The Family Floer Functor is Faithful

8/11/2019 The Family Floer Functor is Faithful

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arXiv:1408

.6794v1

[math.SG]28Aug2014

THE FAMILY FLOER FUNCTOR IS FAITHFUL

MOHAMMED ABOUZAID

Abstract. Family Floer theory yields a functor from the Fukaya category of a symplec-

tic manifold admitting a Lagrangian torus fibration to a (twisted) category of perfectcomplexes on the mirror rigid analytic space. This functor is shown to be faithful by a

degeneration argument involving moduli spaces of annuli.

1. Introduction

Applications of Fukaya categories to symplectic topology require an algebraic model forthese categories: this involves of finding a collection of Lagrangians which generatethe cat-egory in the sense that the Fukaya category fully faithfully embeds in the category of perfectmodules over the corresponding A algebra. For closed symplectic manifolds, the knownstrategies for understanding such categories of modules rely on realising them, in an instanceof homological mirror symmetry, as modules over the endomorphism algebra of (complexesof) coherent sheaves on an algebraic variety, or a non-commutative deformation thereof.Such descriptions are possible in a limited class of examples, which include Calabi-Yau hy-persurfaces in projective space [14, 16] and toric varieties [11, 5]. It reasonable to expectthat these methods will lead to descriptions of Fukaya categories of complete intersectionsin toric varieties[4].

The goal of the family Floer program is to both give a more compelling proof of theseequivalences, and to extend the class of examples for which they can be proved. Keeping

with tradition, we shall call the symplectic side the A-side, and the algebro-geometric sidetheB-side. The current strategies rely on matching computations on the two sides, withouthaving a good reason for the agreement. Moreover, these computations work for a veryspecial class of symplectic structures; in the typical case of the K3 surface, homologicalmirror symmetry is only understood for the restriction of the Fubini-Study form to thequartic hypersurface, whereas the rank of the second cohomology group is 22.

There are essentially only two previous results on family Floer cohomology. In [9], Fukayagave a very general result constructing the local charts of theB-side, which were shown byTu to admit compatible identifications over the overlaps [18]. In[8, Section 6] Fukaya alsooutlined a strategy for assigning to Lagrangians (complexes of) coherent sheaves on themirror, under some convergence assumptions which should yield a complex analytic mirror.

This paper extends these results by (1) constructing a map of morphism spaces from theA-side to the B-side, (2) constructing a map of morphism spaces from the B-side to the

A-side, (3) showing that the composition of these two maps is the identity on the A-side,leading to the main result. The formal results are stated in Section8. The Appendix alsoincludes the construction of anA functor, which should be thought of as the family Floermirror functor.

Date: August 29, 2014.The author was supported by NSF grant DMS-1308179.

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2 M. ABOUZAID

x

y

FqFq

L

L

L

L

LFq LFq

Figure 1.

Remark 1.1. In order to focus on the new ideas, we restrict the setting that we considerby assuming that (1) the ambient symplectic manifold admits a Lagrangian torus fibrationall of whose fibres are smooth and bound no holomorphic discs, and (2) one can choose analmost complex structure for each Lagrangian so that it bounds no holomorphic discs. The

requirement that the Lagrangians bound no holomorphic disc is really only technical, andmeant to avoid discussing foundations of multivalued perturbations in Lagrangian Floertheory [10] (and multiplying the length of the paper by a potentially large factor). Thereader may consult the introduction to [2] for a discussion of the more serious difficultiesone would encounter in the presence of singular fibres.

Since the construction of the homotopy from the composition to the identity uses amoduli space of annuli, faithfulness can be seen as the analogue of the generation criterion[1]. Heuristically, the strategy for the proof is the following: letXbe a symplectic manifoldequipped with a Lagrangian torus fibration over a base Q (we denote the fibre over qQby Fq), and L a Lagrangian in X. Consider moduli spaces of holomorphic discs with 3marked points, on which we impose Lagrangian boundary conditions given byL and a fibre.We shall consider two flavours for this moduli space (see the leftmost diagram in Figure

1): in the first case, one marked point is distinguished as an input mapping to L, and theremaining two are outputs mapping to intersection points x and y ofL with a fibre, while inthe second case, the intersections ofL with a fibre correspond to inputs, while the markedpoint on L is an output.

In the classical versions of Floer theory, one would consider the subcategory of the Fukayacategory ofXwhose objects are fibres, and the Yoneda module over this subcategory as-sociated to L. Upon adding many more marked points, the first of these moduli spacesdefines the map from the Floer cohomologyL to the endomorphism algebra of this Yonedamodule, and the second moduli space defines a map which one could hope to show is a rightinverse by gluing the two triangles to an annulus, and degenerating this annulus to two discsmeeting at an interior point; one of the discs has Lagrangian boundary conditions on anarbitrary fibreand the other has Lagrangian boundary conditions on L and carries the twoboundary marked points. Since the moduli space of discs with boundary on an arbitrary

fibre gives us a copy of the ambient space X, the first type of disc imposes no constraint, sowe are simply considering the moduli space of discs with boundary on L (and two markedpoint). This moduli space represents the identity on Floer cohomology.

Trying to implement this strategy in this setting runs into a convergence problem: sincethe fibres are disjoint, they are Floer theoretically orthogonal, so the Yoneda module definedby L is a direct product of the corresponding modules for all fibres. The map back toFloer theory is not well-defined because it is the sum of infinitely many terms. The correct


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THE FAMILY FLOER FUNCTOR IS FAITHFUL 3

framework for this argument is in fact family Floer cohomology, and the main difficulty thatarises is due to the need to make compatible families of perturbation in defining the Floercohomology ofL with every fibre; in the classical case, one can choose such perturbations

independently for all pairs of objects.We end this introduction with a brief outline of the paper: Sections2 and3introduce

the abstract moduli spaces and the corresponding spaces of maps which will serve to con-struct the mirror to a Lagrangian and the maps back and forth from the Floer complex tothe morphism spaces on the B-side. Familiarity with the ideas explained in [2] may helpunderstand what are otherwise likely to appear as unmotivated constructions.

Section 4 is the heart of the paper, ensuring the compatibility of the Floer theoreticconstructions which yield the maps from theA-side to theB-side and vice versa; in the familyFloer setting, this is particularly delicate because the Lagrangian boundary conditions arevarying over the fibres, and one cannot choose the data arbitrarily for different fibres. Thekey idea it to choose a very fine triangulation of the base of the fibration, make controlledchoices at the vertices of this triangulation, and associate to higher dimensional cells familiesof equations which interpolate between these.

Sections 5 and6 show that the moduli spaces constructed in the previous section canbe glued to obtain a moduli space of annuli parametrised by the base. The main delicatepoint is that it is not possible to perform this construction in such a way that the annuliover every point in the base are obtained by gluing the degenerate annuli corresponding tothat point. This is responsible for the notion of an annulus gluing function introduced inDefinition6.1.

Sections7 and8 are simply a matter of bookkeeping, as all the hard work appears inthe previous four sections. The results of[2], to which the reader is referred, are repeatedlyused. Finally, in the Appendix, the A functor is constructed. This entails giving mapat the linear level between morphisms on the A-side and the B-side, whose construction isquite a bit simpler than the construction in the main part of the paper. The additionalcomplexity of the papers main construction comes from the need to see such a map arisein the boundary of a moduli space of annuli.

Acknowledgments. Discussions with Denis Auroux, Kenji Fukaya, Paul Seidel, and IvanSmith, exploring potential applications of family Floer cohomology, were helpful in justifyingthe development of these techniques.

The author was supported by NSF grant DMS-1308179.

2. Families of Riemann surfaces

2.1. Adamss universal curve.

2.1.1. Adamss family of paths. In [6], Adams constructed a family of paths in the d-simplexfrom the initial to the terminal vertex which are parametrised by the d 1-cube; see Figure2. To give an explicit formula for this family, let Ad= [0, 1)

d1 and Ad = [0, 1]d1, introducethe polytope

(2.1) Td =

(r1, . . . , rd1, s)AdR|0s

ri+ 1


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4 M. ABOUZAID

0 1

2

Figure 2.

and denote by sj the difference s j1

i=1 rj . One model for the Adams map is:

a :

Td

d

R

d+1(2.2)

a(r1, . . . , rd1)(s) =

(1 s1, s1, . . . , 0) 0s1r1((1 s2)(1 r1), (1 s2)r1, s2, . . . , 0) 0s2r2 ((1 sd)

d1i=1 (1 ri), (1 sd)

d1i=2 (1 ri)r1,d1

i=3 (1 ri)r2, . . . , (1 sd)rd1, sd) 0sd1.

(2.3)

LetTd denote the complement inTd of the inverse image under a of the vertices of d;i.e. remove the locus wheres = 0 or some coordinate sj equals 1. Denote byTr the fiberover a pointr Adof the projection mapTd Ad. This is a union of intervals, numberingone more than the coordinates in r which equal 1.

Some compatibility between the spacesTd for different values ofd will be required. Tothis end, letKbe a finite ordered set, and denote by Kthe associated |K|1-dimensionalsimplex which naturally embeds in RK.Definition 2.1. The compactified Adams moduli space of paths AK is the cube on K\{min K, max K}.

The associated partially ordered set consists of pairs of subsets I andJ ofK such that

(2.4) {min K, max K} IJThe partial ordering is such that the pairIJprecedesIJ wheneverIIJJ.Consider also the open subset AK AKcorresponding to the inclusion [0, 1) [0, 1].

There is an alternate description of the boundary strata which is often more useful:introduce the notation

(2.5) Ki ={jK|ji}and Ki ={jK|ii}.Lemma 2.2. The boundary stratum corresponding to I J admits a natural product de-composition

(2.6) AIJ=AJid AJidJid1 AJi1Ji0 AJi0 ,

whereI={min K, i0, i1, . . . , id, max K}.


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6 M. ABOUZAID

0202 012012

0

2

0

2

0

1

2

Figure 3. The moduli spaceU012 over A012.

Definition 2.4. IfR < Si for all i, the R-thick part ofT# S is the image of

(2.16) [R, R][R, R] [R, R].Inductively choose identification of the fibre ofTK over a point r with unions of real

lines

(2.17) R

R Trand neighbourhoods of the boundary strata

(2.18) AIJ AK,together with a gluing mapTfIJr#gI(r) Tr wheneverr lies in AIJ. The existenceof such a gluing map yields a map

(2.19) fIJTK TK|AIJ.Since these spaces of choices are contractible, the commutativity of Diagrams (2.12) and

(2.14) implies that such neighbourhoods and identifications may be chosen so that thereare commuting identifications of fibres whenever I I J J in the following sense:the product decomposition of the boundary strata ofAKyields a map gI\I from AIJ to(0, ]I\I which fits in a commutative diagram of diffeomorphisms(2.20) TfIJr#gI(r)

TfIJfIJr#gI\I(fIJr)#gI(r)

Tr .

Definition 2.5. Theuniversal curve over AK is the map

(2.21) UK TK [0, 1] AK.

The fibresUr are equipped via the choice in Equation (2.17) with identifications(2.22) Ur= R [0, 1]

R [0, 1] . . .R [0, 1],

hence with a complex structure which in the right hand coordinates (s, t) is j s = t oneach component. Denote by

(2.23) U{0}K UK U{1}K


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the two copies ofTK embedded inUKas the product of the real line with the boundary ofthe interval. Denote by BB = R [0, 1] the half-strips:

(2.24) B+ = [0, +) [0, 1] and B= (, 0] [0, 1],and fix maps with disjoint images

min(K) : AK B+ UK(2.25)max(K) : AK B UK(2.26)

whose restrictions to r AK give positive (resp. negative) strip like ends on the fibreUr , which agree up to translation with the natural ones coming from the last (respectivelyfirst) factor in Equation (2.22), and which are compatible with the gluing maps near everyboundary stratum.

Let F:UJ Z be a map fromUJ to a topological space Z. Such a map is constantalong the positive (respectively negative) end if there is a map f: [0, 1]

Z such that

the restriction of F to each fibreUr agrees with f near s = + on the last component(respectively nearon the first component).

The choices of gluing maps for the moduli spaceTKyield a map

(2.27) GIJ: fIJUK UK|AIJ

for all pairs IJlabelling boundary strata ofAK.Definition 2.6. The mapF isobtained by gluing if its restriction to neighbourhoods of allboundary strata yields a commutative diagram

(2.28) fIJUK

UK|AIJ

UK|AIJ Z.

Since the gluing map GIJ is surjective, F is determined, on a neighbourhood of aboundary stratum, by its restriction to the stratum and the gluing map. In addition,continuity implies that its restriction to the boundary is constant along each glued end.

The notion of an R-thick part of the image ofGIJ is inherited from Definition2.4: thisconsists of|I| 1 components, each of which is labelled by a pair of successive elements ofI, and is naturally identified with [R, R] [0, 1]. Let FIJ: AIJ Z be a functionobtained by gluing, and assume Z is a Frechet manifold. We shall say that a section ofFIJT Zis consistent if there is a constant R so that the support is contained in the interiorof the R-thick part and the restriction to the ith component of the R-thick part vanishes

to infinite order along the strata for which the corresponding pair of gluing parameters areinfinite. Aconsistent perturbationofFIJis the image under exponentiation of a consistentsection.

Definition 2.7. A map F:UK Z is obtained byperturbed gluing if its restriction toAIJ for all pairs I Jagrees with a consistent perturbation of a function obtained bygluing.


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8 M. ABOUZAID

2.2. Adams paths in a prism. Equip K {, +} with the total ordering obtained byextending the ordering on K via (+, i) < (, j) for all i, j K. Write every subset ofK {, +} as a union K {} K+ {+}. The constructions of the previous section,applied to this ordered set, yields spaces and maps

(2.29) UK,K+ TK,K+ a

K,K+

AK,K+ ,

and an open subset AK,K+ AK,K+ over which the fibres ofUK,K+ are connected.There are inclusions of the boundaries of each fibreU{i}K,K+ UK,K+ fori = 0, 1.

The additional data of the decomposition yields a natural map

(2.30) K,K+[0, 1].The fibre of

TK,K+ over AK,K+ is therefore equipped with a distinguished component

characterised by the property that its projection to [0, 1] is not constant. To state thecompatibility of this distinguished component with the boundary decomposition ofAK,K+ ,note that such a stratum is labelled by pairsI andJ such that

min K+I+J+K+(2.31)max KIJK.(2.32)

Whenever I are singletons (i.e. respectively consist only of min K+ and max K), thecorresponding boundary stratum is naturally identified with the moduli space AJ,J+ .

Lemma 2.8. The codimension1 boundary strata ofAK,K+ are:

iK\max KAK\i,K+(2.33)

iK+\minK+

AK,K+\i(2.34)

iK\max K

AK,i

AK,i,K+

(2.35)

iK+\minK+

AK,K

+,i

AK+,i

(2.36)

The restriction ofUK,K+ to these strata is naturally isomorphic to the union of pullbacksof the universal curves on each factor.

Using the above description of the boundary strata, inductively choose identifications of

fibres with unions of real lines, compatibly with the choices made in Section 2.1 forUJ fora subset J ofK; note that these moduli spaces appear in the fibres over Equations (2.35)and (2.36), and the compatibility condition is that the restriction of the identification tosuch a boundary stratum agrees with the one previously made. Choose families of positive(respectively negative) strip like ends

: AK,K+ B UK,K+(2.37)


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whose restriction to the boundary strata are compatible with the inductive choices, andthose made in Section2.1.

Finally, in the choices of identification made in Equation (2.17) as part of the gluing

procedure, extend the coordinates of the component distinguished by the map in Equation(2.30), and consider the resulting an open embedding

(2.38) : AK,K+ B UK,K+which is a diffeomorphism over AK,K+ . On each fibre ofUK,K+ , let z1 be the image of(0, 1) under .

Assume now that max K+min K. Letzin denote the boundary marked pointz1. ForeachrK,K+ fix the following positive strip like ends near zin

in : B+B Ur(2.39)(s, t) 1 est

1,(2.40)

where the complex coordinates on B are given by its embedding in C. By construction,

these strip-like ends are compatible with gluing. SinceU{1}K,K+ is naturally ordered via itsidentification with a union of real line, the points preceding or succeedingzin define subsets

of the boundary:

(2.41) UzinK,K+

.

Assuming that max Kmin K+ repeat the same procedure to obtain a marked pointzou on each fibre ofUK,K+ , and pick negative strip-like ends

ou : BB Ur(2.42)whenever rK,K+which are compatible with gluing. The points preceding or succeedingzou yield subsetsUzouK,K+ .

Given IK, introduce the notation(2.43) K

IK

max I

andKI

Kmin I

.

where the sets Ki andKi are as in Equation (2.5). Given a nested pair IK, define the

following subsets ofK {+, }:KinI (KI{}, KI {+})(2.44)KouI (KI{}, KI {+}).(2.45)

Lemma 2.9. The minimal element of KinI is (min K, +), and the maximal element is(max K, ). The minimal element ofKouI is(max I, +)and the maximal element is(min I, ).

2.3. Strips with one input marked point. Let Kbe a nested sequence of totally orderedsets whose maximal element is Kand minimal element is K0.

Definition 2.10. TheAdams moduli space with one input A K;in is the cube

(2.46) A K;in[0, 1]K\K.

The cells ofA K;in are given by pairs of subsets K I J K, and write AI J;infor the corresponding stratum. Define the open subset A K;inA K;in corresponding to theinclusion [0, 1)[0, 1]. It is the union of the strata for which{K}= I.


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10 M. ABOUZAID

max K max J min J min K

Figure 4. A fibre of the universal curve over AKJ ;in

AKJ

.

To define the universal curve over this space, let min K Kdenote the set of minimalelements of subsets ofK lying in K, and max Kthe set of maximal elements. Assign to a

sequence J Ka subset ofKinK0(2.47) in J(max J {}, minJ {+}).The map of posets I Jin I in Jinduces a natural map(2.48) in : A K;in AKinK0 .

Definition 2.11. Theuniversal curve over A K;in is the projection map

(2.49) UK;inin(UKinK0 )A K,in.Similarly defineT K;in. By construction, these spaces are equipped with natural maps

(2.50) UK;in T K;ina

K;in

A K;in,

where, for the sake of consistency, the notation K;inKinK0 is used.By construction, the following diagram commutes:

(2.51) AI J;in

A K;in

AinIinJ

AKinK0

To describe the restriction ofUK;in to the boundary strata, it is convenient to introducemore notation: considering K as a totally ordered subset (with respect to inclusion), the

natural maps from Kto min Kand max Krespectively reverse and preserve ordering. Thisdefines maps min and max from A K to AK, and pullback along these maps yields moduli

spaces over A K

(2.52) minUmin K

maxUmax K.From the choices made in Section 2.1, the fibres are identified with unions of strips. Fixthe induced family of positive strip-like ends + with values in maxUmax K, and negativestrip-like ends with values in minUmin K.

Given an element J K, write KJ and KJ as before for the elements preceding andsucceedingJ. The boundary stratum ofA K;in labelled by KI J is(2.53) AIJ;in

= AIJI0

AJI0

;in,


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where the first factor is the cube on JI0 \ I, and maps by min and max toA

min Imin JI0 AK

I0

(2.54)

Amax Imax JI0 AKI0 ,(2.55)where the fact that K Jis used. The second factor in Equation (2.53) maps by in to(2.56) A

max JI0,min JI0

AI0J0

,I0J0

AI0 inJ0 .

The product of the right hand sides in Equations (2.54)-(2.56) is a stratum ofAKinK0, and

there is a commutative diagram

(2.57) AIJ;in=AIJ

I0

AJI0

;in

A K;in

A

K

I0 A

I0in

J0 A

K

I0

AK

in

K0

The pullback of the universal curves on the three factors of the space at the bottom left isthen naturally isomorphic to the restriction ofUK;in.

Equations (2.33)-(2.36) yield a more explicit description of the codimension 1 strata:

Lemma 2.12. The boundary ofA K;in is covered by the following codimension1 strata:IK\{K}

A K\{I};in(2.58)

IK\{K}

A KI

A KI

;in(2.59)

The restriction of

UK;in to the first stratum is naturally isomorphic to

UK\{

I}

;in, and the

restriction to the second is given bymaxU

max KI

minU

min KI

A KI ;in

A KI

UKI ;in.(2.60)Proof. Equation (2.58) corresponds the case when the coordinate labelled by I vanishes,and the other type of boundary stratum to the case this coordinate equals 1. In Equation

(2.60), the projection from KinK0 toKcan be used in order to identify maxKI {}, as a

subset ofKinK0 with maxKI , and similarly for min K

I .

Pulling back Equations (2.37)-(2.38), defines strip-like ends

: A K,in B UK,in(2.61)as well as an open embedding

(2.62) : A K,in B UK,inwhose restriction to each point in A K,ingives a distinguished component in the correspond-

ing fibre.

Define subsetsU{i}K;in UK;in fori ={0, 1}, and a decomposition ofU{1}K;in

intoUzinK;in corresponding to the points preceding and succeedingzin.


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12 M. ABOUZAID

0 0

0 0 0 2 0

0 2 1 00 2 1 0

Figure 5. The moduli space A K;ou, with vertices labelled by fibres of

UK;ou ( K= 002012).

2.4. Strips with one output marked point. The construction of moduli spaces withoutputs is entirely analogous to that of the moduli space with inputs, replacing the partiallyordered setKinK0 byK

ouK0

in the construction, and a few other minor changes.

Definition 2.13. TheAdams moduli space with one output, A K;ou is the cube

(2.63) A K;ou[0, 1]K\K0 .

The cells ofA K;ou are given by pairs of subsetsK0I J K, and the union of thosestrata for which I=K0 correspond to the open subset A K;ou.

There is a natural map

(2.64) ou : A K;ou AKouK0which assigns to a sequence J Kthe pair (min J {}, maxJ {+}).

Definition 2.14. Theuniversal curve over A K;ou is the projection map

(2.65) UK;ouou(UKouK0 ) A K,ou.

Similarly defineT K;ou. By construction, these spaces are equipped with natural maps

(2.66) UK;ou T K;oua

K;ouKouK0

A K;ou.

By pullback, Equation (2.37) determines families of strip like ends and+, and a familyof distinguished components which are trivialised by a map we still denote .


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As in Equation (2.57), the boundary strata ofA K;ou and their images under ou fit in a

commutative diagram:

(2.67) AIJ;ou = A JI ;ouAIJI

A K;ou

A

IK0AJouI AIK0

AKouK0.

For codimension 1 strata, the boundary ofA K;ou admits a natural decomposition as aunion

IK\K0A K\I;ou(2.68)

IK\K0A KI ;ou

A KI ,(2.69)

and the fibres ofUK;ou are given byUK\I;ou and

A KI ;ou

maxUmax K

I

minU

min KI

UKI ;ou A KI .(2.70)

Pullback defines strip like ends , an embedding ou of a strip into a distinguishedcomponent of each fibre, and subsetsU{i}K;ou UK;ou for i ={0, 1}, corresponding to theboundaries of each fibre over A K;ou. These fibres are equipped with a marked pointzou, and

the ordering onU{1}K;ou yields a decompositionUzou

K;ou which corresponds

to points preceding or succeeding the marked pointzou.

3. Lagrangian Floer theory

3.1. Geometric assumptions. Let (X, ) be a closed symplectic manifold, and : XQa Lagrangian fibration. Let H and D be the space of Hamiltonian diffeomorphisms and alldiffeomorphisms ofX. Let J denote the space of-tame almost complex structures on X.Assume that2(Q) = 0 which excludes the presence of holomorphic spheres in X. For anyJ J, there is a natural isomorphism of vector bundles T X=(T Q) R C. In particular,there is a natural quadratic complex volume form onT Q.

Assume that all Lagrangians are tautologically unobstructedin the sense that

(3.1) there existsJL J so thatL bounds noJL-holomorphic discs.and are graded with respect to , i.e. the mapL RP1 induced by is null-homotopic,and that a lift of this null-homotopy to the universal cover ofRP1 is fixed.

Whenever L and L are both graded Lagrangians, and x

L

L is a transverse inter-

section point, there is a well-defined Maslov index deg(x) Z as explained in [15, Section(12b)].

3.2. The Floer equation. Given a point qQ and a Lagrangian L, pick a Hamiltoniandiffeomorphism so thatL which is transverse to Fq. A familyJ={Jt J}t[0,1] so that(3.2) J1 = (JL)


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14 M. ABOUZAID

yields a holomorphic curve equation onB with Lagrangian boundary conditions

u : BX su= Jttu(3.3)

u(s, 0) Fq u(s, 1) L.(3.4)Given a pair of points (x, y)L Fq, denote byMq(x, y) the Gromov-Floer compact-

ification of the quotient by translation in the R factor of B of the space of solutions toEquations (3.3) and(3.4) which in addition satisfy the asymptotic conditions:

(3.5) lims

u(s, t) = x lims+

u(s, t) = y.

For genericJ,Mq(x, y) is a manifold with boundary such that(3.6) dimRMq(x, y) = deg(x) deg(y) 1,and the codimension 1 strata of the boundary are given by the union

(3.7) x

LFq

Mq(x, z) Mq(z, y).

IfL and Lare graded Lagrangians which are transverse, and which both satisfy Condition(3.1), pick a family Jt of almost complex structures such that J0 = JL and J1 = JL , andwhich is constant in a neighbourhood of the point t = 1/2. To each pairx, yL L, therecorresponds a moduli spaceM(x, y) with boundary conditions given by L along R {0}and L along R {1}. Equation (3.6) holds in this case as does the analogue of Equation(3.7). Finally, defineK(x, y) to be the compactified moduli space of solutions to the Floerequation with one interior marked point lying on the interval R{1/2}. This moduli spacecan be thought of as the continuation moduli space for a constant Hamiltonian family (seethe next Section), and is the product of a closed interval withM(x, y), unless x = y, inwhich it case it is a point, corresponding to the constant solution.

3.3. Continuation maps. LetKbe a totally ordered set.

Definition 3.1. A consistent family of continuation data parametrised byUK is a map(3.8) K= (K, JK, K) :UK H JDsuch that (i)Kis constant along each end of a fibre ofUK, (ii) the mapsK andK, areobtained by gluing, andJKby perturbed gluing, and (iii) for eachz U{1}K(3.9) JK(z) = (K(z) K(z)) JL.

Remark 3.2. Only the restrictions ofK and K toU{1}K shall be used, so the reader mayassume for simplicity that these maps (but not J) factor via the projectionUK TK.

From such consistent families, one can define, for each r AK, a holomorphic curve

equation with moving Lagrangian boundary conditions:u :UrX su(z) = JK(z)tu(z)(3.10)

u(z)Fq ifz U{0}r u(z)K(z)L ifz U{1}r .(3.11)

Given a pair of points xLminK Fq andyLmaxK Fq , denote by(3.12) MK(x, y)AK


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the moduli space of solutions to Equations (3.10)-(3.11), which is addition satisfy the as-ymptotic conditions:

(3.13) lims

u

(s, t) = x lims+

u

+

(s, t) =y.

For a generic family JK, the Gromov-Floer compactificationMq,K(x, y) is a manifoldwith boundary such that

dimRMq,K(x, y) = deg(y) deg(x) + dim(AK)(3.14)= deg(y) deg(x) + |K| 2.(3.15)

The generalisation of Equation (3.7) is that the codimension 1 strata of the boundary comein two families:

iK

zLiFq

Mq,Ki

(x, z) Mq,Ki (z, y)(3.16)

iK\{minK,max K}Mq,K\{i}(x, y).(3.17)

Breaking of Floer strips is incorporated in the first case, corresponding to i = max K ori = min K. So the first strata project to the interior ofAK or to the boundary stratumwhere the cube coordinate corresponding to i vanishes, and the second to the case wherethis coordinate equals 1.

Remark 3.3. Since the choice of data in Definition3.1 takes place for a fixed K, there is aslight abuse of notation in Equations (3.16) and (3.17), since the moduli spaces that appearare defined with respect to the Floer data restricted fromUK. This issue will be addressedin Section4, where such data will be chosen inductively so that there is no ambiguity in thedescription of the boundary strata.

3.4. Continuation in a prism. Let L+

and L be graded Lagrangians which are trans-verse, and which both satisfy Condition (3.1) for almost complex structures JL+ and JL .Let K, K+ be subsets ofK {, +}as in Section2.2.Definition 3.4. A consistent family of continuation data in a prism parametrised byUKis a map

(3.18) K,K+ = (K,K+ , JK,K+ , K,K+ ) :UK,K+ H JDsuch that (i) the restriction to every end is constant, (ii) JK,K+ is obtained by perturbedgluing and the other maps by gluing in a neighbourhood of each boundary stratum, and

(3.19) JK,K+ (z) =

K,K+ (z) K,K+ (z)

JL+ ifz U

z1K,K+ .

We obtain, for each r AK,K+ , a holomorphic curve equation su(z) = J(z)tu(z)with moving Lagrangian boundary conditions

u :UrX u(z) Fq ifz U{0}r(3.20)u(z) (z)L ifz Uz1>r u(z) (z)L+ ifz U

z1

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The Family Floer Functor is Faithful