An invariant of link cobordisms from symplectic Khovanov homology
aa r X i v : . [ m a t h . S G ] F e b An invariant of link cobordisms fromsymplectic Khovanov homology
Jack W. WaldronNovember 15, 2018
Abstract
Symplectic Khovanov homology is an invariant of oriented links defined by Seidel andSmith [1] and conjectured to be isomorphic to Khovanov homology. We define morphisms(up to a global sign ambiguity) between symplectic Khovanov homology groups, corre-sponding to isotopy classes of smooth link cobordisms in R × [0 ,
1] between a fixed pair oflinks. These morphisms define a functor from the category of links and such cobordisms tothe category of abelian groups and group homomorphisms up to a sign ambiguity. This pro-vides an extra structure for symplectic Khovanov homology and more generally an isotopyinvariant of smooth surfaces in R ; a first step in proving the conjectured isomorphism ofsymplectic Khovanov homology and Khovanov homology. The maps themselves are definedusing a generalisation of Seidel’s relative invariant of exact Lefschetz fibrations [2] to exactMorse-Bott-Lefschetz fibrations with non-compact singular loci. Contents
E, π, Ω E , Θ E , J E , D, j ) . . . . . . . . . . . . . . . . . . 263.2 The symplectic associated bundle E × S P ≃ E × C ∗ ( F \
0) . . . . . . . . . . . 293.3 Constructing a regular S –equivariant almost complex structure . . . . . . . . 303.3.1 Regularity over the boundary marked points . . . . . . . . . . . . . . . 313.3.2 Regularity over D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.4 Two calculations of the relative invariant . . . . . . . . . . . . . . . . . . . . . 36 The Jones polynomial V L , introduced in [3], is an invariant of oriented links L ⊂ S , motivatedoriginally by a connection between the Yang-Baxter equation in statistical mechanics and thebraid group. Importantly, V L satisfies skein relations: t − / V + t v/ V + t − V = 0 t v/ V + t / V + tV = 0The pictures in the skein relations above depict a region of a crossing diagram of three orientedlinks, which are identical elsewhere. v is a term, arising as a signed count of certain crossings,which compensates for the fact that there is no canonical choice of orientation on one resolutionof any crossing.Together with the normalisation V unknot = 1, this allows the Jones polynomial to be algo-rithmically computed in a straightforward, though computationally intensive, manner from acrossing diagram of a link. However, the geometric meaning of the Jones polynomial is stillpoorly understood. One hopes that an intrinsically geometric definition of the Jones polynomialwould advance geometric applications to knot theory.Khovanov [4, 5] defines a bigraded abelian group KH ∗ , ∗ ( L ), which categorifies the Jonespolynomial. It is widely known as the Khovanov homology of L , although technically a co-homology theory. The skein relations for the Jones polynomial are replaced by long exactsequences for Khovanov homology (bi-degrees are indicated on the arrows):KH ∗ , ∗ ( ) KH ∗ , ∗ ( ) (0 , − " " ❉❉❉❉❉❉❉❉❉❉❉❉ KH ∗ , ∗ ( )KH ∗ , ∗ ( ) ( − v, − v − o o KH ∗ , ∗ ( ) KH ∗ , ∗ ( ) ( v +1 , v +2) < < ③③③③③③③③③③③③ KH ∗ , ∗ ( ) KH ∗ , ∗ ( ) ( − v +1 , − v +2) " " ❉❉❉❉❉❉❉❉❉❉❉❉ KH ∗ , ∗ ( )KH ∗ , ∗ ( ) ( v, v +1) o o KH ∗ , ∗ ( ) KH ∗ , ∗ ( ) (0 , < < ③③③③③③③③③③③③ A consequence of this is that the Jones polynomial is recovered from a change of variableson the bigraded Poincar´e polynomial of KH ∗ , ∗ ( L ). V L = " P i,j ( − i q j dim(KH i,j ( L ) ⊗ Q ) q + q − q = t / Khovanov homology is also defined in terms of algorithmically computable algebra relatedto crossing diagrams. Its other strengths include, in particular, that it fits into a topological2uantum field theory for knotted-surfaces in R . This property was used by Rasmussen [6]to give a purely combinatorial proof of Milnor’s conjecture on the smooth slice genus of torusknots. The same work can also be used to construct an exotic smooth structure on R .The topological field theory is described as follows. Let Cob be the category whose objectsare oriented links in S and whose morphisms L / / L are isotopy classes of smooth linkcobordisms from L to L in S × [0 , Ab ± be the category of finitely generated, bigradedabelian groups and bigraded homomorphisms defined only up to an overall sign ambiguity. Itis shown [7] that there is a non-trivial functor from Cob to Ab ± , which maps a link L tothe Khovanov homology KH ∗ , ∗ ( L ). To be more precise, it takes L to the Khovanov homologyof the crossing diagram created by projecting a C –small perturbation of L orthogonally to { (0 , , z ) ∈ R } . One often phrases this result as “ Khovanov homology is functorial with respectto smooth link cobordisms ”. Symplectic Khovanov homology KH ∗ symp ( L ) is another invariant of oriented links, due toSeidel and Smith [1]. More recently, it has also been extended to an invariant of tangles byRezazadegan [8]. For links, it is a singly-graded abelian group, defined using Lagrangian Floercohomology in an auxiliary symplectic manifold ( Y m,P , Ω), and Lagrangian submanifolds in Y m,P derived from a presentation of L as a braid closure. Section 4 introduces symplecticKhovanov homology in more detail, as well as extending the definition to work with bridgediagrams of links. In doing so, we ignore orientations of the links, which amounts to replacingthe absolute grading on symplectic Khovanov homology by a relative grading. For the purposesof this paper, we do not require a more precise understanding of the absolute grading. It sufficesto know that the absolute grading always exists, given a choice of orientation.Seidel and Smith conjecture that symplectic Khovanov homology is in fact the same asKhovanov homology. Since KH ∗ symp ( L ) has only one grading, the conjecture is usually phrasedas: Conjecture 1.
There is a canonical isomorphism KH k symp ( L ) ∼ = L i − j = k KH i,j ( L )Here, one has also to be careful with the meaning of canonical . Section 5.6 defines the sym-plectic Khovanov homology of a crossing diagram up to canonical isomorphism, thus allowingone to be precise about the meaning of the above conjecture.The main aim of this paper is to exhibit the same functoriality with respect to smooth linkcobordisms for symplectic Khovanov homology as exists in the setting of Khovanov homology.Recent work of Rezazadegan [9] has independently exhibited similar homomorphisms betweensymplectic Khovanov homology groups (for tangles) corresponding to elementary cap, cup andsaddle cobordisms. He also exhibits some important structure, relevant to Conjecture 1, butdoes not show that these homomorphisms give an invariant of link cobordisms. Theorem 1.1.
Let Ab ± be the category of finitely generated, singly graded abelian groupsand graded homomorphisms, defined only up to an overall sign ambiguity. There is a non-trivial functor from Cob to Ab ± , which maps a link L to the symplectic Khovanov homology KH ∗ symp ( L ) . In particular, some care is needed to define the symplectic Khovanov homology up tocanonical isomorphism for a general link in R . In fact, the same trick of defining symplecticKhovanov homology for crossing diagrams together with a small perturbation of the link workshere too.One can consider Theorem 1.1 as a first step in proving Conjecture 1, since Khovanovhomology is defined in terms of elementary cobordisms between unlinked unions of unknots(and the homomorphisms corresponding to general cobordisms arise directly from these).3ombining the theorem with a simple generalisation of Wehrheim and Woodward’s exactsequence [10] (one has to deal with non-compactness issues similar to those in Section 2) oneshould already be able to exhibit skein exact triangles for symplectic Khovanov homology. Inparticular, Theorem 1.1 should imply the existence of a spectral sequence with Z / Theorem 1.2.
Let L be an unlinked union of links L and L . Then KH ∗ symp ( L ) is the K¨unnethproduct of KH ∗ symp ( L ) and KH ∗ symp ( L ) (i.e. tensor product at the level of chain complexes)up to an overall grading shift. The underlying motivation for the proofs of both results is in the geometry of configurationspaces Conf n ( C ) and Conf n ( C ). In the construction of symplectic Khovanov homology, oneuses the following correspondences to present links in terms of these configuration spaces: • A braid joining a configuration P ∈ Conf n ( C ) of points in the plane to another config-uration Q corresponds to a path P to Q in the configuration space. • Let γ : [0 , / / Conf n ( C ) be a vanishing path ; that is, a smooth path hitting Conf n ( C ) \ Conf n ( C ) transversely, only at γ (0) = (0 , , µ , . . . , µ n − ) such that the µ i are pairwisedistinct. Then γ corresponds to a (2 n − , n ) tangle between configurations γ (0) and γ (1).Suppose we have a map u : D / / Conf n ( C ) intersecting Conf n ( C ) \ Conf n ( C ) transverselyonly on D and only in configurations (0 , , µ , . . . , µ n − ), such that the µ i are pairwise distinct.Then from this one can construct a smooth braid cobordism from the trivial braid at u (1) tothe braid described by u ( ∂ D ). In fact, braid cobordisms up to isotopy, correspond to isotopyclasses of such maps. For details on this correspondence see Section 5.With the further assumption that u is a holomorphic embedding near where it intersectsConf n ( C ) \ Conf n ( C ), there is a natural construction of a singular symplectic fibration over D .These have fibres over ± Y n,u ( ± , in whichKH ∗ symp is defined as a Floer cohomology group. In a manner motivated by Seidel’s relativeinvariant of exact Lefschetz fibrations [2], one would then like to define morphisms betweensymplectic Khovanov homology groups by counting holomorphic sections of these fibrations.In fact, the above describes essentially the method of this paper. The fibrations are allof a particularly nice form (exact Morse-Bott-Lefschetz fibrations) and relative invariants canbe defined in an analogous manner. However, essential non-compactness issues (arising fromnon-compactness of the singular loci) cause problems for convexity, gluing and even definingsymplectic parallel transport in these fibrations.Section 2 develops general tools for studying relative invariants in these fibrations. Inparticular, it is also shown in general how to construct exact Morse-Bott-Lefschetz fibrationstogether with relative invariants from maps of surfaces into singular holomorphic fibrationsof Stein manifolds. Section 3 then develops some specific tools for the calculation of Floercohomology and relative invariants of symplectic associated bundles necessary for proof ofTheorem 1.1 given in the later sections. 4 cknowledgments The author would like to thank the anonymous referee who pointed out an error in an earlierversion of the paper, and also Ivan Smith and Dominic Joyce for important discussions relatingto the correction.
Given a pair L , L ′ of closed, connected Lagrangian submanifolds of a symplectic manifold M satisfying certain conditions, one can define a relatively graded abelian group HF ( L, L ′ ).This is the Floer cohomology, or in our case, the “Lagrangian intersection Floer cohomology”.The cohomology is obtained from a chain complex made of formal sums (over Z ) of transverseintersections of the Lagrangians and a differential defined by counting holomorphic strips withboundary on the Lagrangians which interpolate between intersections. Remark 2.1.
An example of conditions in which the Floer cohomology is defined is given bythe following: • M is a K¨ahler manifold on which the symplectic form is exact and the underlying complexstructure makes M a Stein manifold • c ( M ) = 0 and H ( M ) = 0 • H ( L ) = H ( L ′ ) = 0 • w ( L ) = w ( L ′ ) = 0 (equivalently L , L ′ are spin)These conditions are satisfied where Floer cohomology is used for the definition of symplecticKhovanov homology in [1]. Unless otherwise mentioned these are the conditions under whichwe will use Floer cohomology in this paper.Floer cohomology is defined up to canonical isomorphism, even when each Lagrangian isspecified only up to compactly supported Hamiltonian isotopy. These isotopies and canonicalisomorphisms are used to define the cohomology even when the Lagrangians do not intersecttransversely. Remark 2.2.
The condition that L , L ′ be spin is necessary only to use Z coefficients for HF ( L, L ′ ). Without it Floer cohomology is still defined with Z / L, L ′ are orientable. L , L ′ being spin implies the orientability of the moduli spaces of holomorphic strips (seeLemma 22.11 of [12]), the counting of which defines the differential. The orientation gives aconsistent choice of signs for this counting process.A symplectic vector bundle (cf. [13]) is a vector bundle E / / B with a smooth choice ofskew symmetric bilinear form on each fibre (i.e. a section Ω of Λ E ), which is non-degenerateon each fibre. This is the local (first order) model for a symplectic fibration (where Ω is insteada closed 2–form on the total space).To be more precise, an exact symplectic fibration is a manifold E with corners and a smoothfibration π : E / / B equipped with an exact 2–form Ω = d Θ on E whose restriction to fibresof E is a symplectic form. We shall also require that the corners of E are precisely the boundarypoints of the fibres over ∂B . 5on-degeneracy of Ω on the vertical tangent spaces T E v = ker Dπ means that we candefine horizontal tangents to be T E h = (ker Dπ ) ⊥ Ω . This defines the symplectic connection and symplectic parallel transport over any path γ in the base. As long as points do not flowunder symplectic parallel transport off of the boundary of E , the symplectic parallel transportdefines maps between the fibres over the start and end points of γ . These maps are symplecto-morphisms between the fibres. Isotopic paths in the base yield parallel transport maps whichdiffer by exact symplectomorphisms.In [2] these fibrations are generalised to exact Lefschetz fibrations (over surfaces), by al-lowing complex non-degenerate singularities of π . The monodromy by parallel transport onceanticlockwise around such a singular value in the base is then a Dehn twist σ in the Lagrangianvanishing cycle associated to the singular point. Take an exact Lefschetz fibration over the in-finite strip R × [0 ,
1] which has trivialised symplectic parallel transport over the ends (givingwell-defined fibres at ±∞ ). One assigns to the fibre at + ∞ a pair of exact Lagrangian sub-manifolds L ∞ , L ∞ . Extending these by symplectic parallel transport over the boundaries { } × R , { } × R respectively, Seidel defines a map from the Floer cohomology in the fibre at −∞ to that in the fibre at + ∞ . HF ( L top −∞ = σL top + ∞ , L bottom −∞ ) / / HF ( L top + ∞ , L bottom + ∞ ) L ∞ L ∞ L −∞ = σ ( L ∞ ) L −∞ = L ∞ Figure 1: A basic Lefschetz fibration over the infinite strip with one singular fibre.The orientation indicated on the boundary is such as to make the total monodromy asingle positive Dehn twist σ in the vanishing cycle. The fibres at ±∞ have been iden-tified by symplectic parallel transport along the lower boundary, so the monodromyoccurs entirely on the upper boundary. Working with KH symp , we have a natural construction of singular symplectic fibrations frombraid cobordisms. The singularities that arise in this construction have a slightly more generalform than those of the exact Lefschetz fibrations, considered in [2]. In this section, I describethe corresponding construction for exact Morse-Bott-Lefschetz (MBL) fibrations over surfaces.
Definition 2.3.
An exact
MBL–fibration is a collection (
E, π, Ω , Θ , J , B, j ) such that:(1) E is a smooth, not necessarily compact, manifold with boundary ∂E .(2) B is a Riemann surface with complex structure j , homeomorphic to D with finitely manyboundary points removed.(3) π : E / / B is a smooth map with ∂E = π − ( ∂B ) and such that π | ∂E : ∂E / / ∂B is asmooth fibre bundle. 64) Ω = d Θ is an exact 2–form on E , non-degenerate on T E v := ker Dπ at every point in E .(5) π has finitely many critical values, all in the interior of B .(6) J is an almost complex structure defined on some subset of E which contains a neighbour-hood of the set Crit( π ) of critical points and the complement U of some fibrewise compactsubset of E .(7) π is ( J , j )–holomorphic and Ω( ., J . ) | ( T E v ) ⊗ is everywhere symmetric and positive definite(where J is defined).(8) J preserves T E h on U .(9) Ω is a K¨ahler form for J on some open neighbourhood of Crit( π ).(10) Crit( π ) is smooth and the complex Hessian of π is non-degenerate on complex complementsof T Crit( π ) in T E .Seidel’s exact Lefschetz fibrations have boundary in the fibre direction near which there isa trivialisation of E compatible with Ω and Θ (c.f. [2]). One cannot expect such trivialisationsat boundaries to exist for exact MBL-fibrations, since the singular locus can escape to infinityin a fibre. For this reason, we don’t require there to be trivialisations. We compensate for thisby taking significantly more care with convexity and gluing of exact MBL-fibrations. This isthe main difficult content of Section 2.2.We will consider exact MBL-fibrations with bases B which are of a particular form. Namely B should be a Riemann surface with finite sets I ± of ends (see below), not both empty. Theends may be of two forms: Definition 2.4 (Striplike ends (cf. [2])) . A striplike end e ∈ I ± of a surface B is a properholomorphic embedding γ e : [0 , ∞ ) × [0 , / / B (with the standard complex structure on [0 , ∞ ) × [0 , ⊂ C ) such that γ − e ( ∂B ) = [0 , ∞ ) ×{ , } .An exact MBL–fibration is trivial over the striplike end e if over the image of γ e it is nonsingular and isomorphic as an exact symplectic fibration to [0 , ∞ ) × [0; 1] × E z for some fibre E z . Here one takes Ω and Θ pulled back by the projection to E z , and J split as the sum ofan almost complex structure on the E z –factor and the standard almost complex structure onthe [0 , ∞ ) × [0 , Definition 2.5. A boundary marked point z ∈ ∂B together with a proper holomorphic em-bedding γ e : [0 , ∞ ) × [0 , / / B \ z such that γ − e ( ∂B ) = [0 , ∞ ) × { , } and γ e ( x, t ) / / z as x / / ∞ may also be considered anend. Exact MBL-fibrations are not required to be trivial over these ends.Ends given by boundary marked points can be viewed as striplike ends without the triviali-sation and striplike ends can be completed, by adding a single fibre at infinity , to give boundarymarked points. Switching between these two settings will be important later on. Definition 2.6.
By the fibre at an end e of B we mean: • the fibre over the boundary marked point7 the fibre at infinity of a striplike endWe require also, that the ends of B be pairwise disjoint and that the complement of theends (i.e. of the images of the γ e and any boundary marked points) be compact. This meansin particular that the boundary of B with boundary marked points removed decomposes intoas many open intervals as there are ends.The benefit of striplike ends (with accompanying trivialisations) is that it is easy to compose exact MBL–fibrations at striplike ends. Namely, one forms the composite by gluing oppositelyoriented, but otherwise identical trivialisations of two separate exact MBL–fibrations together.In contrast, the benefit of boundary marked points is twofold. They arise more naturally(see Section 2.2) and holomorphic convexity in the fibre direction is easier to attain.We now define what we mean by exact Lagrangian boundary conditions for an exact MBL-fibration ( E, π ) over a surface B with ends. Definition 2.7.
Let P be the set of boundary marked points of B . An exact Lagrangianboundary condition on ( E, π ) is a subbundle Q of E over ∂B \ P together with a function K Q : Q / / R such that:1. Ω | Q = 02. for any z ∈ ∂B the restriction ( Q z , K Q | Q z ) is a closed, connected exact Lagrangiansubmanifold of E z (i.e. a Lagrangian submanifold such that also d ( K Q | Q z ) = Θ | Q z ).3. ( Q z , K Q | Q z ) extends smoothly, along each component of ∂B \ P , to the fibres over bound-ary marked points. This extension is allowed to depend on the side from which oneapproaches a boundary marked point.4. ( Q z , K Q | Q z ) is constant w.r.t. trivialisations over the striplike endsGiven an exact MBL–fibration over a surface with striplike ends, one can construct anexact MBL–fibration over a surface with boundary marked points. Namely, the base can becompactified by adding a single point at infinity at each end. One then adds to the total spacethe fibres at infinity.Condition (1) implies that Q is preserved by symplectic parallel transport over ∂B and thatΘ | Q = dK Q + π ∗ κ Q for some κ Q ∈ Ω ( ∂B ) (cf. [2] Lemma 1.3). Condition (2) and triviality ofthe striplike ends gives κ Q = 0 there. In fact κ Q = 0 whenever symplectic parallel transportpreserves K Q . Q specifies a pair of Lagrangian submanifolds in the fibre over each marked point and ineach fibre at infinity (i.e. ‘in the fibre at each end’). We will refer to Q as transverse , if thesepairs of Lagrangians are each transverse. Remark 2.8.
Assume we are given a choice of exact Lagrangian submanifold (
L, K L ) in thefibre at infinity or fibre over a boundary marked point at one end of each edge of ∂B . Theneither symplectic parallel transport maps restricted to one of these Lagrangians fail to bedefined over the entire edge on which it lies, or else they are defined and the condition κ Q = 0uniquely specifies a Lagrangian boundary condition. This is the manner in which all exactLagrangian boundary conditions in this paper are constructed.We will be interested in counting compact moduli spaces of holomorphic sections withboundary in Q , so it makes sense to require holomorphic convexity of a neighbourhood in E containing Q as follows: 8 efinition 2.9. An enclosed exact Lagrangian boundary condition ( Q, ρ, R ) is an exact La-grangian boundary condition Q , together with a smooth map ρ : E / / R ≥ and R >
0, suchthat:(i) ρ splits w.r.t. the trivialisations over all striplike ends(ii) ρ − [0 , R ] is fibrewise compact and ρ − [0 , R ) contains Q (iii) ∃ ε > ρ − ( R − ε, R ) • J is defined • ρ is subharmonic w.r.t. J and plurisubharmonic on fibresWe refer to the pair ( ρ, R ) as an enclosure . Remark 2.10.
We do not require J to be integrable. ρ : E / / R ≥ is defined to be sub-harmonic or plurisubharmonic when − d ( dρ ◦ J )( ., J . ) is ≥ >
0, respectively. Given any( i, J ) –holomorphic map from a subset of C into E , the composite ρ ◦ u is subharmonic orplurisubharmonic in the conventional sense.Suppose we extend J to an almost complex structure on E which makes the projection π holomorphic. Then subharmonicity of ρ gives a maximum principle for holomorphic sectionsof E . Namely, they cannot have a maximum of ρ in the range ( R − ǫ, R ). This ensuresthat families of holomorphic sections which are confined to ρ − [0 , R − ǫ ) cannot degenerate tosections which escape this region. The choice of enclosure is important, since in general therewill exist holomorphic sections which leave the region ρ − [0 , R ).It will often be necessary to change enclosures other than just by isotopy through enclosures.For this, we will need a notion of equivalence of enclosures. Definition 2.11.
We say that two enclosures ( ρ, R ) and ( σ, S ), for a given exact Lagrangianboundary condition Q , are equivalent if for some R ′ < R we have: • ρ − [0 , R ′ ) ⊂ σ − [0 , S ) ⊂ ρ − [0 , R ) • ρ is subharmonic on ρ − [ R ′ , R ]or also if they can be related by a sequence of such comparisons, either way round. Forsimplicity, we require that J is fixed throughout these comparisons.In particular this makes enclosures related by isotopy of enclosures equivalent.As defined earlier, an exact MBL–fibration carries a two form which is not necessarilysymplectic on the total space. It is important to observe that this is more flexible than requiringΩ to be an exact symplectic structure on the total space (within a particular enclosure), butit is no weaker for our purposes. Lemma 2.12.
Let ( E, π, Ω , Θ , J , B, j ) be an exact MBL–fibration. Given any enclosure ( ρ, R ) ,there is a canonical choice of isotopy class of exact symplectic form Ω ′ on the total space of ρ − [0 , R ] with: • the same restriction to T E v as Ω • the same symplectic connection as Ω 9 roof.
Let ω be any exact volume form on B compatible with j (which in 2 dimensions meansonly that it induces the correct orientation). Ω restricts to horizontal vectors T z E h for z ∈ E \ Crit( π ) as multiple of π ∗ ω by a value f ( z ). Non-degeneracy of Ω at z is equivalent to f ( z )being non-zero. Furthermore, Ω is non-degenerate at all z in a neighbourhood of Crit( π ) bydefinition.Ω is K¨ahler on some open neighbourhood V of Crit( π ), so f is strictly positive on V \ Crit( π ).The region ( E \ V ) ∩ ρ − [0 , R ) is fibrewise compact and becomes compact when one extendsto the fibres at infinity. f is smooth away from V , so is bounded below. Ω ′ := kπ ∗ ω + Ωsatisfies the necessary axioms for an exact MBL–fibration, and for large enough k ∈ R it isnon-degenerate on ρ − [0 , R ), as required in the lemma. Remark 2.13.
Lemma 2.12 allows one to perform standard holomorphic disc counting con-structions within the enclosure ( ρ, R ) to get invariants of exact Lagrangian boundary conditionsup to Lagrangian isotopy in a similar manner to the way one defines Floer cohomology in exactsymplectic manifolds. See Section 2.3 for more details.
Let E be a Stein manifold with plurisubharmonic function ρ ≥ d Θ := − d ( dρ ◦ i ). Suppose we have a singular holomorphic fibration π : E / / N overa complex manifold N and a smooth map u : B / / N , for some simply connected Riemannsurface B with striplike ends or marked points on the boundary. When the singularities of u ∗ E are of Morse-Bott-Lefschetz type and u is holomorphic near singular points, one can view u ∗ E naturally as an exact MBL–fibration.This section gives a more detailed construction of exact MBL–fibrations in the mannerdescribed above. Most of the content deals with the problems of convexity (for defining en-closed Lagrangian boundary conditions consistently) and composition of fibrations (by gluing trivialisations over striplike ends). Without these trivialisations, the gluing construction forcomposing the relative invariants (cf. Section 2.3) would be difficult. Definition 2.14.
We say a singular value p ∈ N of π is MBL if there is a neighbourhood ofany point in the singular locus Crit( π, p ) fitting into the following commutative diagramCrit( π, p ) × C k × C n E local near Crit( π,p ) o o EN π (cid:15) (cid:15) Crit( π, p ) × C k × C n C × C n ( x , y , z ) ( P y i , z ) (cid:15) (cid:15) C × C n N local biholomorphism near (0 , ) o o Here the top map is a local embedding of exact K¨ahler manifolds defined on a neighbourhoodof Crit( π, p ). However, if we only require it to be holomorphic, we can always deform the exactK¨ahler structure on E to make it an embedding of exact K¨ahler manifolds.We shall denote the set of such critical values by MBL( π ). It is a submanifold of codimension2. The open set of regular values we denote by N reg .This definition can equivalently be expressed in terms of smoothness of critical loci in E and N and non-degeneracy of the Hessian of π on complements to T Crit( π ) ∩ ker( Dπ ) withinker( Dπ ). 10 efinition 2.15. Let (
B, j ) be a simply connected Riemann surface (
B, j ) with striplike endsor marked points on its boundary. Let u : B / / N reg ∪ MBL( π ) be a smooth map with theproperties that: • u is a constant map on each of the striplike ends • u ( ∂B ) ⊂ N reg • u is transverse to MBL( π ) and holomorphic near it.We call such a map admissible . We shall refer to u − (MBL( π )) ⊂ ( B, j ) as the singular values ,since these are the singular values of the pullback fibration u ∗ E .Given an admissible map u , we consider the pullback fibration π : u ∗ E / / B equippedwith u ∗ Ω, u ∗ Θ, u ∗ ρ . There is also a natural choice of almost complex structure ˜ J which agreeswith u ∗ J where u is a holomorphic immersion. Namely, one takes u ∗ J on vertical tangentsand i on horizontal tangents with respect to the symplectic connection induced by u ∗ Ω (and u ∗ J where the fibration is singular). These choices make ( u ∗ M, π, u ∗ Ω , u ∗ Θ , ˜ J , B, j ) an exactMBL–fibration.Furthermore, the isotopy class of u through admissible maps defines an isotopy class ofexact MBL–fibrations. Lemma 2.16.
Given u , B and j as above up to smooth homotopy of u and deformation of j we have an exact MBL–fibration defined up to smooth deformation of the parameters. The rest of this section deals with the deformations needed to construct Lagrangian bound-ary conditions and then ensure holomorphic convexity of a surrounding region (thus makingan enclosed Lagrangian boundary condition). The approach is to approximate B by a treeof embedded holomorphic discs connected at marked points and solve the same problem forembedded holomorphic discs.First we consider the model case, where u is already a holomorphic embedding and B isthe closed unit disc D with finitely many (but at least one) marked points on the boundary. Inthis case the exhausting, plurisubharmonic function ρ : E / / [0 , ∞ ) pulls back to a fibrewise-exhausting, plurisubharmonic function on u ∗ E . Lemma 2.17.
Let ( E, π, Ω , Θ , J , D , j ) be an exact MBL–fibration over the closed unit disc.Assume further that we have ρ : E / / [0 , ∞ ) exhausting (fibrewise), and plurisubharmonicwhere J is defined, such that dρ = Θ ◦ J .For any l ∈ R , one can deform Θ (without changing the restriction of Ω to fibres) insidesome level set ρ max of ρ such that symplectic parallel transport flow lines over paths of lengthat most l in ∂ D do not leave ρ − [0 , ρ max ] .The deformation occurs only over a small open neighbourhood of ∂ D and is well-defined upto isotopy through such deformations of Θ . Furthermore, the deformation may be chosen tohave support disjoint from any particular compact set.Proof. Let A be a small annular neighbourhood of ∂ D not containing any critical values of π . Let ρ be large enough such that for all z ∈ A all critical values of ρ | A are less than ρ .This implies that ρ | − π − ( A ) ( ρ ) is a smooth fibration over A with compact fibre C and hencecarries a flat connection, well-defined up to isotopy. Choose one. It gives a trivialisation overany small open neighbourhood U ⊂ A in the base of the form proj U : C × U / / U . Extend-ing this in the positive time direction by the fibrewise Liouville flow we have a trivialisationproj U : C × [0 , ∞ ) × U / / U . 11n any fibre E z the form Θ restricts to C × { } × z as a contact form Θ ,z on C andrestricts to the whole fibre as e y Θ ,z (here y is the coordinate on [0 , ∞ )). We define Θ C on thetrivialisation C × [0 , ∞ ) × U to have this same restriction to fibres and to vanish on T U . Inparticular Θ C | E z = Θ | E z .Let R be the Reeb vector field on C for the contact form Θ ,z . We can split T E in thetrivialised region as R R ⊕ ker(Θ ,z ) ⊕ R ∂∂y ⊕ T U . Given a vector H in T U the symplectic paralleltransport vector w.r.t. the 2–form d ( e y Θ C ) over it is of the form w R R + W con + w y ∂∂y + H inthat splitting. It has the defining property that for any v R R + V con + v y ∂∂y we have:0 = d ( e y Θ C ) (cid:18) w R R + W con + w y ∂∂y + H, v R R + V con + v y ∂∂y (cid:19) = ( e y d Θ C + e y dy ∧ Θ C ) (cid:18) w R R + W con + w y ∂∂y + H, v R R + V con + v y ∂∂y (cid:19) = e y [ − H (Θ C )( v R R + V con ) + d (Θ ,z )( V con , W con ) + v y w R − v R w y ]Setting v R = 1, V con , v y = 0 gives w y = − H (Θ C )( R ) which by compactness has a finitemaximum over A . i.e. the velocity of this symplectic parallel transport in the y –direction isbounded over compact subsets of the base. This controls the symplectic parallel transport flowlines in large enough regions, so in particular proves the lemma for any deformed Θ whichequals e y Θ C on ρ | − π − ( A ) [ ρ + 1 , K −
1) for large enough K .Now we define a deformation ˜Θ = g Θ + (1 − g )Θ C with a bump function g identically equalto 1 on ρ | − π − ( A ) [ ρ + 1 , K −
1) and zero on the complement of ρ | − π − ( A ) [ ρ , K ). This is therequired deformation of Θ to prove the lemma. Remark 2.18.
Suppose that Crit( π ) is compact. For example this is the case when thesingularities are actually of Lefschetz type. Then, by the same technique, one can contain thesymplectic parallel transport over paths of length at most l in D , not just in the boundary. Remark 2.19.
Suppose we are really just interested in defining symplectic parallel transportmaps over a path γ : [0 , / / N reg , then one can run the same argument in the pullbackfibration γ ∗ E . This gives symplectomorphisms E γ (0) / / E γ (1) defined on any compact subsetof E γ (0) and well-defined up to isotopy within the class of symplectic embeddings (or alsoinclusion, should one enlarge the choice of compact subset).The lemma above allows us to define Lagrangian boundary conditions on such a fibrationsimply by specifying a Lagrangian in a single fibre on each interval of ∂ D and extending to therest of the interval by symplectic parallel transport. Call this Lagrangian boundary condition Q . Furthermore, the region of deformation is contained within a finite level set of ρ , so for alllarge enough R ∈ R ≥ the collection ( Q, ρ, R ) is an enclosed Lagrangian boundary condition.If the Lagrangians in the construction are chosen to be exact, then Q is also exact.Given a surface with striplike ends mapping to D , with edges mapping monotonically to ∂ D ,symplectic parallel transport respects the pullback. Hence, we can similarly control symplecticparallel transport of compact Lagrangian boundary conditions specified in the fibre at infinityover one end of each edge. However, it is not so easy to show these are enclosed.We will now construct enclosures containing these Lagrangian boundary conditions in amodel case where we have deformed the previous fibration over D to have a base with strip-like ends. The enclosures constructed will be defined as deformations of an enclosure ( ρ, R )12erformed together with the deformation which forms the striplike ends. Furthermore, theenclosures will be compatible with the trivialisations of E over the striplike ends, so will becompatible with the construction of gluing fibrations over striplike ends.Label the marked points { z , . . . , z n } ⊂ ∂ D . A neighbourhood of each of these markedpoints becomes a striplike end under the appropriate coordinate change. Namely, we viewsuch a neighbourhood (holomorphically) as a neighbourhood of 0 in the upper half plane H (well-defined up to rescaling of H ) which corresponds to a strip by the map z log z . Withoutloss of generality this model is valid on { z ∈ H : | z | ≤ } and furthermore this region containsonly regular values of π .Locally near each z i we take e D / / D to be the identity away from the z i and near themto be given by: u : e H / / H re iθ h ( r ) e iθ Here e H is H \ h : R ≥ / / R ≥ is a smoothfunction such that: • h ( r ) = 0 for r ≤ • h ( r ) = r for r ≥ • h is increasing rh ( r ) Figure 2: The function h .We now pullback ( E, π ) to ( e E, ˜ π ) which is an exact MBL–fibration over a surface withstriplike ends. e E E ˜ u / / e E e D ˜ π (cid:15) (cid:15) E D π (cid:15) (cid:15) e D D u / / roposition 2.20. Consider ( e E, ˜ π ) as above. Suppose we have a compact connected La-grangian submanifold in the fibre at infinity over one end of each edge of the base. Then wecan construct some ˜ ρ and deformation of the exact MBL–fibration, such that: • L extends by symplectic parallel transport to a Lagrangian boundary condition Q • ( Q, ˜ ρ ) is a well-defined enclosed Lagrangian boundary condition.Furthermore the resulting ( e E, ˜ π ) and ( Q, ˜ ρ ) are well-defined up to deformation through suchchoices. It should be noted here that we may, without loss of generality, use Lemma 2.17 to deform E within some finite level set ρ min of ρ to ensure that Q is defined on e E and contained within u − (cid:0) ρ − [0 , ρ min ) (cid:1) . For the rest of the proof of the proposition we will deform only u and weshall define a ˜ ρ with a convex level set contained in u − (cid:0) ρ − [ ρ min , ρ max ] (cid:1) for any ρ max > ρ min .We will see first how far u ∗ ρ is from making Q an enclosed Lagrangian boundary condition.Let ˜ J be the complex structure on e E (see page 11) and ˜ u ∗ J be the pullback of the complexstructure from E where this is defined. Similarly we have the standard complex structure ˜ j on e D and also the pullback u ∗ j from D where u is an immersion. Let ˜ π ∗ ˜ j , ˜ π ∗ ( u ∗ j ) be thehorizontal lifts D ˜ π | − T e E h ◦ ˜ j ◦ D ˜ π and D ˜ π | − T e E h ◦ u ∗ j ◦ D ˜ π respectively. It should be noted herethat ˜ J and J agree on T e E v and that ˜ J − u ∗ J = ˜ π ∗ ˜ j − ˜ π ∗ ( u ∗ j ).Where u is holomorphic these complex structures agree so u ∗ ρ is plurisubharmonic. Alsowhere u is locally constant − d ( d ( u ∗ ρ ) ◦ ˜ J ) splits as u ∗ Ω on T e E v and zero horizontally, so u ∗ ρ is subharmonic. So far so good. The difficulty arises in dealing with the region t ≤ r ≤ t in e H . Here we have: − d ( d ( u ∗ ρ ) ◦ ˜ J ) = − d ( d ( u ∗ ρ ) ◦ (˜ u ∗ J + ˜ π ∗ (˜ j − u ∗ j )))= ˜ u ∗ Ω | ( T e E v ) ⊗ + ˜ u ∗ Ω | ( T e E h ) ⊗ − d ( d ( u ∗ ρ ) ◦ (˜ π ∗ (˜ j − u ∗ j ))Applied to pairs of vectors ( V, ˜ J V ) the first two terms are positive semi-definite (for thesecond we used that u is nowhere orientation reversing). The third term may not be, but itevaluates to zero on ( T e E v ) ⊗ and we will show how to adjust − d ( d ( u ∗ ρ ) ◦ ˜ J ) by adding a pullback by π of a certain functional e D / / R to ρ to achieve positive semi-definiteness everywhere.Let ˜ R, ˜Θ , R, Θ be horizontal lifts of the vector fields ∂∂r , ∂∂θ to e E, E respectively. Then atany point p we have: ˜ π ∗ (˜ j − u ∗ j ) : r ˜ R h ( r ) − rh ′ ( r ) h ( r ) ˜Θ˜Θ h ( r ) − rh ′ ( r ) rh ′ ( r ) r ˜ R and also: d (˜ u ∗ ρ ) | T p e E h = (cid:0) R ˜ u ( p ) ( ρ )( π ∗ dr ) ˜ u ( p ) + Θ ˜ u ( p ) ( ρ )( π ∗ dθ ) ˜ u ( p ) (cid:1) ◦ D ˜ u p | T p e E h = R ˜ u ( p ) ( ρ ) rh ′ ( r ) (cid:18) ˜ π ∗ drr (cid:19) p + Θ ˜ u ( p ) ( ρ ) (˜ π ∗ dθ ) p Composing the functions defined above yields: d (˜ u ∗ ρ ) ◦ ˜ π ∗ (˜ j − u ∗ j ) p = R ˜ u ( p ) ( ρ )( h ( r ) − rh ′ ( r ))(˜ π ∗ dθ ) p +Θ ˜ u ( p ) ( ρ ) h ( r ) − rh ′ ( r ) h ( r ) (cid:18) ˜ π ∗ drr (cid:19) p − d ( d ( u ∗ ρ ) ◦ ˜ J ) whichwas potentially not positive-semidefinite (see above). − d ( d (˜ u ∗ ρ ) ◦ ˜ π ∗ (˜ j − u ∗ j )) p = R ˜ u ( p ) ( ρ ) r h ′′ ( r )˜ π ∗ (cid:18) drr ∧ dθ (cid:19) p + R ˜ u ( p ) ( R ( ρ )) rh ′ ( r )( rh ′ ( r ) − h ( r ))˜ π ∗ (cid:18) drr ∧ dθ (cid:19) p − Θ ˜ u ( p ) (Θ( ρ ) rh ′ ( r ) − h ( r ) h ( r ) ˜ π ∗ (cid:18) drr ∧ dθ (cid:19) p + d v ( R ˜ u ( p ) ( ρ )) ∧ ( rh ′ ( r ) − h ( r ))˜ π ∗ ( dθ ) p + d v (Θ ˜ u ( p ) ( ρ )) ∧ rh ′ ( r ) − h ( r ) h ( r ) ˜ π ∗ (cid:18) drr (cid:19) p Here d v is the differential d evaluated only in the fibre directions.We do not yet have enough control over these summands, so we consider a one parameterfamily of maps u converging to the identity map on e H . These are defined by replacing thefunction h with the family of functions h t ( r ) = h ( tr ) t for t ∈ [1 , ∞ ).Considering the dependence on t one now has: − d ( d (˜ u ∗ ρ ) ◦ ˜ π ∗ (˜ j − u ∗ j )) p = 1 t R ˜ u ( p ) ( ρ )( rt ) h ′′ ( rt )˜ π ∗ (cid:18) drr ∧ dθ (cid:19) p + 1 t R ˜ u ( p ) ( R ( ρ ))( rt ) h ′ ( rt )(( rt ) h ′ ( rt ) − h ( rt ))˜ π ∗ (cid:18) drr ∧ dθ (cid:19) p − Θ ˜ u ( p ) (Θ( ρ ) ( rt ) h ′ ( rt ) − h ( rt ) h ( rt ) ˜ π ∗ (cid:18) drr ∧ dθ (cid:19) p + 1 t d v ( R ˜ u ( p ) ( ρ )) ∧ (( rt ) h ′ ( rt ) − h ( rt ))˜ π ∗ ( dθ ) p + d v (Θ ˜ u ( p ) ( ρ )) ∧ ( rt ) h ′ ( rt ) − h ( rt ) h ( rt ) ˜ π ∗ (cid:18) drr (cid:19) p With this we can now describe how small − d ( d ( u ∗ ρ ) ◦ ˜ π ∗ (˜ j − u ∗ j )) is in terms of t . To dothis we define || V || for V ∈ T e E v to be ˜ u ∗ Ω( V, ˜ J V ), i.e. the pullback of the metric on fibres of E . Lemma 2.21.
Given any ρ max > ρ min there is some constant K > together with a smoothfunctional α ∈ C ∞ ( e E ) and one-forms β, γ on e E supported over the model neighbourhood { z ∈ e H : | z | ≤ } such that: − d ( d (˜ u ∗ ρ ) ◦ ˜ π ∗ (˜ j − u ∗ j )) = α ˜ π ∗ (cid:18) drr ∧ dθ (cid:19) + β ∧ ˜ π ∗ ( dθ ) + γ ∧ ˜ π ∗ (cid:18) drr (cid:19) and β, γ evaluate to zero horizontally • α, β, γ ≡ where r [ t , t ] • | α | ≤ Kt on ρ − [0 , ρ max ) • For any vertical tangent V ∈ T e E v we have | β ( V ) | , | γ ( V ) | ≤ K t || V || on ρ − [0 , ρ max ) Proof.
We examine the various summands of − d ( d (˜ u ∗ ρ ) ◦ ˜ π ∗ (˜ j − u ∗ j )) p as described above. The expressions in terms of h and rt are all smooth as functions of rt andvanish for 1 ≤ rt ≤
2, so must be bounded independently of t . By compactness R ˜ u ( p ) ( ρ ) and R ˜ u ( p ) ( R ( ρ )) have bounds independent of t . Similarly for d v ( R ˜ u ( p ) ( ρ )).We are interested only in the region r ∈ [ t , t ] and u ( r, θ ) = ( h t ( r ) , θ ), so we restrictattention to ˜ u ∗ ρ in fibres where r ≤ t .By compactness Θ ˜ u ( p ) (Θ( ρ )) and d v (Θ ˜ u ( p ) ( ρ )) are bounded on ρ − [0 , ρ max ) where r ≤ t and take value 0 in the fibre over 0, so vanish to first order as r →
0. Each summand of − d ( d (˜ u ∗ ρ ) ◦ ˜ π ∗ (˜ j − u ∗ j )) p is a product of either of these two terms, or t with bounded terms,hence the result. Corollary 2.22.
Let
C > K + Kt , then over the region t ≤ r ≤ t in e H ω := − d ( d (˜ u ∗ ρ ) ◦ ˜ J ) + ˜ π ∗ (cid:18) C drr ∧ dθ (cid:19) gives a positive semi-definite quadratic form on T e E when applied to pairs of vectors ( X, ˜ J X ) .Proof. Split X = V + H into vertical and horizontal components respectively. We will write || H || for the metric ˜ π ∗ (cid:0) drr ∧ dθ (cid:1) ( , ˜ J ) on T e E h . ω ( X, ˜ J X ) = Ω( V, ˜ J V ) + Ω( H, ˜ J H )+( C + α )˜ π ∗ (cid:18) drr ∧ dθ (cid:19) ( H, ˜ J H )+ β ∧ ˜ π ∗ ( dθ ) ( V, ˜ J H ) + β ∧ ˜ π ∗ ( dθ ) ( H, ˜ J V )+ γ ∧ ˜ π ∗ (cid:18) drr (cid:19) ( V, ˜ J H ) + γ ∧ ˜ π ∗ (cid:18) drr (cid:19) ( H, ˜ J V ) ≥ || V || − Kt || V || || H || + ( C − Kt ) || H || Examination of the discriminant shows this is ≥ X if C > K + Kt > K t + Kt .We are now ready to define ˜ ρ for large enough t and proceed with the proof of Proposition2.20. Proof of Proposition 2.20.
We will construct a family of functions ˜ g t ∈ C ∞ ( e H ) and define˜ ρ := ˜ g t ◦ ˜ π + ˜ u ∗ ρ . For large enough values of t this will have the necessary properties to achieveconvexity on a level set of ˜ ρ contained in ˜ u − ρ − [ ρ min , ρ max ].16t is now convenient to change coordinates by the exponential map e H ←− { z ∈ C : im z ∈ [0 , π ] } Using coordinates z = x + iy on the strip, r = 1 , , x = 0 , log 2 , log 3. Wedefine g to be a scalar function on the strip, such that for some positive ǫ < log 3 − log 22 :(i) g depends only on x and is increasing in x (ii) g ( x, y ) = 0 if x ≥ log 3(iii) g ( x, y ) = ρ min − ρ max if x ≤ − ǫ (iv) ∃ δ > ∂ ∂x g ≥ δ for x ∈ [0 , log 2](v) ∂ ∂x g = 0 for x ∈ [log 2 + ǫ, log 3 − ǫ ](vi) ∂ ∂x g is non negative away from x ∈ [log 3 − ǫ, log 3] ∂ g∂x ≥ δ − ǫ log 2 ǫ +log 2 li n e a r log 3 xg ( x, y ) − ǫ +log 3 g = ρ min − ρ max the only region where ∂ g∂x < Figure 3: The function g in terms of x .These conditions are illustrated in Figure 3. Such a function is easily constructed.Now we extend g to g T , a one-parameter family of functions parametrised by T ∈ [0 , ∞ ].Let L := max (cid:16) − ∂ ∂x g (cid:17) and F := 1 + ( g (log 3 + ǫ ) − g (log 2 + ǫ )) (cid:16) T log 3 − log 2 − ǫ + 2 (cid:17) ρ max − ρ min Then we can define smooth g T by: • g T ( x, y ) = g ( x + T,y )+ ρ max − ρ min F − ρ max + ρ min for x ≤ log 2 + ǫ − T • g T ( x, y ) = g ( x + T,y ) F for x ≥ log 3 − ǫ • g T interpolates linearly in the range x ∈ [log 2 + ǫ − T, log 3 − ǫ ]For some rather messy constant S > T :17 ∂ ∂x g T is non negative away from x ∈ [log 3 − ǫ, log 3] • ∂ ∂x g T ≥ − LF ≥ − SLT • ∂ ∂x g T > δF ≥ SδT for x ∈ [ − T, log 2 − T ]Let ˜ g T be g T pushed forward to e H by the exponential map, then we have: − d ( d ˜ g T ◦ ˜ j ) ( r,θ ) = (cid:18) ∂ ∂x g T (log r, θ ) (cid:19) drr ∧ θ Setting T = log t , we now consider ˜ u ∗ ρ + ˜ π ∗ ˜ g log t on the set ˜ u − ρ − [ ρ min , ρ max ]. For largeenough t we have Sδ log t > K + Kt , so by Corollary 2.22 it is plurisubharmonic over the region r ∈ (cid:2) t , t (cid:3) . Also if t is small enough, then by compactness it is plurisubharmonic over theregion r ∈ (cid:2) t , t (cid:3) . Elsewhere it is subharmonic.The ρ max level set of ˜ u ∗ ρ + ˜ π ∗ ˜ g log t is contained in ˜ u − ρ − [ ρ min , ρ max ]. It agrees with the ρ min level set of ˜ u ∗ ρ near r = 0 and the ρ min level set away from the neighbourhood modelledby e H . Hence it makes our Lagrangian boundary condition enclosed.A careful examination of this procedure shows that the choices involved are all canonicalup to isotopy through such choices. Remark 2.23. As t / / ∞ the above argument gives a deformation of MBL–fibrations fromthe fibration with striplike ends to the original one over D with marked points on the boundary.For each value of t we also have the same Lagrangian boundary condition. With a little care,we also get a deformation of enclosed Lagrangian boundary conditions as t / / ∞ . This willbe important later as it shows the relative invariants from these fibrations are all the same.Now we have shown how to deform our original model fibration over D to have striplikeends, the trivialisations over which are specified entirely by the image in N of the end and thefibre of E / / N over that point. Suppose we now have a pair of holomorphic maps D / / N which agree on one point z ∈ ∂ D , then by deforming these as above to admissible maps fromsurfaces with striplike ends we can glue at the ends corresponding to z . Furthermore, using theprevious lemmas we can still define enclosed Lagrangian boundary conditions on the resultingfibration simply by specifying some Lagrangians in the fibres at infinity. The same works forlarger composites.In order to define an exact MBL–fibration from any admissible map u : B / / N we workwith composites of our model maps (as illustrated in Figure 4). Namely, any admissible u isisotopic through admissible maps to a map obtained as follows. Take an acyclic collection ofholomorphically embedded discs in N joined (without any condition on tangencies) at certainmarked points on their boundaries. By this, we mean that the graph whose vertices are givenby the discs and whose edges correspond to marked points at which the discs are joined, isacyclic. We shall call this a tree construction . Now: • pullback the fibration E / / N over each of these discs • deform them all to have striplike ends instead of each marked point • glue the fibrations at the striplike ends corresponding to the joined marked points18 BL( π ) u Figure 4: An example of a map of a surface B into N given by a tree construction.Trivialisations of the exact MBL–fibration are over lighter shaded regions.Let u : B / / N be such a construction. This specifies for each striplike end e ∈ I a regularpoint z e ∈ N to which it maps and an exact MBL–fibration ( E, π ) over B . The choices involvedin the construction mean that ( E, π ) is well-defined up to isotopy. The same is true if we choosesufficiently many exact Lagrangian submanifolds in the fibres over the points z e and a suitabledeformation (by Lemma 2.17) to define an enclosed exact Lagrangian boundary condition Q .One can always enlarge the support of this deformation such that Q remains defined throughthe relevant isotopies of exact MBL–fibrations (and all subsequent arguments in this section).Much of the tree structure and the positioning of the singular values of π in B by theconstruction does not affect the exact MBL–fibration except by isotopy. Suppose we have asmooth embedding of f : B / / B with following properties (e.g. Figure 5):(i) u ◦ f should be admissible and define the same set { z e } of ends. In particular f ( ∂B )should not contain any critical value of π .(ii) Consider any of the regions of B identified with parts of e H in order to construct thestriplike ends (including those glued together). In each of these regions f ( B ) should bea union of disjoint wedges. This corresponds in the trivialisation of the striplike ends to f ( B ) being a union of ‘substrips’.(iii) f should be isotopic to the identity through smooth embeddings maintaining condition(i) above. Lemma 2.24.
In the above construction, the isotopy of (iii) induces an isotopy of exact MBL–fibrations (and also of any enclosed exact Lagrangian boundary conditions, provided sufficientcare is taken with symplectic parallel transport using Lemma 2.17).
The resulting fibration can be viewed as another potentially very different tree construction.
Proof.
This proof of this is immediate from the definitions above. To construct the accom-panying isotopy of enclosed exact Lagrangian boundary conditions, one simply fixes an exact19 ( B ) B Figure 5: An example restriction of the base of a tree construction used to changethe tree structure (it redistributes the singular values). The regions where striplikeends were glued in the construction of B are indicated by shading.Lagrangian in the fibre over one end of each edge of the base and extends by symplectic par-allel transport, at each stage of the isotopy, to define Q . This works with a sufficiently strongapplication of Lemma 2.17. Lemma 2.25.
Suppose ˜ u : ˜ B / / N is another such construction such that u, ˜ u are isotopicthrough admissible maps fixing the endpoints z e . Then u, ˜ u are related by a finite sequence ofthe following ‘moves’:(a) isotopy of the tree construction of embedded admissible holomorphic discs through suchconstructions(b) decomposition of any of the embedded holomorphic discs into two joined at a new markedpoint (see Figure 6)(c) changing the tree structure by restriction of the base to a surface embedded in it as inLemma 2.24Suppose furthermore we have enclosed exact Lagrangian boundary conditions Q, ˜ Q defined onthe exact MBL–fibrations over u, ˜ u by the same set Lagrangians in the fibres of N over endpoints z e which are not ‘internal’ to the tree constructions. Then the two exact MBL–fibrations,together with enclosed exact Lagrangian boundary conditions, are isotopic.Sketch of proof. We take the base of one of the fibrations and smoothly embed it in the baseof the other as for Lemma 2.24. This can be done such that after application of move (c) thetree constructions are now related only by the remaining two moves. It may help to thinkof decomposing and isotoping the tree structures such that all the discs are complex linearembeddings in our favourite coordinate neighbourhoods of N .20igure 6: An example decomposition of a holomorphic disc with boundary markedpoints into two joined at a new marked point. In this section, we define the relative invariant associated to an exact MBL–fibration withenclosed exact Lagrangian boundary conditions. The invariant’s definition will be largely iden-tical to Seidel’s construction of relative invariants of exact Lefschetz fibrations (cf. [2] sections2.1 and 2.4). We begin by defining a large class of almost complex structures on an exactMBL–fibration (
E, π ) which extend J . Definition 2.26. An almost complex structure on an exact MBL-fibration ( E, π ) over a surface B with striplike ends and boundary marked points P ⊂ ∂B is an almost complex structure J on π − ( B \ P ) such that: • J = J in a neighbourhood of Crit( π ) and on the complement of some compact subset of E • Dπ ◦ J = j ◦ Dπ , i.e π is ( J, j )–holomorphic, • Over striplike ends J is invariant w.r.t. translation in the [0 , ∞ )–direction, • Over embedded curves γ in B ending, with non-zero derivative, at a boundary, markedpoint e , J extends smoothly to an almost complex structure J e,t , in the fibre over e depending smoothly on the angle of approach πt ∈ [0 , π ].The last two conditions ensure that, at each end, J limits to some time dependent almostcomplex structure J t in the fibre over the end. Definition 2.27.
An almost complex structure J on E is compatible relative to j if the 2–formΩ( ., J. ) | ( T E v ) ⊗ is everywhere symmetric and positive definite. We will denote the set of suchalmost complex structures by J ( E, π ).Given a time-dependent almost complex structure J e,t on the fibre at infinity or fibre overthe marked point at each end e of B , the set of those J which agree in the limit with each J e,t will be denoted J ( E, π, { J e,t } e ∈ I ) or simply J ( E, π, { J e } ).For Floer cohomology, compatibility is in general an unnecessarily strong condition. A moredetailed explanation of this is given by Salamon in [14], in particular the discussion surroundingRemark 5.1.1. It suffices instead for a time dependent almost complex structure J t to tame thesymplectic form. This is often an easier condition to use, as it is preserved by small compactlysupported deformations of J t . 21et u : R × [0 , / / M , where M is a K¨ahler manifold. Compatibility in a neighbourhoodof the Lagrangian intersections is still convenient, though not strictly necessary. It ensures thatthe Cauchy-Riemann operator ∂∂s + J t ∂∂t thought of as (the completion of) an operator on the tangent space to the space of paths([0 , → M ) with ends on the Lagrangian submanifolds, can be written as ∂∂s + A with A symmetric near the ends. However, even compatibility near the intersections is notstrictly necessary as A is always asymptotically symmetric at the ends. To avoid discussingthis any further, we will keep the requirement of compatibility near the Lagrangian intersectionsfor the purposes of this paper. Definition 2.28.
An almost complex structure J on E with Lagrangian boundary condition Q is tame relative to j if the 2–form Ω( ., J. ) | ( T E v ) ⊗ is everywhere positive definite and compatiblenear the Lagrangian intersections at the ends. We will denote the set of such almost complexstructures by J tame ( E, π, Q ).Given J e,t ends we define J tame ( E, π, Q, { J e,t } e ∈ I ) in the same way as J ( E, π, { J e,t } e ∈ I ),but with relative compatibility replaced by relative tameness.For a relatively compatible or tame almost complex structure J on an exact MBL–fibration E , we denote the set of smooth sections of E with boundary on Q , which are ( j, J )–holomorphic(except at boundary marked points, where J is not defined) by M J ( Q ). We refer to these simplyas holomorphic sections . If Q is enclosed we will write M J ( Q ) for the subspace of those sectionscontained within the particular enclosure. It will be clear from the context which enclosure isbeing used. Lemma 2.29 (cf. [2], Lemma 2.2) . Let ( E, π, Ω , Θ , J , B, j ) be an exact MBL–fibration over asurface with ends and an enclosed, exact Lagrangian boundary condition ( Q, ρ, R ) . Thet for anyalmost complex structure J on ( E, π, Ω , Θ , J , B, j ) , the space of ( j, J ) –holomorphic sections u with the given Lagrangian boundary condition decomposes into two components: • those with image contained in the fibrewise compact set ρ − [0 , R − ǫ ] • those whose image leaves ρ − [0 , R ] Proof. J = J in a neighbourhood of ρ − ( R ), so ρ is subharmonic w.r.t. J in that region. Thisimplies that ρ ◦ u is subharmonic where it takes values near R , so cannot have a local maximumin the region ρ − [ R − ǫ, R ].In the construction of Floer cohomology, it is necessary to use a regular almost complexstructure. This ensures that the moduli spaces of sections are smooth finite dimensional man-ifolds and, in particular, that the Floer differential squares to zero. Here we have the samerequirement.We say that a relatively tame almost complex structure J on an exact MBL–fibration is regular at u ∈ M J ( Q ) if D u,J is onto. Here D u,J is the extension of the linearised ¯ ∂ J operator atthe J –holomorphic section u to relevant Sobolev completions. For a more detailed descriptionof regularity which applies in this setting, see [2, Section 2]. This condition ensures that M J ( Q )22s a smooth finite dimensional manifold of the correct dimension (given by a Maslov index of u ) near u ∈ M J ( Q ). J is regular with respect to the enclosed Lagrangian boundary condition Q if it is regular for all u ∈ M J ( Q ). We denote the space of such almost complex structuresby J reg tame ( E, π, Q, { J e,t } ) ⊂ J tame ( E, π, Q, { J e,t } ). Corresponding to Lemma 2.20 of [2] and with essentially the same proof, we have genericregularity for complex structures in J tame ( E, π, { J e } ) given any transverse enclosed exact La-grangian boundary condition ( Q, ρ, R ). Lemma 2.30. J reg tame ( E, π, Q, { J e } ) is C ∞ –dense in J tame ( E, π, Q, { J e } ) . More precisely, givena non-empty open subset U ⊂ B which is disjoint from the ends and any J ∈ J tame ( E, π, Q, { J e } ) ,there are J ′ ∈ J reg tame ( E, π, Q, { J e } ) arbitrarily C ∞ -close to J , such that J = J ′ outside π − ( U ) . By bounding the symplectic action, as in [2] one also achieves a compactification M J ( Q )of M J ( Q ) in the Gromov-Floer topology by adding broken sections.Counting isolated holomorphic sections in M J ( Q ) which limit to given sets { x e } e ∈ I ofintersection points in the fibres at the ends (denote these subsets M J ( Q, { x e } )) one defines alinear map on the level of Floer cochain complexes. In fact, by a standard argument, consideringdegenerations of 1–dimensional families of sections, one shows that it is a chain map. O e ∈ I − CF ( Q e , Q e , J e,t ) O e ∈ I + CF ( Q e , Q e , J e,t ) C Φ rel (( E,π ) , ( Q,ρ,R ) ,J ) / / ⊗ e ∈ I − h x e i X { y e } e ∈ I + M J ( Q, { x e } ∪ { y e } )( ⊗ e ∈ I + h y e i ) ✤ / / The above is written out in full detail, since, in particular, the differentials in the Floercochain complexes CF ( L, L ′ , J t ) depend on the choice of almost complex structure. When onechanges this, an argument using continuation maps gives a chain homotopy equivalence to thenew cochain complex which is canonical up to chain homotopy. Similarly, compactly supportedHamiltonian symplectomorphisms of either L or L ′ induce canonical chain homotopy classes ofchain homotopy equivalences. Lemma 2.36 (below) can be viewed as a generalisation of bothof these results in the setting of exact Lagrangian submanifolds of exact symplectic manifolds. Definition 2.31.
The relative invariant Φ rel (( E, π ) , ( Q, ρ, R ) , J ) is defined to be the mapinduced on Floer cohomology by C Φ rel (( E, π ) , ( Q, ρ, R ) , J ). Under the assumption that the Q is spin in any fibre, one can also orient the moduli spaces and perform the count of sectionswith signs (see [15]). This allows the use of Z coefficients. Alternatively, one can instead use Z Z –coefficients. Remark 2.32.
The convention of signs and arrangement of Q e , Q e differs by a 180 ◦ rotationfrom that in [2]. Remark 2.33.
There is a Poincar´e duality arising from trivial fibrations over an infinite stripwith both ends in I − (or both in I + ). Composition with this (see gluing below) allows usto swap ends back and forth between I + and I − or also to view the relative invariant as theelement induced in cohomology by: C Φ rel ∈ O e ∈ I CF ( Q e , Q e )23omposing two exact MBL–fibrations with enclosed Lagrangian boundary conditions canbe done by gluing over a single striplike end (if necessary one uses Poincar´e duality to moveother ends out of the way). Corresponding to Proposition 2.2 of [2] we have: Lemma 2.34.
Gluing two exact MBL–fibrations with enclosed Lagrangian boundary conditionstogether along an oppositely oriented, but otherwise identical striplike end gives the compositionof the Φ rel (( E, π ) , ( Q, ρ, R ) , J ) maps.(It is important that all the data of E, π, Q, J . . . etc agrees where the gluing occurs.)
We shall now show that a variety of changes can be made to ((
E, π ) , ( Q, ρ, R ) , J ) withoutchanging the relative invariant (beyond composing on either side with the canonical isomor-phisms on Floer cohomology). Lemma 2.35. C Φ rel (( E, π ) , ( Q, ρ, R ) , J ) is unchanged when one switches the enclosure ( Q, ρ, R ) for an equivalent enclosure.Proof. Definition 2.11 ensures that the moduli spaces M J ( Q, { x e } ) are unaffected by this. Lemma 2.36.
The map Φ rel (( E, π ) , ( Q, ρ, R ) , J ) depends only on the enclosed region ρ − [0 , R ] .It is invariant (up to composition on either side with the canonical isomorphisms on Floercohomology) under isotopy of the combined data (( E, π ) , ( Q, ρ, R ) , J ) such that:(a) the data remains valid at all stages for the definition of Φ rel (( E, π ) , ( Q, ρ, R ) , J ) ,(b) the induced isotopies at each end E e fix the symplectic form and vary the Lagrangiansubmanifolds only by compactly supported Hamiltonian isotopy.Proof. When one fixes the data over the ends, a standard argument counting sections at allstages of (some perturbation) of this isotopy gives a homotopy of the relative invariant at thelevel of chain complexes. More generally, it gives a homotopy composed with continuationmaps (yielding the canonical isomorphisms) on Floer cohomology groups over the ends.
Remark 2.37.
In fact, the above argument works to describe general deformations of the data(
E, π ), (
Q, ρ, R ) and J . Suppose for example, we do not require the symplectic forms on fibresover the ends to remain fixed (then we most likely also have to vary Q over the ends to ensurethat it remains a Lagrangian boundary condition). The relative invariant then varies by left-and right-composition with the continuation maps from these changes. In some cases (such asdeforming Ω through exact K¨ahler forms with the added condition that components of Q havevanishing first cohomology over R ) these continuation maps are still canonical isomorphisms,but in general one cannot expect that to be the case.Suppose the geometric data defining the exact MBL–fibration and boundary conditionssplits as a some sort of product, then one can often correspondingly split the relative invariant.A simple example of this is demonstrated in the following lemma and a more involved versioncomes in Section 3. Lemma 2.38.
Suppose: • ( E, π ) splits as a smooth fibre product of MBL–fibrations ( E , π ) and ( E , π ) over thesame base ( B, j ) , • the exact Lagrangian boundary condition Q splits as a fibre-product of exact Lagrangianboundary conditions Q , Q in the two factors, ( ρ, R ) , ( ρ , R ) , ( ρ , R ) are enclosures for Q, Q , Q respectively with the property that ρ is C –close to max { ρ , ρ } in some neighbourhood of ρ − ( R ) .Then the relative invariant C Φ rel (( E, π ) , ( Q, ρ, R ) , J ) splits as the product of the relative in-variants on the two factors C Φ rel (( E , π ) , ( Q , ρ , R ) , J ) ⊗ C Φ rel (( E , π ) , ( Q , ρ , R ) , J ) Proof.
This is a simple generalisation of the corresponding product formula for Floer cohomol-ogy (which one has over the ends in the setting of the lemma). (
E, π ) is smooth whenever( E , π ) and ( E , π ) share no singular values in the base.Let J ∈ J reg tame ( E , π , Q ), J ∈ J reg tame ( E , π , Q ) be any regular almost complex structureson E , E for boundary conditions Q , Q respectively. We define the almost complex structure J for the fibre product ( E, π ) by pullback from the embedding E = E × B E / / E × E .Relative tameness of J and J relative to j implies that J on the product restricts well to E and is tame relative to j .The J –holomorphic sections of E are in bijection with the pairs of holomorphic sections of E and E . The regularity of J for Lagrangian boundary condition Q × ∂B Q is an immediateconsequence of the regularity of J and J since the linearisation at any holomorphic sectionsplits as that of J and J , both of which (when extended to the relevant Sobolev spaces) aresurjective. This bijection of moduli spaces identifies the deformation theories of the sections. Remark 2.39.
In the case where ρ , ρ are plurisubharmonic near their respective R -levelsets and the complex structures near there are integrable, there is always a plurisubharmonic C –approximation to max { ρ , ρ } near its R -level set (see [16, Lemma 3.8]). Lemma 2.40.
Trivial MBL–fibrations over the infinite strip (with trivial boundary conditions)induce the identity map on Floer cohomology.Proof.
One may take the complex structure for the fibration from that used in calculating theFloer homology at the striplike ends. The only isolated holomorphic sections (since we do notquotient by the R –action on the base) are the constant sections. These give us the identitymap. In this section, we study the relative invariant on a particular family of twisted products. Thesecalculations are necessary in the setting of symplectic Khovanov homology to describe mapscorresponding to presentations of trivial cobordisms (cf. stabilisation and destabilisation mapsof Section 5.4).The setting will be as follows. Let (
X, J stdX ) be a Stein manifold with exhausting plurisub-harmonic function ρ X and exact K¨ahler form Ω X = − d ( dρ X ◦ J stdX ). Take a pair of compact ex-act Lagrangian submanifolds K, K ′ ⊂ ρ − X [0 , R ) which intersect transversely. These conditionssuffice to define HF ∗ ( K, K ′ ) using an almost complex structure which is a small perturbationof J stdX .Now let ( E, π, Ω E , Θ E , J E , D, j ) be an exact Lefschetz fibration, or more generally an exactMBL–fibration, together with two boundary marked points and enclosed, exact Lagrangianboundary condition Q . Then E × X is an exact MBL–fibration and can be given the Lagrangianboundary condition Q × K on one half of the boundary and Q × K ′ on the other. Lemma 2.3825hows that, at the chain level, the Floer cohomologies at each boundary marked point, andalso the relative invariant, split as tensor products.This aim of this section is to construct the same splitting in a case where the product E × X is replaced by E × S P (see below for more detail on this as a symplectic associated bundle)for some fibre preserving Hamiltonian S –action on E and some circle-bundle P . The maintrick is to make the regular almost complex structure and the transverse Lagrangian bound-ary conditions S –equivariant. This has the consequence that the only isolated holomorphicstrips/sections are those confined to the fixed locus of the action. In general such a trick isnot possible and we will indicate the important points which make it work in the specific caseconsidered in this section. ( E, π, Ω E , Θ E , J E , D, j ) We start by considering the following exact Lefschetz fibration (for some fixed d = 3 x ∈ C ∗ ): C π (cid:15) (cid:15) ( a, b, c ) ❴ (cid:15) (cid:15) C a − ax + bc with the standard exact K¨ahler form on C .Take a closed disc D with two boundary marked points, mapping holomorphically into asmall neighbourhood of the singular value − x in C , such that: • the map is an embedding of D with one with the exception that the two boundary markedpoints map to the same point, • the singular value − x is hit by an interior point of D , − x Figure 7: An illustration of the disc D .We now define the exact Lefschetz fibration E to be the pullback by this smooth map ofthe above Lefschetz fibration C . This has the advantage of canonically identifying the fibresover the boundary marked points with each other.Define ρ E ( a, b, c ) := | a | + | b | + | c | using coordinates on C , and to start with take theexact symplectic form Ω E = d Θ E = − d ( dρ E ◦ J E ), with the standard complex structure J E .We will adjust Θ E and Ω E , below, to give some control over symplectic parallel transport andthe Lagrangian boundary conditions. 26he Hamiltonian S –action is the unit circle part of the holomorphic C ∗ –action ( a, b, c ) ( a, ζb, ζ − c ). It is generated by the Hamiltonian µ ( a, b, c ) = | b | − | c | and has the followingimportant properties: • It preserves fibres of E and of the projection proj a to the a –coordinate. • ρ E , Θ E , Ω E and J E are all S –equivariant.The map proj a , restricts to a Lefschetz fibration on non-singular fibres E z of E with threesingular values. As one approaches a singular fibre of E , two of these singular values collide.Understanding proj a is important as it gives some control over S –equivariant Lagrangianspheres in the fibres: Lemma 3.1.
Any S –equivariant Lagrangian sphere in a regular fibre of E (as above) can beexpressed as µ | − E z (0) ∩ proj a | − E z im γ for some smooth embedded path γ in C between criticalvalues of proj a | E z . Furthermore, the vanishing cycle in for any path in B into the critical valueof π is of this form. In this case γ is a smooth path between the two critical values of p | E z which collide in the critical fibre of π .Proof. Denote the fibre proj a | − E z ( a ) over a ∈ C by E z,a . Consider a Lagrangian sphere L ⊂ E z .By S –equivariance, L intersects fibres E z,a in, possibly empty, unions of circles, except atcritical points of µ (which are precisely the critical points of proj a | E z ). At any regular x ∈ L the intersection T x L ∩ ( T x E z,a ) ⊥ Ω is one-dimensional so we can define a non-vanishing vectorfield locally on T L of these vectors. Projecting these to C by D proj a makes them all tangent,since otherwise we reach a contradiction with S –equivariance and the dimension of L .Hence, locally L is preserved by symplectic parallel transport over certain paths in C . Theonly way for components of L to be closed, embedded spheres is for these paths to end atdistinct critical points.All the critical points satisfy µ = 0, and µ is preserved by symplectic parallel transport, soit suffices to observe that the S –action is transitive on any set of the form µ | − E z,a (0) (these arecircles or points). Remark 3.2.
A consequence of the above lemma is that two equivariant Lagrangian spheresare transverse if and only if they correspond to paths between the critical points of proj a whichintersect only at critical points and which enter the critical points at different angles. Theonly obstruction to being able to make equivariant Lagrangian spheres intersect transverselyby Hamiltonian isotopy through equivariant Lagrangian spheres is topological (from isotopy ofthe paths), because all Lagrangian isotopies in this case are Hamiltonian.We now construct the transverse, exact Lagrangian boundary condition Q on E . In doing,so we will perform a compactly supported deformation of Ω E without changing its restrictionto fibres. To start the construction, we take a pair of S –equivariant exact Lagrangian spheres L out , L ′ out in the fibre at one boundary marked point, say the ‘output’ end. These correspondto embedded paths γ out and γ ′ out under projection to the a –coordinate, by Lemma 3.1. Figure 8shows the choice we shall use in this section, but the same construction can be done with any S –equivariant exact Lagrangian spheres.Extending by symplectic parallel transport over the two components of ∂D defines an exactLagrangian boundary condition and a pair of exact S − –equivariant Lagrangian spheres L in , L ′ in at the input end. Identifying the input and output fibres by symplectic parallel transport overthe component of ∂D going anti-clockwise from the input to the output end of the fibration,27 eighbourhood of x Neighbourhood of − x α proj a ( Q ) = γ ′ out proj a ( Q ) = γ out Figure 8: Projections of Lagrangians to curves γ out , γ ′ out with a single transverseintersection at an endpoint. This describes the Lagrangians at the output end of thefibration.we find that L ′ in and L ′ out are Hamiltonian isotopic and that L in corresponds to γ out acted onby the positive half twist τ α , in the curve α illustrated in Figure 9. To see that this half twistis the correct monodromy, we need only check that it is the monodromy experienced by thecritical points of proj a . Neighbourhood of x Neighbourhood of − x α proj a ( Q − ) = γ ′ in ∼ γ ′ out proj a ( Q − ) = γ in ∼ τ α ( γ out ) Figure 9: The curves of Figure 8 after a half twist τ α in the curve α has been appliedto γ out . This describes the Lagrangians at the input end of the fibration.However, this only identifies the paths γ in , γ ′ in up to isotopy, so we do not necessarily havethat Q is transverse. To correct this, it suffices to perform isotopies of the paths γ in , γ ′ in (to getthe curves in Figure 9. These correspond to S –equivariant Lagrangian isotopies of L in , L ′ in and which are generated by S –equivariant, compactly supported Hamiltonian isotopies sincethe fibres of E are exact K¨ahler. Therefore it suffices to change Θ E and Ω E on compactneighbourhood in E to ensure that Q is transverse. In particular, this change is confined tofibres over a small neighbourhood of ∂D , does not affect the restriction of Θ E , or Ω E to fibres,and preserves S –equivariance of both.Consider again the S –invariant points of E . These are precisely the critical points of proj a .In every fibre E z there are three such points, with the exception of the singular fibre, in whichtwo of these points coincide (in Figures 8 and 9, these are the two points on the right-handside). The pairs of points which collide form a disc over D which projects singularly to D , socannot contain any smooth sections of E .Identify R × [0 ,
1] biholomorphically with D , punctured at both boundary marked points.This is canonical up to the obvious R –action. Symplectic parallel transport maps, over the(closures of the) paths x × [0 ,
1] and R ×{ } are well-defined for a small enough neighbourhood of s ( D ), since symplectic parallel transport preserves the locus s ( D ). Hence, it smoothly trivialisesa neighbourhood M of s ( D ), identifying it with a neighbourhood of { s ( z ) } × D inside E z × D ,28here z is the input boundary marked point. Adjusting Θ in an S –equivariant way near M ,without changing its restriction to fibres, we ensure that symplectic parallel transport respectsthe trivialisation of M . Lemma 3.3. Θ E and consequently Ω E can be deformed S –equivariantly on a compact subspaceof E , avoiding the singular point of π , such that: • their restrictions to fibres of E are unchanged, • the Lagrangian boundary condition Q defined by extending ( L out , L ′ out ) over ∂D is trans-verse, • symplectic parallel transport over all paths in D respects a trivialisation of a neighbourhood M of the only S –equivariant section. • symplectic parallel transport over the section of ∂D passing anti-clockwise from the inputto the output boundary marked point defines a holomorphic map between neighbourhoodsof the S –invariant points in the input and output fibres.Consequently Q | M is also trivial over each part of ∂D .Proof. We have covered all the necessary details above. First we adjust Θ E near the part of ∂D corresponding to R × { } to change the symplectic parallel transport over this path bya Hamiltonian symplectomorphism and ensure that it identifies neighbourhoods of the S –invariant points in the fibres at either boundary marked point by the identity map (these twofibres are already identified as the same fibre in C ). This is holomorphic.Next, we perform the deformation to ensure triviality of M , then (possibly shrinking M )we deform away from M to ensure that Q is transverse elsewhere.For the rest of Section 3, we shall take Ω E to have the properties described in Lemma 3.3. Remark 3.4.
The relative invariant associated to E with exact Lagrangian boundary condition Q is described below. However, the calculation is omitted here as it is covered in more generalityin Section 3.4.Identifying the input with H ∗ ( S ) ≃ Z [ X ] / ( X ) and the output with H ∗ ( { pt } ) ≃ Z , therelative invariant for E and Q as given above is: Z [ X ] / ( X ) / / Z X E × S P ≃ E × C ∗ ( F \ The construction that follows is a recap of the symplectic associated bundle construction inSection 4.3 of [1].We take
X, K, K ′ as at the beginning of this section. Let F X be a holomorphic linebundle with a Hermitian metric and compatible connection, which is a subbundle of the trivial C –bundle over X . Let P ⊂ F be the unit-circle bundle and α ∈ Ω ( P ) be the connectionone-form. The normalisation is such that, if R is the rotational vector field on P (2 π periodic),then α ( R ) ≡
1. Take E to be the exact Lefschetz fibration defined in the previous subsection.The symplectic associated bundle is 29 × S P → X One can check that the 2–form Ω ass = Ω X + Ω E + d ( αµ ) on E × P descends to a well-defined2-form on E × S P → X . Furthermore, on a neighbourhood of µ − (0): • Ω ass is symplectic where Ω E is symplectic; • Ω ass is symplectic on restriction to fibres over D where Ω E .Also in a neighbourhood of µ − (0), symplectic parallel transport over D fixes the projectionto X and is identical to the symplectic parallel transport in E up to an S –ambiguity. Hence, Q × S K , Q × S K ′ , over the two parts of ∂D , form a transverse exact Lagrangian boundarycondition ˜ Q for E × S P → X .Since the S action on E is part of a holomorphic C ∗ –action, we can construct the holo-morphic associated bundle E × C ∗ ( F \ E × S P → X . This givesa natural candidate for a complex structure ˜ J std on E × S P . Seidel and Smith [1] show that,where it symplectic, Ω ass is K¨ahler with respect to ˜ J std .This means that a small neighbourhood of µ − (0) has the structure of an exact Lefschetzfibration with transverse, exact Lagrangian boundary condition ˜ Q . However, enclosures for Q are not yet defined. To define them, it suffices to have an exact, S –equivariant K¨ahler form ˜Ωwhich is equal to Ω ass on some neighbourhood of ˜ Q ⊂ µ − (0) and which is equal to − d ( d ˜ ρ ◦ ˜ J std )on the complement of some slightly larger neighbourhood. The construction of ˜Ω relies uponthe triviality of E × C ∗ ( F \
0) as a holomorphic vector bundle over X (see Remark 3.5 below)and is explained in detail in Section 5.4. Remark 3.5 (a simplified version of [1], Remark 42) . F ⊕ F − is isomorphic to the trivialholomorphic bundle, because X is Stein. As a holomorphic bundle over X , the associatedbundle E × C ∗ ( F \
0) splits by the a, b and c coordinates on E (from C ) respectively as C ⊕ F ⊕ F − . This gives the following identification of holomorphic bundles over X . E × C ∗ ( F \
0) = C ⊕ F ⊕ F − = C ⊕ C It should be noted that the projection to D and S action, do not necessarily fit nicely withthis trivialisation. S –equivariant almost complex structure In order to relate the relative invariant of E × C ∗ ( F \
0) to that of E and the Floer cohomologyof K, K ′ in X , we shall construct a complex structure ˜ J on E × C ∗ ( F \
0) with the followingproperties: • ˜ J ∈ is regular (includes restricting to ends as a regular time-dependent almost complexstructure for the construction of Floer cohomology); • ˜ J is S -equivariant; • proj X is ( ˜ J , J X )–holomorphic for some regular almost complex structure J X on X .This will have the consequence that the moduli of holomorphic sections comes with an S –action, and hence that the only isolated sections are those which are fixed by the S action.With the correct setup, projection to X then identifies these moduli spaces with the modulispaces of holomorphic strips in X with boundary on K, K ′ .30n fact, much of this section can be avoided by considering the S –action on the non-regularmoduli spaces (as Kuranishi spaces [12],[17]) and identifying the S –fixed components of themoduli space with the non-regular moduli space of holomorphic strips in X with boundaryon K, K ′ . In order not to rely on Kuranishi space technology, we shall continue with theequivariant regularity argument.There are two stages to the construction which proceed in a similar manner. Firstly, weconstruct an almost complex structure J e,t at each end, which is regular for Floer cohomologythere (in fact, we may take the same one on both ends). Then we extend these over the interiorand perturb the result to get ˜ J regular for the calculation of the relative invariant. Let z ∈ D be a boundary marked point. We start with ˜ J std restricted to the fibre E z × C ∗ ( F \ S –invariant holomorphic strips u . Weshall write L, L ′ for L in , L ′ in or L out , L ′ out depending on which boundary marked point we areconsidering.The set of S –fixed points of E z consists of just three points, the three critical points ofproj a | E z . Hence, the S –fixed points of E z × C ∗ ( F \
0) form three copies of X , each projectingto one of the critical values of proj a . Consequently, S –equivariant u are specified uniquely byproj X ◦ u and by the critical point hit by proj a ◦ u . Conversely, given a holomorphic strip in X with boundaries on K, K ′ and a choice of critical point of proj a | E z , one gets an S –equivariant,holomorphic u if and only if that critical point is in L in ∩ L ′ in or L out ∩ L ′ out (depending in whichend we are considering). The former contains two of the critical points, the latter contains oneas can be seen from Figures 8 and 9.We would like to know if there are any deformations of S –equivariant holomorphic stripsthrough non- S –equivariant ones. More precisely, is this possible to first order? First somenotation is needed.Let B be the space of smooth maps u : R × [0 , / / E z × C ∗ ( F \
0) satisfying the La-grangian boundary conditions. Let E u, ˜ J std
7→ B be the vector bundle of (0 , D withvalues in u ∗ ( T E z , ˜ J ). Similarly, define B X and E v,J stdX for holomorphic strips v in X . Thesection ¯ ∂ ˜ J std : B / / E u, ˜ J std (extended to the relevant Sobolev and L p –completions) describesthe moduli space of holomorphic strips as its vanishing locus. Its behaviour, to first order, at S –equivariant u can be described by the section ¯ ∂ J stdX : B X / / E u X ,J stdX together with some‘normal data’ (writing u X for proj X ◦ u ).This normal data comes from considering deformations which fix u X . These are deforma-tions of the map b u of the strip into u ∗ X ( E z × C ∗ ( F \ u (note b u is not constant).Because u X is holomorphic, u ∗ X ( E z × C ∗ ( F \ b J . Let b B be the space of holomorphic strips in u ∗ X ( E × C ∗ ( F \ E v, b J b B be the vectorbundle of (0 , D with values in v ∗ ( T ( u ∗ X ( E z × C ∗ ( F \ , b J ). As before, we alsoget a section ¯ ∂ b J .Let D b u, b J , D u, ˜ J std and D u X ,J stdX be the linearisations of ¯ ∂ b J , ¯ ∂ ˜ J std and ¯ ∂ J stdX at b u, u and u X ,respectively. Then we get the following map of short exact sequences:31 E b u, b J / / E b u, b J E u, ˜ J std / / E u, ˜ J std E u X ,J stdX / / E u X ,J stdX / / T b u b B / / T b u b B T u B / / T b u b BE b u, b JD b u, b J (cid:15) (cid:15) T u B T u X B X / / T u BE u, ˜ J std D u, ˜ Jstd (cid:15) (cid:15) T u X B X E u X ,J stdX D uX,JstdX (cid:15) (cid:15) T u X B X / / This has the consequence that ˜ J std is regular at u (i.e. D u, ˜ J std surjects) whenever b J is regularat b u and J stdX is regular at u X . Suppose now, that we change J stdX by a compactly supported C ∞ –small perturbation avoiding K ∩ K ′ , then we can replace ˜ J std in a neighbourhood of the S –fixed locus by the complex structure ˜ J we get from J X and the Hermitian connection on F . With this done generically, for any S -invariant holomorphic u , we have that J X is regularfor u X , so to prove regularity of ˜ J at u , it suffices to show that b J is always regular at b u . Lemma 3.6. ker( D b u, b J ) = 0 . Since the corresponding Maslov index is , this implies regularity.Proof. To do this we identify u ∗ X ( E × C ∗ ( F \ C ⊕ u ∗ X ( F ) ⊕ u ∗ X ( F − ) as in Remark 3.5.We will take coordinates a, b, c in the three summands respectively. With respect to thesecoordinates, the section b u is the constant section at ( a crit , , v ∈ ker( D b u, b J ) is aholomorphic section of C ⊕ u ∗ X ( F ) ⊕ u ∗ X ( F − ) which is everywhere tangent to { a − ad + bc = z } and over the boundary of the strip is tangent to the Lagrangian boundary conditions. Notethat b and c are valued in inverse line bundles, so the product bc is a well-defined complexnumber.At ( a crit , ,
0) the tangents to { a − ad + bc = z } are u ∗ X ( F ) ⊕ u ∗ X ( F − ), since proj a is critical(so no tangent has a component in the a –direction). To first order, the Lagrangian boundaryconditions are of the form { ( zp, e iθ ¯ zp − ) | ( p, p − ) ∈ u ∗ X ( P ) ⊕ u ∗ X ( P − ) } , where θ is fixed. Thevalues of θ are distinct for transverse Lagrangians. Multiplying the b and c coordinates of v gives a holomorphic map of the strip to C = u ∗ X ( F ) ⊗ u ∗ X ( F − ) with upper and lower boundaryin e iθ R for distinct angles θ . Such a map is necessarily zero.This implies that at least one of the b and c coordinates of v vanishes on a set largeenough to apply unique continuation. Hence, one coordinate is identically zero. However, fromthe expression for the tangents to the Lagrangian we see that, where either of the b and c coordinates of v vanish, both vanish, so v ≡ S –invariant holomorphic strips, we have nowproven: Lemma 3.7.
Let J X be any almost complex structure on X which • makes the solutions to Floer’s equation for ( K, K ′ ) regular, • equals J stdX near K ∩ K ′ and outside of a compact neighbourhood.Define ˜ J from J X and the Hermitian connection on F . Then ˜ J is regular for all elements ofthe moduli space M S ˜ J ( E × C ∗ ( F \ , L × S K, L ′ × S K ′ ) of S –invariant holomorphic strips.Furthermore, this forms a component of the moduli space of all holomorphic strips (with thesame boundary conditions) which is canonically identified with (one or two copies of ) the entiremoduli space for X, K, K ′ . emark 3.8. If we allow the use of non-regular ˜ J , we can instead conclude that the moduli of S –equivariant holomorphic strips, as a Kuranishi space, is canonically identified with (one ortwo copies of) the entire moduli space for X, K, K ′ . Then a virtual perturbation preserving the S –action, shows that the only holomorphic strips which count for Floer cohomology are thosearising from the moduli space for X, K, K ′ . Since we avoid going into detail about Kuranishispaces, and also relying on Kuranishi space technology, it is necessary to do the correspondingconstruction with S –equivariant perturbations of almost complex structures.To complete the construction of a regular ˜ J on E z × C ∗ ( F \ Lemma 3.9.
Let J S tame ( E z × C ∗ ( F \ be the space of time-dependent, S –equivariant, almostcomplex structures ˜ J t on E z × C ∗ ( F \ , such that: • ˜ J t tames ˜Ω • ˜ J t = ˜ J std where proj a is near a critical value (in particular, near L × S K ∩ L ′ × S K ′ ) )and outside of a compact neighbourhood • proj a is ( ˜ J t , i t ) –holomorphic for some time dependent almost complex structure i t on C A generic choice of ˜ J t ∈ J S tame ( E z × C ∗ ( F \ is regular for all holomorphic strips with boundaryin L × S K, L ′ × S K ′ .Proof. By Lemma 3.7, ˜ J t ∈ J S tame ( E z × C ∗ ( F \ S –invariantholomorphic strip, so we now consider only non-invariant strips u .The map proj a ◦ u is holomorphic (w.r.t. i t ) and maps the boundary of the strip to thecurves γ, γ ′ to which the Lagrangians L, L ′ project (see Figures 8 and 9). proj a ◦ u must eitherbe constant, or, for the input end, an embedding onto the region surrounded by γ, γ ′ . Thesingular fibres of proj a are biholomorphic to the union of F and F − along the zero section andthe Lagrangians hit these fibres only at the zero section. This means that any holomorphic u with proj a ◦ u constant and with the Lagrangian boundary conditions is necessarily a constantmap and so S –invariant.We may now assume that proj a ◦ u embeds onto a neighbourhood in C . Using this fact, weessentially follow a well-known method of proof (see proof of [2, Lemma 2.4], from which theunderlying argument is copied).Let V ⊂ R × [0 ,
1] be an open neighbourhood with ¯ V disjoint from the boundary of thestrip, which proj a ◦ u maps diffeomorphically onto its image in C . We may assume that V doesnot contain any critical values of proj a , so S –acts freely on a neighbourhood of u ( V ).Let T be the subset of T J S tame ( E z × C ∗ ( F \ Y which vanish except onthe S –orbits of points in a neighbourhood of u ( V ). Now consider the operator: D univu, ˜ J : W u × T 7→ W u,J . Here W u is the W ,p completion of T u B and W u,J is the L p completion of E u,J . D univu, ˜ J is thecompletion of the linearisation of the map ¯ ∂ univu,J : ( u, J ) ¯ ∂ J ( u ). Surjectivity of the operator D univu, ˜ J at all non-invariant, holomorphic u implies the desired result.Suppose that D univu, ˜ J does not surject. Then there exists η = 0, an L p –section of the bundledual to E u,J , which is orthogonal to the image of D univu, ˜ J . It satisfies D ∗ u,J η = 0 and Z R × [0 , h η, Y ◦ Du ◦ j i = 0 for Y ∈ T . η is smooth away from the boundary. Suppose that η ≡ V , then unique continuation proves η ≡ x ∈ V with η x = 0. To derive a contradiction, and conclude the proof, we now construct Y ∈ T forwhich Y ◦ Du at x is an arbitrary ( j, ˜ J )–antilinear map. Multiplying Y by a bump functionsupported near x , this then contradicts R R × [0 , h η, Y ◦ Du ◦ j i = 0. Remark 3.10.
For the following calculation, a quick remark on R –linear maps between com-plex vector spaces is helpful. Suppose A is such a linear map, and J , J are the linear com-plex structures on the domain and range. Then A + = ( A − J AJ ) is complex linear and A − = ( A + J AJ ) is complex anti-linear, with A = A + + A − . We refer to these as thecomplex and linear and anti-linear parts of A , respectively. This generalises straightforwardlyto the case where A is an R –linear map between different complex vector spaces with differentlinear complex structures.Now, to construct Y , as above, it suffices to specify it only at the point u ( x ). Splitting T E z as the sum ker( D proj a ) ⊕ ker( D proj a ) ⊥ ˜Ω , we can describe allowable values for ˜ J , Y, Du as follows: ˜ J = (cid:18) A Bi t (cid:19) where A = − I and B + = 0 ,Y = (cid:18) R ST (cid:19) where S + = A ( BT + RB ) and R + = T + = 0 ,Du = (cid:18) FG (cid:19) where F − = ABG and G − = 0 . Similarly a general complex anti-linear map T ( R × [0 , → T u ( x ) ( E z × C ∗ ( F \ (cid:18) WZ (cid:19) where W + = ABZ and Z + = 0 .G is an isomorphism, because proj a ◦ u is locally a diffeomorphism. This allows us to choose Y as follows. Take R = 0, T = ZG − and S = ABZG − = ABT , then Y ◦ Du = (cid:18) WZ (cid:19) Hence, as claimed, Y ◦ Du can be chosen to be any complex anti-linear map. Corollary 3.11. HF ∗ ( L out × S P | K , L ′ out × S P | K ′ ) ≃ Z ⊗ HF ∗ ( K, K ′ ) HF ∗ ( L in × S P | K , L ′ in × S P | K ′ ) ≃ ( Z ⊕ Z ) ⊗ HF ∗ ( K, K ′ ) Proof.
Using the previous lemma, we choose a regular, S –equivariant, almost complex struc-ture. The only isolated holomorphic strips are those fixed by the S –action. However, theseare in bijection with the isolated holomorphic strips used to calculate HF ∗ ( K, K ′ ), hence theresult. In fact, the same equations hold at the chain level.34 .3.2 Regularity over D We start with a choice of ˜ J endt on the fibres over the boundary marked points, which is regular(for the construction of Floer cohomology at either end). We may in fact take the same almostcomplex structure at both ends. To produce a regular almost complex structure ˜ J on the entirefibration E × C ∗ ( F \ S –invariant sections, then we perturb the almost complex structure away froma neighbourhood of the S –invariant points to ensure regularity for the remaining sections.Recall that the Lagrangians K, K ′ lie in ρ − X [0 , R ), where ρ X is exhausting and plurisubhar-monic. We may assume that the almost complex structure J X , chosen in the previous section,is standard (equal to J stdX ) outside of ρ − X [0 , R ), so all holomorphic strips with boundary on K, K ′ are contained within ρ − X [0 , R ). Definition 3.12.
Let j the standard complex structure on D . We define the family J S tame ( E × C ∗ ( F \ E × C ∗ ( F \ t on D with boundarymarked points removed, by identifying it with R × [0 , J is inthis family if: • ˜ J induces ˜ J endt on the fibres at boundary marked points, • ˜ J is tame relative to j , • ˜ J = ˜ J std outside of some compact set and in a neighbourhood of those Lagrangianintersection points not in M × C ∗ ( F \ • ˜ J makes the S invariant loci complex submanifolds, • ˜ J restricts to the part of the S –invariant locus which may contain sections (a copy of D × X ) as the product of j and J X , • ˜ J is given by ˜ J endt and j on an open neighbourhood of set of S –invariant points in M × C ∗ ( F \ | ρ − X [0 ,R ) with respect to the trivialisation generated by the trivialisation of M .The penultimate point is important as it ensures that the moduli space of S –invariantholomorphic sections is identical to the moduli space of holomorphic strips in ( X, J X ) withboundary on ( K, K ′ ). The final point then ensures that this part of the moduli space ofsections is also regular. It remains to choose ˜ J within this family, which is also regular for allnon-invariant holomorphic sections in order to show that these do not contribute to the relativeinvariant.We follow the same argument as in the previous section, but it is easier here as the complexstructure has more freedom to vary. Lemma 3.13.
Generic ˜ J ∈ J S tame ( E × C ∗ ( F \ is regular.Proof. Pick any ˜ J . Consider the moduli space M non-reg of holomorphic sections u at which ˜ J is not regular. Such u are necessarily non-invariant sections and form a closed subspace of themoduli space of all holomorphic sections. Using the Gromov-Floer topology, and compactifyingthe moduli space by adding broken sections (i.e. a section plus holomorphic strips at one orboth ends), the closure M non-reg of M non-reg is compact. M non-reg also does not contain any S –invariant holomorphic sections, even as broken sections. This is because the almost complex35tructure on the ends is regular and ˜ J is regular at all S –invariant holomorphic sections, sothe gluing theorem shows that any ˜ J is regular at any holomorphic section sufficiently close(in the Gromov-Floer topology) to a broken section involving an S –invariant section.By compactness of M non-reg , it is possible to choose an open neighbourhood W of the S –invariant points, such that for any non-regular, holomorphic section u , there is an openneighbourhood V u in D with ¯ V u ⊂ D \ ∂D and u ( V u ) disjoint from W .The action of S is free at all points which it does not fix and non-invariant sections u aretransverse to S orbits purely by merit of being sections. This means we can construct firstorder deformations Y of ˜ J at a single point in u ( V u ), as in the proof of Lemma 3.9, and extendthese to elements of T J S tame ( E × C ∗ ( F \ T E × C ∗ ( F \
0) as the sum of vertical vectors ker( Dπ ) and horizontalvectors ker( Dπ ) ⊥ ˜Ω , then:˜ J = (cid:18) A Bj (cid:19) where A = − I and B + = 0 ,Y = (cid:18) R S (cid:19) where S + = ARB and R + = 0 ,Du = (cid:18) FI (cid:19) where F − = AB.
A generic complex anti-linear map
T D → u ∗ ( T E × C ∗ ( F \ (cid:18) Z (cid:19) where Z + = 0 , and Y ◦ Du = (cid:18) RF + S (cid:19) . It suffices to choose S − = Z − R ( F + ), because S + cancels with R ( F − ) = RAB = − ARB . In this Section we calculate up to sign the relative invariant in two cases using the fibration E × C ∗ ( F \ Q (see Section 3.2).For the destabilisation map, we modify the Lagrangian boundary condition Q on E in Sec-tion 3.1 and proceed with the same constructions as in Sections 3.2 and 3.3. The curves γ , γ ′ describing the Lagrangian boundary conditions for the stabilisation and destabilisation mapsare illustrated in Figure 10.The Floer cochain complex at input and output ends of E × C ∗ ( F \
0) splits as either of Z ⊗ CF ∗ ( K, K ′ ) , ( Z ⊕ Z ) ⊗ CF ∗ ( K, K ′ ) , depending on whether the curves γ, γ ′ describing the Lagrangian boundary condition intersectat one or both ends. To see this, we simply observe that the Lagrangian intersections formone or two copies of K ∩ K ′ , and that isolated holomorphic strips are S –invariant and thesecorrespond to (one or two copies of) those counted for CF ∗ ( K, K ′ ).36 tabilisationdestabilisation X Figure 10: An illustration of the image under proj a of the S –equivariant La-grangians for the two particular relative invariants. These correspond, as labelled,to parts of the stabilisation and destabilisation maps for symplectic Khovanov ho-mology (see Section 5.4). Generators of the chain complexes for Floer cohomologyare labelled. It should be noted, that the setting for the destabilisation map hasbeen rotated, so that the monodromy clockwise around the base of the correspond-ing fibration gives a positive half twist in a line joining the top and left-hand criticalpoints.To fit in with the notation used in the comparison to Khovanov homology, we prefer towrite Z [ X ] / ( X ) in place of Z ⊕ Z , where X has degree 2. Then: Proposition 3.14.
The relative invariants for the stabilisation and destabilisation map split atthe chain-level as the identity map on CF ∗ ( K, K ′ ) and the following maps on the other factor:destabilisation stabilisation Z / / Z [ X ] / ( X ) Z [ X ] / ( X ) / / Z X X Proof.
Using an S –equivariant regular almost complex structure, we observe that the isolatedholomorphic sections are precisely those which are S –invariant and have constant projectionto X . The remaining holomorphic sections come in non-trivial S –orbits or R –orbits. In E there is only one smooth, S –equivariant section and this is holomorphic. Correspondinglythere is precisely one isolated holomorphic section of E × C ∗ ( F \
0) with boundary on ˜ Q (inthe case of stabilisation or destabilisation) for each point in K ∩ K ′ .It remains only to check the relative Maslov index of the Lagrangian intersections of Q atthe ends of E in order to work out which copy of Z maps to which. This can be done usingproj a , and agrees with the labelling in Figure 10. This section summarises the construction of KH symp with emphasis on those parts of mostrelevance later. The construction is based on the observation that braids on n strands can beviewed as loops in the configuration space Conf n ( C ) of n points in the plane.Suppose we have a symplectic fibration χ : S / / Conf n ( C ) on which symplectic paralleltransport maps are well-defined. Then symplectic parallel transport defines a map π (Conf n ( C ) , P ) / / π (Symp( χ − ( P )))37or any base point P ∈ Conf n ( C ). This is a representation of the braid group. Ideally onewould also require the symplectic parallel transport maps to be compactly supported.This sort of symplectic geometry is not new. Khovanov and Seidel [18] show that sim-ilar symplectic geometry can be used to construct Khovanov’s categorification of the Buraurepresentation of the braid group.We use instead a fibration with subtly different properties. Let Conf n ( C ) be the spaceof configurations with coordinates summing to 0 and let Conf n ( C ) ∼ = C n − be its closure inSym n ( C ). Seidel and Smith [1] define a singular holomorphic fibration of a Stein manifold S n over Conf n ( C ). S n has the property that it pulls back to an exact MBL–fibration over anyholomorphic disc in Conf n ( C ) passing transversely through the locus of Conf n ( C ) \ Conf n ( C )where precisely two coordinates meet (cf. Lemma 27 of [1]).Let D be such a disc with a single singular fibre and let C be a bounded subset of thesingular locus of the pullback fibration over D . Then, for simple enough paths γ (short andlinear in D suffices), the vanishing locus V C to C , in nearby regular fibres over γ , is well-defined(see [1, Section 4.2]). We refer to V C as the relative vanishing cycle to C . If C is open, then V C is a bounded coisotropic, fibring over C with fibres all isotropic spheres.The monodromy of the elementary braid performing a single positive twist in the two chosencoordinates is realised by a loop around the singular value of D . If one is careful to make theloop small, then the monodromy map is well-defined over an open neighbourhood of V C . Bya deformation of the exact symplectic structure, the monodromy can be taken to restrict to aneighbourhood of V C as a fibred Dehn twist in the case that C is open. This is a consequenceof work of Perutz [19]. Non-compactness of the singular locus and of V C are a problem here,so we shall not make direct use of this description, however, it is useful heuristically.The path into the singular value of D corresponds naturally to a ‘singular braid’ of the formof the ( n − , n )–tangle in Figure 11 (right hand side). Heuristically, we think of V C , for largeenough C , as corresponding to this tangle. The loop around the singular value correspondsto the elementary braid (as illustrated on the left hand side). Extending V C over this loopby symplectic parallel transport, one sees already an important invariance result. Namely, V C is unchanged, up to an appropriate isotopy. This corresponds to the ( n − , n )–tangle beingunchanged, up to isotopy, by composition with the braid.Figure 11: An elementary braid (left) and a basic “cap” tangle (right) which isunchanged by the action of that braid. Definition 4.1.
Let S n be the space of degree n monic polynomials with coefficients in thering of 2 by 2 matrices in C [ X ] such that the coefficient of X n − is trace free. The determinantmap χ restricted to S n always gives monic polynomials of degree 2 n with roots summing tozero, so can be thought of as a map to Conf n ( C ) (by identifying monic polynomials with theconfiguration of their roots counting multiplicities). This is an example of the type of fibrationjust discussed.The original definition of S n was as a nilpotent slice of the Lie algebra sl n ( C ), specifically38 local transverse slice to the orbit of the Adjoint action of SL n ( C ) at a nilpotent matrix withprecisely 2 Jordan blocks of size n . This setting is described more explicitly as follows. Definition 4.2.
One defines S n to consist of the following 2 n by 2 n matrices with complexcoefficients: A I ... . . .... IA n where each A i is a 2 by 2 matrix (i.e. in gl ( C )) and A is trace free (in sl ( C )). Then the map χ is defined to give the characteristic polynomial of the matrix.The two definitions are related by the isomorphism: A I ... . . .... IA n X n − n X i =1 X n − i A i which commutes with the map χ on either side. In either setting we shall denote the fibre of S n over a configuration P by Y n,P .Alternatively, it has been shown by Manolescu [20] that, for P ∈ Conf n ( C ), the fibre Y n,P injects holomorphically into the Hilbert scheme Hilb n ( M n ( P )). Here, M n ( P ) is the 2–dimensional (over C ) Milnor fibre associated to the A n singularity. It is explicitly describedas a smooth affine surface by the equation u + v + P ( z ) = 0 in C . The symplectic paralleltransport maps (defined appropriately on compact subsets containing the relevant Lagrangiansubmanifolds) are, in an appropriate sense, lifts of Dehn twists on M n ( P ). This reveals aclose connection between the symplectic geometry of S n with the braid group Br n which actsnaturally by Dehn twists on the A n Milnor fibre.With an appropriate choice of K¨ahler metric [1] uses “rescaled” symplectic parallel transportmaps. For this paper however, it is more convenient to use actual symplectic parallel transport.This has the consequence that we often need to perform some deformation in order to ensurethat the symplectic parallel transport is well-defined on relevant compact subsets. Given anysmooth path in Conf n ( C ) we define the symplectic parallel transport maps on sufficiently largecompact subsets of a given fibre by the deformation described in Lemma 2.17. SpecificallyRemark 2.19 covers this case.The singular locus of S n corresponding to two coordinates in Conf n ( C ) meeting at zero isprecisely MBL( χ ) (see Section 2.2).Furthermore, it is shown that S n − is canonically isomorphic to a singular locus in S n overthe points (0 , , µ , . . . , µ n ) in a manner compatible with the fibrations. Local neighbourhoodshave a particularly nice form. Lemma 4.3 (cf. Lemma 27 of [1]) . Let D be a disc in Conf n ( C ) given by the monocoordinates ( −√ ǫ, √ ǫ, µ , . . . , µ n ) with ǫ small. In terms of monomials, the elements of D have the form ( X − ǫ )Π nk =3 ( X − µ k ) . hen there is a neighbourhood of S n − ⊂ χ − ( D ) ∩ S n on which the fibration χ has the localmodel given by a neighbourhood of Y n − , ( µ ,...,µ n ) × below: D C ( −√ ǫ, √ ǫ,µ ,...,µ n ) ǫ / / χ − ( D ) D χ (cid:15) (cid:15) χ − ( D ) Y n − , ( µ ,...,µ n ) × C local isomorphism near S n − ⊂ χ − ( D ) ∩ S n / / Y n − , ( µ ,...,µ n ) × C C a + b + c (cid:15) (cid:15) In fact, one can drop the requirement that the two coordinates meet at zero (rather thanan arbitrary point in C ) in the case that n ≥ L be a simply connected, compact, exact Lagrangian submanifold of Y n − , ( µ ,...,µ n ) .Then the relative vanishing locus to L (over simple enough paths into the singular value ofthe model fibration in the lemma) gives an exact Lagrangian submanifold of a nearby regularfibre Y n,u ( z ) of S n (cf. [1, Section 4.2]). It is diffeomorphic to L × S and well-defined up toLagrangian isotopy, which, since π ( L × S ) = 0, implies it is well-defined up to compactlysupported Hamiltonian isotopy. Rezazadegan [8] has an approach to the non-compact case,but it is not necessary here.By starting with L = { } = Y , and repeating this construction we can generate compactlysupported Hamiltonian isotopy classes of Lagrangian submanifolds corresponding to isotopyclasses of certain (0 , n )–tangles. Definition 4.4.
Let γ i for i = 1 , , . . . , n be a sequence of vanishing paths in Conf i ( C ) fromnon-singular values z i to singular values w i . Suppose also that, for each i = 1 , , . . . , n − X z i = w i +1 . Then we can consider the composition of the paths X n − i ) γ i as a piecewisesmooth path in Conf n ( C ). We call such a path an iterated vanishing path . Viewed in terms ofconfigurations of points in C , such a path describes a way of bringing together 2 n points intopairs (fixed at 0).Repeating the relative vanishing cycle construction along a short enough iterated vanishingpath, starting with a single point inside { } ∼ = Y , , allows one to construct a Lagrangian iteratedvanishing cycle . Given a longer iterated vanishing path, this construction also works, thoughit becomes necessary to perform a deformation (described in Lemma 2.17 and Remark 2.19)to ensure that symplectic parallel transport of the vanishing loci is always well-defined.It is shown in [1] that iterated vanishing cycles (up to isotopy) are independent of isotopyof the iterated vanishing paths defining them. Definition 4.5. A crossingless matching of P ∈ Conf n ( C ) is an embedded set of n curves in C , such that each coordinate of P is an endpoint of precisely one curve. We denote the set ofcrossingless matchings on P by M nP .It suffices to specify a crossingless matching A ∈ M nP together with an ordering on thecurves in order to determine an iterated vanishing cycle up to isotopy. To construct the iteratedvanishing cycle, one takes any γ : [0 , / / Conf n ( C ) whose coordinates in Conf n ( C ) sweepout the curves of A in C , bringing together endpoints of the curves in the chosen order. Choicesof such γ are all isotopic to each other and these isotopies induce exact isotopies of the iteratedvanishing cycles. In fact, changing the ordering also changes the iterated vanishing cycle [1]only by exact isotopy. 40igure 12: An example crossingless matching for P ∈ Conf ( C ).By [21] any link L can be put into braid position by isotopy, i.e. a position of the formshown in Figure 13 for some braid β on n strands. This position can be thought of as splittinginto two identical crossingless matchings (on some P ∈ Conf n ( C )) and a braid on 2 n strandsconsisting of β on the left-hand n strands and the trivial braid on the others as in the picture. β Figure 13: Braid position of a link, split as two identical crossingless matchings anda braid, trivial on the strands on one side.Let L ⊂ Y n,P be the Lagrangian corresponding to the crossingless matching and L ′ beits image under the parallel transport corresponding to β . Then the symplectic Khovanovhomology is defined as KH symp ( L ) := HF ( L, L ′ ).It follows from general results on Floer cohomology under symplectomorphisms and Hamil-tonian isotopy, that KH symp is independent of braid isotopy. To prove that KH symp did notdepend on the choice of braid representing a given link, Seidel and Smith show that it is invari-ant under the two Markov moves . These are changes to the braid which suffice to pass betweenany two braid positions of the same link.Invariance under conjugation of the braid (
Markov I ) works straightforwardly by a trickinvolving isotopy of crossingless matchings.The
Markov II move (Figure 14) involves adding an extra strand to the braid β and“twisting it in” by either a positive or a negative half twist. The proof of invariance underthis move is more involved and makes use of a local model for the fibration of S n over aneighbourhood of a point in Conf n ( C ) with three coordinates coinciding.Another way in which parts of S m − are locally nested in S m is as follows. Let P ∈ Conf m − ( C ) be a polynomial with pairwise distinct roots, one of which is 0. Then it turnsout that the singular locus of S m over X P is precisely X Y m − ,P . There is also a simpleholomorphic model for neighbourhoods of this locus inside S m . Namely one can describe it asa holomorphic associated bundle over Y m − ,P in the following way. Lemma 4.6 (cf. Lemma 29 of [1]) . Let XP ∈ Conf m − ( C ) be a polynomial with pairwisedistinct roots one of which is . Let D be a small holomorphic bidisc in Conf m ( C ) parametrised Figure 14: The braid position of Figure 13 after application of the Markov II move. by ( d, z ) ( X − Xd + z ) P There is a holomorphic line bundle F over Y m − ,XP such that the following is a local modelfor χ − ( D ) . D C / / χ − ( D ) D χ (cid:15) (cid:15) χ − ( D ) ( F \ × C ∗ C local isomorphism near X Y m − ,XP ∩ χ − (0 , / / ( F \ × C ∗ C C f (cid:15) (cid:15) The map f is induced by the map C / / C taking ( a, b, c, d ) ( d, a − ad + bc ) . The C ∗ –actionon C is given by ( a, b, c, d ) ( a, ζ − b, ζ c, d ) for ζ ∈ C ∗ . We have stated a simple version of the lemma. In fact Lemma 29 of [1] is more general inthat it gives a local model near the fibre over a polynomial P ′ ∈ Conf m ( C ) with root µ ofmultiplicity 3 and the remaining roots pairwise distinct. The difference is that µ is not requiredto be zero. This generality is necessary (though only in some cases).The use of this local model is as follows. One considers the fibration over discs defined bya fixed value of d = 0. C × { d } / / C is an exact Lefschetz fibration with two singular values.These each correspond to moving two of the three roots of ( X − µ ) − ( X − µ ) d + z togetherin different ways. This makes ( F \ × C ∗ C an exact Morse Bott Lefschetz fibration over C .Let K, K ′ in Y m − ,XP be the iterated vanishing cycles constructed for some braid positionof a link. One performs, with each of K, K ′ , the relative vanishing cycle construction in( F \ × C ∗ C to create L, L ′ . This corresponds to adding a new unlinked component to thelink. Then L is carried around the other singularity by parallel transport to τ L , correspondingto twisting the new component into the braid as in Figure 14. When this is done compatiblywith splitting of the fibration from the lemma one finds a bijection of the moduli spaces usedto calculate HF ( K, K ′ ) and HF ( τ L, L ′ ).With an appropriate grading shift, this takes care of invariance under the Markov II move.The important part to notice is that, by Lemma 4.6, the fibration splits in such a way as toseparate the local behaviour of the three coordinates involved from the rest of the link. Thisis studied in more detail later. 42 .2 Symplectic Khovanov homology for links in bridge position Here we generalise the definition of KH symp such that it is defined directly using any bridgeposition of a link, not necessarily just braid positions. This will be useful in defining the mapson KH symp corresponding to smooth closed cobordisms in R between links. These maps willbe covered in the next section.Any link in R = { ( x, y, z ) } is isotopic to one in which the height function (mappingpoints on the link to the value of their z –coordinate) has only non-degenerate critical points.Furthermore we may require all local maxima of z to occur where z > z <
0. We will call such a position of a link admissible . Each half of a link in an admissibleposition (i.e. the parts above and below height z = 0) specifies a path for the iterated vanishingcycle construction and hence a Lagrangian submanifold of the same fibre Y n,P . One shouldnote that these paths need not be embedded or disjoint from each other.An isotopy of a link through admissible positions fixing the point P ∈ Conf n ( C ) causesHamiltonian isotopies of this pair of Lagrangians. On Floer cohomology these give the identityup to canonical isomorphisms (from continuation maps). Although the Floer cohomology mayarise from a different chain complex, before and after the isotopies, it is well-defined up tocanonical isomorphism.Now we extend this definition to bridge diagrams of links and use it to prove that theFloer cohomology one gets for any two bridge diagrams of the same link is always isomorphic(though not with a canonical isomorphism unless one is comparing bridge positions comingfrom isotopic admissible positions).Figure 15: The Hopf link as a bridge diagram and a projection of the admissibleposition the diagram represents ( α –curves are drawn smooth, β –curves dashed). Definition 4.7. An n –bridge diagram ( P, A, B ) of a link consists of a set P of 2 n points inthe plane together with a pair of crossingless matchings A, B (sets of n disjoint curves in theplane which join the points of P in pairs) which intersect transversely away from P . We referto the curves in these matchings as α – and β –curves respectively and to the points of P as the vertices of the diagram.Any bridge diagram is easily turned into an admissible link by pulling the α –curves up outof the plane in which the diagram is drawn and pushing the β –curves down. Such an admissibleposition will project back onto the bridge diagram by the orthogonal projection onto the plane.In fact any two such admissible positions are isotopic by an isotopy of admissible positionsfixing P . Therefore we have a canonical choice of isomorphisms between the Floer homologiesthey induce. Definition 4.8.
Given an admissible position L for a link or bridge diagram ( P, A, B ), wedefine its symplectic Khovanov homology KH symp ( L ) or KH symp ( P, A, B ) to be the LagrangianFloer cohomology HF ( L A , L B ). Here L A , L B are constructed by the iterated vanishing cycleconstruction for a path specified by L , ( P, A, B ) respectively.43rom the above discussion, we see that this gives a well-defined invariant of admissiblepositions, up to isotopy fixing the point P ∈ Conf n ( C ), and also of bridge diagrams up toisotopy fixing P .A regular isotopy of bridge diagrams is an isotopy P ( t ) of P together with isotopies of A andof B through crossingless matchings on P ( t ). The intermediate matchings need not intersecttransversely.A passing move on a crossingless matching is where we take a smooth loop γ disjoint fromthe curves of the matching and enclosing precisely one of the curves and replace another of thecurves with its connect sum with γ .A stabilisation move on an n –bridge diagram is as follows. We mark a closed interval onan α –curve γ which is disjoint from the β –curves and from P . We then add its two endpointsto P , add the interval to B , replace γ in A with the two curves we get by removing the interiorof the interval from γ . The resulting ( P, A, B ) is an ( n + 1)–bridge diagram for the same link.Figure 16: The passing move (above) and the stabilisation move (below).Any bridge diagram D naturally yields a crossing diagram proj( D ) by projecting the con-struction in R × [ − ,
1] to R . There are, however, many different bridge diagrams that yieldthe same crossing diagram by this method. All such diagrams are related by a finite sequenceof stabilisation moves (and destabilisation moves) and regular isotopy fixing the projection. Lemma 4.9.
Finite sequences of the three moves described above applied to any bridge diagram D suffice to perform all Reidemeister moves on proj( D ) . Figure 17: The Reidemeister moves (one way up) after some destabilisation movesare the projections of the above moves.
Proof.
By repeated destabilisation, the neighbourhood in R in which one performs a Reide-meister move can be made to be one of those in Figure 17 (up to a reflection and/or swapping44he α – and β –curves). The illustration makes it clear that isotopy suffices to perform theReidemeister I and II moves and the passing move to perform the Reidemeister III move. Lemma 4.10.
Finite sequences of these three moves suffice to go between any two bridgediagrams of the same link.Proof.
Given D , D , bridge diagrams of the same link, there is a sequence of Reidemeistermoves from proj( D ) to proj( D ). By the previous lemma, there is a finite sequence of movestaking D to some D with proj( D ) = proj( D ). Now a finite sequence of stabilisation anddestabilisation moves relates D and D .Isotopy of P to Q along a path in Conf n ( C ) gives an exact symplectomorphism from(any compact subset of) Y n,P to Y n,Q by symplectic parallel transport. Given an isotopy ofadmissible links which induces this same isotopy P to Q on its intersection with the plane z = 0,this symplectomorphism carries the Lagrangians corresponding to the first admissible link tothose for the second (up to Hamiltonian isotopy). Hence, it gives an isomorphism of symplecticKhovanov homology. Using this, one sees that regular isotopy of bridge diagrams and also thepassing move do not change the isomorphism class of symplectic Khovanov homology. Lemma 4.11.
Stabilisation of a bridge diagram gives a bridge diagram with isomorphic sym-plectic Khovanov homology.
Modified by an isotopy of admissible links, the Markov II + move is just the stabilisationmove (see Figure 18). The proof of invariance under the Markov II + move in [1] can be carriedout in exactly the same manner in this setting since there is no part of it that requires theadmissible links defining the Lagrangians to be of the particular form used in that paper.The only difference is technical. The localisation to coordinates of the form given inLemma 4.6 must be carried out differently if we insist on using the construction of Lemma 2.17instead of rescaled symplectic parallel transport in the iterated vanishing cycle construction.A strictly stronger version of the above result, compatible with the methods of this paper, isproven in Section 5.4, so we will not go into detail here.Figure 18: An illustration of the Markov II + move as seen in a bridge diagram. Inthe first step a new link component is created, then in the second step ‘twisted in’by a positive twist along the thicker line.In conclusion it has been shown in this section that: Theorem 4.12.
The isomorphism class of KH symp of a bridge diagram as a relatively gradedgroup is an invariant of link isotopy. Remark 4.13.
It is worth noting here that the Markov II − move can equally well be usedin the above. Namely the crossingless matchings one gets by replacing the positive twist in45igure 18 by a negative twist differ by an isotopy. In the setting of [1] HF ( L A , L B ) carries anabsolute grading which the Markov II ± moves change by different, but fixed, amounts. Anoverall grading shift from the writhe of the link counteracts this to give an absolute grading onKH symp . To define the same absolute grading for links in bridge position will take a little morecare as these observations show that the absolute grading on HF should not be preserved byall isotopies of admissible links. In fact a correction needs to be made for the passing movesto get the absolute grading on KH symp in this setting. Remark 4.14.
By stretching all the local maxima in the upper half of an admissible linkupwards one can decompose it as a trivial crossingless matching and a braid. Now consideringthe braid group action by diffeomorphisms on the plane containing this trivial matching, onefinds that the upper tangle is isotopic to a tangle whose projection to the ( x, y )–plane has nocrossings. Doing the same with the lower half, the link can be taken by isotopy of the twohalves to one which projects to a bridge diagram. Hence Theorem 4.12 also holds for KH symp of any admissible link.
In the setting of Khovanov homology one has maps between the chain complexes correspond-ing to elementary saddle cobordisms and creation/annihilation (‘cap’/‘cup’) cobordisms asportrayed in a given crossing diagram. In this section we construct analogous maps on KH symp and prove that, up to an overall sign ambiguity, they compose to give maps depending only onthe isotopy class of the composite cobordism. Unless otherwise indicated, isotopies of a cobor-dism must fix the links at either end of the cobordism, since we generally want the symplecticKhovanov homology of the domain and range to be well-defined up to canonical isomorphism.In the case of cobordisms which are themselves isotopies of links through admissible posi-tions there is little difficulty. Namely, corresponding to isotopies of admissible links which fixthe intersection with the plane z = 0 we have continuation maps between Floer cohomologygroups. Suppose we have an isotopy of admissible positions which moves this intersection alonga path γ : R / / Conf n ( C ) which is constant near ±∞ . Then we can construct an admissiblemap u γ : R × [0 , / / R / / Conf n ( C )This defines an exact MBL–fibration (cf. Section 2.2) with two ends. Extending La-grangians along the boundaries of this fibration allows us to calculate a relative invariantmapping between the symplectic Khovanov homologies of the admissible positions at eitherend of the isotopy.In fact this map on KH symp is the same as the map induced by the symplectic paralleltransport along γ , but it is handy to have it given in terms of a symplectic fibration. Remark 5.1.
Deforming the admissible map u γ gives, by Lemma 2.36, the following result.Isotopies which are isotopic through isotopies of admissible positions of links (fixing the linksat either end) give the same map on symplectic Khovanov homology. One should note however,that there exist isotopies from a link to itself which do not give the identity map on Khovanovhomology [7, Theorem 1]. 46 .1 Saddle cobordisms In this section, we construct maps which correspond to certain saddle cobordisms. We beginby outlining how saddle cobordisms naturally arise from admissible maps into Conf n ( C ).Let L be a link in an admissible position which intersects the z = 0–plane in the con-figuration P . Up to some choice, this defines (and is defined by) iterated vanishing cycles γ A , γ B : [0 , / / Conf n ( C ). We shall assume both paths end in a segment of constant pathat P .For the moment, we shall restrict attention to link cobordisms supported close to { z = 0 } .The parts of the trivial cobordism from L to itself away from { z = 0 } can be realised as thefollowing sets: { ( t, z, λ ) ∈ [ − , × [1 , × C : λ is a root of γ A ( z − } , { ( t, z, λ ) ∈ [ − , × [ − , − × C : λ is a root of γ B (1 − z ) } . We now exhibit some link cobordisms by joining these surfaces up with a surface in the missingregion [ − , × [ − , × C .Let u : D / / Conf n ( C ) be an admissible map (see Definition 2.15) with two boundarymarked points ±
1, such that u ( −
1) = P . To make the cobordism we construct be smooth, wealso require u to be a constant map near each of the boundary marked points. Now chooseyour favourite smooth map [ − , × [ − , / / D which is a diffeomorphism away from theboundaries and maps {± } × [ − ,
1] to ± u with this map ˜ u .The following set defines a braid cobordism from the trivial braid to some other braid fixingthe configuration P at the ends: { ( t, z, λ ) ∈ [ − , × [ − , × C : λ is a root of ˜ u ( t, z ) } . Hence, it can be inserted in the above construction to give a link cobordism, starting at L ,which is supported near { z = 0 } .Now we describe how to construct an admissible map corresponding to certain elementarysaddle cobordisms starting at the link L . As input to the construction, take an embedded path δ : [0 , / / C ending at roots of P and otherwise disjoint from them. The link cobordism willbe the elementary braid cobordism which adds a single negative half-twist along the curve δ .For reasons of orientation of holomorphic embeddings in Conf n ( C ), the sign of the half-twistone adds along δ will be necessarily negative. However, this problem is actually an artefact ofthe viewpoint of braid cobordisms. There is no sign associated to elementary saddle cobordismsof links.By fattening the curve δ one can define a smooth map v : D / / C which: • maps ± δ , • otherwise misses all the roots of P , • and is a holomorphic embedding near 0 ∈ D The space of choices of such a smooth map is contractible.47et ˜ P be the product of X − r over all roots r of P which are not ends of δ . Then we definethe admissible map u δ to be the following composite: D Conf ( D ) / / Conf ( D ) Conf n ( C ) / / z X + z ✤ / / ( X − a )( X − b ) ( X − v ( a ))( X − v ( b )) ˜ P ✤ / / One can check that this is admissible. We fix an input boundary marked point at − u ( −
1) is the configuration P . In orderto have this admissible map well-defined up to isotopy fixing the boundary marked points,we shall generally require that the v is constant on the segments of ∂ D between 1 and i andbetween − − i . This causes the output boundary marked point to also be P and the pathin Conf n ( C ) given by u applied to the lower half of ∂ D to be contractible through Conf n ( C ).Hence, the lower half (where z <
0) of the link one has at the output end of the cobordism isthe same as the lower half at the input. This is a convenient simplification when one wishes toexpress these cobordisms in terms of bridge diagrams.Figure 19: An elementary saddle cobordism specified by a curve in a bridge diagram.When projected to a crossing diagram, this example is also the usual elementarysaddle cobordism between resolutions of a crossing.In a bridge diagram (
P, A, B ) for L we draw the curve δ in grey to indicate a cobordismabout to be performed. Figure 19 gives an example of this with an arrow pointing to the bridgediagram for the output link. The simplification mentioned above allows us to fix P and applythe entire half-twist to the curves from A , without changing B . The cobordism illustrated inFigure 19 is an elementary saddle cobordism of the more familiar kind under projection of thebridge diagram to a crossing diagram. From now on we shall view cobordisms “from above”in this manner.It is now a simple matter to define the map f δ that a cobordism specified by a curve δ induces on symplectic Khovanov homology. Definition 5.2.
Given a bridge diagram (
P, A, B ) together with a curve δ , as above, specifyinga cobordism to the bridge diagram ( P ′ , A ′ , B ′ ) we define the map f δ : KH symp ( P, A, B ) / / KH symp ( P, A, B )48o be the relative invariant induced by the admissible map u δ and the pair of Lagrangians usedto define KH symp ( P, A, B ). This relative invariant is well-defined by Lemma 2.36.
Remark 5.3.
The relative invariant of this definition is well-defined up to composition withthe canonical isomorphisms of KH symp of the domain and range. To show this one simplyapplies Lemma 2.36. It is independent of isotopy of δ and of changing of ( P, A, B ) to any otherbridge diagram representing the same admissible link position (i.e. isotopy fixing P and thepassing move).Composing two admissible maps given by elementary saddle cobordisms (by gluing striplikeends of the admissible maps) gives the composite of the maps on symplectic Khovanov homol-ogy. The two singular values for the composite admissible map can be moved past each otherby an isotopy of admissible maps (see Figure 5. Decomposing the result into two admissiblemaps shows that the map on symplectic Khovanov homology can be expressed in different waysas a composite. This is a simple version of Lemma 2.24. It will be vital later to know whichcomposites of saddle cobordisms give the same maps in this way. δ θ δ θ θ θ δ δ Figure 20: Two simple examples of curves δ , θ and θ δ in the plane satisfying theconditions of Lemma 5.4. Lemma 5.4.
Let δ , θ be embedded curves in C \ P , each joining two points of P ∈ Conf n ( C ) .Write θ δ for the result of a positive half twist along δ applied to θ . Then the maps these curvesspecify on any KH symp ( P, A, B ) satisfy the following relation: f θ ◦ f δ = f δ ◦ f θ δ Proof.
One glues the pairs of admissible maps for each composite together and observes thatthe resulting admissible maps are isotopic by an isotopy which moves the singular values pasteach other.This is a minor abuse of notation. For example, the maps f δ have different domain andrange on either side of the equation. However, the composites map between the same (up tocanonical isomorphism) cohomology groups. Corollary 5.5.
Suppose δ and θ do not intersect in C . Then the cobordisms specified by δ and θ can be performed in either order with the same resulting map on KH symp ( P, A, B ) . .2 A semi-canonical splitting of symplectic Khovanov homology In this section, we generalise the splitting of Lemma 4.3 in a way which allows one to split sym-plectic Khovanov homology of a link consisting of multiple unlinked components as a product.This is done in the case of 2 components, where one is an unknot, in Lemma 41 of [1]. We thenstrengthen the result by showing the splitting is well-defined up to canonical isomorphism onone factor and furthermore that one can split certain maps f γ induced by saddle cobordismsin a similar manner.In the following we will use both descriptions of the fibrations S k given in Definitions 4.1and 4.2 and freely switch between them. It should be clear from context which is meant.We start the construction by observing that there is a natural choice of copy of S k in S k + l ,namely X l S k . Similarly X l Conf k ( C ) ⊂ Conf k + l ) ( C ) and these two inclusions are compatiblewith the fibrations. That is, for y in S k we have X l χ ( y ) = χ ( X l y ).Choose any polynomial P = Q i ( X − µ i ) ∈ Conf k ( C ∗ ). Let ξ be the linear subspace of C k +1 given by the equation 2 lz + P ki =1 z i = 0. There is a neighbourhood of X l P ∈ Conf k + l ) ( C )which is biholomorphic to a neighbourhood of ( X l , , P ) in Conf l ( C ) × C × Conf k ( C ) and of( X l , (0 , µ , . . . , µ k )) in Conf l ( C ) × ξ . Explicitly, the biholomorphisms are described (in theopposite order) by l Y i =1 ( X − a i ) , ( z , z , . . . , z k ) ! l Y i =1 ( X − a i ) , z , k Y i =1 ( X − z i − lz k ) ! l Y i =1 ( X − a i − z ) k Y i =1 ( X − z i ) . We will now show that, for y ∈ Y k,P we can locally model any transverse slice to X l y ∈ S k + l on S l × ξ compatibly with the description of a neighbourhood of X l P as Conf l ( C ) × ξ . In factthese models can be chosen in a consistent manner for all y ∈ Y k,P simultaneously: Proposition 5.6 (cf. Lemma 4.3) . There is a tubular neighbourhood of X l Y k,P in S k + l whichis biholomorphic to a neighbourhood of Y k,P in Y k,P × S l × ξ . Furthermore the following diagramof holomorphic maps commutes: S k + l Conf k + l ) ( C ) χ (cid:15) (cid:15) Y k,P × S l × ξ Conf l ( C ) × ξ χ × id ξ (cid:15) (cid:15) Y k,P × S l × ξS k + l local ∼ = defined near X l Y k,P o o Conf l ( C ) × ξ Conf k + l ) ( C ) local ∼ = defined near X l P o o We shall in general actually use the following simpler version (restricting to 0 ∈ ξ ) whichsuffices to describe the behaviour of links with multiple unlinked components. Corollary 5.7.
Given any P ∈ Conf k ( C ∗ ) we have the following local holomorphic model for S k + l defined: • on a neighbourhood of X l Y k,P in χ − ( P Conf l ( C )) ⊂ S k + l • over a neighbourhood of P X l in P Conf l ( C ) ⊂ Conf k + l ) ( C )50 k,P × S l Conf l ( C ) χ (cid:15) (cid:15) χ − ( P Conf l ( C )) P Conf l ( C ) χ (cid:15) (cid:15) Y k,P × S l χ − ( P Conf l ( C )) local ∼ = defined near X l Y k,P o o Conf l ( C ) P Conf l ( C ) local ∼ = defined near P X l o o The map of bases takes Q to P Q . The map of total spaces restricts to Y k,P × { X l } asmultiplication ( y, X l ) yX l . The proof of Proposition 5.6 comes from generalising Lemmas 24 to 27 of [1] and will bepresented in an analogous manner for ease of comparison. The slices S n are thought of here asspaces of 2 n × n matrices. Lemma 5.8.
For y ∈ S k + l , projection to the first l coordinates of C k + l ) restricts to aninjection ker( y l ) / / C l Proof.
For y ∈ S k + l we observe that y l is of the form (cid:18) A IB (cid:19) where A is a 2 l × k matrix and B is a 2 l × l matrix. Hence, if ( v , . . . , v k + l ) ) ∈ ker( y l ), thenwe have: − A v ... v l = v l +1 ... v k + l ) Therefore the first 2 l coordinates determine the rest. Lemma 5.9.
The subspace of y ∈ S k + l such that ker( y l ) is l –dimensional can be canonicallyidentified with S k . The identification is compatible with the adjoint quotient map and theinclusion Conf k ( C ) ⊂ Conf k + l ) ( C ) previously mentioned. Remark 5.10.
This lemma describes the inclusion, written earlier as X l S k ⊂ S k + l . In par-ticular it shows that for P ∈ Conf k ( C ∗ ) the set X l S k can be described as the transverseintersection of S k + l ⊂ sl k + l ( C ) with an orbit of the adjoint action of SL k + l ( C ). This orbit isfurthermore given by the property that its elements have 2 Jordan blocks, each of size l , for theeigenvalue 0. Hence transverse slices to elements of the orbit should contain ( l, l ) –type nilpotentslices , i.e. copies of S l . The proof of Proposition 5.6 will be based upon this observation. Proof.
Let y ∈ S k + l have kernel of dimension 2 l over C . We consider the Jordan blocks for y l for eigenvalue 0. Lemma 5.8 in the case l = 1 says that there are at most two. Hencethere must be precisely 2 and they must both have size at least l . This means the injection ofLemma 5.8 is an isomorphism for such y .We will write vectors v ∈ ( C ) ( k + l ) with coordinates v , . . . , v ( k + l ) ∈ C . Now, observe that y i ( v ) = M + v i +1 where M depends only on v , . . . , v i . We prove, by induction on i , that A k +1 − i = 0. 51n the case i = 1, we have y ( v ) k = A k v . In order for there to be a v ∈ ker( y ) for any choiceof v , we must have A k = 0. For i > A k − i +2 , . . . , A k = 0), we have: y i ( v ) k +1 − i = y (cid:0) y i − ( v ) (cid:1) k +1 − i = A k +1 − i ( y i − ( v ) ) + y i − ( v ) k +2 − i = A k +1 − i ( M + v i ) + A k +2 − i ( y i − ( v ) ) + y i − ( v ) k +3 − i ...= A k +1 − i ( M + v i )Here again M depends only on v , . . . , v i − , so in order for there to be a v ∈ ker( y i ) for anychoice of v , . . . , v i we must have A k +1 − i = 0.We have just shown that the space of y ∈ S k + l with dim ker( y l ) = 2 l is contained in thespace of those y with A k +1 , . . . , A k + l = 0. In fact one can check they are the same. This spaceis canonically isomorphic to S k by the map f : A I ... . . .... IA k A I ... . . . A k . . .0 . . .... I . (1)This has the property that X l χ ( y ) = χ ( f ( y )), so it commutes with the inclusion X l Conf k ( C ) ⊂ Conf k + l ) ( C )Also note that this map takes y to X l y in the polynomial presentation of S k + l .With these results we now prove Proposition 5.6. Proof.
First we choose for every y ∈ X l Y k,P a local affine linear transverse slice varying holo-morphically with y . It suffices (Lemma 5 (i) of [1]) to pick these slices transverse to the adjointorbit of y within S k + l , i.e. from a complement to T y X l Y k,P in T y S k + l . For example one cantake the following. First choose a complement W y to T y Y k,P in T y S k varying holomorphicallywith y. This splitting problem has a positive solution because Y k,P is an affine subvariety of S k .Then take the direct sum of V ⊕ X l W with the orthogonal complement to X l S k as a vectorsubspace of S k + l .Another way to choose a transverse slice to X l y is (by Lemma 8 of [1]) to decompose sl k + l ) ( C ) into eigenspaces of the semisimple part ( X l y ) s of X l y . All the eigenspaces are 1–dimensional with the exception of the 0–eigenspace which is canonically identified with C l byLemma 5.8. This gives a holomorphically varying local transverse slice. X l y s + S l + z Conf l ( C ) × ξ χ × f (cid:15) (cid:15) f is a local biholomorphism z / / C × Conf k ( C ∗ ).Any two local transverse slices at a point are locally isomorphic by an isomorphism thatmoves points only inside their adjoint orbits (Lemma 5 (iii) of [1]) and hence does not affect χ . This gives us the required result so long as this isomorphism can be chosen to vary holo-morphically with y . In fact the isomorphism depends only on a choice of local submanifold K y ⊂ SL k + l ( C ) near the identity element e with tangent space at e complementary to that ofthe adjoint orbit of y s . The choice K y = exp[ sl k + l ( C ) , y s ] suffices (cf. proof of Lemma 27 of[1]).We now demonstrate how one can split symplectic Khovanov homology using appropri-ate localisation arguments. We begin with a useful, if obvious, lemma related to shrinkingcomponents of a link. Lemma 5.11.
Let D be a disc centred at ∈ C and let P = P P ∈ Conf k ( C \ D ) × Conf ( D ) ⊂ Conf k + l ) ( C ) Suppose we have a bridge diagram ( P, A, B ) which splits as the union of bridge diagrams ( P , A , B ) supported on D and ( P , A , B ) supported outside D . See Figure 21 for an exam-ple. Then there is a canonical isotopy of bridge diagrams starting at this bridge diagram andending at one which splits as the union of ( P , A , B ) with ( P , A , B ) , the latter scaled in C by any λ ∈ (0 , . Hence, rescaling defines canonical isomorphisms on KH symp (or canonicalchain homotopy classes of chain maps at the cochain level).One can scale down any admissible map of a surface with boundary marked points into P Conf ( D ) ⊂ Conf k + l ) ( C ) to obtain another admissible map, such that the relative invariant of the two surfaces are relatedby composing on either side by the canonical isomorphisms mentioned above. ( P , A , B ) D ( P , A , B ) Figure 21: A diagram of a link with multiple unlinked parts arranged as specified inLemma 5.11 and Theorem 5.12.
Proof.
The first part is obvious. For the second part one just needs to check that the C ∗ –action on Conf n ( C ), rescaling all roots by a complex number, preserves the property of a53urface being admissible. The result comes from the isotopy of admissible curves one gets bycontinuously performing the scaling and composing on either end by the isotopies from the firstpart. Theorem 5.12.
Let D be a disc centred at ∈ C . Suppose we have a bridge diagram ( P, A, B ) which splits as the union of bridge diagrams ( P , A , B ) supported on D and ( P , A , B ) supported outside D .Then KH symp ( P, A, B ) is isomorphic to the K¨unneth product of KH symp ( P , A , B ) and KH symp ( P , A , B ) . Furthermore, this isomorphism is canonical up to composition with auto-morphisms of the KH symp ( P , A , B ) –factor. Remark 5.13.
We only apply this theorem in cases where one factor has no torsion (e.g. itis an unlinked union of unknots), so the K¨unneth product on cohomology is an honest tensorproduct.We shall denote by Ψ the local isomorphism given in Corollary 5.7 Y k,P × S l Ψ / / χ − ( P Conf l ( C ))The idea is now to localise the calculation of Floer cohomology to the model region Y k,P × S l with Lagrangians and exact symplectic structures which respect the splitting. In very specificcases (where ( P , A , B ) is the simplest bridge diagram for a single unknot, or also a union ofsuch bridge diagrams), one can use the same argument as for Lemma 41 of [1].The proof of Theorem 5.12 below necessarily uses a more complicated argument to controlthe position of the Lagrangian iterated vanishing cycles to ensure they are contained in theregion where Ψ is defined. Otherwise it follows essentially the same argument. We begin byoutlining the steps of the proof.We start with the following information: • Configurations P ∈ Conf k ( C ) and P ∈ Conf l ( C ), • Iterated vanishing paths γ A , γ B in Conf k ( C ) representing the crossingless matching( P , A , B ), • Iterated vanishing paths γ A , γ B in Conf l ( C ) representing the crossingless matching( P , A , B ), • Any choice of exhausting plurisubharmonic functions ρ l , ρ k and ρ k + l on S l , S k , S k + l respectively, such that ρ k is the restriction of ρ k + l to X l S k ⊂ S k + l , • Exact K¨ahler structures Ω k = d Θ k , Ω l = d Θ l , Ω k + l = d Θ k + l defined by the aboveplurisubharmonic functions, • Our favourite choices of Lagrangian iterated vanishing cycles K A , K B ⊂ Y k,P over paths γ A , γ B respectively.Abusing notation, we refer to the restrictions of ρ k , Ω k , Θ k to Y k,P by the same names where itis clear from context which is meant. Similarly, we refer to the restrictions of ρ k + l , Ω k + l , Θ k + l to χ − ( P Conf l ( C )) by the same names.The vanishing paths γ A and P γ A compose to give an iterated vanishing path γ A inConf k + l ) ( C ). We refer to this as the composite . One defines γ B similarly. The symplectic54hovanov homology cochain complex CKH symp ( P, A, B ) may be chosen to be the Floer cochaincomplex for Lagrangian submanifolds L A , L B ⊂ Y k,P defined as iterated vanishing cycles alongthe vanishing paths γ A , γ B respectively.It will be necessary to explicitly specify the deformations performed to control symplecticparallel transport over the paths γ A and γ B . Smoothly varying these choices will then givecontinuation maps between Floer cohomology groups. We may suppose (by isotopy of vanishingpaths) that both γ A and γ B are the composite of smooth paths in Conf l ( C ) with the vanishingpath constructed as a piecewise smooth composite of linear paths X n − Q li = n ( X − ǫ i ) to X n Q li = n +1 ( X − ǫ i ) for some collection of real numbers 0 < ǫ ≪ ǫ ≪ . . . ≪ ǫ l ≪
1. Theiterated vanishing cycle construction (cf. the relative vanishing cycle construction, Lemma 32of [1]) is well-defined using honest (neither deformed, nor rescaled) symplectic parallel transportalong this composite of linear paths when each ǫ i is chosen sufficiently small in terms of ǫ i − and the K¨ahler metric defined near X l ∈ S l .To continue the construction along the rest of γ A and γ B it is in general necessary todeform symplectic parallel transport (see Remark 2.19). We now deform the pullback γ ∗ A Θ l on the fibration γ ∗ A S l . This deformation, performed on fibres away from a small neighbourhoodof 0 ∈ [0 , Y l,P . The same works for γ B .We can view K A , K B as an intermediate step in the iterated vanishing cycle constructionfor γ A , γ B respectively, since these paths split with first sections γ A , γ B . One can continuethe construction with deformed one forms Θ A , Θ B to control symplectic parallel transportover the remainder of the paths (i.e. over P γ A , P γ B ). We denote the resulting Lagrangiansin the fibre Y k + l,P by L A (Θ k + l , Θ A ) , L B (Θ k + l , Θ B ) . In general, for any choice of Θ and necessary deformations of it, the Lagrangians L A (Θ , Θ A ), L B (Θ , Θ B ) depend only on a neighbourhood containing the Lagrangians at each stage of theconstruction. Suppose for some reason that all stages of the construction, starting with K A or K B , are guaranteed to remain entirely within the neighbourhood of X l Y k,P on which Ψ is de-fined, then Θ and the deformations need only be defined in that neighbourhood of X l Y k,P . Ofparticular interest are those Θ which respect the splitting Y k,P × S l together with deformationswhich change only the S l part of Θ.To prove Theorem 5.12 one adjusts the choices of K¨ahler forms (this gives canonical isomor-phisms on cohomology and corresponding homotopy equivalences on the cochain complexes).Then the proof proceeds in two stages. Firstly one deforms Θ k + l and Θ A , Θ B until theLagrangian iterated vanishing cycles are contained inside a holomorphically convex neighbour-hood on which Ψ is defined. This induces a ‘continuation map’ on the Floer cochain complex.Then one performs a deformation of exact symplectic forms (and the various 1–forms), definedonly in the convex region, ending with the pushforward under Ψ of Θ k + Θ l (with deformationsonly of the Θ l part). This gives a second continuation map. The composite with the firstcontinuation map is the cochain homotopy equivalence given in the theorem. Some care isneeded to show the manner in which it is ‘canonical’. Proof of Theorem 5.12.
The slice S k + l can be identified with C k + l ) − (see either definition),with the origin corresponding to X k + l and such that X l S k is a linear subspace. We decompose S k + l ∼ = X l S k ⊕ V , where V is a linear complement to X l S k . Now we choose the exhaustingplurisubharmonic function ρ k on S k and any exhausting plurisubharmonic function ρ V on V .Consider the family of plurisubharmonic functions ρ k + sρ V for s >
0. This restricts as ρ k to X l S k so, taking ρ k + l = ρ k + sρ V for some such s , the Lagrangian submanifolds K A , K B ⊂ Y k,P s . Furthermore, we can choose some R (independent of s )such that X l K A , X l K B lie strictly inside the ρ k + l = R level-set.Choosing s large enough, we can ensure that ρ − k + l [0 , R ] is confined to any particular openneighbourhood of S k in S k + l . Now consider the local submanifolds Ψ( { y } × S l ) for y in Y k,P .These cannot be tangent at X l y to X l S k (they are not tangent to X l Y k,P and other tangenciesto X l S k would imply that their projection (Proposition 5.6) to Conf l ( C ) × ξ varies in the ξ direction). Hence, by compactness of ρ − k + l [0 , R ], we can also ensure that ρ − k + l [0 , R ] ∩ χ − ( P Conf l ( C )) is contained within an arbitrarily small neighbourhood of X l Y k,P . Let s beany number large enough such that this small neighbourhood lies within the range of Ψ.We now choose a candidate ˜ ρ l for the function ρ l on S l which is || z || w.r.t. the naturallinear coordinates one has from entries of the matrices A ∈ sl n ( C ) and A , . . . , A n ∈ gl n ( C )defining S l . This gives an exhausting plurisubharmonic function ˜ ρ split := ρ k + ˜ ρ l on Y k,P × S l .Scaling up ˜ ρ l by a large enough constant factor, we can ensure that the R level-set of ˜ ρ split in Y k,P × S l maps by Ψ to within the 2 R level-set of ρ k + l . This level-set necessarily contains K A × { X l } and K B × { X l } .The enclosures used in the localisation argument will be ( ρ k + l , R ) and (˜ ρ split , R ). In theappropriate setting they will be shown to be equivalent.Now we consider in detail the process by which the Lagrangian iterated vanishing cyclesare defined, starting from K A , K B . By comparison to a region of Y k,P × S l and choice of ρ split = ρ l + ρ k , we show how to deform the exact symplectic structure on the range of Ψ andhow to choose the deformations controlling symplectic parallel transport, such that the iteratedvanishing cycles are contained in the image under Ψ of the R level-set of ˜ ρ split .We begin by considering the paths γ A , γ B in Conf l ( C ) and the iterated vanishing cycleconstruction in S l over them. We start with the exact K¨ahler form˜Ω l := d ˜Θ l := d ( − d ˜ ρ l ◦ i )on S l , and with deformations ˜Θ l,A , ˜Θ l,B of 1–forms over the vanishing paths necessary for theiterated vanishing cycle construction. In general, the Lagrangian iterated vanishing cycles in S l one defines in this way are not confined to the region where Ψ is defined. We fix this byrescaling, using a holomorphic C ∗ –action compatible with the fibration S l . For λ ∈ C ∗ , thisacts on S l as: λ : X n I − n X i =1 X n − i A i / / X n I − n X i =1 X n − i λ i A i and on Conf l ( C ) by multiplying all roots by λ . We will only be interested in sufficiently smallpositive real values of λ .We define a new exhausting plurisubharmonic function ˜ ρ λl as the pushforward by the actionof λ . Similarly, we get ˜Θ λl and deformations ˜Θ l,λ ( A ) , ˜Θ l,λ ( B ) . These deformations are definedover new iterated vanishing paths γ λ ( A ) := λ ( γ A ) , γ λ ( B ) := λ ( γ B ) , representing the scaled crossingless matchings λ ( A ) , λ ( B ) respectively. The Lagrangian iter-ated vanishing cycles also respect this pullback, namely: L λ ( A ) ( ˜Θ λl , ˜Θ l,λ ( A ) ) = λ ( L A ( ˜Θ l , ˜Θ l,A )) L λ ( B ) ( ˜Θ λl , ˜Θ l,λ ( B ) ) = λ ( L B ( ˜Θ l , ˜Θ l,B ))56hoose λ small enough that the new iterated vanishing cycles are contained within so smalla level set that the products K A × L λ ( A ) ( ˜Θ λl , ˜Θ l,λ ( A ) ) and K B × L λ ( B ) ( ˜Θ λl , ˜Θ l,λ ( B ) ) lie withinthe enclosure (˜ ρ split , R ) in Y k,P × S l . Now we define ρ l := ρ λl and ρ split := ρ l + ρ k . Importantly,these are plurisubharmonic for the same complex structure as ˜ ρ l , ˜ ρ split respectively. This alsospecifies Θ l and Θ split = Θ l + Θ k . Similarly, Θ split,A , Θ split,B are given by adding Θ k tothe deformations Θ l,A , Θ l,B . These define relative vanishing cycles to K A , K B respectively,which split the products written above. By careful isotopy of Θ k + l to make it equal to Ψ ∗ Θ split within the image under Ψ of the enclosure (˜ ρ split , R ), we shall control the position of the iteratedLagrangian vanishing cycles in S k + l in the required manner.It is helpful to define one more exact 2–form Ω flat = d Θ flat before defining an isotopy ofexact symplectic forms relating Ω k + l and Ψ ∗ Ω split . For a suitable choice of smooth function h : R / / R , the function ρ flat := h ( ρ k + l ) vanishes where ρ k + l ≤ R , and is plurisubharmonicelsewhere. We define Θ flat = − dρ flat ◦ i .Let g : R / / [0 ,
1] be any smooth function such that g ( t ) = 1 for t ≤ R and g ( t ) = 0for t ≥ R . Then g Ψ ∗ Θ split extends to a 1–form on all of χ − ( P Conf l ( C )). Consider thecompact family d ( r Θ k + l + s Θ flat + t Ψ ∗ Θ split + uǫg Ψ ∗ Θ split )of 2–forms where r, s, t, u ∈ [0 ,
1] and max { r, s, t } ≥ . The space of all K¨ahler forms is convex,so these are K¨ahler for ρ k + l ≤ R (and for trivial reasons for ρ k + l ≥ R ). Non-degeneracy of2–forms is an open condition, so there exists some ǫ > k + l to Ψ ∗ Ω split .Namely, we take the linear isotopies between the following 1–forms (in the given order):(1) Θ k + l , (2) Θ k + l + ǫg Ψ ∗ Θ split , (3) Θ flat + ǫg Ψ ∗ Θ split , (4) Ψ ∗ Θ split + ǫg Ψ ∗ Θ split , (5) Ψ ∗ Θ split . From stage 1 to stage 3, this is an isotopy of exact symplectic forms defined on all of χ − ( P Conf l ( C )) which are equal to Ω k + l where ρ k + l ≥ R . Then it continues to stage 5 asan isotopy of forms defined only where ρ k + l < R .We choose, along with the 1-forms from stages 1 to 3, smoothly varying families of de-formations over the paths P λ ( γ A ) , P λ ( γ B ) to control symplectic parallel transport. Thesedeformations are chosen to be supported where ρ k + l > R . From stage 3 onwards we can usethe deformations Θ split,A , Θ split,B of whichever constant multiple of ǫg Ψ ∗ Θ split we have where ρ k + l < R .The transition to these deformations at stage 3 is no problem. Namely, the deformationsgiven by Θ split,A , Θ split,B force all stages of the construction of the relative vanishing cyclesto K A , K B to be contained in ρ − k + l [0 , R ]. Hence, we can remove the previous deformations(by isotopy) without changing the Lagrangians. In fact, the relative vanishing cycles are It suffices to choose h such that h ( x ) = 0 for x ≤ R and h ′ ( x ) > , h ′′ ( x ) ≥ x > R . Then for anyvector V = 0 we have: − d ( d ( h ◦ ρ ) ◦ i ) ( V, iV ) = h ′′ ( ρ )2 (cid:0) dρ ( V ) + dρ ( iV ) (cid:1) + h ′ ( ρ ) ( − d ( dρ ◦ i )( V, iV )) > K A × L λ ( A ) ( ˜Θ λl , ˜Θ l,λ ( A ) ) and K B × L λ ( B ) ( ˜Θ λl , ˜Θ l,λ ( B ) ) described earlier.Θ flat + ǫg Ψ ∗ Θ split is K¨ahler for the complex structure we have on χ − ( P Conf l ( C )) outsideof the ρ k + l = 3 R level-set, so none of the holomorphic strips defining the differential in theFloer cochain complex leaves the ρ k + l ≤ R locus. For the remaining stages, the forms remainK¨ahler where 3 R ≤ ρk + l ≤ R , so we can get away with calculating Floer cohomology onlywithin this neighbourhood. Also from stage 3 onwards we always have a constant multipleof Ψ ∗ Θ split where ρ k + l < R , so we may continue to use the same deformations to controlsymplectic parallel transport. This means that the Lagrangian relative vanishing cycles do notchange.Finally, once we reach the end of this isotopy of exact symplectic structures, we can comparethe Floer cochain complex to CF ∗ (Ψ − ( K A ) × L λ ( A ) ( ˜Θ λl , ˜Θ l,λ ( A ) ) , Ψ − ( K B ) × L λ ( B ) ( ˜Θ λl , ˜Θ l,λ ( B ) ))within Y k,P × S l . We find that nothing has changed, since all holomorphic strips used to definethe differential here lie within the R level-set of ρ split and this is contained in the pre-imageunder Ψ of the ρ k + l ≤ R region of χ − ( P Conf l ( C )). All the data defining this splits, so thecochain complex splits as a tensor product: CF ∗ (Ψ − ( K A ) , Ψ − ( K B )) ⊗ CF ∗ ( L λ ( A ) ( ˜Θ λl , ˜Θ l,λ ( A ) ) , L λ ( B ) ( ˜Θ λl , ˜Θ l,λ ( B ) ))Using the continuation maps associated to the isotopies of exact symplectic structures andsimultaneous isotopy of the Lagrangians, we find this is isomorphic to the following:CKH symp ( P , A , B ) ⊗ CKH symp ( λ ( P ) , λ ( A ) , λ ( B ))which, by the continuation map induced by varying the rescaling parameter from 1 to λ as inLemma 5.11, is isomorphic toCKH symp ( P , A , B ) ⊗ CKH symp ( P , A , B )It remains only to describe how these isomorphisms relate to the canonical isomorphismson symplectic Khovanov homology (and to give the cochain level version of this). The cochaincomplex CF ∗ (Ψ − ( K A ) , Ψ − ( K B )) is canonically isomorphic to CKH symp ( P , A , B ). How-ever, for ( P , A , B ) the corresponding statement is not quite true. The S l factor in Y k,P × S l is only identified up to some automorphism of S l . Hence, CF ∗ ( L λ ( A ) ( ˜Θ λl , ˜Θ l,λ ( A ) ) , L λ ( B ) ( ˜Θ λl , ˜Θ l,λ ( B ) ))is identified with CKH symp ( P , A , B ) up to an automorphism of cochain complexes.At the other end of the construction we have some choice of CKH symp ( P, A, B ). Any twochoices are related by a chain homotopy equivalence inducing the canonical isomorphisms oncohomology. The isotopy of exact symplectic structures given above induces a well-definedcontinuation mapCKH symp ( P, A, B ) / / CKH symp ( P , A , B ) ⊗ CKH symp ( P , A , B )since the space of choices involved was connected.58 emma 5.14. Let D be a disc centred at ∈ C . Suppose we have a bridge diagram ( P, A, B ) which splits as in Theorem 5.12 as the union of bridge diagrams ( P , A , B ) supported awayfrom and ( P , A , B ) supported inside D .Let u be an admissible map from a disc B into Conf k + l ) ( C ) with image contained in P Conf ( D ) ⊂ Conf k + l ) ( C ) Suppose also that u has a single input marked point mapping to P P and output marked pointmapping to some P P ′ . We write ( P ′ , A ′ , B ′ ) for the bridge diagram given by the output end ofthe saddle cobordisms induced by u . It is the union of ( P , A , B ) with some bridge diagram ( P ′ , A ′ , B ′ ) supported inside D .Then the relative invariant factors up to homotopy through the splittings (as K¨unneth prod-ucts) of KH symp at each end given by Theorem 5.12 KH symp ( P, A, B ) KH symp ( P ′ , A ′ , B ′ ) f u / / KH symp ( P, A, B )KH symp ( P , A , B ) ⊗ KH symp ( P , A , B ) (cid:15) (cid:15) KH symp ( P ′ , A ′ , B ′ )KH symp ( P , A , B ) ⊗ KH symp ( P ′ , A ′ , B ′ ) (cid:15) (cid:15) KH symp ( P , A , B ) ⊗ KH symp ( P , A , B ) KH symp ( P , A , B ) ⊗ KH symp ( P ′ , A ′ , B ′ ) f ⊗ f / / Furthermore, the bottom map splits w.r.t. this product as f ⊗ f where f is the identityon KH symp ( P , A , B ) and f the map on KH symp ( P , A , B ) induced by considering u as amap to Conf ( D ) . Remark 5.15.
We use the same splitting to define, up to an overall sign ambiguity, chainmaps induced by creation and annihilation cobordisms (see Section 5.3). Therefore, up tothis sign ambiguity, the splitting of maps described above in fact holds for general cobordismssupported near 0 ∈ C . Proof.
This is an adaptation of the proof of Theorem 5.12. One begins with a tree construction(as in Section 2.2) representing u .Throughout the argument one must then consider the deformations of one forms over it-erated vanishing paths leading to P = P P and also all discs in the tree construction. Theimmediate problem is that the pullback of d (Θ flat + ǫg Ψ ∗ Θ split ) is not an allowed choice of2–form over any of these discs containing a singular value, since it is not in general K¨ahler nearthe singular locus. The same problem occurs for other of the forms used in the argument.Singular values are isolated in the interior of the discs, and we only need control of sym-plectic parallel transport around the boundaries, so we can perform the following correction.First, choose the deformation controlling symplectic parallel transport to be supportedwithin distance ǫ/ ǫ/ ǫ of theboundary one linearly interpolates (w.r.t. the radial coordinate on the disc) between theoffending 1–form Θ flat + ǫg Ψ ∗ Θ split and the 1–form Θ k + l .The same trick can be performed with Θ k + l replaced by Θ split or any convex combination ofthe two. Hence the entire localisation argument of Theorem 5.12 works for tree constructionsas well. 59uppose P = P ( X ) ∈ Conf n ( C ). We can translate all the roots of P by µ ∈ C by a changeof variables replacing P ( X ) by P ( X − µ ). Using this trick, we can map Conf k ( C \ { (1 + kl ) µ } )into Conf k + l ) ( C ) by P ( X ) ( X − µ ) l P ( X + lk µ )such that the image is precisely the subset of Conf k + l ) ( C ) of polynomials with root µ ofmultiplicity 2 l .There is a map of fibrations S k / / S k + l over this map which has a similar description.Suppose y ( X ) ∈ S k , then y ( X ) is a polynomial of degree k with matrix coefficients. We definethe map of fibrations as y ( X ) ( X − µ ) l y ( X + kl µ )This is an affine linear map (in terms of the entries of all the matrix coefficients of y ( X )).Restricting to fibres of S k over Conf k ( C \ { (1 + kl ) µ } ), this map has image in the singularlocus of the fibres of S k + l which it hits. In fact all the calculations of this section generalisestraightforwardly to this setting. Remark 5.16.
This has the consequence that Theorem 5.12 (and Lemma 5.14) hold whenapplied to unlinked components of a bridge diagram (or cobordism of bridge diagrams) sup-ported on a small disc not necessarily centred at 0 ∈ C . The only extra complication is that,in decomposing a bridge diagram into two, one has to translate the resulting bridge diagramsin C to ensure they are of the form ( P, A, B ) with P ∈ Conf n ( C ). Suppose we have two bridge diagrams D and D ′ which are everywhere identical except that D ′ contains, in some small neighbourhood, an extra unlinked component formed from a singlealpha and beta curve. By Theorem 5.12 there is an isomorphismKH symp ( D ′ ) / / KH symp ( D ) ⊗ H ∗ ( S )which is canonical on the first factor.We now define the creation and annihilation maps explicitly in terms of this splitting.Namely the creation map, corresponding to the elementary cobordism D to D ′ , shall be:KH symp ( D ) / / KH symp ( D ) ⊗ H ∗ ( S ) r r ⊗ H ∗ ( S ) as the ring Z [ X ] / ( X ). The annihilation map corresponding to theelementary cobordism D ′ to D , shall be:KH symp ( D ) ⊗ H ∗ ( S ) / / KH symp ( D ) r ⊗ r ⊗ X r
60n both cases, composing with a (grading preserving) automorphism of the H ∗ ( S ) factorcan only change the map by a sign. Hence they are well defined up to sign.The motivation behind these two definitions is twofold. Firstly, they are simply defined tocopy the corresponding maps between Khovanov homology groups. Secondly it is necessaryin order to make the stabilisation and destabilisation maps of Section 5.4 independent of thevertex of a bridge diagram at which one performs stabilisation. As shown earlier, stabilisation of a bridge diagram yields a new bridge diagram and an iso-morphism between the old and new symplectic Khovanov homologies. In this section, we shallshow that this isomorphism can be realised as the composite of a creation map and a singlesaddle cobordism. A similar construction will also be made for destabilisation. creation map f δ δ ( P + , A + , B + )(P,A,B) ( P stab , A stab , B stab ) Figure 22: An illustration of the stabilisation map locally near a vertex of the bridgediagram.
Definition 5.17.
Let (
P, A, B ) be any bridge diagram. Choose a vertex v and define a newbridge diagram ( P + , A + , B + ) by adding an unlinked unknot component consisting of a single α – and β –curve both supported near v . Let δ be any curve supported near v and joining v toa vertex of the unknot component. Let ( P stab , A stab , B stab ) be the bridge diagram obtained byperforming the saddle cobordism which δ specifies. Figure 22 illustrates these constructionslocally near v .We define the stabilisation map to be the composite of two maps: the creation mapKH symp ( P, A, B ) / / KH symp ( P + , A + , B + )and the map induced by δf δ : KH symp ( P + , A + , B + ) / / KH symp ( P stab , A stab , B stab )We define the destabilisation map as a similar composite. First one performs an elementarycobordism KH symp ( P stab , A stab , B stab ) / / KH symp ( P + , A + , B + ), then the annihilation map.61e begin by describing the iterated vanishing cycle construction in coordinates which fitwith the the model neighbourhood given by Lemma 4.6.Let ( XP, A, B ) be a bridge diagram with 2 m − ∈ C . Let γ A , γ B be iterated vanishing paths in Conf m − ( C ) corresponding to the crossinglessmatchings A, B respectively. With the appropriate choices made, these give iterated vanishingcycles K A , K B ⊂ Y m − ,XP ( X ) . These data are the start point of the constructions of thissection.The base C of the model neighbourhood corresponds locally near (0 ,
0) to a subset ofConf m ( C ) by the map ( d, z ) ( X − Xd + z ) P ( X )In particular, the critical values are the points of the form (3 x , x ) which correspond toconfigurations ( X − x ) ( X + 2 x ) P ( X ) with a root of multiplicity 2 at x .Pick some small x and let d = 3 x . We will discuss how small this x has to be later. We nowextend γ A , γ B by composing with the path t ( X + 2 tx − txm − ) P ( X − txm − ) for t ranging from0 to 1. This is the same as composing X γ A , X γ B with the path ( X − tx ) ( X + 2 tx ) P ( X ),staying in the singular locus. Denote the composite paths by γ A,x , γ
B,x . To finish the iteratedvanishing cycle construction we choose vanishing paths ψ A , ψ B in C for the Lefschetz fibration C π (cid:15) (cid:15) ( a, b, c ) ❴ (cid:15) (cid:15) C a − ad + bc We then compose the paths ( X − Xd + ψ A ) P ( X ) , ( X − Xd + ψ B ) P ( X ) with γ A,x , γ
B,x (considered as paths in Conf m ( C )) respectively to give iterated vanishing paths γ ′ A , γ ′ B inConf m ( C ) ending at some regular value P ′ := ( X − Xd + z ) P ( X ) of χ .Suppose we conclude that x needs to be smaller, then ψ A , ψ B can be scaled down continu-ously by ( d, z ) ( λ d, λ z ) for small real λ . This replaces x by λx .Let S be a disc centred at 0 ∈ C which no roots of XP , except 0, and only one section ofa curve in each of A, B , each ending at 0. The iterated vanishing paths γ ′ A , γ ′ B come from abridge diagram ( P ′ , A ′ , B ′ ) with the following properties: • P ′ = ( X − Xd + z ) P ( X ) has three roots in S . • ( P ′ , A ′ , B ′ ) is identical to ( XP, A, B ) outside of S . • A ′ , B ′ are each intersect S in the end section of one curve and the whole of another.In fact, by considering the effect of moving ψ A , ψ B around the two singularities ± x of theLefschetz fibration, one sees that all ( P ′ , A ′ , B ′ ) with the above properties (for a fixed ( d, z ))are realised in this manner. Also, any two versions of the construction for the same ( P ′ , A ′ , B ′ ),up to isotopy supported on S , are essentially the same. i.e. the pairs ψ A , ψ B are isotopic.Corresponding to a construction as above, we define L A , L B to be the vanishing cycles (ina regular fibre of C → C ), for ψ A , ψ B . Lemma 5.18.
Let ψ be a vanishing path for the Lefschetz fibration C C , as above, withsome fixed value of x ∈ C ∗ .Then, for sufficiently small λ > , the vanishing cycle construction for the vanishing path λ ψ (with x replaced by λx ) is well-defined. Furthermore, it may be confined to an arbitrarilysmall neighbourhood of ∈ C . roof. For fixed d = 0, at a point ( a, b, c ) = ( ± x, , z ) is: ˙ z | a − x | + | b | + | c | a − x )¯ c ¯ b so the locus {| b | = | c | } is preserved by symplectic parallel transport. The closure of thislocus contains the singular points of the fibration, so vanishing cycles must also lie in thislocus. Here, the flow on the a –coordinate is given by: ∂∂t a = 3( a − x )9 | a − x | + 2 | z + 3 ax − a | ˙ z ( t )In particular, one has (cid:12)(cid:12)(cid:12)(cid:12) ∂∂t a (cid:12)(cid:12)(cid:12)(cid:12) ≤ | ˙ z | | a − x | and, as one scales down x λx , the path ψ is replaced by λ ψ , so this bound is sufficient tocontain the vanishing cycles near a = x . This also contains b, c near 0, since | b | = | c | = | bc | = (cid:12)(cid:12) z − a + ad (cid:12)(cid:12) , which vanishes at a = x .The same argument can be applied to constructing an exact Lagrangian boundary conditionon the restriction of C to a disk D with two boundary marked points. Given ψ A , ψ B endingat the input boundary marked point, we get L A , L B in the fibre over that point and, extendingby symplectic parallel transport around, D , we get an exact Lagrangian boundary condition Q . Scaling down the whole construction, as above, suffices to contain Q near 0 ∈ C .In this, section we are interested in two particular constructions of ψ A , ψ B and D . For thestabilisation map ( f δ in Figure 22), we take ψ A = ψ B to be vanishing paths from a regularpoint near z = − x to the singular point z = 2 x , and D to embed (holomorphically) ontoa small neighbourhood of the singular point z = − x . For the destabilisation map, we moveone of ψ A , ψ B around z = − x first. Lemma 5.19.
Let K A , K B be iterated vanishing cycles in a regular fibre Y m − ,XP as above,representing a bridge diagram ( XP, A, B ) . Choose ψ A , ψ B , D describing the stabilisation ordestabilisation move locally near 0.Then, the symplectic Khovanov homology chain complexes split canonically as the tensorproduct of CKH symp ( XP, A, B ) = HF ∗ ( K A , K B ) with CF ∗ ( L A , L B ) of the Lagrangian vanish-ing cycles in C constructed over the same paths. CF ∗ ( L A , L B ) is either Z or Z [ X ]( X ) ≃ Z , withzero differential.The relative invariant splits in the same manner, as the identity map on CKH symp ( XP, A, B ) and the following maps on the remaining factor:destabilisation stabilisation Z / / Z [ X ] / ( X ) Z [ X ] / ( X ) / / Z X X Furthermore, when CF ∗ ( L A , L B ) = Z [ X ]( X ) (a valid choice of the cochain complex for thesimplest unknot diagram), the splitting agrees with that given in Theorem 5.12. roof. The proof uses a localisation argument similar to that in the proof of Theorem 5.12,but somewhat more involved.We shall be using the model neighbourhood of Lemma 4.6, so here it is again as a reminder. D C / / χ − ( D ) D χ (cid:15) (cid:15) χ − ( D ) C × C ∗ ( F \ local isomorphism Ψ near X Y m − ,XP in χ − (0 , / / C × C ∗ ( F \ C f (cid:15) (cid:15) The term F , used on the right-hand side, is a holomorphic line bundle over Y m − ,XP , whichis a subbundle of the trivial C –bundle. C ∗ acts on C by λ : ( a, b, c, d ) ( a, ζb, ζ − c, d ), so C × C ∗ ( F \
0) is isomorphic as a holomorphic line bundle to C ⊕ F ⊕ F − ⊕ C . Because Y m − ,XP is Stein, F ⊕ F − is isomorphic to the trivial C bundle.On Y m − ,XP , we have the exhausting, plurisubharmonic function on S m − , such that K A , K B ⊂ ρ − [0 , R ) for some R ∈ R .We construct an exact K¨ahler form ˜Ω = d ˜Θ on C × C ∗ ( F \
0) with the following properties: • ˜Θ is S –equivariant, • on an open neighbourhood U of 0 × C ∗ ( F \ | ρ − m − [0 ,R ) , symplectic parallel transport fixesthe projection to Y m − ,XP and is given, up to an S –ambiguity by the symplectic paralleltransport in C C with the standard symplectic form,There is a natural choice of exact (1 , C × C ∗ ( F \
0) which we shall callΩ ass = d Θ ass (see Section 3.2 and also the description of symplectic associated bundles in[1, Section 4.3]). This form is non-degenerate, so K¨ahler in a neighbourhood 0 × C ∗ ( F \
0) andon that neighbourhood satisfies the condition on symplectic parallel transport. We shall makeit K¨ahler everywhere by adding a sufficiently positive form.Identifying C × C ∗ ( F \
0) biholomorphically with the trivial C –bundle over Y m − ,XP , wedefine the exhausting plurisubharmonic function ρ split := ρ m + λ || z || . Averaging ρ split overthe S –action, gives another exhausting plurisubharmonic function ρ S split . For large enough λ ∈ R , the neighbourhood ( ρ S split ) − [0 , R ) is small enough in the C –directions that Ω ass isnon-degenerate there.Let h : [0 , ∞ ) / / R be a smooth function, identically zero on [0 , R ) and with h ′ , h ′′ strictlypositive on ( R, ∞ ). By choosing h ′ ( t ) large enough for each t ∈ [2 R, ∞ ], we ensure that: − d ( d ( h ◦ ρ S split ) ◦ i )( V, J V ) > | Ω ass ( V, iV ) | Therefore ˜Θ := − d ( h ◦ ρ S split ) ◦ i + Ω ass is as required.By scaling down any choice of x to λx , symplectic parallel transport applied to K A , K B ,over the path to (3 x , x ) (described earlier) and then along ψ A , ψ B and a section of ∂D ,respectively, can be contained within the neighbourhood U (see Lemma 5.18). Furthermore,no matter how small we choose U , we may still choose x small enough to contain the symplecticparallel transport in this way.Having chosen an exact K¨ahler form on C × C ∗ ( F \
0) which we would like to use, we nowhave to perform a localisation argument to show that we may use it. We start by carefullychoosing the exact K¨ahler form on χ − ( D ). 64 m is biholomorphic to S m − × C where S m − is identified with S m − × { } . This worksbecause S m − is a (complex) affine subspace of S m . Define ρ m on S m to be ρ m − + λ || z || with respect to this splitting. Then, ρ m | S m − = ρ m − and, for large enough λ ∈ R , theneighbourhood ρ − m | χ − ( D [0 , R ), is contained within the neighbourhood on which the localisomorphism with C × C ∗ ( F \
0) is defined.By using h , as above, we can also define a 2–form − d (( dh ◦ ρ m ) ◦ i ), which is identically0 on ρ − m [0 , R ) and K¨ahler elsewhere. We will also need a smooth function g : [0 , ∞ ) / / R which takes value 1 on [0 , R ] and 0 on [3 R, ∞ ).We now choose U , small enough that it corresponds to a subset of ρ − m [0 , R ) under thelocal isomorphism. Then, as described above we choose x small, dependent on U . We are nowready to define an isotopy of exact symplectic forms from a composite of linear between thefollowing 1–forms:(1) Θ m (2) Θ m + ǫg ( ρ m )Ψ ∗ ˜Θ(3) − d ( h ◦ ρ m ) ◦ i + ǫg ( ρ m )Ψ ∗ ˜Θ(4) Ψ ∗ ˜Θ + ǫg ( ρ m )Ψ ∗ ˜Θ(5) Ψ ∗ ˜Θ Remark 5.20.
The following is a summary of the significance of the stages of the isotopy. Theaim is to get from the exact K¨ahler form d Θ m on S m , to the S –equivariant form d Ψ ∗ ˜Θ on themodel neighbourhood. The first half of the isotopy, ending at stage 3, allows the calculation ofFloer homology and the relative invariant to be performed locally on the model neighbourhood.The second half of the isotopy occurs only on that model neighbourhood and ends with theparticular symplectic form, required in order to apply the arguments of Section 3.Until stage 3, the 1–forms are defined on all of χ − ( D ). From stage 3 onwards, theyare defined only on the holomorphically convex region ρ − [0 , R ]. So long as ǫ > K A , K B over the path (0 ,
0) to(3 x , x ) and then along the paths ψ A , ψ B , respectively. At stage 3 and later, this iteratedvanishing cycle construction and the extensions around ∂D do not leave the neighbourhood U .In fact it gives, L A × S F | K A and L B × S F | K B , where F is the unit circle subbundle of F with respect the hermitian metric used to define Ω ass .By convexity of ρ − [0 , R ], holomorphic strips with boundary on these Lagrangians, mustremain within this region. Hence, to study Floer cohomology and the relative invariant, theisotopy of exact symplectic forms need only be defined on this region. At stage 5, a similarconvexity argument would show that we can work with the total space of C × C ∗ ( F \ ρ − [0 , R ]). However, it is not necessary to do this.Proposition 3.14 concludes the proof that the Floer cochain complexes and the relativeinvariants split as described.It remains only to show that the splitting of Floer cochain complexes agrees with thatof Theorem 5.12 where it describes the addition of a single unlinked unknot component to( XP, A, B ). The splittings actually agree at the chain level, which can be shown as follows.65ne constructs ψ A = ψ B to be short enough that the Floer cohomology calculation can besimultaneously localised in both ways with the same Y m − ,XP factor. Namely, the neighbour-hood to which one localises for Theorem 5.12 can be found within C × C ∗ ( F \ Corollary 5.21.
The stabilisation and destabilisation maps on symplectic Khovanov homologyat a given vertex are inverses of each other.
Figure 23: Two ways of performing the same cobordism of admissible links locallywith non-isotopic curve δ (indicated by grey lines).The two cobordisms specified locally by Figure 23 are isotopic as link cobordisms. However,the curves specifying them are not isotopic. They are related by a simple “switching move”simultaneously sliding the curve δ along the two α –curves with which it shares vertices. Proposition 5.22.
The two cobordisms specified locally by Figure 23 induce the same maps(up to sign) on KH symp . We start with a weaker result in a very special case.
Lemma 5.23.
In the case specified locally by Figure 24, the two cobordisms induce the samemaps (up to sign) on the Z summand in the bottom degree of cohomology. Figure 24: A trivial case of the switching move.66 roof.
Suppose first that we are dealing with the case specified globally by Figure 24.The domain of the maps is H ∗ ( S ) ⊗ which has a single Z summand in the top degree of thecohomology. Both maps are induced by the saddle cobordism part of different stabilisations,so by the previous section we know that there is, for each, a splitting in which they can bewritten (up to sign) as: H ∗ ( S ) ⊗ Z [ X ] / ( X ) / / H ∗ ( S ) ⊗ Z a ⊗ X a ⊗ a ⊗ Z in thebottom degree of cohomology on both sides. Up to a sign ambiguity there is only one suchisomorphism, so we are done.In the slightly less trivial case where Figure 24 is only the local model we denote by D , thebridge diagram for the rest of the link. The maps on symplectic Khovanov homology split (usingLemma 5.14) as the identity on KH symp ( D ) tensored with the maps H ∗ ( S ) ⊗ / / H ∗ ( S )compared in the lemma.Now, to prove Proposition 5.22, we refer to Figure 25. The figure has four numbered rows,each of which describes a cobordism composed of three elementary saddle cobordisms. Thethicker grey lines indicate the cobordism to be performed as you pass to the next diagramon that row. Several of the diagrams have two cobordisms marked, but only when the linesspecifying them do not intersect, so by Lemma 5.4 it does not matter in which order they areperformed.We start with row 1. The first map is the composite of two of the saddle cobordisms usedin stabilisation. The symplectic Khovanov homology splits as KH symp ( D ) ⊗ H ∗ ( S ) ⊗ and allterms map to zero, with the exception of KH symp ( D ) ⊗ ⊗ symp ( D ) of the next diagram. The second map in thisrow is the cobordism for which we would like to prove the proposition. The final diagram inrow 1 represents the output end of that cobordism. We have taken the liberty of choosingan admissible position which is not quite a bridge diagram, purely because it looks simpler(observe there is a crossing between two α –curves).Row 2 is an application of Lemma 5.4 to swap the order of the cobordisms in row 1. Hence,row 2 gives the same map on symplectic Khovanov homology.Row 3 is the same as row 2 except that the switching move has been applied to the firstcobordism. By Lemma 5.23 rows 2 and 3 give the same map (up to sign) on KH symp ( D ) ⊗ ⊗ symp ( D ) ⊗ ⊗ symp ( D ) ⊗ ⊗ In this section, we explain how KH symp of a crossing diagram is well-defined, with canonicalisomorphisms, up to a possible overall sign ambiguity. To do this one defines KH symp of acrossing diagram to be KH symp ( D ) for any bridge position which projects to that crossing67 Figure 25: Four related cobordisms used to prove the invariance under the switchingmove.diagram. The difficulty is in specifying a consistent choice of canonical isomorphism betweenthe symplectic Khovanov homologies of any two bridge diagrams with the same projection.Two such bridge diagrams are related by • a sequence of stabilisations and destabilisations preserving the projection • isotopy sliding the α – and β –curves and vertices along the projection.We take the stabilisation and destabilisation maps and the maps induced by these isotopies tobe the canonical isomorphisms. Lemma 5.24.
For a crossing diagram in which each component of the link is involved in atleast one crossing the symplectic Khovanov homology is well-defined (up to sign). Namely, theisomorphisms mentioned above are consistent. roof. The condition on components being involved in crossings means that the diagram canbe decomposed into edges (by cutting both strands at each crossing) without any closed loopsremaining. Any bridge diagram projecting to the crossing diagram will have some number ofvertices on each edge (none are possible at the crossings). Any two such bridge diagrams areisotopic through such diagrams if and only if these numbers agree on each edge. Moreover, anytwo such isotopies are isotopic (the crossing condition is vital here) so the maps they induceare the same.Only the stabilisation and destabilisation maps can change the numbers of vertices on anedge. By applying the switching move (Proposition 5.22) to the saddle cobordism of a stabil-isation it is immediate that it makes no difference (up to sign) at which vertex one stabilisesor destabilises. It therefore suffices to show that a stabilisation followed by a destabilisation atthe same vertex gives the identity map. This is covered by Corollary 5.21.Careful consideration of the switching move shows also that the isotopy which moves twovertices from one edge past a crossing to another using the passing move is the same as themap that destabilises one edge and stabilises the next. Hence we can add these isotopies to thechoice of canonical isomorphisms. This allows the condition that each component is involvedin at least one crossing to be lifted.A consequence is that, given a neighbourhood in which an elementary saddle cobordismis performed to a crossing diagram, it does not matter which of the curves representing asaddle cobordism one chooses on a bridge diagram to define the map on symplectic Khovanovhomology. Hence, the maps from creation, annihilation and saddle cobordisms (and similarlyisotopy) are well-defined (up to sign) on the symplectic Khovanov homology of a crossingdiagram.
Given any smooth cobordism between links in R × [ − , t –coordinate on [ − , symp .It is possible that a more intrinsically geometric definition of the maps induced by cap/cupcobordisms would deal with the remaining moves. An alternative definition of the maps inducedby cobordisms was recently given by Rezazadegan [9]. However, it defines the maps inducedby cap/cup cobordisms in essentially the same manner.69e now prove invariance under those movie moves, for which invariance is not immediatelyobvious by the above.Figure 26: A non-trivial position of the trivial cobordism.As a warm-up, there is movie move 9, which relates the cobordism in Figure 26 to the trivialone. This is realised by a stabilisation map. Hence, by definition of the canonical isomorphisms,it gives the identity map.Moves 10 and 12 are more interesting. They are illustrated in Figures 27 as sequences ofbridge diagrams with thicker lines marking the saddle cobordism still to be performed. Theselines are removed once the cobordism is performed. The cobordisms should be read from topto bottom in each column.Figure 27: Movie moves 10 (between the two movies on the left) and 12 (betweenthe two movies on the right) in terms of bridge diagrams. Each movie should be readfrom to to bottom. The thicker lines indicate the cobordisms still to be performedin that movie. 70ovie move 10 is covered by isotopies of admissible links together with simultaneous isotopyof the curve δ defining a saddle cobordism, which, by the discussion in Section 5.1 provesinvariance of the induced map (see Figure 27). Movie move 12 requires, on top of that, the useof the switching move, Proposition 5.22 (see the first part of the movie).We should also prove invariance under reflected and reversed movie moves, but the proofsare identical.As well as being invariant under the movie moves, an invariant of cobordisms up to isotopymust be invariant under the commuting of ‘distant’ cobordisms (i.e. supported on disjointneighbourhoods of the diagram). This is immediate from Lemma 5.4.In conclusion, the maps (up to sign) on the symplectic Khovanov homology of a link,induced by specific positions of link cobordisms, are actually invariants of the isotopy class ofsmooth cobordism. This concludes the proof of Theorem 1.1. References [1] P. Seidel and I. Smith, “A link invariant from the symplectic geometry of nilpotentslices,”
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