ R = ∅ , and • L is transverse to φ H ( L ).The first point implies that no element of Γ H stops or starts on R and the secondimplies that Γ H is a finite set. When we want to emphasize these conditions wewill refer to the Floer data as being admissible , but since we generally imposeadmissibility we will usually not mention it.Associated to this data, let CF( ι ) = CF( ι ; H , J ) = Z · Γ H denote the Z -vector space formally generated by the Hamiltonian chords. Intuitively, if H t = H is time-independent and C -small, then CF( ι ) will have two gener-ators for each self-intersection point and one generator for each critical pointof the function H ◦ ι (see Lemma 2.9). Hence it is useful to imagine that4F( ι ) = CM( L ) ⊕ CM( R ), where CM denotes the Morse complex. Following[21], we define A ∞ operations m k : CF( ι ) ⊗ k → CF( ι ) for all k ≥ H , J . δ := m counts holomorphic strips and is the Floerdifferential. The first main results are: Theorem 1.4 (See Sections 6.1 and 6.2) . The Floer differential is well-definedand satisfies δ = 0 . Hence the Floer cohomology HF( ι ) := Ker δ/ Im δ is well-defined. Moreover, the cohomology is independent of the Floer data H , J . Theorem 1.5 (See Theorem 7.7 and Section 7.5) . The operations m k satisfy the A ∞ relations. More generally, ι can be admitted as an object of the exact Fukayacategory (as defined in [21]), and the quasi-isomorphism class of ι is independentof the choices needed to include it as an object in the Fukaya category. Inparticular, the homotopy type of the A ∞ algebra (CF( ι ) , { m k } ) is an invariantof ι . Unlike the embedded case, it is not necessarily true that HF( ι ) = 0; seeSection 1.3 for an example. Also, it is not necessarily true that an exact de-formation of an immersion ι is a Hamiltonian deformation. A one-parameterfamily of immersions { ι t } t with ι = ι is an exact deformation if the one form ι ∗ t ω ( ∂ι t /∂t, · ) is exact; it is a Hamiltonian deformation if ι t = φ t ◦ ι for some fam-ily of Hamiltonian diffeomorphisms φ t . If { ι t } t is a Hamiltonian deformationthen it is exact, but the converse is not true if ι is only an immersion. Con-dition 1.2 is preserved under Hamiltonian deformation but not under generalexact deformation (although if E ( p, q ) = 0 for all ( p, q ), it is preserved for smallenough exact deformation). If ι is exact, then { ι t } t is an exact deformation ifand only if each ι t is exact. Theorem 1.6 (See Theorem 7.11) . Let { ι t } t be a (connected) family of ex-act graded Lagrangian immersions such that each ι t satisfies Condition 1.2.Moreover assume that each ι t has only transverse double points. Then any twoimmersions in the family are quasi-isomorphic in the Fukaya category. This theorem gives a partial answer to Question 13.18 in [3], which wenow explain. Exact deformations have an interesting interpretation in termsof Legendrians. Observe that M × R is a contact manifold with contact form α = dz − σ , where z is the R -coordinate. If e L ⊂ M × R is an immersed Leg-endrian then the projection to M is an immersed exact Lagrangian; conversely,an immersed exact Lagrangian lifts to an immersed Legendrian via the immer-sion ι × f L : L → M × R . The self-intersection points of L correspond to Reebchords of e L . Moreover, e L is embedded if and only if E ( p, q ) = 0 for all ( p, q ) ∈ R .Question 13.18 in [3] asks if immersed Floer cohomology is invariant under anexact deformation which lifts to an isotopy of embedded Legendrians (in general,exact deformation only lifts to isotopy of immersed
Legendrians). Theorem 1.6gives a partial affirmative answer to this: if ι satisfies the positivity conditionand lifts to an embedded Legendrian, and { ι t } t lifts to an isotopy of embedded5egendrians, then each ι t must also satisfy the positivity condition; hence ifthe immersed Lagrangians have only transverse double points then the theoremsays that the Floer cohomology is invariant. (Note that Theorem 1.6 may stillapply even if the isotopy of Legendrians is not an embedded isotopy.)Now we discuss the clean self-intersection case. See Definition 8.1 for aprecise definition; essentially the adjective clean means that the self-intersection(in L ) is a submanifold of M , with tangent space equal to the intersection ofthe tangent spaces of the branches. Also, singular points are still only doublepoints. We define R and R in the same way as before; these are now smoothmanifolds instead of just finite sets of points. In particular, we can still ask ifthe positivity condition is satisfied. In case it is, we can define the Floer theoryin the same way as before. Theorem 1.7 (See Section 8) . Let ι be a graded, exact, clean immersion sat-isfying Condition 1.2. Then ι can be included as an object of the exact Fukayacategory, and the quasi-isomorphism class of ι is independent of the choicesneeded to include it. In the previous theorem, if L = R then admissible Floer data does not existbecause the endpoints of Hamiltonian chords necessarily intersect R . To dealwith this, we need to alter the definition of Γ H slightly: we should also includelifts to L of the start and end points of the chords, see Definition 8.7. Also,if L = R , then L → L is a double cover and L is exact. In this situation, wedo not need to require that ι has only double points, so consider ι : L → L any covering space, say r -fold. Then ι defines a local system on L with fiber( Z ) r (see prior to Theorem 8.10), call it E ι , and ( L, E ι ) can be thought of as anobject of the embedded, exact Fukaya category with local systems. We suspectthe following theorem is essentially already known to many people. Theorem 1.8 (See Theorem 8.10) . ι and ( L, E ι ) are quasi-isomorphic objectsin the Fukaya category. The Floer-cohomology is isomorphic to the singularcohomology (with Z -coefficients) of L ∐ R = L × L L . It is interesting to note that if L is an embedded exact Lagrangian then L × L L = L and the Floer cohomology is isomorphic to the singular cohomologyof L × L L . More generally, if L and L are embedded, exact Lagrangians,then their fiber product is L × M L = L ∩ L and there is a spectral sequencecomputing the Floer cohomology of the pair ( L , L ) which starts at the singularcohomology of L ∩ L (assuming they intersect cleanly). A similar spectralsequence should exist in the immersed case. (We do not investigate it in thispaper.)The immersions which are covering spaces L → L have an obvious classof exact deformations. Take a generic function f on L , and deform ι in thedirection of − J ∇ f . These deformations may or may not satisfy the positivitycondition; if they do then the result is a quasi-isomorphic object of the Fukayacategory. 6 heorem 1.9 (See Theorem 8.13) . Let { ι t } | t |≤ ǫ be the deformation describedabove. Assume each ι t satisfies Condition 1.2, and each ι t for t = 0 has trans-verse double points. Then any immersion in the family is quasi-isomorphic to ι . In particular, choosing different permissible functions f could lead to quasi-isomorphic objects with differing numbers of self-intersection points. Thesewould be examples of exact deformations qualitatively different than those ap-pearing in Theorem 1.6.In Section 9 we discuss the case of immersions with general clean self-intersection. Our theory works for these more general immersions as long asthey satisfy the positivity condition and the image of the immersion is totallygeodesic with respect to some metric on M . An example of such an immersionis one in which all self-intersections are transverse but not necessarily doublepoints (i.e., they can be triple points, etc.). The totally geodesic restriction islikely a technical restriction that could be removed with a more elaborate setup. The general framework we follow is the one devised by Seidel in [21] for ex-act embedded Lagrangians. The definition of the Floer cohomology and the A ∞ structure exactly parallels that given in [21]. However, because our La-grangians are immersed and not embedded, holomorphic discs with corners atthe self-intersection points might exist. These in principle could prevent the A ∞ relations from holding, so the extra work we need to do is to show that thesediscs do not not cause problems.To deal with this, we define moduli spaces of curves with two types of bound-ary marked points, which we call Type I and Type II marked points. Type Imarked points are the kind appearing in [21] and must converge to Hamiltonianchords. Type II marked points converge to self-intersection points.We also define two types of moduli spaces: holomorphic polygons, and holo-morphic discs. Holomorphic polygons have Type I and possibly Type II markedpoints and satisfy an inhomogeneous Cauchy-Riemann equation. The holo-morphic polygons with only Type I marked points are used to define the A ∞ structure. Holomorphic discs have only Type II marked points and satisfy ahomogeneous Cauchy-Riemann equation; they appear attached to polygons inthe compactification of the moduli space of polygons and serve only an auxiliarypurpose; in particular, they do not enter into the definition of the A ∞ structure.The preliminary extra details needed beyond [21] are a description of theanalytic and Fredholm setup of the moduli spaces of polygons (including oneswith Type II points), a description of the Gromov compactness of the two typesof moduli spaces, and a statement on regularity of the moduli spaces of holomor-phic polygons. These mostly follow from standard facts already well representedin the literature. With these in hand, the main type of statement to prove whichis not needed in [21] is the following: a zero- or one-dimensional moduli spaceof holomorphic polygons with no Type II points can be compactified without7ntroducing any Type II points. The positivity condition, Gromov compactness,and regularity are the essential ingredients needed to prove this statement. The first example is an immersed sphere in C n with one self-intersection point,which is described in Example 13.12 in [3]. The sphere can be described asfollows: Start with the curve C = { s + it ∈ C | t = s − s } ; C is an immersedcircle in C which looks like an ∞ symbol, and has one self-intersection point atthe origin. Let L = { ( λx , . . . , λx n ) | λ ∈ C, x , . . . , x n ∈ R , x + · · · + x n = 1 } .Then L is an immersed sphere inside C n with a single self-intersection point atthe origin. The indices of the two elements of R are − n + 1. A simplecalculation shows that the index n + 1 branch jump has positive energy. (Thisis easy for the n = 1 case; for the higher dimensional case just consider one ofthe holomorphic discs bounded by C × { } n − ⊂ L .) Thus the Lagrangian L satisfies the positivity condition when n ≥
2. Since every compact Lagrangianin C n is displaceable, the Floer cohomology is 0.The next examples are Lagrangian spheres in (smoothings of) A N surfaces,described in [4]. The A N surface is the hypersurface X = { xy + f N ( z ) = 0 } ⊂ C , where f N ( z ) is a degree N polynomial with no repeated roots. As symplecticmanifolds, A ∼ = C and A ∼ = T ∗ S . The map X → C , ( x, y, z ) z is aLefschetz fibration with critical values at the roots of f N . Given an embeddedpath in the base C that starts and stops at the same critical value and doesn’tintersect any other critical values, the union of all the vanishing cycles over allthe points in the base path is an immersed sphere with a single self-intersectionpoint. The indices of the branch jumps are 3 and −
1, and they satisfy thepositivity condition. We refer readers to [4] for details; also, the case N = 2 isessentially the affine version of an example studied by Auroux in [5]. In [4], apearly version of the Floer theory described in this paper is used to calculate theFloer cohomology. The result is that the cohomology has rank 4 if the interiorof the region bounded by the base path contains at least one critical value, andis 0 otherwise. All of the Lagrangians in the N = 1 case fall into the lattercategory; they essentially coincide with the Lagrangian in C described in theprevious class of examples. In [2], Akaho defines Floer cohomology and the product for immersed La-grangians that satisfy the condition π ( M, L ) = 0. This a priori rules outall disc bubbles. Such Lagrangians automatically satisfy Condition 1.2.In [3], Akaho and Joyce define Floer theory for arbitrary (oriented, spin) im-mersed Lagrangians with only transverse double points. Their theory is similarto the theory of Fukaya, Oh, Ohta, and Ono for embedded Lagrangians [12], andrequires using the Novikov ring Λ ,nov , and the virtual perturbation techniqueof Kuranishi spaces to deal with transversality issues.8ur theory does not require using Kuranishi spaces, and this makes it simplerand more computable (of course, the expense is loss in generality). In additionto the virtual perturbation technique, another difficulty of the general theory isthat it involves a combination of geometric and algebraic constructions to arriveat the final A ∞ algebra; for example, the homological perturbation lemma isused several times along the way. Also, at some point one has to deal withinfinite dimensional chain complexes. The methods of Seidel [21] that we em-ploy are more direct and purely geometric, and the chain complexes are finitedimensional from the beginning.A local Floer cohomology for two cleanly intersecting embedded Lagrangianswas studied by Pozniak [18]. It seems likely that a Morse-Bott version of [3] canbe defined to deal with Lagrangians that are cleanly immersed by mimickingthe Morse-Bott version of Floer theory for two cleanly intersecting embeddedLagrangians in [12].In the language of [12] and [3], Lagrangians satisfying Condition 1.2 areautomatically unobstructed; this is stated as Proposition 13.10 in [3]. Thisproposition has a weaker hypothesis than the positivity condition: it is simplyrequired that there are no self-intersection points of index 2. In the next sectionwe will discuss the prospect of bringing our results in line with this.We learned a great deal about immersed Floer theory from the work ofSheridan in [24] and [23]. The example that Sheridan deals with does notexactly fit into any of the frameworks discussed so far; rather, Sheridan dealswith bubbling and compactness issues by working in a covering of the symplecticmanifold in which the immersed Lagrangian lifts to an embedded Lagrangian.Theorem 1.9 is inspired by a similar result of Sheridan for his example. We expect that Condition 1.2 can be relaxed to the following: if Ind( p, q ) = 2then E ( p, q ) ≤
0. The reason is that the virtual dimension of the moduli spaceof holomorphic discs with one (Type II) marked point of index less than 2 isnegative. Hence, generically, one expects no such discs to exist, and hence theyshouldn’t cause a problem with disc bubbling. The index 2 discs are moreproblematic because (in the terminology of [3]) they contribute to m (1) andcan be an obstruction to Floer cohomology being well-defined. Thus they needto still be explicitly ruled out with a positivity condition. In this paper wenever address the regularity of holomorphic discs (we only require the modulispaces of inhomogeneous holomorphic polygons to be regular) and thereforeneed Condition 1.2. However, we expect that the results of Ekholm, Etnyre,and Sullivan [8], [9] on Legendrian contact homology can be adapted to oursetting to give the necessary regularity arguments. We plan to investigate thisin future work.From a computational point of view, it would be beneficial to have a Morse-Bott version of the theory that does not require perturbing the Lagrangianby a Hamiltonian. In this theory, one would take a Morse function on L ∐ R and define the Floer cochain complex to be the Morse complex of L plus the9orse complex of R . Then the Floer differential, and more generally the A ∞ structure, would be defined by counting “pearly curves”, which are combinationsof holomorphic discs and Morse trajectories. In the case of embedded monotoneLagrangians, such a theory for Floer cohomology is developed by Biran andCornea in [6]. Sheridan also develops such a theory for his immersed Lagrangiansin [23] and [24]. In [4], the first author described how such a theory worksfor Floer cohomology in the context discussed in this paper, and we plan toaddress the full A ∞ structure in future work. We remark that in [4], a strongerpositivity condition was imposed; namely that if E ( p, q ) > p, q ) ≥ (dim L + 3) /
2. This stronger condition is needed in the pearly case to rule outdegenerations resulting from nodal holomorphic curves which contain a constantdisc component that maps to a self-intersection point; we do not know if it can berelaxed to the condition used in this paper. (Such problematic degenerations arenot a problem in this paper because they are ruled out by the use of Hamiltonianperturbations; see Remark 3.6.)
Here is an outline of the paper. First, a few general points. Before Section 7we focus solely on Floer cohomology and restrict attention to the case that ι has transverse double points. In particular, in Section 6 we explain why theFloer differential is well-defined, and the Floer cohomology is invariant underthe choice of Floer data. The proofs are standard, but we still explain themcarefully in order to show why the positivity condition is needed. In Section7 we consider the full A ∞ structure, and in Section 8 we consider the cleanimmersion case. Fewer details are given in these sections because the mainpoints are similar to those discussed in detail in Section 6.Regarding moduli spaces of curves, we will consider two types of curveswhich we call polygons and discs. In the literature these two terms are oftensynonymous, but for us they will have different meanings: Polygons will be usedto define the A ∞ structure, and discs will be used to describe bubbling. BeforeSection 7 the only polygons we will consider are strips because we will onlydiscuss Floer cohomology and not the full A ∞ structure. The term curve is acatch-all term that can mean a strip, or other polygon, or a disc.We begin in Section 2 by discussing the notion of marked strips and markeddiscs, and the Maslov index. In Section 3 we describe how the space of stripsand discs with W ,p regularity can be given the structure of a Banach manifold,and we define the moduli spaces we will use. In Sections 4 and 5 we discusstransversality and compactness of the moduli spaces. Marked polygons (otherthan strips) will be introduced in Section 7 when we talk about the full A ∞ structure.In Section 9 we explain how we can relax the condition that self-intersectionpoints are double points. An appendix on asymptotic analysis is included forcompletion. 10 Marked curves: topological definitions
In general, we will consider two different types of curves, which we will call poly-gons and discs. Holomorphic polygons will be used to define the A ∞ structurefor a Lagrangian, and holomorphic discs will be used to describe disc bubbling.Strips are special types of polygons (they can be thought of as 2-gons), andthey are the curves used to define Floer cohomology. We will be concernedonly with Floer cohomology through Section 6, so we postpone the discussionof general marked polygons to Section 7.2. This section discusses the definitionsand topological properties of strips and discs. Holomorphic strips and discs willbe considered in Section 3.Let ˆ S denote a closed Riemann surface with boundary. In particular, ˆ S hasa complex structure j , which is suppressed from the notation. We will be inter-ested in two types of marked boundary points which we call Type I and TypeII points. Let Σ = ( z , . . . , z k ) ⊂ ∂ ˆ S denote an ordered list of Type I markedpoints, and ∆ = ( z ∆1 , . . . , z ∆ m ) ⊂ ∂ ˆ S denote an ordered list of Type II markedpoints. Assume that all marked points are distinct. Moreover, assume thateach marked point is labeled as incoming ( − ) or outgoing (+); this determinesdecompositions Σ = Σ − ∐ Σ + and ∆ = ∆ − ∐ ∆ + . Given ˆ S and Σ, let S be the(non-compact) Riemann surface S = ˆ S \ Σ. When we write this we really meanthat ˆ S, Σ is part of the data of S .One of the features of holomorphic curves with boundary on an immersedLagrangian is that the curve can jump branches at a self-intersection point. Todeal with this type of behavior (and for other reasons), boundary lifts to L needto be included as additional data of the curve. The following definition will playan important role in describing the behavior of the lifts at the branch points. Definition 2.1.
Let γ : ( − ǫ, ǫ ) → L be a path such that γ (0) ∈ R . Let˜ γ : ( − ǫ, ∪ (0 , ǫ ) → L be a lift of γ defined away from 0, and let p = lim s → − ˜ γ ( s ) , q = lim s → + ˜ γ ( s ) . If p = q then ( p, q ) ∈ R and we say that the path ˜ γ has a branch jump of type ( p, q ). If p = q then ˜ γ extends continuously over 0 and γ does not have a branchjump at 0. Note that ˜ γ is unique if it exists and γ is not constant on ( − ǫ,
0) or(0 , ǫ ). Let ˆ S = D = { | z | ≤ } , Σ − = { − } , Σ + = { } and S = ˆ S \ Σ. Identify S with R × [0 ,
1] and Σ with { ±∞ } ; we will call this Riemann surface the strip and denote it by Z . Explicitly, use the standard identification z = s + it ∈ Z = R × [0 , ie πz + 1 ie πz − ∈ S, ±∞ 7→ ± . Fix a Hamiltonian H = { H t } t . Recall that Γ H is the set of time-1 chordson L . 11 efinition 2.2. A C -marked strip u = ( u, ∆ , α, ℓ ) connecting γ ± ∈ Γ H con-sists of • a continuous map u : ( Z, ∂Z ) → ( M, L ) such that lim s →±∞ u ( s, t ) = γ ± ( t )uniformly in t , • a list of Type II marked points ∆ ⊂ ∂Z , all of which are consideredoutgoing, and are ordered starting from left to right along the bottom ofthe strip and continuing right to left along the top, • a map α : { , . . . , | ∆ | } → R , and • a continuous map ℓ : ∂Z \ ∆ → L such that ι ◦ ℓ = u | ∂Z \ ∆.Moreover, ℓ has a branch jump at each z ∆ i ∈ ∆ of type α ( i ) ∈ R (in the senseof Definition 2.1; moving counterclockwise around ∂Z corresponds to increasing s in the notation of the definition). Denote by B ( Z, ∆ , α ) the space of all C -marked strips with Type II points ∆ ( H is suppressed from the notation), and B ( Z, α ) = ` ∆ B ( Z, ∆ , α ) × { ∆ } . Let ˆ S = D = { | z | ≤ } . Definition 2.3. A C -marked disc u = ( u, ∆ , α, ℓ ) consists of • a continuous map u : ( D, ∂D ) → ( M, L ), • a list of Type II marked points ∆ ⊂ ∂D which are ordered in counter-clockwise order, along with a decomposition ∆ = ∆ + ∐ ∆ − into outgoingand incoming points, • a map α : { , . . . , | ∆ | } → R , and • a continuous map ℓ : ∂D \ ∆ → L such that ι ◦ ℓ = u | ( ∂D \ ∆).Moreover, moving in the counterclockwise direction, ℓ has a branch jump ateach z ∆ i ∈ ∆ + of type α ( i ) ∈ R ; and, moving in the clockwise direction, ℓ hasa branch jump at each z ∆ i ∈ ∆ − of type α ( i ) ∈ R . See Definition 2.1 for thedefinition of a branch jump type.Denote by B disc ( D, ∆ , α ) the space of all C -marked discs, and B disc ( D, α ) = ` ∆ B disc ( D, ∆ , α ) × { ∆ } . Let G ( M, ω ) → M be the fiber bundle of Lagrangian planes in T M . GivenΛ , Λ ∈ G ( M, ω ) over the same point in M , with Λ ∩ Λ = { } , and θ , θ ∈ R such that Det M (Λ j ) = e πiθ j for j = 0 ,
1, define the index of the pair of gradedplanes to beInd((Λ , θ ) , (Λ , θ )) = n + θ − θ − · Angle(Λ , Λ ) . (3)12ere n = dim M/ , Λ ) = α + · · · + α n , where α i ∈ (0 , ) aredefined by requiring that there exists a unitary (with respect to J M and ω ) basis u , . . . , u n of Λ such thatΛ = Span R { e πiα u , . . . , e πiα n u n } . From Equation (3) we can see thatInd((Λ , θ ) , (Λ , θ )) = n − Ind((Λ , θ ) , (Λ , θ )) . Alternatively, the index can be defined in the following way. Let λ : [0 , →G ( M, ω ) be a continuous path of Lagrangians all lying over the same point in M .Furthermore, assume λ (0) = Λ , λ (1) = Λ , and assume there exists a continu-ous family of real numbers θ t from θ to θ such that Det M ( λ ( t )) = e πiθ t . ThenInd((Λ , θ ) , (Λ , θ )) = n/ µ ( λ, Λ ) . The µ on the right-hand side denotesthe Maslov index of a path of Lagrangians with respect to a fixed Lagrangian,as defined in [19]. Equivalently, if λ pos denotes the path of Lagrangians whichis the positive definite rotation from Λ to Λ , then Ind((Λ , θ ) , (Λ , θ )) = µ ( λ λ pos ). Here the right-hand side denotes the Maslov index of a loop.There is yet another definition of the index which connects it to the indextheory of Cauchy-Riemann operators. To describe this, let p ∈ M be the pointwith Λ , Λ ⊂ T p M . Consider the unit disc D with trivial symplectic vectorbundle E = D × T p M . Fix a marked point z ∈ ∂D (which should be thoughtof as an incoming marked point) and fix a parametrization (0 , ∼ = ∂D \ { z } (with the arc given the counterclockwise orientation). Define a Lagrangian sub-bundle F of E over this arc by F t = λ ( t ) (if necessary, first homotope λ slightlyso that it is constant near t = 0 and t = 1). Let ¯ ∂ ( E,F ) denote the standardCauchy-Riemann operator of this bundle pair; the domain is W , sections of E with F boundary conditions, and the target is L sections of Λ , D ⊗ E . ThenInd((Λ , θ ) , (Λ , θ )) is equal to the Fredholm index of the operator ¯ ∂ ( E,F ) . Seeequation (11.20) and Lemma 11.11 in [21] for a proof.We now describe how to define an index for Hamiltonian chords. Let H = { H t } t be an admissible Hamiltonian, Γ H be the set of time-1 Hamiltonianchords, and φ H t be the flow of the Hamiltonian vector field X H . For γ ∈ Γ H ,let Λ t ∈ G ( M, ω ), θ t ∈ R be continuous in t ∈ [0 ,
1] and such that(Λ , θ ) = ( T γ (0) L, θ L ( ι − ( γ (0)))) , D ( φ H t ) · Λ = Λ t , Det M (Λ t ) = e πiθ t . In other words, (Λ , θ ) is the graded plane determined by the Lagrangian at γ (0), and the linearized flow Dφ H t is used to move it to a graded plane (Λ , θ )contained in T γ (1) M . Definition 2.4.
The index of γ ∈ Γ H isInd γ = Ind((Λ , θ ) , ( T γ (1) L, θ L ( ι − ( γ (1))))) . This coincides with the definition given in Section (12b) of [21]. To motivateit, consider the Lagrangians L = φ H ( L ) and L = L . Hamiltonian chords on13 correspond to points of L ∩ L via γ ↔ γ (1). The Lagrangian L inheritsa grading from the grading of L and the isotopy t φ H t ( L ), 0 ≤ t ≤ , θ ) is the graded Lagrangian plane determined by thetangent space of L at γ (1).Following [3], the grading of L also allows us to assign an index to elementsof R . Definition 2.5.
Let ( p, q ) ∈ R . Then the index of ( p, q ) isInd( p, q ) = Ind(( ι ∗ T p L, θ L ( p )) , ( ι ∗ T q L, θ L ( q ))) . Now we define the Maslov index for marked curves. Let u = ( u, ∆ , α, ℓ )be a marked curve with domain S = ˆ S \ Σ and Type II points ∆. Let S ′ = S ∪ Σ × [0 ,
1] be the compact surface with corners obtained from ˆ S by replacingeach point in Σ with a unit interval; topologically S ′ is a closed disc. Forinstance, if S = Z = R × [0 , S ′ = [ −∞ , ∞ ] × [0 ,
1] which can bethought of as a compact rectangle. Also, { z } × [0 , ⊂ ∂S ′ is assumed to beparameterized in such a way that motion from 0 to 1 in { z } × [0 ,
1] correspondsto counterclockwise motion along ∂S ′ if z is an outgoing point, and clockwisemotion if z is an incoming point. (Note that with the strip −∞ is an incomingpoint and + ∞ is an outgoing point.) Since u converges to chords at points ofΣ, it can be thought of as a continuous map into M with domain S ′ . Over S ′ , u ∗ T M defines a symplectic vector bundle. On the boundary of S ′ and awayfrom ∆ and Σ × [0 , ι ∗ ℓ ∗ T L defines a Lagrangian sub-bundle. Extend thesub-bundle over Σ × [0 , ∐ ∆ ⊂ ∂S ′ as follows (in the following, a positivedefinite rotation is the rotation t e tJ for appropriate ω -compatible J ): • For an outgoing Type I point z + ∈ Σ that converges to γ ∈ Γ H , we wantthe Lagrangian subbundle over { z + } × [0 , ⊂ ∂S ′ to be the path ofLagrangians that starts with T γ (0) L , then moves along the linearized flow Dφ H t until a plane in T γ (1) M is reached, and then is followed by a positivedefinite rotation from this plane to T γ (1) L . Forward progress along thispath should correspond to moving in the counterclockwise direction along ∂S ′ . More precisely, we do the following. Homotope u ∗ T M slightly so thatit is constant in a small neighborhood of { z + }×{ } ∈ { z + }× [0 , ⊂ S ′ .This gives a a trivialization in a neighborhood of this point. Let ǫ > t ∈ [0 , − ǫ ], the fiber of the subbundle over { z + }×{ t } ∈{ z + } × [0 ,
1] is Dφ H t ( T γ (0) L ). For 1 − ǫ ≤ t ≤
1, move in the positivedefinite direction until T γ (1) L is reached. The result is that the Lagrangiansub-bundle has been extended over { z + } × [0 , ⊂ ∂S ′ . We emphasizethat since z + is an outgoing point, moving in the forward direction ofthe interval { z + } × [0 ,
1] corresponds to moving in the counterclockwisedirection around ∂S ′ . • For an incoming Type I point z − ∈ Σ, the same procedure is applied.The difference in the resulting bundle is that forward movement along { z − } × [0 ,
1] corresponds to clockwise movement around ∂S ′ . (Note that14his is automatically achieved by the way we identified { z − } × [0 ,
1] witha portion of the boundary of S ′ .) • For an outgoing Type II marked point of type ( p, q ) ∈ R implant a coun-terclockwise path moving in the positive definite direction from ι ∗ T p L to ι ∗ T q L . • For an incoming Type II marked point of type ( p, q ) ∈ R implant a coun-terclockwise path moving in the negative definite direction from ι ∗ T q L to ι ∗ T p L (equivalently, a clockwise path moving in the positive definitedirection from ι ∗ T p L to ι ∗ T q L ). Definition 2.6.
The Maslov index µ ( u ) of u is defined to the be the Maslovindex of the bundle pair defined above. Lemma 2.7.
Let u = ( u, ∆ , α, ℓ ) ∈ B ( Z, ∆ , α ) be a marked strip connecting γ ± ∈ Γ H . Then the Maslov index µ ( u ) of u is µ ( u ) = Ind γ − − Ind γ + − X ≤ i ≤| ∆ | Ind α ( i ) . Lemma 2.8.
Let u = ( u, ∆ , α, ℓ ) ∈ B disc ( D, ∆ , α ) be a marked disc with ∆ =( z ∆1 , . . . , z ∆ k ) . Then the Maslov index µ ( u ) of u is µ ( u ) = X z ∆ i ∈ ∆ − Ind α ( i ) − X z ∆ i ∈ ∆ + Ind α ( i ) . It is instructive to consider the indices in the following situation. Let H t = H be a C -small time independent admissible Hamiltonian. Furthermore, assumethat H L := H ◦ ι is a Morse function on L , and the critical points of H L aredisjoint from ι − ( R ). Then there exists ǫ > < ǫ < ǫ there exists a canonical bijection Γ ǫH ∼ = Crit( H L ) ∐ R . Note that φ ǫH = φ Hǫ . Tosee the bijection, first note that the set of chords can be broken into two disjointsets, Γ ǫH = Γ L ∐ Γ R , based on the distance between ι − ( γ (0)) and ι − ( γ (1))in L . If the distance is small, γ is in Γ L , and if the distance is large, γ is inΓ R . The chords in Γ R jump branches of L and correspond to points of R . If x ∈ R , and ι − ( x ) = { p, q } , then there will be a chord that jumps from the p to the q branch and a chord that jumps from the q to the p branch. The chordthat jumps from the p to the q branch is characterized by the property that ι − ( γ (0)) is near p and ι − ( γ (1)) is near q ; a similar statement holds for theother chord with p and q reversed. Thus these two chords correspond to ( p, q )and ( q, p ) in R , respectively. The chords in Γ L correspond to the critical pointsof H L . This can be deduced from the fact that X H ( p ) is tangent to L if andonly if p ∈ Crit H L .Now we examine the indices of these chords. For p ∈ Crit H L , let γ p ∈ γ L denote the corresponding chord, and for ( p, q ) ∈ R let γ ( p,q ) denote thecorresponding chord. 15 emma 2.9. Ind γ p = dim L − Ind p H L = Ind p ( − H L ) , where Ind p H L denotesthe Morse index (number of negative eigenvalues of the Hessian of H L at p ), and Ind γ ( p,q ) = Ind( p, q ) , where the index on the right-hand side is from Definition2.5.Proof. First consider the case of γ p . In this case there is no essential differencebetween an embedded or immersed Lagrangian, so we start by examining thesimple example of L = R ⊂ C and H ( x, y ) = ± x /
2, with Ω M = dz and ω = dx ∧ dy . Equip L with any constant grading θ L ∈ Z . Then φ Ht ( x, y ) =( x, y ∓ tx ), so φ ǫH ( L ) is a line of slope ∓ ǫ , and this Lagrangian inherits agrading by continuing θ L along the Hamiltonian isotopy. The only chord is theconstant chord γ ( t ) = (0 , , H L . By the discussion after Definition 2.4, Ind γ is equal to the index ofthe pair of graded planes ( T (0 , φ ǫH ( L ) , T (0 , L ). The result is Ind γ = 0 if H = − x and Ind γ = 1 if H = + x . Thus the lemma holds in the simple case R ⊂ C , H = ± x . For more general H (depending on x and y ) the calculationis essentially the same as long as H L has a single critical point at x = 0 and ǫ is chosen small enough ( γ will no longer just be a constant chord though).The general result can then be deduced from this simple example by using theWeinstein neighborhood theorem and the Morse lemma to reduce the problemto a direct sum of one-dimensional problems.Now consider the case of γ ( p,q ) with ( p, q ) ∈ R . In this case, Ind γ ( p,q ) is calculated by moving the graded plane T γ ( p,q ) (0) L along the linearized flow Dφ ǫHt until a graded plane Λ contained in T γ ( p,q ) (1) M is reached, and then theindex of the chord is the index of the pair of graded planes (Λ , T γ ( p,q ) (1) L ). If ǫ > ι ∗ T p L, ι ∗ T q L ) in such a way that the planes stay transversethroughout the homotopy. Thus Ind γ ( p,q ) = Ind( p, q ). In this section we define Banach manifolds of marked strips with W ,p -regularity.We also define moduli spaces of (inhomogeneous) holomorphic strips, and alsoholomorphic discs, and describe the Fredholm theory of the strips. The analyticsetup and Fredholm theory of the discs is not needed. The constructions andpropositions are essentially standard and can be seen as a combination of [21]and [3]. One place where we give a few details is Proposition 3.7. This propo-sition deals with the regularity of holomorphic strips. It is an important pointto address because Type II points are not present in the embedded Lagrangiancase of [21], and also our setup has a Hamiltonian term and a domain dependentcomplex structure, which is different than [3].Fix a metric g fix on M that agrees with ω ( · , J M · ) outside of a compactsubset and is such that L is totally geodesic. This can be done by the followinglemma. 16 emma 3.1. Let L be an immersed Lagrangian such that all singular pointsare transverse double points. Then there exists a metric g on M such that L istotally geodesic.Proof. First, suppose L is a smooth submanifold of M . Let E ⊂ T M | L be avector bundle over L such that E ⊕ T L = T M | L . Let g be any metric on M such that E is the orthogonal complement of T L in T M | L . L is totally geodesicif and only if the second fundamental form vanishes, and this is equivalentto g ( Z, ∇ X X ) = 0 for all Z ∈ E x , X ∈ T x L , x ∈ L . Let ˜ X, ˜ Z be vectorfields in T M defined on a neighborhood of x ∈ L such that [ ˜ Z, ˜ X ]( x ) = 0and ˜ Z ( x ) = Z , ˜ X ( x ) = X . Then g ( Z, ∇ X X ) = − ( ˜ Z ( g ( ˜ X, ˜ X )))( x ). Thus wehave the following characterization of totally geodesic with given normal bundle:Given L and E , and a metric g on M such that T L and E are orthogonal, L istotally geodesic if and only if ˜ Z ( g ( ˜ X, ˜ X ))( x ) = 0 for all x ∈ L , X ∈ T x L , and Z ∈ E x .Now suppose embedded L and E are given, and g is a metric so that T L and E are orthogonal and L is totally geodesic. Let φ be a smooth non-negativefunction on M so that ( Zφ ) | L = 0 for any vector Z in E . Then for any Z ∈ E x , X ∈ T x L we have ˜ Z (( φg )( ˜ X, ˜ X ))( x ) = ( Zφ )( x ) · g ( X, X )( x ) + φ ( x ) · ( ˜ Z ( g ( ˜ X, ˜ X )))( x ) = 0 on L . Thus, at points where φg is a metric, L is totallygeodesic with respect to φg and still has normal bundle E . Similarly, if g and g are metrics such that E is orthogonal to T L for both and L is totally geodesicwith respect to both, then the same is true of g + g .Using these facts we can construct metrics on M such that embedded L istotally geodesic in the following way. First, fix any E so that E ⊕ T L = T M | L .Given a small open set U in M , it is clear that there exists some metric on U such that E | L ∩ U is orthogonal to T L | L ∩ U , and L ∩ U is totally geodesic in U . Then use a partition of unity { φ i } on M which satsifies ( Zφ i )( x ) = 0 for Z ∈ E x , x ∈ L to piece together these local metrics to get a global metric on M . L will be totally geodesic with respect to this global metric, and also E and T L will be orthogonal.Now suppose ι : L → L ⊂ M is a Lagrangian immersion with transversedouble points. In a small neighborhood U of a double point, we can find adiffeomorphism from U to a small open ball in R n such that L maps to theunion of two n -planes. Take the standard metric on R n , and pull it back to U in M . We have thus constructed a metric on M near the double points of L such that L is totally geodesic. Choose subbundle E ⊂ ι ∗ T M so that E isthe orthogonal complement of T L near the double points of L with respect tothe constructed metric. Since L is embedded away from the double points, theprevious arguments can be used to extend the constructed metric to all of M in such a way that L is totally geodesic and E is the normal bundle.Let Z + = [0 , ∞ ) × [0 ,
1] and Z − = ( −∞ , × [0 , S = ˆ S \ Σand Type II marked points ∆. A choice of a strip-like end for a marked point z ∈ Σ ± ∐ ∆ ± of Type I or Type II is a proper holomorphic embedding ǫ : Z ± → ˆ S \ { z } (4)17hat satisfies lim s →±∞ ǫ ( s, t ) = z and is such that the image of ǫ is disjoint fromΣ ∐ ∆. Fix an admissible Hamiltonian H . First consider the case of fixed Type IImarked points ∆ = ∆ + = ( z ∆1 , . . . , z ∆ m ) ⊂ ∂Z (∆ = ∅ is allowed). Fix a choiceof strip like ends, ǫ i for each Type II marked point z ∆ i ∈ ∆, such that the imagesof the strip-like ends do not overlap.The strip-like ends determine a metric g on Z , up to equivalence, as follows.First, on ǫ i ([1 , ∞ ) × [0 , s + it the standard coordinates on [1 , ∞ ) × [0 , ⊂ Z + , g = ds + dt . Second, on the complement of the images of the strip-likeends, with s + it the standard coordinates on Z , g = ds + dt . Finally, on theremaining part, the metric can be chosen in any way that smoothly matchesup with the previous choices. Different choices of strip-like ends will result inequivalent metrics. Call such a metric a compatible metric. The metric definesa measure (volume form) on Z \ ∆.Let γ ± ∈ Γ H and suppose u = ( u, ∆ , α, ℓ ) is a marked strip satisfying u is constant in a neighborhood of each Type II marked point, u ( s, t ) = γ ± ( t ) for s near ±∞ , and (5) u is smooth. Definition 3.2.
With u as above, let W ,p ( u ∗ T M ) denote the set of all sections ξ of u ∗ T M over Z \ ∆ such that ξ is in W ,p and has boundary values in ι ∗ ℓ ∗ T L .The norm is defined by using the metric on Z \ ∆ determined by the strip likeends and the fixed metric g fix on M . We assume that p > U u,ρ be the subset of W ,p ( u ∗ T M ) consisting of all sections ξ such that k ξ k W ,p < ρ for all z ∈ Z \ ∆. DefineΦ u,ρ : U u,ρ → B ( Z, ∆ , α ) , ξ u ξ = ( u ξ , ∆ , α, ℓ ξ ) , (6)where u ξ = exp u ξ and ℓ ξ = exp ℓ ( ξ | ∂Z \ ∆). For the last equation we use thefact that L is totally geodesic for g fix and ξ | ∂Z \ ∆ can be lifted to a vectorfield on T L using ℓ and ι . For small enough ρ , Φ u,ρ is injective. Definition 3.3.
Let B ,p ( Z, ∆ , α ) = [ u,ρ> Φ u,ρ ( U u,ρ ) , where the union is over all u that satisfy (5). The maps Φ u,ρ with ρ > B ,p ( Z, ∆ , α ) the structure of a smooth Banach manifold.Now consider the case where the Type II marked points ∆ are allowed tovary. LetConfig m ( Z ) = { ∆ | | ∆ | = m, ∆ is ordered as in Definition 2.2 } (7) ⊂ ( ∂Z ) m \ (fat diagonal)18enote the configuration space of Type II marked points. Definition 3.4.
For α : { , . . . , m } → R , let B ,p ( Z, α ) = a ∆ ∈ Config m ( Z ) B ,p ( Z, ∆ , α ) × { ∆ } . B ,p ( Z, α ) is given the structure of a C -Banach manifold with coordinate chartsΨ u,ρ : U u,ρ × C → B ,p ( Z, α ) , ( ξ, ∆) (Φ u,ρ ( ξ ) ◦ υ ∆ , ∆) , (8)where C is an open neighborhood of some ∆ ∈ Config m ( Z ), Φ u,ρ is the map(6) for ∆ , and υ ∆ is a diffeomorphism from Z \ ∆ → Z \ ∆ which is abiholomorphism in a neighborhood of ∆. The manifold structure is only C because of the appearance of υ ∆ (which appears because the underlying domainsof the maps are not all the same). We now define moduli spaces of (inhomogeneous) holomorphic strips and holo-morphic discs.
Definition 3.5.
Fix admissible Floer data H , J as in Definition 1.3. Given γ ± ∈ Γ H and α : { , . . . , m } → R , where if m = 0 we require γ − = γ + (seeRemark 3.6), let f M ( γ − , γ + ; α ) = f M ( γ − , γ + ; α ; H , J )denote the set of all u = ( u, ∆ , α, ℓ ) ∈ B ( Z, α ) such that • u restricted to Z \ ∆ is smooth, • lim s →±∞ u ( s, t ) = γ ± ( t ) uniformly in t , • the energy of u is finite: E ( u ) = Z Z \ ∆ (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂s (cid:12)(cid:12)(cid:12)(cid:12) dsdt < ∞ , • and u restricted to Z \ ∆ satisfies Floer’s equation ∂u∂s + J t ( u ) (cid:18) ∂u∂t − X H t ( u ) (cid:19) = 0 . (9)Let M ( γ − , γ + ; α ) = M ( γ − , γ + ; α ; H , J ) = f M ( γ − , γ + ; α ) / R , where R acts on f M ( γ − , γ + ; α ) by translation. Note that besides translating u and ℓ , R also translates ∆ in the obvious way. If ∆ = ∅ (so α = ∅ : ∅ → R )define f M ( γ − , γ + ) = f M ( γ − , γ + ; ∅ ) , M ( γ − , γ + ) = M ( γ − , γ + ; ∅ ) . M ( γ − , γ + ; α ) is called a moduli space of strips .19 emark 3.6. If γ − = γ + =: γ then the constant strip u ( s, t ) = γ ( t ) is asolution of Floer’s equation. If ∆ = ∅ then this curve is the only solution,as can be seen by equations (12) and (13). In this case it is unstable andshould not be considered, so we define M ( γ, γ ) = ∅ . If ∆ = ∅ then in principleconstant strips are allowed; however, under the assumption that H is admissible, M ( γ, γ ; α ) necessarily cannot contain constant strips because R is disjoint fromthe endpoints of all Hamiltonian chords. This will be important when we studytransversality in Section 5. Proposition 3.7.
If the Floer data H , J is admissible, then f M ( γ − , γ + ; α ) ⊂ B ,p ( Z, α ) . Proof.
This proposition is mostly a standard fact. Inhomogeneous holomorphiccurves are smooth away from the marked points and hence have regularity W ,ploc away from the marked points. It is well-known that the strips actually have ex-ponential convergence near ±∞ . The strips also have exponential convergencenear the Type II marked points. This fact is slightly less standard because thecomplex structure and Hamiltonian are not translation invariant on the strip-like ends for the Type II points. One could pass to the graph construction (asin Chapter 8 of [16]) to make the domain dependence of J and the Hamiltonianterm disappear and then try to apply [20]; however, one would then have todeal with non-transverse Lagrangian boundary conditions (because the bound-ary condition becomes R × L in the graph). Instead, we give an alternative proofof exponential convergence in Theorem A.4, which in turn is based on W ,p esti-mates from [14]. The proposition then follows because exponential convergencenear the marked points is enough to put the strips into B ,p ( Z, α ).We also need to consider moduli spaces of discs that are holomorphic withrespect to a fixed (that is, domain independent) almost complex structure J on M . Let ∆ be a list of Type II marked points, and α : { , . . . , | ∆ | } → R a choiceof branch jumps. Since ι : L → M is exact, a non-constant holomorphic discmust necessarily have at least one branch point. Therefore we assume | ∆ | ≥ Definition 3.8.
Given J , ∆, and α as above let f M disc (∆ , α ; J ) denote the setof all u = ( u, ∆ , α, ℓ ) ∈ B disc (∆ , α ) such that • u restricted to D \ ∆ is smooth, • R D \ ∆ u ∗ ω < ∞ , and • u satisfies the homogeneous Cauchy-Riemann equation ∂u∂s + J ( u ) ∂u∂t = 0 , where s + it are holomorphic coordinates on D \ ∆. Calling this solution constant makes more sense if u is viewed as a map from R into thespace of paths in M with boundary on L . J and α , let f M disc ( α ; J ) = a ∆ f M disc (∆ , α ; J ) × { ∆ } ⊂ B disc ( α ) . Here the union is over all ∆ such that | ∆ | is constant (the constant is determinedby the domain of α ). Let M disc ( α ; J ) = f M disc ( α ; J ) / Aut( D ) , where Aut( D ) = PSL(2 , R ) acts in the obvious way. M disc ( α ; J ) is called a moduli space of discs . Remark 3.9.
For certain α , M disc ( α ; J ) consists of (equivalence classes of)marked discs u = ( u, ∆ , α, ℓ ) such that u is constant. However, ℓ can havedifferent values on different components of ∂D \ ∆. The map u is constant ifand only if R u ∗ ω = 0, which by exactness is actually a condition on α . We onlyconsider these moduli spaces if | ∆ | ≥ This section explains how to exhibit f M ( γ − , γ + ; α ) as the zero set of a section ofa Banach bundle over B ,p ( Z, α ), and the resulting Fredholm theory. Fix Floerdata H , J , chords γ ± ∈ Γ H , and α . This determines B ,p ( Z, α ) as in Section3.1.Let E ,p ( Z, α ) → B ,p ( Z, α ) be the Banach bundle with fiber E ,p ( Z, α ) ( u, ∆) = L p (Λ , ⊗ C u ∗ T M ) . Here the right hand side consists of L p -sections over Z \ ∆, and the L p -norm isdefined in a way similar to Definition 3.2. The complex structure of u ∗ T M over( s, t ) ∈ Z is J t . The bundle comes equipped with a C -section¯ ∂ H , J ,α : B ,p ( Z, α ) → E ,p ( Z, α ) (10)defined by ¯ ∂ H , J ,α ( u ) = 12 (cid:20) ∂u∂s + J t ( u ) (cid:18) ∂u∂t − X H t (cid:19)(cid:21) ⊗ ( ds − idt ) . By Proposition 3.7, f M ( γ − , γ + ; α ) = ¯ ∂ − H , J ,α (0 section ). Restricting to the charts U u ,ρ × C given by (8), ¯ ∂ H , J ,α becomes a smooth section and the linearizationat a holomorphic strip is a Fredholm operator. We summarize these importantaspects with the following proposition, which is standard. Proposition 3.10. If ( u, ∆) ∈ U u ,ρ × C and ¯ ∂ H , J ,α ( u ) = 0 , then the lineariza-tion of ¯ ∂ H , J ,α at ( u, ∆) can naturally be identified with a Fredholm operator D ¯ ∂ H , J ,α = D ( u, ∆) ¯ ∂ H , J ,α : W ,p ( u ∗ T M ) × T ∆ C → L p (Λ , ⊗ C u ∗ T M ) . he formula for the first component is ( ξ, (cid:26) [ ∇ s ξ + J t ( ∇ t ξ − ∇ ξ X H t )] − J t ∇ ξ J t (cid:20) ∂u∂s − J t (cid:18) ∂u∂t − X H t (cid:19)(cid:21)(cid:27) ⊗ C ( ds − idt ) . The index is
Ind D ¯ ∂ H , J ,α = Ind γ − − Ind γ + − X Ind α ( i ) + | ∆ | . (11) Fix admissible Floer data H , J and γ ± ∈ Γ H with γ − = γ + . The ultimate goalof this section is to describe the Gromov compactification of the moduli space M ( γ − , γ + ) of strips connecting γ − to γ + .First we need to describe the Gromov compactification of J -holomorphicdiscs with one incoming Type II marked point. To this end, let J be any(domain independent) almost complex structure (later J will be J or J where J = { J t } t ). Let α : { } → R be given. We describe the compactification of M disc ( α ; J ). Definition 4.1.
With J and α as above, the compactified moduli space of discs M disc ( α ; J ) consists of equivalence classes of pairs ( T, { [ u v ] } v ∈ V er ( T ) ) where: • T is a planar tree with a distinguished root vertex v ∈ V er ( T ). • For each vertex v ∈ V er ( T ), [ u v ] = [( u v , ∆ v , α v , ℓ v )] ∈ M disc ( α v ; J ) is astable marked disc with | ∆ v | = (cid:26) valency( v ) v = v valency( v ) + 1 v = v and ∆ − v = { z ∆1 } (i.e., on each component the first marked point is in-coming and the rest are outgoing). • The branch jump types among the discs are compatible in the sense thatif two vertices v and v are connected by an edge, then the correspond-ing branch jump types agree. (More precisely: The fact that T has adistinguished root vertex implies that the edges can be canonically ori-ented in the outward direction from the root. If we label the incomingedge at a vertex as the first edge, then because T is planar each otheredge gets a unique number by proceeding in counterclockwise order. Ifan outgoing edge at v is the i th edge and it connects to v , we requirethat α v ( i ) = α v (1).) In particular, this means that the domain discscan be glued together along the marked points and the resulting map iscontinuous on the glued domain. This means that if u v is constant then | ∆ v | ≥
3. See also Remark 3.9. T, { [ u v ] } v ∈ V er ( T ) ) and ( T ′ , { [ u ′ v ] } v ∈ V er ( T ′ ) ) are equivalent if thereexists an isomorphism φ : T → T ′ of rooted planar trees such that [ u v ] = [ u ′ φ ( v ) ]for all vertices v of T . We denote elements of M disc ( α ; J ) using bold letters,for example u = [( T, { [ u v ] } v ∈ V er ( T ) )]. Proposition 4.2. M disc ( α ; J ) as described above is the Gromov compactifica-tion of M disc ( α ; J ) .Proof. This is Gromov’s compactness theorem for discs u on immersed La-grangians with boundary lifts ℓ ; see for example [3], [14]. Note that due toexactness of L , an a priori bound for the symplectic area is given by Z u ∗ ω = − f L ( p ) + f L ( q ) , where ( p, q ) = α (1); hence the space M disc ( α ; J ) is compact with respect to theusual Gromov topology. See also the proof of Proposition 4.4 for some remarkson the bubbling off analysis.Now we move on to strips. Briefly, the compactification of the moduli spaceof strips consists of broken strips with trees of disc bubbles (in the sense ofDefinition 4.1) attached to Type II marked boundary points. Here are thedetails. Definition 4.3.
Fix admissible Floer data H , J , and γ ± ∈ Γ H with γ − = γ + .The compactified moduli space of strips M ( γ − , γ + ) = M ( γ − , γ + ; H , J ) consistsof all tuples ([ u ] , . . . , [ u N ] , v , . . . , v , v , . . . , v )such that • each u i = ( u i , ∆ i , α i , ℓ i ) is a marked strip connecting γ i, − to γ i, + , • each strip is stable in the sense that if γ i, − = γ i, + then ∆ i = ∅ , • the total number of Type II marked points on the bottom of the strips is k , and the total number on the top is k , • γ − = γ , − , γ i, + = γ i +1 , − , and γ N, + = γ + , • v is an element of M disc ( α i ; J ) where α i : { } → R is the branchjump type of the i th Type II marked point occurring along the bottomof the broken strips (starting from the left side and moving right); themarked point is considered incoming for v (and outgoing for the strip,the element of R is the same for both), • v is an element of M disc ( α i ; J ) where α i : { } → R is the branch jumptype of the i th Type II marked point occurring along the top of the brokenstrips (starting from the right side and moving left); the marked point isconsidered incoming for v . 23e denote elements of M ( γ − , γ + ) using bold letters, for example u = ([ u ] , . . . , [ u N ] , v , . . . , v , v , . . . , v ) . Proposition 4.4. M ( γ − , γ + ) as described above is the Gromov compactifica-tion of M ( γ − , γ + ) .Proof. By definition (see Section (8g) of [21] for details), the energy of a holo-morphic strip u is E ( u ) = Z (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂s (cid:12)(cid:12)(cid:12)(cid:12) ds ∧ dt. (12)By exactness and Stokes’ theorem, the energy of each strip in M ( γ − , γ + ) is equalto A ( γ − ) − A ( γ + ), where A is the action of a chord, defined by the formula A ( γ ) = − Z ( γ ∗ σ + H t ( γ ( t )) dt ) − f L ( γ (0)) + f L ( γ (1)) . (13)Thus there is an energy bound and Gromov compactness can be applied in theusual way, for example as explained in [17], with a few modifications needed forthe immersed case.The key difference in the immersed case is that when the rescaling procedureis applied to find a disc bubble, the resulting bubble is only smooth on the discminus a point (equivalently, on the upper half-plane with the missing point beingthe point at infinity). The same consideration applies to the strip component:a priori, after the bubbling off, the remaining strip is only smooth away fromthe points where the bubbles appeared. If the Lagrangian is embedded, theremovable singularities theorem implies that the maps extend smoothly overthe missing points. The removable singularities theorem does not apply inthe immersed case; instead, we need to appeal to Appendix A to concludethat the curves extend continuously over the missing points, and moreover theconvergence is exponential in strip-like coordinates. See also the discussion inthe proof of Proposition 3.7. Exactness implies that there can be no disc bubbleswithout branch jumps, and no sphere bubbles.Note that having the boundary lifts ℓ for the curves is necessary to be ableto apply the results of Appendix A. Otherwise we would have to deal with thepossibility of curves switching branches arbitrarily many times (and possiblyinfinitely many times in the limit). See [7] for a discussion of compactness (orlack thereof) in this case. The goal of this section is to prove that for generic Floer data H , J , the modulispaces f M ( γ − , γ + ; α ; H , J ) are regular for all α and all γ ± . Regularity meansthat the linearized operators D ( u, ∆) ¯ ∂ H , J ,α : W ,p ( u ∗ T M ) × T ∆ C → L p (Λ , ⊗ u ∗ T M )24f Proposition 3.10 are surjective for all holomorphic strips in ( u, ∆) in the mod-uli spaces f M ( γ − , γ + ; α ; H , J ). Given an admissible H (that is, H satisfies theconditions following Definition 1.3), a time-dependent almost complex structure J = { J t } t is said to be regular if for all branch jump types α , and all chords γ ± ∈ Γ H , all the moduli spaces f M ( γ − , γ + ; α ; H , J ) are regular.The existence of regular almost complex structures follows from the usualmethods. We give an outline here to highlight that the immersed case does notpresent any difficulties. Also, we remark that achieving regularity for all α isnot problematic. First, the following “somewhere injectivity-like” result followsfrom the proof of Theorem 4.3 in [11]. (The theorem in [11] is stated for curves u : R → M which satisfy Floer’s equation and are 1-periodic in a certain sensein the second variable. However, it is easy to check that the proof holds forany inhomogeneous holomorphic curve u : R × (0 , → M which converges tochords at ±∞ . Lagrangian boundary conditions are not even needed.) Proposition 5.1.
Let u ∈ f M ( γ − , γ + ; α ) and assume u is nonconstant; that is u ( s, t ) = γ − ( t ) for some ( s, t ) . Let Reg( u ) be the set of ( s, t ) ∈ R × (0 , suchthat • ∂u∂s ( s, t ) = 0 , • u ( s, t ) = γ ± ( t ) , and • u ( s, t ) / ∈ u (( R \ { s } ) × { t } ) .Then Reg( u ) is an open dense subset of R × (0 , . Note that by Remark 3.6 any u ∈ f M ( γ − , γ + ; α ) is nonconstant as long as H is admissible.The next step is to consider the universal moduli spaces. Namely, let J be some Banach space completion of the set of smooth time-dependent ω -compatible almost complex structures. Let f M ( γ − , γ + ; α ; H , J ) = a J ∈J f M ( γ − , γ + ; α ; H , J ) × { J } . As in Section 3.3, there exists a functional analytic framework in which thismoduli space can be exhibited as the zero set of a C -section¯ ∂ H , J ,α : B ,p ( Z, α ) × J → E ,p ( Z, α ; J ) . The analogue of Proposition 3.10 is the following: B ,p ( Z, α ) × J admits Ba-nach manifold charts of the form U u,ρ × C × J , and there are correspondingtrivializations of E ,p ( Z, α ; J ) over these charts, and ¯ ∂ H , J ,α is smooth whenrestricted to these charts.Now cover B ,p ( Z, α ) × J by countably many of the charts U := U u,ρ ×C × J from above. Proposition 5.1 and standard techniques imply that thelinearization of the universal section ¯ ∂ H , J ,α is surjective at every element of25 M ( γ − , γ + ; α ; H , J ) ∩ U . Now let J reg ( γ − , γ + ; α ) be the intersection (over all U ) of the sets of regular values of the projections f M ( γ − , γ + ; α ; H , J ) ∩ U → J . By the Sard-Smale theorem and the fact that we only use countably many U ’s, J reg ( γ − , γ + ; α ) is a Baire set. Let J reg0 be the intersection of all J reg ( γ − , γ + ; α )over all γ ± and all α . Using an argument of Taubes, there exists a subset J reg ⊂ J reg0 such that J reg consists of smooth almost complex structures andis a Baire set in the C ∞ topology (see the proof of Proposition 3.1.5 in [16]).Then we have: Proposition 5.2.
Let H be admissible. Then J reg is a Baire set consisting ofsmooth time-dependent regular almost complex structures. In particular, for J ∈J reg , each moduli space M ( γ − , γ + ; α ; H , J ) is a smooth manifold of dimension Ind γ − − Ind γ + − m X i =1 Ind α ( i ) + m − where m is the number of Type II marked points. Our treatment of Floer cohomology follows the standard lines. The Floer dif-ferential is defined by couting rigid strips which satisfy Floer’s equation, just asin the embedded case. However, because our Lagrangians are only immersed,extra details are needed to show that disc bubbling does not cause problems.In general, disc bubbles attached to Type II points could appear in the Gromovcompactification of the moduli space of strips, see Definition 4.3. We use thepositivity condition to show that generically this does not happen for the stripswe are interested in.Recall that for now we are assuming that ι has transverse double points. Let H , J be admissible Floer data with J ∈ J reg . The Floer cochain complex isCF( ι ; H , J ) = CF( ι ) = M k CF k ( ι ) , CF k ( ι ) = M γ : Ind γ = k Z · γ. (14)The Floer complex is Z -graded by Definition 2.4. The Floer differential is δ : CF( ι ) → CF( ι ) , γ + X γ − : Ind γ − =Ind γ + +1 |M ( γ − , γ + ) | · γ − . (15)The Floer cohomology is HF( ι ) = H(CF( ι ) , δ ) . (16)In the next subsection we will show that δ is well-defined and δ = 0. Inthe subsequent subsection we will show that the Floer cohomology HF( ι ) isindependent of the choice of ( H , J ). 26or simplicity, discussion is limited to the case of a single Lagrangian ι .HF( ι ) should be interpreted as the self-Floer cohomology of ι and could also bedenoted as HF( ι, ι ). We could also consider two Lagrangian immersions ι and ι ′ and define HF( ι, ι ′ ) using the same techniques.More generally, the immersions can be viewed as objects in the exact Fukayacategory; this will be discussed in Section 7.5. With this point of view, we willshow that ι and ι ′ are quasi-isomorphic objects in the Fukaya category if ι ′ =Φ ◦ ι for some Hamiltonian diffeomorphism Φ. In fact, we will prove a strongerstatement, that if ι ′ is an exact deformation of ι that meets certain requirements,then ι and ι ′ are quasi-isomorphic objects. For embedded Lagrangians, thenotion of exact deformation and Hamiltonian deformation agree; for immersedLagrangians, Hamiltonian deformation is strictly stronger. ( ι ; H , J ) is a chain complex The positivity Condition 1.2, the compactness result (Proposition 4.4) and theregularity result (Proposition 5.2) are the main tools that will be used in thissection,First, we show δ is well-defined. Lemma 6.1. If γ ± ∈ Γ H with Ind γ − = Ind γ + +1 then M ( γ − , γ + ) is a compact0-dimensional manifold. Hence δ is well-defined.Proof. Let [ u ′ n ] ∈ M ( γ − , γ + ) be a sequence of strips. We need to show thatit has a convergent subsequence. By Proposition 4.4, it has a subsequenceconverging to a broken strip with disc bubbles, call it u = ([ u ] , . . . , [ u N ] , v , . . . , v , v , . . . , v ) ∈ M ( γ − , γ + ) . Here, u i = ( u i , ∆ i , α i , ℓ i ) ∈ f M ( γ i, − , γ i, + ; α i )are strips with γ − = γ , − , γ i, + = γ i +1 , − , and γ N, + = γ + ; and v are trees of J -holomorphic discs attached to the Type II marked points of the strips onthe bottom boundary; and v are trees of J -holomorphic discs attached to theType II marked points of the strips on the the top boundary.The total symplectic area of a holomorphic disc v ij is positive because ithas one non-nodal Type II marked point and hence must be nonconstant. ThusCondition 1.2 implies the branch jump at which v ij attaches, call it ( p ij , q ij ), mustsatisfy Ind( p ij , q ij ) ≥ . (17)By Proposition 5.2, each [ u i ] belongs to a smooth moduli space and hence ≤ dim f M ( γ i, − , γ i, + ; α i ) = Ind γ i, − − Ind γ i, + − | ∆ i | X j =1 Ind α i ( j ) + | ∆ i | . (18) Note that constant strips are prohibited (see Remark 3.6) and hence R translation is afree action on the parametrized moduli spaces, so these moduli spaces have dimension at leastone. α i ( j ) ≥ i, j . Thus1 = Ind γ − − Ind γ + = X i (Ind γ i, − − Ind γ i, + ) ≥ N + X i, ≤ j ≤| ∆ i | Ind α i ( j ) − X i | ∆ i | ≥ N + X i | ∆ i | . (19)This implies N = 1 and ∆ = ∅ .Thus u = ([ u ]); that is, [ u ′ n ] has a subsequence converging to [ u ]. Lemma 6.2. δ = 0 .Proof. The standard proof of this involves two parts, gluing and compactness.We prove the compactness part here; the gluing works in the same way as inthe embedded case. The relevant compactness result we need to prove is thefollowing: If Ind γ − = Ind γ + + 2, then M ( γ − , γ + ) = M ( γ − , γ + ) a a Ind γ =Ind γ − − M ( γ − , γ ) × M ( γ, γ + ) . (20)In other words, the moduli space can be compactified by adding in brokentrajectories consisting of two strips and no disc bubbles.To prove this, arguing as in the proof of Lemma 6.1, let [ u ′ n ] ∈ M ( γ − , γ + )be a sequence. It has a subsequence that converges to some u ∈ M ( γ − , γ + ) asbefore. The inequality (19) in this case becomes2 ≥ N + X i | ∆ i | . Since N ≥
1, the only possibilities are N = 1 or 2 and all ∆ i = ∅ . By regularity,if N = 2 then necessarily Ind γ , + = Ind γ − −
1. This proves the decomposition(20) and hence the lemma.
HF( ι ) In this subsection we show that the Floer cohomology does not depend on thechoice of generic Floer data H , J . We use the standard continuation methodto prove this. Again, the difference from the embedded case is we need to showthat disc bubbles, which might appear in the Gromov compactification attachedto a Type II point, do not cause a problem. The positivity Condition 1.2 againplays a key role.First we introduce the notion of continuation data. For the rest of thissection, let H , J and H , J be two sets of admissible Floer data with J , J both regular. Smooth families { H s } s = { H s,t } s,t , { J s } s = { J s,t } s,t , ≤ s, t ≤ H s = H , J s = J for s near 0 and H s = H , J s = J for s near 1, and H s,t = 0 and J s,t = J M near the boundary of M .We can view H s , J s as being defined for all s ∈ R .Our goal is to prove the following proposition, which says the the Floercohomology does not depend on the choice of Floer data. Proposition 6.3.
A generic choice of continuation data induces a chain map
Φ : CF( ι ; H , J ) → CF( ι ; H , J ) (22) which induces an isomorphism on cohomology. Before defining Φ we first introduce some moduli spaces. Given γ ∈ Γ H and γ ∈ Γ H , and branch jump types α , let M ( γ , γ ; α ; { H s } s , { J s } s ) (23)denote the set of all pairs ( u, ∆) where u = ( u, ∆ , α, ℓ ) satisfies ∂u∂s + J s,t ( u ) (cid:18) ∂u∂t − X H s,t (cid:19) = 0 , (24) Z (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂s (cid:12)(cid:12)(cid:12)(cid:12) < ∞ , lim s →−∞ u ( s, t ) = γ ( t ) , lim s →∞ u ( s, t ) = γ ( t ) (uniformly in t ) . Analogously to Section 3.3, this moduli space can be set up in a standardfunctional analytic framework. The relevant result is that the moduli space isthe zero set of an operator ¯ ∂ { H s } s , { J s } s ,α with Fredholm indexInd ¯ ∂ { H s } s , { J s } s ,α = Ind γ − Ind γ − X i Ind α ( i ) + | ∆ | , (25)and the linearization of ¯ ∂ { H s } s , { J s } s ,α at a solution has the same form as inProposition 3.10.The map Φ is defined by counting rigid elements of M ( γ , γ ; { H s } s , { J s } s )(i.e., (23) with α = ∅ so there are no Type II points), see equation (26). Forthis to make sense, the moduli space needs to be regular, which we discussnow. First, we remark that constant strips pose a problem for regularity. Theyare easy to avoid though—they can only exist if X H s,t is independent of s onsome subset of M containing a chord. We call { H s } s = { H s,t } s,t regular ifit does not admit constant strips; generic { H s } s are regular. Since { J s } s = { J s,t } s,t is essentially a domain dependent almost complex structure, standardmethods imply that there exists a Baire set of regular almost complex structures.Then, for regular { H s } s , { J s } s , we have: for each γ , γ , α , the moduli space M ( γ , γ ; α ; { H s } s , { J s } s ) is a smooth manifold with dimension given by (25).Note that the moduli space is not invariant under R translation.29ow defineΦ = Φ { H s } s , { J s } s : CF( ι ; H , J ) → CF( ι ; H , J )by the formulaΦ( γ ) = X Ind γ =Ind γ |M ( γ , γ ; { H s } s , { J s } s ) | · γ . (26)We show that Φ is well-defined; that is, M ( γ , γ ; { H s } s , { J s } s ) is compactwhen Ind γ = Ind γ . To this end, consider a sequence of elements. The energyof a strip u is E ( u ) = Z (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂s (cid:12)(cid:12)(cid:12)(cid:12) ds ∧ dt. (27)It is straightforward to show that this is bounded by a constant (depending on { H s } s ) plus A ( γ ) − A ( γ ), where A is defined as in (13). Hence there exists asubsequence which converges to an element of M ( γ , γ ; { H s } s , { J s } s ). Here,the compactification is similar to that described in Definition 4.3. Precisely,elements of the compactification consist of up to three different types of brokenstrips, with disc bubbles, which connect together: • Possibly an element u ∈ M ( γ , γ ′ ; H , J ). • An element u ∈ M ( γ ′ , γ ′ ; α ; { H s } s , { J s } s ) where α : { , . . . , m } → R for some m , along with stable discs v ij ∈ M disc ( J s j ,i , α ij ) that attach tothe strip at branch jumps. Here i is 0 or 1 depending whether the discattaches to the bottom or top boundary of the strip, s j is the real part ofthe corresponding Type II marked point, and α ij : { } → R is the typeof the branch jump at which the disc attaches. • Possibly an element u ∈ M ( γ ′ , γ ; H , J ).If the limit has a u component, then by regularity of H , J and an argumentsimilar to the proof of Lemma 6.1 we haveInd γ − Ind γ ′ ≥ . Likewise if u is a component thenInd γ ′ − Ind γ ≥ . By the regularity of { H s } s , { J s } s ,0 ≤ Ind γ ′ − Ind γ ′ − m X j =1 Ind α ( j ) + m, hence Ind γ ′ − Ind γ ′ ≥ X Ind α ( j ) − m ≥ m. γ = Ind γ , we must have m = 0 and there is not a u or u com-ponent. That is, the limit is an element of M ( γ , γ ; { H s } s , { J s } s ) and com-pactness is proved.To show that Φ is a chain map, consider the compactification of the modulispaces M ( γ , γ ; { H s } s , { J s } s )when Ind γ = Ind γ + 1. Arguing as above, we see that for the limit of asequence, the only possibilities are that m = 0, and there is either no u , u components or just one such component. Furthermore, from the proof of Lemma6.1, if one of these components exists then it must not have any Type II markedpoints. In other words, M ( γ , γ ; { H s } s , { J s } s ) can be compactified by addingin a Ind γ ′ =Ind γ − M ( γ , γ ′ ; H , J ) × M ( γ ′ , γ ; { H s } s , { J s } s )and a Ind γ ′ =Ind γ +1 M ( γ , γ ′ ; { H s } s , { J s } s ) × M ( γ ′ , γ ; H , J ) . Along with a standard gluing result, this proves that Φ is a chain map.Next, we want to show that Φ induces an isomorphism on cohomology.To do this we first need to consider homotopies of continuation data. Let { H ,s } s , { J ,s } s and { H ,s } s , { J ,s } s be two sets of continuation data from H , J to H , J . A homotopy between them consists of families { H r,s } r,s = { H r,s,t } r,s,t , { J r,s } r,s = { J r,s,t } r,s,t such that H r, = H , H r, = H and J r, = J , J r, = J for 0 ≤ r ≤
1, and H r,s,t = 0 and J r,s,t = J M near the boundary of M . In other words, for eachfixed r , { H r,s } s , { J r,s } s is a continuation from H , J to H , J .The homotopy { H r,s } r,s , { J r,s } r,s induces a chain homotopyΨ : CF( ι ; H , J ) → CF( ι ; H , J )between Φ { H ,s } s , { J ,s } s and Φ { H ,s } s , { J ,s } s , defined as follows. Consider theparametrized moduli spaces M ( γ , γ ; α ; { H r,s } r,s , { J r,s } r,s ) = a 31e claim that Ψ is well-defined and gives a chain homotopy between thetwo chain maps induced by the different sets of continuation data. Let us justsketch how to prove that Ψ is well-defined; the fact that Ψ is a chain homotopycan be proved in a similar way by identifying the boundary components ofthe moduli space of dimension 1. To prove that Ψ is well-defined we need toshow that M ( γ , γ ; { H r,s } r,s , { J r,s } r,s ) is compact when Ind γ = Ind γ − u n , r n ) be a sequence of elements; we need to show it has a convergentsubsequence. The energy is bounded for the same reason that (27) is bounded.By regularity of continuation data { H r,s } s , { J r,s } s for r = 0 , 1, the sequence r n must be bounded away from 0 and 1. Thus we may assume it converges to r with 0 < r < 1. We may then assume that u n converges to an element of M ( γ , γ ; { H r,s } s , { J r,s } s ). As before, elements of this compactified modulispace consists of broken strips plus disc bubbles. Thus the limit of u n consistsof • Possibly a component u ∈ M ( γ , γ ′ ; H , J ). • A central component u ∈ M ( γ ′ , γ ′ ; α ; { H r,s } s , { J r,s } s ) plus disc bubbles v ij attached to all the branch jumps determined by α : { , . . . , m } → R . • Possibly a component u ∈ M ( γ ′ , γ ; H , J ).As before, if u is present then Ind γ − Ind γ ′ ≥ u is present thenInd γ ′ − γ ≥ 1. By regularity of the parametrized moduli space we also have0 ≤ Ind γ ′ − Ind γ ′ − m X i =1 Ind α ( i ) + m + 1 , and thus Ind γ ′ − Ind γ ′ ≥ − m. Since we assumed that Ind γ − Ind γ = − 1, the only possibility is that m = 0and u , u are not present. Thus the limit curve is an element of the modulispace and compactness is proved.Finally to complete the proof that the chain homotopy Φ induced by con-tinuation data induces an isomorphism on cohomology, we appeal to Floer’soriginal argument. This argument goes as follows. Suppose given three sets ofFloer data H , J H , J H , J and two sets of continuation data { H s } ≤ s ≤ , { J s } ≤ s ≤ { H ′ s } ≤ s ≤ , { J ′ s } ≤ s ≤ . For large r define the concatenated continuation data { H H ′ r,s } s , { J J ′ r,s } s by H H ′ r,s = H s + r s ≤ − r + 1 H − r + 1 ≤ s ≤ r + 1 H ′ s − r s ≥ r + 1 , J J ′ r,s is defined in a similar way.The concatenated continuations are clearly homotopic for different valuesof r , hence they all induce homotopic chain maps. For large r , a gluing argu-ment shows that the chain map induced by the concatenation is equal to thecomposition of the chain maps induced by the continuations { H s } s , { J s } s and { H ′ s } s , { J ′ s } s .Now given a continuation { H s } s , { J s } s let { H ′ s } s , { J ′ s } s be the contin-uation run backwards. Composition gives a chain mapCF( ι ; H , J ) → CF( ι ; H , J ) . From above, this composition is chain homotopic to the map induced by the con-catenation. The concatenation is clearly homotopic to a small perturbation ofthe identity continuation, which induces the identity map on cohomology. Thusthe maps Φ induced by continuations must induce isomorphisms on cohomology.This completes the proof of Proposition 6.3. In this section we make some simple remarks on the Euler characteristic ofHF( ι ). Lemma 6.4. Assume L is orientable and let n = dim L . If n is odd then χ (HF( ι )) = 0 , and if n is even then χ (HF( ι )) = χ ( L ) + ( − n/ I ( ι ) , where I ( ι ) is the algebraic number of self-intersection points of ι .Proof. First observe that choosing a Hamiltonian as in Lemma 2.9 shows that χ (HF( ι )) = χ ( L ) + P ( p,q ) ∈ R ( − Ind( p,q ) . Since Ind( p, q ) + Ind( q, p ) = n , if n isodd then χ (HF( ι )) = χ ( L ) + 0 = χ ( L ) = 0.Assume now that n is even and let x ∈ R with ι − ( x ) = { p, q } . Definesign( x ) = ± ι ∗ T p L ⊕ ι ∗ T q L = ± T x M as oriented vectorspaces. Note that ι ∗ T p L ⊕ ι ∗ T q L = ι ∗ T q L ⊕ ι ∗ T p L as oriented vector spaces, andchanging the orientation of L does not change the orientation of this direct sum.Thus sign( x ) is well-defined as long as L is orientable, and does not depend onthe actual orientation. By definition I ( ι ) = P x ∈ R sign( x ). From Definition2.5 it follows that sign( x ) = ( − Ind( p,q )+ n ( n − / = ( − Ind( p,q )+ n/ (since n iseven). Thus P ( p,q ) ( − Ind( p,q ) = ( − n/ P x sign( x ) = ( − n/ I ( ι ).Generally speaking, whenever ι : L → C n is a regular immersion of an ori-ented n -manifold, it is shown in [15] that 2 I ( ι ) = − ( − n ( n − / χ ( ν ), where χ ( ν ) is the Euler characteristic of the oriented normal bundle ν of the immer-sion. If ι is a Lagrangian immersion then ν = ( − n ( n − / T L as an orientedvector bundle. It is interesting to note that this puts topological restrictions on The equation proved in [15] has a different sign due to different orientation conventions. C as a Lagrangian. Moregenerally, we have Lemma 6.5. Let ι : L → M be an orientable Lagrangian immersion (not neces-sarily graded or exact) and let n = dim L be even. Then χ ( L ) + ( − n/ I ( ι ) =[ L ] · [ L ] . In particular, if H n ( M, Q ) = 0 then this number is .Proof. Perturbing L slightly by a Hamiltonian of the type in Lemma 2.9 andarguing as in the proof of the previous lemma shows that the self-intersectionnumber (as a homology class) of [ L ] = ι ∗ [ L ] is [ L ] · [ L ] = χ ( L ) + ( − n/ I ( ι ).Since [ L ] · [ L ] depends only the homology class [ L ] ∈ H n ( M, Q ), if H n ( M, Q ) = 0then [ L ] · [ L ] = 0. A ∞ -structure In this section we describe the A ∞ structure on CF( ι ). The A ∞ structureconsists of maps m k : CF( ι ) ⊗ k → CF( ι ) for k ≥ A ∞ relations.As in [21], the maps are defined by counting inhomogeneous holomorphic curves.In our terminology this means that curves contain only Type I points, which arerequired to converge to Hamiltonian chords. We call these curves holomorphicpolygons. To prove that the maps m k are well-defined and satisfy the A ∞ relations we need to introduce some moduli spaces which are not needed inthe embedded Lagrangian case. These auxilliary moduli spaces, which we stillcall holomorphic polygons, are allowed to contain Type I and Type II points.However, regularity and the positivity condition allow us to prove the followingkey statement: a zero- or one-dimensional moduli space of holomorphic polygonswithout Type II marked points can be compactified without introducing anyType II marked points. This is analogous to the case of strips (2-gons) used inSection 6 to define the Floer cohomology. Let R d +1 be the moduli space of discs with d +1 Type I marked boundary points, R d +1 its Deligne-Mumford-Stasheff compactification obtained by adding nodaldiscs, S d +1 → R d +1 the universal bundle, and S d +1 → R d +1 the correspondinguniversal bundle. Readers are referred to Section 9 of [21] for details. The fiberover r ∈ R d +1 is a boundary punctured disc S d +1 r = D \ Σ, where Σ is a list of d +1 marked points representing the class r ∈ R d +1 . We view the marked pointsΣ = ( z , . . . , z d ) as Type I points, with Σ − = { z } and Σ + = { z , . . . , z d } . Wealso assume that the marked points are cyclically counterclockwise oriented.Fix a consistent universal choice of strip-like ends (see Lemma 9.3 in [21])and fix Floer data H = { H t } t , J = { J t } t . A choice of perturbation dataconsists of, for each d , 34 a fiber-wise one-form K d +1 ∈ Ω S d +1 / R d +1 ( C ∞ c ( M )) such that for any r ∈ R d +1 and V a vector tangent to the boundary of S d +1 r , the function K d +1 ( V ) ∈ C ∞ c ( M ) vanishes on L , • and a domain-dependent almost complex structure J d +1 = { J d +1 z } z ∈S d +1 .On strip like ends, it is also required that K d +1 = H t dt and J d +1 z = J t where z = s + it is the strip-like coordinate. We always assume the choice of perturbationdata is consistent in the sense of Lemma 9.5 in [21]. Consistent perturbationdata induces perturbation data over S d +1 , which we will continue to denote as K d +1 , J d +1 .Now we incorporate Type II marked points. All Type II marked points areoutgoing. Let R d +1 ,m be the set R d +1 ,m = { ( r, ∆) ∈ R d +1 × Config m ( D ) | ∆ is valid Type II points for S d +1 r } . By valid Type II points we mean that ∆ is disjoint from the set of Type Ipoints, and also they are labelled in counterclockwise order with the first pointbeing the one that appears first after the zeroeth Type I marked point. Thereis a corresponding universal bundle S d +1 ,m → R d +1 ,m . We do not considercompactified versions (indeed, as will be seen later, these moduli spaces playan auxiliary role). Perturbation data K d +1 , J d +1 induces perturbation data forthe moduli spaces with Type II points simply by taking it to be constant inthe Config m ( D ) direction. We denote this perturbation data using the samesymbols, so K d +1 ∈ Ω S d +1 ,m / R d +1 ,m ( C ∞ c ( M )) , K d +1 |S d +1 ,mr × { ∆ } := K d +1 |S d +1 r , and J d +1 = { J d +1( z, ∆) } ( z, ∆) ∈S d +1 ,m , J d +1( z, ∆) := J d +1 z . For each d, m we can cover R d +1 ,m by countably many open sets such thatall the fibers over the points in each open set can be given strip-like ends for theType II points in a smooth way. The end result is that all the uncompactifiedmoduli spaces of curves can be covered by countably many open sets, and overeach open set the universal bundle can be trivialized in such a way that thestrip-like ends for Type I and Type II points are trivialized in the sense ofequation (9.1) in [21]. The moduli spaces containing only Type I marked pointsagree with the corresponding moduli spaces in [21]. In this section we set up the space of maps in which the holomorphic polygonsused to define the A ∞ operations will lie in. The maps we will study willgeneralize the notion of a marked strip (see Section 2.1) by allowing more TypeI points. To distinguish from a marked disc (see Section 2.2), which has noType I points, we will call this generalization a marked polygon .35irst we discuss the domain of a marked polygon. Let Σ = ( z , . . . , z d ) ⊂ D be a list of Type I points, cyclically counterclockwise oriented, d ≥ 2. LetΣ − = { z } , Σ + = { z , . . . , z d } . Fix some choice of strip like ends for thesemarked points. Definition 7.1. Fix a Hamiltonian H . A C -marked polygon u = ( u, ∆ , α, ℓ )with domain D \ Σ connecting γ , . . . , γ d ∈ Γ H consists of the following data. • A continuous map u : ( D \ Σ , ∂ ( D \ Σ)) → ( M, L ) such thatlim s j →±∞ u ( s j , t j ) = γ j ( t j ) , uniformly in t j . Here s j + it j are strip-like coordinates on the strip-like ends for the TypeI marked points z j , and the limit condition ±∞ is chosen depending onwhether z j is incoming (i.e., j = 0) or outgoing (i.e., 1 ≤ j ≤ d ). • ∆ = ( z ∆1 , . . . , z ∆ m ) ⊂ ∂ ( D \ Σ) is a list of Type II marked points, alloutgoing. Also, the points are ordered counterclockwise in such a waythat z ∆1 is the first point of ∆ that appears after z in the counterclockwisedirection. • A map α : { , . . . , m } → R . • A continuous map ℓ : ∂ ( D \ Σ ∐ ∆) → L such that ι ◦ ℓ = u | ∂ ( D \ Σ ∐ ∆) . Moreover, ℓ has a branch jump at each z ∆ i ∈ ∆ of type α ( i ).The Maslov index of a marked polygon is defined as in Section 2.3. Theanalogue of Lemma 2.7 is: Lemma 7.2. The Maslov index of the marked polygon u = ( u, ∆ , α, ℓ ) connect-ing γ , . . . , γ d is µ ( u ) = Ind γ − d X i =1 Ind γ i − m X i =1 α ( i ) . Let R d +1 ,m be as in Section 7.1, with d + 1 ≥ , m ≥ 0. In particular, thisimplies a choice of strip-like ends has been chosen for the Type I points Σ foreach fiber S d +1 ,m ( r, ∆) = S d +1 r =: D \ Σ of the universal bundle. Definition 7.3. Let B ( S d +1 ,m , α ) denote the set of all tuples ( u, r, ∆) such that • ( r, ∆) ∈ R d +1 ,m , and • u = ( u, ∆ , α, ℓ ) is a C -marked polygon with domain S d +1 r .36s in Section 3.1, define the subset B ,p ( S d +1 ,m , α ) ⊂ B ( S d +1 ,m , α ) consist-ing of elements of regularity W ,p with respect to a volume form which is ds ∧ dt on each strip-like end. This requires trivializing the domains of the marked poly-gons in both the Type I and Type II directions; we use the trivializations of S d +1 ,m discussed at the end of Section 7.1 to do this. The process is similar tothe case of marked strips.Given perturbation data K d +1 , J d +1 , there exists a Banach bundle and sec-tion ¯ ∂ K d +1 , J d +1 ,α : B ,p ( S d +1 ,m , α ) → E ,p ( S d +1 ,m , α ) . (28)As usual, this is a C -section of a C -Banach manifold; there are special trivi-alizations with respect to which the section becomes a Fredholm operator. Theindex is Ind γ − d X i =1 Ind γ i − m X i =1 Ind α ( i ) + d + m − . (29) Now we define the moduli spaces of holomorphic polygons. The holomorphicpolygons with only Type I points will be used to define the A ∞ structure, justas in [21]. Suppose given Floer data H , J , perturbation data K d +1 , J d +1 , andstrip-like ends for S d +1 → R d +1 . This induces perturbation data K d +1 , J d +1 on S d +1 ,m → R d +1 ,m for all m ≥ Definition 7.4. Let γ , . . . , γ d ∈ Γ H and α : { , . . . , m } → R be given. Then M ( γ , . . . , γ d ; α ) = M ( γ , . . . , γ d ; α ; K d +1 , J d +1 )denotes the set of all ( u, r, ∆) ∈ B ( S d +1 ,m , α ) such that • u |S d +1 r \ ∆ is smooth, • u connects the chords γ , . . . , γ d (in the sense of the first bullet point ofDefinition 7.1), • u satisfies the inhomogeneous Cauchy-Riemann equation( du − X K d +1 ) (0 , = 0 , • and u has finite energy: E ( u ) = 12 Z S d +1 r \ ∆ | du − X K d +1 | < ∞ . (30)Here X K d +1 denotes the 1-form on S d +1 r with values in u ∗ T M defined by X K d +1 ( z )( ξ ) = X K d +1 ( z )( ξ ) ( u ( z )) , ξ ∈ T z S d +1 r . K d +1 ( z )( ξ ) is a Hamiltonian function. In case m = 0 we simplydenote this space as M ( γ , . . . , γ d ) = M ( γ , . . . , γ d ; K d +1 , J d +1 ) . Proposition 7.5. For each d ≥ , there exists a generic set of consistent pertur-bation data K d +1 , J d +1 such that the moduli spaces M ( γ , . . . , γ d ; α ) are regularfor all γ , . . . , γ d and all α : { , . . . , m } → R . In particular, M ( γ , . . . , γ d ; α ) is a smooth manifold of dimension Ind γ − d X i =1 Ind γ i − m X i =1 Ind α ( i ) + d + m − . Proof. Theorem A.4 implies that elements of the moduli space have W ,p -regularity near Type II points. It is standard that they have similar regularitynear Type I points. Thus M ( γ , . . . , γ d ; α ) coincides with the zero-set of thesection in (28). To prove regularity, we need to show that, for each d ≥ J d +1 and K d +1 can be found so that the operators for all m ≥ K d +1 , the moduli spaces cannotcontain maps which are constant away from the strip-like ends. Away fromthe strip-like ends, J d +1 is a domain dependent complex structure. For each m , there is thus a Baire set of K d +1 , J d +1 which makes (28) transverse to thezero-section. Taking the intersection over all m of these Baire sets then resultsin a Baire set which makes all the operators transverse. The index calculationis a combination of those appearing in [21] and [3].Now we discuss Gromov compactness of M ( γ , . . . , γ d ) (Gromov compact-ness for α = ∅ is not needed). As in Section 4, the difference from the em-bedded Lagrangian case in [21] is that now disc bubbles attached to Type IIpoints can appear in the compactification. First we remark that every elementof M ( γ , . . . , γ d ) has an a priori energy bound (energy defined by (30)). In fact,more generally, an element ( u, r, ∆) ∈ M ( γ , . . . , γ d ; α ) has energy E ( u ) = A ( γ ) − X i ≥ A ( γ i ) − X j E ( α ( j )) − Z S d +1 r R K d +1 ( u ) . (31)This is a generalization of equation (8.12) in [21] and follows from Stokes’ theo-rem, the conditions imposed on the perturbation data K d +1 in Section 7.1, andthe fact that the perturbation data K d +1 ,m on moduli spaces of domains withType II points is just the pull back under the forgetful map of the perturbationdata K d +1 defined on moduli spaces of domains with no Type II points. In thelast integral, R K d +1 is a 2-form on S d +1 r with values in compactly supportedfunctions on M ; it can be interpreted as the curvature of a Hamiltonian fibra-tion, see Section (8g) of [21]. If, at the point z ∈ S d +1 r , R K d +1 ( z ) is the form Hds ∧ dt , then R K d +1 ( u )( z ) denotes the form H ( u ( z )) ds ∧ dt . In particular, thismakes sense even at Type II points, where u is continuous but not smooth. Also, R K d +1 ( u ) vanishes on the strip-like ends of S rd +1 . Thus the curvature integral38n (31) can be bounded independently of u . Since the perturbation data on themoduli space of domains is chosen to be consistent with the compactificationof the moduli space of domains (Lemma 9.5 in [21]), the integral can also bebounded independently of r ∈ R d +1 (because the bound varies continuouslywith r , and by consistency the bound can be extended continuously for r in thecompactified moduli space). Thus (31) can be bounded independently of u .This takes care of the energy bound. Gromov compactness of polygons isthen a generalization of the compactification of strips described in Definition4.3. Briefly, elements can be described as trees of holomorphic polygons withdisc bubbles attached at Type II points. The disc bubbles are the feature notpresent in the embedded Lagrangian case. The precise statement is Proposition 7.6. Elements of the compactified moduli space M ( γ , . . . , γ d ) consist of (equivalence classes of ) tuples ( T, { (( u i , r i , ∆ i ) , F i , v i1 , . . . , v im i ) } i ∈ V er ( T ) ) where • T is a planar tree with a distinguished root vertex and a distinguishedordering , , . . . of the edges connected to the root. The distinguishedordering respects the planar structure in the sense that the order proceedsin counterclockwise direction. • ( u i = ( u i , ∆ i , α i , ℓ i ) , r i , ∆ i ) ∈ M ( γ i , . . . , γ id i ; α i ) for each vertex i of T , • for each nonroot vertex i ∈ V er ( T ) , F i ⊂ { , . . . , d i } ; and for the rootvertex r ∈ V er ( T ) , F r ⊂ { , . . . , d r } with ∈ F r ; moreover | F i | +valency( i ) = d i + 1 for all vertices i ∈ V er ( T ) ( F i encodes which chordsof a component are among the original set of chords γ , . . . , γ d ), • v i1 , . . . , v im i are elements of the compactified moduli spaces of holomorphicdiscs with one incoming Type II marked point (see Definition 4.1) that at-tach to the branch jump points of u i as specified by α i : { , . . . , m i } → R ;moreover, each v ij is J z ∆ i,j holomorphic, where J z ∆ i,j is domain independentand determined by the perturbation data J d i +1 in the obvious way.Moreover, the chords are required to be compatible in the following sense: • P i ∈ V er ( T ) | F i | = d + 1 . • The fact that T has a distinguished root vertex implies that the edges canbe canonically oriented in the outward direction from the root. If we la-bel the incoming edge at a non-root vertex as edge 0, then because T isplanar each other edge gets a unique number, starting at , by proceedingin counterclockwise order. The outgoing edges from the root get a numberfrom the first bullet point above. If an outgoing edge at i ∈ V er ( T ) isedge number a ≥ , and it connects to vertex j ∈ V er ( T ) , we require that γ ib = γ j , where b is the a th number in the set { , . . . , d i } \ F i . • The external chords, namely the ones which aren’t matched up in pairs bythe edges of T , in counterclockwise order are precisely γ , . . . , γ d . Also, if r ∈ V er ( T ) is the root, then γ r = γ . n particular, this means that the domain discs can be glued together along themarked points as specified by the tree, and the resulting map is continuous onthe glued domain, and there are d + 1 remaining Type I marked boundary pointswhich converge to γ , . . . , γ d .Sketch of proof. First we briefly explain how the compactness works in the em-bedded Lagrangian case from [21], where discs attached to Type II points arenot needed. Suppose given a sequence of elements from the moduli space. Wemay take a subsequence of maps such that the underlying sequence of moduliof domains is convergent with limit r ∞ ∈ R d +1 . Each map in the sequence hasdomain which is the fiber S d +1 r n over r n of the universal bundle S d +1 → R d +1 .Since r n → r ∞ , the gluing procedure explained in Section (9e) of [21] allowsone to identify any compact subset of S d +1 r ∞ which is disjoint from the markedpoints and the nodal points with a similar subset of S d +1 r n , for n large enough.Now take a sequence of such compact subsets which exhausts S d +1 r ∞ . If, on thecompact subsets, the sequence of maps have a gradient bound (gradient withrespect to a metric on the domain which is standard in strip-like ends and thethin part of the thick-thin decomposition from Section (9e) of [21]) then thereexists a convergent subsequence. This gives a limit map on the domain S d +1 r ∞ .The limit map satisfies the correct inhomogeneous Cauchy-Riemann equation,namely the one determined by the perturbation data K d +1 |S d +1 r ∞ , because theperturbation data was chosen to be consistent with compactification (of modulispace of domains). Bubbling off analysis and exactness of the Lagrangian canbe used to show that a gradient bound must exist. The remaining thing to con-sider is accumulation of energy in the thin part of the domains. This results inthe thin part breaking into a strip (or a sequence of strips) which satisfy Floer’sequation. The strips may be internal in that they connect different polygoncomponents, or external in the sense that they connect a polygon to a chord. Ineither case, they are part of the compactification described by the proposition:they correspond to polygon components with d i = 1 in the second bullet point.Now we consider the immersed case. The proof proceeds in the same way asbefore, except that now in the bubbling off analysis it is possible for a holomor-phic disc bubble to appear. A bubble must attach to a holomorphic polygon ata branch point, i.e. a Type II point. The last thing to check is that the limitcurve satisfies the correct inhomogeneous holomorphic curve equation. This fol-lows because the perturbation data K d i +1 ,m i for each polygon component ofthe limit domain is the pull back under the forgetful map of the perturbation K d i +1 for the corresponding domain with no Type II points. (In other words:the limit curve satisfies the correct inhomogeneous holomorphic curve equationfor the same reason that the limit curve in the embedded case does.) A ∞ -structure In this subsection we define the A ∞ structure using the moduli spaces of holo-morphic polygons with only Type I points. The definition is the same as the one40iven in [21] for embedded Lagrangians. However, to show that the maps arewell-defined and satisfy the A ∞ relations, we need to show that disc bubblingdoes not cause a problem. Similar to the case of Floer cohomology considered inSection 6, the key tools needed to show this are regularity, Gromov compactness,and the positivity condition.Given regular Floer data H , J , the underlying Z -vector space for the A ∞ algebra is CF( ι ) = CF( ι ; H , J ) = M γ ∈ Γ H Z · γ. The grading is given by the index of the chords as defined in Definition 2.4. For k ≥ 1, define operations m k : CF( ι ) ⊗ k → CF( ι )by m k : γ ⊗ · · · ⊗ γ k X Ind γ =Ind γ + ··· γ k +2 − k |M ( γ , . . . , γ k ) | · γ . (32)These operations are said to satisfy the A ∞ relations if, for each k ≥ x , . . . , x k ∈ CF( ι ), X i,j m k − j +1 ( x , . . . , x i , m j ( x i +1 , . . . , x i + j ) , x i + j +1 , . . . , x k ) = 0 . Theorem 7.7. m k are well-defined and satisfy the A ∞ relations.Proof. We give a sketch of the proof following the usual lines. The proof relieson standard gluing results, the regularity Proposition 7.5, and the compactnessProposition 7.6; it can be seen as a generalization of the arguments used inSection 6. The difference from [21] is that we need to rule out disc bubbling;that is, we need to show that zero- and one-dimensional moduli spaces involvingonly Type I marked points have compactifications which still only involve TypeI marked points.First consider well-definedness: we need to show that M ( γ , . . . , γ k ) is com-pact when Ind γ = P Ind γ i + 2 − k . Consider a sequence of elements, wemay assume they converge to an element of the compactified moduli space de-scribed in Proposition 7.6. Call this element ( T, { u i , v i1 , . . . , v im i } i ∈ V er ( T ) ).Let u i = ( u i , ∆ i , α i , ℓ i ) connect the chords γ i, , . . . , γ i,k i . By regularity of u i ,for each i we haveInd γ i, − k i X j =1 Ind γ i,j − m i X j =1 α i ( j ) + k i + m i − ≥ . Combining this inequality with the positivity Condition 1.2, we getInd γ i, − k i X j =1 Ind γ i,j ≥ m i X j =1 α i ( j ) − k i − m i + 2 ≥ m i − k i + 2 . i ∈ V er ( T ) then gives2 − k = Ind γ − k X j =1 Ind γ j ≥ | V er ( T ) | X i =1 m i + 2 | V er ( T ) | − | V er ( T ) | X i =1 k i = 2 | V er ( T ) | X i =1 m i + 2 | V er ( T ) | − ( | V er ( T ) | − k )= 2 | V er ( T ) | X i =1 m i + | V er ( T ) | + 1 − k. Thus 1 ≥ | V er ( T ) | X i =1 m i + | V er ( T ) | . (33)The only possibility is that T has one vertex and all m i = 0; that is, the limitcurve actually lies in M ( γ , . . . , γ k ) and compactness is proved.To prove the A ∞ relations, consider the moduli spaces M ( γ , . . . , γ k ) whichare one dimensional; that is, with Ind γ = Ind γ + · · · Ind γ k +2 − k +1. Arguingas before, inequality (33) now becomes2 ≥ | V er ( T ) | X i =1 m i + | V er ( T ) | . Since | V er ( T ) | ≥ 1, all m i = 0 and T consists of 1 or 2 vertices. The elementsthat have trees with 2 vertices form the boundary of the moduli space; eachsuch tree corresponds to a term in an A ∞ relation. A gluing argument gives theconverse, and hence the A ∞ relations follow by the fact that the boundary of a1-dimensional manifold consists of an even number of points. A ∞ categories, and invariance In this section we explain why the A ∞ algebra of ι is independent, up to equiv-alence, of the choices used to construct it, such as H , J and the perturbationdata and strip-like ends on each S d +1 . Two A ∞ algebras are equivalent if thereis an A ∞ morphism between them that induces an isomorphism on cohomology.Such a morphism is actually a homotopy equivalence; conversely, a homotopyequivalence induces an isomorphism on cohomology.Perhaps the best way to approach this is to view the Lagrangian immersionas an object of the Fukaya category and proceed as in Section 10 of [21]. Wepresent a sketch of the argument here for the convenience of the reader. First,note that previous arguments can be used to define more general A ∞ structuremaps m k : CF( L , L ) ⊗ · · · ⊗ CF( L k − , L k ) → CF( L , L k ) , where each L i is an immersed (or embedded) exact Lagrangian satisfying As-sumption 1.2. m k is defined by counting inhomogeneous holomorphic polygons42n the usual way. To ensure that the m k ’s satisfy the A ∞ relations, the per-turbation data for the holomorphic polygons needs to be chosen in a consistentway. This includes Floer data for each pair of Lagrangians. We refer the readersto [21] for details. The end result is that we get an A ∞ category in which theimmersion ι is an object.Suppose that we have chosen Floer data and perturbation data and strip-likeends. We can think of another choice of this data, for the same ι , as a differentobject which we will call ι ′ for clarity. To show that the A ∞ algebras for ι and ι ′ are equivalent we need to find elements f ∈ CF( ι, ι ′ ) and g ∈ CF( ι ′ , ι ) such that m ( f, g ) is cohomologous to the unit in HF( ι, ι ) and m ( g, f ) is cohomologous tothe unit in HF( ι ′ , ι ′ ). Indeed, existence of such f and g implies that the inclusionof the category with only object ι (or ι ′ ) into the category with objects ι and ι ′ is an equivalence. Here, the categories we mean are the honest categoriesobtained by taking cohomology of the A ∞ categories. The equivalence of the A ∞ algebras for ι and ι ′ then follows from Theorem 2.9 in [21].The units of HF( ι, ι ) and HF( ι ′ , ι ′ ) can be constructed in the following way.Consider ι . Define an element e ι ∈ CF( ι, ι ) by e ι = X γ :Ind γ =0 |M ( γ ) | · γ. (34)Here, M ( γ ) is the space of inhomogeneous holomorphic polygons with one in-coming Type I marked point that converges to γ . In Definition 7.1 when wedefined the notion of holomorphic polygon we assumed that the number of TypeI marked points was greater than or equal to 3. In the present case there is onlyone Type I marked point; this is dealt with by assuming that the position of theType I marked point is fixed, say z = − 1, and then the rest of the definitionis the same. The inhomogeneous equation should agree with Floer’s equationfor the Floer data for ι on a strip-like end for the marked point, and shouldbe generic elsewhere. Methods very similar to Section 6.2 show that m e ι = 0,hence [ e ι ] ∈ HF( ι, ι ), and also [ e ι ] is a unit with respect to the multiplicationinduced on HF( ι, ι ) by m .Likewise, define f = e ι,ι ′ ∈ CF( ι, ι ′ ) using the same equation, except thaton the strip-like end the inhomogeneous equation should be Floer’s equation forthe Floer data for the pair ι, ι ′ . Similarly, define g = e ι ′ ,ι . The same methodsthen show that [ m ( f, g )] = [ e ι ] ∈ HF( ι, ι ) and [ m ( g, f )] = [ e ι ′ ] ∈ HF( ι ′ , ι ′ ).Thus the A ∞ algebras for ι and ι ′ are equivalent.By incorporating moving Lagrangian boundary conditions, the same tech-nique can be applied to show that ι and ι ′ = φ ◦ ι are quasi-isomorphic objects ofthe Fukaya category when φ is a Hamiltonian diffeomorphism (we may assume φ is the identity outside of a compact subset). Moving Lagrangian bound-ary conditions are explained in Section (8k) of [21], the immersed case worksthe same way as the embedded case. Here is a brief overview. As before, define f ∈ CF( ι, ι ′ ) by counting inhomogeneous holomorphic discs with one fixed TypeI marked point and that satisfy Floer’s equation near the marked point. Since ι = ι ′ , the discs satisfy a moving Lagrangian boundary condition, determined by43icking some path of Hamiltonian diffeomorphisms { φ t } t from the identity to φ ,and choosing some identification of the parameter space { t } with the boundaryof the disc outside the strip-like end. Then the Lagrangian boundary conditionat t is φ t ◦ ι . This means that the boundary lift ℓ of the disc u = ( u, ∆ , α, ℓ )should satisfy φ t ◦ ι ◦ ℓ ( t ) = u ( t ) for t ∈ ∂D \ { z } . The perturbation dataneeds to have slightly different boundary conditions in the moving Lagrangiancase, see formula (8.21) of [21]. The end result is that [ f ] ∈ HF( ι, ι ′ ) is an iso-morphism, hence ι and ι ′ are quasi-isomorphic objects in the Fukaya category.In particular, the abstract theory (Theorem 2.9 in [21]) implies that the A ∞ algebras CF( ι ) and CF( ι ′ ) are equivalent. One could also copy Section (10c) of[21] to construct concrete higher order maps that would give an A ∞ morphismbetween CF( ι ) and CF( ι ′ ). Remark 7.8. There exists a set { φ z } of Hamiltonian diffeomorphisms of M parameterized by z ∈ D such that if u : D → M is a map contributing to the f ∈ CF( ι, ι ′ ) defined above, then the map z φ z ( u ( z )) is a disc with non-movingLagrangian boundary conditions that satisfies an inhomogeneous holomorphiccurve equation. See formula (8.22) of [21] for more details. Thus the analyticdetails of the moving Lagrangian boundary condition can be reduced to the non-moving case. Later on we will consider deformations which are exact but notHamiltonian. In this case, the procedure just outlined does not work. Another(less serious) difficulty is that formula (8.21) in [21] no longer makes sense.Nevertheless, in some cases we will still be able to construct a quasi-isomorphism f . In summary: Theorem 7.9. Different choices of Floer data and perturbation data will resultin quasi-isomorphic A ∞ algebras for the immersed Lagrangian ι . Also, if φ isa Hamiltonian diffeomorphism, then ι and φ ◦ ι are quasi-isomorphic objects ofthe Fukaya category. In particular, they have quasi-isomorphic A ∞ algebras. Remark 7.10. Also, independence on J M can be shown in the following way.Let J ′ M be another valid choice. Let ˜ M be the non-compact manifold obtainedby adding a (symplectically) conical end to M . There exists complex structures J and J ′ which agree outside a compact set, are cylindrical outside of a compactset, are compatible with the contact form σ on ∂M (far enough out on theconical end), and satisfy J | M = J M and J ′ | M = J ′ M . Then working inside ˜ M the arguments of this section can be applied to show that using J M and J ′ M lead to quasi-isomorphic objects.Next, we consider invariance under certain exact deformations. First, somegeneral remarks. Let { ι t } t be a family of Lagrangian immersions. We call thefamily an exact deformation if, for each t , the one form β t on L defined by β t = ι ∗ t ω (cid:18) ∂ι t ∂t , · (cid:19) 44s exact. If one of the immersions, say ι , is exact, then the family is an exactdeformation if and only if all of the immersions are exact. Indeed, if df = ι ∗ σ and dh t = β t , then f t = f + Z t (cid:18) h s + σ (cid:18) ∂ι s ∂s (cid:19) ◦ ι s (cid:19) ds (35)satisfies df t = ι ∗ t σ . A family of immersions is called a Hamiltonian deformationif there exists a family of Hamiltonian diffeomorphisms { φ t } t with φ = id suchthat ι t = φ t ◦ ι . For immersed Lagrangians, not every exact deformation is aHamiltonian deformation. This differs from the embedded case where the twoconcepts coincide.Now consider the energy and indices of branch jumps. Suppose { ι t } t is afamily of exact immersed Lagrangians, and suppose all the immersions have onlytransverse double point self-intersections. Let R t = { ( p, q ) ∈ L × L | ι t ( p ) = ι t ( q ) , p = q } . There are canonical bijections R t ∼ = R for all t and we can viewbranch jumps ( p t , q t ) ∈ R t as varying continuously with t . Since the index of abranch jump takes values in Z , it follows that Ind( p t , q t ) = Ind( p , q ) for all t .The energy however can vary. By definition and equation (35), E ( p t , q t ) = − f t ( p t ) + f t ( q t ) (36)= − f ( p t ) + f ( q t ) + Z t (cid:18) − h s ( p t ) + h s ( q t ) − σ (cid:18) ∂ι s ∂s ( p t ) (cid:19) + σ (cid:18) ∂ι s ∂s ( q t ) (cid:19)(cid:19) ds. Note that if the deformation is Hamiltonian, then p t = p , q t = q for all t , and h s and ∂ι s ∂s are the restrictions of a function and a vector field defined globallyon M . Hence E ( p t , q t ) = E ( p , q ) for all t .A consequence of this behavior is that the positivity condition may hold for t = 0 but may fail for t far enough away from 0. As observed in [3] Section 13.4,this can be interpreted as a type of wall-crossing behavior. In the language of[12] and [3], Lagrangians satisfying the positivity condition are automaticallyunobstructed. If such a Lagrangian is deformed to the point where the positivitycondition fails, it may become obstructed. Thus Lagrangian immersions sepa-rated by a “wall” can have very different Floer theoretic properties. It wouldbe interesting to find an explicit example of this.The proof of Theorem 7.9 does not work for general exact deformationsbecause Gromov compactness (from [14]) does not hold for moving Lagrangianboundary conditions that contain an instant where the immersed Lagrangianhas non-transverse self-intersection. If we rule this case out, then we can proveinvariance under exact deformation. Theorem 7.11. Let { ι t } a ≤ t ≤ b be a family of exact immersions. Suppose thateach ι t has only transverse double points. Moreover, suppose that each ι t satisfiesCondition 1.2. Then all the different ι t are quasi-isomorphic objects in theFukaya category. roof. We will show that given any t ∈ [ a, b ], there exists an open neighborhood U of t in [ a, b ] such that if t ∈ U then ι t and ι t are quasi-isomorphic objects.The theorem will then follow immediately.The idea of the proof is to exhibit a bijection between the moduli spacesfor various ι t used to define m and m . We appeal to regularity and Gromovcompactness to prove this bijection. Once this is done, we note that there areobvious bijections between Hamiltonian chords with endpoints on the ι t , andhence given a unit e t ,t ∈ CF( ι t , ι t ) there are corresponding e t,t ′ ∈ CF( ι t , ι t ′ )for each t, t ′ near t . The fact that the moduli spaces used to define m and m are bijective for the various ι t then implies that e t,t ′ is a quasi-isomorphism.Without loss of generality assume t = 0. We need to find δ > ≤ t < δ then ι and ι t are quasi-isomorphic. Choose regular Floer data H , J for ι , and let φ = φ H be the time-1 flow of H . Let Γ H ( ι t , ι t ′ ) denotethe set of time-1 chords of H that start on ι t and stop on ι t ′ . Choose δ > H ( ι , ι ) and Γ H ( ι t , ι t ′ ) forany 0 ≤ t, t ′ < δ . We may assume that the endpoints of the chords always missthe self-intersection points of ι t and ι t ′ , since this is true for t = t ′ = 0.We denote by γ i chords in Γ H ( ι , ι ) and γ t,t ′ i chords in Γ H ( ι t , ι t ′ ). Gen-erally, any undecorated notation will be with respect to ι , and notation dec-orated with t, t ′ , . . . will be with respect to ι t , ι t ′ , . . . . A ∞ operations dependupon choices of Floer data and perturbation data, which will be specified later.First, choose some perturbation data K d +1 , J d +1 for ι (for all d ), so the A ∞ algebra for ι is defined. Consider the moduli spaces used to define m and m , which are M ( γ , γ ; H , J ) , Ind γ = Ind γ + 1 , M ( γ , γ , γ ; K , J ) , Ind γ = Ind γ + Ind γ , respectively. Since these moduli spaces are regular, and compact and zero di-mensional, and there are only finitely many of them, any holomorphic curve inone of these moduli spaces will persist under slight perturbation of its boundaryconditions. In particular, by shrinking δ if necessary, there are injective maps M ( γ , γ ; H , J ) → M ( γ t,t ′ , γ t,t ′ ; H , J ) , (37) M ( γ , γ , γ ; K , J ) → M ( γ t,t ′′ , γ t,t ′ , γ t ′ ,t ′′ ; K , J )for 0 dimensional moduli spaces and 0 ≤ t, t ′ , t ′′ < δ . Note that we do notyet change the perturbation data for the moduli spaces involving ι t , ι t ′ , ι t ′′ .Elements from all the moduli spaces have an a priori energy bound by equations(31), (13), (35) and (36). We claim that Gromov compactness implies that, forsmall enough δ , these maps are surjective as well. We explain this just for thefirst map. If not, there exists sequences of numbers t n , t ′ n → u n ] ∈ M ( γ t n , γ t ′ n ; H , J ) that are not in the image of thefirst map. Applying Gromov compactness (the compactness theorem proved in[14] works in this setting since ι t → ι and ι has transverse self-intersections),46e get a stable strip with boundary on ι . Since its index is 0, this stable stripconsists of a single strip, that is, it is an element [ u ] ∈ M ( γ , γ ; H , J ). Since M ( γ , γ ; H , J ) is a finite set, the implicit function theorem implies that [ u ]maps to [ u n ] under the first map for large enough n , a contradiction. This alsoimplies M ( γ t,t ′ , γ t,t ′ ; H , J ) and M ( γ t,t ′′ , γ t,t ′ , γ t ′ ,t ′′ ; K , J ) are regular.The above argument does not say anything about the spaces the spaces M ( γ t,t ′ , γ t,t ′ ; α ; H , J ) and M ( γ t,t ′′ , γ t,t ′ , γ t ′ ,t ′′ ; α ; K , J ) for α = ∅ . We needthese to be regular too, which can be achieved by a slight perturbation of thethe perturbation data. So we now choose perturbation data and Floer datafor each ι t with 0 ≤ t < δ , and also each tuple of such Lagrangians, so thatwe have a well-defined Fukaya category which includes all ι t as objects. Wemay assume (by first fixing a set of chords and fixing t, t ′ , t ′′ and looking atsmall deformations of Floer data and perturbation data) that Floer data andperturbation data are chosen sufficiently close to H , J , K , J so that there arecanonical bijections M ( γ t,t ′ , γ t,t ′ ; H , J ) ∼ = M ( γ t,t ′ , γ t,t ′ ; H t,t ′ , J t,t ′ ) , (38) M ( γ t,t ′′ , γ t,t ′ , γ t ′ ,t ′′ ; K , J ) ∼ = M ( γ t,t ′′ , γ t,t ′ , γ t ′ ,t ′′ ; K t,t ′ ,t ′′ , J t,t ′ ,t ′′ )of 0 dimensional moduli spaces. (Again, part of the reason this can be done isthat these moduli spaces contain only finitely many elements and there are onlyfinitely many such moduli spaces.)Since all the different sets Γ H ( ι t , ι t ′ ) are bijective to each other, the sameis true for all Γ H t,t ′ ( ι t , ι t ′ ) after shrinking δ if necessary. Thus all the vectorspaces CF( ι t , ι t ′ ) are isomorphic to each other in a natural way, and furthermorethe bijections (38) imply that the A ∞ operations m t and m t,t ′ are all identifiedunder these isomorphisms. In particular, if e , ∈ CF( ι , ι ) is a (cohomological)unit, then the corresponding e t,t ′ ∈ CF( ι t , ι t ′ ) are quasi-isomorphisms. Remark 7.12. Alternatively, a proof can be given using moving Lagrangianboundary conditions, which we now sketch. First, construct an element e ∈ CF( ι a , ι b ) (which will turn out to be a quasi-isomorphism) in the same way aswas done for Hamiltonian deformations prior to Remark 7.8. Namely, e is acount of rigid inhomogeneous holomorphic discs with one fixed incoming Type Imarked point and moving Lagrangian boundary conditions. Take perturbationdata K on the disc which, for simplicity, is 0 near the boundary of the disc.For Hamiltonian moving Lagrangian boundary conditions, we took K to satisfy d ( K ( ξ ) | L z ) = ω ( ∂ ξ L z , · ) for ξ ∈ T z ∂D (see formula (8.21) in [21]); this doesn’tmake sense in the non-Hamiltonian case. The slight difficulty this causes is thatthe energy equation (31), which is E ( u ) = A ( γ ) − R D R K ( u ) in this case, nolonger holds for a curve u converging to the chord γ on ι a , ι b . To see this, startwith E ( u ) = 12 Z D | du − X K | d vol D = Z D (cid:18) u ∗ ω + d ( K ( u )) − R K ( u ) (cid:19) . 47e want to use Stokes’ theorem on the right-hand side. To this end, parame-terize the portion of ∂D with moving Lagrangian boundary conditions as [ a, b ],so the Lagrangian for t ∈ [ a, b ] ⊂ ∂D is ι t . Let Φ : [ a, b ] × L → M be the mapΦ( t, x ) = ι t ( x ), and f : [ a, b ] × L → M be f ( t, x ) = f t ( x ), where df t = ι ∗ t σ . ThenΦ ∗ σ = df − hdt , where h : [ a, b ] × L → R is the function such that d ( h ( t, · )) = β t (see prior to equation (35)). Indeed, using equation (35), we have( df )( t, · ) = df t + ˙ f t dt = ι ∗ t σ + ˙ f t dt = Φ ∗ σ − h ∂ t ι t , σ i dt + ( h t + h ∂ t ι t , σ i ) dt = Φ ∗ σ + h t dt. Using Stokes’ theorem, ω = dσ , and the definition of action (13), we then get E ( u ) = A ( γ ) − Z D R K ( u ) − Z [ a,b ] h ( t, ℓ ( t )) dt (39)where ℓ is the boundary lift to L of u . This still gives an a priori energy bound.To show that e ∈ CF( ι a , ι b ) is well-defined, we need to show that Gromovcompactness holds for a sequence of elements from this moduli space. The en-ergy bound was provided above. It remains to consider disc bubbling. Using agraph construction (see Section 8 of [16]), we can view u as an honest holomor-phic section (with respect to an appropriate J ) of the bundle M × D → D . Theboundary conditions becomes the fixed totally real immersion L × ∂D → M × D ,( x, z ) ( ι z ( x ) , z ) with ι z the immersion for z ∈ ∂D . The immersion now hasa clean self-intersection instead of a transverse self-intersection (here is wherewe use ι t has transverse self-intersection for all t ). The Gromov compactnessproved in [14] still holds in this setting. The result is that in the limit disc bub-bles may appear in fibers of the section, i.e. lying on specific ι z , and must attachto the main disc component at a branch jump point. This is the expected be-havior, and it matches what happens in the Hamiltonian deformation case. Thepositivity condition and regularity can then be used to rule out unwanted discbubbling. The next section, where we consider cleanly immersed Lagrangians,contains the necessary analytic results for this.The rest of the proof proceeds in the same way as the Hamiltonian defor-mation case outlined before Remark 7.8. Up to this point we have assumed ι has transverse double points. In this sectionwe explain why the theory works equally well if the double points are cleaninstead of transverse, in the following sense. Definition 8.1. An immersion ι : L → L ⊂ M has clean self-intersection , or isa clean immersion , if it has the following properties: • The preimage of any point in L consists of 1 or 2 points.48 The set R = { x ∈ L | | ι − ( x ) | = 2 } is a closed submanifold of M . • For any p, q ∈ L with ι ( p ) = ι ( q ) =: x and p = q , we have ι ∗ T p L ∩ ι ∗ T q L = T x R .These conditions imply that the set ι − ( R ) is a closed submanifold of L ,and if p ∈ ι − ( R ) then ι ∗ T p ι − ( R ) = T ι ( p ) R . R may have several differentcomponents of different dimensions. The preimage of a component may ormay not be connected. For a discussion of a more general definition of cleanimmersion, see Section 9. Remark 8.2. This definition allows the case that ι : L → L is a 2-1 coveringspace. However, this case needs to be handled slightly differently than the othercases. In order to streamline the discussion we thus assume that ι is not acovering; in other words, R is not all of L . We will return to this exceptionalcase later on.As before, let R = { ( p, q ) ∈ L × L | ι ( p ) = ι ( q ) , p = q } be the set of allbranch jump types, and we will use α to denote functions α : { , . . . , m } → R .We continue to assume that L is exact with primitive f L : L → R and gradedwith grading θ L : L → R .The notion of Lagrangian angle (see Section 2.3) can be extended to the caseof clean intersection as follows. Suppose given two Lagrangian planes Λ , Λ in asymplectic vector space ( V, ω ), and let Λ = Λ ∩ Λ . Choose a path of Lagrangianplanes Λ t from Λ to Λ such that • Λ ⊂ Λ t ⊂ Λ + Λ for all t , and • the image Λ t of Λ t inside the symplectic vector space (Λ + Λ ) / Λ is thepositive definite path from Λ to Λ .Let θ t be a path of real numbers such that e πiθ t = Det M (Λ t ). Then definethe angle between Λ and Λ by the formula 2 · Angle(Λ , Λ ) := θ − θ .With this definition in hand we extend the notion of the index of a branchjump to the clean intersection case by using the same formula as the transversecase. The index of ( p, q ) ∈ R is defined to beInd( p, q ) = dim L + θ L ( q ) − θ L ( p ) − · Angle( ι ∗ T p L, ι ∗ T q L ) . Now assume that positivity Condition 1.2 holds; that is if ( p, q ) ∈ R and E ( p, q ) = − f L ( p ) + f L ( q ) > 0, then Ind( p, q ) ≥ 3. Let us analyze this conditiona little bit. Note that both E and Ind are locally constant as functions R → R .Let C ⊂ R be a component such that the preimage consists of two components C , C . Then E ( p, q ) = − f L ( p ) + f L ( q ) is independent of ( p, q ) ∈ R with p ∈ C and q ∈ C . Hence define E ( C , C ) := − f L ( p ) + f L ( q ). Likewise, Ind( p, q ) isindependent of ( p, q ) and we can define Ind( C , C ) := Ind( p, q ).If the preimage of C consists of a single component C (and hence C → C is a non-trivial double cover) then more can be said. Since any ( p, q ) ∈ R with p, q ∈ C can be connected to ( q, p ) in R , we must have E ( p, q ) = E ( q, p ).49ince E ( p, q ) = − E ( q, p ) from the formula, we have E ( p, q ) = 0. Thus we de-fine E ( C, C ) = 0. Similar considerations apply to the index: Since Ind( p, q ) +Ind( q, p ) = dim L + dim C , we define Ind( C, C ) := Ind( p, q ) = Ind( q, p ) = (dim L + dim C ). C vacuously satisfies the positivity assumption because E ( C, C ) = 0. Thus the positivity condition can be rephrased as Condition 8.3. If C is a component of R such that the preimage consists oftwo distinct components C and C and E ( C , C ) > C , C ) ≥ We continue to call Floer data H , J that satisfies Definition 1.3 admissible. Re-call from Remark 8.2 that for now we do not allow R to be all of L , hence admissi-ble Floer data exists. Assume, in addition to Floer data, that we are given strip-like ends and perturbation data K d +1 , J d +1 for the families S d +1 ,m → R d +1 ,m for all d + 1 ≥ , m ≥ 0, as in Section 7.1.Given any fixed α : { , . . . , m } → R define analogs of the Banach mani-folds B ,p ( Z, α ), B ,p ( S d +1 ,m , α ), and B ,pdisc ( D, α ) that we defined before, andcorresponding Banach bundles over these spaces along with Cauchy-Riemannsections. The main difference is that now weighted Sobolev spaces need to beused in order to get a good Fredholm theory due to the non-transversality ofthe self-intersections. To do this, pick some small δ > 0. A δ -weighted L p normis a norm on a strip (or half infinite strip) that is of the form || ξ || L p ; δ = Z | ξ | p e δ | s | dsdt, where s + it are strip-like coordinates. In the setup of our Banach manifolds wenow require the regularity to be W ,p ; δ . For instance, Definition 3.2 is modifiedby requiring ξ to be in W ,p ; δ over each strip-like end. For δ > H , J ). Theorem A.4 implies that bounded L energy of holomorphic curvesis enough to get W ,p ; δ regularity near Type II marked points; it is standardthat the same statement holds near Type I marked points. One minor technicalpoint that needs to be addressed is that we need to choose the reference metric g fix on M in such a way that L is totally geodesic. This can be done by thefollowing lemma. Lemma 8.4. Let L be cleanly immersed. Then there exists a metric g on M so that L is totally geodesic.Proof. As the proof will make evident, we may assume R consists of a singlecomponent. By the definition of cleanly immersed, a tubular neighborhood of R inside M can be identified with a neighborhood of the zero-section in thenormal bundle N = N R,M in such a way that L maps to (open neighborhoodsin) subbundles E and E of N . The intersection of E and E is the zero-sectionand the corank of E ⊕ E equals the dimension of R . Choose another subbundle50 ′ so that E ⊕ E ⊕ E ′ = N . Choose a metric on N of the form h ⊕ h ⊕ h ′ ,and choose a metric on the manifold R . These metrics induce a metric g on thetotal space of N (i.e., for each x ∈ N , g gives a map g x : T x N ⊗ T x N → R ).Then the following diffeomorphisms N → N are all isometries with respect to g :( e , e , e ′ ) ( − e , − e , − e ′ ) , ( e , e , e ′ ) ( e , − e , − e ′ ) , ( e , e , e ′ ) ( − e , e , − e ′ ) . The fixed point set of the first map is R , the fixed point set of the second is E , and the fixed point set of the third is E . Thus R , E and E are totallygeodesic with respect to the metric g . Now pull the metric g back to the tubularneighborhood of R in M . This defines g near the singular points of L . Then thearguments in the proof of Lemma 3.1 can be applied to show that an extensionof g to all of M can be found that makes L totally geodesic.The addition of the weights makes maps in the spaces converge to the valuesof ι ◦ α on either side of the Type II points. We want to allow these valuesto vary; that is, we want to allow α to vary continuously. To describe this wefirst set up some notation. Let R comp be the set of all ( C , C ) such that forsome ( p, q ) ∈ R , C is the component of ι − ( R ) containing p and C is thecomponent containing q . Note that any α : { , . . . , m } → R induces a map e α : { , . . . , m } → R comp . Given e α , define B ,p ; δ ( Z, e α ) = a α B ,p ; δ ( Z, α ) × { α } where the union is over all α that induce the given e α . Similarly we can define B ,p ; δ ( S d +1 ,m , e α ) and B ,p ; δdisc ( D, e α ). To incorporate the varying of α into theBanach space setup we enlarge the local model for the coordinate charts byadding on an open subset of R d i for each 1 ≤ i ≤ m , where d i = dim e α ( i ).These spaces also have Banach bundles and Cauchy-Riemann sections.We now define moduli spaces M ( γ , . . . , γ d ; e α ) as in Definition 7.4. Thesemoduli spaces coincide with the zero-sets of Cauchy-Riemann sections¯ ∂ : B ,p ; δ ( S d +1 , e α ) → E ,p ; δ ( S d +1 ,m , e α ) . Similarly, given any J we can define the moduli space of discs M disc ( e α ; J ).The linearization of the Cauchy-Riemann operator at an element ( u, r, ∆ , α ) of M ( γ , . . . , γ d ; e α ) has the form D ¯ ∂ : W ,p ; δ ( u ∗ T M ) × T ( r, ∆) R d +1 ,m × T α R → L p (Λ , ⊗ C u ∗ T M ) , where T α R := L mi =1 T ι ( α ( i )) R . In case d + 1 = 2 (so the domain is a strip),replace the T ( r, ∆) R d +1 ,m term with T ∆ Config m ( Z ) (see equation (7)). Thisoperator is Fredholm and has indexInd D ¯ ∂ = Ind γ − d X i =1 Ind γ i − m X i =1 Ind e α ( i ) + m + (cid:26) d − d ≥ d = 1 . (40)The same methods as before prove the following proposition.51 roposition 8.5. Generic perturbation data is regular. For regular perturba-tion data, all the spaces M ( γ , . . . , γ d ; e α ) are smooth manifolds whose dimen-sions are determined by (40) (in case d = 1 we subtract 1 from the index to getthe dimension of the moduli space because of the R action).Furthermore, the moduli spaces M ( γ , . . . , γ d ) have compactifications similarto those described in Proposition 7.6. A ∞ structure Condition 8.3 can be phrased in the following way: If ( C , C ) ∈ R comp and E ( C , C ) > C , C ) ≥ 3. Thus a clean immersion satisfying thisassumption formally resembles the transverse self-intersection case under Con-dition 1.2, with α replaced with e α and R replaced with R comp . In particular,notice the similarity between equation (40) and equations (11), (29).Now define the A ∞ structure in the same way as before: pick generic Floerdata H , J and perturbation data, take CF( ι ) to be generated by the chords on ι , and then define the m k by counting inhomogeneous holomorphic polygons. Itfollows that the proofs used to establish existence of Floer cohomology and A ∞ structure for the transverse self-intersection case carry over immediately to theclean intersection case. Theorem 8.6. Let ι : L → L be an immersed Lagrangian with clean self-intersection in the sense of Definition 8.1. Assume that R = L and the im-mersion satisfies Condition 8.3. Then, for generic Floer data ( H , J ) andgeneric perturbation data, the Floer cohomology HF( ι ) and the A ∞ algebra (CF( ι ) , { m k } k ≥ ) can be defined as in Sections 6 and 7.4. Moreover, the Floercohomology does not depend on the choice of Floer data or perturbation data,and the same is true of the A ∞ algebra up to homotopy equivalence. The theorem also holds when R = L . This will be proved in the next section. R = L : Equivalence with local systems We now consider the case when R = L ; in other words, L is an embeddedLagrangian submanifold of M and ι : L → L is a covering space. This in factimplies that L is exact also, so this case turns out to be easier than the non-covering space case. In Definition 8.1 we assumed that immersed points are atmost double points. This however is not necessary, so assume now only that L → L is a d -fold covering space and L is connected. Actually since L is compactit is automatically a finite cover. Given Hamiltonian H , recall that Γ H is theset of (time 1) Hamiltonian chords that stop and start on L . In Definition 1.3we required that none of the chords started or stopped on R ; this can never betrue now. To account for this, we change the definition of Γ H . Definition 8.7. Let Γ H denote the set of triples ˜ γ = ( γ, p , p ) such that • γ : [0 , → M is a Hamiltonian chord that starts and stops on L , and52 p , p ∈ L are such that ι ( p ) = γ (0) , ι ( p ) = γ (1).We now say that a Hamiltonian is admissible as long as φ H ( L ) is transverseto L . If H is admissible then Γ H is a finite set with d elements for each pointof φ H ( L ) ∩ L . In this case the Floer cochain complex is defined to beCF( ι ; H , J ) = M ˜ γ ∈ Γ H Z · ˜ γ. (41)The grading of ι defines a grading of chords ˜ γ ∈ Γ H in much the same way asbefore; the only difference is that in Definition 2.4, T γ (0) L needs to be replacedwith ι ∗ T p L and T γ (1) L with ι ∗ T p L . (Actually, Lemma 8.9 implies that L canbe graded and the grading on ι is the pull back of a grading on L , so the indexof a chord ˜ γ is in fact determined by T γ ( i ) L , i = 0 , ι ; H , J ).We define the moduli spaces of strips M (˜ γ − , ˜ γ + ; α ), polygons M (˜ γ , . . . , ˜ γ k ; α )and discs M disc ( α, J ) much in the same way as before. The difference is thatnow the boundary lift ℓ of a holomorphic curve must agree on either side of aType I marked point with the lifts p , p of the endpoints γ (0) , γ (1) of the chord˜ γ = ( γ, p , p ) corresponding to the marked point. The order in which they agreeshould depend on whether the point is incoming or outgoing. The same conven-tion as for Type II marked points applies: If ℓ − ∈ L denotes the limiting valueof ℓ before the Type I point (with respect to counterclockwise orientation), and ℓ + the value after, then if the marked point is outgoing ℓ − = p , ℓ + = p . Forincoming, ℓ − = p , ℓ + = p .The regularity theory of Section 5 carries over immediately. Thus Floer data H , J and perturbation data that make all the uncompactified moduli spaces ofstrips and polygons regular exists. The compactness theory of Section 4 holdsas well with the obvious modification: the matching condition at a Type I nodalpoint of a broken curve needs to include matching of the chord plus matchingof the lifts of the endpoints of the chord. This is automatically kept track ofwith the new notation since a chord γ is replaced with ˜ γ = ( γ, p , p ).We would now like to conclude that the A ∞ algebra (CF( ι ) , { m k } ) for ι iswell-defined and, up to equivalence, is independent of the various choices usedto construct it. However, there is one point that still needs addressing. Thecondition [ φ H ( L ) ∪ ( φ H ) − ( L )] ∩ R = ∅ in Definition 1.3 is used to rule out theexistence of constant strips with disc bubbles attached. In general, such stripswill not be regular; see Proposition 5.1 and the preceding discussion. Since thecovering space is finite, the map ι ∗ : H ( L, R ) → H ( L, R ) is injective, and thus L is exact. Any non-constant element of M disc ( ι ; α, J ) corresponds to a non-constant holomorphic disc with boundary on L ; since this latter type of disc doesnot exist M disc ( ι ; α, J ) consists only of constant discs for any J . In particular,if α : { } → R (i.e., there is only one marked point) then M disc ( ι ; α, J ) = ∅ because there are no constant discs with only one marked point. Thus theGromov compactness of the moduli spaces M (˜ γ , . . . , ˜ γ k ) takes a particularlystrong form; namely, elements of the compactified moduli space never contain53isc bubble components. It follows that constant strips never enter into thecompactified moduli spaces needed to define the A ∞ structure, and hence theydo not cause a problem. Thus our previous arguments go through to prove thefollowing theorem. Theorem 8.8. Suppose ι : L → L is a covering space and L is compact. Thenthe A ∞ algebra (CF( ι ) , { m k } ) is well-defined and independent of the choicesinvolved, up to homotopy equivalence. Here, CF( ι ) is defined as in (41) and m k is defined using the moduli spaces M (˜ γ , . . . , ˜ γ k ) . Now we want to relate the above construction to the construction of Floertheory with coefficients in a local system. First, we sketch the general theoryas it applies to L . Since L is an exact embedded Lagrangian, the A ∞ alge-bra for L is well-defined in the usual way (note that by this A ∞ algebra wedo not mean CF( ι )). Let E → L be a Z local system of rank r over L . Ex-plicitly, pick a basepoint x ∈ L and view E as a Z vector bundle with fiberisomorphic to E x = V = ( Z ) ⊕ r (with r finite). Let Hol( c ) : V → V denoteparallel translation over the loop c in L with basepoint x . Then the map ρ = ρ E : π ( L ) = π ( L, x ) → Aut Z ( V ) defined by ρ ( c ) v = (Hol( c )) − v isa representation of π ( L ). Conversely, given a representation ρ , we can con-struct a bundle E = E ρ → L by setting E = e L × ρ V = e L × V / ∼ , where( g · x, ρ ( g ) v ) ∼ ( x, v ). Here, e L is the universal cover of L and g · x denotes theleft action of π ( L ) on e L . (Our convention is that multiplication in π ( L ) isconcatenation from left to right of loops, so gh means first g and then h . If e L isthought of as homotopy classes of paths in L with basepoint x , then g · x is thepath which is first g and then x .) These two constructions are inverse to eachother, in the sense that given ρ , we have ρ E ρ ∼ = ρ ; and given E , we have E ρ E ∼ = E .Next, suppose given H such that φ H ( L ) is transverse to L . Define Γ H tobe the set of all chords γ starting and stopping on L . Then defineCF( L, E ) = M γ ∈ Γ H Hom Z ( E γ (0) , E γ (1) ) . If we assume L is graded, then CF( L, E ) can be graded in the usual way, namelythe summand Hom Z ( E γ (0) , E γ (1) ) is given the grading Ind( γ ). One could per-haps define a slightly more general notion of grading but this will suffice for ourpurposes. Since we assume that ι is graded, the next lemma implies that L isin fact also graded. Lemma 8.9. L can be graded if and only if ι can be graded. In particular, anygrading on ι is invariant with respect to the group of deck transformations ofthe covering space L → L .Proof. If L is graded, then ι can be graded by pulling back the grading functionon L . The pullback is obviously invariant with respect to deck transformations.Conversely, suppose that ι has a given grading. Then ι ∗ ( π ( L )) is containedin the kernel of the map (Det M ) ∗ : π ( L ) → π ( S ). Since | π ( L ) : ι ∗ ( π ( L )) | =54 L : L | < ∞ , it follows that (Det M ) ∗ is the zero map, and hence L can begraded. By the first part of the proof there is thus an invariant grading on ι . Since any two gradings on ι differ by an integer, the given grading on ι isinvariant also. A ∞ structure maps are defined in the following way: For x i ∈ Hom( E γ i (0) , E γ i (1) ),define m k ( x , . . . , x k ) = X γ X [ u ] ∈M ( γ ,...,γ k ) Hol( u ; x , . . . , x k ) . Here the outer sum is over all γ such that Ind γ = Ind γ + · · · + Ind γ k + 2 − k . Hol( u ; x , . . . , x k ) is defined in the following way. For y ∈ E γ (0) , paralleltranslate y over the boundary arc of u between the marked points z and z ;call the result y ∈ E γ (0) . Then x ( y ) ∈ E γ (1) . Parallel translate this vectorover the boundary arc of u between z and z to get y ∈ E γ (0) . Apply x toget x ( y ) ∈ E γ (1) . Now repeat the process until the clockwise side of z isreached. The resulting vector is Hol( u ; x , . . . , x k )( y ) ∈ E γ (1) , the image of y under the holonomy map.The result is that (CF( L, E ) , { m k } ) is an A ∞ algebra; more generally ( L, E )can be viewed as an object of the Fukaya category. Note that for any exactembedded Lagrangian L ′ we can take E ′ = L ′ × Z and then L ′ and ( L ′ , E ′ ) arethe same object of the Fukaya category.Now consider ι : L → L . Pick a basepoint x ∈ L and let B = ι − ( x ).By parallel transport, for each loop c ∈ π ( L ), the covering space ι defines abijection Hol( c ) : B → B . Let ρ : π ( L ) → Aut Set ( B ) be the homomorphismdefined by ρ ( c ) = (Hol( c )) − . Let V be the Z vector space formally defined tohave basis B . Then we can view Aut Set ( B ) ⊂ Aut Z ( V ), and hence view ρ asa representation into Aut Z ( V ). Thus we get a local system with fiber V whichwe denote as E ι . Theorem 8.10. Let ι : L → L be a covering space and assume that ι (andhence L ) is exact. Then ι and ( L, E ι ) are isomorphic objects of the Fukayacategory. In fact, their A ∞ algebras are isomorphic in an obvious way if thesame perturbation data is used to define both. Moreover, HF( ι ) ∼ = HF( L, E ι ) ∼ = H ∗ sing ( L, E ∗ ι ⊗ E ι ) ∼ = H ∗ sing ( L ∐ R, Z ) as graded vector spaces. Here the second to last group denotes singular cohomol-ogy with coefficients in the local system E ∗ ι ⊗ E ι . Note also that L ∐ R ∼ = L × L L by equation (2) .Proof. Assume that the same H is used to define CF( ι ) and CF( L, E ι ). Thenthe map ˜ γ = ( γ, p , p ) p ∗ ⊗ p ∈ Hom( E ι,γ (0) , E ι,γ (1) )induces a bijection CF( ι ) ∼ = CF( L, E ι ). (Here we are using the fact that, for each x ∈ L , ι − ( x ) can be canonically identified with a basis of the fiber E ι,x . Notealso that E ι ∼ = E ∗ ι .) Moreover, if the same perturbation data is used to define55oth A ∞ algebras then this bijection intertwines the two sets of A ∞ operations.Hence the A ∞ algebras are isomorphic.It follows that HF( ι ) ∼ = HF( L, E ι ). The isomorphismHF( L, E ι ) ∼ = H sing ( L, E ∗ ι ⊗ E ι )is well-known and follows because L is exact. We sketch a proof here; actually,we will sketch HF( ι ) ∼ = H sing ( L, E ∗ ι ⊗ E ι ).First, consider just L with no local system. Take a C -small time indepen-dent Hamiltonian H and let h = H | L , and let φ be the time-1 flow of X H . Let J be a time independent almost complex structure. For H small enough, theHamiltonian chords on L correspond to the intersection points φ ( L ) ∩ L (seethe paragraph after Definition 2.4), and these in turn correspond to the criticalpoints of h . If in addition h has non-degenerate critical points, then the intersec-tion will be transverse. In this situation, Floer showed in [10] that holomorphicstrips which satisfy Floer’s equation (9) correspond to gradient flow lines of h with respect to the metric ω ( · , J · ). The correspondence is easy to describe: thegradient flow line x ( s ) (which satisfies ˙ x ( s ) = ∇ h ( x ( s )) = JX H ( x ( s ))), corre-sponds to the strip u ( s, t ) = x ( s ). It follows that the Morse cohomology of L with respect to the function − h is isomorphic to the Floer cohomology of L withrespect to H . Indeed, if p and p are critical points of − h , then the matrixelement h p , δ Morse p i for the Morse cohomology differential equals the numberof −∇ ( − h ) = ∇ h flow lines from p to p . By the above correspondence, thesame is true of the Floer differential. By Lemma 2.9, this isomorphism is agraded isomorphism.Now we consider the Floer cohomology of ι , with J, H as above, and thesingular cohomology of the local system E ∗ ι ⊗ E ι on L . We will use the fact that L ∐ R can be identified with the fiber product L × L L , see equation (2). Firstnote that the generators of CF( ι ) can be identified with the critical points ofthe function ˜ h = h ◦ ι : L × L L → R via the chord ˜ γ = ( γ, p , p ) corresponds tothe critical point ( p , p ) ∈ L × L L . Second, strips satisfying Floer’s equation(9) with boundaries on ι (i.e. the ones used to compute HF( ι )) are nothingmore than strips on L with bottom and top boundary lifts to L . The strip on L is a gradient flow line of h . The bottom and top boundary lifts together canbe thought of as giving a lift to L × L L ; if the bottom lift is ℓ and the toplift is ℓ , think of this as the lift ( ℓ , ℓ ) to L × L L . It follows that HF( ι ) isisomorphic to the Morse cohomology of − ˜ h . It is a graded isomorphism becauseLemma 8.9 implies that the index of ˜ γ = ( γ, p , p ) equals the index of γ , andLemma 2.9 implies that the index of γ equals the index of the correspondingcritical point of − h , which in turn equals the index of the critical point of − ˜ h .The Morse cohomology is isomorphic to the singular cohomology of L × L L . Asimple adaptation of Example 3H.2 in [13] shows that the singular cohomologyof L × L L is isomorphic to the singular cohomology of L with coefficients in E ∗ ι ⊗ E ι .We now examine the invariance of ι under exact, non-Hamiltonian deforma-tions arising from Morse functions. To begin with, let h : L → R be a function.56iven h , define h R : R → R by h R ( p, q ) = − h ( p ) + h ( q ), and assume that h issuch that h R is Morse. Let { ι t } t be an exact deformation of ι := ι that satisfies dh = ι ∗ t ω ( ∂ι t ∂t , · ), so { ι t } t is a deformation of ι in the direction of − J ∇ h (mod-ulo reparametrization of the domain). Let f t : L → R be as in equation (35), soit satisfies df t = ι ∗ t σ . The condition that h R is Morse implies that ι t has onlytransverse double points for small t = 0. Moreover, for any fixed t = 0, there isa bijective correspondence between R t = { ( p, q ) ∈ L × L | ι t ( p ) = ι t ( q ) , p = q } and the critical points of h R .The immersions ι t for t = 0 may or may not satisfy the positivity condition.We will show that if they do, then they are all quasi-isomorphic objects of theFukaya category; in particular, they are all quasi-isomorphic to ι . This is quiteinteresting because the topology of ι and ι t are different; for instance, by takingdifferent functions it could be possible to construct immersions with differentnumbers of self-intersection points.Before giving the proof we analyze the positivity condition a little further.First, consider the index of self-intersection points of ι t . Lemma 8.11. Let L = R ⊂ C and L t = e it · L . Then { L t } t is a deformationof L corresponding to the function h : L → R , h ( x ) = − x . Equip C withthe complex volume form dz and give L a grading θ , and then grade L t with θ t by continuation. Then for t > , Ind(( T L , θ ) , ( T L t , θ t )) = 1 .If instead L t = e − it · L , then the function is h ( x ) = x and Ind(( T L , θ ) , ( T L t , θ t )) = 0 for t > .Proof. This is a simple calculation using equation (3). Corollary 8.12. For the deformation { ι t } t , it follows that, for t > , Ind( p t , q t ) = Ind( h R , ( p , q )) , where the right-hand side denotes the Morse index of the critical point ( p , q ) ∈ R which corresponds to ( p t , q t ) ∈ R t .Proof. Consider first the simple example of L = R ∐ R , L = R ⊂ C . Let p, q ∈ L denote the two preimages of 0 ∈ L , and let x denote the coordinate on the firstcopy of R in L (the one containing p ), and y the coordinate on the second copy.Take h : L → R to be h ( x ) = 0 , h ( y ) = y / 2. The component { ( x, y ) | x = y } of R contains ( p, q ), and on this component h R = x / y / 2. The flow of − i ∇ h fixes the first branch of L and rotates the second branch in the clockwisedirection. Thus, assuming both branches have the same grading to begin with,after perturbation the index of ( p t , q t ) will be 0 by Lemma 8.11, which is alsothe Morse index of h R at ( p, q ). Taking instead h ( x ) = 0 , h ( y ) = − y / L = R n ∐ R n and M = C n , assuming that both branches of L have the same57rading to begin with. For general L and M , again assuming that branches of L have the same grading, the result follows from the local calculation by takinga Weinstein neighborhood.It remains to show that all branches of L have the same grading; that is,that θ L ( p ) = θ L ( q ) whenever ι ( p ) = ι ( q ). This follows from Lemma 8.9.Next consider the energy E ( p t , q t ) of ( p t , q t ) ∈ R t . Recall that E ( p , q ) = 0since ι is a covering of an embedded exact Lagrangian. By equation (36) ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 E ( p t , q t ) = − df ( ˙ p ) + df ( ˙ q ) − h ( p ) + h ( q ) − σ (cid:18) ∂ι t ∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 ( p ) (cid:19) + σ (cid:18) ∂ι t ∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 ( q ) (cid:19) = − σ (cid:18) Dι ( ˙ p ) + ∂ι t ∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 ( p ) (cid:19) + σ (cid:18) Dι ( ˙ q ) + ∂ι t ∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 ( q ) (cid:19) − h ( p ) + h ( q )= h R ( p , q ) . Here we have used ι ∗ σ = df and ι t ( p t ) = ι t ( q t ) for all t (or rather, the derivativeat t = 0 of this identity). Since h R ( p, q ) = − h R ( q, p ), it follows that the globalmaximum of h R is positive and the global minimum is negative. The Morseindex of the global minimum is 0, therefore the corresponding self-intersectionpoint of ι t has index 0 for small t > 0. By the above calculation, it has negativeenergy. Likewise, the self-intersection point corresponding to the global maxi-mum of h R has index n and positive energy. Hence, for small t > 0, this pointwill satisfy the positivity condition provided n = dim L ≥ 3. Similar considera-tions apply to deforming for time t < 0: the self-intersection points of ι t corre-sponding to the global min and max of h R will automatically satisfy the posi-tivity condition provided n ≥ 3. This can be seen by replacing h with h ′ = − h ,because moving in negative time for h is the same as moving in positive timefor h ′ . Also, the global min of h R becomes the global max of h ′ R and vice versa,and the Morse indices are related by Ind( h R , ( p , q )) = n − Ind( h ′ R , ( p , q )).More generally, for small t = 0, E ( p t , q t ) > E ( q − t , p − t ) > 0; andInd( p t , q t ) = Ind( q − t , p − t ). Thus ι t satisfies the positivity condition if and onlyif ι − t does.We now prove the theorem mentioned before. Note that this theorem canprofitably be combined with Theorem 7.11. Theorem 8.13. Let h be as above and let { ι t } t be the corresponding familyof immersions. Assume that for small t = 0 , all the ι t satisfy Assumption 1.2.Then there exists ǫ > such that ι t and ι are quasi-isomorphic when | t | < ǫ .Proof. Let us change notation and denote t by τ , so { ι τ } τ is the family ofLagrangian immersions. We do this so as not to confuse τ with the t in families { H t } t and { J t } t of time-dependent Floer data.The idea of the proof is to count curves with moving Lagrangian bound-ary conditions to get elements e ,τ ∈ CF( ι, ι τ ) and e τ, ∈ CF( ι τ , ι ) which are58uasi-isomorphisms. This is the standard method when { ι τ } τ is a Hamiltoniandeformation, and we also used it in Remark 7.12 to give an alternative proof toTheorem 7.11. The difficulty in the present case is that the family of Lagrangianimmersions undergoes a change in topology at τ = 0. In particular, the anglesbetween the branches at a self-intersection point go to 0 as τ → 0. The com-pactness theorem of [14] does not apply in this situation. To get around this,we will lift curves to T ∗ L in order to prove compactness. We emphasize thatthe main non-standard point of the proof is the compactness.The first step is to define a Lagrangian lift L τ of L τ to T ∗ L . To do this,fix a Weinstein neighborhood W of L inside M ; W may also be viewed as aneighborhood of L inside T ∗ L . The covering space map ι : L → L induces alocal symplectomorphism I : T ∗ L → T ∗ L ; let W ′ = I − ( W ). Choose ǫ > L τ = ι τ ( L ) is contained inside the Weinstein neighborhood W for | τ | ≤ ǫ .Let L τ be the graph of − τ dh . Then I restricts to give an immersion L τ → M .Let Π : T ∗ L → L be the projection, and let Π τ = Π | L τ : L τ → L . Withoutloss of generality (by Theorem 7.11), we may assume that ι τ = I ◦ Π − τ . So L τ = ι τ ( L ) = I ( L τ ), and L τ is a lift of L τ to T ∗ L .The second step is to define Floer data for each pair ( ι τ , ι τ ′ ) of Lagrangianimmersions in M , and corresponding Floer data for each pair of embeddedLagrangians ( L τ , L τ ′ ) in T ∗ L , with | τ | , | τ ′ | < ǫ . First, fix some Floer data H , J for the pair of Lagrangians ( ι, ι ), with H = 0 outside a compact subset of W .For the purposes of making the setup easy to visualize, we choose H as follows.Take H = { H = H t } t to be time independent and C -small. Also, near L inthe Weinstein neighborhood W , take H to be constant in the direction normalto L . Then the flow of X H will move in the direction normal to L , and φ H ( L )will be the graph of an exact 1-form over L . We may assume that the time-1chords of X H from L to L are constant chords with images equal to the pointsof φ H ( L ) ∩ L . Take ǫ to be small compared to H , and assume H is chosen sothat the intersection points φ H ( L ) ∩ L are away from the self-intersection pointsof L τ for τ = 0. Then, for each pair of numbers τ, τ ′ , perturb J slightly to get J τ,τ ′ which makes H , J τ,τ ′ regular Floer data for the pair ( ι τ , ι τ ′ ).Next define Floer data for each pair of Lagrangians ( L τ , L τ ′ ) inside T ∗ L bylifting the Floer data for the pair ( ι τ , ι τ ′ ) as follows. Let H ′ = H ◦ I : W ′ → R .Since H vanishes near the boundary of W , H ′ can be extended by 0 to all of T ∗ L . Then take H ′ = { H ′ t } t as the time-independent Hamiltonian on T ∗ L .The complex structure J τ,τ ′ can also be lifted to a complex structure on W ′ .Extend it in any reasonable way to all of T ∗ L (for example, make it contacttype outside of a compact subset), call the result J ′ τ,τ ′ . The result is that weget Floer data H ′ , J τ,τ ′ for each pair of Lagrangians ( L τ , L τ ′ ) in T ∗ L .The third step is to define some moduli spaces. For the rest of the proofwe assume τ > τ < S = D \ { z = − } ; view z = − λ be a coordinate for ∂S ∼ = R such that λ increases in the counterclockwise direction around ∂S . Let χ : ∂S → [0 , τ ] besmooth, increasing, and surjective with ˙ χ ( λ ) = 0 for | λ | large. Pick some generic59erturbation data K ,τ and complex structure J ,τ (so K ,τ is a 1-form on S with values in the Hamiltonian functions on M , and J ,τ is an S -dependentalmost complex structure on M ) that agrees with the Floer data H , J ,τ onsome strip-like coordinates near z . We may choose K ,τ so that K ,τ = 0outside the Weinstein neighborhood W , and also K ,τ ( ξ ) | L ≡ ξ ∈ T ∂S .A generator of CF( ι , ι τ ) is a tuple ˜ γ = ( γ, p , p ) with γ a time-1 X H chord connecting L to L τ , and p , p ∈ L with ι ( p ) = γ (0), ι τ ( p ) = γ (1).For a generator ˜ γ , let M (˜ γ ; { ι χ ( λ ) } λ ) consist of equivalence classes of tuples( u, ℓ ) with u : D → M a finite energy map such that ( du − X K ,τ ) , = 0, u converges to the chord γ at the point z , and u satisfies the moving Lagrangianboundary conditions u ( λ ) ∈ L χ ( λ ) for λ ∈ ∂S . The lift ℓ : ∂S → L satisfies u ( λ ) = ι χ ( λ ) ( ℓ ( λ )), ℓ ( −∞ ) = p and ℓ (+ ∞ ) = p .Next, define a similar moduli space M ( γ ′ ; { L χ ( λ ) } λ ) of curves into T ∗ L with moving Lagrangian boundary conditions. For perturbation data, use data K ′ ,τ and J ′ ,τ which in W ′ is a lift through I of the data K ,τ , J ,τ . Near themarked point z , the data should agree with the previously defined Floer data H ′ , J ′ ,τ . Since L and L τ are embedded, γ ′ simply denotes a time-1 X H ′ chordfrom L to L τ , and the lifts ℓ are not needed as part of the data.Next, we define two moduli spaces of strips with non-moving Lagrangianboundary conditions, analagous to Definition 3.5 but with no Type II points.The first moduli space is M (˜ γ − , ˜ γ + ; ι , ι τ ). It consists of pairs ( u, ℓ ), where u is a strip into M with bottom boundary on L and top boundary on L τ , and ℓ is a lift to L of the boundary conditions. The map u satisfies Floer’s equationwith the Floer data H , J ,τ for the pair ( ι , ι τ ). The second moduli space is M ( γ ′− , γ ′ + ; L, L τ ). It consists of strips u ′ into T ∗ L with bottom boundary on L and top boundary on L τ . The map u ′ satisfies Floer’s equation with the Floerdata H ′ , J ′ ,τ .The fourth step is to define bijections between the moduli spaces for ( ι, ι τ )and those for ( L, L τ ). Consider M (˜ γ ; { ι χ ( λ ) } λ ) and M ( γ ′ ; { L χ ( λ ) } λ ) first. Fixa chord ˜ γ = ( γ, p , p ) and consider an element ( u, ℓ ) from the moduli space M (˜ γ ; { ι χ ( λ ) } λ ). By an analog of equation (39), the energy of u is E ( u ) = 12 Z | du − X K ,τ | = A (˜ γ ) − Z S R K ,τ ( u ) (42) − Z R h ( χ ( λ ) , ℓ ( λ )) ˙ χ ( λ ) dλ = − Z ( γ ∗ σ + H ( γ ( t ))) dt − f ( p ) + f τ ( p ) − Z S R K ,τ ( u ) − Z R h ( χ ( λ ) , ℓ ( λ )) ˙ χ ( λ ) dλ. Here f , f τ : L → R are the primitives for ι ∗ σ, ι ∗ τ σ defined by (35). By con-struction of H , the chord γ is short and approximately equal to a constantchord. Also, because L is exact, f is the pullback through ι of a function on L .Moreover, f τ is a deformation of f . Thus E ( u ) can be made arbitrarily small60y taking ǫ, H , K small. By standard Gromov monotonicity, if u leaves theWeinstein neighborhood W , then it must have energy larger than some fixedconstant E > K ,τ is 0 outside of W ). Thus,by taking the data small enough, u cannot leave W . Let γ ′ be the time-1 X H ′ chord in W ′ ⊂ T ∗ L which maps to γ under the projection W ′ → W and satisfies γ ′ (1) = Π − τ ( p ). Since the image of u lies inside W , and u converges to γ at z , u can be lifted to a map u ′ : S → W ′ which converges to γ ′ at z . By construc-tion, Iu ′ ( λ ) = u ( λ ) = ι χ ( λ ) ( ℓ ( λ )) = I Π − χ ( λ ) ( ℓ ( λ )). Since I is a covering map and u ′ (+ ∞ ) = Π − χ (+ ∞ ) ( p ) = Π − χ (+ ∞ ) ( ℓ (+ ∞ )), it follows that u ′ ( λ ) = Π − χ ( λ ) ( ℓ ( λ ))for all λ . In particular, u ′ ( λ ) ∈ L χ ( λ ) . We thus get u ′ ∈ M ( γ ′ ; { L χ ( λ ) } λ ).Conversely, suppose given a curve u ′ ∈ M ( γ ′ ; { L χ ( λ ) } λ ). An energy calcu-lation analogous to the one above shows that u ′ has small energy, and henceby monotonicity cannot leave the neighborhood W ′ . Let p = Π u ′ ( −∞ ), p =Π u ′ (+ ∞ ) for ±∞ ∈ ∂S , and let ˜ γ = ( Iγ, p , p ). Then ˜ u = ( u = Iu ′ , ℓ = u ′ | ∂S )is an element of the moduli space M (˜ γ ; { ι χ ( λ ) } λ ). The lifting and pushingdown constructions are inverses to each other, and thus we get a bijection M (˜ γ ; { ι χ ( λ ) } λ ) ∼ = M ( γ ′ ; { L χ ( λ ) } λ ).Next consider the moduli spaces of strips. A bijection can be constructedin a similar way, but with one minor tweak, as follows. Let M ( γ ′− , γ ′ + ; L, L τ ) + be the set of pairs ( u ′ , p + , ) with u ′ ∈ M ( γ ′− , γ ′ + ; L, L τ ) and p + , ∈ L satisfying ι ( p + , ) = ι ( γ ′ + (0)). Consider an element ( u, ℓ = ℓ ∐ ℓ ) from M (˜ γ − , ˜ γ + ; ι , ι τ ).Write ˜ γ ± = ( γ ± , p ± , , p ± , ) with γ ± chords in M from L to L τ , and p ± , , p ± , ∈ L satisfying ι ( p − , ) = γ − (0) , ι ( p + , ) = γ + (0) , ι τ ( p − , ) = γ − (1) , ι τ ( p + , ) = γ + (1). Let γ ′± be the lift of γ ± with γ ′± (1) = Π − τ ( p ± , ). Then u can belifted to get the element u ′ ∈ M ( γ ′− , γ ′ + ; L, L τ ). Note that it is not necessarilytrue that γ ′− (0) = p − , and γ ′ + (0) = p + , (this is the reason for encoding theextra data p + , ). The correspondence M (˜ γ − , ˜ γ + ; ι , ι τ ) → M ( γ − , γ + ; L, L τ ) + is given by ( u, ℓ ) ( u ′ , p + , ).Conversely, given an element ( u ′ , p + , ) ∈ M ( γ ′− , γ ′ + ; L, L τ ) + , let u = Iu ′ , ℓ ( s ) = Π u ′ ( s, ℓ be defined by ℓ (+ ∞ ) = p + , and ιℓ ( s ) = ιu ′ ( s, γ − = ( Iγ ′− , ℓ ( −∞ ) , ℓ ( −∞ )) and ˜ γ + = ( Iγ ′ + , p + , , ℓ (+ ∞ )). Then ˜ u =( u, ℓ = ℓ ∐ ℓ ) is an element of the moduli space M (˜ γ − , ˜ γ + ; ι, ι τ ). We thus getbijections M ( γ ′− , γ ′ + ; L, L τ ) + ∼ = M (˜ γ − , ˜ γ + ; ι, ι τ ).The fifth step is to show that the moduli spaces M (˜ γ ; { ι χ ( λ ) } λ ) are regularfor generic data, and to explain why Gromov compactness holds. (For themoduli spaces of strips both of these things are standard because there are nomoving Lagrangian boundary conditions.) For regularity, note that an element( u, ℓ ) in the moduli space will be smooth up to the boundary, even at the pointin the boundary where the corresponding Lagrangian boundary condition ι χ ( λ ) collapses onto ι , because of the existence of ℓ . Hence the moduli space can beset up in the usual functional analytic framework and standard methods implythat it will be regular. In particular, it will be a smooth manifold of dimensionInd ˜ γ . Note that the moduli space M ( γ ′ , { L χ ( λ ) } λ ) will also be regular. Thereason is that a curve is regular if and only if the image of the linearized operatoris surjective; the map I induces an obvious isomorphism between the linearized61perators of corresponding curves in the two moduli spaces, hence a curve isregular if and only if its corresponding curve is regular.Now consider Gromov compactness. Let ( u n , ℓ n ) be a sequence of curves in M (˜ γ, { ι χ ( λ ) } λ ). They can be lifted to a sequence of curves u ′ n in M ( γ ′ , { L χ ( λ ) } λ ).By usual Gromov compactness for embedded Lagrangians, the sequence u ′ n hasa subsequence which converges to an element u ′∞ ∈ M ( γ ′ , { L χ ( λ ) } λ ) along witha broken strip (with bottom boundary on L , top boundary on L τ ) connecting γ ′ to γ ′ . By exactness there are no disc components. The curve u ′∞ can be pusheddown to give a curve ( u ∞ , ℓ ∞ ) ∈ M (˜ γ ; { ι χ ( λ ) } λ ). Similarly, the broken stripcan be pushed down to get a broken strip in M on ( ι, ι τ ). Note that in thissituation there is no ambiguity for the choice of the lift to L of the bottom of thebroken strip, because the right-most point of the lift needs to equal ℓ ∞ ( −∞ ).The sixth and final step is to complete the proof. Since the rest of the de-tails are standard (with the help of the positivity condition), we will only saya few words. Define e ,τ ∈ CF( ι , ι τ ) analagously to equation (34) by countingelements of the moduli spaces M (˜ γ ; { ι χ ( λ ) } λ ) with Ind ˜ γ = 0, and similarly e τ, ∈ CF( ι τ , ι ). (The definition of e τ, requires setting up moduli spaces sim-ilar to the ones explained above, but with the curves connecting generators ofCF( ι τ , ι ) instead of CF( ι, ι τ ).) By Gromov compactness and regularity for 0dimensional moduli spaces, e ,τ and e τ, are well-defined. By Gromov com-pactness and regularity for 1 dimensional moduli spaces, these elements are m closed. To deduce that m ( e ,τ , e τ, ) is cohomologous to the unit e , ∈ CF( ι, ι ),one uses a gluing theorem to conclude that m ( e ,τ , e τ, ) is a count of elementsfrom moduli spaces M (˜ γ ; { ι χ ( λ ) } λ ), with χ determined by concatenation ofprevious boundary conditions, and satisfying χ ( ±∞ ) = 0. Then one consid-ers families of moduli spaces M (˜ γ ; { ι χ r ( λ ) } λ ) parameterized by r ∈ [0 , χ ≡ 0. The moduli space with r = 0 defines the unit e , , and hence the familyof moduli spaces can be used to show that m ( e ,τ , e τ, ) is cohomologous to e , .A similar construction shows that the unit e τ,τ ∈ CF( ι τ , ι τ ) is cohomologous to m ( e τ, , e ,τ ). In this case, χ ≡ τ , and χ r ( ±∞ ) = τ for all r .This is essentially the end of the proof but we need to point out a fewsubtleties about the moduli spaces M (˜ γ ; { ι χ r ( λ ) } λ ) before we are done. Thefirst point is that we need to use the lifting procedure described in step fourabove in order to prove Gromov compactness (again, this is because of movingLagrangian boundary conditions). In the case where the moduli space describescurves attached to a generator of CF( ι, ι ), there is no difficulty in uniquelylifting a curve. However, in the case that a curve ( u, ℓ ) attaches to a generatorof CF( ι τ , ι τ ), it is not immediately clear that a lift u ′ which, near z , has topboundary on L τ will also have bottom boundary on L τ . The reason this is notclear is that I − ( L τ ) contains L τ as a component, but also has several othercomponents. However, the existence of ℓ and the contractibility of the domainof u imply that this will not be a problem.The second point is that in order to apply the lifting procedure (to proveGromov compactness) we need to ensure that curves in M (˜ γ ; { ι χ r ( λ ) } λ ) stayinside the Weinstein neighborhood W . Before, we showed this was the case byarguing that the energy given by equation (42) could be made small by taking62he parameters to be small. In the present situation, the term h ( χ ( λ ) , · ) ˙ χ ( λ )appearing in the last integral of (42) is replaced by a function j : L × ∂S → R satisfying d ( j ( λ, · )) = ι ∗ χ r ( λ ) ω ( ∂∂λ ( ι χ r ( λ ) ) , · ). Thus, by defining χ r appropriately,the energy can still be made small. In Section 8.3 we showed that when ι : L → L is a covering space, the Floertheory of ι can be interpreted as the Floer theory of ( L, E ι ), where L is viewedas an embedded exact Lagrangian submanifold of M and E ι is a local systemcorresponding to the cover ι . In particular, we did not require that immersedpoints are only double points.In fact, a similar consideration applies to any clean immersion ι ; in otherwords, we do not need to require that the preimage of a point x ∈ L consists ofonly one or two points. A more general definition of clean immersion than thatgiven in Definition 8.1 is the following: ι : L → L ⊂ M is a clean immersion if ι is an immersion and the fiber product L × L L = { ( p, q ) ∈ L × L | ι ( p ) = ι ( q ) } is a smooth submanifold of L × L such that ι ∗ ( T ( p,q ) L × L L ) = ι ∗ T p L ∩ ι ∗ T q L whenever ( p, q ) ∈ L × L L . Note that L × L L always contains a distinguisheddiagonal component which is diffeomorphic to L (regardless of whether or notthe fiber product is a manifold). The union of the other components is R .The Gromov compactness theorem proved in [14] holds for this more generaldefinition of clean immersion.First consider the case that R is 0-dimensional. The proof of Lemma 3.1can be easily modified to show that L is totally geodesic for some metric. Thenall other proofs go through in the same way to show that the A ∞ algebra(CF( ι ) , m k ) is well-defined. Note that if x ∈ L is a singular point with d branchesgoing through it, then R has d ( d − 1) elements which map to x under ι .If R is not 0-dimensional and is more complicated than the type consideredin Section 8 it is not clear that L is totally geodesic for some metric. Thus morework needs to be done to set up the Banach spaces of maps; perhaps it can bedone in some way by using families of metrics which are domain dependent sothat only certain branches of L need to be totally geodesic at any given time.For any given immersion, if this difficulty can be overcome then (CF( ι ) , m k ) willbe well-defined. A Asymptotic Analysis In this section, we prove some asymptotic estimates that show holomorphiccurves have appropriate decay on the strip-like ends for Type II marked points.The main result is Theorem A.4, which is used in Propositions 3.7 and 4.4. Theresult is also needed for the corresponding results in the clean intersection case(Section 8). The hard analysis needed in the proof of Theorem A.4 is a W ,p -estimate for holomorphic curves with boundary on an immersed Lagrangian;63uch an estimate is proved in [14].We begin with a lemma that relates estimates on a disc to estimates on astrip. Let S + = R + × [0 , ⊂ C be the half strip, and let D + = { w ∈ C | Im w ≥ , | w | ≤ } be the upper half unit disc. Let ϕ : S + → D + \{ } be thebiholomorphic map defined by w = ϕ ( z ) = e − πz . Then ϕ − ( w ) = − π log( w ) . Lemma A.1. Suppose a function f : S + → R satisfies (cid:13)(cid:13) f ◦ ϕ − (cid:13)(cid:13) W ,p ( D + ) < ∞ for some p > . Assume f ◦ ϕ − (0) := lim z → f ◦ ϕ − ( z ) = 0 . Then k f k W ,p ; δ ( S + ) < ∞ for any δ ∈ (0 , ( p − π ) .Proof. By the Sobolev embedding theorem, f ◦ ϕ − ∈ C ,µ ( D + ) with µ =1 − p > 0. Since f ◦ ϕ − (0) = 0 , one has (cid:12)(cid:12) f ◦ ϕ − ( w ) (cid:12)(cid:12) ≤ C | w | µ for someconstant C > . Hence | f ( z ) | ≤ C | ϕ ( z ) | µ = Ce − πµs , where s = Re z. Then Z S + | f ( z ) | p e δs · √− dz ∧ d ¯ z ≤ Z S + Ce ( − πpµ + δ ) s · √− dz ∧ d ¯ z < ∞ as long as − πpµ + δ < 0; that is, δ < πpµ = ( p − π. Also Z S + (cid:12)(cid:12)(cid:12)(cid:12) ∂f∂z (cid:12)(cid:12)(cid:12)(cid:12) p e δs · √− dz ∧ d ¯ z = Z S + (cid:12)(cid:12)(cid:12)(cid:12) ∂∂w ( f ◦ ϕ − ) ∂ϕ∂z (cid:12)(cid:12)(cid:12)(cid:12) p e δs · √− dz ∧ d ¯ z ≤ C ′ Z D + (cid:12)(cid:12)(cid:12)(cid:12) ∂∂w ( f ◦ ϕ − ) (cid:12)(cid:12)(cid:12)(cid:12) p | w | p − − δπ · √− dw ∧ d ¯ w ≤ C ′ (cid:13)(cid:13) f ◦ ϕ − (cid:13)(cid:13) pW ,p ( D + ) as long as p − − δπ > 0. A similar calculation holds for the L p ; δ ( S + )-norm of ∂f∂ ¯ z . Lemma A.2. There exists a constant C > such that the following is true:For any C function f : S + → R , p > and δ > , we have | f ( s, t ) | ≤ Ce − δ ( s − / ( p +2) k f k L p ; δ ( S + ) (1 + k∇ f k C ( S + ) ) for all s ≥ .Proof. If k∇ f k C = ∞ then the statement is obvious, so we may assume that ∇ f is C bounded. Let U = [0 , be the unit square in R . Consider a C function g : U → R . Suppose g is not constant. Let x ∈ U satisfy | g ( x ) | = k g k C > K = k∇ g k C > . Then for x ∈ U with | x − x | < | g ( x ) | K , we have | g ( x ) − g ( x ) | ≤ K | x − x | ≤ | g ( x ) | . Thus | g ( x ) | ≥ | g ( x ) | = k g k C . It followsthat Z U | g | p ≥ Z U ∩ B (cid:16) x , | g ( x | K (cid:17) p k g k pC ≥ p k g k pC min (cid:26) π , π (cid:18) | g ( x ) | K (cid:19) (cid:27) = 12 p k g k pC min (cid:26) π , π k g k C K (cid:27) . U ∩ B ( x , | g ( x ) | K ) contains at least a quarter circle ofradius either 1 or | g ( x ) | / K .) Hence if 0 < k∇ g k C = K ≤ K ′ , we have either k g k /pC ≤ /p · π /p ( K ′ ) /p k g k L p ≤ K ′ ) /p k g k L p or (43) k g k C ≤ · /p π /p k g k L p ≤ k g k L p . If g is constant (so k∇ g k C = 0), we have k g k C = k g k L p . (44)Now consider g n ( s, t ) := f ( s + n, t ) | U . By assumption, there exists K ′ := k∇ f k C ( S + ) < ∞ such that k∇ g n k C ≤ K ′ for all n . Also k g n k pL p ( U ) = Z [ n,n +1] × [0 , | f | p · √− dz ∧ d ¯ z ≤ e − δn Z [ n,n +1] × [0 , | f | p e δs · √− dz ∧ d ¯ z ≤ e − δn k f k pL p ; δ ( S + ) . Thus k g n k L p ( U ) ≤ e − δnp k f k L p ; δ ( S + ) and (43) and (44) (with g replaced by g n )give either k g n k p C ≤ K ′ ) /p e − δnp k f k L p ; δ ( S + ) or k g n k C ≤ e − δnp k f k L p ; δ ( S + ) . Thus, k g n k C ≤ C ′ (1 + K ′ ) e − δnp +2 k f k L p ; δ ( S + ) , and hence | f ( s, t ) | ≤ Ce − δ ( s − p +2 k f k L p ; δ ( S + ) (1 + k∇ f k C ) . Lemma A.3. There exists a constant C > such that the following is true:For any C function f : S + → R , p > and δ > we have: |∇ f ( s, t ) | ≤ Ce − δ ( s − / ( p +2) k∇ f k L p ; δ ( S + ) (1 + (cid:13)(cid:13) ∇ f (cid:13)(cid:13) C ( S + ) ) . Proof. Apply Lemma A.2 with f replaced with ∇ f. We now formulate the main result. Assume ι has clean self intersection (seeDefinition 8.1). Let u : S + → M be a map such that v = u ◦ ϕ − satisfies v x + J ( w, v )( v y − X K ( v )) = 0 , (45)65here w = x + √− y is the coordinate for D + \{ } . Here X K is a domain ( D + )dependent Hamiltonian vector field as in Section 7.3 (where we called it X K d +1 ).In particular, it is smooth at 0. We assume u also comes with a boundary lift l , i.e. ι ◦ l | ( R + × { j } ) = u | ( R + × { j } ), for j = 0 , 1. Notice that equation (45) isthe equation that the main component of a holomorphic polygon satisfies neara Type II marked point. Theorem A.4. Let u be as above. Then the following are equivalent.a) u has finite L -energy; that is, R S + | du − X K | < ∞ .b) There exists constants ǫ > and C such that k u k C ([ s, ∞ ) × [0 , ≤ Ce − ǫs . c) For some δ > , u ∈ W ,p ; δ ( S + , M ) for all p ≥ .Proof. Assume a). The proof of Theorem A in [20] shows that lim s →∞ u ( s, t ) = m for some m ∈ M , uniformly in t , and also lim s →∞ | ∂ s u ( s, t ) | = 0 uniformlyin t . Notice that in the proof we only need the standard Gromov’s Monotonicity(see Lemma 3.4 in [7] for example). Also Gromov’s graph trick can be used toget rid of the Hamiltonian term and make the complex structure be domainindependent (as in Chapter 8 of [16]).Let v = u ◦ φ − . By looking at the graph of v again we can get rid of theHamiltonian term X K and also make J domain independent. Then by Theorem1.4 in [14], v ∈ W ,p ( D + ) for some p > 2. Then Lemma A.1, Lemma A.2, andLemma A.3 give us b).b) implies c) is obvious, and c) implies a) is also clear because X K is smoothat 0 and hence R S + | X K | = R D + | X K | < ∞ . References [1] Mohammed Abouzaid, On the Fukaya categories of higher genus surfaces ,Adv. Math. (2008), no. 3, 1192–1235. 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