Traceless Character Varieties, the Link Surgeries Spectral Sequence, and Khovanov Homology
SSYMPLECTIC INSTANTON HOMOLOGY, THE LINK SURGERIESSPECTRAL SEQUENCE, AND KHOVANOV HOMOLOGY
HENRY T. HORTON
Abstract.
We further develop the symplectic instanton homology defined in our previous article[Hor16] by investigating its behavior under Dehn surgery. In particular, we prove a exact trianglerelating the symplectic instanton homology of ∞ -, 0-, and 1-surgeries along a framed knot in aclosed 3-manifold. More generally, we show that for any framed link in a closed 3-manifold, thereis a spectral sequence with E -page a direct sum of symplectic instanton homologies of all possiblecombinations of 0- and 1-surgeries on the components of the link converging to the symplecticinstanton homology of the ambient 3-manifold. As an application of our results, we compute thesymplectic instanton homology of many infinite families of 3-manifolds and also, given a link L ⊂ S ,establish a spectral sequence with E -page isomorphic to the mod 2 reduced Khovanov homologyof the mirror of L converging to the mod 2 symplectic instanton homology of the branched doublecover of L . Contents
1. Introduction 12. Review: Symplectic Instanton Homology 43. Symplectic Instanton Homology of Nontrivial SO(3)-Bundles 74. Functoriality for Nontrivial Bundles 155. Exact Triangle for Surgery Triads 196. Instanton L -Spaces 327. A Spectral Sequence for Link Surgeries 368. Khovanov Homology and Symplectic Instanton Homology of Branched Double Covers 429. Nontrivial Bundles on Branched Double Covers 47References 501. Introduction
Starting from the seminal work of Floer on instanton homology [Flo95] (see also [BD95]), it hasbeen noted that Floer homology invariants of closed 3-manifolds behave in a controlled way underDehn surgery. More precisely, let Y be a closed, oriented 3-manifold and let ( K, λ ) be a framedknot in Y . Letting Y λ ( K ) denote the result of λ -surgery on K (and similarly defining Y λ + µ ( K ) for µ a meridian of K ), there are standard cobordisms W : Y −→ Y λ ( K ), W λ : Y λ ( K ) −→ Y λ + µ ( K ), a r X i v : . [ m a t h . G T ] F e b HENRY T. HORTON and W λ + µ : Y λ + µ ( K ) −→ Y , each of which consists of a single 2-handle attachment. For a “Floerhomology theory” A : Bord −→ F - Vect , there should be an exact triangle of the form A ( Y ) A ( W ) (cid:47) (cid:47) A ( Y λ ( K )) A ( W λ ) (cid:119) (cid:119) A ( Y λ + µ ( K )) A ( W λ + µ ) (cid:102) (cid:102) Exact triangles of this form have been established for many types of Floer homologies of 3-manifolds:instanton homology [Flo95], monopole Floer homology [KMOS07], and Heegaard Floer homology[OS04] are some prominent examples.In our previous article [Hor16], we introduced a new construction of a Floer homology invariantfor 3-manifolds Y , which we call symplectic instanton homology and denote SI( Y ). Thissymplectic instanton homology is conjecturally related to the framed instanton homology consideredby Kronheimer and Mrowka [KM11b] and is constructed roughly as follows (see Section 2 for acomplete recollection of the definition). Fix a Heegaard splitting Y = H α ∪ Σ g H β of the closed,oriented 3-manifold Y . If we additionally fix a basepoint z ∈ Σ g on the Heegaard surface, we mayconsider a standardly embedded θ -graph in a neighborhood of z in Y , embedded in such a way thatthere is exactly one vertex is each handlebody and each of the three edges of the graph intersectthe Heegaard surface transversely in a single point. Write Y θ = H θα ∪ Σ g, H θβ for the induceddecomposition of the complement of the θ -graph in Y . We may then consider the traceless SU(2) -character variety of each piece of the decomposition, meaning the space of SU(2)-representationsof the fundamental group of the complement of the θ -graph that sends the meridian of each edgeof the θ -graph to a traceless SU(2) matrix. If M g, denotes the traceless character variety ofthe thrice-punctured Heegaard surface and L α , L β denote the images of the traceless charactervarieties of H θα , H θβ in M g, under restriction to the boundary, then one may show that M g, is amonotone symplectic manifold and L α , L β are monotone Lagrangian submanifolds. Therefore wemay consider the Lagrangian Floer homology of ( L α , L β ) in M g, , and we defineSI( Y ) = HF( L α , L β ) . In [Hor16] we showed that SI( Y ) is independent of the choice of Heegaard splitting used to defineit, so that it is in fact an invariant of Y .In this article, we aim to establish a surgery exact triangle for symplectic instanton homology of theform described above. As was the case for Floer’s classical instanton homology, we will be requiredto extend our definition of symplectic instanton homology to incorporate additional structure on Y for the exact triangle to possibly hold. While Floer’s invariant needed to be generalized fromSU(2)-bundles to SO(3)-bundles on the 3-manifold Y , our symplectic instanton homology needs totake into account SO(3)-bundles on Y . In dimension 3, such bundles are classified by their secondStiefel-Whitney class, and by Poincar´e duality we may therefore think of this data as coming froma mod 2 homology class ω ∈ H ( Y ; F ).With the above stated, we may describe the major results of this article. The first is the constructionof an extension of SI( Y ) that takes into account the additional data of a mod 2 homology class ω ∈ H ( Y ; F ). Theorem 1.1.
For any closed, oriented -manifold Y and homology class ω ∈ H ( Y ; F ) , there isa well-defined symplectic instanton homology group SI(
Y, ω ) that is an invariant of the pair ( Y, ω ) . YMPLECTIC INSTANTON HOMOLOGY, SPECTRAL SEQUENCES, AND KHOVANOV HOMOLOGY 3
By a generalization of the techniques in [Hor16], we may also define homomorphisms between these“twisted” symplectic instanton homology groups that are induced by 4-dimensional cobordisms W equipped with a mod 2 homology class Ω ∈ H ( W, ∂W ; F ). Theorem 1.2.
Given a compact, connected, oriented cobordism W : Y −→ Y (cid:48) of closed, con-nected, oriented -manifolds and a mod homology class Ω ∈ H ( W, ∂W ; F ) , there is an inducedhomomorphism SI( W, Ω) : SI(
Y, ∂ Ω | Y ) −→ SI( Y (cid:48) , ∂ Ω | Y (cid:48) ) that is an invariant of the ( W, Ω) . Furthermore, the assignment ( W, Ω) (cid:55)→ SI( W, Ω) is functorialin the sense that (1) SI( Y × [0 , , ω × [0 , SI(
Y,ω ) . (2) SI( W (cid:48) , Ω (cid:48) ) ◦ SI( W, Ω) = SI( W ∪ Y (cid:48) W (cid:48) , Ω + Ω (cid:48) ) . With the invariance and functoriality of SI(
Y, ω ) in place, we can now properly state the surgeryexact triangle for symplectic instanton homology:
Theorem 1.3.
For any framed knot ( K, λ ) in a closed, oriented -manifold Y , there is an exacttriangle SI(
Y, ω K ) SI( W, Ω K ) (cid:47) (cid:47) SI( Y λ ( K )) SI( W λ ) (cid:119) (cid:119) SI( Y λ + µ ( K )) SI( W λ + µ , Ω (cid:48) K ) (cid:103) (cid:103) where ω K is the mod homology class in Y represented by the knot K and Ω K is the relative mod homology class in W represented by the core of the -handle attached to K (similarly for Ω (cid:48) K in W λ + µ ). Note that the surgery exact triangle is for Dehn surgery on a knot . For general links , the surgeryexact triangle must be replaced with a spectral sequence.
Theorem 1.4.
For any framed link ( L, λ ) in a closed, oriented -manifold Y , there is a spectralsequence whose E -page is a direct sum of symplectic instanton homologies of all possible combina-tions of λ k - and ( λ k + µ k ) -surgeries on the components L k of L that converges to SI(
Y, ω L ) . As a special application of the above link surgeries spectral sequence, we establish a relationshipbetween Khovanov homology and symplectic instanton homology.
Theorem 1.5.
For any link L ⊂ S , there is a spectral sequence with E -page isomorphic to Kh( m ( L ); F ) converging to SI(Σ( L )) . The outline of this article is as follows. In Section 2, we review the construction of symplecticinstanton homology as given in [Hor16]. In Section 3, we show how to generalize this constructionto incorporate a homology class ω ∈ H ( Y ; F ) to produce an invariant SI( Y, ω ) of the pair (
Y, ω ).The functoriality of this “twisted” symplectic instanton homology with respect to cobordisms W equipped with a homology class Ω ∈ H ( W, ∂W ; F ) is established in Section 4. In Section 5,we prove the surgery exact triangle for symplectic instanton homology. Section 6 gives severalinfinite families of 3-manifolds whose symplectic instanton homology is of minimal rank. The linksurgeries spectral sequence for symplectic instanton homology is proven in Section 7. In Section 8we explore the connection between Khovanov homology of the symplectic instanton homology ofbranched double covers, and in Section 9 we consider further the case of non-trivial SO(3)-bundleson branched double covers. HENRY T. HORTON
Remark 1.6.
During the course of this work, similar results were obtained in the Ph.D thesis ofGuillem Cazassus ([Caz16a], see also [Caz16b] and [Caz17]) for a different version of symplecticinstanton homology defined by Manolescu and Woodward [MW12]. In particular, Cazassus deriveda surgery exact triangle for the Manolescu-Woodward invariant using similar techniques to thoseused in the present article.
Acknowledgements.
We thank Paul Kirk, Dylan Thurston, and Christopher Woodward foruseful conversations on various aspects of this work. This article consists of material which makesup part of the author’s Ph.D. thesis, written at Indiana University.2.
Review: Symplectic Instanton Homology
Here we recall the construction of symplectic instanton homology from [Hor16].2.1.
Smooth and Symplectic Topology of Moduli Spaces.
Let Σ g, denote the surface ofgenus g with three punctures, which we will think of as all lying in a small disk embedded in thesurface. Let M g, denote the moduli space of flat connections on the trivial SU(2)-bundle over Σ g, ,subject to the condition that the holonomy of the connection around any of the three punctures isa traceless SU(2) matrix.If we write C ( i ) for the conjugacy class of traceless matrices in SU(2), M g, admits the holonomydescription M g, = (cid:40) A , . . . , A g , B , . . . , B g ∈ SU(2) , C , C , C ∈ C ( i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g (cid:89) k =1 [ A k , B k ] = C C C (cid:41) (cid:44) conj.The following properties of M g, are well-known. Proposition 2.1. M g, is a smooth, closed manifold of dimension g . There is a naturally definedsymplectic form ω on M g, which is monotone with monotonicity constant τ = . Cerf Decompositions of -Manifolds. The most flexible definition of the symplectic in-stanton homology of a closed, oriented 3-manifold Y is in terms of a Cerf decomposition of Y . Wewill also want to use Cerf decompositions for the definition of symplectic instanton homology fornontrivial SO(3)-bundles, so we recall the relevant definitions and results here.Suppose X − and X + are two closed, oriented surfaces. A bordism from X − to X + is a pair( Y, φ ) consisting of a compact, oriented 3-manifold Y and a orientation-preserving diffeomorphism φ : ∂Y −→ X − (cid:113) X + . Two bordisms ( Y, φ ) and ( Y (cid:48) , φ (cid:48) ) from X − to X + are equivalent if thereexists an orientation-preserving diffeomorphism ψ : Y −→ Y (cid:48) with φ ◦ ψ | ∂Y = φ (cid:48) . We will denote theequivalence class of the bordism ( Y, φ ) by [
Y, φ ], or simply [ Y ] when the boundary parametrization φ does not need to be explicitly mentioned.The (2+1) -dimensional connected bordism category , Bord , is the category whose objects areclosed, oriented, connected surfaces and whose morphisms are equivalence classes of 3-dimensional,compact, oriented, and connected bordisms. Composition in Bord is defined by gluing togetherbordisms using the boundary parametrizations.Morphisms in Bord may be factored into simple pieces with the help of certain Morse functions.A Morse datum for a bordism (
Y, φ ) is a pair ( f, t ) consisting of a Morse function f : Y −→ R and a strictly increasing tuple of real numbers t = ( t , . . . , t m ) subject to the following: YMPLECTIC INSTANTON HOMOLOGY, SPECTRAL SEQUENCES, AND KHOVANOV HOMOLOGY 5 (i) min f ( y ) = t and f − ( t ) = φ − ( X − ) ( i.e. the minimum of f is t and this minimum isattained at all points of the incoming boundary of Y and nowhere else), and similarly themax f ( y ) = t m and f − ( t m ) = φ − ( X + ).(ii) f − ( t ) is connected for all t ∈ R .(iii) Critical points and critical values of f are in one-to-one correspondence, i.e. f : Crit( f ) −→ f (Crit( f )) is a bijection.(iv) t , . . . , t m are all regular values of f and each interval ( t k − , t k ) contains at most one criticalpoint of f .A bordism ( Y, φ ) is an elementary bordism if it admits a Morse datum with at most one criticalpoint; if it admits a Morse datum with no critical points, then it is called a cylindrical bordism .Note that for any Morse datum ( f, t ), f − ([ t k − , t k ]) is an elementary bordism. Hence a Morsedatum induces a decomposition Y = Y ∪ X Y ∪ X · · · ∪ X m − Y m where Y k = f − ([ t k − , t k ]) , X k = f − ( t k ) . We call any decomposition of Y into elementary bordisms arising from a Morse datum in this waya Cerf decomposition of (
Y, φ ).Note that a Cerf decomposition of (
Y, φ ) induces a factorization of the morphism [
Y, φ ] ∈ Bord ( X − , X + ):[ Y φ ] = [ Y , φ ] ◦ · · · ◦ [ Y m , φ m ] , where for k = 1 , . . . , m , [ Y k , φ k ] ∈ Bord ( X k − , X k ). We will also call this induced factorization a Cerf decomposition of [
Y, φ ]. Given two Cerf decompositions[
Y, φ ] = [ Y , φ ] ◦ · · · ◦ [ Y m , φ m ] , [ Y, φ ] = [ Y (cid:48) , φ (cid:48) ] ◦ · · · ◦ [ Y (cid:48) m , φ (cid:48) m ]of an equivalence class [ Y, φ ], we say they are equivalent if there exist orientation-preservingdiffeomorphisms ψ k : X k −→ X (cid:48) k ( k = 0 , . . . , m , where X = X − and X m = X + ) such that ψ = id X − , ψ m = id X + , and[ Y k , φ k ] = [ Y (cid:48) k , ( ψ k − (cid:113) ψ k ) ◦ φ (cid:48) k ] for all k = 0 , . . . , m. There are several natural modifications one may make to a given Cerf decomposition of a morphism.
Definition 2.2. (Cerf Moves) Let [(
Y, φ )] ∈ Hom
Bord n +1 ( X − , X + ) be an equivalence class ofbordisms and suppose we have a Cerf decomposition[( Y, φ )] = [( Y , φ )] ◦ · · · ◦ [( Y m , φ m )] . By a
Cerf move we mean one of the following modifications made to [( Y , φ )] ◦ · · · ◦ [( Y m , φ m )](below we omit the boundary parametrizations φ k to simplify notation):(a) ( Critical point cancellation ) Replace · · · ◦ [ Y k ] ◦ [ Y k +1 ] ◦ · · · with · · · ◦ [ Y k ∪ Y k +1 ] ◦ · · · if[ Y k ∪ Y k +1 ] is a cylindrical bordism.(b) ( Critical point switch ) Replace · · · ◦ [ Y k ] ◦ [ Y k +1 ] ◦ · · · with · · · ◦ [ Y (cid:48) k ] ◦ [ Y (cid:48) k +1 ] ◦ · · · , where Y k , Y k +1 , Y (cid:48) k , and Y (cid:48) k +1 satisfy the following conditions: Y k ∪ Y k +1 ∼ = Y (cid:48) k ∪ Y (cid:48) k +1 , and for somechoice of Morse data ( f, t ), ( f (cid:48) , t (cid:48) ) inducing the two Cerf decompositions and a metric on Y , the attaching cycles for the critical points y k and y k +1 of f in X k and y (cid:48) k and y (cid:48) k +1 of f (cid:48) in X (cid:48) k are disjoint; in X k − = X (cid:48) k − , the attaching cycles of y k and y (cid:48) k +1 are homotopic;and in X k +1 = X (cid:48) k +1 the attaching cycles of y k +1 and y (cid:48) k are homotopic. See Figure 1 foran example of this move. HENRY T. HORTON y k y k +1 X k X k − X k +1 Y k Y k +1 X k +1 X k X k − Y k Y k +1 y k +1 y k Figure 1.
A 3-dimensional critical point switch of two 2-handles.(c) (
Cylinder creation ) Replace · · · ◦ [ Y k ] ◦ · · · with · · · ◦ [ Y (cid:48) k ] ◦ [ Y (cid:48)(cid:48) k ] ◦ · · · where Y k ∼ = Y (cid:48) k ∪ Y (cid:48)(cid:48) k and one of [ Y (cid:48) k ], [ Y (cid:48)(cid:48) k ] is cylindrical.(d) ( Cylinder cancellation ) Replace · · · ◦ [ Y k ] ◦ [ Y k +1 ] ◦ · · · with · · · ◦ [ Y k ◦ Y k +1 ] ◦ · · · wheneverone of [ Y k ], [ Y k +1 ] is cylindrical.It turns out that the four Cerf moves described above are sufficient to relate any two Cerf decom-positions of a given morphism [ Y, φ ]: Proposition 2.3.
Any two Cerf decompositions of a given morphism [ Y, φ ] are related by a finitesequence of Cerf moves. Symplectic Instanton Homology.
Now let Y be a closed, oriented 3-manifold and write Y (cid:48) for the result of removing two open balls from Y . We may then choose a Cerf decomposition Y (cid:48) = ( Y (cid:48) , . . . , Y (cid:48) m ), which is induced by some Morse function f : Y (cid:48) −→ R which, among otherconditions, satisfies f − (min f ) = ∂Y − and f − (max f ) = ∂Y + . Pick a point z ∈ Y (cid:48) such that thegradient flow line γ z of f through z connects ∂Y − and ∂Y + (and hence does not pass through anycritical points or their attaching cycles).Remove a small tubular neighborhood of γ z , and replace it with a cylinder with a trivial 3-strandedtangle removed. Then we obtain a new 3-manifold Y θ , which is Y with two open balls and a trivial3-stranded tangle connecting them removed (homotopy equivalent to Y minus a neighborhood ofa trivially embedded θ graph, hence the notation). The chosen Cerf decomposition of Y (cid:48) inducesa Cerf decomposition Y θ = ( Y θ , . . . , Y θm ) where each Y θk is just Y (cid:48) k with a trivial 3-stranded tangleconnecting the two boundary components removed. The intermediate surfaces X θk in this Cerfdecomposition are just closed surfaces with exactly 3 punctures.To each punctured surface X θk , we may associate the traceless character variety M g ( X k ) , . Since ∂Y θk = X θk − (cid:113) X θk , we may define a subset L ( Y θk ) ⊂ M − g ( X k − ) , × M g ( X k ) , by restricting SU(2)-representations of π ( Y θk ) with traceless holonomy around the strands of the trivial tangle to ∂Y θk .It is easily proven that L ( Y θk ) is in fact a Lagrangian submanifold of M − g ( X k − ) , × M g ( X k ) , , orequivalently a Lagrangian correspondence M g ( X k − ) , −→ M g ( X k ) , . YMPLECTIC INSTANTON HOMOLOGY, SPECTRAL SEQUENCES, AND KHOVANOV HOMOLOGY 7
We define the symplectic instanton homology of Y to be the quilted Floer homology groupSI( Y ) = HF( L ( Y θ ) , . . . , L ( Y θm )) . One checks that this quilted Floer group is unchanged under Cerf moves on ( Y θ , . . . , Y θm ) (whichare equivalent to Cerf moves on ( Y (cid:48) , . . . , Y (cid:48) m ), since the trivial tangle has no interaction with criticalpoints), so that it is an invariant of the homeomorphism type of Y .3. Symplectic Instanton Homology of Nontrivial
SO(3) -Bundles
The symplectic instanton homology SI( Y ) recalled in the previous section can be thought of as usingthe trivial SO(3)-bundle on Y in its definition. More generally, one may wish to define symplecticinstanton homology using a nontrivial SO(3)-bundle on Y , and it will indeed be necessary to havesuch a generalization in order to state the surgery exact triangle for symplectic instanton homology.3.1. Moduli Spaces of Flat
SO(3) -Bundles.
On a compact, orientable manifold Y of dimensionat most 3, principal SO(3)-bundles P −→ Y are classified by their second Stiefel-Whitney class w ( P ) ∈ H ( Y, ∂Y ; F ). In particular, if Y is a compact, orientable 3-manifold, then by Poincar´e-Lefschetz duality an SO(3)-bundle P −→ Y is classified by the homology class ω = PD( w ( P )) ∈ H ( Y ; F ). The following proposition, which describes how the homology class ω appears in theholonomy description for the moduli space of flat connections on P −→ Y , is well-known. Proposition 3.1.
Let Y be a compact, orientable -manifold and P −→ Y a principal SO(3) -bundle on Y . Let ω = PD( w ( P )) and also let ω denote a knot in Y representing this homologyclass, by abuse of notation. Then the moduli space of flat connections on P has the holonomydescription { ρ : π ( Y \ ω ) −→ SU(2) | ρ ( µ ω ) = − I } / conjugation , where µ ω is a meridian of ω . Cerf Decompositions with Homology Class.
In analogy with the definition of SI( Y )(which corresponds to ω = 0) we wish to define a “twisted” version of symplectic instanton homologyusing moduli spaces of flat connections on the nontrivial bundle P −→ Y by looking at the pieces ofa Cerf decomposition of Y . To do this, we will incorporate P into our Cerf decompositions throughthe homology class ω = PD( w ( P )), by virtue of Proposition 3.1. Definition 3.2.
Let Y be a compact, oriented 3-manifold and ω ∈ H ( Y ; F ) be a mod 2 homologyclass in Y . A Cerf decomposition of (
Y, ω ) is a decomposition[(
Y, ω )] = [( Y , ω )] ◦ · · · ◦ [( Y m , ω m )] , where [ Y ] = [ Y ] ◦ · · · ◦ [ Y m ] is a Cerf decomposition of Y in the usual sense and for k = 1 , . . . , m , ω k ∈ H ( Y k ; F ) and these classes satisfy the condition that ω + · · · + ω m = ω in H ( Y ; F ) (wherewe conflate ω k with its image in H ( Y ; F ) under inclusion ).In order to decide which Cerf decompositions determine the same pair ( Y, ω ), we need Cerf movesfor these Cerf decompositions with homology class. The Cerf moves in this context just requireslight modifications made to the usual moves where the homology class is not considered.
Definition 3.3. (Cerf Moves) Let [(
Y, φ )] ∈ Hom
Bord ( X − , X + ) be an equivalence class of bordismsand ω ∈ H ( Y ; F ). Suppose we have a Cerf decomposition[( Y, ω )] = [( Y , ω )] ◦ · · · ◦ [( Y m , ω m )] . By a
Cerf move we mean one of the following modifications made to [( Y , ω )] ◦ · · · ◦ [( Y m , ω m )]: HENRY T. HORTON (a) (
Critical point cancellation ) Replace · · · ◦ [( Y k , ω k )] ◦ [( Y k +1 , ω k +1 )] ◦ · · · with · · · ◦ [( Y k ∪ Y k +1 , ω k + ω k +1 )] ◦ · · · if [ Y k ◦ Y k +1 ] is a cylindrical bordism.(b) ( Critical point switch ) Replace · · · ◦ [( Y k , ω k )] ◦ [( Y k +1 , ω k +1 )] ◦ · · · with · · · ◦ [( Y (cid:48) k , ω (cid:48) k )] ◦ [( Y (cid:48) k +1 , ω (cid:48) k +1 )] ◦ · · · , where Y k , Y k +1 , Y (cid:48) k , and Y (cid:48) k +1 satisfy the following conditions: Y k ∪ Y k +1 ∼ = Y (cid:48) k ∪ Y (cid:48) k +1 , and for some choice of Morse data ( f, t ), ( f (cid:48) , t (cid:48) ) inducing the two Cerfdecompositions and a metric on Y , the attaching cycles for the critical points y k and y k +1 of f in X k and y (cid:48) k and y (cid:48) k +1 of f (cid:48) in X (cid:48) k are disjoint; in X k − = X (cid:48) k − , the attaching cyclesof y k and y (cid:48) k +1 are homotopic; and in X k +1 = X (cid:48) k +1 the attaching cycles of y k +1 and y (cid:48) k arehomotopic. See Figure 1 for an example of this move. Furthermore, the homology classesinvolved should satisfy ω k + ω k +1 = ω (cid:48) k + ω (cid:48) k +1 .(c) ( Cylinder creation ) Replace · · · ◦ [( Y k , ω k )] ◦ · · · with · · · ◦ [( Y (cid:48) k , ω (cid:48) k )] ◦ [( Y (cid:48)(cid:48) k , ω (cid:48)(cid:48) k )] ◦ · · · where Y k ∼ = Y (cid:48) k ∪ Y (cid:48)(cid:48) k , one of [ Y (cid:48) k ], [ Y (cid:48)(cid:48) k ] is cylindrical, and ω k = ω (cid:48) k + ω (cid:48)(cid:48) k .(d) ( Cylinder cancellation ) Replace · · · ◦ [( Y k , ω k )] ◦ [( Y k +1 , ω k +1 )] ◦ · · · with · · · ◦ [( Y k ◦ Y k +1 , ω k + ω k +1 )] ◦ · · · whenever one of [ Y k ], [ Y k +1 ] is cylindrical.(e) ( Homology class swap ) Replace · · · ◦ [( Y k , ω k )] ◦ [( Y k +1 , ω k +1 )] ◦ · · · with · · · ◦ [( Y k , ω (cid:48) k )] ◦ [( Y k +1 , ω (cid:48) k +1 )] ◦ · · · whenever ω k + ω k +1 = ω (cid:48) k + ω (cid:48) k +1 and at least one of [ Y k ], [ Y k +1 ] iscylindrical.The following fundamental theorem follows effortlessly from the usual Cerf theory. Theorem 3.4. If [( Y, φ )] is a connected bordism of dimension at least three and ω ∈ H ( Y ; F ) ,then any two Cerf decompositions of [( Y, φ, ω )] are related by a finite sequence of Cerf moves. Definition via Cerf Decompositions.
As before, given a closed, oriented 3-manifold Y ,we can remove a pair of open balls from Y to get a 3-manifold Y (cid:48) with a Cerf decomposition( Y (cid:48) , . . . , Y (cid:48) m ), and then we can remove a trivial 3-stranded tangle connecting the boundary com-ponents and missing all critical points get get a 3-manifold Y θ with induced Cerf decomposition( Y θ , . . . , Y θm ).We wish to incorporate the data of a nontrivial SO(3)-bundle ω into our Lagrangian correspon-dences. Abusing notation, let ω ∈ H ( Y ; F ) be the Poincar´e dual to the second Stiefel-Whitneyclass of our SO(3)-bundle. We may represent ω by a smoothly embedded unoriented curve in Y ,which furthermore may be assumed disjoint from the 3-balls we remove, so that we get an inducedcurve ω (taking our abuse of notation further) in Y θ that is geometrically unlinked with the trivial3-stranded tangle.In terms of the Cerf decomposition Y θ = ( Y θ , . . . , Y θm ), a generic embedded representative of ω intersects each intermediate surface X θk in an even number of points (since they are all separat-ing). We may then eliminate the intersection points in pairs without changing the homology classrepresented by the resulting curve; indeed, the two curves are homologous via a saddle cobordism.As a result, for each k = 1 , . . . , m , we have an unoriented curve ω k (possibly empty, but we mayassume it is connected up to homology by using saddle moves to merge components) contained onthe interior of Y θk , and ω + · · · + ω m is homologous to our original ω .With this data fixed, associate to each piece Y θk of the Cerf decomposition the moduli space M ( Y θk , ω k ) = (cid:26) ρ : π ( Y θk \ (3-tangle ∪ ω k )) −→ SU(2) (cid:12)(cid:12)(cid:12)(cid:12) ρ (meridian of tangle strand) is traceless ρ (meridian of ω k ) = − I (cid:27) (cid:30) conj.Denote the image of M ( Y θk , ω k ) in M − g ( X k − ) , × M g ( X k ) , under restriction to the boundary by L ( Y θk , ω k ). Proposition 3.5. L ( Y θk , ω k ) defines a smooth Lagrangian correspondence M g ( X k − ) , −→ M g ( X k ) , . YMPLECTIC INSTANTON HOMOLOGY, SPECTRAL SEQUENCES, AND KHOVANOV HOMOLOGY 9
Proof. L ( Y θk , ω k ) is an isotropic submanifold of M − g ( X k − ) , × M g ( X k ) , for the same reason L ( Y θk )is (the proof applies verbatim). L ( Y θk , ω k ) is clearly 3( g ( X k − ) + g ( X k )) = dim( M − g ( X k − ) , × M g ( X k ) , )-dimensional, hence it is Lagrangian. (cid:3) Now we verify that the generalized Lagrangian correspondence ( L ( Y θ , ω ) , . . . , L ( Y θm , ω m )) doesn’tdepend on the choice of Cerf decomposition. Theorem 3.6.
The morphism ( L ( Y θ , ω ) , . . . , L ( Y θm , ω m )) ∈ Hom
Symp ( M , , M , ) is unchangedunder Cerf moves on (( Y θ , ω ) , . . . , ( Y θm , ω m )) .Proof. The proof of invariance under Cerf moves (a)-(d) are just trivial modifications of the ω = 0case proved in [Hor16, Theorem 3.5]. It remains to check invariance under the new Cerf moveappearing when homology classes are introduced, the homology class swap. We wish to compare( . . . , L ( Y θk , ω k ) , L ( Y θk +1 , ω k +1 ) , . . . ) and ( . . . , L ( Y θk , ω (cid:48) k ) , L ( Y θk +1 , ω (cid:48) k +1 ) , . . . ), where at least one of Y θk , Y θk +1 are cylindrical and ω k + ω k +1 = ω (cid:48) k + ω (cid:48) k +1 . Without loss of generality, suppose Y θk +1 iscylindrical. Then Y θk ∪ Y θk +1 ∼ = Y θk and the associated Lagrangian correspondences are composable,and L ( Y θk , ω k ) ◦ L ( Y θk +1 , ω k +1 ) ∼ = L ( Y θk , ω k + ω k +1 )= L ( Y θk , ω k + ω (cid:48) k +1 ) ∼ = L ( Y θk , ω (cid:48) k ) ◦ L ( Y θk +1 , ω (cid:48) k +1 ) . It follows that ( . . . , L ( Y θk , ω k ) , L ( Y θk +1 , ω k +1 ) , . . . ) and ( . . . , L ( Y θk , ω (cid:48) k ) , L ( Y θk +1 , ω (cid:48) k +1 ) , . . . ) are iden-tical morphisms in Hom Symp ( M , , M , ). (cid:3) We define the symplectic instanton homology of (
Y, ω ) to be the quilted Floer homology groupSI(
Y, ω ) = H ∗ (CF( L ( Y θ , ω ) , . . . , L ( Y θm , ω m )) . By Theorem 3.6, SI(
Y, ω ) depends only on the diffeomorphism type of Y and the homology class ω ∈ H ( Y ; F ).3.4. Definition via Heegaard Diagrams.
For computations, it is best to use a nice, symmetrickind of Cerf decomposition. A genus g Heegaard splitting of Y , Y = H α ∪ Σ g H β , will induce aCerf decomposition with 2 g pieces, one for each handle of H α and H β . We can explicitly visualizethese handle attachments using a Heegaard diagram. Recall that a genus g (pointed) Heegaarddiagram is a tuple (Σ g , α , β , z ) consisting of a closed surface Σ g , two g -tuples of simple closedcurves α = { α , . . . , α g } and β = { β , . . . , β g } , and a basepoint z ∈ Σ g \ ( α ∪ β ) satisfying thefollowing conditions: • The α -curves are all pairwise disjoint and the β -curves are all pairwise disjoint. (An α -curveis permitted to intersect a β -curve, however) • The α -curves are linearly independent in H (Σ g ; R ), and the β -curves are linearly indepen-dent in H (Σ g ; R ).A Heegaard diagram (Σ g , α , β , z ) is assigned to a Heegaard splitting Y = H α ∪ Σ g H β by identifyingthe Heegaard surface with the abstract surface Σ g , taking the α -curves to be disjoint, homologi-cally independent curves bounding disks in H α , taking the β -curves to be disjoint, homologicallyindependent curves bounding disks in H β , and taking z to be any point disjoint from the chosen α - and β -curves. There are many choices for α - and β -curves satisfying these conditions, but theyare related by a finite sequence of the following moves: (1) (Isotopies) Any α -curve can we changed by an isotopy that misses the other α -curves andthe basepoint z . Any β -curve can we changed by an isotopy that misses the other β -curvesand the basepoint z .(2) (Handleslides) If γ , γ , and γ are disjoint simple closed curves in a surface Σ which togetherbound a embedded pair of pants (called the “handleslide region”) in Σ, we say that γ is ahandleslide of γ over γ (the order of the curves is irrelevant for this definition). We canreplace any α -curve α k by a curve α (cid:48) k that is a handleslide of α k over another α -curve α j , j (cid:54) = k , as long as the handleslide region does not contain any other α -curve or the basepoint z . We can replace any β -curve β k by a curve β (cid:48) k that is a handleslide of β k over another β -curve β j , j (cid:54) = k , as long as the handleslide region does not contain any other β -curve orthe basepoint z .By the Reidemeister-Singer theorem, all Heegaard splittings of a given 3-manifold Y differ up toisotopy only by (de)stabilizations, which amounts to connect summing with the standard genus 1Heegaard splitting for S (or removing such a connect summand). In terms of Heegaard diagrams,this amounts to the following move:(3) (Stabilization) Let H = (Σ , α , β ) be the (unpointed) genus 1 Heegaard diagram with α the standard meridian of Σ and β the standard longitude. Then the stabilization H (cid:48) ofa Heegaard diagram H = (Σ g , α , β , z ) is the connect sum of this diagram with the diagram H . In other words, H (cid:48) = (Σ g +1 , α ∪ { α } , β ∪ { β } , z ). The connect sum is performed ina small neighborhood of the basepoint z .All possible Heegaard diagrams representing a Heegaard splitting of a 3-manifold Y are related bya finite sequence of isotopies, handleslides, and (de)stabilizations. Hence to prove that an invariantof Heegaard diagrams induces an invariant of 3-manifolds, we need only check that the invariant isunchanged under moves (1)-(3). A Heegaard diagram (Σ g , α , β , z ) for a 3-manifold Y gives a Cerf decomposition with trivial 3-tangle for Y θ by the following process. Start with Σ g × [ − , g × {− } along the curves α k × {− } to obtain the α -handlebody minus a 3-ball, H (cid:48) α , and attach 2-handlesto Σ g × { } along the curves β k × { } to obtain the β -handlebody minus a 3-ball, H (cid:48) β ; we then have Y (cid:48) := Y \ (two 3-balls) = H (cid:48) α ∪ (Σ g × [ − , ∪ H (cid:48) β . Since the α -curves (resp. β -curves) are pairwisedisjoint, the order of the handle attachments for each handlebody is irrelevant. We can dually thinkof H (cid:48) α as being obtained from 1-handle attachments, so that we have a Cerf decomposition Y (cid:48) = Y α ∪ Σ Y α ∪ Σ · · · ∪ Σ g − Y α g ∪ Σ g Y β g ∪ Σ g − · · · ∪ Σ Y β , where each Y α k : Σ k − −→ Σ k is a 1-handle cobordism induced by the curve α k , and each Y β k :Σ k −→ Σ k − is a 2-handle cobordism induced by the curve β k . Since the basepoint z ∈ Σ g isdisjoint from all attaching cycles, we can fix a Morse function inducing this Cerf decompositionand consider the gradient flow line γ z passing through z ; this flow line necessarily connects thetwo boundary 2-spheres of Y (cid:48) . We may therefore remove a trivial 3-stranded tangle from a regularneighborhood of γ z get a Cerf decomposition Y θ = Y θα ∪ Σ , Y θα ∪ Σ , · · · ∪ Σ g − , Y θα g ∪ Σ g, Y θβ g ∪ Σ g − , · · · ∪ Σ , Y θβ . Given a mod 2 homology class ω ∈ H ( Y ; F ), we can write ω = ω α + · · · + ω α g + ω β g + · · · + ω β ∈ H ( Y α ; F ) ⊕· · ·⊕ H ( Y α g ; F ) ⊕ H ( Y β g ; F ) ⊕· · ·⊕ H ( Y β ; F ) . We have already proved the more general statement that SI(
Y, ω ) is an invariant of (
Y, ω ) by defining it in termsof Cerf decompositions and showing it is unchanged under Cerf moves.
YMPLECTIC INSTANTON HOMOLOGY, SPECTRAL SEQUENCES, AND KHOVANOV HOMOLOGY 11
Clearly these induce well-defined mod 2-homology classes in Y θ , which we will still denote by ω , ω α k , and ω β k . We therefore get a Cerf decomposition with homology class(3.1) ( Y θ , ω ) = ( Y θα , ω α ) ∪ Σ , · · · ∪ Σ g − , ( Y θα g , ω α g ) ∪ Σ g, ( Y θβ g , ω β g ) ∪ Σ g − , · · · ∪ Σ , ( Y θβ , ω β ) . We wish to simplify the generalized Lagrangian correspondence associated to this Cerf decompo-sition. The first thing to note is that we can take almost all of the homology classes to be zero.Indeed, by applying the Mayer-Vietoris sequence to the Heegaard splitting Y = H α ∪ Σ g H β , onesees that both i α ∗ : H ( H α ; F ) −→ H ( Y ; F ) and i β ∗ : H ( H β ; F ) −→ H ( Y ; F ) are surjective, sothat one may take ω to be represented by a circle lying entirely in just one of the handlebodies,say H α . Furthermore, i ∗ : H (Σ g ; F ) −→ H ( H α ; F ) is also a surjection, so we can further restrictto the case where ω is represented by a curve in a collar neighborhood of the boundary of H α .This implies that we can insert a cylinder (Σ g, × [0 , , ω ) between ( Y θα g , ω α g ) and ( Y θβ g , ω β g ) in theCerf decomposition (3.1) and set all the other homology classes equal to zero. The Lagrangiancorrespondence associated to (Σ g, × [0 , , ω ) turns out to have a simple description: Lemma 3.7.
Let W θ = Σ g, × [0 , be a trivial cobordism with a trivial -stranded tangle removed,and suppose ω ∈ H ( W θ ; F ) . Then L ( W θ , ω ) = { ([ ρ ] , [ ρ (cid:48) ]) ∈ M g, × M g, | ρ (cid:48) ( γ ) = ( − γ · ω ρ ( γ ) for each γ } , where γ is any smooth curve in Σ g, and γ · ω is the mod intersection number of γ with theprojection of a geometric representative for ω α to Σ g, .Proof. L ( W θ , ω ) consists of pairs of (gauge equivalence classes of) connections ( A , A ) ∈ M g, × M g, which are the boundary values of a connection A on W θ that has traceless holonomy aroundeach strand of the trivial tangle and holonomy − I around the meridian of the curve ω . For anybased curve γ in Σ g, (with basepoint missing ω ), consider γ × [0 , ⊂ W θ as a rectangle. Thisrectangle intersects ω exactly γ · ω times. Therefore the holonomy of A around the boundary ofthis square is ( − I ) γ · ω . This holonomy is also equal to Hol γ ×{ } ( A ) Hol γ ×{ } ( A ) − , and thereforeHol γ ×{ } ( A ) = ( − γ · ω Hol γ ×{ } ( A )for any curve γ in Σ g, . (cid:3) The following theorem shows that in the current setup, we can associate genuine Lagrangian sub-manifolds of M g, to H α and H β , not just generalized Lagrangian correspondences pt −→ M g, . Theorem 3.8.
The geometric compositions L ωα = L ( Y θα ) ◦ · · · ◦ L ( Y θα g ) ◦ L (Σ g, × [0 , , ω ) and L ωβ = L (Σ g, × [0 , , ω ) ◦ L ( Y θβ g ) ◦ · · · ◦ L ( Y θβ ) are embedded.Proof. The proof that the geometric composition L α = L ( Y θα ) ◦ · · · ◦ L ( Y θα g )is embedded is already known from [Hor16, Section 4]. L α has the holonomy description L α = { [ ρ ] ∈ M g, | ρ ( α k ) = I for each k } . Then it is clear by Lemma 3.7 that L ωα = L α ◦ L (Σ g, × [0 , , ω )= { [ ρ ] ∈ M g, | ρ ( α k ) = ( − α k · ω I for each k } and that this composition is embedded. The analogous statement for L ωβ follows in the sameway. (cid:3) The holonomy description for L ωα (and the obvious analogue for L ωβ ) is useful in its own right, sowe record it separately here: Proposition 3.9.
Given ω ∈ H ( Y ; F ) and a Heegaard splitting Y = H α ∪ Σ g H β , represent ω bya smooth loop in a bicollar neighborhood of the Heegaard surface, which we may assume lies on aparallel copy of Σ g . Then L ωα = { [ ρ ] ∈ M g, | ρ ( α k ) = ( − α k · ω I for each k } ,L ωβ = { [ ρ ] ∈ M g, | ρ ( β k ) = ( − β k · ω I for each k } , where α k · ω is the mod intersection number of α k with the projection of the geometric representativefor ω to Σ g , and similarly for β k · ω . As a consequence of the above, when SI(
Y, ω ) is computed from a Heegaard splitting, it can be com-puted as a classical Lagrangian Floer homology group for two Lagrangians in M g, , rather than thequilted Floer homology group of a pair of generalized Lagrangian correspondences. Furthermore,we may always think of ω ∈ H ( Y ; F ) as represented by a curve in a parallel copy of the Heegaardsurface, and incorporate it either into the Lagrangian for the α -handlebody or the β -handlebody.Hence we may compute the symplectic instanton homology asSI( Y, ω ) = HF( L ωα , L β ) = HF( L α , L ωβ ) , so that only one of the Lagrangians is different from the ω = 0 case in [Hor16], and the way it isdifferent is described exactly as in Proposition 3.9.Note that Theorem 4.2 of [Hor16] (which says that L α is a copy of ( S ) g for any set of attachingcurves α ) applies equally well to L ωα , so that L ωα is a Lagrangian ( S ) g in M g, , and furthermorethis identification is achieved by conjugating [ A , . . . , B g , C , C , C ] ∈ L ωα to [ A (cid:48) , . . . , B (cid:48) g , i , j , − k ](it turns out that the ( A (cid:48) , . . . , B (cid:48) g ) describe a g -fold product of 3-spheres when varying over all[ A , . . . , B g , C , C , C ] ∈ L α , see [Hor16, Theorem 4.2]). This identification says that we canconsider L ωα as SU(2)-representations of the α -handlebody minus ω sending the meridian of ω to − I , without modding out by conjugation. This in turn leads to the following interpretation of L ωα ∩ L β : Theorem 3.10.
For any two sets of attaching curves α and β and homology class ω ∈ H ( Y ; F ) ,we have that L ωα ∩ L β ∼ = { ρ : π ( Y \ ω ) −→ SU(2) | ρ ( µ ω ) = − I } . Proof.
Let H α and H β denote the α - and β -handlebodies, respectively. Then the discussion aboveshows that L α = { ρ : π ( H α \ ω ) −→ SU(2) | ρ ( µ ω ) = − I } , L β = Hom( π ( H β ) , SU(2)) , where we do not mod out by the action of conjugation. Furthermore, wee see that (cid:40) [ A , B , . . . , A g , B g , C , C , C ] ∈ M g, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g (cid:89) k =1 [ A k , B k ] = I (cid:41) = Hom( π (Σ g ) , SU(2)) , and L ωα , L β always lie entirely inside this subset of M g, . Therefore by the Seifert-Van Kampentheorem an intersection point of L ωα and L β corresponds to an element of { ρ : π ( Y \ ω ) −→ SU(2) | ρ ( µ ω ) = − I } . (cid:3) YMPLECTIC INSTANTON HOMOLOGY, SPECTRAL SEQUENCES, AND KHOVANOV HOMOLOGY 13
We also remark that the proof of naturality for SI( Y ) in [Hor16, Section 6] directly translates to aproof of naturality for SI( Y, ω ), resulting in the following:
Theorem 3.11.
SI(
Y, ω ) is a natural invariant of the pair ( Y, ω ) , in that one can pin down SI(
Y, ω ) as a concrete group as opposed to an isomorphism class of groups. Computations for Nontrivial Bundles.
Many of the computations for SI( Y ) from [Hor16,Section 7] can be generalized to nontrivial SO(3)-bundles. Proposition 3.12.
For any closed, oriented -manifold Y and any SO(3) -bundle ω on Y , χ (SI( Y, ω )) = (cid:40) | H ( Y ; Z ) | , if b ( Y ) = 0 , , otherwise. We will postpone the proof of Proposition 3.12 until Section 6.
Proposition 3.13.
As a Z / -graded unital algebra, we have that SI( S × S , ω ) = (cid:40) H −∗ ( S ) , if ω is trivial, , if ω is nontrivial.Proof. For ω = 0, this is Proposition 7.3 of [Hor16]. For ω nontrivial, we proceed as follows. Thenonzero element ω ∈ H ( S × S ; F ) ∼ = F can be represented by { pt } × S . Consider the genus1 Heegaard diagram (Σ , α, β, z ) for S × S , where α = β is the meridian of the torus. In thisHeegaard diagram, ω can be realized as the standard longitude of the torus. Pushing this into the α -handlebody, we see that L α = { [ − I, A, i , j , − k ] } , L β = { [ I, A, i , j , − k ] } . Then L α ∩ L β = ∅ , so that we necessarily have SI( S × S , ω ) = 0. (cid:3) Proposition 3.14.
Let p and q be relatively prime positive integers and ω be any SO(3) -bundle on L ( p, q ) . Then SI( L ( p, q ) , ω ) ∼ = Z p .Proof. For ω = 0, this is Proposition 7.4 of [Hor16]. For ω (cid:54) = 0, the computation proceeds in thesame way: if we fix the standard genus 1 pointed Heegaard diagram (Σ , α, β, z ) for L ( p, q ), where α is the standard meridian m and β = qm + p(cid:96) in π (Σ ) ∼ = Z (cid:104) m (cid:105) ⊕ Z (cid:104) (cid:96) (cid:105) , then intersection pointsof L ωα and L β correspond to p th roots of ± I in SU(2). For either choice of sign, we have one of thefollowing descriptions of L ωα ∩ L β : • If p is odd, L ωα ∩ L β consists of an isolated point (the trivial representation) and ( p − S . • If p is even, L ωα ∩ L β consists of an isolated point (again the trivial representation) and ( p −
2) copies of S .In either case, Po´zniak’s Morse-Bott spectral sequence [Poz99] has E -page H ∗ ( L ωα ∩ L β ) ∼ = Z p and converges to HF( L ωα , L β ) ∼ = SI( L ( p, q ) , ω ). Since χ (SI( L ( p, q ) , ω )) = p by Proposition 3.12, thespectral sequence must collapse immediately, so that SI( L ( p, q ) , ω ) ∼ = Z p . (cid:3) Connected Sums.
The K¨unneth principle satisfied by the symplectic instanton homology oftrivial SO(3)-bundles has a straightforward generalization to the nontrivial case.
Theorem 3.15.
For any closed, oriented -manifolds Y and Y (cid:48) equipped with SO(3) -bundles ω and ω (cid:48) , the symplectic instanton chain complex for ( Y Y (cid:48) , ω ω (cid:48) ) satisfies the following K¨unnethprinciple: CSI( Y Y (cid:48) , ω ∪ ω (cid:48) ) ∼ = CSI( Y, ω ) ⊗ CSI( Y (cid:48) , ω (cid:48) ) . Proof.
Given Heegaard diagrams H = (Σ g , α , β , z ) for Y and H (cid:48) = (Σ g (cid:48) , α (cid:48) , β (cid:48) , z (cid:48) ) for Y (cid:48) , we geta Heegaard diagram H H (cid:48) = (Σ g + g (cid:48) , α ∪ α (cid:48) , β ∪ β (cid:48) , z (cid:48)(cid:48) ) for Y Y (cid:48) , where we perform the connectsum by removing neighborhoods of z and z (cid:48) , and z (cid:48)(cid:48) ∈ Σ g + g (cid:48) is a point in the connect sum region.Note that in H H (cid:48) , the attaching regions for the α - and β -handles are completely disjoint from theattaching regions for the α (cid:48) - and β (cid:48) -handles. Therefore by a sequence of critical point switches, wecan construct Y Y (cid:48) by attaching handles in the following order: α -handles, β -handles, α (cid:48) -handles, β (cid:48) -handles (if the attaching regions were not disjoint, in general we could only attach ( α ∪ α (cid:48) )-handles and then ( β ∪ β (cid:48) )-handles). Furthermore, the embedded loops ω and ω (cid:48) can be chosen tolie in the parts of the Cerf decomposition involving the α - and α (cid:48) -handle attachments, respectively.Therefore we get a sequence of Lagrangian correspondences M , L ωα −−−→ M g, L β −−−→ M , L ω (cid:48) α (cid:48) −−−→ M g (cid:48) , L β (cid:48) −−−→ M , , which by Theorem 3.6 is geometrically equivalent to the sequence given by the Heegaard diagram H H (cid:48) : M , L ω ∪ ω (cid:48) α ∪ α (cid:48) −−−−−→ M g + g (cid:48) , L β ∪ β (cid:48) −−−−−→ M , . Hence we have an identification of quilted Floer complexesCF( L ω ∪ ω (cid:48) α ∪ α (cid:48) , L β ∪ β (cid:48) ) ∼ = CF( L ωα , L β , L ω (cid:48) α (cid:48) , L β (cid:48) ) . On the other hand, L β and L ω (cid:48) α (cid:48) have embedded geometric composition, and since M , = { pt } , itis clear that L β ◦ L ω (cid:48) α (cid:48) ∼ = L β × L ω (cid:48) α (cid:48) ⊂ M g, × M g (cid:48) , . ThereforeCF( L ω ∪ ω (cid:48) α ∪ α (cid:48) , L β ∪ β (cid:48) ) ∼ = CF( L ωα , L β × L ω (cid:48) α (cid:48) , L β (cid:48) )= CF( L ωα × L β (cid:48) , L β × L ω (cid:48) α (cid:48) ) ∼ = CF( L ωα , L β ) ⊗ CF( L ω (cid:48) α (cid:48) , L β (cid:48) ) . (cid:3) On ( n S ) g − n S × S ) . Let H = (Σ g , β , γ ) denote the standard Heegaard diagram for( n S ) g − n S × S ), with the first n β - and γ -curves corresponding to the S summands.Write L (cid:48) β (resp. L (cid:48) γ ) for the Lagrangian in M n, corresponding to the first n β - (resp. γ -) handleattachments. Let L βγ denote the Lagrangian correspondence M n, −→ M g, corresponding toattaching the remaining g − n handles (which are the same for β and γ ).For the Heegaard diagram H above, we have L β ∩ L γ ∼ = ( S ) g − n and this intersection is clean.There is a Hamiltonian isotopy ϕ of M g, taking L γ to another Lagrangian ˜ L γ such that L β ∩ ˜ L γ consists of 2 g − n points, and in fact this isotopy can be chosen so that L β ∩ ˜ L γ = { [ I, I, . . . , I, I, I, ( − (cid:15) I, . . . , I, ( − (cid:15) g − n I ] ∈ M g, | ( (cid:15) , . . . , (cid:15) g − n ) ∈ { , } g − n } . Under the isomorphism of unital algebras CF( L β , ˜ L γ ) ∼ = SI( g − n S × S ) ∼ = H −∗ ( S ) ⊗ ( g − n ) , theelement Θ βγ ∈ CF( L β , ˜ L γ ) corresponding to the intersection point with ( (cid:15) , . . . , (cid:15) g − n ) = (1 , . . . , H −∗ ( S ) ⊗ ( g − n ) . YMPLECTIC INSTANTON HOMOLOGY, SPECTRAL SEQUENCES, AND KHOVANOV HOMOLOGY 15 L β L γ ~ L βγ M n; M g; Figure 2.
The quilt defining the map CF( L (cid:48) β , L (cid:48) γ ) −→ CF( L (cid:48) β , L Tβγ , L βγ , L (cid:48) γ ) : θ (cid:55)→ θ × Θ βγ .Now, write ˜ L βγ for the image of the Lagrangian correspondence L βγ under the Hamiltonian isotopyid × ϕ of M n, × M g, (with ϕ as in the previous paragraph), and consider the quilt map picturedin Figure 2. Because Θ βγ is the unit of CF( L β , ˜ L γ ) ∼ = CF( L (cid:48) β , L Tβγ , L βγ , L (cid:48) γ ), it is easy to see thatthe relative invariant defined by this quilt is simply the map θ (cid:55)→ θ × Θ βγ .More generally, if L (cid:48) β ∩ L (cid:48) γ = { [ ρ ] } for some representation ρ (not necessarily the trivial repre-sentation), then the quilt map of Figure 2 is given by ρ (cid:55)→ ρ × Θ ρ , where in this case we defineΘ ρ similarly to Θ βγ , except that on Σ n \ pt ⊂ Σ g , it should agree with ρ instead of the trivialrepresentation. This fact will be useful later on.4. Functoriality for Nontrivial Bundles
Now we turn to the topic of functoriality of symplectic instanton homology with respect to cobor-disms in the presence of nontrivial SO(3)-bundles, i.e. assigning to each bundle cobordism (
W, P ) :( Y , P ) −→ ( Y , P ) a homomorphism between the symplectic instanton homologies of the bound-ary components.4.1. The Statement.
By a classic theorem of Dold and Whitney [DW59], SO(3)-bundles on acompact 4-manifold W are classified by pairs ( w , p ) ∈ H ( W ; Z / × H ( W ; Z ) (the Stiefel-Whitney and Pontryagin classes of the bundle) such that the Pontryagin square of w is the mod4 reduction of p . Since H ( W ) = 0 for ∂W (cid:54) = ∅ , we therefore see that SO(3)-bundles overa cobordism between non-empty 3-manifolds are simply classified by their second Stiefel-Whitneyclass. By Poincar´e-Lefschetz duality, we may then think of such a bundle P −→ W as correspondingto a relative mod 2 homology class Ω ∈ H ( W, ∂W ; F ). By naturality of Poincar´e-Lefschetz duality,we furthermore have that if ∂ Ω = ω ⊕ ω ∈ H ( Y ; F ) ⊕ H ( Y ; F ) ∼ = H ( ∂W ; F ) (where ∂ is theconnecting homomorphism in the long exact sequence of the pair ( W, ∂W )), then ω i is the mod 2homology class representing the SO(3)-bundle P i = i ∗ Y i P . In this way, we eliminate direct referenceto SO(3)-bundles in this section.Let W : Y −→ Y (cid:48) be a compact, connected, oriented cobordism of closed, connected, oriented 3-manifolds. Given a mod 2 relative homology class Ω ∈ H ( W, ∂W ; F ), write ω ⊕ ω (cid:48) ∈ H ( Y ; F ) ⊕ H ( Y (cid:48) ; F ) ∼ = H ( ∂W ; F ) for its image under the boundary map in the long exact sequence for thepair ( W, ∂W ). Let use write ( W, Ω) : (
Y, ω ) −→ ( Y (cid:48) , ω (cid:48) ) for such a combination of a cobordismwith a mod 2 homology class. We claim the following: Theorem 4.1.
To any compact, connected, oriented cobordism with homology class ( W, Ω) :(
Y, ω ) −→ ( Y (cid:48) , ω (cid:48) ) we may associate a well-defined homomorphism SI( W, Ω) : SI(
Y, ω ) −→ SI( Y (cid:48) , ω (cid:48) ) that is functorial in the following sense: (1) For any closed, connected, oriented -manifold Y , SI( Y × [0 , , ω × [0 , SI(
Y,ω ) . (2) For any other compact, connected, oriented cobordism with homology class ( W (cid:48) , Ω (cid:48) ) : ( Y (cid:48) , ω (cid:48) ) −→ ( Y (cid:48)(cid:48) , ω (cid:48)(cid:48) ) , we have SI( W ∪ Y (cid:48) W (cid:48) , Ω ∪ Y (cid:48) Ω (cid:48) ) = SI( W (cid:48) , Ω (cid:48) ) ◦ SI( W, Ω) . The homomorphism SI( W, Ω) is constructed similarly to the Ω = 0 case treated in [Hor16], definedfirst for individual handle attachments and then extended to arbitrary ( W, Ω) by Kirby calculus.4.2. 1 -Handle Attachments.
First, suppose ( W , Ω) : (
Y, ω ) −→ ( Y (cid:48) , ω (cid:48) ) consists of a single 1-handle attachment. Then we necessarily have that Y (cid:48) ∼ = Y S × S ). A Mayer-Vietoris argumentshows that in this case, H ( W ; F ) ∼ = H ( Y ; F ), and therefore H ( W , ∂W ; F ) ∼ = H ( Y ; F ). Itfollows that for 1-handle attachments, we necessarily have ω (cid:48) = ω ∪ ∈ H ( Y ; F ) ⊕ H ( S × S ; F ) ∼ = H ( Y (cid:48) ; F ), and the handle attachment map must be of the formCSI( W , Ω) : CSI(
Y, ω ) −→ CSI(
Y, ω ) ⊗ CSI( S × S , . Recall that CSI( S × S , ∼ = H −∗ ( S ) as a unital algebra; let Θ denote its unit. Then we definethe 1-handle attachment map in the same way as the Ω = 0 case:CSI( W , Ω)( ξ ) = ξ ⊗ Θ . If W consists of m W = W , ∪ · · · ∪ W ,m anddefine CSI( W , Ω) = CSI( W ,m , Ω) ◦ · · · ◦ CSI( W , , Ω) , where the homology classes Ω are all the same by the above discussion. We may then proceedexactly as in the Ω = 0 case [Hor16, Theorem 8.2] to obtain the following: Theorem 4.2.
The map
SI( W , Ω) : SI(
Y, ω ) −→ SI( Y (cid:48) , ω (cid:48) ) is invariant under the ordering of the -handles of W and handleslides amongst them, and therefore is an invariant of the pair ( W , Ω) . -Handle Attachments. The situation for 3-handles is dual to the 1-handle case. If ( W , Ω) :(
Y, ω ) −→ ( Y (cid:48) , ω (cid:48) ) consists of a single 3-handle attachment, then Y ∼ = Y (cid:48) S × S ) and H ( W, ∂W ; F ) ∼ = H ( Y (cid:48) ; F ). Hence ω = ω (cid:48) ∪ ∈ H ( Y (cid:48) ; F ) ⊕ H ( S × S ; F ), and the handle attachment map isof the form CSI( W , Ω) : CSI( Y (cid:48) , ω (cid:48) ) ⊗ CSI( S × S , −→ CSI( Y (cid:48) , ω (cid:48) ) . We may then define the 3-handle attachment map in the same way as the Ω = 0 case:CSI( W , Ω)( ξ ⊗ η ) = (cid:40) ξ, if η = Θ , , if η (cid:54) = Θ . If W consists of m W = W , ∪ · · · ∪ W ,m anddefine CSI( W , Ω) = CSI( W ,m , Ω) ◦ · · · ◦ CSI( W , , Ω) , where the homology classes Ω are all the same by the above discussion. We may then proceedexactly as in the Ω = 0 case [Hor16, Theorem 8.5] to obtain the following: Theorem 4.3.
The map
SI( W , Ω) : SI(
Y, ω ) −→ SI( Y (cid:48) , ω (cid:48) ) is invariant under the ordering of the -handles of W and handleslides amongst them, and therefore is an invariant of the pair ( W , Ω) . YMPLECTIC INSTANTON HOMOLOGY, SPECTRAL SEQUENCES, AND KHOVANOV HOMOLOGY 17 -Handle Attachments.
Finally, consider the case where ( W , Ω) : (
Y, ω ) −→ ( Y (cid:48) , ω (cid:48) ) con-sists of a single 2-handle attachment. A 2-handle is attached to Y × { } ⊂ Y × [0 ,
1] along aframed link L = ( L, λ ) to obtain Y (cid:48) ∼ = Y ( L ). H ( Y ; F ) and H ( Y (cid:48) ; F ) differ only possibly in thehomology class represented by the meridian of the link L ; write ω L for the mod 2 homology classof L in Y and write ω (cid:48) L for the mod 2 homology class of L in Y (cid:48) . It is easy to see that the imageof any Ω ∈ H ( W , ∂W ; F ) in H ( Y ; F ) ⊕ H ( Y (cid:48) ; F ) under the boundary map in the long exactsequence for the pair ( W , ∂W ) must have the form( i ∗ ω + (cid:15)ω L , i (cid:48)∗ ω + (cid:15) (cid:48) ω (cid:48) L ) ∈ H ( Y ; F ) ⊕ H ( Y (cid:48) ; F ) , where (cid:15), (cid:15) (cid:48) ∈ F , ω is a mod 2 homology class in Y \ L , and i : Y \ L (cid:44) → Y and i (cid:48) : Y \ L (cid:44) → Y (cid:48) arethe inclusion maps.In order to define the 2-handle maps, we follow constructions of Ozsv´ath and Szab´o and useHeegaard splittings which are subordinate to the framed link L in a suitable sense. Definition 4.4. A bouquet for the framed link L ⊂ Y of n components is a 1-complex B ( L )embedded in Y with • n + 1 0-cells given by a basepoint y ∈ Y \ L and basepoints y i ∈ L i . • n L i and n paths δ i ⊂ Y satisfying δ i (0) = y , δ i (1) = y i , and δ i ([0 , ∩ L = ∅ .Clearly a regular neighborhood of a bouquet B ( L ) is a genus n -handlebody and L is unknottedinside this handlebody. This handlebody may not give a Heegaard splitting of Y , but there willbe some genus g ≥ n Heegaard splitting of Y with one of the handlebodies containing this regularneighborhood. Hence we introduce the following definition. Definition 4.5.
A Heegaard triple (Σ g , α , β , γ , z ) is said to be subordinate to the bouquet B ( L ) if the following conditions are satisfied: • Attaching 2-handles along { α i } gi =1 and { β i } gi = n +1 gives the complement of B ( L ) in Y . • γ i = β i for i = n + 1 , . . . , g . • After surgering out β n +1 , . . . , β g , both β i and γ i lie in the obvious punctured torus T i ⊂ Σ g corresponding to L i for i = 1 , . . . , n . • For i = 1 , . . . , n the β i are meridians for L i and the γ i are the longitudes of L i specified by λ i .Note that for such a Heegaard triple, H αβ = (Σ g , α , β , z ) is a Heegaard diagram for Y , H αγ =(Σ g , α , γ , z ) is a Heegaard diagram for Y ( L ), and H βγ = (Σ g , β , γ , z ) is a Heegaard diagram for g − n ( S × S ).We may now proceed to define 2-handle cobordism maps. Let Y be a closed, connected, oriented3-manifold and L a framed link in Y , write W for the cobordism corresponding to attaching a2-handle to Y along L , and let Ω ∈ H ( W , ∂W ; F ) be any homology class. Recall from the abovethat ∂ Ω = ( i ∗ ω + (cid:15)ω L , i (cid:48)∗ ω + (cid:15) (cid:48) ω (cid:48) L ) ∈ H ( Y ; F ) ⊕ H ( Y (cid:48) ; F ) . Now, fixing a bouquet B ( L ) for L and a Heegaard triple H = (Σ g , α , β , γ , z ) subordinate to B ( L ),we can represent the mod 2 homology classes i ∗ ω + (cid:15)ω L and i (cid:48)∗ ω + (cid:15) (cid:48) ω (cid:48) L by a knot lying in the α -handlebody of Y and Y ( L ). It is clear that L β ∩ L γ ∼ = ( S ) g − n is a clean intersection, andtherefore HF( L β , L γ ) = SI( S , ⊗ n ⊗ SI( S × S , ⊗ ( g − n ) ∼ = H −∗ ( S ) ⊗ ( g − n )8 HENRY T. HORTON M g + n; L γ L β L β L i ∗ ! + γ ! L α L i ∗ ! + γ ! L α Θ βγ Figure 3.
Triangles counted by the 2-handle map CSI( W , Ω).as a unital algebra; write Θ βγ for its unit. We then define the 2-handle attachment map as atriangle map for the Lagrangians L i ∗ ω + (cid:15)ω L α = L i (cid:48)∗ ω + (cid:15) (cid:48) ω (cid:48) L α , L β , and L γ (see Figure 3):CSI( W , Ω) : CSI(
Y, i ∗ ω + (cid:15)ω L ) −→ CSI( Y ( L ) , i (cid:48)∗ ω + (cid:15) (cid:48) ω (cid:48) L ) ,ξ (cid:55)→ µ αβγ ( ξ, Θ βγ ) . With the effect of Ω relegated to the Lagrangian for the α -handlebody, proving that SI( W , Ω) isindependent of the choice of bouquet for L , the ordering of the components of L , and handleslidesamong the components of L proceeds exactly as usual, as the proofs for the Ω = 0 case only usedthe algebra structure of HF( L β , L γ ). Hence we conclude the following. Theorem 4.6.
The map
SI( W , Ω) : SI(
Y, ω ) −→ SI( Y (cid:48) , ω (cid:48) ) induced by adding -handles to aframed link L in Y is independent of the choice of bouquet B ( L ) used to define it. Furthermore,if L and L are two framed links in Y and Ω k ∈ H ( W ( L k ) , ∂W ( L k ); F ) ( k = 1 , are mod homology classes with Ω | Y ( L ) = Ω | Y ( L ) , then SI( W ( L ∪ L ) , Ω ∪ Ω ) = SI( W ( L ) , Ω ) ◦ SI( W ( L ) , Ω ) and if L (cid:48) differs from L by handleslides amongst the components of L , then SI( W ( L (cid:48) ) , Ω) =SI( W ( L ) , Ω) . General Cobordisms.
Now, given an arbitrary 4-dimensional cobordism W : Y −→ Y (cid:48) , wemay represent it as a relative handlebody built on Y , with handles of index 1, 2, and 3 added inorder of increasing index. Hence we may write W = W ∪ W ∪ W , where each W k is a cobordismconsisting entirely of k -handles. Furthermore, given a homology class Ω ∈ H ( W, ∂W ; F ), thereexist homology classes Ω k ∈ H ( W k , ∂W k ; F ) such thatΩ = ( i ) ∗ Ω + ( i ) ∗ Ω + ( i ) ∗ Ω . (4.1)We then define SI( W, Ω) = SI( W , Ω ) ◦ SI( W , Ω ) ◦ SI( W , Ω ) . To prove that SI( W, Ω) : SI(
Y, ω ) −→ SI( Y (cid:48) , ω (cid:48) ) as defined above is an invariant of ( W, Ω), wemust show that Kirby moves on the handles of W as well as different choices of the Ω k satisfyingEquation (4.1) result in the same homomorphism.First we show independence of the decomposition of Ω as in Equation (4.1). Recall from thediscussion of 1- and 3-handle maps that we necessarily have Ω = ω × [0 ,
1] for ω = Ω | Y andΩ = ω (cid:48) × [0 ,
1] for ω (cid:48) = Ω | Y (cid:48) . This forces the decomposition of Equation (4.1) to be unique, sothat it causes no trouble for the well-definedness of SI( W, Ω).
YMPLECTIC INSTANTON HOMOLOGY, SPECTRAL SEQUENCES, AND KHOVANOV HOMOLOGY 19
As for invariance under Kirby moves, we have already checked those Kirby moves which involvehandles all of the same index, and it only remains to show that cancelling pairs of handles does notchange SI( W, Ω). The proofs of the corresponding results when Ω = 0 [Hor16, Lemmata 8.15 and8.16] apply word-for-word here, so we omit them.
Lemma 4.7.
Let W : Y −→ Y (cid:48) be a cobordism corresponding to a single -handle attachment,and let W : Y (cid:48) −→ Y (cid:48)(cid:48) be a cobordism corresponding to a -handle attachment along a framedknot K in Y (cid:48) that cancels the -handle from W . Then for any Ω ∈ H ( W ∪ W , ∂ ( W ∪ W ); F ) , SI( W ∪ W , Ω) is the identity map. Lemma 4.8.
Let W : Y −→ Y (cid:48) be a cobordism corresponding to attaching a -handle along aframed knot K in Y , and let W : Y (cid:48) −→ Y (cid:48)(cid:48) be a cobordism corresponding to attaching a -handleto Y (cid:48) along some -sphere such that the -handle cancels the -handle from W . Then for any Ω ∈ H ( W ∪ W , ∂ ( W ∪ W ); F ) , SI( W ∪ W , Ω) is the identity. As a result of the work done thus far in this section, we conclude the following.
Theorem 4.9.
For any compact, connected, oriented cobordism W : Y −→ Y (cid:48) of closed, connected,oriented -manifolds Y and Y (cid:48) and any homology class Ω ∈ H ( W, ∂W ; F ) , there is a well-definedhomomorphism SI( W, Ω) : SI( Y, Ω | Y ) −→ SI( Y (cid:48) , Ω | Y (cid:48) ) that is an invariant of the pair ( W, Ω) . For our final effort of this section, we show that the assignment ( W, Ω) (cid:55)→ SI( W, Ω) is functorial, inthe following sense:
Theorem 4.10.
Suppose W : Y −→ Y (cid:48) and W (cid:48) : Y (cid:48) −→ Y (cid:48)(cid:48) are two compact, connected, orientedcobordisms of closed, connected, oriented -manifolds and fix homology classes Ω ∈ H ( W, ∂W ; F ) and Ω (cid:48) ∈ H ( W (cid:48) , ∂W (cid:48) ; F ) such that Ω | Y (cid:48) = Ω (cid:48) | Y (cid:48) . Then SI( W ∪ W (cid:48) , Ω ∪ Ω (cid:48) ) = SI( W (cid:48) , Ω (cid:48) ) ◦ SI( W, Ω) . Proof.
To prove the theorem, we must show that 1- and 2-handle maps commute with one another,and that 2- and 3-handle maps commute with one another. Once again, the proof for the Ω = 0case carries over directly, and we refer the reader to [Hor16, Theorem 8.18]. (cid:3) Exact Triangle for Surgery Triads
Now that we have extended symplectic instanton homology to take into account nontrivial SO(3)-bundles, we can properly state and prove the exact triangle for Dehn surgery on a knot.5.1. 3 -Manifold Triads and the Statement.
Suppose (cid:101) Y is a compact, oriented 3-manifold withtorus boundary. Given three oriented simple closed curves γ , γ , and γ in ∂ (cid:101) Y satisfying γ ∩ γ ) = γ ∩ γ ) = γ ∩ γ ) = − , we may form three closed, oriented 3-manifolds Y , Y , and Y by Dehn filling (cid:101) Y along γ , γ , and γ , respectively. If a triple of closed, oriented 3-manifolds ( Y, Y , Y ) arises in this way from some (cid:101) Y , we call ( Y, Y , Y ) a surgery triad .It is easy to see that ( Y, Y , Y ) is a surgery triad if and only if there is a framed knot K = ( K, λ )in Y such that Y = Y λ and Y = Y λ + µ , where µ is the meridian of K . Note that surgery triads arecyclic, if that if ( Y, Y , Y ) is a surgery triad, then so are ( Y , Y , Y ) and ( Y , Y, Y ). If we want toemphasize this particular framed knot K , we will say that ( Y, Y , Y ) is a surgery triad relative to ( K, λ ). If K = ( K, λ ) is a framed knot in a closed, oriented 3-manifold Y with meridian µ and p, q areintegers, we write Y p/q ( K ) = Y pµ + qλ ( K )for the result of removing νK from Y and gluing back in a solid torus such that the curve pµ + qλ in ∂ ( νK ) bounds a disk.If K is a nullhomologous knot in a closed, oriented 3-manifold Y , there is a unique framing λ Seifert for K that is nullhomologous in Y \ νK which is called the Seifert framing of K . In this case,we just write Y p/q ( K ) for Y p,q ( K, λ
Seifert ), where the lack of framing in the notation means we areusing the Seifert framing by default.We can form several interesting families of surgery triads:
Example 5.1.
Given a framed knot K in a closed, oriented 3-manifold Y and any integer n , both( Y, Y n ( K ) , Y n +1 ( K )) and ( Y ( K ) , Y / ( n +1) ( K ) , Y /n ( K )) are surgery triads. Example 5.2.
More generally, given a framed knot K in a closed, oriented 3-manifold Y andrelatively prime integers p , q , B´ezout’s identity allows us to find integers p , q satisfying p q − p q = 1. If we write p = p + p and q = q + q , then ( Y p /q ( K ) , Y p /q ( K ) , Y p /q ( K )) is asurgery triad. Example 5.3.
Let L be a link in S and fix a planar diagram D for L . Given a crossing in D ,we may resolve it in one of two ways (see Figure 4) to obtain links L and L differing from L only at the chosen crossing in the prescribed manner. Then the branched double covers of thesethree links, (Σ( L ) , Σ( L ) , Σ( L )) form a surgery triad. We will discuss surgery triads of this formin more detail in Section 8. LL L Figure 4.
0- and 1-resolutions of a crossing in a link.In general, in Floer theories for 3-manifolds one expects an exact triangle relating the Floer ho-mologies of any three 3-manifolds fitting into a surgery triad. For symplectic instanton homology,we will prove the following:
Theorem 5.4.
For any -manifold triad ( Y, Y , Y ) relative to a framed knot ( K, λ ) in Y and SO(3) -bundle P −→ Y , there is an exact triangle of symplectic instanton homology groups: SI(
Y, ω + ω K ) (cid:47) (cid:47) SI( Y , ω ) (cid:120) (cid:120) SI( Y , ω ) (cid:103) (cid:103) Here ω , ω , and ω are the mod homology classes Poincar´e dual to the bundles induced by P on Y , Y λ , and Y λ + µ : ω = PD( w ( P )) ∈ H ( Y ; F ) , ω λ = i ∗ PD( i ∗ Y \ νK w ( P )) ∈ H ( Y ; F ) ,ω = i ∗ PD( i ∗ Y \ νK w ( P )) ∈ H ( Y ; F ) . YMPLECTIC INSTANTON HOMOLOGY, SPECTRAL SEQUENCES, AND KHOVANOV HOMOLOGY 21
Seidel’s Exact Triangle and Quilted Floer Homology.
Although the surgery exacttriangle will essentially be an application of Seidel’s exact triangle for symplectic Dehn twists[Sei03, WW15, MW16] in the setting of quilted Floer homology for monotone Lagrangians, werecall some relevant ideas in the proof, as they will be needed later when we establish the linksurgeries spectral sequence in Section 7.In what follows, we fix: • Two monotone symplectic manifolds M and M (cid:48) with the same monotonicity constant. • A monotone Lagrangian submanifold L : pt −→ M (which we think of as a Lagrangiancorrespondence). • A monotone Lagrangian correspondence L : M −→ M (cid:48) . • Two monotone Lagrangian submanifolds L , V : M (cid:48) −→ pt, where V is topologically asphere.Our version of Seidel’s exact triangle will be constructed from a sequence of mapsCF( L , L, V ) ⊗ CF( V T , L ) C Φ −−−−→ CF( L , L, L ) C Φ −−−−→ CF( L , L, τ V L ) . On the chain level, the maps appearing in the exact sequence are defined as follows. The first map C Φ : CF( L , L, V ) ⊗ CF( V T , L ) −→ CF( L , L, L )is the relative invariant counting quilted pseudoholomorphic triangles as pictured in Figure 5. Thesecond map C Φ : CF( L , L, L ) −→ CF( L , L, τ V L )is the relative invariant counting pseudoholomorphic sections of the quilted Lefschetz fibration inFigure 6. M L VL MV T L Figure 5.
Quilts counted bythe map C Φ . M L L M τ V L τ V L Figure 6.
Quilted Lefschetzfibration defining C Φ .There is a distinguished Floer chain c ∈ CF( L , τ V L ) ,µ ( c ) = 0defined as follows. Let E −→ D be the standard Lefschetz fibration over the disk with regularfiber M (cid:48) and vanishing cycle V . By removing a point z c ∈ ∂ D , we get a Lefschetz fibration withstrip-like end E c −→ D c . Denote the strip-like end by (cid:15) z c : R + × [0 , −→ D c . We equip E c −→ D c with the following moving Lagrangian boundary conditions: For s (cid:29)
0, identify the fibers over (cid:15) z c ( s,
0) with L . As we leave the strip-like end and travel along ∂ D c , we carry L along by paralleltransport. Once we reach the other side of the strip-like end, our parallel-transported Lagrangianwill be isotopic to τ V L ; the boundary condition along (cid:15) z c ( s, s (cid:29) c : k : CF( V, L ) −→ CF(
V, τ V L ) ,µ ( k ( · )) + k ( µ ( · )) + µ ( · , c ) = 0 . M L V τ V τ V L τ V L τ V L V M Figure 7.
The 1-parameter family of Lefschetz fibrations defining the chain null-homotopy k .The construction of k is pictured schematically in Figure 7. The meaning of the left and rightsurfaces with strip-like ends should be evident; the dotted lines connecting boundary componentsindicate gluings of components using sufficiently long gluing lengths, so that the associated relativeinvariant of the glued fibrations corresponds to the composition of the individual relative invariants.In particular, the left-hand surface represents µ ( · , c ), and the right-hand surface represents atriangle glued with a Floer chain which is equal to zero (see [WW15, Corollary 4.31], for example).There is a clear 1-parameter family of Lefschetz fibrations E k,r −→ S k,r (0 ≤ r ≤
1) over the strip,equal to the left of Figure 7 for r = 0 and equal to the right of Figure 7 for r = 1, achieved bymoving the position of singular fiber. The nullhomotopy k is then defined by counting isolatedpoints of the parametrized moduli space of pseudoholomorphic sections of E k,r −→ S k,r .Finally, we may use k to construct the map h : CF( L , L, V ) ⊗ CF( V T , L ) −→ CF( L , L, τ V L ) ,h ( x ⊗ y ) = ˜ µ ( x, k ( y )) + ˜ µ ( x, y, c ) , where ˜ µ (resp. ˜ µ ) is the quilted triangle (resp. quilted rectangle) map pictured in Figure 8(resp. Figure 9). By a routine calculation, one may show that h defines a chain nullhomotopy of C Φ ◦ C Φ .The maps C Φ , C Φ , and h turn out to fit into a general homological algebraic framework thatallow one to establish the existence of an exact triangle. We recall some terminology necessary forthe statement of the next result. An R -graded Λ -module is an Λ-module M with a direct sumdecomposition M = (cid:77) s ∈ R M s . YMPLECTIC INSTANTON HOMOLOGY, SPECTRAL SEQUENCES, AND KHOVANOV HOMOLOGY 23 M τ V L VL MV T L Figure 8.
Quilted trianglemap ˜ µ . M VL ML L τ V L Figure 9.
Quilted rectanglemap ˜ µ .The support of M is the set of real numberssupp( M ) = { s ∈ R | M s (cid:54) = 0 } . Given an interval I ⊂ R , we say that M has gap I if s, t ∈ supp( M ) implies that | s − t | / ∈ I .Finally, a Λ-linear map f : M −→ M (cid:48) of R -graded R -modules has order I if f ( M s ) ⊂ (cid:77) t ∈ I M (cid:48) s + t for all s ∈ R . Lemma 5.5. (Double Mapping Cone Lemma [Per08, Lemma 5.4] [WW15, Lemma 5.9])
Let ε > and suppose that ( E , δ ) , ( E , δ ) , and ( E , δ ) are free, finitely generated chain complexes of R -graded Λ -modules. Suppose we have chain maps f : E −→ E and g : E −→ E and a chainnullhomotopy h : E −→ E of g ◦ f . Assume that this data satisfies the following conditions: (1) δ , δ , and δ all have order [2 ε, ∞ ) . (2) We have decompositions f = f + f , g = g + g , and h = h + h such that f has order [0 , ε ) , g and h have order , and f , g , and h all have order [2 ε, ∞ ) . (3) 0 −→ E f −−→ E g −−→ E −→ is a short exact sequence of abelian groups.Then ( h, g ) : Cone( f ) −→ E is a quasi-isomorphism. To relate this to our intended application, we need to explain how to give the Floer chain complexes R -gradings. Given a ring R (which will be Z or Z / R denote the ringΛ R = (cid:40) n (cid:88) k =1 r k q s k | r k ∈ R, s k ∈ R (cid:41) , where q is a formal variable and the ring multiplication is defined on monomials by( rq s )( r (cid:48) q s (cid:48) ) = ( rr (cid:48) ) q s + s (cid:48) . Λ R admits the obvious R -grading Λ R = (cid:77) s ∈ R { rq s | r ∈ R } . Given two simply connected, transversely intersecting monotone Lagrangian submanifolds L , L of a monotone symplectic manifold M , we define the Lagrangian Floer chain complex with Λ R coefficients to have the obvious chain groupsCF( L , L ; Λ R ) = (cid:77) x ∈ L ∩ L Λ R (cid:104) x (cid:105) but the modified differential defined on generators by ∂ Λ R x = (cid:88) y ∈ L ,L u ∈ M ( x,y ) o ( u ) q E ( u ) y, where o ( u ) is the local orientation of M ( x, y ) at u ( ± R = Z , 1 if using R = Z / E ( u ) is the energy of the pseudoholomorphic strip u . There is an obvious extension of the Floerchain complex with Λ R coefficients to the setting of quilted Floer homology, where the energy of apseudoholomorphic quilt is simply the sum of the energies of each of its components.The maps C Φ , C Φ , k , and h all have obvious extensions to Λ R coefficients by incorporatingthe energy of the quilted pseudoholomorphic maps/sections they count into their definitions. Onemay show that these modified maps satisfy the hypotheses of Lemma 5.5, and it follows that( h, C Φ ) : Cone( C Φ ) −→ CF( L , L, τ V L ; Λ R ) is a quasi-isomorphism. Since mapping conesnaturally fit into exact triangles, we obtain the desired exact triangle in Floer homology with Λ R coefficients.To obtain the desired result for R = Z or R = Z / L , L ; Λ R ) / ( q − −→ HF( L , L )is an isomorphism (cf. [WW15, Remark 4.3]). Therefore we conclude the following: Proposition 5.6.
Suppose L is a monotone Lagrangian submanifold of the monotone symplecticmanifold M , L and V are monotone Lagrangian submanifolds of the monotone symplectic manifold M (cid:48) with V a sphere, and that L is a monotone generalized Lagrangian correspondence L : M −→ M (cid:48) . Then there is an exact triangle of quilted Floer homology groups HF( L , L, V ) ⊗ HF( V T , L ) Φ (cid:47) (cid:47) HF( L , L, L ) Φ (cid:118) (cid:118) HF( L , L, τ V L ) (cid:106) (cid:106) The Surgery Exact Triangle.
We now use Seidel’s exact triangle, in the form of Proposition5.6, to establish the surgery exact triangle in symplectic instanton homology.Let Y be a closed, oriented 3-manifold. Given a framed knot K = ( K, λ ) in Y , we may choose aHeegaard triple (Σ g +1 , α , β , γ , z ) subordinate to a bouquet for K (cf. [Hor16, Section 8.3]). Wemay also choose a Heegaard triple (Σ g +1 , α , β , δ , z ) subordinate to ( K, λ + µ ). The quadruplediagram H = (Σ g +1 , α , β , γ , δ , z ) satisfies the following conditions: YMPLECTIC INSTANTON HOMOLOGY, SPECTRAL SEQUENCES, AND KHOVANOV HOMOLOGY 25 • β g +1 is a meridian for K . • γ g +1 represents the framing λ of K and δ g +1 represents the framing λ + µ of K . • (Σ g +1 , { α , . . . , α g +1 } , { β , . . . , β g } , z ) represents the complement of K in Y . • γ k = β k and δ k = β k for k (cid:54) = g + 1. • H αβ represents Y , H αγ represents Y λ , and H αδ represents Y λ + µ .In particular, H αβ , H αγ , and H αδ induce length 2 g + 2 Cerf decompositions of Y θ where we firstattach ( g + 1) 1-handles with attaching cycles in Σ g +1 given by α , and then we attach ( g + 1)2-handles with attaching cycles in Σ g +1 given by β , γ , or δ (depending on which diagram we areconsidering). The conditions on β , γ , and δ imply that these three Cerf decompositions differ onlyin the final 2-handle attachment.The final 2-handle attachments may be represented by Lagrangian submanifolds L β g +1 , L γ g +1 , and L δ g +1 of M , . Now note that we in fact have δ g +1 = τ β g +1 γ g +1 , so that if V = { [ ρ ] ∈ M , | ρ ( β g +1 ) = − I } , then on the level of Lagrangians L δ g +1 = τ V L γ g +1 . Writing L for the Lagrangian correspondence coming from the first g β - (equivalently, γ - or δ -)handle attachments, Seidel’s exact triangle readsHF( L α , L, V ) ⊗ HF( V T , L γ g +1 ) Φ (cid:47) (cid:47) HF( L α , L, τ V L γ g +1 ) Φ (cid:117) (cid:117) HF( L α , L, L γ g +1 ) (cid:106) (cid:106) Two of these Floer homology groups may be immediately identified:HF( L α , L, L γ g +1 ) = SI( Y λ ) , HF( L α , L, τ V L γ g +1 ) = SI( Y λ + µ ) . The remaining Floer homology group is identified thusly:HF( L α , L, V ) ⊗ HF( V T , L γ g +1 ) = SI( Y, ω K ) ⊗ Z = SI( Y, ω K ) , where HF( L α , L, V ) = SI( Y, ω K ) since β g +1 is a meridian of K and V precisely represents pairs offlat connections in M − , × M , which simultaneously extend to flat connections on the nontriv-ial SO(3) bundle over the bordism Σ , −→ Σ , whose second Stiefel-Whitney class is Poincar´edual to K , and HF( V T , L γ g +1 ) ∼ = Z because the only intersection point of the two Lagrangians is[ − I, ± I, i , j , − k ] ∈ M , (where the ± is determined by the framing λ ).It follows that we have a surgery exact triangleSI( Y, ω K ) (cid:47) (cid:47) SI( Y λ ) (cid:121) (cid:121) SI( Y λ + µ ) (cid:102) (cid:102) In fact, given a nontrivial SO(3)-bundle P −→ Y , we get three induced homology classes Poincar´edual to the induced bundles on Y , Y λ , and Y λ + µ : ω = PD( w ( P )) ∈ H ( Y ; F ) , ω λ = i λ ∗ PD( i ∗ Y \ νK w ( P )) ∈ H ( Y λ ; F ) ,ω λ + µ = i λ + µ ∗ PD( i ∗ Y \ νK w ( P )) ∈ H ( Y λ + µ ; F ) . These may all be represented by embedded curves disjoint from νK (indeed, they can be repre-sented by curves lying entirely in the α -handlebody), so in the above setup we may incorporate nontrivial bundles into the generalized Lagrangian correspondence L without any modification tothe argument. Hence we have more generally an exact triangleSI( Y, ω + ω K ) (cid:47) (cid:47) SI( Y λ , ω λ ) (cid:118) (cid:118) SI( Y λ + µ , ω λ + µ ) (cid:104) (cid:104) Remark 5.7.
Recall that symplectic instanton homology SI(
Y, ω ) = HF( L ωα , L β ) admits a Z / L ωα and L β intersect transversely, andfix orientations of L ωα , L β , and M g, , so that for any x ∈ L ωα ∩ L β , the local intersection number( L ωα · L β ) x ∈ {± } is well-defined. Considered as a generator of CF( L ωα , L β ), the Z / x is 0 if ( L ωα · L β ) x = +1 and 1 otherwise. The maps in Seidel’s exact triangle have well-definedgradings; in our context the gradings are as indicated below:SI( Y, ω + ω K ) [0] (cid:47) (cid:47) SI( Y λ , ω λ ) [0] (cid:118) (cid:118) SI( Y λ + µ , ω λ + µ ) [1] (cid:104) (cid:104) From Pseudoholomorphic Sections to Pseudoholomorphic Polygons.
It remains toidentify the maps in the triangle with the homomorphisms induced by the relevant 2-handle attach-ments. Since one of the maps involves counting pseudoholomorphic sections of a Lefschetz fibration,we will need to show that this is equal to the count of pseudoholomorphic triangles defining ourcobordism maps. Furthermore, it will also be useful for us later to interpret the chain homotopy h appearing in the proof of the exact triangle as a count of certain pseudoholomorphic rectangles.We will continue considering the setup from the previous subsection that was induced by theHeegaard quadruple H = (Σ g +1 , α , β , γ , δ , z ), although some of the results here apply in greatergenerality. L T γ g +1 L δ g +1 τ β g +1 L T γ L δ τ β L T LE E Figure 10.
Two quilted Lefschetz-Bott fibrations.All results in this subsection rely on the following technical lemma:
Lemma 5.8.
Let E and E (cid:48) be the two quilted Lefschetz-Bott fibrations pictured in Figure 10.Then the relative invariant Φ E is identified with the relative invariant Φ E (cid:48) under the isomorphism HF( L Tγ g +1 , L T , L, L δ g +1 ) ∼ = HF( L Tγ g +1 ◦ L T , L ◦ L δ g +1 ) = HF( L Tγ , L δ ) . In other words, the following YMPLECTIC INSTANTON HOMOLOGY, SPECTRAL SEQUENCES, AND KHOVANOV HOMOLOGY 27 diagram commutes:
HF( L Tγ g +1 , L T , L, L δ g +1 ) geometriccomposition (cid:15) (cid:15) HF(pt) Φ E (cid:54) (cid:54) Φ E (cid:48) (cid:40) (cid:40) HF( L Tγ , L δ ) Proof.
First, we determine Φ E (cid:48) (pt). Equip E (cid:48) with a horizontal almost complex structure. Of allpossible horizontal sections of E (cid:48) , only the one corresponding to Θ γδ has index zero. By monotonic-ity, any non-horizontal section of E (cid:48) will have strictly positive index and hence does not contributeto Φ E (cid:48) (pt). Therefore Φ E (cid:48) (pt) = Θ γδ . (5.1) L T γ g +1 L δ g +1 τ β g +1 L T τ β g +1 L T γ g +1 L δ g +1 L LL T E E E Figure 11.
Breaking the quilted Lefschetz fibration S into simpler pieces.To determine Φ E (pt), we consider E as a gluing of two quilted Lefschetz fibrations E and E as indicated in Figure 11. For sufficiently long gluing lengths, C Φ E (pt) = ( C Φ E ◦ C Φ E )(pt)on the chain level, so this decomposition allows us to count pseudoholomorphic sections on eachpiece of the gluing separately. By reasoning analogous to the computation of Φ E (cid:48) (pt), we have thatΦ E (pt) = θ ∈ HF( L Tγ g +1 , L δ g +1 ). On the other hand, by the discussion in Section 3.7, Φ E ( θ ) = θ × Θ γδ ∈ HF( L Tγ g +1 , L T , L, L δ g +1 ). ThereforeΦ E (pt) = θ × Θ γδ . (5.2)To complete the proof, simply note that under geometric composition, θ × Θ γδ maps to Θ γδ . Hencethe desired triangle commutes. (cid:3) With the above lemma in place, we can begin interpreting the chain maps C Φ and C Φ in thesurgery exact triangle as counts of pseudoholomorphic triangles with certain specified vertices, andwe can also show that the chain homotopy h can be considered as a count of pseudoholomorphicrectangles with certain specified vertices. Let us start with C Φ . A basic manipulation of quilted M ; L γ g +1 L M g +1 ; L δ g +1 τ β g +1 M ; L γ g +1 L L δ g +1 τ β L α M g +1 ; M ; L α M g +1 ; M g +1 ; L γ L δ τ β L α ∼ = geometriccomposition τ β g +1 Figure 12.
Identifying C Φ with a classical triangle map.surfaces (see Figure 12) shows that there is a commutative diagramCF( L α , L, L γ g +1 ) ∼ = (cid:15) (cid:15) C Φ (cid:47) (cid:47) CF( L α , L, L δ g +1 ) ∼ = (cid:15) (cid:15) CF( L α , L γ ) µ ( · ,c (cid:48) ) (cid:47) (cid:47) CF( L α , L δ )where the vertical maps are induced by the geometric compositions L ◦ L γ g +1 = L γ and L ◦ L δ g +1 = L δ . Now c (cid:48) ∈ CF( L γ , L δ ) is the unit for the algebra CF( L γ , L δ ) ∼ = H −∗ (pt; Z ) ⊗ H −∗ ( S ; Z ) ⊗ g , andis hence identified with Θ γδ . Furthermore, µ ( · , Θ γδ ) is precisely the map induced by the standard2-handle cobordism from Y λ ( K ) to Y λ + µ ( K ). We conclude the following: Proposition 5.9.
The map C Φ in the surgery exact triangle may be identified with the cobordismmap F W λ + µ : CF( L α , L γ ) −→ CF( L α , L δ ) , where W λ + µ is the standard -handle cobordism between Y λ ( K ) and Y λ + µ ( K ) . In other words, there is a commutative diagram CF( L α , L, L γ g +1 ) ∼ = (cid:15) (cid:15) C Φ (cid:47) (cid:47) CF( L α , L, L δ g +1 ) ∼ = (cid:15) (cid:15) CF( L α , L γ ) F Wλ + µ (cid:47) (cid:47) CF( L α , L δ ) where the vertical maps are induced by geometric composition. YMPLECTIC INSTANTON HOMOLOGY, SPECTRAL SEQUENCES, AND KHOVANOV HOMOLOGY 29 M ; VL M g +1 ; L γ g +1 M ; VLL α M g +1 ; M ; L γ g +1 M g +1 ; M g +1 ; L ! K β L γ ∼ = geometriccomposition L α L α [ ρ ] ρ ρ × Θ ρ Θ ρ Figure 13.
Identifying C Φ with a classical triangle map.Now C Φ is already obviously a (quilted) triangle map, but a priori it is not of the same formas one of the classical triangle maps associated to a 2-handle cobordism and it also seems tolack a fixed vertex. Note that the upper left corner of the triangle in Figure 5 must be mappedto an intersection point of L γ g +1 and V T in M , . But these Lagrangians intersect only at therepresentation [ ρ ] = [ − I, ± I, i , j , − k ] (where the ± depends on the framing λ of the knot K ⊂ Y ).Therefore the given vertex is forced to be fixed.We still wish to perform some quilt manipulations similar to those done for C Φ to pass from aquilted triangle to just a triangle mapping to M g +1 , . The required manipulations are pictured inFigure 13. The first indicated move sufficiently stretches the quilt so that we may consider it as acomposition of the two pictured quilt maps in the upper right corner of the figure. The first quiltin the composition is the map ρ (cid:55)→ ρ × Θ ρ discussed at the end of Section 3.7. The input for theupper left corner of the second quilt in the composition is forced to be ρ × Θ ρ since V T ∩ L γ g +1 = { [ ρ ] } . Finally, under the geometric composition CF( V T , L T , L, L γ g +1 ) ∼ = CF( L ω K β , L γ ), ρ × Θ ρ ismapped to Θ ρ . Therefore we can conclude the following: Proposition 5.10.
The map C Φ in the surgery exact triangle may be identified with the cobordismmap F W λ : CF( L α , L ω K β ) −→ CF( L α , L γ ) , where W λ is the standard -handle cobordism between Y and Y λ ( K ) . In other words, there is a commutative diagram CF( L α , L, V ) ⊗ CF( V T , L γ g +1 ) ∼ = (cid:15) (cid:15) C Φ (cid:47) (cid:47) CF( L α , L, L γ g +1 ) ∼ = (cid:15) (cid:15) CF( L α , L ω K β ) F Wλ (cid:47) (cid:47) CF( L α , L γ ) where the vertical maps are induced by geometric composition. Recall that the map h : CF( L α , L, V ) ⊗ CF( V T , L γ g +1 ) −→ CF( L α , L, L δ g +1 ) is defined by h ( x, y ) = ˜ µ ( x, k ( y )) + ˜ µ ( x, y, c ) , where c ∈ CF( L γ g +1 , L δ g +1 ) is the Floer chain defined by the standard Lefschetz fibration over thedisk with monodromy τ β g +1 (see Section 5.2), and k : CF( V T , L γ g +1 ) −→ CF( L β g +1 , L δ g +1 ) is thechain nullhomotopy of µ ( · , c ) described in Figure 7 above.The first thing to notice is that the first term, ˜ µ ( x, k ( y )) is identically zero in this context. Thisis because k is a map of degree 1, but the Z / V T , L γ g +1 ) ∼ = Z (cid:104) ρ (cid:105) andCF( L γ g +1 , L δ g +1 ) ∼ = Z (cid:104) c (cid:105) are both supported only in degree 1 mod 2. Hence we simply have h ( x, y ) = ˜ µ ( x, y, c ) . The fact that CF( V T , L γ g +1 ) ∼ = Z (cid:104) ρ (cid:105) also forces h to depend only on x ∈ CF( L α , L ω K β ), so that h ( x ) = ˜ µ ( x, ρ , c ) . At this point, showing that h corresponds to a classical rectangle map follows by a combination ofthe proofs that C Φ and C Φ correspond to classical triangle maps, as suggested in Figure 14. Proposition 5.11.
The map h used in the proof of the surgery exact triangle may be identifiedwith the rectangle map µ ( · , Θ βγ , Θ γδ ) : CF( L α , L ω K β ) −→ CF( L α , L δ ) . In other words, there is acommutative diagram CF( L α , L, V ) ∼ = (cid:15) (cid:15) h (cid:47) (cid:47) CF( L α , L, L δ g +1 ) ∼ = (cid:15) (cid:15) CF( L α , L ω K β ) µ ( · , Θ βγ , Θ γδ ) (cid:47) (cid:47) CF( L α , L δ ) where the vertical maps are induced by geometric composition. It remains to identify the connecting map SI( Y λ + µ ( K )) −→ SI(
Y, ω K ) in the surgery exact triangleas being induced by a triangle map. Since the construction of this map was indirect (by homologicalalgebra methods), instead of trying explicitly determine this map, we will show that it can be replaced by a triangle map while still preserving exactness of the surgery triangle. Nevertheless, westill conjecture that the map (before replacement) can be shown to be a triangle map.The argument relies on basic Kirby calculus and the cyclic symmetry of surgery triads. Figure 15depicts the framed knots that induce the standard 2-handle cobordisms between manifolds in thesurgery triad ( Y, Y λ ( K ) , Y λ + µ ( K )), i.e. attaching a 2-handle to Y along K with framing λ gives Y λ ( K ), attaching a 2-handle to Y λ ( K ) along N with framing − Y λ + µ ( K ), and attaching a2-handle to Y λ + µ along C with framing − Y . Here we think of N as the boundary of anormal disk to K (using λ to think of the normal bundle to K as K × D ), given a framing of − C is a small meridian of N which is also given a framing of − N . YMPLECTIC INSTANTON HOMOLOGY, SPECTRAL SEQUENCES, AND KHOVANOV HOMOLOGY 31 M ; VL M g +1 ; L γ g +1 τ β g +1 M ; VLL α M g +1 ; M ; L γ g +1 L α M g +1 ; ∼ = geometriccomposition L δ g +1 L δ g +1 M g +1 ; L ! K β L α L γ τ β L δ M g +1 ; ρ ρ × Θ ρ [ ρ ] Θ ρ τ β g +1 Figure 14.
Identifying h with a classical rectangle map.Since ( Y, Y λ ( K ) , Y λ + µ ( K )) is a surgery triad, so are ( Y λ ( K ) , Y λ + µ ( K ) , Y ) and ( Y λ + µ ( K ) , Y, Y λ ( K )).Hence we have two exact sequencesSI( Y λ ( K ) , ω N ∪ ω N ∪ ω K ) SI( W λ , Ω ) −−−−−−−−→ SI( Y λ + µ ( K ) , ω N ∪ ω K ) SI( W λ + µ , Ω ) −−−−−−−−−→ SI(
Y, ω N ∪ ω K ) , SI( Y λ + µ ( K ) , ω C ∪ ω C ∪ ω K ) SI( W λ + µ , Ω ) −−−−−−−−−→ SI(
Y, ω C ∪ ω K ) SI( W, Ω ) −−−−−−−→ SI( Y µ ( K ) , ω C ∪ ω K ) , where we use the nontrivial bundle ω N ∪ ω K (resp. ω C ∪ ω K ) throughout in the first (resp. second)exact sequence. Note that ω N ∪ ω N ∪ ω K = ω C = 0 ∈ H ( Y λ ( K ); F ), ω N ∪ ω K = ω C ∪ ω C ∪ ω K =0 ∈ H ( Y λ + µ ( K ); F ), and ω N = ω C = 0 ∈ H ( Y ; F ), so that the two exact sequences above canbe simplified to SI( Y λ ( K )) SI( W λ ) −−−−−→ SI( Y λ + µ ( K )) SI( W λ + µ , Ω (cid:48) K ) −−−−−−−−−−→ SI(
Y, ω K )(5.3) Figure 15.
Framed knots inducing cobordisms in a surgery triad.SI( Y λ + µ ( K )) SI( W λ + µ , Ω (cid:48) K ) −−−−−−−−−−→ SI(
Y, ω K ) SI( W, Ω K ) −−−−−−−→ SI( Y µ ( K ))(5.4)Exactness of sequences (5.3) and (5.4) imply thatker(SI( W λ + µ , Ω (cid:48) K )) = im(SI( W λ )) , im(SI( W λ + µ , Ω (cid:48) K )) = ker(SI( W, Ω K )) , and therefore the triangle SI( Y, ω K ) SI( W, Ω K ) (cid:47) (cid:47) SI( Y µ ( K )) SI( W λ ) (cid:119) (cid:119) SI( Y λ + µ ( K )) SI( W λ + µ , Ω (cid:48) K ) (cid:103) (cid:103) is exact. 6. Instanton L -Spaces Computation of χ (SI( Y, ω )) . Recall that Proposition 3.12 claims the Euler characteristic ofsymplectic instanton homology is given by χ (SI( Y, ω )) = (cid:40) | H ( Y ; Z ) | , if b ( Y ) = 0 , , otherwise,Now that we have established the surgery exact triangle, we will use it to prove this statement. Proof of Proposition 3.12.
Scaduto’s computation of the Euler characteristic of framed instantonhomology in [Sca15, Corollary 1.4] essentially applies word-for-word here; we reproduce his argu-ment. The proof proceeds by considering cases of increasing generality.
Case 1.
First, suppose that Y is an integral homology 3-sphere. Then H ( Y ; F ) = 0, so that wenecessarily have ω = 0 and χ (SI( Y )) = 1 by [Hor16, Theorem 7.1]. Case 2.
Next, suppose that Y is a rational homology 3-sphere obtained by integral surgery onan algebraically split link L = L ∪ · · · ∪ L m . Hence Y = S p ,...,p m ) ( L ) for some integers p k ∈ Z , k = 1 , . . . , m and furthermore | H ( Y ; Z ) | = | p · · · p m | since L is algebraically split. Suppose theEuler characteristic formula holds for rational homology spheres M which are obtained by integralsurgery on an algebraically split link and satisfy | H ( M ; Z ) | < | p · · · p m | . As Case 1 establishes the YMPLECTIC INSTANTON HOMOLOGY, SPECTRAL SEQUENCES, AND KHOVANOV HOMOLOGY 33 base case for this induction, we may assume Y is not an integral homology 3-sphere and thereforewithout loss of generality p >
1. Note that( S p − ,p ,...,p m ) ( L ) , S p ,p ,...,p m ) ( L ) , S ∞ ,p ,...,p m ) ( L ))is a surgery triad (cf. Example 5.1), and therefore we get a surgery exact triangleSI( S p − ,p ,...,p m ) ( L ) , ω ∪ ω L ) [0] (cid:47) (cid:47) SI( S p ,p ,...,p m ) ( L ) , ω ) [0] (cid:116) (cid:116) SI( S ∞ ,p ,...,p m ) ( L ) , ω ) [1] (cid:106) (cid:106) where [ · ] denotes the mod 2 degree of each map and ω ∈ H ( Y ; F ) is an arbitrary homology classin Y , which induces homology classes in the other two spaces. By exactness and the degree of themaps, we have that χ (SI( Y, ω )) = χ (SI( S p − ,p ,...,p m ) ( L ) , ω ∪ ω L )) + χ (SI( S ∞ ,p ,...,p m ) ( L ) , ω ))= | ( p − p · · · p m | + | p · · · p m | = | p · · · p m | , where in the second line we used the inductive hypothesis. Case 2 therefore is true by induction. Case 3.
Now suppose Y is any rational homology 3-sphere. By [Oht96, Corollary 2.5], there existsan algebraically split link L ⊂ S such that there is an integral framing ( p , . . . , p m ) with S p ,...,p m ) ( L ) ∼ = Y L ( n , · · · L ( n k , n , . . . , n k . Note that both S p ,...,p m ) ( L ) and L ( n , · · · L ( n k ,
1) fallunder Case 2, so that the K¨unneth principle for connected sums implies that χ (SI( Y, ω )) = χ (SI( S p ,...,p m ) ( L ) , ω ∪ ω ∪ · · · ∪ ω k )) χ (SI( L ( n , · · · L ( n k , , ω ∪ · · · ∪ ω k ))= | H ( S p ,...,p m ) ( L ); Z ) || H ( L ( n , · · · L ( n k , Z ) | = | H ( Y ; Z ) | . Therefore the Euler characteristic formula holds for all rational homology 3-spheres.
Case b ( Y ) > . Now suppose the 3-manifold Y is such that b ( Y ) >
0. Then we wish to showthat χ (SI( Y, ω )) = 0 for all ω ∈ H ( Y ; F ). Note that in this case, there is a 3-manifold M with b ( M ) = b ( Y ) − K, λ ) in M such that Y ∼ = M λ ( K ). Since ( M λ + µ ( K ) , M, Y )is a surgery triad, for any ω ∈ H ( Y ; F ) we have a surgery exact triangleSI( M λ + µ ( K ) , ω + ω K ) [0] (cid:47) (cid:47) SI(
M, ω ) [0] (cid:121) (cid:121) SI(
Y, ω ) [1] (cid:104) (cid:104) where again [ · ] indicates the mod 2 degree of the map. Then it follows that χ (SI( Y, ω )) = χ (SI( M λ + µ ( K ) , ω + ω K )) − χ (SI( M, ω )) . Applying induction and using this equation will allow us to make our conclusion. For the basecase, if b ( Y ) = 1, then M and M λ + µ ( K ) are rational homology spheres with | H ( M ; Z ) | = | H ( M λ + µ ( K ); Z ) | , and therefore Case 3 implies that χ (SI( Y, ω )) = 0. Now suppose that the desired statement is true for all 3-manifolds with first Betti number less than n . Then if b ( Y ) = n ,the inductive hypothesis implies that χ (SI( M λ + µ ( K ) , ω + ω K )) = χ (SI( M, ω )) = 0 and the resultfollows. (cid:3)
Instanton L -Spaces: Definition and Examples. Proposition 3.12 suggests that for arational homology sphere Y , SI( Y, ω ) is as “simple” as possible ( i.e. minimal rank) if it is freeabelian of rank | H ( Y ; Z ) | . Note that by Proposition 3.14, lens spaces L ( p, q ) satisfy this propertyfor any ω ∈ H ( L ( p, q ); F ), since SI( L ( p, q ) , ω ) ∼ = Z p . In analogy with Heegaard Floer theory, wemake the following definition. Definition 6.1. An instanton L -space is a rational homology 3-sphere Y such that SI( Y, ω ) isfree abelian of rank | H ( Y ; Z ) | for all ω ∈ H ( Y ; F ).The reason we require the property to hold for all ω ∈ H ( Y ; F ) is because families of instanton L -spaces should be generated from certain surgery triads via the exact triangle, and in generalthere is an unavoidable twisting along the core of the surgery appearing in one of the terms of theexact triangle.There is a useful “two out of three” principle for surgery triads containing instanton L -spaces. Proposition 6.2.
Suppose ( Y, Y , Y ) is a surgery triad such that | H ( Y ; Z ) | = | H ( Y ; Z ) | + | H ( Y ; Z ) | (this condition may always be achieved by cyclically permuting the elements of the triad, which givesanother triad). If Y and Y are instanton L -spaces, then so is Y .Proof. Fix some class ω ∈ H ( Y ; F ) and let ω , ω denote the corresponding classes in Y and Y .We first claim that the map SI( Y , ω ) −→ SI( Y , ω ) in the exact triangle for ( Y, Y , Y ) must bethe zero map. To see this, assume for the sake of contradiction that the map is nonzero. Thenexactness and the rank-nullity theorem imply the strict inequalityrk SI( Y, ω + ω K ) < rk SI( Y , ω ) + rk SI( Y , ω ) = χ (SI( Y, ω + ω K )) , which is not possible. Therefore in the situation of the Proposition, we in fact have a short exactsequence 0 −→ SI( Y , ω ) −→ SI(
Y, ω + ω K ) −→ SI( Y , ω ) −→ , from which it follows that SI( Y, ω + ω K ) is free abelian of rank | H ( Y ; Z ) | + | H ( Y ; Z ) | = | H ( Y ; Z ) | .By varying ω ∈ H ( Y ; F ), we conclude that Y is an instanton L -space. (cid:3) Using Proposition 6.2, we can generate many families of examples:
Example 6.3. (Large Surgeries) Suppose K is a knot in S and n > S n ( K ) is an instanton L -space. Then for any integer m > n , S m ( K ) is also an instanton L -space,as can be seen by repeatedly applying the surgery exact triangles for surgery triads of the form( S , S m − ( K ) , S m ( K )).A concrete example of this is given by taking K to be the torus knot T p,q and n = pq −
1. Then S pq − ( T p,q ) ∼ = L ( p, q ), and hence S s ( T p,q ) is an instanton L -space for all m ≥ pq − P ( − , , L (18 , S ( P ( − , , ∼ = L (19 ,
7) and S n ( P ( − , , n >
19. In any case, we see that S n ( P ( − , , L -space for all n ≥
18, and this gives an infinite family of hyperbolic examples.
YMPLECTIC INSTANTON HOMOLOGY, SPECTRAL SEQUENCES, AND KHOVANOV HOMOLOGY 35
Example 6.4. (Plumbing Graphs) By a weighted graph we mean a pair (
G, m ) where G is agraph (possibly with multiple connected components) and m (which stands for “multiplicity”) is aninteger-valued function on the vertices of G . We can form a 4-manifold with boundary, W ( G, m ),by associating to each vertex v of G the disk bundle over S with Euler number m ( v ) and plumbingtogether bundles whose vertices are connected by an edge. Let Y ( G, m ) denote the boundary of W ( G, m ). For a vertex v , let d ( v ) (the degree of v ) denote the number of edges of G containing v .We claim that if ( G, m ) satisfies(1) G is a disjoint union of trees;(2) d ( v ) ≤ m ( v ) with the inequality being strict for at least one vertex v ;then Y ( G, m ) is an instanton L -space. To see this, we use induction on the number of vertices andsubinduction on the multiplicity of the new vertex added in the inductive step. First, if G = { v } with m ( v ) ≥ Y ( G, m ) is the lens space L ( m ( v ) , L -space. Now suppose the claim holds for all disjoint unions oftrees with at most n vertices, and suppose ( G, m ) is a disjoint union of trees with n + 1 vertices.Choose a leaf v ∗ of G , i.e. a vertex with d ( v ∗ ) = 1; we now subinduct on m ( v ∗ ). If m ( v ∗ ) = 1,then Y ( G, m ) = Y ( G (cid:48) , m (cid:48) ) where G (cid:48) = G \ { v ∗ } and m (cid:48) agrees with m except on the vertex v (cid:48) thatis connected to v ∗ in G ; we set m (cid:48) ( v (cid:48) ) = m ( v (cid:48) ) −
1. ( G (cid:48) , m (cid:48) ) satisfies our hypotheses and has n vertices, so by the inductive hypothesis Y ( G, m ) = Y ( G (cid:48) , m (cid:48) ) is an instanton L -space.Now, for the subinductive hypothesis, suppose the claim holds for all ( G, m ) with up to n +1 verticeswhere all leaves of G have multiplicity at most k . For the subinductive step, let ( G, m ) be a weightedgraph with n + 1 vertices, and suppose v ∗ is a leaf of G with m ( v ∗ ) = k + 1 and all other leaves havemultiplicity at most k . We can form two weighted graphs from ( G, m ): ( G , m ), with G = G \{ v ∗ } and m ( v ) agreeing with m ( v ) for v (cid:54) = v ∗ ; and ( G , m ), with G = G and m ( v ∗ ) = k , but with m otherwise agreeing with m . It is clear that ( Y ( G, m ) , Y ( G , m ) , Y ( G , m )) is a surgery triadand that | H ( Y ( G, m ); Z ) | = | H ( Y ( G , m ); Z ) | + | H ( Y ( G , m ); Z ) | . Since Y ( G , m ) and Y ( G , m ) are instanton L -spaces by the subinductive hypothesis, Proposition6.2 implies that Y ( G, m ) is as well.
Example 6.5. (Branched Double Covers of Quasi-Alternating Links) The set of quasi-alternatinglinks Q is the smallest set of links satisfying the following properties:(1) The unknot is in Q .(2) If L is any link admitting some diagram with a crossing such that the associated 0- and1-resolutions L and L (see Figure 4) satisfy – L and L are in Q , – det( L ) , det( L ) (cid:54) = 0, – det( L ) = det( L ) + det( L ),then L is also in Q . Quasi-alternating links were introduced and studied by Ozsv´ath and Szab´o inthe context of Heegaard Floer theory [OS05]. They showed that all alternating knots are quasi-alternating, but gave an example of a quasi-alternating knot that is not alternating.To see that branched double covers of quasi-alternating links are instanton L -spaces, we proceedby induction on the crossing number of the link. The base case certainly holds, as Σ( U ) = S is aninstanton L -space. Now suppose all branched double covers of quasi-alternating links with crossingnumber at most n are instanton L -spaces, and suppose L is a quasi-alternating link with crossingnumber n + 1. Choose a diagram for L with a distinguished crossing as in item (2) above, so thatin particular the 0- and 1-resolutions are quasi-alternating and satisfy det( L ) = det( L ) + det( L ). Since L and L are quasi-alternating with crossing number at most n , the inductive hypothesisimplies that Σ( L ) and Σ( L ) are instanton L -spaces. Then Proposition 6.2 implies that Σ( L ) isalso an instanton L -space. Therefore the branched double cover of any quasi-alternating link is aninstanton L -space. Example 6.6. (Connected Sums) If Y and Y (cid:48) are instanton L -spaces, then so is their con-nected sum Y Y (cid:48) . This is because | H ( Y Y (cid:48) ; Z ) | = | H ( Y ; Z ) | · | H ( Y (cid:48) ; Z ) | and by the K¨unnethprinciple for symplectic instanton homology of connected sums, SI( Y Y (cid:48) ) ∼ = SI( Y ) ⊗ SI( Y (cid:48) ) ∼ = Z | H ( Y ; Z ) |·| H ( Y (cid:48) ; Z ) | . 7. A Spectral Sequence for Link Surgeries
We now show how to generalize the exact triangle for Dehn surgery on a knot to a spectral se-quence for Dehn surgeries on a link. Our approach is heavily inspired by proofs of similar spectralsequences for Heegaard Floer homology [OS05], singular instanton homology [KM11a], monopoleFloer homology [Blo11], and framed instanton homology [Sca15].7.1.
The Link Surgeries Complex.
Let Y be a closed, oriented 3-manifold and L = ( L ∪ · · · ∪ L m , λ ) be an oriented link with an enumeration of its components. An element v = ( v , . . . , v m ) ∈{ , , ∞} m will be called a multi-framing of L . Y v will denote the 3-manifold obtained from Y by performing v k -surgery on L k for eack k = 1 , . . . , m (with respect to the framing λ ).Recall that for each multi-framing v of L , we can construct a genus g + m Heegaard triple H v =(Σ g + m , α , β , η ( v ) , z ) satisfying the following conditions: • Attaching handles to { α , . . . , α g + m } and { β , . . . , β g } gives the complement of L in Y . • β g + k is a meridian for L k . • η ( v ) g + k represents the v k -framing of L k with respect to the given framing λ . • η ( v ) k = β k for all k = 1 , . . . , g . • (Σ g + m , α , η ( v ) , z ) is a Heegaard diagram for Y v .The same choice of curves α and β can be made for all v ∈ { , , ∞} m , so that the various triplediagrams H v differ only in the framing curves η ( v ). For any two multi-framings v, w , the Heegaarddiagram { Σ g + m , η ( v ) , η ( w ) , z } represents a connected sum of copies of S × S (the number n vw ofwhich depends on how many components v and w differ in), so that we have as usual a distinguishedelement of maximal degree Θ vw ∈ HF( L η ( v ) , L η ( w ) ) ∼ = H −∗ ( S ) ⊗ n vw which acts as a unit for thetriangle product.For notational convenience, we will conflate the indices ∞ and − {− , , } m , with the understanding that to obtain Y v , we perform ∞ -surgery on L k if v k = −
1. With this change in notation, it now makes sense to define the weight of v by the formula | v | = m (cid:88) k =1 | v k | . In other words, | v | is the number of entries of v not equal to 0. Note that {− , , } m is a latticewith the natural ordering given by v ≤ w ⇐⇒ v k ≤ w k for all k = 1 , . . . , m. When | w − v | = 1 and v ≤ w , we say that w is an immediate successor of v . YMPLECTIC INSTANTON HOMOLOGY, SPECTRAL SEQUENCES, AND KHOVANOV HOMOLOGY 37 If w is an immediate successor of v , then v and w differ in just one of the components’ framings, and Y w is obtained from Y v by attaching a single 2-handle. In this case, write ∂ vw : CSI( Y v ) −→ CSI( Y w )for the map induced by this 2-handle attachment. More generally, when | w − v | = k + 1 and v ≤ w ,given a sequence v < u < · · · < u k < w of immediate successors, define ∂ v w ,define ∂ vw ≡ (cid:101) C ( Y, L ) = (cid:77) v ∈{− , , } m CSI( Y v ) , ∂ = (cid:88) v ≤ w ∂ vw . Proposition 7.1. ∂ ≡ , so that ( (cid:101) C ( Y, L ) , ∂ ) is indeed a chain complex.Proof. The idea is to show that for any ξ ∈ (cid:101) C ( Y, L ), the quantity ∂ ξ is precisely a count ofdegenerations of pseudoholomorphic polygons which coincides with an A ∞ associativity relationthat is known by general theory to be zero. The relevant A ∞ relation here is the A k relation, (cid:88) ≤ i 2: For the rest of cases we proceed by induction. Note that for any path v < u < · · · < u k < w through immediate successors, there will always be a torus connect summand of Σ g + m where η ( v ) = η ( u ) = · · · = η ( u k ) and η ( w ) meets these curves transversely in a single point. We willshow that the polygon count from this particular summand is zero, which will force the overallpolygon count to be zero by the K¨unneth principle.When k = 2, let v < u < u < w be a path from v to w through immediate successors. By addingan auxiliary Lagrangian L η ( u − ) which is a small Hamiltonian isotope of L η ( v ) , we see that µ (Θ η ( v ) η ( u ) ⊗ Θ η ( u ) η ( u ) ⊗ Θ η ( u ) η ( w ) ) = µ ( µ (Θ η ( u − ) η ( v ) ⊗ Θ η ( v ) η ( u ) ) ⊗ Θ η ( u ) η ( u ) ⊗ Θ η ( u ) η ( w ) ) . On the other hand, the A ∞ relations tell us that µ ( µ (Θ η ( u − ) η ( v ) ⊗ Θ η ( v ) η ( u ) ) ⊗ Θ η ( u ) η ( u ) ⊗ Θ η ( u ) η ( w ) )= µ (Θ η ( u − ) η ( v ) ⊗ µ (Θ η ( v ) η ( u ) ⊗ Θ η ( u ) η ( u ) ) ⊗ Θ η ( u ) η ( w ) )+ µ (Θ η ( u − ) η ( v ) ⊗ Θ η ( v ) η ( u ) ) ⊗ µ (Θ η ( u ) η ( u ) ⊗ Θ η ( u ) η ( w ) )) . It is clear that µ (Θ η ( v ) η ( u ) ⊗ Θ η ( u ) η ( u ) ) = Θ η ( v ) η ( u ) , µ (Θ η ( u ) η ( u ) ⊗ Θ η ( u ) η ( w ) ) = Θ η ( u ) η ( w ) , and therefore the two terms on the right-hand side above are equal since L η ( u j ) is a small Hamilton-ian isotope of L η ( u (cid:96) ) for any − , , . . . , k . Hence with F coefficients, µ (Θ η ( v ) η ( u ) ⊗ Θ η ( u ) η ( u ) ⊗ Θ η ( u ) η ( w ) ) = 0. (As usual, one can establish this result with Z coefficients as well if appropriatesigns are added)For the inductive step, suppose that we have established that µ k (Θ η ( v ) η ( u ) ⊗ · · · ⊗ Θ η ( u k ) η ( w ) ) = 0 . We want to show that µ k +1 (Θ η ( u − ) η ( v ) ⊗ Θ η ( v ) η ( u ) ⊗ · · · ⊗ Θ η ( u k ) η ( w ) ) = 0 , where L η ( u − ) is a small Hamiltonian isotope of L η ( v ) . Let M ( k + 3) denote the moduli space ofconformal structures on the ( k + 3)-gon; this space is homeomorphic to R k . For any homotopy classof Whitney ( k + 3)-gons φ ∈ π (Θ u − v , Θ vu , . . . , Θ u k w , Θ wv ) , there is a forgetful map G : M ( φ ) −→ M ( k + 3)which keeps track of the conformal class of the domain. By Gromov compactness, this map isproper. YMPLECTIC INSTANTON HOMOLOGY, SPECTRAL SEQUENCES, AND KHOVANOV HOMOLOGY 39 Figure 16. Schematic of polygons counted by µ k +1 (Θ η ( u − ) η ( v ) ⊗ Θ η ( v ) η ( u ) ⊗ · · · ⊗ Θ η ( u k ) η ( w ) ), k = 3.Note that the Lagrangian boundary conditions for the ( k + 3)-gon we are considering involveexactly k non-convex corners, each of which contributes a 1-parameter family of deformations ofany pseudoholomorphic polygon by varying the length of the “slit” at the corner, or equivalentlyvarying the conformal structure on the domain. It follows that M ( φ ) at dimension at least k ,since any pseudoholomorphic representative u of φ automatically lies in a k -parameter family ofrepresentatives.Suppose dim M ( φ ) is exactly k . Then for any conformal structure c ∈ M ( k + 3), G − ( c ) is a finiteset of points, since G is proper. Near the ends of the moduli space M ( φ ), the standard gluingtheory for pseudoholomorphic polygons implies that the count G − ( c ) will correspond to a countof degenerate pseudoholomorphic polygons where the appropriate “slits” in the image have becomedeep enough to meet another Lagrangian. In particular, by making a deep slit at the appropriatecorner, we degenerate into a ( k + 2)-gon of the type we have already considered (in the inductivehypothesis) and a triangle (see Figure 17 for a schematic). The inductive hypothesis thereforeimplies that G − ( c ) is even for c near the appropriate end of M ( k + 3), and hence must be evenfor any c ∈ M ( k + 3).Now note that the only way to have a rigid pseudoholomorphic polygon coming from the entireHeegaard ( k + 3)-diagram under consideration is to have the other connect summands determinethe conformal structure on the special torus connect summand investigated above, and there willonly be a discrete set of pseudoholomorphic polygons contributed by this connect summand for afixed conformal structure if the dimension of the relevant moduli space is exactly k . Since we justdetermined that the count of such polygons is zero modulo 2, the K¨unneth principle then implies Figure 17. Schematic of the limiting pairs of polygons obtained by the two possibleslits at a non-convex corner.that the overall count is zero modulo 2, so that µ k +1 (Θ η ( u − ) η ( v ) ⊗ Θ η ( v ) η ( u ) ⊗ · · · ⊗ Θ η ( u k ) η ( w ) ) = 0 . The desired result hence follows by induction. (cid:3) Write C ( Y, L ) = (cid:77) v ∈{ , } m CSI( Y v ) ⊂ (cid:101) C ( Y, L ) . We call the subcomplex ( C ( Y, L ) , ∂ ) of ( (cid:101) C ( Y, L ) , ∂ ) the link surgeries complex of ( Y, L ).We may define a filtration on C ( Y, L ) by F i C ( Y, L ) = (cid:77) | v |≥ i CSI( Y v ) . Since ∂ is a sum of ∂ vw ’s with v ≤ w , it is clear that ∂ preserves the filtration: ∂ F i C ( Y, L ) ⊆ F i C ( Y, L ). Hence ( C ( Y, L ) , ∂ ) is a filtered chain complex and we obtain an associated spectralsequence E rp,q ( Y, L ) with E = (cid:77) v ∈{ , } m SI( Y v ) , d = (cid:88) v ≤ w | w − v | =1 ∂ vw . We will call the spectral sequence E rp,q ( Y, L ) the link surgeries spectral sequence for ( Y, L ).7.2. Convergence of the Spectral Sequence. Our goal now is to identify the E ∞ page of thelink surgeries spectral sequence E rp,q ( Y, L ) with the symplectic instanton homology SI( Y ). Weachieve this by the technique of “dropping a component,” which we explain in detail below.For i = − , , 1, define ( C i , ∂ i ) to be the link surgeries complex for ( Y i ( L ) , L \ L ), where Y i ( L )denotes the result of i -surgery on L ⊂ Y and we recall that if i = − 1, this means ∞ -surgery.Note that for v (cid:48) , w (cid:48) ∈ {− , , } m − , we have that ( ∂ i ) v (cid:48) w (cid:48) = ∂ vw , where v = ( i, v (cid:48) ) and w = ( i, w (cid:48) ).Therefore we may consider ( C , ∂ ) and ( C , ∂ ) as subcomplexes of ( C ( Y, L ) , ∂ ). ( C − , ∂ − ) isthe complex corresponding to “dropping a component,” since no surgery is performed on L in thiscomplex. YMPLECTIC INSTANTON HOMOLOGY, SPECTRAL SEQUENCES, AND KHOVANOV HOMOLOGY 41 Define a map f : C −→ C , f = (cid:88) v,w ∈{ , } m v =0 ,w =1 ∂ vw . f is simply the sum of components of ∂ involving surgery on the component L . f is a (anti-)chainmap for the same reason ∂ = 0, and by construction it is clear that ( C ( Y, L ) , ∂ ) = Cone( f ), i.e. C ( Y, L ) = C ⊕ C , ∂ = (cid:18) ∂ f ∂ (cid:19) . Now consider the map F : C − −→ C ( Y, L ) , F = (cid:88) v = − w ∈{ , } ∂ vw . F is a chain map (anti-chain map if we work with Z -coefficients), by a variant of the proof that ∂ = 0: F ∂ − + ∂ F = (cid:88) v = u = − w ∈{ , } ∂ uw ∂ vu + (cid:88) v = − u ,w ∈{ , } ∂ uw ∂ vu = 0 . Consider the filtrations on ( C , ∂ ) and ( C − , ∂ − ) given by the sum of all surgery coefficients exceptthe one for L , (cid:80) mi =2 | v i | . F respects these filtrations, and in grading p on the E -page of the inducedspectral sequence, F restricts to a map of the form F p : (cid:77) v = − (cid:80) mi =2 | v i | = p CSI( Y v ) −→ (cid:77) v ∈{ , } (cid:80) mi =2 | v i | = p CSI( Y v ) . The restriction of F p to a single direct summand has the form F p | CSI( Y v ) = ∂ vv ⊕ ∂ vv where v and v differ from v only in the first component: v = 0 and v = 1 ( v = − C − ). But in homology ∂ vv is the same as the map Φ in the surgery exact triangle(by Proposition 5.10), and ∂ vv is the same as the map h which serves as a chain nullhomotopyof C Φ ◦ C Φ (by Proposition 5.11). It therefore follows from Lemma 5.5 that F is a quasi-isomorphism, and hence so is F : C − −→ C ( Y, L ) . In other words, the complex C − obtained from C ( Y, L ) obtained by “dropping a component” of L ( i.e. not performing surgery on the component L ) is in fact quasi-isomorphic to C ( Y, L ).The above argument readily generalizes to dropping an arbitrary number of components of L . Bydropping all m components of L , we see that C ( Y, L ) is quasi-isomorphic to CSI( Y, ω L ), and weconclude the following: Theorem 7.2. The link surgeries spectral sequence E r ( Y, L ) for an m -component link L in a closed,oriented -manifold Y converges by the m th page to the symplectic instanton homology SI( Y, ω L ) . As was the case for the surgery exact triangle, a nontrivial SO(3)-bundle on Y may be incorporatedinto the link surgeries spectral seqeuence without changing the argument in any way. Indeed, any ω ∈ H ( Y ; F ) may be represented by a knot lying entirely in the α -handlebody of the Heegaardsplittings used in the proof of the link surgeries spectral sequence, and for any multiframing v ∈ { , , ∞} m , ω induces an obvious homology class ω v ∈ H ( Y v ; F ). In this case, the link surgeriesspectral sequence has E -page E = (cid:77) v ∈{ , } m SI( Y v , ω v )and converges by the m th page to SI( Y, ω ∪ ω L ).8. Khovanov Homology and Symplectic Instanton Homology of Branched DoubleCovers As an application of the link surgeries spectral sequence, we exhibit a spectral sequence fromthe Khovanov homology of (the mirror of) a link in S to the symplectic instanton homologyof its branched double cover. The method of proof is inspired by Ozsv´ath-Szab´o’s work on thecorresponding spectral sequence in Heegaard Floer theory [OS05].8.1. Khovanov Homology. Let us give a definition of Khovanov homology suitable for our ap-plications. Suppose X ⊂ R is a disjoint union of k simple closed curves, X = S ∪ · · · ∪ S k . Define Z ( X ) to be the F -vector space generated by the S i , and write V ( X ) = ∧ ∗ Z ( X )for the corresponding exterior algebra.Suppose X (cid:48) = S (cid:48) ∪ · · · ∪ S (cid:48) k − is obtained from X by merging two components S i and S j . Then Z ( X (cid:48) ) = Z ( X ) / ( S j − S i ) and there are natural isomorphisms α : ( S j − S i ) ∧ V ( X ) ∼ = −−→ V ( X (cid:48) ) , β : V ( X (cid:48) ) ∼ = −−→ V ( X ) / (( S j − S i ) ∧ V ( X )) . We may define a multiplication map by m : V ( X ) −→ V ( X (cid:48) ) ,m ( ξ ) = α (( S j − S i ) ∧ ξ )and a comultiplication map ∆ : V ( X (cid:48) ) −→ V ( X ) , ∆( ξ ) = ( S j − S i ) ∧ β ( ξ ) . Now, given a link L in S , choose an oriented link diagram D for L . Write n for the total number ofcrossings in D , and n + (or n − ) for the number of positive (or negative) crossings of D . Furthermore,choose an enumeration (1 , . . . , n ) of the crossings. For any I = ( (cid:15) , . . . , (cid:15) n ) ∈ { , } n , let D ( I ) denotethe result of resolving all crossings of D , where the i th crossing is replaced with its (cid:15) i -resolution(where 0- and 1-resolutions are as defined in Figure 4). D ( I ) is then a disjoint union of circles inthe plane, so we can define Z ( D ( I )) and V ( D ( I )) for all I ∈ { , } n . The i th Khovanov chaingroup is given by CKh( D , i ) = (cid:77) I ∈{ , } n | I | + n + = i V ( D ( I ))The Khovanov differential d : CKh( D , i ) −→ CKh( D , i − d | V ( D ( I )) = (cid:88) I (cid:48) is an immediatesuccessor of I d I
Kh( D ) = H ∗ (cid:32)(cid:77) i CKh( D , i ) , d (cid:33) is defined. Kh( D ) is independent of the diagram D of L up to isomorphism, and therefore we willwrite Kh( L ) to denote this isomorphism class of groups.We will actually need a reduced variant of Khovanov homology. Fix a basepoint p ∈ L . Then ageneric diagram D for L will have an induced basepoint, also denoted p , away from all crossingsin D . It follows that for any I ∈ { , } n (where n is the number of crossings in D ), there is adisinguished component S I ∈ Z ( D ( I )) containing p . Let us write˜ V ( D ( I )) = S I ∧ V ( D ( I )) . Then the i th reduced Khovanov chain group is then defined byCKhr( D , i ) = (cid:77) I ∈{ , } n | I | + n + = i ˜ V ( D ( I )) , and the reduced Khovanov homology is the homology of this subcomplex:Khr( D ) = H ∗ (cid:32)(cid:77) i CKhr( D , i ) , d (cid:33) . Khr( D ) (with F coefficients) is independent of the choice of diagram D for L as well as the choiceof basepoint p ∈ L up to isomorphism, and we will write Khr( L ) for this isomorphism class ofgroups.8.2. Branched Double Covers of Unlinks. It turns out that the branched double cover of anylink in S is obtained from the branched double cover of some unlink by adding finitely many2-handles. In this subsection, we explain why this is true and give a natural identification of thesymplectic instanton homology of the branched double cover of an unlink.Let L ⊂ S be any link and fix a diagram D for L . A Conway sphere for L is an embedded2-sphere S in S intersecting L transversely in exactly 4 points. Such a sphere divides S into two3-balls; let us suppose that one of these 3-balls, B , contains exactly one crossing of the diagram D .Write ˜ Y for the branched double cover of S \ B branched over L \ ( L ∩ B ). Note that the brancheddouble cover of a 3-ball branched over two arcs is a solid torus (via the hyperelliptic involution),and therefore ˜ Y is Σ( L ) with a solid torus removed. The meridian γ of this torus (in Σ( L )) isprecisely the branched double cover of the pair of arcs from L .Now consider the 0- and 1-resolutions L and L of L (as pictured in Figure 4) at the singlecrossing inside B . Again, ˜ Y is Σ( L ) minus a solid torus and also Σ( L ) minus a solid torus.However, the meridians γ (resp. γ ) of these solid tori correspond to the branched double coversof the pair of arcs L ∩ B (resp. L ∩ B ) in Σ( L ) (resp. Σ( L )). It is easily checked that γ ∩ γ ) = γ ∩ γ ) = γ ∩ γ ) = − 1. It therefore follows that (Σ( L ) , Σ( L ) , Σ( L )) is asurgery triad, as claimed in Example 5.3. It is not necessary to use a diagram for L in this subsection, but we do it for simplicity and also because we willeventually relate to Khovanov homology, which does use a link diagram in its chain-level definition. By resolving all the crossings of some diagram D for L ⊂ S , we obtain an unlink with k + 1components, for some non-negative integer k . The particular choice of resolutions tells us how toattach 2-handles to Σ( L ) to obtain Σ( U (cid:113)· · ·(cid:113) U ). The branched double cover of the two componentunlink is S × S , and more generally the branched double cover of the k + 1-component unlink is k S × S . We already know the symplectic instanton homology of k S × S (as a unital algebra,no less), but we wish to have a geometric interpretation of it related to the fact that k S × S is the branched double cover of the k + 1 component unlink. The following proposition establishessuch an interpretation. Proposition 8.1. SI( k S × S ) is a free Λ ∗ H ( k S × S ) -module of rank , generated by theelement Θ k corresponding to the usual highest index intersection point of the relevant Lagrangians.Furthermore, the Λ ∗ H ( k S × S ; F ) -module structure of SI( k S × S ) is natural with respectto -handle cobordism maps, meaning the following: (1) If K is a knot dual to one of the pt × S ’s in a summand of k S × S , then -surgery on K gives k − S × S , where the component whose S × S summand K was dual to hasbeen removed. If π : H ( k S × S ) / [ K ] −→ H ( k − S × S ) denotes the identificationnaturally induced by this -surgery and W : k S × S −→ k − S × S denotes the -handle cobordism induced by the -surgery, then F W ( ξ · Θ k ) = π ( ξ ) · Θ k − for all ξ ∈ Λ ∗ H ( k S × S ) . (2) If K is the unknot in k S × S and W : k S × S −→ k +1 S × S is the -handlecobordism induced by -surgery on K , then F W ( ξ · Θ k ) = ( ξ ∧ [ K W ]) · Θ k +1 for all ξ ∈ Λ ∗ H ( k S × S ) , where [ K W ] is the generator of the kernel of i ∗ : H ( k S × S ) −→ H ( W ) .Proof. We already know that SI( k S × S ) ∼ = H −∗ ( S ) ⊗ k as a unital algebra. We will write H −∗ ( S ) = F (cid:104) Θ , θ (cid:105) , with Θ the unit. H ( k S × S ; F ) has as basis the homology classesof the pt × S ’s in each connect summand; denote these classes by X , . . . , X k . We define aΛ ∗ H ( k S × S ; F )-module structure on F (cid:104) Θ , θ (cid:105) ⊗ k as follows. On Θ ⊗ k , the action is X i · Θ ⊗ k (cid:55)→ Θ ⊗ ( i − ⊗ θ ⊗ Θ ⊗ ( k − i − , ( X i ∧ X j ) · Θ ⊗ k (cid:55)→ Θ ⊗ ( i − ⊗ θ ⊗ Θ ⊗ ( j − i − ⊗ θ ⊗ Θ ⊗ ( k − j − i − ( i < j )...( X ∧ · · · ∧ X k ) · Θ ⊗ k (cid:55)→ θ ⊗ k . We see that every element η ∈ F (cid:104) Θ , θ (cid:105) ⊗ k can be written as ξ η · Θ ⊗ k for a unique ξ η ∈ Λ ∗ H ( k S × S ; F ). We extend the action to all of F (cid:104) Θ , θ (cid:105) ⊗ k by setting ξ · η = ( ξ ∧ ξ η ) · Θ ⊗ k . It is clear that with this action, F (cid:104) Θ , θ (cid:105) ⊗ k ∼ = Λ ∗ H ( k S × S ; F ) (cid:104) Θ ⊗ k (cid:105) as a Λ ∗ H ( k S × S ; F )-module. Now we prove the naturality with respect to 2-handle cobordism maps, as in statements(1) and (2) of the Proposition.(1) Thinking of the fundamental group of the torus as generated by the standard meridian andlongitude µ and λ , consider the Heegaard triple diagrams H = (Σ , λ, λ, λ ) and H (cid:48) = (Σ , λ, λ, µ ).It is then easy to see that the triple diagram k − H H (cid:48) represents 0-surgery in k S × S along YMPLECTIC INSTANTON HOMOLOGY, SPECTRAL SEQUENCES, AND KHOVANOV HOMOLOGY 45 the knot K = pt × S coming from the S × S summand corresponding to the α - and β -curves of H (cid:48) . Direct inspection then reveals that F W ( ξ · Θ k ) = π ( ξ ) · Θ k − ;if ξ contains a wedge factor of X k then F W ( ξ · Θ k ) = 0 for Maslov index reasons, and if ξ does notcontain a wedge factor of X k then the claimed formula holds by considering F W being a closestpoint map on the factor corresponding to H (cid:48) .(2) Continuing the notation above, consider a third Heegaard triple H (cid:48)(cid:48) = (Σ , µ, λ, λ ). Then thetriple diagram k H H (cid:48)(cid:48) represents 0 surgery on k S × S along the unknot K in the S summandcorresponding to the α - and β - curves of H (cid:48)(cid:48) . Similar arguments to the above establish that F W ( ξ · Θ k ) = ( ξ ∧ [ K W ]) · Θ k +1 ; F W can be thought of as inducing a closest point map on all factors, and the [ K W ] appears becausethe factor H (cid:48)(cid:48) always gives the trivial connection θ . (cid:3) A Spectral Sequence from Khovanov Homology. We now explain how Khovanov ho-mology relates to the symplectic instanton homology of k S × S and deduce that the E -page ofa certain link surgery spectral sequence associated to a branched double cover Σ( L ) is isomorphicto Khr( m ( L ); F ).Let L ⊂ S be a link and fix an n -crossing diagram D for the mirror link m ( L ). As before, givenany I ∈ { , } n , we may construct the associated resolved diagram D ( I ). Proposition 8.2. For any I ∈ { , } n , there is an isomorphism Ψ I : ˜ V ( D ( I )) −→ SI(Σ( D ( I ))) which is natural in the following sense: if I (cid:48) ∈ { , } n is an immediate successor of I and F I
SI(Σ( D ( I (cid:48) ))) Proof. The proof is really just a matter of understanding the relationship between merging/splittingunlinks and surgeries on the branched double cover, and then applying Proposition 8.1.Fix a basepoint p ∈ L and write D ( I ) = S ( I ) (cid:113)· · ·(cid:113) S ( I ) k ( I ) , where we order the circle componentsso that S ( I ) contains the basepoint p , and similarly write D ( I (cid:48) ) = S ( I (cid:48) ) (cid:113)· · ·(cid:113) S ( I (cid:48) ) k ( I (cid:48) ) . For con-venience, we will assume further that the components are ordered so that the Khovanov differential d I
Merging two circles and the effect on homology.Suppose D ( I (cid:48) ) differs from D ( I ) by the merging of S ( I ) and S ( I ) into S ( I (cid:48) ) (see Figure 18 for aschematic). Then it is clear that X (cid:48) , X (cid:48) ∈ H (Σ( D ( I (cid:48) )); F ) are homologous, since after merging,both γ and γ will have an endpoint on S ( I (cid:48) ) . One then sees that Σ( D ( I (cid:48) )) is obtained fromΣ( D ( I )) by 0-surgery along the branched double cover of the pictured arc δ , which is just an unknotin Σ( D ( I )). It follows from case (2) of Proposition 8.1 that in this case, F I
Theorem 8.3. For any link L in S , there is a spectral sequence with E page given by Khr( m ( L ); F ) abutting to SI(Σ( L ); F ) .Proof. Fix a diagram D for m ( L ) with n crossings and a basepoint p . At the i th crossing of D , let µ i ∈ H (Σ( L ); F ) be the mod 2 homology class represented by the branched double cover of thearc in S connecting the two strands of L at that crossing. Write µ = µ + · · · + µ n ∈ H (Σ( L ); F ).Given any I ∈ { , } n , we have that I -framed surgery along the link µ = µ (cid:113) · · · (cid:113) µ n results inΣ( D ( I )), for the same reason that (Σ( J ) , Σ( J ) , Σ( J )) is a surgery triad for any link J .From the previous paragraph and the link surgeries spectral sequence for symplectic instantonhomology, it follows that we have a spectral sequence with E = (cid:77) I ∈{ , } n SI(Σ( D ( I ))) , d | SI(Σ( D ( I ))) = (cid:88) I (cid:48) an immediatesuccessor of I F I
The oriented resolution of a crossing.To show that the desired surface exists, choose an orientation of L and let D (cid:48) be the diagramobtained from D by performing a oriented resolution (see Figure 19) of each crossing. D (cid:48) willconsist of N oriented circles S (cid:48) , . . . , S (cid:48) N in the plane. To each S (cid:48) k , assign two signs a k and b k : • a k = +1 if S (cid:48) k is oriented counterclockwise; a k = − • b k = ( − M , where M is the number of circles in D (cid:48) that surround S (cid:48) k .Fill in each disk bounded by a S (cid:48) k for which a k b k = +1 (one may wish to think as each S (cid:48) k as lyingin the plane z = k in R , as some regions will be colored multiple times). To get the original link L back, we attach bands with a half twist to connect the circles where we originally performed theoriented resolutions. The condition that S (cid:48) k be filled only if a k b k = +1 ensures that for any twocircles joined by a band, exactly one of them has its interior filled, so that we may fill the appropriatehalf of the band to obtain our desired surface, which has boundary consisting of segements from D and for each crossing of D , an arc connecting the two pieces of L near that crossing.The lift of the surface in S constructed in the previous paragraph to the branched double coverΣ( L ) is a surface whose boundary is precisely µ , and therefore µ is nullhomologous. It follows thatSI(Σ( L ) , µ ) = SI(Σ( L )), and therefore our spectral sequence converges to SI(Σ( L )), as claimed. (cid:3) As a quick corollary, we obtain a rank inequality for the symplectic instanton homology of abranched double cover. Corollary 8.4. For any link L in S , | det( L ) | ≤ rk SI(Σ( L ); F ) ≤ rk Khr( m ( L ); F ) .Proof. This follows from the fact that χ (SI(Σ( L ))) = | H (Σ( L )) | = | det( L ) | and that there is aspectral sequence Khr( m ( L )) ⇒ SI(Σ( L )). (cid:3) Nontrivial Bundles on Branched Double Covers The spectral sequence for a branched double cover discussed in the previous section can also bestudied for nontrivial SO(3)-bundles using a homology related to a “twisted” version of Khovanovhomology, as first explained in the gauge theory context by Scaduto and Stoffregen [SS16].9.1. Two-Fold Marking Data. Let L = L (cid:113) · · · (cid:113) L N ⊂ S be a link. A two-fold markingdatum for L is a function ω : { L , . . . , L N } −→ Z / such that ω ( L ) + · · · + ω ( L N ) ≡ ω as assigning a collection of points p = { p , . . . , p m } to L , m < N , with one point on L k if ω ( L k ) = 1, such that the total number ofpoints is even. The pointed link ( L, p ) is easily seen to correspond to an element of H (Σ( L ); F ):if one takes a collection of arcs in S whose interiors are disjoint from L and whose endpoints areprecisely the basepoints p , then the lift of this collection of arcs to Σ( L ) will be a well-defined mod2 homology class. In fact, all elements of H (Σ( L ); F ) arise from some such collection of basepoints p ⊂ L , so that we have the following: Proposition 9.1. There is a one-to-one correspondence between two-fold marking data ω for L and elements of H (Σ( L ); F ) . In particular, each two-fold marking datum ω corresponds to a nontrivial SO(3)-bundle on Σ( L ),which by abuse of notation we will also refer to as ω .9.2. Twisted Khovanov Homology and Dotted Diagram Homology. We now wish to in-troduce a variant of Khovanov homology which takes into account two-fold marking data. To dothis, we will first need to introduce a compatibility relation between the two-fold marking data andthe link diagrams we use. Let L ⊂ S be a link and D be a diagram for L . An arc of D is a strandof D that descends to an edge of the associated 4-valent graph. Let Γ denote the set of all arcs in D , and given a component L k of L , let Γ( L k ) denote the set of all arcs contained in the image of L k in D . Given a two-fold marking datum ω for L , we say that an assignment ˇ ω : Γ −→ Z / compatible with ω if (cid:88) γ ∈ Γ( L k ) ˇ ω ( γ ) ≡ ω ( L k ) mod 2 . The pair ( D , ˇ ω ) will be called a two-fold marked diagram . Note that the 0- and 1-resolutions ofany crossing in D have well-defined induced two-fold markings, giving two-fold marked diagrams( D , ˇ ω ) and ( D , ˇ ω ).Given a two-fold marked link ( L, ω ), fix a compatible two-fold marked diagram ( D , ˇ ω ) with n crossings, and fix an auxiliary basepoint p ∈ D (which has nothing to do with ˇ ω ). For any I = ( (cid:15) , . . . , (cid:15) n ) ∈ { , } n , we then obtain a resolved diagram ( D ( I ) , ˇ ω ( I )) consisting of k ( I ) pointedcircles: ( D ( I ) , ˇ ω ( I )) = ( S ( I ) , ˇ ω ( I ) ) ∪ · · · ∪ ( S ( I ) k ( I ) , ˇ ω ( I ) k ( I ) ) . Exactly one of these circles, which we will denote by S I , contains the auxiliary basepoint p . As inthe untwisted case, we define CKh( D , ˇ ω ) = (cid:77) I ∈{ , } n S I ∧ V ( D ( I )) , i.e. CKh( D , ˇ ω ) as a chain group is just the reduced Khovanov chain group with respect to thebasepoint p , and we incorporate ˇ ω into the differential as follows. The differential d on CKh( D , ˇ ω )is a sum of “horizontal” and “vertical” differentials, d = d h + d v . The horizontal differential d h is just the usual Khovanov differential, defined in terms of merge/splitmaps. The vertical differential d v is defined by its restriction to a direct summand: d v ξ = k ( I ) (cid:88) j =1 ˇ ω ( I ) j S ( I ) j ∧ ξ for ξ ∈ S I ∧ V ( D ( I )) . One may check that d = 0, and the twisted Khovanov homology of ( L, ω ) is defined asKh( L, ω ) = H ∗ (CKh( D , ˇ ω ) , d ) . YMPLECTIC INSTANTON HOMOLOGY, SPECTRAL SEQUENCES, AND KHOVANOV HOMOLOGY 49 Kh( L, ω ) may be thought of as a generalization or deformation of Khr( L ): if ω is the trivialtwo-fold marking data (which assigns 0 to all components of L ), then Kh( L, ω ) ∼ = Khr( L ).The horizontal and vertical differentials on CKh( D , ˇ ω ) are easily seen to commute, and thereforeCKh( D , ˇ ω ) admits the structure of a double complex. The homology with respect to the horizontaldifferential d h is the reduced Khovanov homology of L , and the homology with respect to the verticaldifferential d v results in the subcomplex of (CKh( D , ˇ ω ) , d h ) consisting of summands correspondingto I ∈ { , } n with all ˇ ω ( I ) j ’s even. We call the homology of this subcomplex with respect to d h the dotted diagram homology of ( D , ˇ ω ) and denote it byHd( D , ˇ ω ) = H ∗ ( H ∗ (CKh( D , ˇ ω ) , d v ) , d h ) . We remark that Hd( D , ˇ ω ) is not an invariant of ( L, ω ), but it naturally appears as the E -page ofa link surgeries spectral sequence for a nontrivial SO(3)-bundle on the branched double cover of L , as we explain in the next subsection. Note that the spectral sequence for the double complex(CKh( D , ˇ ω ) , d v , d h ) gives another spectral sequence Hd( D , ˇ ω ) ⇒ Kh( L, ω ). Remark 9.2. It is easy to see that d v | ˜ V ( D ( I )) is an isomorphism if ˇ ω ( I ) is nonzero on any componentof D ( I ), and zero otherwise. It follows that H ∗ (CKh( D , ˇ ω ) , d v ) is just the subcomplex of CKhr( D )consisting of I -summands with ˇ ω ( I ) ≡ Spectral Sequence for Nontrivial Bundles on Branched Double Covers. We now ex-plain the relevance of dotted diagram homology to the symplectic instanton homology of nontrivialSO(3)-bundles over branched double covers of links. Theorem 9.3. Let ( L, ω ) be a two-fold marked link in S , and suppose ( D , ˇ ω ) is a compatibletwo-fold marked diagram for the mirror ( m ( L ) , ω ) . Then there is a spectral sequence with E -pageisomorphic to the dotted-diagram homology Hd( D , ˇ ω ) converging to SI(Σ( L ) , ω ) , where ω denotesthe SO(3) -bundle on Σ( L ) induced by the two-fold marking data ω .Proof. Let µ denote the n -component link in Σ( L ) obtained by lifting the n arcs in S connectingthe two local components of each crossing of D . As in the proof of Theorem 8.3, for any I ∈ { , } n ,the result of I -framed surgery on µ = µ (cid:113) · · · (cid:113) µ n results in Σ( D ( I )). Letting ω ∈ H (Σ( L ); F )denote the mod 2 homology class corresponding to the two-fold marking data ω (by abuse ofnotation), we get induced homology classes ω ( I ) ∈ H (Σ( D ( I )); F ) for each I ∈ { , } n . The linksurgery spectral sequence for (Σ( L ) , ω ) and the link µ then has E = (cid:77) I ∈{ , } n SI(Σ( D ( I )) , ω ( I )) , d | SI(Σ( D ( I )) ,ω ( I )) = (cid:88) I (cid:48) an immediatesuccessor of I F I
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