Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

  • Upload
    swertyy

  • View
    223

  • Download
    0

Embed Size (px)

Citation preview

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    1/32

    arXiv:0905

    .1369v2

    [math.SG]13Aug2010

    PSEUDOHOLOMORPHIC QUILTS

    Katrin Wehrheim and Chris T. Woodward

    We define relative Floer theoretic invariants arising from quilted pseudoholomorphic sur-

    faces: Collections of pseudoholomorphic maps to various target spaces with seam conditions

    in Lagrangian correspondences. As application we construct a morphism on quantum homology

    associated to any monotone Lagrangian correspondence.

    Contents

    1. Introduction 12. Invariants for surfaces with strip-like ends 3

    3. Invariants for quilted surfaces 11

    4. Independence of quilt invariants 22

    5. Geometric composition and quilt invariants 23

    6. Application: Morphism between quantum homologies 26

    References 31

    1. Introduction

    Lagrangian Floer cohomology associates to a pair of Lagrangian manifolds a chain complex whosedifferential counts pseudoholomorphic strips with boundary values in the given Lagrangians. Theseform a receptacle for relative invariants defined from surfaces with strip-like ends, see e.g. [15]. Thesimplest instance of this invariant proves the independence of Floer cohomology from the choicesof almost complex structure and perturbation data. More complicated cases of these invariantsconstruct the product on Floer cohomology, and the structure maps of the Fukaya category.

    This paper is one of a sequence in which we generalize these invariants to include Lagrangian cor-respondences. The first papers in the sequence were [19], [17]; however, we have tried to make thepaper as self-contained as possible. Recall that if (M0, 0) and (M1, 1) are symplectic manifolds,then a Lagrangian correspondence from M0 to M1 is a Lagrangian submanifold of M

    0 M1, where

    M0 := (M0, 0). Whereas Lagrangian manifolds form elliptic boundary conditions for pseudoholo-morphic curves, Lagrangian correspondences form elliptic seam conditions. For a pair of curves withboundary in M0 and M1, a seam condition L01 roughly speaking requires corresponding boundaryvalues to pair to a point on L01 M

    0 M1. (Differently put, a neighbourhood of the seam can be

    folded up to a pseudoholomorphic curve in M

    0 M1 with boundary values on L01.) Using suchseam conditions between pseudoholomorphic strips, we defined in [19] a quilted Floer cohomologyHF(L01, L12, . . . , L(k1)k) for a cyclic sequence of Lagrangian correspondences L(1) M

    1 M

    between symplectic manifolds M0, M1, . . . , M k = M0. In this paper we construct new Floer typeinvariants arising from quilted pseudoholomorphic surfaces. These quilts consist of pseudoholo-morphic surfaces (with boundary and strip-like ends in various target spaces) which satisfy seamconditions (mapping certain pairs of boundary components to Lagrangian correspondences) andboundary conditions (mapping other boundary components to simple Lagrangian submanifolds).Similar moduli spaces appeared in the work of Perutz [8] and have been considered by Khovanovand Rozansky [5] under the name of pseudoholomorphic foams. The present paper provides a generallanguage, invariance and gluing theorems for pseudoholomorphic quilts. Moreover, we establish an

    1

    http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2http://arxiv.org/abs/0905.1369v2
  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    2/32

    2 KATRIN WEHRHEIM AND CHRIS T. WOODWARD

    invariance of relative quilt invariants under strip shrinking and geometric composition of Lagrangiancorrespondences. The geometric composition of two Lagrangian correspondences L01 M

    0 M1,

    L12 M

    1 M2 isL01 L12 :=

    (x0, x2) M0 M2

    x1 : (x0, x1) L01, (x1, x2) L12.In general, this will be a singular subset of M0 M2, with isotropic tangent spaces where smooth.However, if we assume transversality of the intersection L01 M1 L12 :=

    L01 L12

    M0

    M1 M2

    , then the restriction of the projection 02 : M0 M1 M

    1 M2 M

    0 M2 to

    L01 M1 L12 is automatically an immersion. If in addition 02 is injective, then L01 L12 is asmooth Lagrangian correspondence and we will call it an embedded geometric composition. If thecomposition L(1) L(+1) is embedded, then under suitable monotonicity assumptions there is acanonical isomorphism

    (1) HF(. . . , L(1), L(+1), . . .) = H F(. . . , L(1) L(+1), . . .).

    For the precise monotonicity and admissibility conditions see [19] or Section 4. The proof in [17, 19]proceeds by shrinking the strip in M

    to width zero and replacing the two seams labeled L

    (1)and

    L(+1) with a single seam labeled L(1) L(+1). In complete analogy, Theorem 5.1 shows thatthe same can be done in a more general quilted surface: The two relative invariants arising from thequilted surfaces with or without extra strip are intertwined by the above isomorphism (1) of Floercohomologies.

    A first application of pseudoholomorphic quilts is the proof of independence of quilted Floerhomology from the choices of strip widths in [19]. As a second example we construct in Section 6a morphism on quantum homologies L01 : H F(M0) HF(M1) associated to a Lagrangiancorrespondence L01 M

    0 M1. This is in general not a ring morphism, simply for degree reasons,

    and even on the classical level. However, using the invariance and gluing theorems for quilts, weshow that it factors L01 = L01 L01 into a ring morphism : HF(M0) HF(L01, L01) anda morphism : HF(L01, L01) HF(M1) satisfying

    (x y) (x) (y) = x TL01 y,

    where the right hand side is a composition on quilted Floer cohomology with an element TL01 HF(Lt01, L01, L

    t01, L01) that only depends on L01. We hope this will be useful in studying Ruans

    conjecture [4] that quantum cohomology rings are invariant under birational transformations withgood properties with respect to the first Chern class.

    In the sequel [18] we construct a symplectic 2-category with a categorification functor via pseu-doholomorphic quilts. In particular, any Lagrangian correspondence L01 M

    0 M1 gives rise

    to a functor (L01) : Don#(M0) Don

    #(M1) between (somewhat extended) Donaldson-Fukayacategories. Given another Lagrangian correspondence L12 M

    1 M2, the algebraic composition

    (L01) (L12) : Don#(M0) Don

    #(M2) is always defined, and if the geometric compositionL01 L12 is embedded then (L01) (L12) = (L01 L12). In other words, embedded geometriccomposition is isomorphic to the algebraic composition in the symplectic category, and categorifi-cation commutes with composition.

    We thank Paul Seidel and Ivan Smith for encouragement and helpful discussions, and the refereefor most valuable feedback.

    1.1. Notation and monotonicity assumptions. One notational warning: When dealing withfunctors we will use functorial notation for compositions, that is 0 1 maps an object x to1(0(x)). When dealing with simple maps like symplectomorphisms, we will however stick to thetraditional notation (1 0)(x) = 1(0(x)).

    For Lagrangian correspondences and (quilted) Floer cohomology we will use the notation devel-oped in [19]. Moreover, we will frequently refer to assumptions on monotonicity, Maslov indices,and grading of symplectic manifolds M and Lagrangian submanifolds L M. We briefly summarizethese here from [19].

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    3/32

    PSEUDOHOLOMORPHIC QUILTS 3

    (M1): (M, ) is monotone, that is [] = c1(T M) for some 0.

    (M2): If > 0 then M is compact. If = 0 then M is (necessarily) noncompact but satisfies bounded

    geometry assumptions as in [15, Chapter 7].1

    (L1): L is monotone, that is 2

    u = I(u) for all [u] 2(M, L). Here I : 2(M, L) Z is theMaslov index and 0 is (necessarily) as in (M1).

    (L2): L is compact and oriented.

    (L3): L has (effective) minimal Maslov number NL 3. Here NL is the generator of I({[u] 2(M, L)|

    u > 0}) N.

    When working with Maslov coverings and gradings we will restrict our considerations to thosethat are compatible with orientations as follows.

    (G1): M is equipped with a Maslov covering LagN(M) for N even, and the induced 2-fold Maslovcovering Lag2(M) is the one described induced by the orientation of M.

    (G2): L is equipped with a grading NL : L LagN(M), and the induced 2-grading L Lag2(M) is

    the one given by the orientation of L.Finally, note that our conventions differ from Seidels definition of graded Floer cohomology in

    [13] in two points which cancel each other: The roles ofx and x+ are interchanged and we switchedthe sign of the Maslov index in the definition of the degree.

    2. Invariants for surfaces with strip-like ends

    We begin with a formal definition of surfaces with strip-like ends analogous to [ 14, Section 2.4] (inthe exact case) and [12] (in the case of surfaces without boundary). To reduce notation somewhatwe restrict to strip-like ends, i.e. punctures on the boundary. One could in addition allow cylindricalends by adding punctures in the interior of the surface, see Remark 2.11.

    Definition 2.1. A surface with strip-like ends consists of the following data:

    a) A compact Riemann surface S with boundary S = C1 . . . Cm and dn 0 distinct pointszn,1, . . . , zn,dn Cn in cyclic order on each boundary circle Cn = S

    1. We will use the indiceson Cn modulo dn, index all marked points by

    E = E(S) =

    e = (n, l)n {1, . . . , m}, l {1, . . . , dn},

    and use the notation e 1 := (n, l 1) for the cyclically adjacent index to e = (n, l). We denoteby Ie = In,l Cn the component ofS between ze = zn,l and ze+1 = zn,l+1. However, S mayalso have compact components I = Cn = S1.

    b) A complex structure jS on S := S\ {ze | e E }.c) A set ofstrip-like ends for S, that is a set of embeddings with disjoint images

    e : R [0, e] S

    for all e E such that e(R {0, e}) S, lims(e(s, t)) = ze, and e

    jS = j0 isthe canonical complex structure on the half-strip R [0, e] of width2 e > 0. We denotethe set of incoming ends e : R [0, e] S by E = E(S) and the set of outgoing endse : R+ [0, e] S by E+ = E+(S).

    1More precisely, we consider symplectic manifolds that are the interior of Seidels compact symplectic manifolds

    with boundary and corners. We can in fact deal with more general noncompact exact manifolds, such as cotangentbundles or symplectic manifolds with convex ends. For that purpose note that we do not consider cylindrical ends

    mapping to noncompact symplectic manifolds, so the only technical requirement on exact manifolds is that thecompactness in Theorem 3.9 holds, see the footnote in the proof there.

    2Note that here, by a conformal change of coordinates, we can always assume the width to be e = 1. The freedomof widths will only become relevant in the definition of quilted surfaces with strip-like ends.

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    4/32

    4 KATRIN WEHRHEIM AND CHRIS T. WOODWARD

    d) An ordering of the set of (compact) boundary components of S and orderings E =(e1 , . . . , e

    N

    ), E+ = (e+1 , . . . , e

    +N+

    ) of the sets of incoming and outgoing ends. Here ei =

    (ni , l

    i ) denotes the incoming or outgoing end at zei .

    Elliptic boundary value problems are associated to surfaces with strip-like ends as follows. Let Ebe a complex vector bundle over S and F = (FI)I0(S) a tuple of totally real subbundles FI E|Iover the boundary components I S. Suppose that the sub-bundles FIe1 , FIe are constant andintersect transversally in a trivialization of E near ze.

    3 Let

    DE,F : 0(S, E; F) 0,1(S, E)

    be a real Cauchy-Riemann operator acting on sections with boundary values in FI over each com-ponent I S. Transversality on the ends implies that the operator DE,F is Fredholm. If S = Shas no strip-like ends, and S0 S denotes the union of components without boundary, we denoteby I(E, F) the topological index

    I(E, F) = deg(E|S0) + I0(S)

    I(FI),

    where I(FI) is the Maslov index of the boundary data determined from a trivialization of E|(S\S0) =(S\S0) Cr. The index theorem for surfaces with boundary [6, Appendix C] implies

    (2) Ind(DE,F) = rankC(E)(S) + I(E, F).

    Note that the topological index I(E, F) is automatically even if the fibers of F are oriented, sinceloops of oriented totally real subspaces have even Maslov index and complex bundles have evendegree.

    A special case of these oriented totally real boundary conditions will arise from orientedLagrangian submanifolds. We fix a compact, monotone (or noncompact, exact) symplectic man-ifold (M, ) satisfying (M1-2) and let M be equipped with an N-fold Maslov covering satisfying(G1). For every boundary component I 0(S) let LI M be a compact, monotone, graded

    Lagrangian submanifold satisfying (L1-2) and (G2). We will also write Le := LIe for the Lagrangianassociated to the noncompact boundary component Ie = R between ze1 and ze.

    These assumptions ensure that the grading |x| ZN on the Floer chain groups HF(Le, Le)induces a Z2-grading (1)|x| which only depends on the orientations of (Le)eE(S). Moreover, wesay that the tuple (LI)I0(S) is relatively spin if all Lagrangians LI are relatively spin with respectto one fixed background class b H2(M, Z2), see [20] for more details.

    With these preparations we can construct moduli spaces of pseudoholomorphic maps from thesurface S. For each pair (Le1, Le) for e E+ resp. (Le, Le1) for e E choose a regular pair(He, Je) of Hamiltonian and almost complex structure (as in [19]) such that the graded Floer coho-mology HF(Le1, Le) resp. HF(Le, Le1) is well defined. Here Je : [0, e] J(M, ) is a smoothfamily in the space of -compatible almost complex structures on M. Let Ham(S; (He)eE ) denotethe set of C(M)-valued one-forms KS 1(S, C(M)) such that KS |S = 0 and S,eKS = Hedt

    on each strip-like end. Let YS 1(S, Vect(M)) denote the corresponding Hamiltonian vector field

    valued one-form, then S,eYS equals to XHedt on each strip-like end. We denote by J(S; (Je)eE )the subset ofJS C (S, J(M, )) that equals to the given perturbation datum Je on each strip-likeend.

    We denote by I the set of tuples X = (xe )eE with x

    e I(Le, Le1) and by I+ the set of

    tuples X+ = (x+e )eE+ with x+e I(Le1, Le). For each of these tuples we denote by

    MS(X, X+) :=

    u : S M

    (a) (d)3As pointed out to us by the referee, one can avoid the use of trivializations here by assuming that the bundles

    FI are defined over the closure I S (which is diffeomorphic to a closed interval) and intersect transversely at the

    points ofS S.

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    5/32

    PSEUDOHOLOMORPHIC QUILTS 5

    the space of (JS , KS)-holomorphic maps with Lagrangian boundary conditions, finite energy, andfixed limits, that is

    a) J,Ku := JS(u) (du YS(u)) (du YS(u)) jS = 0,b) u(I) LI for all I 0(S),c) EKS(u) :=

    S

    u + d(KS u)

    < ,

    d) lims u(S,e(s, t)) = xe (t) for all e E.

    Remark 2.2. 1) For any map u : S M that satisfies the Lagrangian boundary conditions (b)and exponential convergence to the limits in (d) (and hence automatically also satisfies (c)) oneconstructs a linearized operator Du as in [6] with a minor modification to handle the boundaryconditions: For any section of Eu = uT M satisfying the linearized boundary conditionsin Fu =

    (u|I)T LI

    I0(S)

    set u() = u()1J,K(expu()). For the moduli space to be

    locally homeomorphic to the zero set of u, we need to define the exponential map exp suchthat exp(T LI) LI, and u() has to be parallel transport of (0, 1)-forms from base point u tobase point expu(). To satisfy the first requirement we choose metrics gI on M that make the

    Lagrangians LI totally geodesic and define the exponential map exp : S T M M by usingmetrics varying along S, equal to gI near the boundary components I S. Restricting to asufficiently small neighbourhoodU T M of the zero section, we can rephrase this as exp(z, ) =exp0( + Q(z, )) in terms of a standard exponential map exp0 : T M M using a fixed metricand a quadratic correction Q : S U T M satisfying Q(, 0) = 0 and dQ(, 0) = 0. In fact,the quadratic correction is explicitly given by Q(z, TpM U) := (exp

    0p)

    1(exp(z, )) .To satisfy the second requirement we define the parallel transport u() by the usual complexlinear connection constructed from Levi-Civita connections of fixed metrics, but we do paralleltransport along the paths s expu(z)(z,s(z)) = exp

    0u(s + Q(s)). Note that the derivative

    at s = 0 of these paths still is , independent of the quadratic correction.The operator u is well-defined for any p > 2 as a map from the W

    1,p-closure of 0(S, uT M)to the Lp-closure of 0,1(S, uT M). We denote by Du := du(0) its derivative at zero. Thelinearized operator constructed in this way takes the same form as in [ 6] it is in fact inde-

    pendent of the choice of quadratic correction. So it takes the form Du = DEu,Fu of a realCauchy-Riemann operator as discussed in the linear theory above. When u is holomorphic,the operator Du is the linearized operator of (a) defined independently from the choice ofconnection and metric.

    2) If the tuple of Lagrangians (LI)I0(S) is monotone in the sense of [19], then elementsu MS(X, X+) (and more generally maps u as in (1)) satisfy the energy-index relation

    (3) 2EKS (u) = Ind(Du) + c(X, X+).

    The identity is seen by fixing one element v0 MS(X, X+) but interpreting it as map

    v0 : S M defined on the surface S = (S, j) with reversed orientation. Now given anyu MS(X, X

    +) we glue it with v0 (by reparametrization that makes them constant near theends) to a map w = u#v0 on the compact doubled surface S#S, in which each end of S isglued to the corresponding end of S. Since KS|S = 0 and KS u = KS v0 in the limit onthe strip-like ends, the Hamiltonian terms for u and v0 cancel and we have

    S#Sw =

    S

    u +

    S

    v0 = EKS (u) + EKS (v0).

    On the other hand, a linear gluing theorem as in [ 12, Section 3.2] provides

    Ind(Du) + Ind(Dv0) = Ind(Dw) = I(wT M, (wT LI)I0((S#S))) +

    dimM2 (S#(S))

    Here the equality to the Maslov index follows from (2). Finally, we have 2S#S w

    =

    I(wT M, (wT LI)I0((S#S))) by the monotonicity assumption. This proves (3) with

    c(X, X+) = Ind(Dv0) dimM

    2(S#S) 2EKS(v0).

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    6/32

    6 KATRIN WEHRHEIM AND CHRIS T. WOODWARD

    3) If the Lagrangians (LI)I0(S) are all oriented, then the index Ind(Du) of a map u as in (1)is determined mod 2 by the surface S and limit conditions X, X+. This is since in the above

    index formula a change ofu with fixed ends is only reflected in a change of the topological indexI(wT M, (wT LI)I0(S#S)). This index however is always even since by (L2) the totallyreal subbundles wT LI are oriented.

    z1

    z3

    z4

    z2

    z0

    u

    L2

    L1L0

    M

    L4L3

    x

    y

    p

    q

    r

    Figure 1. A holomorphic curve u MS((x, y), (p,q,r)) for a surface S with endsE = {2, 0} and E+ = {4, 1, 3}

    Theorem 2.3. Suppose that(LI)I0(S) is a monotone tuple of Lagrangian submanifolds satisfying(L1-2) and (M1-2), and that regular perturbation data (He, Je) are chosen for each end e E.Then for any HS Ham(S; (He)eE ) there exists a dense comeagre

    4 subset Jreg(S; (Je)eE ; HS) J(S; (Je)eE ) such that for any tuple (X, X

    +) I I+ the following holds.

    a) MS(X, X+) is a smooth manifold.b) The zero dimensional component MS(X, X

    +)0 is finite.c) The one-dimensional component MS(X, X+)1 has a compactification as a one-manifold with

    boundary

    MS(X, X+)1 =

    eE, yI(Le,Le1)

    M(xe , y)0 MS(X|xe y, X+)0

    eE+, yI(Le1,Le)

    MS(X, X+|x+e y)0 M(y, x

    +e )0,

    where the tuple X|xey is X with the intersection point xe replaced by y.d) If (LI)I0(S) is relatively spin then there exist a coherent set of orientations on the zero

    and one-dimensional moduli spaces so that the inclusion of the boundary in (c) has the signs

    (1)

    f

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    7/32

    PSEUDOHOLOMORPHIC QUILTS 7

    By items (c),(d), the maps CS are chain maps and so descend to a map of Floer cohomologies

    (4) S : eE

    HF(Le, Le1) eE+

    H F(Le1, Le).

    In order for the Floer cohomologies to be well defined, we have to assume in addition that allLagrangians in (LI)I0(S) satisfy (L3). Floers argument using parametrized moduli spaces carriesover to this case to show that S is independent of the choice of perturbation data, complex structure

    j on S, and the strip-like ends. We sketch this argument in Section 4 for the more general quiltedcase.

    Remark 2.4. Recall that M is equipped with an N-fold Maslov covering and each Lagrangiansubmanifold LI M is graded. Suppose that S is connected. Then the effect of the relativeinvariant S on the grading is by a shift in degree of

    |S| =1

    2dim M(#E+ (S)) mod N.

    That is, the coefficient ofCS

    eExe

    in front of

    eE+x+e is zero unless the degrees |x

    e | =

    d(NLe(xe ),

    NLe1(x

    e )) and |x

    +e | = d(

    NLe1(x

    +e ),

    NLe(x

    +e )) satisfy

    eE+

    |x+e | eE

    |xe | =12 dim M

    #E+ (S)

    mod N.

    Here #E+ is the number of outgoing ends of S. So, for example, S preserves the degree if S is adisk with one outgoing end and any number of incoming ends.

    To check the degree identity fix paths e : [0, 1] LagN(TxeM) from

    NLe1

    (xe) to NLe(xe)

    for each end e E(S), and denote their projections by e : [0, 1] TxeM. Let DTxeM,e be theCauchy-Riemann operator in TxeM on the disk with one incoming strip-like end and with boundaryconditions e. Then (see [19]) we have

    |xe | = Ind(DTxeM,1e

    ), e E

    with the reversed path 1e from NLe(x

    e ) to

    NLe1

    (xe ) in case e E. In this case we have

    Ind(DTxeM,1e

    ) + Ind(DTxeM,e) =12 dim M mod N

    since gluing the two disks gives rise to a Cauchy-Riemann operator on the disk with bound-ary conditions given by the loop 1e #e, which lifts to a loop in Lag

    N(TxeM) and hence hasMaslov index 0 mod N. Now consider an isolated solution u MS(X, X

    +)0. For each ende E(S) we can glue the operator DTxeM,

    1e

    on the disk to the linearized Cauchy-Riemann oper-

    ator DuTM,(uTLI )I0(S) on the surface S. This gives rise to a Cauchy-Riemann operator on the

    compact surface S with boundary conditions given by Lagrangian subbundles (given by uT LI forcompact boundary components I S and composed of uT Le and 1e for noncompact compo-nents Ie) that lift to loops in Lag

    N(M) (given by LI u|I resp. composed ofLe u|Ie and (e)1).

    In a trivialization of uT M their Maslov indices are hence divisible by N, and so the index of the

    glued Cauchy-Riemann operator is12 dim M (S) = Ind(DuTM,(uTLI)I0(S)) +

    eE(S)

    Ind(DTM,1e ) mod N

    = 0 +

    eE+(S)

    (12 dim M |x+e |) +

    eE(S)

    |xe | mod N.

    Example 2.5 (Strip Example). If S is the strip R [0, 1] (i.e. the disk with one incoming andone outgoing puncture) then we can choose perturbation data that preserve the R-invariance of theholomorphic curves. Then any nonconstant solution comes in a 1-dimensional family and hence Sis generated by the constant solutions. The same holds for the disks S and S with two incoming

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    8/32

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    9/32

    PSEUDOHOLOMORPHIC QUILTS 9

    Note that the trace does not depend on the choice of the signs i in (5). On the other hand,let #

    ee+ (S) denote the surface obtained by gluing together the ends e, and choose an ordering of

    the boundary components and strip-like ends. (There is no canonical choice for this general gluingprocedure.) The glued surface #ee+ (S) can be written as the geometric trace

    SE+\{e+} S

    e+,e0

    #

    Se+ Se0

    #

    S Se0

    #

    Se Se0

    #

    SE\{e} S

    e,e0

    ,

    where S0#S1 denotes the gluing of all incoming ends of S0 to the outgoing ends of S1 (which mustbe identical and in the same order). Here superscripts indicate the indexing of the ends of the

    surfaces, so e.g SE+\{e+} is a product of strips R [0, 1] with both incoming and outgoing ends

    indexed by E+ \ {e+}. The surfaces Se are the products of strips with incoming ends indexed

    by (E \ {e}, e) resp. E+ and outgoing ends indexed by E resp. (E+ \ {e+}, e+) (in the orderindicated). The relative invariants associated to the surfaces in this geometric trace are exactly theones that we compose in the definition (8) of the algebraic trace. In fact, the standard Floer gluingconstruction implies the following analogue of the gluing formula [14, 2.30] in the exact case.

    Id

    Id

    S Id

    e+ Id

    e Id

    Figure 2. Gluing example for a connected surface S

    Theorem 2.7 (Gluing Theorem). LetS be a surface with strip-like ends and(Le)eE(S) Lagrangiansas in Theorem 2.3, satisfying in addition (L3) and (G1-2). Suppose that e E(S) such thatLe+1 = Le and Le+ = Le1. Then

    #ee+ (S)= (S,#ee+ (S)

    )dim(M)/2 Tre,e+(S),

    where S,#

    e

    e+ (S)

    = 1 is a universal sign depending on the surfaces, that is, the ordering of boundary

    components etc.

    Unfortunately it seems one cannot make the sign more precise, since there is no canonical con-vention for ordering the boundary components etc. of the glued surface.

    Sketch of Proof: By Theorem 4.1, the relative invariant for the geometric trace can be computedusing a surface with long necks between the glued surfaces. Solutions (of both the linear andnon-linear equation) on this surface are in one-to-one correspondence with pairs of solutions onthe two separate surfaces; counting the latter exactly corresponds to composition. The one-to-one correspondence is proven by an implicit function theorem (using the fact that the linearizedoperator as well as its adjoint are surjective in the index 0 case) and a compactness result (using

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    10/32

    10 KATRIN WEHRHEIM AND CHRIS T. WOODWARD

    monotonicity to exclude bubbling). Details for the analogous closed case can be found in [12, Chapter5.4]. Universality of the gluing sign for the case of simultaneous gluing of all ends is proved in [ 20].

    By definition, our algebraic trace is the composition of the relative invariants of SE+\{e+}

    Se+,e0

    ,Se+ S

    e0 , S S

    e0 , Se S

    e0 , and S

    E\{e} S

    e,e0 , see Lemma 2.6.

    In [20] we determine more explicitly the gluing signs in two special cases: Gluing a surface withone outgoing end to the first incoming end of another surface, and gluing the ends of a surface withsingle incoming and outgoing ends (which lie on the same boundary component).

    Theorem 2.8. If S is a disjoint union S = S0 S1 with ze S0, ze+ S1, and if S1 has asingle outgoing end E+(S1) = {e+}, we define canonical orderings as follows: Suppose that e is thelast incoming end of S0 and the boundary components containing ze resp. ze+ are last in S0 resp.

    first inS1. Then we order the boundary components and ends of the glued surface (S0)#ee+ (S1) :=

    #ee+ (S0 S1) by appending the additional boundaries and incoming ends of S1 to the ordering for

    S0. With these conventions we have

    (S0)#ee+ (S1) = S0 (1E(S0)\{e}

    S1),

    where = 1 if n = 12 dim M is even or the number b1 of boundary components of S1 is odd, and ingeneral

    = (1)n(b1+1)

    eE(S0)\{e}

    (n|xe|).

    In our concrete situations the surfaces will usually have one boundary component, b1 = 1, andone outgoing end, hence the gluing sign will be = +1.

    Theorem 2.9. IfS is connected with exactly one incoming and one outgoing end E = {e+, e} lyingon the same, first boundary component, we define canonical orderings as follows: The glued surface#

    ee+ (S) has no further ends but two new compact boundary components, which we order by taking

    the one labelled L1 first and that labelledL2 second, where (L1, L2) denote the ordered boundaryconditions at the outgoing ende+. (Then the ordered boundary conditions ate are (L

    2, L1).) After

    these new components we order the remaining boundary components in the order induced byS. Withthat convention we have

    (9) #ee+

    (S) : 1 Tr(S) =i

    (1)|xi|CS(xi), xi,

    the (graded) sum over the xi coefficients of CS(xi).

    Example 2.10. We compute the invariants for closed surfaces as follows:

    a) (Disk) IfS is the disk with boundary condition L, then S is the number of isolated perturbedJ-holomorphic disks with boundary in L. Because of the monotonicity assumption, and sincewe do not quotient out by automorphisms of the disk, each component of the moduli space ofsuch disks has at least the dimension of L, hence S = 0.

    b) (Annulus) Let A = #S denote the annulus, obtained by gluing along the two ends of the infinitestrip S = R [0, 1] with boundary conditions L0 and L1. Let the boundary components be

    ordered like (L0, L1), as in Theorem 2.9. Then the gluing formula producesA = Tr(Id) = rank HF

    even(L0, L1) rank HFodd(L0, L1).

    The same result can be obtained by decomposing the annulus into cup and cap and computingthe universal sign to A = .

    c) (Sphere with holes) Let S denote the sphere with g + 1 disks removed and boundary conditionL over each component. S can be obtained by gluing together g 1 copies of the surface S0,which is obtained by removing a disk from the strip R [0, 1]; see Figure 3. The latter definesan automorphism S0 on HF(L, L), and the gluing formulas give

    S = Tr(g1S0

    ) =

    (1)|xi|g1S0 (xi), xi.

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    11/32

    PSEUDOHOLOMORPHIC QUILTS 11

    Figure 3. Gluing copies of S0

    Remark 2.11. If M is compact, then one can also allow surfaces to have incoming or outgoingcylindrical ends, equipped with a periodic Hamiltonian perturbation. In this case the relative invari-ant acts on the product of Floer cohomology groups with a number of copies of the cylindricalFloer cohomology HF(Id), isomorphic to the quantum cohomology QH(M) of M. For instance, adisk with one puncture in the interior gives rise to a canonical element L HF(Id). Splitting theannulus into two half-cylinders glued at a cylindrical end gives rise to the identity

    rank HFeven(L0, L1) rank H Fodd(L0, L1) = L0 , L1HF(Id).

    By considering a disk with one interior and one boundary puncture one obtains the open-closed mapHF(L, L) HF(Id), which is discussed by Albers [2, Theorem 3.1].

    3. Invariants for quilted surfaces

    Quilted surfaces are obtained from a collection of surfaces with strip-like ends by sewing together

    certain pairs of boundary components. We give a formal definition below, again restricting to strip-like ends, i.e. punctures on the boundary. One could in addition allow cylindrical ends by addingpunctures in the interior of the surface, see Remark 3.12.

    Definition 3.1. A quilted surface S with strip-like ends consists of the following data:

    a) A collection S = (Sk)k=1,...,m of patches, that is surfaces with strip-like ends as in Defini-tion 2.1 (a)-(c). In particular, each Sk carries a complex structures jk and has strip-like ends(k,e)eE(Sk) of widths k,e > 0 near marked points lims k,e(s, t) = zk,e Sk. We denoteby Ik,e Sk the noncompact boundary component between zk,e1 and zk,e.

    b) A collection S =

    {(k, I), (k , I)}S

    of seams, that is pairwise disjoint 2-element subsets

    mk=1

    {k} 0(Sk),

    and for each S, a diffeomorphism of boundary components

    : Sk I I Sk

    that isi) real analytic: Every z I has an open neighbourhood U Sk such that |UI extends

    to an embedding z : U Sk with zjk = jk . In particular, this forces to reverse

    the orientation on the boundary components. 6

    6In order to drop the real analyticity condition, one would have to extend each part of the analysis for pseu-

    doholomorphic curves with Lagrangian boundary conditions (e.g. linear theory, nonlinear regularity, and removal ofsingularities) to the case of maps (u0, u1) : (H2, H2) (M0M1, L01) on the half space with Lagrangian boundary

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    12/32

    12 KATRIN WEHRHEIM AND CHRIS T. WOODWARD

    ii) compatible with strip-like ends : IfI (and hence I) is noncompact, i.e. lie between marked

    points, I = Ik,e and I = Ik ,e , then we require that matches up the end e with

    e 1 and the end e 1 with e

    . That is

    1

    k ,e k,e1 maps (s, k,e1) (s, 0)if both ends are incoming, or it maps (s, 0) (s, k ,e ) if both ends are outgoing. Wedisallow matching of an incoming with an outgoing end, and the condition on the other pairof ends is analogous, see Figure 4.

    c) As a consequence of (a) and (b) we obtain a set of remaining boundary components Ib Skbthat are not identified with another boundary component ofS. These true boundary componentsof S are indexed by

    B =

    (kb, Ib)bB

    :=

    mk=1

    {k} 0(Sk) \S

    .

    Moreover, we can read off the quilted ends e E(S) = E(S) E+(S) consisting of a maximalsequence e = (ki, ei)i=1,...,ne of ends of patches with boundaries ki,ei(, ki,ei)

    = ki+1,ei+1(, 0)identified via some seam i . This end sequence could be cyclic, i.e. with an additional identifi-

    cation kn,en(, kn,en) = k1,e1(, 0) via some seam n . Otherwise the end sequence is noncyclic,i.e. k1,e1(, 0) Ib0 and kn,en(, kn,en) Ibn take value in some true boundary componentsb0, bn B. In both cases, the ends ki,ei of patches in one quilted end e are either all incom-ing, ei E(Ski), in which case we call the quilted end incoming, e E(S), or they are alloutgoing, ei E+(Ski), in which case we call the quilted end incoming, e E+(S).

    d) Orderings of the patches and of the boundary components of each Sk as in Definition 2.1 (d).There is no ordering of ends of single patches but orderings E(S) = (e

    1 , . . . , e

    N(S)

    )

    and E+(S) = (e+1 , . . . , e

    +N(S)

    ) of the quilted ends. Moreover, we fix an ordering e =(k1, e1), . . . (kne , ene)

    of strip-like ends for each quilted end e. For noncyclic ends, this order-

    ing is determined by the order of seams as in (c). For cyclic ends, we need to fix a first strip-likeend (k1, e1) to fix this ordering.

    0 0 0 00 0 0 01 1 1 11 1 1 10 0 0 00 0 0 01 1 1 11 1 1 10 0 0 00 0 0 01 1 1 11 1 1 10 0 0 00 0 0 01 1 1 11 1 1 1

    Sk ee 1

    e 1e

    I

    I

    Sk

    z

    z

    Figure 4. Totally real seam compatible with strip-like ends

    A picture of a quilt is shown in Figure 5. Ends at the top resp. bottom of a picture will alwaysindicate outgoing resp. incoming ends. The alternative picture is that in which the ends are neigh-bourhoods of punctures and we indicate by arrows whether the ends are outgoing or incoming. Herewe draw the quilted surface as a disc, but in general this could be a more general surface. Also, thecomplex structures on the patches are not necessarily induced by the embedding into the plane, as

    conditions such that each component ui is pseudoholomorphic with respect to a different complex structure ji on H2,that is, Ji dui = dui ji for i = 0, 1. This seems doable, but would require a lot of hard technical work. However,

    none of this work is necessary if one restricts to real analytic seams. In that case, the only fundamental difficulty isresolved by Lemma 3.4 establishing homotopies between quilted surfaces of the same combinatorial type.

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    13/32

    PSEUDOHOLOMORPHIC QUILTS 13

    drawn. Ignoring complex structures, the two views of the quilted surface in Figure 5 are diffeomor-phic. In this example the end sequences are (2, 0), (1, 0) and (2, 1) for the incoming ends (at the

    bottom), and (2, 2), (3, 0), (2, 3), (1, 1) for the outgoing end (at the top). (Our only choice here iswhich marked point on the boundary circles of Si to label by 0.)

    0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0

    1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1

    0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0

    1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1

    0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 00 0 0 0 0 0 0 0

    1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 11 1 1 1 1 1 1 1

    0 0

    0 0

    0 0

    0 0

    0 0

    0 0

    1 1

    1 1

    1 1

    1 1

    1 1

    1 1

    0 0 0 0 0 0

    0 0 0 0 0 0

    0 0 0 0 0 0

    0 0 0 0 0 0

    0 0 0 0 0 0

    0 0 0 0 0 0

    0 0 0 0 0 0

    0 0 0 0 0 0

    0 0 0 0 0 0

    0 0 0 0 0 0

    0 0 0 0 0 0

    0 0 0 0 0 0

    0 0 0 0 0 0

    0 0 0 0 0 0

    1 1 1 1 1 1

    1 1 1 1 1 1

    1 1 1 1 1 1

    1 1 1 1 1 1

    1 1 1 1 1 1

    1 1 1 1 1 1

    1 1 1 1 1 1

    1 1 1 1 1 1

    1 1 1 1 1 1

    1 1 1 1 1 1

    1 1 1 1 1 1

    1 1 1 1 1 1

    1 1 1 1 1 1

    1 1 1 1 1 1

    0 0 0

    0 0 0

    0 0 0

    0 0 0

    0 0 0

    0 0 0

    0 0 0

    0 0 0

    1 1 1

    1 1 1

    1 1 1

    1 1 1

    1 1 1

    1 1 1

    1 1 1

    1 1 1

    S1S2

    S3S1

    S2

    S3

    Figure 5. Two views of a quilted surface

    Remark 3.2. a) Every compact boundary component S1 = I Sk of a surface with strip-like ends has a uniformizing neighbourhood UI Sk such that (UI, jk) = (S

    1 [0, ), i),where S1 = R/Z and i denotes the standard complex structure on R R = C. Due to thestrip-like ends, the same holds for noncompact boundary components R = I Sk, where(UI, jk) = (R [0, ), i).

    b) Seams that are compatible with the strip-like ends are automatically real analytic on these endssince extends to (s, k,e t) (s, t) resp. (s, t) (s, k,e t).

    c) In the uniformizing neighbourhoods of (a), the condition that a seam : I I be real

    analytic is equivalent to : R R (resp. : S1 S1 in the compact case) being realanalytic. As a consequence, the extensions z match up to an embedding : UI Skwith jk = jk for some possibly smaller uniformizing neighbourhood UI

    = R [0, ) with > 0.

    Remark 3.3. Equivalently, we can define a quilted surface Sby specifying a single surface with strip-like ends S and a finite number of connected, non-intersecting, real analytic submanifolds I S.(The real analytic condition means that every point z I has a uniformizing neighbourhoodS U C that maps I to R C.) We only need to impose a compatibility condition between the

    seams I and the strip-like ends of S: On every end e : R [0, e] S, the seams are given by

    straight lines, i.e. 1e (I) consists of none, one, or two lines R {t} for pairwise disjoint t [0, e].

    The patches Sk of the corresponding quilted surface S are then determined by the closures of theconnected components of S \ S I. The seams and seam maps are given by the submanifoldsI and their embedding into the two (not necessarily different) adjacent connected components.The true boundary components of S are exactly the boundary components of S, and the incomingand outgoing quilted ends E(S) E+(S) are exactly the incoming and outgoing ends of S. Henceorderings of boundary components and ends of S induce the necessary orderings for S, with theexception that we need to specify an ordering of the patches.

    Eventually, we will see that the relative quilt invariants only depend on the combinatorial structureof a quilted surface, and the specific choice of complex structure, strip-like ends, and seam maps isimmaterial. (So it will suffice to define quilted surfaces by pictures as in Figure 5.) The followinglemma provides the homotopies for the proof in Section 4.

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    14/32

    14 KATRIN WEHRHEIM AND CHRIS T. WOODWARD

    Lemma 3.4. LetS0 and S1 be two quilted surfaces of same combinatorial type, i.e. with the samepatches (Sk)k=1,...,m (as oriented 2-dimensional manifolds with boundary), in- and outgoing marked

    points (zk,e), seams S, and orderings, but possibly different complex structures ji

    k, strip-likeends ik,e, and seam maps i fori = 0, 1. Then there exists a smooth homotopy(S

    )[0,1] of quiltedsurfaces, connecting the two. The homotopy is smooth in the sense that jk ,

    k,e, and

    depend

    smoothly on .

    Proof. Since the orientations and combinatorial compatibility with the ends are fixed, we can choosehomotopies of diffeomorphisms connecting the given seam maps. (This is since Diff+(R) = {f C(R,R) | f(x) > 0 x R, limx f(x) = } is convex.) We can moreover choose homotopiesof embeddings k,e|R{0} on one boundary component and homotopies

    k,e > 0 of widths for each

    marked point. Then the compatibility condition for seams fixes embeddings k,e|R{k,e} on all other

    boundary components near marked points, except for those that are true boundary components ofSi for which we can choose the homotopy freely. Now, with fixed limits and boundary values, wecan find a homotopy of embeddings k,e for each marked point. This fixes the complex structures j

    k

    in neighbourhoods of each marked point. By construction, the seams are automatically real analyticin these neighbourhoods.Next, recall that we have (anti-holomorphic) embeddings i : U

    i Sk extending the seam maps

    i : I Sk . We can pick the neighbourhoods to be the same Ui = U and sufficiently small such

    that no two neighbourhoods or images thereof intersect each other. Then we choose homotopiesof embeddings : U Sk extending the seam maps

    , of standard form ((s, t) (s, t)

    resp. (s, t) (s, t)) on the strip-like ends, connecting the given i, and preserving the emptyintersections. Now we can find homotopies of the complex structures jk |U on each of the fixed seamneighbourhoods. This choice induces complex structures jk |(U) = (

    )j

    k

    on each image of

    a seam neighbourhood. This construction ensures that the homotopy of seam maps becomes ahomotopy of real analytic seams, compatible with the strip-like ends. Finally, we can extend thecomplex structures to the complement of the strip-like ends and neighbourhoods of seams to findthe full homotopy of complex structures jk on each patch Sk.

    Elliptic boundary value problems are associated to quilted surfaces with strip-like ends as follows.Suppose that E = (Ek)k=1,...,m is a collection of complex vector bundles over the patches Sk, andF is a collection of totally real sub-bundles

    F = (F)S (Fb)bB, F Ek

    |I (Ek |I ), Fb Ekb |Ib .

    Here we write E as short hand for the complex vector bundle with reversed complex structure,(E, J) = (E, J). Let

    0(S, E; F) m

    k=10(Sk , Ek)

    denote the subspace of collections of sections k (Ek) such that (k , k ) maps I to Ffor every seam = {(k, I), (k, I

    )} S and kb maps Ib to Fb for every boundary component

    b = (kb, Ib) B.

    Lemma 3.5. Suppose that, in a suitable trivialization of the bundles Ek near each end zk,ethe subbundles F and Fb are constant and transverse in the sense that for each quilted ende = (ki, ei)i=1,...,n E(S) we have

    (10) Fb0 F1 . . . Fn1 Fbn Ek1 Ek2 . . . Ekn .

    (Here i S are the seams between the ends andb0, bn B are the remaining boundary componentsof the quilted end, resp. n is the additional seam in the case of a cyclic end, in which case wereplace Fbn Fb0 above with Fn .) Then the direct sum of Cauchy-Riemann operators

    DE,F = mk=1DEk :

    0(S, E; F) 0,1(S, E) :=m

    k=10,1(Sk, Ek)

    is Fredholm as a map between the W1,p and Lp Sobolev completions for any 1 < p < .

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    15/32

    PSEUDOHOLOMORPHIC QUILTS 15

    Proof. The Fredholm property follows from local estimates for DE,F and its formal adjoint.For sections k (Ek) compactly supported in the interior of Sk we have kW1,p(Sk)

    C(DEkkLp

    (Sk) + kLp

    (Sk)) as in the standard theory, e.g. [6]. To obtain local estimatesnear a seam point z I we use the embedding z : Sk U Sk and consider the section

    := (k , zk ) of E

    k

    |U zEk . It satisfies Lagrangian boundary conditions (U I) Fand, due to the reversal of complex structures in both Ek and by

    z , the operators DEk k and

    DEkk combine to a Cauchy-Riemann operator on with respect to the complex structure jk

    on U. Hence, if k is supported in U and k is supported in z(U), then we obtain an estimate

    kW1,p + kW1,p C

    DEk kLp + DEkkLp + kLp + kLp

    .

    Finally, given sections (k (Ek))k=1,...,m supported sufficiently close to the marked points ina quilted end e = (ki, ei)i=1,...,n, we can trivialize the bundles Eki |im ki,ei

    = R [0, ki,ei ]

    (Vi, Ji) and boundary conditions Fi Vi Vi+1 resp. Fb0 V1, Fbn Vn. Then we view the

    sections as tuple =

    ki |im ki,ei : R [0, ki,ei ] Vii=1,...,n

    of maps on strips, extended trivially

    to the complement of im ki,ei

    R [0, ki,ei

    ]. The (trivialized) Cauchy-Riemann operator onthese sections can now be rewritten DE,F(k)k=1,...,m = (s + D), where D =

    ni=1Jit is a self-

    adjoint operator on ni=1L2([0, ki,ei ], Vi) whose domain is given by W

    1,2-maps = (i)i=1,...,n thatsatisfy the matching conditions (i(ki,ei), i+1(0)) Fi resp. (n(kn,en), 0(0)) Fbn Fb0 . Thetransversality assumption (10) implies that D is invertible and hence, as in [10], s + D is a Banachspace isomorphism, i.e. for (k) supported near e as above

    (k)W1,2(S) :=k

    kW1,2(Sk) = W1,2

    C(s + D)L2 = Ck

    DEkkL2(Sk) =: CDE,F(k)L2(S).

    This estimate generalizes to W1,p and Lp by the same construction as in [11, Lemma 2.4]. Puttingthe estimates together we obtain (k)W1,p(S) C(DE,F(k)Lp(S) + K(k)Lp(S) for all (k)

    0(S, E; F). Here K : W1,p(S) Lp(S) is the restriction operator to some compact subsets ofthe patches Sk followed by the standard Sobolev inclusion, hence it is a compact operator. Thisestimate shows that DE,F has finite dimensional kernel and closed image. The analogous estimatefor the adjoint operator shows that DE,F has finite dimensional cokernel and hence is Fredholm.

    In the case that S has no strip-like ends, we define a topological index I(E, F) as follows. Foreach patch Sk with boundary we choose a complex trivialization of the bundle Ek = Sk Crk . Eachbundle Fb Ekb |Ib

    = Ib Crkb has a Maslov index I(Fb) depending on the trivialization of Ekb .

    Similarly,F E

    k

    |I (Ek |I )

    = I (Crk ) Crk

    has a Maslov index I(F) depending on the trivializations ofEk and Ek . Let S0 S be the unionof components without boundary and define

    I(E, F) := deg(E|S0) + S

    I(F) +bB

    I(Fb).

    We leave it to the reader to check that the sum is independent of the choice of trivializations. Boththe topological and analytic index are invariant under deformation, and by deforming the seamconditions F to those of split form (F F with F E

    k

    |I and F

    (Ek |I) one obtains

    from (2) an index formula

    Ind(DE,F) =i

    rankC(Ei)(Si) + I(E, F).

    We now construct moduli spaces of pseudoholomorphic quilted surfaces. Let

    M =

    (Mk, k)k=1,...,m

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    16/32

    16 KATRIN WEHRHEIM AND CHRIS T. WOODWARD

    be a collection of symplectic manifolds. A Lagrangian boundary condition for (S, M) is a collection

    L = (L)S (Lb Mkb)bB, L Mk

    Mk , Lb Mkb

    of Lagrangian correspondences and Lagrangian submanifolds associated to the seams and boundarycomponents of the quilted surface. We will indicate the domains Mk, the seam conditions L, andthe true boundary conditions Lb by marking the surfaces, seams and boundaries of the quiltedsurfaces as in figure 6.

    0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0

    1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1

    0 0

    0 0

    0 0

    0 00 00 0

    1 1

    1 1

    1 1

    1 11 11 1

    0 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

    0 0 0 0 0 0 0

    0 0 0 0 0 0 0

    0 0 0 0 0 0 0

    0 0 0 0 0 0 0

    0 0 0 0 0 0 0

    0 0 0 0 0 0 0

    0 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

    1 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 1

    1 1 1 1 1 1 1

    1 1 1 1 1 1 1

    1 1 1 1 1 1 1

    1 1 1 1 1 1 1

    1 1 1 1 1 1 1

    1 1 1 1 1 1 1

    1 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 1

    M2

    M1

    L1L12

    L2

    M3

    L2

    L23

    Figure 6. Lagrangian boundary conditions for a quilt

    We say that the tuples M and L are graded if each Mk is equipped with an N-fold Maslov coveringfor a fixed N N and each Lagrangian L Mk Mk and Lb Mkb is graded with respect to therespective Maslov covering. Moreover, we assume that the gradings are compatible with orientationsin the sense of (G1-2).

    We say that L is relatively spin if all Lagrangians in the tuple are relatively spin with respect to

    one fixed set of background classes bk H2(Mk,Z2), see [20].

    Definition 3.6. We say that the tuple of Lagrangian boundary and seam conditions L for S ismonotone if the sequences Le in (11) resp. (12) below for each end e E(S) are monotone for

    Floer theory (in the sense of [19]) and the following holds: Let S#S denote the quilted surfaceobtained by gluing a copy S of S with reversed complex structure (and hence reversed ends)to S at all corresponding ends. This quilted surface has components (Sk#S

    k )k=1,...,m, compact

    seams : I I = S

    1 for S (one for each noncompact and two for each compact seamof S), compact boundary components Ib = S

    1 for b B (one for each noncompact and two foreach compact boundary component of S), but no strip-like ends. Then for each tuple of mapsu : S#S M (that is uk : S

    k #Sk Mk) that takes values in L over the seams and boundary

    components (that is (uk uk )(I) L and uk(Ib) Lb) we have the action-index relation

    2mk=1

    ukk = I(Eu, Fu)

    for Eu = (ukT Mk)k=1,...,m and Fu =

    (uk uk )

    T LS

    (ukbT Lb)bB.

    Remark 3.7. Note the following analogue of the monotonicity Lemma in [19]: If each Mk ismonotone in the sense of (M1) and each L and Lb is monotone in the sense of (L1), all with thesame constant 0, and each 1(L) 1(M

    k

    Mk) and 1(Lb) 1(Mkb) has torsion image,then L is monotone. This is because any tuple u in Definition 3.6 is homotopic to the union of adisc (D,D) (Mk Mk , L) at each seam, a disc (D,D) (Mkb , Lb) at each true boundarycomponent of the quilt, and a closed curve k Mk for each patch.

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    17/32

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    18/32

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    19/32

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    20/32

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    21/32

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    22/32

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    23/32

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    24/32

    24 KATRIN WEHRHEIM AND CHRIS T. WOODWARD

    On the level of chain complexes the maps e are simply the identity. Therefore it suffices to showthat the maps CS and CS [dknk] are equal for sufficiently small width. See (15) for an explanation

    of the grading shift. The bijection between pseudoholomorphic quilts follows from essentially thesame degeneration argument in [17] which was also used in [19]. However, in this case the adjointof the linearized operator is honestly surjective. (There is no translational symmetry, so elements ofthe zero-dimensional moduli space have linearized operators of index zero, not one.) The surfacesto the left and right of the shrinking strip are arbitrary quilted surfaces, but this is of no relevancein the proof. It suffices to work with the uniformizing neighbourhoods R (, 0] S0 andR [0, ) S2 in which the seam maps are the identity on R.

    Remark 5.3. In [17] a strip degeneration argument is used to establish a canonical isomorphism

    (20) HF(L0, L01, L12, L2) HF(L0, L02, L2)

    for embedded composition of monotone Lagrangians. The natural alternative approach to definingan isomorphism, or even just a homomorphism is to try and interpolate the middle strip to zerowidth in the relative invariant that is used in [19] to prove the independence of H F(L0, L01, L12, L2)from the choice of width 1 > 0. The resulting quilted surface might look as indicated on the left inFigure 8. This approach was pioneered by Matthias Schwarz and might be realized as a jumpingboundary condition as investigated in [1]. However, this analytic setup is at present restricted toa highly specialized class of Lagrangians and almost complex structures. In essence, this approachwould construct a canonical element of the Floer homology HF(L02, (L01, L12)) associated to thethree seams coming together. Our approach also induces such a canonical element, given by theidentity 1L02 HF(L02, L02) and the strip shrinking isomorphism to HF(L02, (L01, L12)). So wereplace the picture of seams coming together by one where the seams corresponding to L01, L12,L02 run into an infinite cylindrical end, as on the right in Figure 8, where the inner circle is meantto represent a cylindrical end. This picture defines a relative invariant

    : HF(L02, (L01, L12)) H F(L0, L01, L12, L2) HF(L0, L02, L2).

    Now the isomorphism (20) can then alternatively be described by (T, ), where T HF(L02, (L01, L12))is the morphism corresponding to the identity 1L02 H F(L02, L02), see Corollary 5.4.

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0

    1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    L2

    L12

    L01L02

    L0

    L2

    L12

    L01

    L02

    L0

    TM1

    M2

    M0

    M1

    M2

    M0

    Figure 8. Alternative approaches to a homomorphism

    As a first application of Theorem 5.1 we prove the claim of Remark 5.3. For that purpose wedenote by

    : HF(L0, L01, L12, L2) HF(L0, L02, L2),

    : H F(L02, (L01, L12)) HF(L02, L02)

    the isomorphisms (1) from [19]. Then we have the following alternative description of (which a

    priori depends on L0 and L2) in terms of and the identity morphism 1L02 H F(L02, L02), definedin [18].

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    25/32

    PSEUDOHOLOMORPHIC QUILTS 25

    Corollary 5.4. Let

    : HF(L02, (L01, L12)) HF(L0, L01, L12, L2) HF(L0, L02, L2)

    denote the relative invariant associated to the quilted surface on the right in Figure 8. Then we have for allf HF(L0, L01, L12, L2)

    (f) = (1(1L02) f).

    Moreover, in the notation of[18], we have (Tf) = T(L0)f, where T : (L02) (L01, L12)is a natural transformation of functors associated to any T HF(L02, (L01, L12)).

    0 0 0 0 0 0 0 0 0 0

    0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

    1 1 1 1 1 1 1 1 1 1

    1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1

    M0

    M1

    M2L02

    L01

    L12

    L0

    T TMa

    MbL

    Lab

    Lab

    Figure 9. Natural transformation associated to a Floer cohomology class: Generalcase from [18] and the present special case.

    Proof. We apply Theorem 5.1 to = S, where the quilted surface S contains one simple strip in

    M1. (The other surfaces are triangles.) This implies (Tf) = S((T)(f)), where the quiltedsurface S is obtained by replacing this strip with a seam condition in L01 L12 = L02. To calculate

    this for (T) = 1L02 we use the gluing formula (18) to obtain (1

    (1L02)f) = S((f)), whereS is the surface that is obtained by gluing the quilted cap of Figure 7 into S. Since S is a simpledouble strip (with seam condition L02 and boundary conditions L0, L2), and we do not quotient outby translation, the relative invariant S is the identity, as in Example 2.5. This proves the firstclaim.

    The second claim follows from a deformation of the quilt S to the glued quilt that corresponds, by(18), to the composition of the natural transformation T T(L0) HF((L0, L02), (L0, L01, L12))(given by the quilt in Figure 9) with the pair of pants product from HF((L0, L02), (L0, L01, L12)) HF((L0, L01, L12), L2) to HF((L0, L02), L2) defined in [18]. Figure 10 gives a picture summary ofthese arguments.

    0

    =(T)=1L02

    T

    L2 L02

    L12L01

    L0

    f

    =

    f

    L2 L0L02

    T

    L12L01

    L2 L0

    (f)

    L2 L02 L0

    L02

    (f)

    L02(T)

    Figure 10. Proof by pretty picture

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    26/32

    26 KATRIN WEHRHEIM AND CHRIS T. WOODWARD

    6. Application: Morphism between quantum homologies

    Given a Lagrangian correspondence L01 M0 M1 a natural question is whether analogous to a

    symplectomorphism or certain birational equivalences as in [4] it induces a ring morphism (L01) :HF(M0) HF(M1) between the quantum homologies. A quilted cylinder indeed defines amorphism, which however generally does not intertwine the product structures. Conjecturally, itis a Floer theoretic equivalent of the slant product8 on cohomology H(M0) H(M1) with thePoincare dual of the fundamental class of L01. Even in this classical case, one generally only obtainsa ring homomorphism if L01 is the graph of a continuous map. Moreover, both the classical andFloer theoretic morphism will shift the grading unless dim M0 = dim M1. While Albers-Schwarzare working on refined versions of this morphism, we here use the framework of quilts to clarify thealgebraic structure and behaviour under geometric composition of Lagrangian correspondences.

    Remark 6.1. The quilted Floer cohomology H F(M) of the diagonal by definition is nothingelse but Hamiltonian periodic Floer cohomology HF(M) = HF(H) for a generic HamiltonianH : S1 M R. The latter is naturally isomorphic to the quantum homology of M, see [9].

    Theorem 6.2. LetL01 M0 M1 be a Lagrangian correspondence between symplectic manifolds

    M0, M1 satisfying (M12) and (L12), and assume that the pair (L01, L01) is monotone for Floertheory.9 Then the quilted cylinder in Figure 11 defines a natural map

    L01 : H F(M0) HF(M1).

    M0

    M1

    L01

    M0 M1

    L01

    Figure 11. Two views of the quilted cylinder defining L01 . It consists of two halfcylinders (the domains of maps to M0 and M1 in this order to fix orientations),with a seam identifying the two circle boundaries (giving rise to a seam conditionin L01). Arrows indicate incoming and outgoing ends, the outside circle is alwaysan outgoing end, and indicates incoming ends.

    Moreover, this map factors L01 = L01 L01 according to Figure 12 into maps

    L01 : HF(M0) HF(L01, L01), L01 : HF(L01, L01) HF(M1).Here L01 is a ring morphism, whereas L01 satisfies

    L01(x y) L01(x) L01(y) = x TL01 y

    8On de Rham cohomology, the product k(M0 M1) (M0)

    k+dim(M0)(M1) is given by integrationover one factor, (, )

    M0

    ( ).9The latter monotonicity requires an action-index relation for annuli in M0 M1 with boundaries on L01. All these

    monotonicity assumptions can be replaced by any other set of assumptions which ensure that all Floer homologies

    involved are well defined and disk bubbles on L01 cannot appear in 0- or 1-dimensional moduli spaces of holomorphicquilts.

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    27/32

    PSEUDOHOLOMORPHIC QUILTS 27

    for allx, y HF(L01, L01) and a fixed element TL01 HF(Lt01, L01, L

    t01, L01) that only depends on

    L01. For the latter we assume monotonicity of the sequence (Lt01, L01, Lt01, L01) and the composition

    10

    on the right hand side as well as TL01 are defined by relative quilt invariants as indicated in Figure 15.

    L01

    L01M1

    M0M0

    L01

    M1

    L01

    Figure 12. Two views of the splitting L01 = L01 L01 . The quilted surfacedefining L01 consists of a disk labeled M0 with one interior puncture (incomingend) and one boundary puncture (outgoing end), a disk labeled M1 with one bound-ary puncture (outgoing end), and a seam labeled L01 identifying the two boundarycomponents. The quilted surface defining L01 is the same with M0, M1 and incom-ing/outgoing ends interchanged.

    Remark 6.3. As pointed out by the referee, the composition of pull-back L01 with push-forwardL01 is not the one that classically preserves multiplicative structures. However, the classical pro-

    jection formula f!(x f(y)) = f!(x) y relating multiplicative structures under pull-back f andpush-forward f! has a Floer theoretic version as follows.

    We view LT01 : HF(M1) HF(LT01, L

    T01)

    = HF(L01, L01) as push-back and L01 again as

    push-forward, then the Floer theoretic projection formula holds:L01

    x LT01(y)

    = L01(x) y x HF(L01, L01), y HF(M1).

    The proof of this identity is a recommended exercise in applying the gluing and deformation lawsfor holomorphic quilts. Hint: Read the proof of Theorem 6.2 for very similar arguments.

    Remark 6.4. a) One can check with Remark 3.10 that the degree (modulo the minimal Maslovnumber NL01) of L01 agrees with the degree dL01 :=

    12 (dim(M1) dim(M0)) of the slant

    product. Indeed, the outgoing cylindrical end counts as #E+(S1) = 2. To see this, alternativelyadd a seam connecting the outgoing end to itself, labeled with the diagonal of M1. Then wehave two patches mapping to M1 with one outgoing end each and total Euler characteristic 1.

    b) By concatenating the morphism L01 : HF(M0) HF(M1) with the PSS-isomorphismsH(M0;Z) HF(M0) and HF(M1) H(M1;Z) from [9] one can see why this should bethe slant product: The concatenation is a map H(M0;Z) H(M1;Z) on Morse homology.

    Between two critical points x0 Crit(f0) M0 and x1 Crit(f1) M1 the coefficient ofthe map is given by counting holomorphic disks in 2(M

    0 M1, L01) with a marked point in

    the interior mapping to the product Wu(x0, f0) Ws(x1, f1) of unstable and stable manifolds.Here the holomorphic curve equation is perturbed by Hamiltonian vector fields and uses almost

    10Strictly speaking, the composition is defined by viewing x and y as elements in the trivially isomorphic quiltedFloer cohomologies x HF(M1 , (L

    t01, L01)) and y HF((L

    t01, L01),M1 ) here in the notation for the pair of

    generalized Lagrangian correspondences M1 and (Lt01, L01) from M1 to M1. Then x TL01 y is the composi-

    tion constructed in [18] in the refined Donaldson-Fukaya category Don#(M1,M1) of correspondences from M1 to

    M1. In fact, it is the composition of morphisms M1x (Lt01, L01)

    TL01 (Lt01, L01)

    y M1 , taking values in

    HF(M1 ,M1), which again is trivially identified with HF(M1).

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    28/32

    28 KATRIN WEHRHEIM AND CHRIS T. WOODWARD

    complex structures parametrized by the disk. One can homotope the Hamiltonian perturbationsto zero, not affecting the map on homology.

    Next, recall that due to the grading ambiguity of NL01 on L01, all gradings on Floer andMorse homology above should be taken modulo NL01. Hence, restricted to a fixed gradingk = 0, . . . , dim M0 the morphism L01 induces a map

    kL01 : Hk(M0;Z) N0

    Hk+dL01+NL01 (M1;Z).

    Here the contributions for each N0 arise from holomorphic disks as above, of Maslov indexNL01 . For the leading part of this map, the contributions for = 0 are those of zero energy,that is exactly the constant maps to L01. This is essentially the same calculation as in [3,Remark 1.3].

    With this one can check that the leading part of kL01 is indeed the Morse theoretic version

    of the slant product. If one could moreover achieve transversality with S1-invariant almostcomplex structures, then (w.l.o.g. putting the marked point in the center of the disk) the only

    isolated solutions would be constant disks. This would identify L01 with a Morse theoreticversion of the slant product.

    Proof of Theorem 6.2. Our assumptions guarantee that Theorem 3.9 applies to the moduli spacesof pseudoholomorphic maps from the quilted cylinder S indicated in Figure 11 to M0 and M1, withseam condition on L01. (In fact, doubling this quilted surface exactly amounts to an annulus mappingto M0 M1 with both boundary conditions in L01. Monotonicity for that surface is given by themonotonicity for Floer theory.) Hence these moduli spaces define a map L01 on Floer cohomologyas in (14). Strictly speaking, the cyclic generalized correspondences associated to the two ends ofS are the empty sequences as trivial correspondence from Mi to Mi. However, quilted Floercohomology for these is trivially identified with quilted HF(Mi), which is also trivially identifiedwith the usual Floer cohomology for the pair H F(Mi , Mi). All of these count pseudoholomorphiccylinders in Mi with a Hamiltonian perturbation.

    The map L01 : HF(M0) HF(M1) is natural in the sense that, by Theorems 4.1 and

    4.2 it is independent of the choice of complex structures and seam maps on S, almost complexstructures, and Hamiltonian perturbations. The map is hence determined purely by L01 and thechosen combinatorial structure of the quilted cylinder.

    Next, the splitting L01 = L01 L01 follows from Theorem 3.13 applied to the gluing of quiltedsurfaces indicated in Figure 12. The gluing sign is +1 since there are no further incoming ends(see Theorem 2.8 for the sign), and if we fix the order of patches for L01 as induced by L01.(For L01 the order of patches is irrelevant for orientations since each patch has even deficiency see Remark 3.11.) Monotonicity for these quilts as well as the pair of pants quilts below followsfrom the monotonicity for disks and spheres ensured by (M2) and (L2) since all Hamiltonian orbitsresp. Hamiltonian chords generating the Floer cohomologies can be contracted.11 To establishthe interaction of these maps with the product structures recall that the product on HF(Mi) isthe relative invariant Spop defined (see Section 2) by the pair of pants surface (a sphere with twoincoming and one outgoing puncture). As usual, we draw this surface as a disk with two inner disksremoved their boundaries are the incoming ends, while the outer circle is the outgoing end. Theproduct on HF(L01, L01) is in the setting of Section 2 given by the half pair of pants surface (a diskwith two incoming and one outgoing puncture on the boundary). However, since the disk maps toM0 M1, we can also view it as two disks mapping to M0 and M1 respectively (where one carries theopposite orientation), with three boundary punctures each, all three boundary components identifiedand satisfying the seam condition in L01. Viewed as quilted surface, this is a pair of pants surface

    11Strictly speaking, we either work with C1-small Hamiltonian perturbations whose orbits resp. chords are auto-matically contractible, or we restrict the map to contractible orbits resp. chords. The isomorphism of Floer coho-

    mologies for different Hamiltonian perturbations automatically maps contractible to contractible generator, and henceshows that the noncontractible generators do in fact not contribute to the cohomology.

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    29/32

    PSEUDOHOLOMORPHIC QUILTS 29

    L01L01 L01

    M0

    M1

    M1

    M0

    L01

    L01

    L01

    Figure 13. Two views of the quilted pair of pants defining the product on HF(L01, L01)

    M0 M0

    M1M1

    L01

    L01

    =

    Figure 14. Gluing and homotopy of quilts proving that L01 is a ring homomorphism

    on which three seams connect each pair of ends, see Figure 13. If we view H F(L01, L01) as quiltedFloer cohomology for the sequence (L01, Lt01), then the relative invariant Spop associated to this

    quilted pair of pants is exactly the product. We can hence use this definition to calculate for allf, g HF(M0)

    L01(f) L01(g) =

    Spop (L01 L01)

    (f, g) =

    L01 Spop

    (f, g) = L01(f g).

    Here the first and third equality is by definition. The middle identity Spop (L01 L01) =

    L01 Spop follows from gluing the quilted surfaces in different order (using Theorem 3.13) andhomotoping (using Theorem 4.1) between the two resulting quilted pair of pants surfaces that bothhave one seam connecting the outgoing end to itself and the two incoming ends on M0, see Figure 14.The gluing signs are +1, in the first case since the incoming patches have one boundary component,in the second case since the outgoing patch has only one incoming end, c.f. Theorem 2.8. This provesthat L01 is indeed a ring morphism.

    Similarly, we can write for all x, y HF(L01, L01)

    L01(x) L01(y) L01(x y) =

    Spop (L01 L01) L01 Spop

    (x, y)

    = S3p

    S2p(x, y)

    =

    S3comp

    Id (S1 S0) Id

    (x, y)

    = S3comp(x, TL01, y) = x TL01 y.

    Again, the first equality is by definition and the second identity follows from the gluing Theorem 3.13.In this case, however, we obtain two different quilted surfaces: The first composition results inthe quilted pair of pants S3p with two seams between three patches (and the M0-patches orderedaccording to order of incoming ends), whereas the second composition results in the quilted pair ofpants S2p with two seams between two patches (with the M0-patch ordered before the M1-patch), seeFigure 15. Again, the gluing signs are +1 since the incoming patches have one boundary component,c.f. Theorem 2.8. These two quilted surfaces are connected only by a singular homotopy which lets

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    30/32

    30 KATRIN WEHRHEIM AND CHRIS T. WOODWARD

    M1S3p

    y xL01M0 M0L01

    y x

    M1

    L01

    L01

    M0

    S2p

    S0

    x=

    S3comp

    TL01

    where TL01 =

    S1

    L01M0 L01

    L01

    L01

    L01

    y

    M0

    L01M0

    L01

    L01

    M1M1

    M0

    M1

    M1

    M0

    Figure 15. Manipulation of quilts proving the product identity for L01 . Each ofthe two quilts S3p and S2p in the first row can also be obtained by gluing one ofthe two quilts S1 and S0 in the definition of TL01 into the middle end of the quiltS3comp in the second row.

    the two seams intersect at a point and then connects them differently. Within our framework wecan express this as follows: Both S3p and S2p are obtained from the same quilted surface S3comp

    (consisting of one M1 disk with three boundary and one interior puncture and two M0 disks withtwo boundary punctures) by gluing different quilted disks into the middle end (which has four seamsending in it, hence each disk must have two seams connecting these ends). (The induced orderingon S3comp has the two M0-patches according to incoming ends before the M1-patch.) To obtainS3p we glue in the quilted disk S1 which has one M1 strip connecting different ends of the M1patch in S3comp to produce an annulus, which is the M1 patch in S3p. To obtain S2p we use thequilted disk S0 with one M0 strip connecting the two M0 patches in S3comp to provide a single M0patch in S2p. (The ordering of patches on S1 and S0 is immaterial since only the middle patch ineach has odd deficiency. The gluing signs are +1 for each patch since the incoming patches haveone boundary component. Note however that the middle patch gets glued at two ends. For S0these are two composition gluings, joining different surfaces, but for S1 the second gluing joins asurface to itself. Signs for this gluing are discussed in [20].) The smooth part of these homotopies(using Theorem 4.1) together with again the gluing Theorem 3.13 prove the third identity. Since

    both quilted disks S1 and S0 just have an outgoing end, they define an element of quilted Floercohomology

    TL01 := S1 S0 HF(Lt01, L01, L

    t01, L01).

    (Here we need a separate (e.g. monotonicity) assumption to ensure the Floer homology is welldefined. Then the quilts S0, S1 are monotone since their double can be glued and homotoped to anannulus with boundaries on L01. By a similar homotopy, or using the gluing identities, we deducemonotonicity for S3comp.) With this definition, the fourth identity again is just by convention, andsince we define the triple composition x T y by the relative quilt invariant

    S3comp : HF(L01, L01) H F(Lt01, L01, L

    t01, L01) H F(L01, L01) HF(M1),

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    31/32

    PSEUDOHOLOMORPHIC QUILTS 31

    this proves the product identity for L01. Finally, the quilted Floer cohomology class TL01 onlydepends on L01 and the (given) simple combinatorial type of the quilted surfaces S1 and S0, by

    Theorems 4.1 and 4.2.

    Unlike the previous theorem, which directly generalizes to any version of Floer cohomology withNovikov rings, virtual fundamental classes, or abstract deformations, the following quantizationcommutes with composition result requires the full strict monotonicity assumptions due to thepotential figure eight bubbling described in [17].

    Theorem 6.5. LetL01 M0 M1 and L12 M

    1 M2 be Lagrangian correspondences satisfying

    (L12) between symplectic manifolds M0, M1, M2 satisfying (M12) with a fixed monotonicity con-stant 0. Assume that both pairs (L01, L01) and (L12, L12) are monotone for Floer theory andthat the geometric composition L01 L12 is embedded. Then

    L01 L12 = L01L12 .

    Proof. While the geometric composition L01 L12 of two monotone Lagrangian correspondencesmay not automatically satisfy (L2), the monotonicity for Floer theory transfers to the geometriccomposition (see [19]), and then also implies (L2). Hence L01 L12 also satisfies the assumptions ofTheorem 6.2 and L01L12 is well defined.

    Now, by the gluing Theorem 3.13 (with +1 sign since no further incoming ends are involved)we have L01 L12 = (L01,L12), where the latter is the relative quilt invariant associated to thequilted surface on the left in Figure 16: two disks with interior puncture mapping to M0 resp.M2, one annulus A without puncture mapping to M1, and two seams connecting the boundary ofeach disk to one of the annulus boundary components, and with seam condition in L01 and L12respectively. In the definition of (L01,L12) we can pick any complex structure on (in particular)the annulus, e.g. A = R/Z [0, ] C/Z with the standard complex structure for any > 0. Forthe other patches we pick the same complex structures and perturbations as some regular choice inthe definition of L01L12 . We claim that this provides regular moduli spaces for (L01,L12) as well,

    and that the 0-dimensional spaces of solutions are in fact bijective to those for L01L12. For theproof of this bold claim we refer to [17], where the same analysis (just somewhat harder due to thenoncompactness of the shrinking domain) is carried out in the case where A is replaced by a stripR [0, ] betweeen other strips of fixed width. The crucial part of this proof is to exclude bubblingeffects: As 0, one may obtain sphere bubbles in M0, M1, or M2 as well as disk bubbles in(M0 M1, L01), (M

    2 M2, L12), or (M

    0 M2, L01 L12), and finally a novel type of bubble, called

    the figure eight bubble, see [17, 19] for details. For the latter we are currently lacking a removablesingularity theorem, so cannot exclude it by reasons such as a trivial homotopy class of figure eightmaps or high codimension (in fact, we expect cases where these bubbles appear generically). Oursolution is to use the full power of the monotonicity assumption together with an (indirectly proven)energy quantization for the figure eight bubble: If it appears then the remaining main componentis a holomorphic quilt contributing to L01L12 which has less energy than the index 0 solutionscontributing to (L01,L12). However, the energy for index 0 solutions is the same for both relative

    quilt invariants, so by strict monotonicity the remaining solution would have negative index whichis excluded by regularity of the moduli spaces. This proves the equality (L01,L12) = L01L12 andthus finishes the proof of the Theorem.

    References

    [1] A. Abbondandolo and M. Schwarz. Floer homology of cotangent bundles and the loop product. arXiv:0810.1995.

    [2] P. Albers. On the extrinsic topology of Lagrangian submanifolds. IMRN, 38:23412371, 2005.

    [3] P. Albers. Erratum for On the extrinsic topology of Lagrangian submanifolds [mr2180810]. Int. Math. Res.Not. IMRN, (7):13631369, 2010.

    [4] T. Coates and Y. Ruan. Quantum cohomology and crepant resolutions: A conjecture, 2007. arXiv.org:0710.5901.

  • 8/3/2019 Katrin Wehrheim and Chris T. Woodward- Pseudoholomorphic Quilts

    32/32

    32 KATRIN WEHRHEIM AND CHRIS T. WOODWARD

    M2

    L12

    M1M0

    =

    M0 M2L01

    L12

    L01

    Figure 16. Annulus shrinking proving the equivalence of algebraic and geometric composition

    [5] M. Khovanov and L. Rozansky. Topological Landau-Ginzburg models on the world-sheet foam. Adv. Theor. Math.

    Phys., 11(2):233259, 2007.

    [6] D. McDuff and D.A. Salamon. J-holomorphic curves and symplectic topology, volume 52 of American Mathe-

    matical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2004.

    [7] Y.-G. Oh. Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I. Comm. Pure Appl.

    Math., 46(7):949993, 1993.

    [8] T. Perutz. Lagrangian matching invariants for fibred four-manifolds. I. Geom. Topol., 11:759828, 2007.

    [9] S. Piunikhin, D.A. Salamon, and M. Schwarz. Symplectic Floer-Donaldson theory and quantum cohomology. InContact and symplectic geometry (Cambridge, 1994), volume 8 of Publ. Newton Inst., pages 171200. CambridgeUniv. Press, Cambridge, 1996.

    [10] J. Robbin and D.A. Salamon. The spectral flow and the Maslov index. Bull. London Math. Soc., 27(1):133,1995.

    [11] D.A. Salamon. Lectures on Floer homology. In Symplectic geometry and topology (Park City, UT, 1997), volume 7of IAS/Park City Math. Ser., pages 143229. Amer. Math. Soc., Providence, RI, 1999.

    [12] M. Schwarz. Cohomology Operations from S1-Cobordisms in Floer Homology,. PhD thesis, ETH Zurich, 1995.www.math.uni-leipzig.de/schwarz/.

    [13] P. Seidel. Graded Lagrangian submanifolds. Bull. Soc. Math. France, 128(1):103149, 2000.

    [14] P. Seidel. A long exact sequence for symplectic Floer cohomology. Topology, 42(5):10031063, 2003.

    [15] P. Seidel. Fukaya categories and Picard-Lefschetz theory. Zurich Lectures in Advanced Mathematics. EuropeanMathematical Society (EMS), Zurich, 2008.

    [16] K. Wehrheim. Energy quantization and mean value inequalities for nonlinear boundary value problems. J. Eur.Math. Soc. (JEMS), 7(3):305318, 2005.

    [17] K. Wehrheim and C. Woodward. Floer cohomology and geometric composition of Lagrangian correspondences.arXiv:0905.1368.

    [18] K. Wehrheim and C. Woodward. Functoriality for Lagrangian correspondences in Floer theory. to appear in

    Quantum Topology. arXiv:0708.2851.

    [19] K. Wehrheim and C. Woodward. Quilted Floer cohomology. Geometry & Topology, 14:833902, 2010.

    arXiv:0905.1370.

    [20] K. Wehrheim and C. Woodward. Orientations for pseudoholomorphic quilts. 2009 preprint.

    Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139. E-mailaddress: [email protected]

    Department of M athematics, Rutgers University, Piscataway, NJ 08854. E-mail address: [email protected]

    K.W. was partially supported by NSF CAREER grant DMS0844188. C.W. was partially supported by NSF grant

    DMS0904358.