TTRISECTIONS AND OZSV ´ATH-SZAB ´O COBORDISM INVARIANTS
WILLIAM E. OLSEN
Abstract.
Given a smooth, compact four-manifold X viewed as a cobordism from the empty setto its connected boundary, we demonstrate how to use the data of a trisection map π : X → R to compute the induced cobordism maps on Heegaard Floer homology associated to X . Contents
1. Introduction 12. Trisections of four-manifolds 33. Background on Heegaard Floer homology 64. Trisections and Ozsv´ath-Szab´o cobordism invariants 8Bibliography 201.
Introduction
A new tool in smooth four-manifold topology has recently been introduced under the name of trisected Morse -functions (or trisections for short) by Gay and Kirby [GK16]. Recent develop-ments in this area demonstrate rich connections and applications to other aspects of four-manifoldtopology, including a new approach to studying symplectic manifolds and their embedded sub-manifolds [LMS20; Lam19; LM18], and to surface knots (embedded in S and other more general4-manifolds) [MZ17; MZ18] along with associated surgery operations [GM18; KM20].Of particular interest to the trisection community is the construction of new [KT18; Cas+19]and the adaptation of established invariants in the trisection framework. In this article, we areconcerned with the latter as we endeavor to demonstrate a technique for computing the HeegaardFloer cobordism maps from the data of a relative ( g, k ; p, b )-trisection map (see Sections 3 and 2for definitions). Our main result can be summarized as follows (see Section 4 for more precisestatements): Theorem A.
Fix a smooth, connected, oriented, compact four-manifold X with connected boundary ∂X = Y , and let π : X → R be a (relative) trisection map. Using π as input data, one can recoverthe induced cobordism maps in Heegaard Floer homology F ◦ X, s : HF ◦ ( S ) → HF ◦ ( Y, s | Y ) , (1.1) where X is viewed as a cobordism from S to Y after removing the interior of a small ball, and ◦ ∈ { + , − , ∞ , ∧} are the variants defined in [OS04b]. Theorem A has a few interesting characteristics and implications that may be worth mentioning.The first is that Ozsv´ath and Szab´o prove that their induced maps are smooth invariants of theunderlying cobordism [OS06, Theorem 1.1] which implies that Theorem A may be used in thedetection of exotic phenomena. For example, following the usual Mayer-Vietoris strategy found
Date : February 2, 2021. a r X i v : . [ m a t h . G T ] F e b WILLIAM E. OLSEN in Floer homology theories, it is theoretically possible to use Theorem A to recover the mixedinvariants [OS06, Theorem 9.1] of a closed four-manifold X with b +2 ( X ) >
1. However, we warnthe reader that, practically speaking, using Theorem A to compute the mixed invariants by hand inany particular example remains a daunting task due to the general unruliness of pseudo-holomorphiccurves.The second feature we’d like to highlight, and perhaps most important in the author’s opinion,is that Theorem A makes no reference to a handle decomposition of the underlying four-manifold.Instead, the theorem takes as input data a (definite) broken fibration (which has been isotoped intoa special form) and manages to return the Heegaard Floer cobordism maps as output. As such,Theorem A may give some insight into how one might compare the relative invariants arising indifferent Floer homology theories–most notably a comparison between the Oszv´ath-Szab´o mixedinvariants and Perutz’s Lagrangian matching invariants [Per07; Per08].With these preliminary remarks in place, we quickly summarize the proof of Theorem A. Inbrief, a trisection map π : X → R is a singular fibration over the disk whose singular set is of aprescribed type (assuming X has non-empty boundary, the singular set has indefinite folds/cusps,none of which intersect the boundary). The central fiber (preimage of (0 ,
0) under π ) is a genus g surface with b > π induces anopen book decomposition of the boundary 3-manifold Y = ∂X whose monodromy can be recoveredby flowing a regular fiber once around the boundary of the disk (after choosing the appropriateauxiliary data, such as a metric and compatible connection). By starting at the central fiber,flowing to the boundary, once around, and then back to the center , one obtains a Heegaard triplewhich is slide-equivalent to one which is subordinate to a bouquet for a framed link as in [OS06]–diagrammatically, the result is a closed surface decorated by three complete sets of attaching curvesand a canonical choice of basepoint. Theorem A then follows via the usual naturality considerations. Organization.
This note is organized as follows. In Sections 2 and 3, we briefly review the neces-sary background behind trisected Morse-2 functions and the induced cobordism maps in HeegaardFloer homology. The heart of the paper is found in Section 4 where we give the details of theconstruction outlined above for how to use the data of a relative trisection map to compute theOzsv´ath-Szab´o cobordism maps. In the last section we comment on the role of the contact class[OS05; HKM09] in our setup.
Acknowledgements.
This work would not have been possible without the insight and generoussupport of my Ph.D. advisor, David Gay. Also I’d like to thank Juanita Pinz´on-Caicedo and JohnBaldwin for their interest in this project. Finally, I would like to thank the Max Planck Institute forMathematics in Bonn, Germany, for hostimg me while I worked towards completing this project. Although we don’t directly study holomorphic sections of π , the author was deeply inspired by the constructions of Lagrangian boundary conditions found in [Sei08] and [Per07]. The approach taken here is meant to be reminiscent ofthese constructions.
RISECTIONS AND OZSV ´ATH-SZAB ´O COBORDISM INVARIANTS 3 Trisections of four-manifolds
The literature is rich with helpful and insightful constructions of the trisection theory. For thisreason, we only briefly review its foundational material and point the interested reader elsewherefor a less terse introduction. For a general overview of trisections and direct comparisons withthe more familiar description of four-manifolds via handle decompositions and Kirby calculus, werecommend the original [GK16] and the more recent survey [Gay19]. For interesting examples oftrisections and their diagrams, including descriptions for various surgery operations such as theGluck twist and its variants, we recommend [KM20; LM18; GM18; AM19] and [Koe17]. For abroader perspective on stable maps from four-manifolds to surfaces, including details about how tosimplify the topology of such maps, we suggest [GK15; GK12; BS17] and the references therein.2.1.
Heegaard splittings and the essentials of diagrammatic representations of mani-folds.
In this first section, we establish a vocabulary and notation for discussing the diagrammaticrepresentations of 3- and 4-manifolds which are prevalent throughout. When possible, we closelyfollow the terminology and notation of [GM18, Section 2.2] and [JTZ12, Section 2].We start with the essentials: • Σ g,b is a compact, oriented, connected surface of genus g with b boundary components. • δ = { δ , . . . , δ g − p } ⊂ Σ g,b is a genus p cut system , i.e. a collection of disjoint simple closedcurves which collectively Σ g,b into a connected genus p surface. In symbols, Σ g,b \∪ i δ i ∼ = Σ p,b . • Two genus p cut systems δ and δ (cid:48) are said to be slide-equivalent if there exists a sequence ofhandle-slides taking δ to δ (cid:48) (see [JTZ12, Section 2] for a picture of a handle-slide). Moreover, δ and δ (cid:48) are said to be slide-diffeomorphic if there exists a diffeomorphism φ : Σ → Σ suchthat φ ( δ ) is slide-equivalant to δ (cid:48) .An important fact that we’ll use repeatedly is the following : a genus p cut system on Σ g,b determines (up to diffeomorphism rel. boundary) a compression body C δ which is the cobordismobtained from I × Σ g,b by attaching three-dimensional 2-handles to { } × Σ g,b along the curves { } × δ . Moreover, any two such compression bodies C δ , C δ (cid:48) are diffeomorphic rel. boundary ifand only if δ , δ (cid:48) are slide-diffeomorphic. Definition 2.1.
Let Y be a connected, oriented 3-manifold which may or may not have boundary.An embedded genus- g Heegaard diagram for Y is a triple (Σ g,b , α , β ) where α , β are genus p cutsystems on Σ g,b which respectively bound compressing disks on either side of Σ g,b . If ∂Y = ∅ , then p = g .The relevance of the above definitions is that if (Σ g,b , α (cid:48) , β (cid:48) ) is another abstract diagram with α ∼ α (cid:48) and β ∼ β (cid:48) slide-equivalent, then (Σ g,b , α (cid:48) , β (cid:48) ) is also a Heegaard diagram for Y . Everyoriented, connected 3-manifold Y admits an embedded Heegaard diagram.In fact, up to diffeomorphism rel. Σ g,b , an abstract diagram (Σ g,b , α , β ) determines a 3-manifold Y via the following procedure: start with [ − , × Σ g,b and attach 2-handles to {− } × Σ g,b alongthe curves {− } × α , and similarly attach 2-handles to { } × Σ g,b along { } × β . For more detailsabout the precise relationship between abstract diagrams, embedded diagrams, and their associatedclasses of 3-manifolds, we refer the reader to [JTZ12, Section 2]. Strictly speaking, we should be more careful about distinguishing genus p cut systems from isotopy classes of such–this point of view is taken in [JTZ12]. The interested reader is also pointed there for more details about (sutured)compression bodies. There is a great deal of subtlety when comparing ‘abstract’ diagrams, by which we mean a picture of a surfacedecorated with colored collections of curves, and ‘embedded’ diagrams, as we’ve defined above. For more on thissubtlety and how its relevant to the Heegaard Floer computatinos which come later, we recommend [JTZ12] for anexcellent discussion and examples.
WILLIAM E. OLSEN
Definition 2.2. A Heegaard triple is a 4-tuple (Σ g , α , β , γ ) where Σ g is a closed, oriented surfaceand each of α , β , and γ are genus g cut systems on Σ g .Just as (embedded) Heegaard diagrams determine a smooth 3-manifold up to diffeomorphismrel. Σ, a Heegaard triple determines (up to diffeomorphism) a smooth 4-manifold. To this end, let H = (Σ , α , β , γ ) be a Heegaard triple. In [OS04b, Section 8], Ozsv´ath and Szab´o associate to H afour-manifold X α , β , γ via X α , β , γ := (cid:16) (Σ × ∆) ∪ ( U α × e α ) ∪ ( U β × e β ) ∪ ( U γ × e γ ) (cid:17) / ∼ (2.1)where ∆ is a triangle with edges labeled e α , e β , and e γ clockwise, and ∼ is the relation determinedby gluing U τ × e τ to Σ × ∆ along Σ × e τ for each τ ∈ { α , β , γ } using the natural identification.We note that if H = (Σ , α , β , γ ) is a general Heegaard triple, with no conditions on the pair-wise cut systems, then the four-manifold X α , β , γ constructed in equation (2.1) has three boundarycomponents ∂X α , β , γ = − Y α , β (cid:116) − Y β , γ (cid:116) Y α , γ (2.2)given by the three Heegaard splittings (Σ , α , β ), (Σ , β , γ ), and (Σ , γ , α ).However, if H = (Σ , α , β , γ ) is required to be a trisection diagram, so that we have(Σ , α , β ) ∼ = (Σ , β , γ ) ∼ = (Σ , γ , α ) ∼ = k S × S , then it follows (again from Laudenbach-Poenaru [LP72]) that we can fill in these three boundarycomponents and obtain a closed four-manifold.2.2. Trisections as singular fibrations.
The theory of trisections arose from the study of genericsmooth maps from four-manifolds to surfaces [GK12; GK15; GK16], and while the diagrammaticconsequences of the trisection theory are certainly interesting, we’ll maintain the historical perspec-tive and view stable maps π : X → D as the primary object of interest. Along the way we explainhow the familiar notions of connection, parallel transport, and vanishing cycles can be importedinto this setting. To be clear, none of what’s presented in this section is original, our main sourcesbeing the excellent work [Hay14; BH12; BH16; Beh14].Fix X to be a compact, oriented, connected smooth 4-manifold with corners, and let D be theunit disk in the plane. A stable map π : X → D is one whose critical locus and critical imageadmit local coordinate descriptions of the following two types :(1) Indefinite fold model : in local coordinates, π is equivalent to:( t, x, y, z ) (cid:55)→ ( t, x + y − z ) (2.3)(2) Indefinite cusp model : in local coordinates, π is equivalent to( t, x, y, z ) (cid:55)→ ( t, x + 3 tx + y − z ) (2.4)Since we’re interested in four-manifolds with non-empty boundary, we impose the followingadditional constraints on π near the boundary :(3) The boundary of X decomposes into two codimension zero pieces, the horizontal part ∂ h X and the vertical part ∂ v X , so that ∂X = ∂ v X ∪ ∂ h X . The vertical part of the boundary isdefined to be ∂ v X = π − ( ∂ D ), and the horizontal part is defined to be the closure of itscomplement.An important aspect of the behavior of π near the horizontal part of the boundary is the followingregularity condition. We recommend [Hay14] for more details on stable maps. See [BS17, Remark 4.5] for comments on how to extend the homotopy techniques of indefinite fibrations to manifoldswith boundary.
RISECTIONS AND OZSV ´ATH-SZAB ´O COBORDISM INVARIANTS 5 (4) The stable map π restricts to smooth fibration on ∂ h X .The tangent space at any point p ∈ X splits into T X p = T p X h ⊕ T p X v (2.5)where the vertical tangent space is defined to be T p X v = ker( Dπ ), and the horizontal tangent space T p X h is its orthogonal complement with respect to a chosen metric on X (compare [Hay14]).(5) If x lies in ∂ h X , then the horizontal part of the tangent space lies in T x ∂ h X .Our main concern in introducing the assumptions (1) - (5) is so that we can import the technologyof [Hay14; Beh14; BH16] into our setting where X has boundary–namely, so that we can use thetools of π -compatible connections and parallel transport. For example, the splitting (2.5) defines a π -compatible connection , as in [BH16]. Moreover, since T h X is parallel to ∂ h X , these π -compatibleconnections have well-defined parallel transport maps (which are only partially defined on thefibers) in our case when X has non-empty boundary. For more details about connections andparallel transport maps in the context of singular fibrations, we recommend [BH16]. Definition 2.3.
Let X be a compact, oriented, connected smooth 4-manifold with connectedboundary Y = ∂X . A ( g, k ; p, b ) -trisection map π : X → D is a stable map satisfying theboundary conditions (1) – (5) above and whose critical image is shown in Figure 1 below. Figure 1.
The image of the critical value set in a ( g, k ; p, b )-trisection map. Thecentral fiber Σ := π − (0 ,
0) is an oriented genus g surface with b boundary compo-nents. The fibers over the purple boundary of the unit disk are the pages of an openbook decomposition of the boundary 3-manifold Y –the page has genus p . Each ofthe blue boxes denotes a Cerf box , as in [GK16, Figure 17].From a ( g, k ; p, b )–trisection map π : X → D one can recover the standard decomposition of X into three pieces as described in [CGP18b]. Clearly, the three dotted line segments in Figure 1decompose the image of π into three sectors D , D , and D . Define Z i := π − ( D i ), and note thatthe local models described in equations (2.3)–(2.4) imply that X = Z ∪ Z ∪ Z is naturally a ( g, k ; p, b )–trisection of X –see [CGP18a] for more details.In [GK16], Gay and Kirby show that such a stable map on a four-manifold induces an open bookdecomposition of its boundary 3-manifold. WILLIAM E. OLSEN
Theorem 2.4 ([GK16]) . A relative ( g, k ; p, b ) -trisection map π : X → D induces an open bookdecomposition on the boundary three-manifold. Importantly, the monodromy diffeomorphism of the open book decomposition on the boundary3-manifold can be recovered combinatorially from the a trisection diagram (Σ , α , β , γ ) [CGP18a,Theorem 5]. This is discussed in more detail in Section 4.1.3. Background on Heegaard Floer homology
This sections provides a brief review of those aspects of Heegaard Floer homology that willbe most important to us: the Heegaard Floer chain complexes, the chain maps induced by four-dimensional cobordisms, and the definition of the ‘mixed’ invariants for closed four-manifolds X with b +2 ( X ) >
1. We assume the reader is familiar with the Heegaard Floer canon [OS04b; OS06;Lip06].3.1.
Heegaard Floer chain complexes.
Fix a closed, connected, oriented three-manifold Y ,and denote by Spin c ( Y ) the space of Spin c structures on Y . Given a pointed Heegaard splitting H = (Σ , α , β , w ) of Y , Ozsv´ath and Szab´o [OS04b; OS04a] study the Lagrangian Floer cohomologyof the two tori T α = α × · · · × α g T β = β × · · · × β g inside the symmetric product Sym g (Σ g ). To review their construction, fix s ∈ Spin c ( Y ) and recallthe map s w : T α ∩ T β → Spin c ( Y ). Assuming the Heegaard splitting is admissable for the Spin c -structure s (see [OS04b] for more details), and after choosing a (generic) family J of almost complexstructures on Sym g (Σ g ), define the chain complex CF ∞ ( H , s ) to be freely generated over F bypairs [ x, i ] where x ∈ T α ∩ T β satisfies s w ( x ) = s , i is an integer, F is the field of two elements,and H denotes the pair H = ( H, J ).The differential on CF ∞ ( H , s ) is given by ∂ [ x , i ] = (cid:88) y ∈ T α ∩ T β s w ( y )= s (cid:88) φ ∈ π ( x , y ) µ ( φ )=1 M ( φ ) · [ y , i − n w ( φ )] , (3.1)where π ( x , y ) is the space of homotopy classes of Whitney disks connecting x to y , µ ( φ ) is theMaslov index, M ( φ ) is the moduli space of J -holomorphic disks in the class φ (modulo the action of R ), and n w ( φ ) is the algebraic intersection number of φ with the divisor { w }× Sym g − (Σ). The chaingroups CF ∞ ( H , s ) come equipped with an F [ U, U − ]-action, where U acts by U · [ x , i ] = [ x , i − −
2. With the infinity complex ( CF ∞ ( H , s ) , ∂ ) inhand, one obtains other complexes CF + , CF − , and (cid:100) CF by restricting attention to pairs [ x, i ] with i ≥ , i <
0, and i = 0, respectively. The subsequent complexes have an induced F [ U ]-action,which is trivial in the case of (cid:100) CF .Clearly, the plus, minus, and infinity variations are related by a short exact sequence0 → CF − ( H , s ) → CF ∞ ( H , s ) → CF + ( H , s ) → · · · → HF + ∗ +1 ( Y, s ) δ −→ HF −∗ ( Y, s ) → HF ∞∗ ( Y, s ) → · · · (3.3)where δ : HF ∗ +1 ( Y, s ) → HF −∗ ( Y, s ) is a connecting homomorphism. Last, the reduced HeegaardFloer homology groups HF ± red ( Y, s ) are defined as HF − red ( Y, s ) := ker (cid:0) ι ∗ : HF − ( Y, s ) → HF ∞ ( Y, s ) (cid:1) (3.4)and HF + red ( Y, s ) := coker (cid:0) π ∗ : HF ∞ ( Y, s ) → HF + ( Y, s ) (cid:1) . (3.5) RISECTIONS AND OZSV ´ATH-SZAB ´O COBORDISM INVARIANTS 7
The connecting homomorphism δ induces an isomorphism from HF + red ( Y, s ) to HF − red ( Y, s ). Unlike HF ± , the modules HF ± red ( Y, s ) are always finite-dimensional over F .3.2. Maps associated to cobordisms.
In addition to defining the F [ U ]-modules HF ◦ ( Y, s ),Ozsv´ath and Szab´o show that four-dimensional cobordisms between 3-manifolds induce F [ U ]-equivariant maps between the respective Floer homology groups. We briefly highlight the mainpoints of this construction. Our exposition closely follows that of [LMW08, Section 4].Consider a smooth, connected, oriented four-dimensional cobordism W from Y − to Y + , where Y ± are also closed, connected, and oriented, and fix a Spin c structure s on W . Let f be a self-indexingMorse function on W , and consider the associated handle decomposition of W : Y − W −−→ Y W −−→ Y W −−→ Y + where the cobordism W i contains only index- i handles.Given the data ( W = (cid:83) i W i , f ), Ozsv´ath and Szab´o [OS06] associate to ( W, s ) an induced map F ◦ W, s : HF ◦ ( Y − , s Y − ) → HF ◦ ( Y + , s Y + ) between the Floer homologies of Y − and Y + by first definingmaps F ◦ W i , s | Wi for each i = 1 , , F ◦ W, s to be their composition. We now review thedefinitions of these three maps. One- and three-handle maps.
Suppose that W is a cobordism from Y − to Y which consists entirelyof 1-handle additions, and let s be a Spin c -structure on W . Since W consists only of 1-handles,it follows that Y ∼ = Y − S × S ) n where n is the number of 1-handles added by W . By [OS06],there is a non-canonical identification of the Floer homology of Y and the module HF ◦ ( Y , s Y − s ) ∼ = HF ◦ ( Y − , s Y − ) ⊗ H ∗ ( T n ; Z ) (3.6)where H ∗ ( T n ; Z ) is the usual singular homology of the n -torus T n with integer coefficients. Let Θ + denote the generator (we’re working over F ) of the top-graded part of H ∗ ( T n , Z ). The cobordismmap F ◦ W , s is defined on generators to be F ◦ W , s : HF ◦ ( Y , s Y ) → HF ◦ ( Y , s Y ) [ x , i ] (cid:55)→ [ x ⊗ Θ + , i ] (3.7)It is proved in [OS06, Section 4.3] that, up to composition with canonical isomorphisms, F ◦ W , s does not depend on the choices made in its construction. For brevity, we will usually denote the1-handle map by F .Next, if W is a cobordism which can be built using only 3-handles, then for s ∈ Spin c ( W ) themap F ◦ W , s is defined as the dual of F ◦ W , s by F ◦ W , s : HF ◦ ( Y , s Y ) → HF ◦ ( Y + , s Y + ) [ x ⊗ Θ − , i ] (cid:55)→ [ x , i ]where Θ − is the generator of the lowest-graded part of H ∗ ( T n , Z ). Moreover, F ◦ W , s ([ x ⊗ ξ, i ]) = 0for any homogeneous generator ξ which does not lie in the bottom degree. Again, the map isindependent of the choices made in its construction (e.g. choice of splitting (3.6)). Two-handle maps.
Suppose now that W consists only of 2-handle additions. In [OS06, Definition4.2] Ozsv´ath and Szab´o describe such cobordisms with a special type of Heegaard triple diagrams,as we now describe. Since W from Y to Y consists only of 2-handle additions, the cobordismcorresponds to surgery on some framed link L ⊂ Y . Denote by (cid:96) the number of components of L , and fix a basepoint in Y . Let B ( L ) be the union of L with a path from each component tothe basepoint. The boundary of a regular neighborhood of B ( L ) is a genus (cid:96) surface, which has asubset identified with (cid:96) punctured tori F i , one for each link component. WILLIAM E. OLSEN
Definition 3.1.
A Heegaard triple (Σ , α , β , γ , w ) is said to be subordinate to a bouqet B ( L ) forthe framed link L if( B1) (Σ , { α , . . . , α g } , { β , . . . , β g − (cid:96) } ) describes the complement of B ( L ).( B2) { γ , . . . , γ g − (cid:96) } , are small isotopic translates of { β , . . . , β g − (cid:96) } ( B3)
After surgering out the { β , . . . , β g − (cid:96) } , the induced curves β i and γ i , for i = g − (cid:96) + 1 , . . . , g ,lie on the punctured torus F i .( B4)
For i = g − (cid:96) + 1 , . . . , g , the curves β i represent meridians for the link components, disjointfrom all γ j for i (cid:54) = j , and meeting γ i in a single transverse point.( B5) for i = g − (cid:96) + 1 , . . . , g , the homology classes of the γ i correspond to the framings of thelink components.The following lemma shows that one can represent the cobordism W ( L ) via a Heegaard triplesubordinate to a bouquet for the framed link L . For a proof, see for example [Zem15, Lemma 9.4]or [OS06, Proposition 4.3]. Lemma 3.2.
Suppose (Σ , α, β , γ , w ) is a Heegaard triple that is subordinate to a bouquet for aframed link L in Y . After filling in the boundary component Y β , γ with - and -handles, we obtainthe handle cobordism W ( Y, L ) . We now define the cobordism maps for 2-handle cobordisms. Suppose L ⊂ Y is a framed linkin Y , and B ( L ) is a bouquet. Let (Σ , α , β , γ , w ) be a Heegaard triple subordinate to B ( L ). LetΘ ∈ T β ∩ T γ denote the intersection point in top Maslov grading [OS06, Section 2.4].If s ∈ Spin c ( W ( Y, L )), the 2-handle map F − L , s : CF − (Σ , α , β , w, s | Y ) → CF − (Σ , β , γ , w, s | Y ( L ) )is defined as a count of holomorphic triangles F − L , s ([ x , i ]) := (cid:88) y ∈ T α ∩ T γ (cid:88) ψ ∈ π ( x , Θ β, γ , y µ ( ψ )=0 s w ( ψ )= s M ( ψ ) · [ i − n w ( ψ )] , (3.8)where π ( x , Θ , y ) is the set of homotopy classes of Whitney triangles with vertices x , Θ , y , and M ( ϕ ) is the moduli space of holomorphic representatives of ϕ .Throughout Section 4, we will be interested in studying the holomorphic triangle map (3.8) fordiagrams which are not a priori subordinate to a bouquet for a framed link. We address this issuethere. 4. Trisections and Ozsv´ath-Szab´o cobordism invariants
In this section we prove Theorem A and demonstrate how one can use the data of a relativetrisection map π : X → R , along with some auxiliary data, to compute the induced cobordismmaps in Heegaard Floer homology.4.1. Constructing Heegaard triples from relative trisection diagrams.
Fix X to be acompact, oriented, connected, smooth 4-manifold. The input data we require is a tuple ( π, (cid:104)· , ·(cid:105) , H )consisting of a ( g, k ; p, b )-trisection map π : X → R , a Riemannian metric (cid:104)· , ·(cid:105) on X , and a π -compatible connection H . Equipped with such data, we may choose three reference arcs η α , η β , η γ :[0 , → D as in Figure 2 below. As discussed in Section 2.2, associated to these reference arcs arethree Morse functions f α , f β , f γ defined on the compression bodies U α , U β , and U γ respectively.For τ ∈ { α , β , γ } , these Morse functions satisfy: • f τ : U τ → [0 ,
3] is a Morse function with f − τ (0) = Σ and f − τ (3) = Σ τ the surface obtainedby doing surgery on Σ along the τ -curves; and, RISECTIONS AND OZSV ´ATH-SZAB ´O COBORDISM INVARIANTS 9
Figure 2.
For τ ∈ { α , β , γ } , we have reference arcs η τ : [0 , → D for which f τ : U τ → [0 ,
3] is a Morse function. Drawn in pink is the descending manifold foran index 2 critical point for f α whose intersection with Σ is an α curve drawn inred. • f τ has g − p index two critical points whose descending manifolds intersect Σ along the τ curves.We define the surface Σ α to be the fiber f − α (3) and we fix an identification of Σ α with Σ p,b .Next, endow Σ α with a model collection of pairwise disjoint arcs { a , . . . , a n } which constitute abasis for H (Σ α ; ∂ Σ α ), as in Figure 3 below. We call such a collection the standard arc basis , andnote that n can be computed as n = 2 p + b − a a a p − a p a p +1 a p + b − • • • Figure 3.
The standard arc basis of H (Σ α , ∂ Σ α ; Z ).Next, define { b , . . . , b n } ⊂ Σ α and { c , . . . , c n } ⊂ Σ α to be two additional arc bases whichsatisfy the following criteria (see Figure 4):(1) The arc bases { a , . . . , a n } , { b , . . . , b n } and { c , . . . , c n } are isotopic (not relative to theendpoints) by a small isotopy;(2) For each i = 1 , . . . , n , a i has a single positive transverse intersection with b i , where theorientation of b i is inherited from a i .(3) For each i = 1 , . . . , n , b i has a single positive transverse intersection with c i , where theorientation of the c i is inherited from the b i .(4) For each i = 1 , . . . , n , a i has a single positive intersection with c i .Our next step is to use the gradient vector field ∇ f α of the Morse function f α to flow thearcs { a , . . . , a n } ⊂ Σ α onto Σ. We’ll denote the images of { a , . . . , a n } under this flow by a = a i b i c i w∂ Σ α Figure 4.
A zoomed in picture near the boundary of Σ α . { a , . . . , a n } ⊂ Σ. Note that generic choices ensure that the a i are pairwise disjoint form each otherand from the original α -curves { α , . . . , α g − p } ⊂ Σ. Note, however, that the images { a , . . . , a n } are only well-defined up to handle-slides over the original α -curves.With the data of (Σ , α , β , γ ; a ) in hand (along with the additional data ( g, (cid:104)· , ·(cid:105) , H ) that westarted with), we’re ready to implement the monodromy algorithm [CGP18a, Theorem 5] of Gay-Castro-Pinz´on-Caiced´o to obtain two new collections of arcs b = { b , . . . , b n } and c = { c , . . . , c n } which, when taken all together with a , encode the monodromy diffeomorphism µ : Σ α → Σ α of theopen book on the boundary 3-manifold Y = ∂X (see [CGP18a] for more details).To obtain b , perform a sequence of handle-slides of a arcs over α curves until a ∩ β = ∅ ; theresulting collection of arcs is b = { b , . . . , b n } . Next, we obtain c by performing another sequenceof handle-slides of b arcs over β curves until b ∩ γ = ∅ , and denote the resulting collection of arcsby c = { c , . . . , c n } . By construction, the data D = (Σ , α , β , γ ; a , b , c ) constitute an arced (relative)trisection diagram of X [GM18, Definition 2.12].We now describe how to glue together the above data to construct a Heegaard triple, in thesense of Ozsv´ath-Szab´o [OS04b, Section 8.1], which encodes the cobordism X : ∅ → Y . LetΣ be the surface obtained by gluing the boundaries of Σ and − Σ α via an orientation reversingdiffeomorphism (see Figure 5 below) Σ := Σ ∪ ∂ − Σ α . (4.1)Note that the genus of Σ is g (Σ) = g + p + b − Figure 5.
Constructing Σ by identifying the boundary of the central surface Σ(left-hand-side) with that of a page of the open book (right-hand-side) on Y whichcomes decorated with the parallel arc bases { a } , { b } , and { c } , drawn in red, blue,and green respectively. RISECTIONS AND OZSV ´ATH-SZAB ´O COBORDISM INVARIANTS 11
Next, we define three new handlebodies U α , U β , and U γ , each bounded by Σ, by specifying theirattaching curves. The U α handlebody is determined by the curves { α , . . . , α g + p + b − } where α i = (cid:40) α i ≤ i ≤ | α | a i ∪ ∂ a i | α | + 1 ≤ i ≤ g (Σ) (4.2)For the β -handlebody U β , we define β i = (cid:40) β i ≤ i ≤ | β | b i ∪ ∂ b i | β | + 1 ≤ i ≤ g (Σ) (4.3)Finally, the γ -handlebody U γ is determined by γ i = (cid:40) γ i ≤ i ≤ | γ | c i ∪ ∂ c i | γ | + 1 ≤ i ≤ g (Σ) (4.4) Example 4.1.
Consider for example the relative trisection diagram for X = B in the left-hand-side of Figure 5. After performing the procedure described above, the resulting Heegaard triplelooks like Figure 6 shown below. Figure 6.
A Heegaard triple produced by applying the procedure described aboveto the relative trisection diagram for X = B . The closed surface Σ is obtained bygluing the central surface Σ to − Σ α along their boundaries, and the closed curves areobtained by taking a union of the original closed curves from the trisection diagram(Σ , α , β , γ ) with those obtained by ‘doubling’ the arc bases using the gradient vectorfields of f α , f β , and f γ , respectively.Thus far, we have described how, given a relative trisection diagram D = (Σ , α , β , γ ) which iscompatible with a given ( g, k ; p, b )-trisection map f : X → D , to construct a new Heegaard triple D = (Σ , α , β , γ ). However, it is not at all clear how the original 4-manifold X , as described bythe diagram D , and the potentially new 4-manifold X , as described by the diagram D , are related.The remainder of this section clarifies this relationship via a technique which we call a trisector’scut .Our strategy for relating X and X involves a series of intermediate manifolds which we nowdescribe. Starting with X , which comes equipped with the decomposition X = X ∪ X ∪ X ,consider a collar neighborhood of the boundary of X , denoted ν ( ∂X ). Figure 7.
A collar neighborhood of the boundary X .After rounding corners we parametrize this collar neighborhood via ϕ : [0 , × k S × S → ν ( ∂X ) , where ∂X is embedded in ν ( ∂X ) as { } × k S × S . For a chosen basepoint z ∈ π − (1) ∼ = Σ α ,let η : [0 , → ν ( ∂X )be a short arc connecting z to its image in { } × π − (1). This being done, delete from X thecomplement of ν ( ∂X ) union a tubular neighborhood of η . Figure 8.
A schematic for deleting the complement of ν ( ∂X ) union a tubularneighborhood of η .In symbols, delete the following subset from X : (cid:0) X \ ν ( ∂X ) (cid:1) ∪ ν ( η ) (4.5)We give the resulting 4-manifold a name, X , and its importance is demonstrated in Proposition4.2 below. Proposition 4.2.
Let X α , β , γ be the four-manifold constructed as in equation (2.1) from the Hee-gaard triple (Σ , α , β , γ ) , and define X to be the smooth four-manifold obtained from X α , β , γ afterfilling in the boundary components − Y α , β and − Y β , γ with (cid:92) k i +2 p + b − S × B , i = 1 , , respectively.Then the four-manifolds X and X are diffeomorphic.Remark. The author would like to warmly thank David Gay and Juanita Pinz´on-Caicedo for helpfulsuggestions during the development of this proof.
RISECTIONS AND OZSV ´ATH-SZAB ´O COBORDISM INVARIANTS 13
Proof.
The essential point of the argument is showing how to embed the spine X α , β , γ of X into X . To do so, we need to identify the surface Σ and the handlebodies it bounds U α , U β , and U γ as the appropriate submanifolds of X . The result will then quickly follow from the uniquenesstheorem of Laudenbach-Poenaru [LP72].To begin, notice that after making the modifications to X ⊂ X as in Figure 7, the base diagramis now reminiscent of the familiar keyhole contour which we parametrize as B = [ − π/ , π/ × [0 , θ ∈ [ − π/ , π/
6] and t ∈ [0 ,
1] are coordinates.Following Behrens [Beh14, Section 3.2], we say that a parametrization κ : B → [ − π/ , π/ × [0 , compatible with π if the critical image C κ := κ ◦ π ( Crit ( π )) is in the following standard position: • All cusps point to the right (i.e. in the positive t -direction). • Each R θ := { θ } × [0 ,
1] meets C κ in exactly g − p points, and each intersection is either ata cusp or meets transversely in a fold point. • For a fixed small ε >
0, there exists a 2 ε -neighborhood N ε of ∂ θ B := [ − π/ , π/ × { , } such that κ ◦ π ( Crit ( π )) ∩ N ε = ∅ . Figure 9.
An impressionistic picture of the trisector’s cut. The green arc in thebase represents η γ , and above it lies the relative compression body U γ viewed as arelative cobordism from the central surface Σ to Σ γ . Analogous statements can bemade for the red arc in the base which represents η α .Fix a π -compatible parametrization κ : B → [ − π/ , π/ × [0 ,
1] of the base, and consider thereference arcs η α := {− π/ } × [0 , η β := { π } × [0 , η γ := { π/ } × [0 , U α := π − ( η α ) U β := π − ( η β ) U γ := π − ( η γ ) are each relative compression bodies. It’s well-known that one can round the corners of thesecompression bodies and obtain honest 3-dimensional handlebodies. To be explicit, we refer toLemma 8.4 of [JZ18] where the reader can also find a proof. Lemma 4.3 (Lemma 8.4 of [JZ18]) . Let U α be the relative compression body formed by attaching -dimensional -handles to I × Σ along the curves { } × α . After rounding corners, we can view U α as a handlebody (in the usual sense) of genus | α | − χ (Σ α ) + 1 and boundary (cid:0) { } × Σ (cid:1) ∪ ∂ Σ α . Furthermore, a set of compressing disks for U α can be obtained by taking | α | compressing disks D α with boundary { } × α for α ∈ α , as well as disks of the form D c ∗ i := I × c ∗ i for pairwise disjoint,embedded arcs c ∗ , . . . , c ∗ b Σ α in Σ that avoid the α curves, and form a basis of H (Σ α , ∂ Σ α ) . Thesecut U α into a single -ball. Applying Lemma 4.3 to the three sutured compression bodies U α , U β , and U γ above, we obtainthree 3-dimensional handlebodies with ∂U τ = Σ τ for each τ ∈ { α , β , γ } . We take as the centralsurface in our spine-decomposition of X to be Σ := ∂U α . Clearly, Σ bounds the U α handle-bodydescribed in equation (4.2). Notice, however, that the U β and U γ handlebodies are completelydisjoint from Σ. To remedy this, we isotope the attaching circles for the β - and γ -handlebodiesonto Σ, and it is via this isotopy that we see how the monodromy of the open book decompositionof Y naturally arises. After isotoping the attaching curves onto the same central surface Σ, we willhave completed the proof that the spine X α , β , γ of X embeds into X .Now, we’ll construct an isotopy for the attaching circles for the handlebodies U β and U γ . Todo so, recall from the beginning of this section that we have a chosen π -compatible connection H . The first step is to thicken the surface Σ β := ∂U β to Σ β × [0 , ε ] using the inward pointingnormal direction coming from the boundary. Since we have an π -compatible parametrization of thebase, the attaching circles on Σ β × { ε } are isotopic to those of Σ β = Σ β × { } . Next, we use the π -compatible connection H to transport the attaching circles on Σ β × { ε } onto to Σ α × { ε } .Since the β attaching circles have been isotoped onto Σ using a π -compatible connection, itfollows by the monodromy algorithm of [CGP18a, Theorem 5] that the resulting curves are preciselythose for U β –see Figure 10 below. Next, we repeat the above process using the γ attaching circlesand arrive at the same conclusion for U γ . Thus, we’ve shown that the spine X α , β , γ of X embeds into X , and the proposition follows after applying the uniqueness theorem of [LP72] to the remainingboundary components. (cid:3) Remark.
The boundary of X is Y k S × S ), and after filling in the k S × S , we recoverthe original 4-manifold X . Corollary 4.4.
In the Heegaard triple (Σ , α , β , γ , w ) constructed above, we have that (Σ , α , β ) , (Σ , β , γ ) and (Σ , α , γ ) are Heegaard diagrams for the three-manifolds (cid:96) S × S , (cid:96) S × S , and Y k S × S ) where (cid:96) i = k i + 2 p + b − .Proof. The statements for (Σ , α , β ) and (Σ , β , γ ) follow from a combination of two facts; the firstbeing that (Σ , α , β , γ ) is a relative trisection, so that to begin with the pairwise tuples yield connectsums of S × S ; and the second being that the monodromy of the open book can be trivializedover one sector at a time. (cid:3) Holomorphic triangles and cobordism maps.
Fix X to be a smooth, oriented, compactfour-manifold with connected boundary, and equip X with a ( g, k ; p.b )-trisection map π : X → D ,a metric (cid:104)· , ·(cid:105) , and a π -compatible connection H as in Section 4.1 above. If (Σ , α , β , γ ) is a RISECTIONS AND OZSV ´ATH-SZAB ´O COBORDISM INVARIANTS 15
Figure 10.
A schematic for visualizing how the attaching curves for U β and U γ areisotoped onto Σ using parallel transport. On the right, there is a filled-in red squarewhich is meant to represent the relative compression body U α and its boundary isΣ–similarly for the blue and green squares. Once the blue square is ‘pushed in’,one can flow the attaching curves clockwise onto a copy of Σ which is also slightlypushed in. Finally, one employs the same strategy to the γ curves. The generalimpression should be that of a nautilus shell.(relative) trisection diagram associated to these data, we show how the holomorphic triangle map(3.8) applied to the pointed Heegaard triple H = (Σ , α , β , γ , w ) computes the induced cobordismmap of Ozsv´ath and Szab´o. Proposition 4.5.
Let H = (Σ , α , β , γ , w ) be a pointed Heegaard triple constructed using the pre-scription described in subsection 4.1 above, and let X α , β , γ be its associated four-manifold spinewhich we view as a cobordism from (cid:96) S × S to Y after filling in − Y β , γ with (cid:92) (cid:96) S × B anda -handle. Then H is slide-equivalent to another Heegaard triple H (cid:48) which is subordinate to abouquet for a framed link L ⊂ (cid:96) S × S for which the -handle cobordism W ( (cid:96) S × S , L ) isdiffeomorphic to X α , β , γ as cobordisms from (cid:96) S × S to Y .Proof. Recall from [MSZ16, Definition 4.5] that a disk D γ properly embedded in U γ is primitive in U γ with respect to U β (cid:48) if there exists a compression disk D β (cid:48) i satisfying the condition | D γ ∩ D β (cid:48) i | = 1.Since (Σ , β (cid:48) , γ ) is a genus g = g + p + b − (cid:96) S × S , it follows from[MSZ16, Theorem 2.7] that U γ admits an ordered collection of compression disks { D γ (cid:48) i } where thecorresponding attaching circles γ (cid:48) i = ∂D γ (cid:48) i satisfy(1) For i = 1 , . . . , g − k − p , γ (cid:48) i satisfies | γ (cid:48) i ∩ β (cid:48) i | = 1 and | γ (cid:48) i ∩ β j | = 0 for i (cid:54) = j .(2) For i = g − k − p + 1 , . . . , g + p + b − γ (cid:48) i is parallel to β (cid:48) i .We remark that since γ (cid:48) and γ are cut systems for the same handlebody U γ , it follows from [Joh06]that γ ∼ γ (cid:48) . This being done, it follows from [KM20, p.5] (see, in particular [KM20, Figure 2]) that for i = 1 , . . . , g − k − p , γ (cid:48) i can be interpreted as the framed attaching sphere for a 2-handle cobordism,where each γ (cid:48) i is given the surface framing.Finally, we exhibit a bouquet for the framed attaching link L = { γ (cid:48) , . . . , γ (cid:48) g − p − k } and check that(Σ , α (cid:48) , β (cid:48) , γ (cid:48) ) is subordinate to it. For each γ (cid:48) i ∈ { γ (cid:48) , . . . , γ (cid:48) g − k − p } , choose a properly embedded arc η i ⊂ U β which has one endpoint on γ (cid:48) i and the other on w , the fixed basepoint. Then the union of η i comprise a bouquet for the link L . Furthermore, (Σ , { α , . . . , α (cid:48) g + p + b − } , { β (cid:48) g − p − k +1 , . . . , β (cid:48) g + p + b − } is a Heegaard diagram for the complement of L in (cid:96) S × S . Next, taking a thin tubularneighborhood of β (cid:48) i ∪ γ (cid:48) i constitutes a punctured torus for each i = 1 , . . . , g − k − p . Last, theconditions that β (cid:48) i constitute a meridian and that γ (cid:48) i constitute a longitude are self evident afterusing the surface framing to push γ (cid:48) i into U β handlebody. Thus, the conditions ( B1) – (
B5) aresatisfied. (cid:3)
The remainder of the proof of Theorem A is an application of various naturality results whichare standard in the Heegaard Floer theory. We recapitulate some of the details here, but we claimno originality to them–see, for example, [OS04b, p.360] for what is essentially the same argumentand [JTZ12] for more details concerning naturality issues in Heegaard Floer homology. Following[JTZ12], we make a notational definition.
Definition 4.6.
Let (Σ , α , β , β (cid:48) ) be an admissable triple diagram. If β ∼ β (cid:48) are handle-slideequivalent, then we’ll write Ψ αβ → β (cid:48) for the map F ◦ α , β , β (cid:48) ( − ⊗ Θ β , β (cid:48) ) : HF ◦ (Σ , α , β ) → HF ◦ (Σ , α , β (cid:48) ) (4.6)Similarly, if α (cid:48) ∼ α , then let Ψ α (cid:48) → αγ denote the map F ◦ α (cid:48) , α , γ (Θ α (cid:48) , α ⊗ − ) : HF ◦ (Σ , α (cid:48) , γ ) → HF ◦ (Σ , α , γ ) (4.7)We take a moment to compare Spin c -structures on X to those on X . Observe that there is anatural restriction map r : Spin c ( X ) → Spin c ( X ) (4.8)The restriction map r is surjective, and conversely, a Spin c -structure s on X admits a uniqueextension to X if it is isomorphic to the unique torsion Spin c -structure s in a neighborhood of k S × S . Proposition 4.7.
Fix a
Spin c -structure s ∈ Spin c ( X ) . Let H = (Σ , α , β , γ , w ) be the pointed s -admissable Heegaard triple constructed as above, and let H (cid:48) = (Σ , α (cid:48) , β (cid:48) , γ (cid:48) , w ) be a Heegaardtriple which is strongly equivalent to H and which is subordinate to a bouquet for a framed link L as in Proposition 4.5 above. Then in the diagram below HF ◦ (Σ , α , β , s ) HF ◦ (Σ , α , γ , s α , γ ) HF ◦ (Σ , α (cid:48) , β (cid:48) , s ) HF ◦ (Σ , α (cid:48) , γ (cid:48) , s α (cid:48) , γ (cid:48) ) Ψ α → α (cid:48) β → β (cid:48) F ◦ α , β , γ , s Ψ α → α (cid:48) γ → γ (cid:48) F ◦ L , s we have the following equality F ◦ L , s ◦ Ψ α → α (cid:48) β → β (cid:48) (Θ α , β ) = Ψ α → α (cid:48) γ → γ (cid:48) ◦ F α , β , γ , s (Θ α , β ) (4.9) RISECTIONS AND OZSV ´ATH-SZAB ´O COBORDISM INVARIANTS 17
Proof.
Similar results are common in the literature, so we’ll be brief (cf. [OS06, p.360]). Byassumption, the cut systems α ∼ α (cid:48) , β ∼ β (cid:48) , and γ ∼ γ (cid:48) are related by sequences of isotopies andhandleslides. Start by considering the sequence α ∼ α (cid:48) , which yields the following diagram: HF ◦ (Σ , α , β , s ) HF ◦ (Σ , α , γ , s α , γ ) HF ◦ (Σ , α (cid:48) , β , s ) HF ◦ (Σ , α (cid:48) , γ , s α (cid:48) , γ ) F ◦ α , β , γ , s Ψ α → α (cid:48) β Ψ α → α (cid:48) γ F ◦ α (cid:48) , β , γ , s Figure 11.
The commutative square associated to the sequence of isotopies andhandle slides connecting α to α (cid:48) .By [JTZ12, Proposition 9.10] we have that both Ψ α → α (cid:48) β and Ψ α → α (cid:48) γ are isomorphisms, and by[JTZ12, Lemma 9.4] we have that HF ◦ top (Σ , α , β , s ) ∼ = F (cid:104) Θ α , β (cid:105) and HF ◦ top (Σ , α (cid:48) , β , s ) ∼ = F (cid:104) Θ α (cid:48) , β (cid:105) .It is now immediate that Ψ α → α (cid:48) β (Θ α , β ) = Θ α (cid:48) , β .Using [JTZ12, Lemma 9.5], we may assume that (Σ , α (cid:48) , α , β , γ , w ) has also been made admiss-able, so we can apply the associativity theorem for holomorphic triangles [OS04b, Theorem 8.16]and conclude that F ◦ α (cid:48) , α , γ (cid:0) Θ α (cid:48) , α ⊗ F ◦ α , β , γ , s (Θ α , β ⊗ Θ β , γ ) (cid:1) = F ◦ α (cid:48) , β , γ , s (cid:0) F ◦ α (cid:48) , α , β (Θ α (cid:48) , α ⊗ Θ α , β ) ⊗ Θ β , γ (cid:1) (4.10)Clearly, equation (4.10) shows that the diagram in Figure 11 commutes for the generator Θ α , β .Having handled the sequence α ∼ α (cid:48) , we consider next the sequence of isotopies and handleslidesamongst the β -curves. In a similar fashion, we consider the following diagram HF ◦ (Σ , α (cid:48) , β , s ) HF ◦ (Σ , α (cid:48) , γ , s α (cid:48) , γ ) HF ◦ (Σ , α (cid:48) , β (cid:48) , s ) HF ◦ (Σ , α (cid:48) , γ , s α (cid:48) , γ ) F ◦ α (cid:48) , β , γ , s Ψ α (cid:48) β → β (cid:48) F ◦ α (cid:48) , β (cid:48) , γ , s Figure 12.
The commutative square associated to the sequence of isotopies andhandle slides connecting β to β (cid:48) .The proof that Figure 12 is commutative, however, is slightly different than that for Figure 11, sowe include the proof here. As before, we apply [JTZ12, Lemma 9.5] to justify that (Σ , α (cid:48) , β , β (cid:48) , γ , w )is admissable. Applying the associativity theorem for holomorphic triangles, we see that F ◦ α (cid:48) , β (cid:48) , γ (cid:0) F ◦ α (cid:48) , β , β (cid:48) (Θ α (cid:48) , β ⊗ Θ β , β (cid:48) ) ⊗ Θ β (cid:48) , γ (cid:1) = F ◦ α (cid:48) , β , γ (cid:0) Θ α (cid:48) , β ⊗ F ◦ β , β (cid:48) , γ (Θ β (cid:48) , β ⊗ Θ β (cid:48) , γ ) (cid:1) (4.11)By again applying [JTZ12, Proposition 9.10] and [JTZ12, Lemma 9.4], we observe that F ◦ β , β (cid:48) , γ (Θ β , β (cid:48) ⊗ Θ β (cid:48) , γ ) = Θ β , γ (4.12)which turns equation (4.12) into F ◦ α (cid:48) , β (cid:48) , γ (cid:0) F ◦ α (cid:48) , β , β (cid:48) (Θ α (cid:48) , β ⊗ Θ β , β (cid:48) ) ⊗ Θ β (cid:48) , γ (cid:1) = F ◦ α (cid:48) , β , γ (Θ α (cid:48) , β ⊗ Θ β , γ ) (4.13)It is immediate from equation (4.13) that Figure 12 commutes for the generator Θ α (cid:48) , β . Having studied the sequences α ∼ α (cid:48) and β ∼ β (cid:48) , we leave it to the reader to build an analogouscommutative diagram for the sequence γ ∼ γ (cid:48) and top generator Θ α (cid:48) , γ . The proof that it iscommutative follows as for the sequence β ∼ β (cid:48) .To demonstrate the assertion made in the proposition, we observe that after stacking Figures 11and 12 on top of the appropriate diagram for the γ ∼ γ (cid:48) sequence, we arrive at a new commutativediagram which is equivalent to equation (4.9). This is so for two reasons: first, by Definition themaps F ◦ α (cid:48) , β (cid:48) , γ (cid:48) , s and F ◦ L , s are equivalent, and second, by [JTZ12, Proposition 9.10] we haveΨ α → α (cid:48) β → β (cid:48) = Ψ α (cid:48) β → β (cid:48) ◦ Ψ α → α (cid:48) β and Ψ α → α (cid:48) γ → γ (cid:48) = Ψ α (cid:48) γ → γ (cid:48) ◦ Ψ α → α (cid:48) γ . (cid:3) Theorem 4.8.
In the diagram below, HF ◦ ( S ) HF ◦ ( Y, s ) HF ◦ (Σ , α , β , s ) HF ◦ (Σ , α , γ , s α , γ ) F F ◦ X, s F ◦ α , β , γ , s α , β , γ F the following equality holds F ◦ F ◦ α , β , γ , s α , β , γ ◦ F (Θ) = F ◦ X, s (Θ) (4.14) Proof.
This follows immediately after combining the construction of X with Proposition 4.7, thedefinitions of the 1- and 3-handle cobordism maps, and the classic results of [OS06] which showthat F X, s is independent of the handle decomposition of X . (cid:3) Remarks on the contact class.
Fix X to be a smooth, oriented, compact four-manifold withconnected boundary ∂X = Y . As discussed in Section 2.1, a ( g, k ; p, b )-trisection map π : X → D induces an open book decomposition on its boundary 3-manifold Y . Given the data of such anopen book, Honda-Kazez-Matic [HKM09] define a class c ( ξ ) ∈ HF + ( − Y, s ξ ) and show that c + ( ξ )agrees with the Ozsv´ath-Szab´o contact invariant [OS05] associated to ( Y, ξ ), where ξ is a contactstructure supported by the given open book. In this section, we initiate a study of the relationshipbetween c + ( ξ ) and relative trisection maps π inducing an open book which supports ξ .To begin, fix a ( g, k ; p.b )-trisection map π : X → D , and let (Σ , α , β , γ ) be its associateddiagram. Next, construct the pointed Heegaard triple (Σ , β , γ , α , w ) as in Subsection 4.1 above.Following [Bal13, Section 2.2], define for each i = g − p + 1 , . . . , g (Σ) = g + p + b − θ i , x i , and y i , as shown in Figure 13 below, θ i = β i ∩ γ i ∩ Σ α x i = γ i ∩ α i ∩ Σ α (4.15) y i = β i ∩ α i ∩ Σ α and let Θ, x , and y be the corresponding intersection points Θ = { Θ (1) β , γ , . . . , Θ ( g − p ) β , γ , θ g − p +1 , . . . , θ g + p + b − } ∈ T β ∩ T γ x = { Θ (1) γ , α , . . . , Θ ( g − p ) γ , α , x g − p +1 , . . . , x g + p + b − } ∈ T γ ∩ T α (4.16) y = { Θ (1) β , α , . . . , Θ ( g − p ) β , α , y g − p +1 , . . . , y g + p + b − } ∈ T β ∩ T α We have intentionally flipped the roles of α , β , and γ in this construction, as will be apparent momentarily. RISECTIONS AND OZSV ´ATH-SZAB ´O COBORDISM INVARIANTS 19
To describe the symbols Θ ( i ) ξ , ζ , for i = 1 , . . . , g − p and ξ , ζ ∈ { α , β , γ } , recall that by the connectsum formula [OS04a] and Corollary 4.4, it follows that HF + (Σ , β , α , s ) ∼ = HF + (Σ , γ , β , s ) ∼ = Λ ∗ ( H ( k +2 p + b − S × S )) ⊗ F [ U, U − ] /U · F [ U ] (4.17)and HF + (Σ , γ , α , s s ) ∼ = HF + ( Y, s ) ⊗ HF + ( k S × S ; s ) (4.18)With these observations in mind, we choose the Θ ( i ) ξ , ζ so that they represent the top-degree homologyclass in these decompositions. wx i y i θ i Σ α Σ Figure 13.
A local picture of the intersection points θ i , x i , and y i .Given the above familiar setting, we’d like to make a few remarks: • As in [HKM09], the generator [ x ,
0] is a cycle in CF + (Σ , γ , α , w ), and its image in homologyis mapped to c + ( Y, ξ ) ∈ HF + ( − Y, s ξ ) under the 3-handle cobordism map. That is, HF + ( − Y , s ξ s ) HF + ( − Y, s ξ )[ x , c + ( ξ ) F In particular, the image of c + ( Y, ξ ) under F + X, s coincides with the image of [ x ,
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