Augmentations and immersed Lagrangian fillings
AAUGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS
YU PAN AND DAN RUTHERFORD
Abstract.
For a Legendrian link Λ ⊂ J M with M = R or S , immersed exact Lagrangianfillings L ⊂ Symp( J M ) ∼ = T ∗ ( R > × M ) of Λ can be lifted to conical Legendrian fillingsΣ ⊂ J ( R > × M ) of Λ. When Σ is embedded, using the version of functoriality for Legendriancontact homology (LCH) from [30], for each augmentation α : A (Σ) → Z / (cid:15) (Σ ,α ) : A (Λ) → Z /
2. With Σ fixed, the set of homotopyclasses of all such induced augmentations, I Σ ⊂ Aug (Λ) / ∼ , is a Legendrian isotopy invariant ofΣ. We establish methods to compute I Σ based on the correspondence between Morse complexfamilies and augmentations. This includes developing a functoriality for the cellular DGA from[31] with respect to Legendrian cobordisms, and proving its equivalence to the functoriality forLCH. For arbitrary n ≥
1, we give examples of Legendrian torus knots with 2 n distinct conicalLegendrian fillings distinguished by their induced augmentation sets. We prove that when ρ (cid:54) = 1and Λ ⊂ J R every ρ -graded augmentation of Λ can be induced in this manner by an immersedLagrangian filling. Alternatively, this is viewed as a computation of cobordism classes for anappropriate notion of ρ -graded augmented Legendrian cobordism. Contents
1. Introduction 12. Immersed DGA maps and cobordisms 53. Immersed DGA maps and augmentations 144. Immersed maps and the cellular DGA 185. Computations via Morse complex families 276. Examples 377. Every augmentation is induced by an immersed filling 40Appendix A. Isomorphism between the cellular and immersed LCH functors 51Appendix B. A geometric model for Legendrian cobordisms 61References 67 Introduction
A fundamental holomorphic curve invariant of a Legendrian submanifold, Λ, is the Legendriancontact homology (LCH) dg-algebra (DGA), denoted A (Λ). As part of the symplectic fieldtheory package, the LCH algebra is functorial for an appropriate class of cobordisms. In thisarticle we consider 1-dimensional Legendrian links in the 1-jet spaces, J M with M = R or S ,and exact Lagrangian cobordisms in the symplectization, Symp ( J M ) = R × J M ; throughout,our coefficient field is Z /
2. For Λ − , Λ + ⊂ J M , such a cobordism, L : Λ − → Λ + , cylindricalover Λ − and Λ + at the negative and positive ends of Symp ( J M ), equipped with a Z /ρ -valuedMaslov potential induces a Z /ρ -graded DGA map f L : A (Λ + ) → A (Λ − ), cf. [13, 15]. Inparticular, when L is an exact Lagrangian filling, i.e. a cobordism L : ∅ → Λ, the induced map (cid:15) L : A (Λ) → Z / ρ -graded augmentation which by definition is a unital ring homomorphism that satisfies (cid:15) L ◦ ∂ = 0 and preserves a Z /ρ -grading on A (Λ).A natural question is: a r X i v : . [ m a t h . S G ] J un YU PAN AND DAN RUTHERFORD
Question 1.1.
Which augmentations come from exact Lagrangian fillings?While orientable exact Lagrangian fillings have been constructed for several classes of Leg-endrian knots [15, 21, 36, 38], there are many augmentations that cannot be induced by anyorientable filling as obstructions to such fillings arise from the Thurston-Bennequin number ofΛ and from the linearized homology of the augmentation; see [7, 13, 11]. The main result ofthis article shows that if one extends the setting to allow immersed cobordisms with doublepoints then the algebra more closely matches the geometry. When ρ is even, every ρ -gradedaugmentation can be induced by an orientable immersed exact Lagrangian filling. An extension of the functoriality for LCH to immersed Lagrangian cobordisms is implementedin [30] by working with a class of Legendrian cobordisms as follows. Applying a symplectomor-phism
Symp ( J M ) ∼ = T ∗ ( R > × M ) an exact immersed Lagrangian cobordism, L ⊂ Symp ( J M ),can be lifted to a Legendrian Σ ⊂ J ( R > × M ) with the cylindrical ends of L translating toconical ends for Σ and double points of L becoming Reeb chords of Σ. See Section 2.4. WhenΣ is embedded and equipped with a Z /ρ -valued Maslov potential, such a conical Legendriancobordism , Σ : Λ − → Λ + , induces a diagram of ρ -graded DGA maps(1.1) A (Λ + ) f Σ → A (Σ) i Σ ← (cid:45) A (Λ − )where A (Σ) is generated by A (Λ − ) and the Reeb chords of Σ. Diagrams of the above formare referred to in [30] as immersed DGA maps . There, a notion of homotopy for immersedDGA maps is introduced, and the homotopy type of the immersed map (1.1) is shown to be aninvariant of the conical Legendrian isotopy type of Σ. When Σ : ∅ → Λ is a conical Legendrianfilling equipped with a choice of ρ -graded augmentation, α : A (Σ) → Z /
2, we can then definean induced augmentation (cid:15) (Σ ,α ) : A (Λ) → Z / (cid:15) (Σ ,α ) = α ◦ f Σ . Thisgeneralizes the construction of induced augmentations from embedded Lagrangian fillings.We can now state our first main result, where in the following we write (cid:15) (cid:39) (cid:15) (cid:48) to indicate thattwo augmentations are DGA homotopic. Theorem 1.2.
Let Λ ⊂ J R have the Z /ρ -valued Maslov potential µ where ρ ≥ , and let (cid:15) : A (Λ) → Z / be any ρ -graded augmentation. (1) If ρ (cid:54) = 1 , there exists a conical Legendrian filling Σ of Λ with Z /ρ -valued Maslov potentialextending µ together with a ρ -graded augmentation α : A (Σ) → Z / such that (cid:15) (cid:39) (cid:15) (Σ ,α ) .Moreover, if ρ is even, then Σ is orientable. (2) If ρ = 1 , then there exists a conical Legendrian cobordism Σ : U → Λ where U is thestandard Legendrian unknot with tb ( U ) = − together with a -graded augmentation α : A (Σ) → Z / such that (cid:15) (cid:39) α ◦ f Σ . The algebra of the standard Legendrian unknot, U , is generated by a single Reeb chord b ofdegree 1. In the case where ρ = 1, if the restriction of α to A ( U ) ⊂ A (Σ) sends b to 0, then byconcatenating with the standard filling of U we see that (cid:15) can be induced by a pair (Σ (cid:48) , α (cid:48) ) whereΣ (cid:48) is a conical Legendrian filling. However, in the case that α restricts to the augmentation of A ( U ) that maps b to 1, we are not sure whether (cid:15) can be induced by a Legendrian filling. Infact, we conjecture that this is not possible. Conjecture 1.3.
There is no conical Legendrian filling Σ of the Legendrian unknot U with -graded augmentation α : A (Σ) → Z / such that the induced augmentation (cid:15) (Σ ,α ) maps theunique Reeb chord b to . For the case of embedded Lagrangians, induced augmentations provide an effective meansof distinguishing Lagrangian fillings of a given Legendrian knot. The results of [30] show thatthese induced augmentations are actually invariants of the associated conical Legendrian fill-ings. (Note that conical Legendrian isotopy appears to be a much less restrictive notion ofequivalence than isotopy of the corresponding exact Lagrangians, since during the course of a
UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 3
Legendrian isotopy any number of double points can be added and removed from the Lagrangianprojections.)A key new feature in the immersed case is that a single Legendrian filling, Σ, can inducemore than one augmentation of Λ due to the dependence of the construction on the choiceof augmentation α of A (Σ). As a result, to obtain an invariant of Σ we should consider the set of (DGA homotopy classes of) induced augmentations I Σ ⊂ Aug ρ (Λ) / ∼ . After developingmethods to compute this invariant induced augmentation set , we demonstrate that I Σ canbe effective for distinguishing conical Legendrian fillings in the following theorem. Theorem 1.4.
For each n ≥ , there exists n distinct conical Legendrian fillings, Σ , . . . , Σ n of the max-tb Legendrian torus knot T (2 , n + 1) such that (i) the Σ i are all orientable with genus n − and have Z -valued Maslov potentials, (ii) each Σ i has a single Reeb chord of degree , (iii) and the induced augmentation sets satisfy I Σ i (cid:54) = I Σ j when i (cid:54) = j . Note that the Lagrangian projections L , . . . , L n are immersed Lagrangian fillings of T (2 , n +1) with a single double point. Moreover, the dg-algebras A (Σ i ) are all isomorphic to one another,and thus do not distinguish the Σ i on their own.1.1. Cobordism classes of augmented Legendrians.
In the context of relative symplecticfield theory (SFT) type invariants [19], it is natural to consider Lagrangian cobordisms betweenLegendrian submanifolds in the symplectization of a contact manifold (or more generally insymplectic cobordisms with concave and convex ends modeled on the negative and positiveends of symplectizations). In the following discussion, we refer to exact Lagrangian cobordismsin
Symp ( J M ) with cylindrical ends as SFT-cobordisms . The relation defined by embeddedSFT-cobordisms, or even SFT-concordances, is not symmetric [8, 28], and hence does not definean equivalence relation on Legendrian links in J M . In fact, it is a major open question [10] inthe field whether or not SFT-cobordisms define a partial order on the set of Legendrian isotopyclasses in J R .The lack of a readily available symmetry is visible after lifting an SFT-cobordism to a conicalLegendrian cobordism Σ ⊂ J ( R > × M ) = T ∗ ( R > × M ) × R z in the difference in behavior at thetwo ends of Σ: as Σ approaches 0 (resp. + ∞ ) along the R > factor the cotangent coordinatesand the z -coordinates of Σ appear as those of Λ − (resp. Λ + ) but shrinking (resp. expanding).However, when one allows for SFT-cobordisms to be immersed it becomes possible to reversetheir direction as an expanding end can be modified to a shrinking end (and vice-versa) atthe expense of creating some additional Reeb chords. Thus, the relation of conical Legendriancobordism is equivalent to another standard notion of Legendrian cobordism in 1-jet spacesintroduced by Arnold, cf. [1, 2, 3]. In Arnold’s definition, a Legendrian cobordism between twoLegendrians Λ , Λ ⊂ J M is a compact Legendrian, Σ ⊂ J ([0 , × M ), whose restriction to J ( { i }× M ) is Λ i . Seminal results in this theory of Legendrian (and also Lagrangian) cobordismswere achieved by Audin, Eliashberg, and Vassiliev in the 1980’s, including homotopy theoreticcharacterizations of various Lagrange and Legendre cobordism groups. See [17, 4, 39].For an alternate perspective on our main theorem, we can incorporate augmentations intoan Arnold-type cobordism theory. Define a ρ -graded augmented Legendrian to be a Leg-endrian submanifold Λ ⊂ J M equipped with a Z /ρ -valued Maslov potential and a ρ -gradedaugmentation (cid:15) : A (Λ) → Z /
2. For compact Legendrian cobordisms Σ ⊂ J ([0 , × M ) sat-isfying a suitable Morse minimum boundary condition (see Section 2.5), the LCH dg-algebraof Σ is defined as in [16], and contains A (Λ ) and A (Λ ) as sub-dg-algebras. We then declaretwo ρ -graded augmentation Legendrians, (Λ , (cid:15) ) and (Λ , (cid:15) ), to be cobordant if there existsa pair (Σ , α ) consisting of a Legendrian cobordism equipped with a ρ -graded augmentation of A (Σ) whose restriction to A (Λ i ) is homotopic to (cid:15) i for i = 0 ,
1. As a variant on Theorem 1.2we obtain the following.
YU PAN AND DAN RUTHERFORD
Theorem 1.5.
Let ρ ≥ be a non-negative integer, and let (Λ , (cid:15) ) be a ρ -graded augmentedLegendrian in J R . (1) If ρ (cid:54) = 1 , then (Λ , (cid:15) ) is cobordant to ∅ . (2) If ρ = 1 , then (Λ , (cid:15) ) is either cobordant to ∅ or ( U, (cid:15) ) where U is the standard Legendrianunknot and (cid:15) : A ( U ) → Z / satisfies (cid:15) ( b ) = 1 on the unique Reeb chord of U . As in Conjecture 1.3, we expect that (
U, (cid:15) ) is not null-cobordant, so that there should beexactly one cobordism class when ρ (cid:54) = 1 and exactly two cobordism classes when ρ = 1.In a further article [29], we give a complete classification of cobordism classes of augmentedLegendrians in J S . Interestingly, in J S it is often the case that cobordant Legendriansmay become non-cobordant once they are equipped with augmentations. We note that for longLegendrian knots in R a concordance group of similar spirit, incorporating quadratic at infinitygenerating families as additional equipment rather than augmentations, is considered recentlyin the work of Limouzineau, [25].1.2. Methods and outline.
Our approach is based on working with certain algebraic/combinatorialstructures equivalent to augmentations called Morse complex families (MCFs) that were intro-duced by Pushkar (unpublished) and studied by Henry in [22]. MCFs can be viewed as com-binatorial approximations to generating families of functions, as they consist of formal Morsecomplexes and handleslide data assigned to a Legendrian submanifold. For 1-dimensional Leg-endrians, the work of Henry [22, 24] establishes a bijection between equivalence classes of MCFsand homotopy classes of augmentations. More recently, a correspondence between MCFs andaugmentations for 2-dimensional Legendrians was obtained in [34] using the cellular DGA whichis a cellular model for LCH developed in the work of the second author and Sullivan [31, 32, 33].The strategy of the present article is to extend these methods to the case of Legendrian cobor-disms, and then apply Morse complex families to compute induced augmentation sets. In moredetail, to prove Theorem 1.2 we accomplish the following tasks (1)-(3). Note that (1) and (2)may be of some independent interest.(1) We develop a functoriality of the cellular DGA for Legendrian cobordisms, and we extendthe equivalence with LCH from [31, 32, 33] to an isomorphism between the immersedLCH functor from [30] and its cellular analog.Although it seems crucial for this construction to be carried out in the broader setting ofLegendrian cobordisms and immersed DGA maps, this allows the cellular DGA to be appliedjust as well for working with embedded SFT-cobordisms.(2) We extend the correspondence between MCFs and augmentations to the case of 2-dimensional Legendrian cobordisms. In particular, we give a method based on MCFsfor computing the induced augmentation sets of Legendrian cobordisms and fillings.In Section 6, we illustrate this method with examples and use it to prove Theorem 1.4. Makinguse of (1) and (2), Theorem 1.2 is then reduced to the following construction.(3) Given a 1-dimensional Legendrian knot, Λ ⊂ J R , with a ρ -graded MCF C Λ with ρ (cid:54) = 1(resp. ρ = 1), we produce a Legendrian filling Σ of Λ (resp. a Legendrian cobordismΣ : U → Λ) with a 2-dimensional ρ -graded MCF C Σ extending C Λ .In Section 7 we provide such a construction via an induction on the complexity of the frontprojection of Λ. This construction is simplified by making use of a standard form for MCFs(the “ SR -form”) introduced by Henry.The remainder of the paper is organized as follows. In Section 2 we collect background from[30] on immersed DGA maps, conical Legendrian cobordisms, and the immersed LCH functor, F . In addition, we discuss Morse minimum cobordisms and show that Theorem 1.5 follows from UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 5
Theorem 1.2. In Section 3, we establish some basic properties of induced augmentation setsassociated to immersed DGA maps including a composition formula and Legendrian isotopyinvariance. In Section 4, we review the cellular DGA and define a cellular LCH functor, F cell .We state an isomorphism between the cellular LCH functor and the immersed LCH functor.The proof of the isomorphism is an extension of [32, 33], and is postponed to the AppendicesA and B. In Section 5, we consider augmentations of the cellular DGA, and after reviewing thedefinition, we extend the correspondence between Morse complex families and augmentationsto the cobordism case and establish an MCF characterization of the induced augmentationset. In Section 6, we make use of A -form MCFs to compute the induced augmentation setsfor several examples of Legendrian fillings, and we prove Theorem 1.4. In particular, we giveexplicit examples of augmentations that can be induced by Legendrian fillings but cannot beinduced by any embedded fillings. Finally, the main body of the article concludes with Section7 that, after recalling Henry’s SR -form MCFs, provides the proof of Theorem 1.2.1.3. Acknowledgements.
DR is partially supported by grant 429536 from the Simons Foun-dation. YP is partially supported by the NSF Grant (DMS-1510305).2.
Immersed DGA maps and cobordisms
In this section we review the algebraic and geometric setup from [30], and discuss two classesof Legendrian cobordisms. Section 2.1 recalls a class of DGAs relevant for Legendrian contacthomology, and records in Proposition 2.1 a method for producing stable tame isomorphisms.In Section 2.2 we review the concept of immersed DGA maps and immersed homotopy thendiscuss two equivalent characterizations of composition. In addition, we recall the category,
DGA ρ im , of Z /ρ -graded DGAs with immersed maps constructed in [30], and its connection withthe ordinary homotopy category of DGAs, DGA ρ hom ; see Proposition 2.7.Next, we turn to the geometric side with a brief review of Legendrians and LCH in thesetting of 1-jet spaces in 2.3. Section 2.4 discusses conical Legendrian cobordisms and, afterobserving their connection with immersed Lagrangian cobordisms, recalls the construction ofimmersed DGA maps from conical Legendrian cobordisms as in [30]. The construction is nicelyencoded in a functor, F , called the immersed LCH functor that plays a central role in thisarticle. Finally, Section 2.5 discusses Morse minimum Legendrian cobordisms, connects themwith conical Legendrian cobordisms, and provides a proposition that allows for Morse minimumcobordisms to be used for computations with the immersed LCH functor. The section concludesby defining the equivalence relation of ρ -graded augmented Legendrian cobordism and showingthat Theorem 1.5 follows from Theorem 1.2.2.1. Differential graded algebras.
We work in the algebraic context of [30, Sections 2 and 3]that we will now briefly review. In this article, differential graded algebras (abbr.
DGAs ),( A , ∂ ), are defined over Z / Z /ρ for some fixed ρ ≥
0. The differential, ∂ , has degree − ρ ). We restrict attention to based DGAs where the Z / A = Z / (cid:104) x , . . . , x n (cid:105) is free associative (non-commutative) with identity element and is equippedwith a choice of (finite) free generating set { x , . . . , x n } ; generators have degrees | x i | ∈ Z /ρ .Subalgebras (resp. 2-sided ideals) generated by a subset Y ⊂ A are notated as Z / (cid:104) Y (cid:105) (resp. I ( Y )). Such a DGA ( A , ∂ ) is triangular if (with respect to some ordering of the generatingset) we have ∂x i ∈ Z / (cid:104) x , . . . , x i − (cid:105) for all 1 ≤ i ≤ n . The coproduct of based DGAs, A ∗ B ,is the based DGA whose generating set is the union of the generating sets of A and B andwhose differential extends the differentials of A and B . A stabilization of a DGA A is a DGAof the form A ∗ S where S has generating set of the form { a , b , . . . , a r , b r } with differential ∂a i = b i , 1 ≤ i ≤ r . DGA morphisms are unital, algebra homomorphisms that preserve the Z /ρ -grading and commute with differentials. A stable tame isomorphism from A to B is aDGA isomorphism ϕ : A ∗ S → B ∗ S (cid:48) between stabilizations of A and B that is tame, i.e. it is a YU PAN AND DAN RUTHERFORD composition of isomorphisms that have a certain form on generators; see [30, Section 2.2]. Wesay DGAs are equivalent if they are stable tame isomorphic. Two DGA maps f, g : A → B are
DGA homotopic if they satisfy f − g = ∂ B ◦ K + K ◦ ∂ A for some ( f, g )-derivation, K : A → B , where an ( f, g ) -derivation is a degree 1 (mod ρ ) linearmap satisfying K ( xy ) = K ( x ) g ( y ) + ( − | x | f ( x ) K ( y ). When ϕ : A ∗ S → B ∗ S (cid:48) is a stableisomorphism there is an associated DGA homotopy equivalence h : A → B given by h = π (cid:48) ◦ h ◦ ι where ι : A → A ∗ S and π (cid:48) : B ∗ S (cid:48) → B are inclusion and projection.The following proposition is contained in [30, Propositions 2.3 and 2.5]. Proposition 2.1.
Let ( A , ∂ ) be a DGA that is triangular with respect to the ordered generatingset { x , . . . , x n } , and suppose that ∂x i = x j + w where w ∈ Z / (cid:104) x , . . . , x j − (cid:105) . (1) Then, A /I where I = I ( x i , ∂x i ) is a triangular DGA with respect to the generating set { x , . . . , (cid:98) j, . . . , (cid:98) i, . . . , x n } , and there is a stable tame isomorphism ϕ : A /I ∗ S → A with S = Z / (cid:104) y, z (cid:105) . Moreover, ϕ = g ∗ h where h ( y ) = x i , h ( z ) = ∂x i , and g : A /I → A is DGA homotopy inverse to the quotient map p : A → A /I via g ◦ p − id A = ∂ ◦ H + H ◦ ∂ where H : A → A is the ( g ◦ p, id A ) -derivation satisfying H ( x j ) = x i and H ( x l ) = 0 for l (cid:54) = j . (2) If B ⊂ A is a based sub-DGA generated by Y ⊂ { x , . . . , x n } and x i , x j / ∈ Y , then ϕ ( x k ) = x k for all x k ∈ Y . (3) For ϕ − , the associated DGA homotopy equivalence A ϕ − → A /I ∗ S π → A /I has π ◦ ϕ − = p where p : A → A /I is the quotient map. Definition 2.2. A ρ -graded augmentation to Z / Z /ρ -graded DGA, ( A , ∂ ), is a DGAmorphism (cid:15) : ( A , ∂ ) → ( Z / ,
0) where the grading on Z / ρ .I.e., (cid:15) is a Z / A to Z / (cid:15) ◦ ∂ = 0 , (cid:15) (1) = 1 , and (cid:15) ( x ) (cid:54) = 0 implies that | x | = 0 ∈ Z /ρ .2.2. Immersed DGA maps.
The main construction of [30] extends the functoriality of theLegendrian contact homology DGA to the case where the domain category consists of Legen-drians with a class of immersed
Lagrangian cobordisms. To accomplish this it is natural to alsoenlarge the class of morphisms in the target category of DGAs, and this is done by introducing immersed
DGA maps with a suitable notion of homotopy. Here, we recall these notions whichwill be central in the remainder of the article.
Definition 2.3.
Let ( A , ∂ ) and ( A , ∂ ) be triangular DGAs. An immersed DGA map , M , from A to A is a diagram of DGA maps M = (cid:16) A f → B i ← (cid:45) A (cid:17) where B is a triangular DGA and f and i are DGA maps such that i is an inclusion induced byan inclusion of the generating set of A into the generating set of B .Two immersed DGA maps M = (cid:16) A f → B i ← (cid:45) A (cid:17) and M (cid:48) = (cid:18) A f (cid:48) → B (cid:48) i (cid:48) ← (cid:45) A (cid:19) are im-mersed homotopic if there exists a stable tame isomorphism ϕ : B ∗ S → B ∗ S (cid:48) such that • ϕ ◦ i = i (cid:48) and • ϕ ◦ f (cid:39) f (cid:48) (DGA homotopy). UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 7
That is, the left (resp. right) half of the diagram
B ∗ S ϕ (cid:15) (cid:15) A f (cid:59) (cid:59) f (cid:48) (cid:35) (cid:35) A i (cid:99) (cid:99) i (cid:48) (cid:123) (cid:123) B (cid:48) ∗ S (cid:48) is commutative up to DGA homotopy (resp. fully commutative). Definition 2.4.
The composition M ◦ M of two immersed DGA maps M k = (cid:0) A k f k → B k i k ← (cid:45) A k +1 (cid:1) , for k = 1 , M ◦ M = (cid:0) A f → B i ← (cid:45) A (cid:1) , where B = B ∗ B / I ( { i ( x ) − f ( x ) | x ∈ A } ) is the categorical push out of i and f and f = p ◦ f , i = p ◦ i with p k : B k → B the projection maps. The algebra B is triangular with respect to the generatingset obtained as the union of the generators of B and B with the generators of A removed.This is summarized by the diagram: BB p (cid:61) (cid:61) B p (cid:97) (cid:97) A f (cid:62) (cid:62) A i (cid:96) (cid:96) f (cid:62) (cid:62) A i (cid:96) (cid:96) There is an alternate characterization of composition of immersed maps, up to immersedhomotopy, as follows. Again, let M k = (cid:0) A k f k → B k i k ← (cid:45) A k +1 (cid:1) , k = 1 , a , . . . , a m denote the generators of A , and let (cid:98) A have free generating set (cid:98) a , . . . , (cid:98) a m withdegree shift | (cid:98) a i | = | a i | + 1. Proposition 2.5 (Proposition 3.6 of [30]) . Suppose that ( D , ∂ ) is a triangular DGA such that • D = B ∗ (cid:98) A ∗ B , • for k = 1 , , ∂ | B k = ∂ B k , and • for ≤ i ≤ m , (2.1) ∂ (cid:98) a i = i ( a i ) + f ( a i ) + γ i where γ i ∈ I ( (cid:98) a , . . . , (cid:98) a m ) .Then, there is an immersed homotopy M ◦ M (cid:39) (cid:16) A j ◦ f → D j ◦ i ← (cid:45) A (cid:17) where j k : B k → B ∗ (cid:98) A ∗ B is the inclusion. Note that differentials of the form (2.1) always exist on D . For instance, one can define(2.2) ∂ (cid:98) a i = i ( a i ) + f ( a i ) + Γ ◦ ∂a i where Γ : A → D is the unique ( i , f )-derivation satisfying Γ( a i ) = (cid:98) a i . Definition 2.6.
For ρ ∈ Z ≥ fixed, we define a category DGA ρ im whose objects are triangu-lar DGAs graded by Z /ρ and whose morphisms are immersed homotopy classes of immersedDGA maps. Define a related category DGA ρ hom to have the same objects as DGA ρ im but withmorphisms given by DGA homotopy classes of ordinary DGA maps. YU PAN AND DAN RUTHERFORD
Proposition 2.7 (Propositions 3.9 and 3.10 of [30]) . For any ρ ∈ Z ≥ , DGA ρ im is a categorywith identity morphisms given by the homotopy classes of the immersed maps (cid:16) A id → A id ← (cid:45) A (cid:17) . Moreover, there is a functor I : DGA ρ hom → DGA ρ im that is the identity on objects and has I (cid:16) [ A f → A ] (cid:17) = (cid:104)(cid:16) A f → A id ← (cid:45) A (cid:17)(cid:105) , and I is injective on all hom-spaces. We emphasize that the triangularity condition is used crucially in [30] in establishing thewell-definedness of compositions in
DGA ρ im .2.3. The Legendrian contact homology DGA and exact Lagrangian cobordisms.
Re-call that the 1-jet space, J E , of an n -dimensional manifold, E , has its standard contact struc-ture ξ = ker α where α = dz − (cid:80) y i dx i in coordinates ( x , . . . , x n , y , . . . , y n , z ) ∈ T ∗ E × R = J E arising from local coordinates ( x , . . . , x n ) on E . A Legendrian submanifold , Λ, is an n -dimensional submanifold that is tangent to ξ everywhere. In this article, we will only need toconsider Legendrian submanifolds of dimension n = 1 or 2, i.e. Legendrian knots and surfaces.We use the notations π x : J E → E and π xz : J E → J E = E × R for the base and frontprojections . Generically, outside of a codimension 1 subset Λ cusp ⊂ Λ consisting of cusp points(resp. cusp edge and swallowtail points) the front projection, π xz | Λ , of a Legendrian knot (resp.surface) is an immersion. Moreover, any p ∈ Λ \ Λ cusp has a neighborhood W ⊂ Λ that is the1-jet, j f , of some local defining function f : U → R defined in a neighborhood U ⊂ E of π x ( p ). In particular, π xz ( W ) agrees with the graph of f . Suppose that E has boundary and N ⊂ ∂E is a boundary component. If Λ intersects π − x ( N ) transversally (this is equivalent tothe map π x | Λ being transverse to N ), then the restriction of Λ to J N is the Legendrian sub-manifold Λ | N ⊂ J N that is the image of Λ ∩ π − x ( N ) under the restriction map π − x ( N ) → J N .The front projection of Λ | N is the intersection of π xz (Λ) ⊂ E × R with N × R .The standard Reeb vector field on J E is ∂∂z , and Reeb chords of Λ, i.e. trajectories of ∂∂z having endpoints on Λ, correspond to critical points of local difference functions , f i,j = f i − f j , where f i , f j are local defining functions for Λ with f i > f j . For ρ ∈ Z ≥ , a Z /ρ -valued Maslov potential µ for a 1 or 2 dimensional Legendrian Λ is a locally constant map µ : Λ \ Λ cusp → Z /ρ , such that near each cusp point or edge, the value of µ at the upper sheetof the cusp is one more than the value of µ at the lower sheet; such a Maslov potential exists ifand only if ρ is a divisor of the Maslov number of Λ which is 2 rot (Λ) when Λ is 1-dimensional,cf. [30, Section 4]. When Λ is equipped with a choice of Z /ρ -valued Maslov potential, eachReeb chord, c , is assigned a Z /ρ -grading by | c | = µ ( c u ) − µ ( c l ) + ind ( f i,j ) − ∈ Z /ρ where c u and c l are the upper and lower endpoints of c and ind ( f i,j ) is the Morse index of c whenviewed as a critical point of the local difference function f i,j = f i − f j (where f i , f j are definingfunctions for Λ near c u and c l .) In the case that Λ is connected, the grading of Reeb chords isindependent of the choice of Maslov potential, but this is not true in the multi-component case.In order to have a well defined grading of Reeb chords (and also of the Legendrian contacthomology algebra) we will always work with Legendrians equipped with a choice of Maslovpotential. Definition 2.8.
For ρ ∈ Z ≥ , a ρ -graded Legendrian is a pair (Λ , µ ) consisting of a Legen-drian submanifold Λ ⊂ J E together with a choice of Z /ρ -valued Maslov potential, µ . Remark 2.9.
When the base space E is oriented a Z / Z / UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 9
Maslov potential, µ : Λ \ Λ cusp → Z /
2, to be 0 (resp. 1) at points where the base projectionof Λ to E is orientation preserving (resp. reversing). Moreover, this procedure can be reversedto produce an orientation from a Z / Z / Z /ρ -graded Legendrian with ρ even has a welldefined orientation (from reducing the Maslov potential mod 2).For a ρ -graded Legendrian submanifold Λ in J E , symplectic field theory gives a Floer typeinvariant called the Legendrian contact homology DGA (aka. the Chekanov-Eliashbergalgebra) [9, 18, 14] that we will denote by ( A (Λ) , ∂ ). It is a Z /ρ -graded triangular, based DGAover Z / Z /ρ -grading arising from the choice of Maslovpotential, µ . The differential ∂ is defined by counting holomorphic disks either in T ∗ M , cf.[14], or Symp ( J M ), cf. [11], with boundary on either the cotangent projection of Λ or theLagrangian cylinder R × Λ. According to [12], one can alternatively compute the differential bycounting gradient flow trees (abbrv.
GFTs ) which are certain trees whose edges parametrizeflowlines of gradients of local difference functions, −∇ f i,j , with f i,j >
0; see also [32]. The Z /ρ -graded DGA ( A (Λ) , ∂ ) is an invariant of (Λ , µ ) up to stable tame isomorphism.The LCH DGA is functorial for exact Lagrangian cobordisms, and we will restrict our at-tention to the case of cobordisms in the symplectization of J M . An exact Lagrangiancobordism L from Λ − to Λ + is an embedded surface in Symp ( J M ) := ( R t × J M, d ( e t α )) (asshown in Figure 1) that • agrees with the cylinder R × Λ + (resp. R × Λ − ) when t is very positive (resp. verynegative; and • there is a function g : L → R such that ( e t α ) | L = dg and g is constant for t near ±∞ .Such a function g is called a primitive .When Λ − is empty, we say that L is an exact Lagrangian filling of L . t Λ − Λ + L Figure 1.
An exact Lagrangian cobordism from Λ − to Λ + .According to [15, 13, 16], an exact Lagrangian cobordism L from Λ − to Λ + induces a DGAmap f L from A (Λ + ) to A (Λ − ). (For now, we suppress discussion of the grading.) When twosuch cobordisms are isotopic through exact Lagrangian cobordisms, their induced DGA mapsare DGA homotopic. Moreover, when two exact Lagrangian cobordisms L i , i = 1 , i toΛ i +1 are concatenated together to form L ◦ L (by translating L in the positive t -direction,truncating the two cobordisms, and gluing along a region of the form ( a, b ) × Λ ), the inducedDGA map f L ◦ L is DGA homotopic to f L ◦ f L . In summary, we have a functor between asuitably defined category of Legendrian knots with morphisms exact Lagrangian cobordismsand the homotopy category of DGAs.2.4. Conical Legendrian cobordisms.
In [30], functoriality for the LCH DGA was general-ized to a class of immersed exact Lagrangian cobordisms by working with conical Legendriancobordisms. We now review relevant results from [30].
Let M be a 1-manifold. In J ( R > × M ), we denote the R > coordinate by s . Definition 2.10.
Let Λ be a Legendrian link in J M that is parametrized by θ (cid:55)→ ( x ( θ ) , y ( θ ) , z ( θ )),let f : I → R > where I ⊂ R > is an interval, and let A ∈ R be a constant. Define j ( f ( s ) · Λ + A ) ⊂ J ( I × M )to be the Legendrian that is parametrized by( s, θ ) (cid:55)→ ( s, x ( θ ) , f (cid:48) ( s ) z ( θ ) , f ( s ) y ( θ ) , f ( s ) z ( θ ) + A )= ( s, x, u, y, z ) ∈ J ( I × M )where ( s, x ) are the coordinates on I × M , ( u, y ) are the coordinates on the cotangent fibers,and z is the R coordinate on J ( I × M ) = T ∗ ( I × M ) × R .Note that the front projection of j ( f ( s ) · Λ + A ) is obtained from π xz (Λ) by forming thecylinder in the s -direction, multiplying z -coordinates by f ( s ), and then shifting them by A . Definition 2.11. A conical Legendrian cobordism Σ from Λ − to Λ + is an embedded Leg-endrian surface in J ( R > × M ) (see Figure 2) such that • Σ has conical ends, i.e, when s > s + (resp. s < s − ) for some positive number s ± , theLegendrian surface Σ is j ( s · Λ ± + A ± ) for some constant A ± ; and • the intersection Σ ∩ J ([ s − , s + ] × M ) is compact.Two conical Legendrian cobordisms from Λ − to Λ + are conical Legendrian isotopic if theyare isotopic through conical Legendrian cobordisms from Λ − to Λ + .Λ − Λ + Σ z x s Figure 2.
A sketch of a conical Legendrian cobordism in the front projection to J ( R > × M ).Note that there is a contactomorphism between J ( R > × M ) and ( Symp ( J M ) × R w , dw + e t α )(see [30] for details). Consider the image of Σ in Symp ( J M ) × R and then take the Lagrangianprojection to Symp ( J M ). The resulting surface L is an (immersed) exact Lagrangian surfacein Symp ( J M ), and we call surfaces obtained in this manner good Lagrangian cobordisms from Λ − to Λ + . Note that the conical ends condition on Σ implies that L has cylindrical endsand that the primitive is constant on top and bottom cylinders. As long as the immersedLagrangians are equipped with such primitives, the construction is reversible, so we have abijection between conical Legendrian cobordisms and good Lagrangian cobordisms; see eg. [30,Proposition 4.9]. Thus, conical Legendrian cobordisms generalize exact Lagrangian cobordismsfrom the embedded case to the immersed case. Conical Legendrian cobordisms can be con-catenated in a way that generalizes the concatenation of exact Lagrangian cobordisms; see [30,Section 4.4].The functoriality of the LCH DGA extends to conical Legendrian cobordisms provided theinduced maps are allowed to be immersed DGA maps. Theorem 2.12 ([30]) . A conical Legendrian cobordism Σ from Λ − to Λ + induces an immersedDGA map M Σ = (cid:0) A (Λ + ) f → A (Σ) i ← (cid:45) A (Λ − ) (cid:1) , UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 11 satisfying: (1)
The DGA A (Σ) is generated by Reeb chords of Σ and Reeb chords of Λ − . (2) When Λ ± and Σ are ρ -graded such that the Z /ρ -valued Maslov potential of Σ restrictsto the Maslov potentials on Λ ± , all the DGAs A (Λ + ) , A (Σ) , and A (Λ − ) inherit Z /ρ -gradings that are preserved by f and i . (3) When two conical Legendrian cobordisms Σ and Σ (cid:48) from Λ − to Λ + are conical Legendrianisotopic, their induced immersed DGA maps are immersed homotopic. (4) When Σ : Λ → Λ and Σ : Λ → Λ are concatenated the immersed maps satisfy M Σ ◦ Σ (cid:39) M Σ ◦ M Σ (immersed homotopy) . (5) When Σ has no Reeb chords, i.e., when it corresponds to an embedded exact Lagrangiancobordism L , we have that A (Σ) = A (Λ − ) and the f map is DGA homotopic to theinduced map f L from [15] . The construction of the immersed DGA map is summarized as follows. Given a conicalLegendrian cobordism Σ from Λ − to Λ + , we construct a Morse cobordism (cid:101) Σ (as in [15]) byreplacing the conical ends j ( s · Λ ± + A ± ) with standard Morse ends j ( h ± ( s ) · Λ ± + B ± ), where h − (resp. h + ) are positive Morse functions with a single minimum (resp. maximum). The setof Reeb chords of (cid:101) Σ is in bijection with the set of Reeb chords of Σ and Λ ± . The DGA A (Σ)is generated by the Reeb chords of (cid:101) Σ that correspond to Reeb chords of Σ and Λ − , and thedifferential is defined by the usual count of GFTs with respect to a suitable choice of metric g on R > × M having the form g R × g ± near the critical points of h ± where g R is the Euclideanmetric on R > and g ± are regular metrics used for computing A (Λ ± ). Moreover, the DGA A (Λ − ) is a sub-DGA of A (Σ) and the map i is the natural DGA inclusion map. Finally, theDGA map f is defined by counting GFTs with positive puncture at one of the Λ + Reeb chordsof (cid:101)
Σ and with image to the left of the local maximum of h + .In [30, Section 6.3], the construction of Theorem 2.12 is formulated as a functor F : Leg ρim → DGA ρim called the immersed LCH functor . The category
Leg ρim has objects (Λ , g, µ ) consisting ofan (embedded) ρ -graded Legendrian knot, Λ ⊂ J M , with Maslov potential, µ , equipped witha choice of regular metric, g , on M . Morphisms from (Λ − , g − , µ − ) to (Λ + , g + , µ + ) are conicalLegendrian isotopy classes of ρ -graded conical Legendrian cobordisms, Σ : Λ − → Λ + , withMaslov potentials, µ , extending the Maslov potentials µ − and µ + . Corollary 2.13.
The correspondence (Λ , g, µ ) (cid:55)→ A (Λ , g, µ ) , (Σ , µ ) (cid:55)→ [ M Σ ] where [ M Σ ] denotes the immersed homotopy class of the immersed map M Σ from Theorem 2.12defines a contravariant functor F : Leg ρim → DGA ρim . Morse minimum cobordisms.
Let [ a − , a + ] ⊂ R be a closed interval. A compact Leg-endrian Σ min ⊂ J ([ a − , a + ] × M ) is called a Morse minimum cobordism from Λ − to Λ + ifthere are neighborhoods U − = [ a − , a − + δ ) and U + = ( a + − δ, a + ], for some δ >
0, such thatΣ min has the form j ( h ± ( s ) · Λ ± + B ± ) in J ( U ± × M ) where h ± : U ± → R > are positive Morsefunctions with unique critical points that are non-degenerate local minima at a ± .As in [16], the LCH DGA of a Morse minimum cobordism A (Σ min ) is well-defined and canbe computed using GFTs with respect to a regular metric g on [ a − , a + ] × M having the form Here, regular means that (i) there are no GFTs for Λ of negative formal dimension and (ii) all 0-dimensionalGFTs for Λ are transversally cut out. See [12]. g R × g ± in a neighborhood of { a ± } × M where g ± are regular metrics for Λ ± . Moreover, thereare DGA inclusions i ± : A (Λ ± ) (cid:44) → A (Σ min )obtained via identifying Reeb chords of Λ ± with the Reeb chords of Σ min located above { a ± }× M .See also [30, Section 5.3]. Construction 2.14.
Given Σ min one can form an associated conical Legendrian cobordismΣ conic ⊂ J ( R > × M ) by shifting the interval [ a − , a + ] into [ a − + s , a + + s ] ⊂ R > , andmodifying the ends to have the form j ( (cid:98) h ± ( s ) · Λ ± + B ± ) with (cid:98) h − : (0 , a − + s + δ ) → R > , and (cid:98) h + : ( a + + s − δ, + ∞ ) → R > chosen as follows: • The function (cid:98) h − has no critical points, is increasing on (0 , a − + s + δ ), and satisfies (cid:98) h − ( s ) = s, for s ∈ (0 , a − + s ] , (cid:98) h − ( s ) = h − ( s − s ) , near s = a − + s + δ . • The function (cid:98) h + has a unique critical point that is a non-degenerate local minimum at s = a + + s , and satisfies (cid:98) h + ( s ) = h + ( s − s ) , for s ∈ ( a + + s − δ, a + + s ] , (cid:98) h + ( s ) = s, for s (cid:29) g min for Σ min defined on [ a − , a + ] × M and having the form g R × g ± near { a ± } × M we construct a metric g conic on R > × M by shifting g min by s in the R > -direction and then extending to agree with g R × g − on (0 , a − + s ] × M and g R × g + on[ a + + s , + ∞ ) × M .Note that the conditions on (cid:98) h − can be arranged with an appropriate choice of s . See Figure3. Σ min Σ conic (cid:101) Σ conic Σ Σ Σ
Figure 3.
A Morse minimum cobordism, an associated conical cobordism, andthe cobordism (cid:101) Σ conic used in the proof of Proposition 2.15. Proposition 2.15.
When Σ min ⊂ J ([ a − , a + ] × M ) is a Morse minimum cobordism from Λ − to Λ + and Σ conic is an associated conical cobordism as in Construction 2.14. Then, the immersedLCH functor satisfies F (Σ conic ) = (cid:20) A (Λ + ) i + → A (Σ min ) i − ← (cid:45) A (Λ − ) (cid:21) . Proof.
A Morse cobordism (cid:101) Σ conic (with a Morse minimum at the negative end and a Morsemaximum at the positive end) that can be used to compute the induced immersed map M Σ conic = (cid:16) A (Λ + ) f → A (Σ conic ) i ← (cid:45) A (Λ − ) (cid:17) from Theorem 2.12 is obtained as follows: UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 13 (1) At the negative end of Σ conic , replace j ( (cid:98) h − ( s ) · Λ − + B − ) with j ( h − ( s − s ) · Λ − + B − ).This matches the negative end of Σ min shifted by s .(2) At the postive end, replace the function (cid:98) h + with some (cid:101) h + : ( a + + s − δ, s + δ ] → R > .Here, s is chosen large enough so that (cid:98) h + ( s ) = s for s ≥ s − δ and we require • (cid:101) h + ( s ) = (cid:98) h + ( s ) for s ∈ ( a + + s − δ, s ], • (cid:101) h + is increasing on [ s , s + δ ] and has a single critical point on this interval that isa non-degenerate local maximum at s + δ .See Figure 3.By definition, the DGA A (Σ conic ) is generated by those Reeb chords of (cid:101) Σ conic that appear in theregion where s < s + δ . These Reeb chords are the same as the Reeb chords of Σ min but shiftedby s in the s direction. Moreover, using metrics of the form g min and g conic as in Construction2.14 to compute GFTs, because of the Morse minima all of the GFTs with positive puncturesat these chords are contained in the region [ a − + s , a + + s ] × M and therefore coincide (up tothe shift in the s direction) with the GFTs of Σ min ; see eg. [16]. Thus, A (Σ conic ) = A (Σ min )and the map i agrees with i − by definition. The GFTs that define the map f (by definition)have their unique positive punctures at the Reeb chords located at { s + δ } × M , and (becauseof the local minimum of (cid:101) h + at s = a + + s ) have there images contained in ( a + + s , s + δ ) × M .In this region, (cid:101) Σ conic = j ( (cid:101) h + ( s ) · Λ + + B + ) and (cid:101) h + strictly increases, so the computation ofsuch GFTs is as in the case of the identity cobordism from Λ + to itself found in [30, Proposition6.15]. For each Reeb chord, b , of Λ + there is a single gradient trajectory that connects the Reebchords of (cid:101) Σ conic at s = s + δ corresponding to b to the Reeb chord of (cid:101) Σ conic at s = a + + s corresponding to b , and these are the only rigid GFTs. It follows that f = i + . (cid:3) As discussed in the introduction, Morse minimum cobordisms may be used to define anequivalence relation on augmented Legendrians. Refer to a triple (Λ , µ, (cid:15) ) consisting of a ρ -graded Legendrian Λ ⊂ J M with Maslov potential, µ , and a ρ -graded augmentation, (cid:15) : A (Λ) → Z /
2, as a ρ -graded augmented Legendrian . Definition 2.16.
Two ρ -graded augmented Legendrians (Λ i , µ i , (cid:15) i ), i = 0 ,
1, in J M are cobordant if there exists a triple (Σ , µ, α ) consisting of a Morse minimum cobordism Σ ⊂ J ([0 , × M ) from Λ to Λ together with a Maslov potential µ extending the µ i and a ρ -graded augmentation α : A (Σ) → Z / α | A (Λ i ) (cid:39) (cid:15) i (DGA homotopy).It is straightforward to see that cobordism defines an equivalence relation on ρ -graded aug-mented Legendrians.We can now show that Theorem 1.2 implies Theorem 1.5 from the introduction. Proof of Theorem 1.5.
Given a ρ -graded augmented Legendrian (Λ , µ, (cid:15) ), assuming Theorem1.2, there exists a conical Legendrian cobordism with Z /ρ -valued Maslov potential,Σ : ∅ → Λ (if ρ (cid:54) = 1) or Σ : U → Λ (if ρ = 1) , together with a ρ -graded augmentation α : A (Σ) → Z / (cid:15) (cid:39) α ◦ f Σ . Now, a Legendrianisotopy that is compactly supported in the conical ends of Σ modifies Σ to have the form Σ conic for some Morse minimum cobordism Σ min as in Construction 2.14. Then, from Theorem 2.12there is an immersed DGA homotopy M Σ (cid:39) M Σ conic that (after using Proposition 2.15 toevaluate M Σ conic and replacing the stable tame isomorphism ϕ : A (Σ min ) ∗ S → A (Σ min ) ∗ S (cid:48) with its associated homotopy equivalence) gives rise to a DGA homotopy commutative diagram: A (Σ min ) h (cid:15) (cid:15) A (Λ + ) i + (cid:57) (cid:57) f Σ (cid:37) (cid:37) A (Σ)Then, we can compute (cid:15) (cid:39) α ◦ f Σ (cid:39) ( α ◦ h ) ◦ i + , so that (Σ min , α ◦ h ) provides the cobordism of ρ -graded augmented Legendrians from ∅ (if ρ (cid:54) = 1) or ( U, (cid:15) ) (if ρ = 1) to (Λ , (cid:15) ) as in the statement of Theorem 1.5 where (cid:15) = α ◦ h ◦ i − . (cid:3) Remark 2.17. (1) When ρ is even, the cobordism Σ and Legendrians are canonically ori-ented by the Maslov potential µ . If ρ is odd, Σ may be orientable or not. In the oddcase, a refined relation of oriented cobordism for ρ -graded augmentation Legendriansarises from requiring that the Λ i and Σ are additionally equipped with orientations.We leave the computation of such oriented, odd-graded cobordism classes of augmentedLegendrians in J R as an open problem.(2) Without augmentations, Legendrian cobordism classes in J R are computed as follows;see eg. [3, Section 5.1]. Two Legendrians in J R are oriented cobordant if and only ifthey have the same rotation number, while any two Legendrians in J R are non-orientedcobordant. In particular, Theorem 1.5 implies the well known result of Sabloff, see [35],that if Λ ⊂ J R has an ρ -graded augmentation with ρ even, then rot (Λ) = 0.(3) As Legendrians that admit augmentations exhibit significantly more rigid behavior thangeneral Legendrians, we do not see any a priori reason to expect that cobordism classesof ρ -graded augmented Legendrians should closely match the classical cobordism classesof Legendrians. In fact, in the case of J S we will show in [29] that there are manyexamples of non-cobordant augmented Legendrians that become cobordant if one ignoresthe augmentations.3. Immersed DGA maps and augmentations
A DGA morphism f : A → B contravariantly induces a map on homotopy classes of aug-mentations, f ∗ : Aug ( B ) / ∼ → Aug ( A ) / ∼ . In Section 3.1, we consider analogous constructionsfor immersed DGA maps focusing on the induced augmentation set of an immersed DGA map, M , that is a subset I ( M ) ⊂ Aug ( B ) / ∼ × Aug ( A ) / ∼ . We show that the induced augmentationset induced by a conical Legendrian cobordism Σ is a Legendrian invariant of Σ. In addition,we make some observations about the form of the augmentation set in the case of Legendrianfillings and embedded Legendrian cobordisms. In Section 3.2 we show that immersed augmen-tation sets compose as relations, and we record the effect of concatenating a conical Legendriancobordism with the (invertible) Legendrian cobordism arising from a Legendrian isotopy.3.1. Induced augmentation sets.
We work with Z /ρ -graded DGAs, with ρ ≥ understoodto be fixed . As such, when the grading does not need to be emphasized we may refer to ρ -gradedaugmentations simply as augmentations. We denote by Aug ρ ( A ) = Aug ( A ) the set of all ( ρ -graded) augmentations of A to Z /
2, and we write
Aug ( A ) / ∼ for the set of all DGA homotopyclasses of augmentations. In the case that A = A (Λ) is the DGA of some ρ -graded Legendrianknot or cobordism, we may shorten these notations to Aug (Λ) and
Aug (Λ) / ∼ . A DGA map f : A → B induces a pullback map f ∗ : Aug ( B ) → Aug ( A ) , f ∗ (cid:15) = (cid:15) ◦ f, UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 15 and this gives a well-defined map on DGA homotopy classes also denoted as f ∗ : Aug ( B ) / ∼ → Aug ( A ) / ∼ . Let M = (cid:0) A f → B i ← (cid:45) A (cid:1) be an immersed DGA map. Then, the pullback constructionresults in maps Aug ( A ) f ∗ ← Aug ( B ) i ∗ → Aug ( A ) and Aug ( A ) / ∼ f ∗ ← Aug ( B ) / ∼ i ∗ → Aug ( A ) / ∼ . The latter diagram is equivalent to the map i ∗ × f ∗ : Aug ( B ) / ∼ → ( Aug ( A ) / ∼ ) × ( Aug ( A ) / ∼ )that we call the augmentation map induced by M . Definition 3.1.
The induced augmentation set , I ( M ), of an immersed DGA map M = (cid:0) A f → B i ← (cid:45) A (cid:1) is the image of the augmentation map, I ( M ) = Im( i ∗ × f ∗ ) ⊂ Aug ( A ) / ∼ × Aug ( A ) / ∼ . The induced augmentation set is an invariant of the immersed homotopy class of M . Proposition 3.2.
Suppose that M = (cid:0) A f → B i ← (cid:45) A (cid:1) and M (cid:48) = (cid:0) A f (cid:48) → B (cid:48) i (cid:48) ← (cid:45) A (cid:1) areimmersed DGA maps that are immersed homotopic. Then, there is a bijection h ∗ : Aug ( B (cid:48) ) / ∼ ∼ = → Aug ( B ) / ∼ fitting into a commutative diagram (3.1) Aug ( B ) / ∼ f ∗ (cid:119) (cid:119) i ∗ (cid:39) (cid:39) Aug ( A ) / ∼ Aug ( A ) / ∼ Aug ( B (cid:48) ) / ∼ h ∗ ∼ = (cid:79) (cid:79) ( f (cid:48) ) ∗ (cid:103) (cid:103) ( i (cid:48) ) ∗ (cid:55) (cid:55) . In particular, the induced augmentation sets satisfy I ( M ) = I ( M (cid:48) ) .Proof. There exists a diagram(3.2)
B ∗ S ϕ ∼ = (cid:15) (cid:15) A f (cid:59) (cid:59) f (cid:48) (cid:35) (cid:35) A i (cid:99) (cid:99) i (cid:48) (cid:123) (cid:123) B (cid:48) ∗ S (cid:48) where ϕ is a DGA isomorphism such that ϕ ◦ f (cid:39) f (cid:48) and ϕ ◦ i = i (cid:48) . Let h = π (cid:48) ◦ ϕ ◦ ι , where ι : B → B ∗ S and π (cid:48) : B (cid:48) ∗ S (cid:48) → B (cid:48) are the inclusions and projections, bethe associated homotopy equivalence from B to B (cid:48) . Then, since the maps such as f : A → B ∗ S in (3.2) are implicitly understood to mean ι ◦ f , the diagram (3.2) leads to a similar homotopycommutative diagram (3.2)’ with the vertical map replaced with h : B → B (cid:48) .Now, the association A (cid:32) Aug ( A ) / ∼ ,g : A → B (cid:32) g ∗ : Aug ( B ) / ∼ → Aug ( A ) / ∼ gives a well-defined contravariant functor from the category DGA ρ hom (where morphisms areDGA homotopy classes of maps) to the category of sets. In particular, since h is a homotopy equivalence, h ∗ is a bijection, and since (3.2)’ is homotopy commutative, the diagram (3.1) isindeed fully commutative. (cid:3) When Σ : Λ − → Λ + is a conical Legendrian cobordism with immersed DGA map, M Σ , asin Theorem 2.12, we write I Σ = I ( M Σ ) ⊂ Aug (Λ − ) / ∼ × Aug (Λ + ) / ∼ and refer to I Σ as the induced augmentation set of Σ. Corollary 3.3.
Suppose that Σ , Σ (cid:48) : Λ − → Λ + are conical Legendrian cobordisms related by aconical Legendrian isotopy. Then, there is a commutative diagram (3.3) Aug (Σ) / ∼ f ∗ Σ (cid:119) (cid:119) i ∗ Σ (cid:39) (cid:39) Aug (Λ + ) / ∼ Aug (Λ − ) / ∼ Aug (Σ (cid:48) ) / ∼ h ∗ ∼ = (cid:79) (cid:79) f ∗ Σ (cid:48) (cid:103) (cid:103) i ∗ Σ (cid:48) (cid:55) (cid:55) i.e., h ∗ is a bijection and i ∗ Σ (cid:48) × f ∗ Σ (cid:48) = ( i ∗ Σ × f ∗ Σ ) ◦ h ∗ .In particular, the induced augmentation set I Σ = Im ( i ∗ Σ × f ∗ Σ ) ⊂ Aug (Λ − ) / ∼ × Aug (Λ + ) / ∼ , is an invariant of Σ . Remark 3.4. (1) To provide a more refined invariant of Σ, one can take multiplicities intoaccount when considering I Σ . Eg., the function Aug (Λ − ) / ∼ × Aug (Λ + ) / ∼ → Z ≥ , ([ (cid:15) − ] , [ (cid:15) + ]) (cid:55)→ (cid:12)(cid:12) ( i ∗ Σ × f ∗ Σ ) − (cid:0) ([ (cid:15) − ] , [ (cid:15) + ]) (cid:1)(cid:12)(cid:12) is a conical Legendrian invariant of Σ. A similar invariant (using a normalized count ofaugmentations rather than homotopy classes) is studied for 1-dimensional Σ in [37].(2) The set Aug (Λ ± ) can be equipped with the additional structure of an A ∞ -categorywhose moduli space of objects is Aug (Λ ± ) / ∼ , cf. [5, 26], and one could hope for afurther refinement of the augmentation map i ∗ Σ × f ∗ Σ to take this structure into account.In the case of embedded cobordisms, such a refinement is made in [28].3.1.1. The embedded case.
When the conical Legendrian cobordism Σ : Λ − → Λ + correspondsto an embedded Lagrangian cobordism, L , i.e. when Σ has no Reeb chords, A (Σ) = A (Λ − )and i Σ = id A (Λ − ) . Therefore, in this case the augmentation map i ∗ Σ × f ∗ Σ is determined by theinduced augmentation set(3.4) I Σ = { ([ (cid:15) ] , f ∗ Σ [ (cid:15) ]) | [ (cid:15) ] ∈ Aug (Λ − ) / ∼} ⊂ Aug (Λ − ) / ∼ × Aug (Λ + ) / ∼ which is simply the function f ∗ Σ viewed as a relation.More generally, when M = ( A f → A id ← (cid:45) A ) is the image of an ordinary DGA map f : A → A under the functor from Proposition 2.7, the induced augmentation set for M isjust the graph of f ∗ .3.1.2. Induced augmentations and immersed fillings.
Let Σ be a conical Legendrian filling, orequivalently a good immersed Lagrangian filling. Since Λ − = ∅ , we have A (Λ − ) = Z /
2. Thus,
Aug (Λ − ) = Aug (Λ − ) / ∼ consists of a single element, so that we can view I Σ as a subset of Aug (Λ + ). To emphasize the analogy with the case of embedded Lagrangian fillings, given anaugmentation α : A (Σ) → Z /
2, we use the notation (cid:15) (Σ ,α ) : A (Λ + ) → Z / , (cid:15) (Σ ,α ) = α ◦ f Σ UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 17 and refer to (cid:15) (Σ ,α ) as the augmentation induced by Σ via α . Thus, the induced augmentationset I Σ ⊂ Aug (Λ + ) consists of those augmentations of Λ + that can be induced by some choice ofaugmentation for Σ.3.2. Concatenation and induced augmentation sets.
Let
Rel denote the category whoseobjects are sets and morphisms are relations, i.e. a morphism R ⊂ Hom
Rel ( X, Y ) is just asubset R ⊂ X × Y . Given relations R ⊂ X × Y and S ⊂ Y × Z , there composition is S ◦ R = { ( x, z ) ∈ X × Z | ∃ y ∈ Y such that ( x, y ) ∈ R and ( y, z ) ∈ S } . Observation 3.5. (1) Any function f : X → Y defines a relation Γ f = { ( x, f ( x )) | x ∈ X } ⊂ X × Y .(2) Given any R ⊂ X × Y and g : Y → Z , we haveΓ g ◦ R = { ( x, g ( y )) | ( x, y ) ∈ R } = ( id × g )( R ) . (3) If f : X → Y is a bijection, and S ⊂ Y × Z ,then S ◦ Γ f = ( f − × id )( S ) . Proposition 3.6.
The induced augmentation set construction gives a contravariant functor
DGA ρ im → Rel , A (cid:32) Aug ( A ) / ∼ [ M ] (cid:32) I ( M ) . In particular, the induced augmentation sets for a pair of conical Legendrian cobordisms, Σ i :Λ i +1 → Λ i , i = 1 , , satisfy (3.5) I Σ ◦ Σ = I Σ ◦ I Σ . Proof.
Let M = (cid:0) A f → B i ← A (cid:1) and M = (cid:0) A f → B i ← A (cid:1) be immersed maps. The set I ( M ◦ M ) only depends on the immersed homotopy class of M ◦ M . Thus, we can computeit using the immersed map A f → D i ← A as in Proposition 2.5 where D = B ∗ (cid:98) A ∗ B withdifferential as in (2.2). That I ( M ◦ M ) = I ( M ) ◦ I ( M ) then follows from: Claim:
Let X be the set of triples ( α , α , K ) such that α i ∈ Aug ( B i ) and K : A → Z / i ∗ α to f ∗ α , i.e. a ( i ∗ α , f ∗ α )-derivation with i ∗ α − f ∗ α = K ◦ ∂ . There is a bijection X → Aug ( D ) , ( α , α , K ) (cid:55)→ α where α : D → Z / α | B i = α i and α ( (cid:98) a i ) = K ( a i ) for all generators a i ∈ A .To verify the claim note that ( α ◦ ∂ ) | B i = 0 if and only if B i is an augmentation. In addition,since ∂ ( (cid:98) a i ) = i ( a i ) + f ( a i ) + Γ ◦ ∂a i where Γ( a i ) = (cid:98) a i is an ( i , f )-derivation, the equation α ◦ ∂ ( (cid:98) a i ) = 0 is equivalent to i ∗ α ( a i ) + f ∗ α ( a i ) = K ◦ ∂a i . (cid:3) In the case that the corresponding exact Lagrangian cobordism is embedded, we get a simplerstatement.
Proposition 3.7.
Suppose
Σ : Λ − → Λ + is a conical Legendrian cobordism with embeddedLagrangian projection, and let Σ (cid:48) : Λ (cid:48)− → Λ − and Σ (cid:48)(cid:48) : Λ + → Λ (cid:48)(cid:48) + . (1) Then, I Σ ◦ Σ (cid:48) = ( id × f ∗ Σ )( I Σ (cid:48) )(2) If f ∗ Σ : Aug (Λ − ) / ∼ → Aug (Λ + ) / ∼ is a bijection, then I Σ (cid:48)(cid:48) ◦ Σ = (( f ∗ Σ ) − × id )( I Σ (cid:48)(cid:48) ) . Proof.
As discussed in 3.1.1, when Σ is embedded I Σ is the relation Γ f ∗ Σ associated to the function f ∗ Σ . Thus, the formulas from Observation 3.5 can be applied. (cid:3) An important case where (2) of Proposition 3.7 applies is when Σ is induced by a Legendrianisotopy. We now briefly review a version of this construction.Let Φ = { Λ s } s ∈ R > be a Legendrian isotopy from Λ − to Λ + so that Λ s : (cid:116) ci =1 S → J M for s ∈ R > is a Legendrian embedding satisfying Λ s = Λ − for 0 < s ≤ s = Λ + for s (cid:29) s ( θ ) = ( x ( s, θ ) , y ( s, θ ) , z ( s, θ ))there is a conical Legendrian cobordismΣ Φ : R > × ( (cid:116) ci =1 S ) → J ( R > × M ) = { ( s, x, u, y, z ) } from Λ − to Λ + associated to Φ that is parametrized byΣ Φ ( s, θ ) = (cid:18) s, x ( s, θ ) , z ( s, θ ) + s · ∂z∂s ( s, θ ) − s · y ( s, θ ) · ∂x∂s ( s, θ ) , s · y ( s, θ ) , s · z ( s, θ ) (cid:19) . It can be shown that after reparametrizing by an appropriate orientation preserving diffeomor-phism, α : R > → R > , the conical Legendrian cobordism corresponding to Φ (cid:48) = { Λ α ( s ) } willnot have Reeb chords. Indeed, with our setup, one can take α ( s ) = s a with a > Symp ( J M ). Corollary 3.8.
Let
Φ = { Λ s } s ∈ R > be a Legendrian isotopy parametrized so that the conicalLegendrian cobordism Σ Φ does not have Reeb chords. Then, f ∗ Σ Φ : Aug (Λ − ) / ∼ → Aug (Λ + ) / ∼ is a bijection. In particular, we have I Σ Φ ◦ Σ (cid:48) = ( id × f ∗ Σ Φ )( I Σ (cid:48) ) and I Σ (cid:48)(cid:48) ◦ Σ Φ = (( f ∗ Σ Φ ) − × id )( I Σ (cid:48)(cid:48) ) . whenever Σ (cid:48) and Σ (cid:48)(cid:48) are composable with Σ Φ .Proof. Let Φ − be be the inverse Legendrian isotopy { Λ /s } s ∈ R > reparametrized, if necessary,to ensure that Σ Φ − has no Reeb chords. Then, the immersed LCH functor satisfies F (Σ Φ ) = [ M Σ Φ ] = I ( f Σ Φ ) and F (Σ Φ − ) = [ M Σ Φ − ] = I ( f Σ Φ − )where I : DGA ρ hom → DGA ρ im is the functor from Proposition 2.7. There is a clear conicalLegendrian isotopy between Σ Φ ◦ Σ Φ − (resp. Σ Φ − ◦ Σ Φ ) and the identity cobordism j ( s · Λ + )(resp. j ( s · Λ − )). From functoriality, it follows that I ( f Σ Φ ◦ f Σ Φ − ) and I ( f Σ Φ − ◦ f Σ Φ ) areidentity morphisms in DGA ρ im , and since I is injective on hom-sets we conclude that f Σ Φ and f Σ Φ − are homotopy inverses. (cid:3) Immersed maps and the cellular DGA
In [31, 32, 33], a cellular version of Legendrian contact homology is introduced and shown tobe equivalent to the usual LCH DGA in the case of closed Legendrian surfaces. In Sections 4.1and 4.2, we briefly review the cellular DGA and extend its definition to the case of (compact)Legendrian cobordisms. The DGA of the identity cobordisms is computed in Section 4.3 andshown to be a mapping cylinder DGA. In Section 4.4 a cellular version, F cell , of the immersedLCH functor is defined with a domain category consisting of ρ -graded Legendrians equippedwith some additional data and with compact Legendrian cobordisms as morphisms. Finally, inSection 4.5 we state in Proposition 4.9 a precise relationship between the immersed and cellularLCH functors, F and F cell , that will allow us to work with F cell in place of F when consideringinduced augmentation sets; see Corollary 4.12. Specifically, after unifying the domain categoriesby precomposing with suitable functors, F and F cell become isomorphic. The proof of theisomorphism is an extension of the isomorphism between the cellular and LCH DGAs for closedLegendrian surfaces from [31, 32, 33], and the details are postponed until the Appendices A andB. UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 19
Review of the cellular DGA.
The cellular DGA construction requires as input a Leg-endrian knot or surface equipped with a suitable polygonal decomposition of its base projection.Recall that for a generic 1-dimensional Legendrian knot Λ ⊂ J M the singularities of the frontprojection, π xz (Λ) ⊂ M × R , are cusp points and crossings points (i.e. transverse double points).Generically, front projections of Legendrian surfaces have crossing arcs and cusp edges as codi-mension 1 singularities and triple points (intersection of three smooth sheets of Λ), cusp-sheetintersections (where a smooth sheet intersects a cusp edge), and swallowtail points as codimen-sion 2 singularities. See eg. [3]; the front singularities for surfaces are illustrated in [31, Section2.2]. We say that Λ has generic front and base projections if the front singularities aregeneric, and the base projections (to M ) of the different classes of front singularities are all selftransverse and transverse to one another. Definition 4.1.
Let Λ ⊂ J M be a closed Legendrian submanifold with dim Λ = 1 or 2 havinggeneric front and base projections. A compatible polygonal decomposition for Λ is apolygonal decomposition, E = { e dα } , of the base projection of Λ, (cid:116) d =0 (cid:116) α e dα = π x (Λ) ⊂ M, (where the superscript 0 ≤ d ≤ π xz (Λ) (crossings, cusps, swallow tail points, etc.) is contained in the1-skeleton of E . In addition, we require that:(1) Each 1-cell is assigned an orientation.(2) Each 2-cell is assigned an initial and terminal vertex, v and v . If v = v , then aprefered direction around the boundary of the 2-cell is also chosen.(3) At each swallowtail point, s , the two polygonal corners that border the crossing arcnear its endpoint at s are labelled as S and T .Let (Λ , E ) be a pair consisting of a closed Legendrian knot or surface, Λ ⊂ J M , togetherwith a choice of compatible polygonal decomposition. The cellular DGA of (Λ , E ) will bedenoted A (Λ , E ) or A cell (Λ , E ), and is defined as follows. Algebra:
Given a cell, e dα , denote by { S αp } the set of sheets of Λ above e dα . By definition, sheetsabove e dα are those components of Λ ∩ π − x ( e dα ) not contained in any cusp edge. (Note: (i) Sheetsare subsets of Λ, not π xz (Λ), so that, eg., a crossing arc of π xz (Λ) above a 1-cell correspondsto two sheets. (ii) A swallowtail point is considered to be a sheet above a 0-cell.) The algebra A (Λ , E ) is the free unital, associative Z / e dα , S αi , S αj ) where S αi and S αj are sheets above e dα such that the inequality z ( S αi ) > z ( S αj ) holdspointwise above e dα . We denote the generator associated to ( e dα , S αi , S αj ) as a αi,j , b αi,j , or c αi,j when the dimension of e dα is 0, 1, or 2 respectively. Grading: A Z /ρ -grading on A (Λ , E ) arises from a choice of Z /ρ -valued Maslov potential, µ ,for Λ. The Z /ρ -grading of generators is | a αi,j | = µ ( S αi ) − µ ( S αj ) − , | b αi,j | = µ ( S αi ) − µ ( S αj ) , | c αi,j | = µ ( S αi ) − µ ( S αj ) + 1 . (If S αi is a swallowtail point above a 0-cell, then take µ ( S αi ) to be the value of µ on the twosheets that cross near S αi .) Differential:
The differential ∂ : A (Λ , E ) → A (Λ , E ) is characterized on the generators of A (Λ , E ) by matrix formulas whose precise form depends on the dimension of the associated cell e dα ∈ E . We review these formulas here in the case that Λ does not have swallowtail points . Seealso [31, 34]. By polygonal decomposition, we mean a CW-complex decomposition such that the boundary of each 2-cellconsists of a sequence of vertices and edges (with repeats allowed). • For -cells: We choose a linear ordering of the sheets of Λ above e α so that the z -coordinates appear in non-increasing order z ( S αι (1) ) ≥ z ( S αι (2) ) ≥ · · · ≥ z ( S αι ( n ) )and use it to place the generators, a αp,q , into a matrix, A , whose ( i, j )-entry is a αι ( i ) ,ι ( j ) when z ( S αι ( i ) ) > z ( S αι ( j ) ) and is 0 otherwise. When ∂ is applied entry-by-entry to A wehave ∂A = A . • For -cells: After a choice of linear ordering of sheets as above, we place the generators, b αp,q , associated to a 1-cell, e α , into an n × n matrix, B . In addition, we form n × n boundary matrices , A − and A + , associated to the initial and terminal vertices, e − and e , of e α . The sheets above e ± are identified with a subset of the sheets above e α (since every sheet of e ± belongs to the closure of a unique sheet of e α ). Using thisidentification, we place the e ± generators, a ± p,q , into the corresponding rows and columnsof the n × n matrices A ± . Whenever two sheets of e α , S αk and S αk (cid:48) , meet at a cusp pointabove e ± , we place a 2 × N = (cid:20) (cid:21) on the diagonal at the location of the two (possibly non-consecutive) rows and columnsof A ± that correspond to S αk and S αk (cid:48) with respect to the linear ordering of sheets of e α .All other entries of A ± are 0. The differential on the b αp,q satisfies ∂B = A + ( I + B ) + ( I + B ) A − . • For -cells: The sheets above a 2-cell, e α , are already linearly ordered by descending z -coordinate, and using this ordering we place the generators c αi,j into an n × n -matrix, C . We form n × n boundary matrices, A v and A v , associated to the initial and terminalvertices, v and v , for e α following the same procedure as in the 1-cell case. Additionalboundary matrices B , . . . , B l are associated to the 1-cells that appear around the bound-ary of C . We assume that the numbering is such that B , . . . , B j (resp. B j +1 , . . . , B l ) arethe boundary matrices for the sequence of edges that appear along the path γ + (resp. γ − ) where γ ± are the two paths that travel around the boundary of e α (in the domain ofthe characteristic map) from v to v . When v = v one of γ ± is constant (as specifiedby the choice of preferred direction around ∂e α ). The B i are formed using the sameprocedure as for the boundary matrices associated to vertices except that the 2 × N blocks that correspond to pairs of cusp sheets are replaced with 2 × ∂C = A v C + CA v + ( I + B j ) η j · · · ( I + B ) η + ( I + B l ) η l · · · ( I + B j +1 ) η j +1 where η i is +1 (resp. −
1) when the orientation of the B i edge agrees (resp. disagrees)with the orientation (from v to v ) of the corresponding path γ ± . Note that since the B i are strictly upper triangular, ( I + B i ) − = I + B i + B i + · · · + B n − i . Remark 4.2.
To define ∂ in the case that Λ has swallowtail points, the following additionsshould be made. • The definition of the boundary matrices A ± , A v , and A v needs to be adjusted whenthe vertex is a swallowtail point of Λ. • Whenever a 2-cell contains one of the corners labeled S or T at a swallowtail point,an additional matrix need to be inserted into the product ( I + B j ) · · · ( I + B ) or ( I + B l ) · · · ( I + B j +1 ).These details may be found in [31] or [34] and are not needed for the arguments that follow. UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 21 e α e β { }× e α { }× e α (0 , × e α { }× e β { }× e β (0 , × e β v v Figure 4.
Notation for cells in E (left) and E (cid:48) (right). Observation 4.3.
Whenever E (cid:48) ⊂ E is a (CW) sub-complex, the collection of generatorsassociated to cells of E (cid:48) form a sub DGA of A cell (Σ , E ). Moreover, if E (cid:48) is a decomposition ofa curve or curve segment C ⊂ M such that the restriction of Σ to C (as in Section 2.3) is a1-dimensional Legendrian knot Λ ⊂ J C , then this sub-DGA is precisely A cell (Λ , E (cid:48) ).4.2. The cellular DGA for compact Legendrian cobordisms.
In extending the definitionof the cellular DGA to Legendrian cobordisms, it is most natural to consider compact cobordismsrather than conical cobordisms. The following definition coincides precisely with a standardnotion of Legendrian cobordism introduced by Arnold, cf. [1, 2, 3].
Definition 4.4.
Given an interval I ⊂ [0 ,
1] and Λ ⊂ J M , we write j (1 I · Λ) for the
Leg-endrian cylinder on Λ in J ( I × M ). It is defined to be the product of Λ ⊂ J M with the0-section in T ∗ I , i.e.(4.1) j (1 I · Λ) = 0 T ∗ I × Λ ⊂ T ∗ I × J M ∼ = J ( I × M ) . The notation is chosen to be consistent with that of Definition 2.10 with 1 I : I → R > denotingthe constant function 1.Let Λ , Λ ⊂ J M . A compact Legendrian cobordism from Λ to Λ , written Σ : Λ → Λ , is a Legendrian surface Σ ⊂ J ([0 , × M ) that, for some (cid:15) >
0, Σ agrees with the Legendriancylinder j (1 [0 ,(cid:15) ] · Λ ) in J ([0 , (cid:15) ] × M ) and agrees with j (1 [1 − (cid:15), · Λ ) in J ([1 − (cid:15), , × M ).When Σ : Λ → Λ is a compact Legendrian cobordism, we modify the definition of compatiblepolygonal decomposition for Σ to require that π x (Σ) ∩ ( { } × M ) and π x (Σ) ∩ ( { } × M ) are(CW) sub-complexes of E , that we denote E and E . Then, the definition of the cellularDGA extends immediately to give DGAs A (Σ , E ) when Σ is a compact Lagrangian cobordism.Moreover, since E and E may be viewed as polygonal decompositions for Λ and Λ , as in theObservation 4.3, we have inclusion maps(4.2) i : A (Λ , E ) → A (Σ , E ) and i : A (Λ , E ) → A (Σ , E ) . The DGA of a product cobordism.
Given a 1-dimensional Legendrian with compatiblepolygonal decomposition, (Λ , E ), we now compute the DGA of the product cobordism Σ = j (1 [0 , · Λ) ⊂ J ([0 , × M ). This will be a useful ingredient in a few later arguments.As a preliminary, use E to form the product decomposition , E (cid:48) , for π x (Σ) = [0 , × π x (Λ)as follows: For each d -cell e dα ∈ E (here, d = 0 or 1), decompose[0 , × e dα = { } × e dα (cid:116) (0 , × e dα (cid:116) { } × e dα so that { k } × e dα are d -cells in E (cid:48) , for k = 0 ,
1, and (0 , × e dα is a ( d + 1)-cell in E (cid:48) . Orient 1-cellsof the form { k } × e β in the same direction as e β , and those of the form (0 , × e α using thestandard orientation of (0 , , × e β choose the initial and terminal vertices v and v to be { } × e − and { } × e where e − and e are the initial and terminal vertices of e β . See Figure 4.Let us fix notation for the generators of the cellular DGA A (Σ , E (cid:48) ). Observe that for any e dα ∈ E , the sheets of Λ above e dα are in bijection with the sheets of Σ above any of the { k } × e dα or (0 , × e dα cells of E (cid:48) , so the generators associated to these cells are also in bijection withthose of e dα . • For a 0-cell e α ∈ E with corresponding generators a αi,j we notate the generators corre-sponding to { } × e α , { } × e α , and (0 , × e α as i ( a αi,j ), i ( a αi,j ), and b αi,j . • For a 1-cell e β ∈ E with corresponding generators b βi,j we notate the generators corre-sponding to { } × e β , { } × e β , and (0 , × e β as i ( b βi,j ), i ( b βi,j ), and c βi,j .In Proposition 4.5, the DGA of the product cobordism Σ is described as a mapping cylinderDGA. We now briefly review the relevant definitions, referring the reader to [30, Section 2] fora more thorough treatment in the present algebraic setting of triangular DGAs over Z /
2. Let f : ( A , ∂ A ) → ( B , ∂ B ) be a DGA map between based DGAs. The standard mapping cylinderDGA of f is ( A ∗ (cid:98)
A ∗ B , ∂ Γ ) where (cid:98) A has generators { (cid:98) a i } in correspondence with the generators { a i } of A but with the degree shift | (cid:98) a i | = | a i | + 1. The differential, ∂ Γ , satisfies ∂ Γ | A = ∂ A , ∂ Γ | B = ∂ B , and ∂ Γ ( (cid:98) a i ) = f ( a i ) + a i + Γ ◦ ∂ A ( a i )where Γ : A → A ∗ (cid:98)
A ∗ B is the unique ( f, id A )-derivation satisfying Γ( a i ) = (cid:98) a i . Proposition 4.5.
Given (Λ , E ) with Λ ⊂ J M and dim Λ = 1 , let (Σ , E (cid:48) ) be the productcobordism, Σ = j (1 [0 , · Λ) ⊂ J ([0 , × M ) equipped with the product decomposition, E (cid:48) . (1) For k = 0 , , the maps i k : A (Λ , E ) → A (Σ , E (cid:48) ) that extend the correspondence ofgenerators are DGA isomorphisms from A (Λ , E ) onto the sub-DGAs A k ⊂ A (Σ , E (cid:48) ) associated to the subcomplexes { k } × E ⊂ E (cid:48) . (2) Identifying the sub-algebra of A (Σ , E (cid:48) ) generated by the b αi,j and c βi,j with (cid:99) A using thegrading preserving bijection (cid:92) i ( a αi,j ) ↔ b αi,j and (cid:92) i ( b βi,j ) ↔ c βi,j gives a DGA isomorphism ϕ : ( A (Σ , E (cid:48) ) , ∂ ) ∼ = → ( A ∗ (cid:99) A ∗ A , ∂ Γ ) with the standard mapping cylinder DGA of the map i ◦ i − : A → A .Proof. (1) is obvious. For (2), ϕ is clearly an algebra isomorphism which is the identity on the A k , so we just to check that ∂ Γ ◦ ϕ ( x ) = ϕ ◦ ∂ ( x ) when x = b αi,j or c βi,j . From the definition of ∂ Γ , we have ∂ Γ (cid:92) i ( a αi,j ) = i ( a αi,j ) + i ( a αi,j ) + Γ ◦ ∂ Λ a αi,j (4.3) ∂ Γ (cid:92) i ( b βi,j ) = i ( b βi,j ) + i ( b βi,j ) + Γ ◦ ∂ Λ b βi,j (4.4)where ∂ Λ is the differential on A (Λ , E ) and Γ : A (Λ , E ) → A ∗ (cid:99) A ∗ A is the ( i , i )-derivationsatisfying Γ( x ) = (cid:91) i ( x ) on generators.Using matrix notation, this allows us to compute ϕ ◦ ∂ ( B α ) = ϕ ( i ( A α )( I + B α ) + ( I + B α ) i ( A α ))= i ( A α ) + i ( A α ) (cid:92) i ( A α ) + i ( A α ) + (cid:92) i ( A α ) i ( A α )= i ( A α ) + i ( A α ) + Γ( A α )= ∂ Γ ( (cid:92) i ( A α )) = ∂ Γ ◦ ϕ ( B α ) , UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 23 and ∂ Γ ◦ ϕ ( C β ) = ∂ Γ ( (cid:92) i ( B β )) = i ( B β ) + i ( B β ) + Γ( ∂B β )= i ( B β ) + i ( B β ) + Γ( A + ( I + B β ) + ( I + B β ) A − )= i ( B β ) + i ( B β ) + Γ( A + )( I + i ( B β )) + i ( A + ) (cid:92) i ( B β )+ (cid:92) i ( B β ) i ( A − ) + ( I + i ( B β ))Γ( A − )= ϕ (cid:18) i ( B β ) + i ( B β ) + B + ( I + i ( B β )) + i ( A + ) C β + C β i ( A − ) + ( I + i ( B β )) B − (cid:19) = ϕ (cid:18) i ( A + ) C β + C β i ( A − ) + ( I + B + )( I + i ( B β )) + ( I + i ( B β ))( I + B − ) (cid:19) = ϕ ◦ ∂ ( C β ) . At the fourth equality, it should be observed that when A − and A + are the boundary matricesfor e β associated to the vertices e − and e , the boundary matrices B − and B + for (0 , × e β associated to the edges (0 , × e − and (0 , × e indeed satisfy ϕ ( B ± ) = Γ( A ± ). [Note thatwhen a (cid:20) (cid:21) block appears on the diagonal of A ± due to two sheets of e β meeting at a cuspabove e ± , since Γ(1) = 0 (any derivation has this property), the appropriate (cid:20) (cid:21) block willappear in Γ( A ± ).] (cid:3) Immersed DGA maps from cobordisms via the cellular DGA.
Our aim is to nowdefine a cellular version of the immersed LCH functor F : Leg ρim → DGA ρ im from Corollary 2.13.Recall that the domain category for F has ρ -graded Legendrians in J M equipped with regularmetrics as objects and has (conical Legendrian isotopy classes of) conical Legendrian cobordismsin J ( R > × M ) (equivalently, good immersed Lagrangian cobordisms in Symp ( J M )) as mor-phisms. For the cellular construction we instead work with a category of compact cobordisms.With M = R or S and ρ ≥ cellular Legendrian cobordism category ,denoted Leg ρ cell , whose objects are triples (Λ , E , µ ) consisting of a 1-dimensional Legendrianlink, Λ ⊂ J M , together with a choice, E , of compatible polygonal decomposition, and a choice, µ , of Z /ρ -valued Maslov potential. Morphisms from (Λ , E , µ ) to (Λ , E , µ ) are equivalenceclasses of compact Legendrian cobordisms Σ ⊂ J ([0 , × M ) from Λ − to Λ + having genericbase and front projection and equipped with a Z /ρ -valued Maslov potential extending µ and µ . Here, two cobordisms are considered equivalent if their front and base projections are combinatorially equivalent . That is, Σ and Σ (cid:48) are equivalent if there is a homeomorphism φ : ([0 , × M ) × R → ([0 , × M ) × R with φ ( π xz (Σ)) = φ ( π xz (Σ (cid:48) )) that is a composition of (i)a homeomorphism, φ , of the [0 , × M factor, and (ii) a homeomorphism, φ , that preservesthe [0 , × M factor. Moreover, φ and φ should be isotopic to the identity and equal to theidentity in a neighborhood of the boundary. Remark 4.6. (1) As with the category
Leg ρ im , in the definition of morphisms Σ is notequipped with any additional structure beyond a choice of Maslov potential, eg. Σ isnot equipped with a polygonal decomposition.(2) In contrast to Leg ρ im , we do NOT allow general Legendrian isotopies of Σ in the equiva-lence relation used to define morphisms. This is because from the initial definition of ourcellular LCH functor we will not check directly that the assignment of immersed DGAmaps to Legendrian cobordisms factors through general Legendrian isotopies. However, it will later be established in Corollary 4.11 that the induced immersed DGA maps(considered up to immersed homotopy) are indeed Legendrian invariants of Σ.The next proposition defines the cellular LCH functor , F cell . Proposition 4.7.
There is a well-defined contravariant functor F cell : Leg ρ cell → DGA ρim givenby (Λ , E ) (cid:32) A cell (Λ , E ) , Σ : (Λ , E ) → (Λ , E ) (cid:32) (cid:2) A cell (Λ , E ) i → A cell (Σ , E (cid:48) ) i ← (cid:45) A cell (Λ , E ) (cid:3) where E (cid:48) is any choice of compatible polygonal decomposition for Σ that restricts to E k on { k } × M , k = 0 , , and the maps i and i are as in (4.2).Proof. To see that F cell is well-defined it is enough to verify independence of the choice of E (cid:48) .(Since modifying (Σ , E (cid:48) ) in a manner that preserves the combinatorics of the front and baseprojections does not affect the cellular DGA.)In [31, Section 4.2-4.4], it is shown in the case when Σ is a closed surface that any compatiblepolygonal decompositions E (cid:48) and E (cid:48)(cid:48) for Σ can be made the same after some sequence of thefollowing modifications and their inverses (see [31] for the precise meanings).(1) Subdivide a 1-cell.(2) Subdivide a 2-cell.(3) Delete a 1-valent edge.(4) Switch the S and T decorations at a swallowtail point.The proof extends with minor adjustments to the case of a compact cobordism. Moreover, if E (cid:48) and E (cid:48)(cid:48) agree above ∂ ([0 , × M ), then the polygonal decomposition of ∂ ([0 , × M ) can beleft unchanged throughout the sequence of modifications. [To see this, follow the proof of [31,Theorem 4.1], but treat all 0-cells (resp. 1-cells) in ∂ ([0 , × M ) as if they belong to what isnotated there as Σ (resp. Σ ).]If E (cid:48) and E (cid:48)(cid:48) are related by one of the modifications (1)-(4), then [31, Theorems 4.2-4.5] providestable tame isomorphisms ϕ : A cell (Σ , E (cid:48) ) ∗ S (cid:48) → A cell (Σ , E (cid:48)(cid:48) ) ∗ S (cid:48)(cid:48) . Moreover, in all cases ϕ canbe seen to restrict to the identity on the sub-algebras A cell (Λ k , E k ), k = 0 ,
1. [For (1)-(3), ϕ isdefined using the construction of Proposition 2.1, and generators from A cell (Λ k , E k ) are neveramong the generators that are canceled. For (4), ϕ is the identity on all generators except for1-cells with endpoints at swallowtail points.] Thus, we will have identities ϕ ◦ i k = i k , k = 0 , ϕ provides the required immersed DGA homotopy.To verify that F cell preserves composition, suppose that E (cid:48) and E (cid:48)(cid:48) are compatible polygonaldecompositions for a pair of composable (compact) cobordisms, Σ (cid:48) : (Λ , E ) → (Λ , E ) andΣ (cid:48)(cid:48) : (Λ , E ) → (Λ , E ). Since E (cid:48) and E (cid:48)(cid:48) agree with E along Λ , they can be glued to forma well-defined compatible polygonal decomposition, E (cid:48) ∪ E (cid:48)(cid:48) , on Σ (cid:48)(cid:48) ◦ Σ (cid:48) . Moreover, when thisdecomposition is used for computing it is readily verified from Definition 2.4 that F cell (Σ (cid:48)(cid:48) ◦ Σ (cid:48) ) = F cell (Σ (cid:48) ) ◦ F cell (Σ (cid:48)(cid:48) ).Finally, we check that F cell preserves identities. The identity morphism in Hom
Leg ρ cell ((Λ , E ) , (Λ , E ))is the product cobordism, Σ = j (1 [0 , · Λ), and for computing F cell (Σ) we equip it with theproduct decomposition, E (cid:48) . The computation from Proposition 4.5 shows that the morphism F cell (Σ) is the immersed homotopy class of the immersed DGA map D = (cid:0) A (Λ , E ) i → ( A ∗ (cid:99) A ∗ A , ∂ Γ ) i ← (cid:45) A (Λ , E ) (cid:1) where A and A are two copies of A (Λ , E ) and ( A ∗ (cid:99) A ∗ A , ∂ Γ ) is the standard mappingcylinder DGA for the identity map on A (Λ , E ). On the other hand, the identity morphism in DGA ρ im is just [ I ] where I is the immersed DGA map I = (cid:0) A (Λ , E ) id → A (Λ , E ) id ← A (Λ , E ) (cid:1) . UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 25
Note that the immersed map D is obtained from two copies of I precisely as in the statement ofProposition 2.5, which therefore shows that D is immersed homotopic to I ◦ I = I as required. (cid:3) Isomorphism between the cellular and immersed LCH functors.
In this section,we state a precise relationship between the immersed and cellular LCH functors, F and F cell ,with the proof as well as some technical details postponed until the Appendices A and B.The functors F : Leg ρ im → DGA ρim and F cell : Leg ρ cell → DGA ρ im have slightly different domaincategories. In Leg ρim (resp.
Leg ρ cell ), the objects are (closed, 1-dimensional) ρ -graded Legendri-ans in J M equipped with a suitable choice of metric (resp. polygonal decomposition), whilemorphisms are equivalence classes of conical (resp. compact) ρ -graded Legendrian cobordisms.For comparing the two functors, it is convenient to work with a third domain category, thatagain consists of Legendrians equipped with some additional data. Definition 4.8.
With ρ ∈ Z ≥ and 1-dimensional M fixed, define the transverse Legendriancategory , Leg ρ (cid:116) , to have objects (Λ , E (cid:116) , µ ) where Λ ⊂ J M is Legendrian, µ is a Z /ρ -valuedMaslov potential, and E (cid:116) is an admissible transverse decomposition for Λ as defined inAppendix A, Definition A.1. Morphisms are compact ρ -graded Legendrian cobordisms up tocombinatorial equivalence of front and base projections (as in Section 4.4).An admissible transverse decomposition of Λ is a cellular decomposition of a neighborhoodof the base projection π x (Λ) ⊂ M , having cells transverse to the projection of the singular setof Λ, subject to some technical restrictions that will not be relevant in the main body of thearticle. At this point, it is sufficient to note that any 1-dimensional Legendrian Λ ⊂ J M withgeneric front and base projections admits an admissible transverse decomposition.In Appendix A, two functors are constructed G std : Leg ρ (cid:116) → Leg ρ im , and G || : Leg ρ (cid:116) → Leg ρ cell , (Λ , E (cid:116) ) (cid:55)→ ( (cid:101) Λ , (cid:101) g ) (Λ , E (cid:116) ) (cid:55)→ (Λ , E || )where (cid:101) Λ, called the standard geometric model for Λ, is Legendrian isotopic to Λ with frontand base projections combinatorially equivalent to those of Λ. The actions on morphisms areas follows: • The morphism spaces in
Leg ρ (cid:116) and Leg ρ cell only depend on Λ and are the same in bothcategories, and G || acts as the identity on morphisms. • Given a compact cobordism Σ from Λ to Λ , we can choose any conical Legendriancobordism Σ (cid:48) ⊂ J ( R > × M ) from G std (Λ ) to G std (Λ ) whose front and base projectionsare combinatorially equivalent to Σ (after truncating the conical ends). The conicalLegendrian isotopy type of Σ (cid:48) depends only on Σ, and we have G std ([Σ]) = [Σ (cid:48) ].The standard geometric model, G std (Λ , E (cid:116) ) = ( (cid:101) Λ , (cid:101) g ), is constructed so that the Reeb chordsof (cid:101) Λ closely match the generators of the cellular DGA, A (Λ , E || ). In fact, A ( (cid:101) Λ , (cid:101) g ) is isomorphicto a stabilization of A (Λ , E || ), and a particular homotopy equivalence q Λ : A ( (cid:101) Λ , (cid:101) g ) → A (Λ , E || )that we call the canonical quotient map is identified in Section A.4. As q Λ is invertible inthe DGA homotopy category DGA ρ hom , it follows that the immersed map Q Λ = (cid:2) A ( (cid:101) Λ , (cid:101) g ) q Λ → A (Λ , E || ) id ← (cid:45) A (Λ , E || ) (cid:3) ∈ DGA ρ im (that is the image of q Λ under the functor from Proposition 2.7) is an isomorphism in DGA ρ im . Proposition 4.9.
The canonical quotient map construction, (Λ , E (cid:116) ) ∈ Leg ρ (cid:116) (cid:55)→ Q Λ ∈ Hom
DGA ρ im ( A ( (cid:101) Λ , (cid:101) g ) , A (Λ , E || )) gives an isomorphism (invertible natural transformation) from the functor that is the composition Leg ρ (cid:116) G std → Leg ρim F → DGA ρ im . to the functor Leg ρ (cid:116) G || → Leg ρ cell F cell → DGA ρ im . That is, the Q Λ are invertible, and for any compact Legendrian cobordism Σ : (Λ − , E (cid:116) − ) → (Λ + , E (cid:116) + ) with generic base and front projection we have a commutative diagram in DGA ρ im , (4.5) A ( (cid:101) Λ + , (cid:101) g + ) F ◦ G std (Σ) (cid:47) (cid:47) Q Λ+ (cid:15) (cid:15) A ( (cid:101) Λ − , (cid:101) g − ) Q Λ − (cid:15) (cid:15) A cell (Λ + , E || + ) F cell ◦ G || (Σ) (cid:47) (cid:47) A cell (Λ − , E ||− ) . In summary, the following diagram is commutative up to a canonical isomorphism of functors:
Leg ρim F (cid:37) (cid:37)
Leg ρ (cid:116) G std (cid:59) (cid:59) G || (cid:35) (cid:35) DGA ρ im Leg ρ cell F cell (cid:57) (cid:57) Remark 4.10.
An arbitrary Legendrian Λ ⊂ J M is related to one in the image of G std by aLegendrian isotopy (that preserves the combinatorial appearance of the front projection). ThusProposition 4.9 allows the cellular DGA to be used for computing immersed maps F (Σ) inducedby a conical Legendrian cobordism Σ : Λ − → Λ + after possibly changing the Legendrians atthe ends from Λ ± to Λ (cid:48)± by Legendrian isotopy. Composing Σ with cobordisms Σ + : Λ + → Λ (cid:48) + and Σ − : Λ (cid:48)− → Λ − induced by the isotopies modifies Σ to a cobordism Σ (cid:48) : Λ (cid:48)− → Λ (cid:48) + to whichthe cellular computation of Proposition 4.9 can be applied directly. Moreover, if one wants towork with the original Λ ± , then the maps F (Σ − ) and F (Σ + ) can be explicitly computed as in[15, Section 6].The following corollary of Proposition 4.9 shows that the immersed DGA maps associated toa compact cobordism Σ by the cellular functor, F cell , only depend on the Legendrian isotopytype of Σ. Corollary 4.11.
Given (Λ , E ) , (Λ , E ) ∈ Leg ρ cell and compact cobordisms Σ and Σ (cid:48) . If Σ and Σ (cid:48) are Legendrian isotopic rel. boundary, then F cell (Σ) = F cell (Σ (cid:48) ) in DGA ρ im .Proof. Case 1.
Suppose (Λ k , E k ) = G || (Λ k , E (cid:116) k ) for some admissible transverse decompositions E (cid:116) k for Λ k , k = 0 , DGA ρ im , F cell (Σ) = Q Λ ◦ F ( G std (Σ)) ◦ Q − = Q Λ ◦ F ( G std (Σ (cid:48) )) ◦ Q − = F cell (Σ (cid:48) ) . [At the second equality we used that F ( G std (Σ)) only depends on the conical Legendrian isotopytype of G std (Σ).] Case 2.
For general E and E .In this case, fix some choice of admissible transverse decompositions E (cid:116) k for Λ k , k = 0 ,
1. Let I k denote the product cobordism j (1 [0 , · Λ k ) viewed as a morphism in Hom
Leg ρ cell ((Λ k , E k ) , (Λ k , G || ( E (cid:116) k )), UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 27 and let Σ (cid:116) and Σ (cid:48) (cid:116) denote Σ and Σ (cid:48) viewed as morphisms in
Hom
Leg ρ cell ((Λ , G || ( E (cid:116) )) , (Λ , G || ( E (cid:116) )).Using Case 1, we compute F cell (Σ) = F cell ( I − ◦ Σ (cid:116) ◦ I ) = F cell ( I − ) ◦ F cell (Σ (cid:116) ) ◦ F cell ( I )= F cell ( I − ) ◦ F cell (Σ (cid:48) (cid:116) ) ◦ F cell ( I ) = F cell (Σ (cid:48) ) . (cid:3) As in Section 3.1, given an immersed DGA map M = (cid:16) A f → B i ← (cid:45) A (cid:17) we write I ( M ) = Im ( i ∗ × f ∗ ) ⊂ Aug ρ ( A ) / ∼ × Aug ρ ( A ) / ∼ for the induced augmentation set of M . Accordingto Proposition 3.2, I ( M ) only depends on the immersed homotopy class of M . Corollary 4.12.
Let Σ ⊂ J ([0 , × M ) be a compact Legendrian cobordism from Λ to Λ . Let E (cid:116) k be an admissible transverse decomposition for Λ k , k = 0 , , and write G std (Λ k , E (cid:116) k ) = ( (cid:101) Λ k , (cid:101) g k ) and G || (Λ k , E (cid:116) k ) = (Λ k , E || k ) . Then, the induced augmentation sets I Σ := I ( F ( G std (Σ))) and I cell Σ := I ( F cell ( G || (Σ))) are related by (4.6) ( q ∗ Λ × q ∗ Λ ) (cid:0) I cell Σ (cid:1) = I Σ where q ∗ Λ × q ∗ Λ : Aug ρ (Λ , E || ) / ∼ × Aug ρ (Λ , E || ) / ∼ ∼ = → Aug ρ ( (cid:101) Λ , (cid:101) g ) / ∼ × Aug ρ ( (cid:101) Λ , (cid:101) g ) / ∼ is the bijection induced by the canonical quotient maps, q Λ k : A ( (cid:101) Λ k , (cid:101) g k ) → A (Λ k , E || k ) .Proof. Applying the functor from Proposition 3.6 to (4.5) produces the diagram of relations(4.7)
Aug ( (cid:101) Λ ) , (cid:101) g ) / ∼ Aug ( (cid:101) Λ , (cid:101) g ) / ∼ I Σ (cid:111) (cid:111) Aug (Λ , E || ) / ∼ I ( Q Λ1 ) (cid:79) (cid:79) Aug (Λ , E ) / ∼ I cell Σ (cid:111) (cid:111) I ( Q Λ0 ) (cid:79) (cid:79) . As noted in 3.1.1, the induced augmentation sets associated to Q Λ k are just the graphs of q ∗ Λ k ,so Observation 3.5 (2) and (3) can be applied to arrive at (4.6). (cid:3) Computations via Morse complex families
With Corollary 4.12 we have seen that for a compact Legendrian cobordism, Σ, the twoversions of the induced augmentation sets, I cell Σ and I Σ , defined via the cellular and ordinaryversions of the LCH functor, F cell and F , are equivalent invariants of Σ (in a canonical way).In this section, we focus on obtaining methods for computing I cell Σ . Augmentations of thecellular DGA can be conveniently viewed as chain homotopy diagrams (abbr. CHDs) whichare certain homotopy commutative diagrams of chain complexes that have their form specifiedby the base and front projections of a Legendrian knot or surface. CHDs for 2-dimensionalLegendrians were introduced in [34] where they were shown to be closely connected with Morsecomplex families (abbr. MCFs) which are combinatorial objects motivated by Morse theory. Inthe 1-dimensional case, Morse complex families have been studied in the works of Henry andRutherford, [22, 23, 24]. In particular, for a Legendrian knot, Λ ⊂ J R , [22, 24] establishes abijection between Aug (Λ) / ∼ and equivalence classes of MCFs for Λ.In Section 5.1, we briefly review the correspondence between augmentations and CHDs inthe context of compact Legendrian cobordisms. In Sections 5.2-5.3, we review MCFs and theirconnection with CHDs in the context of cobordisms. In Section 5.4, Proposition 5.19 establishesa characterization of induced augmentation sets in terms of MCFs. The section concludesin Section 5.5 with a discussion of A -form MCFs, as defined by Henry, that are particularlyconvenient for computing induced augmentation sets in explicit examples. Augmentations of the cellular DGA.
We briefly recall the definition of chain homotopydiagrams as in [34], and extend the definition to compact Legendrian cobordisms.Let Λ ⊂ J M be either a Legendrian knot or surface, i.e. dim(Λ) = dim( M ) = 1 or 2,equipped with a Z /ρ -valued Maslov potential, µ , and let E be a compatible polygonal decom-position for Λ. We allow the case that Λ is a compact Legendrian cobordism, and in this case E also restricts to a polygonal decomposition for the boundary components of Λ. To simplifythe definition, when dim(Λ) = 2 we assume Λ has no swallowtail points. To each d -cell e dα ∈ E , d = 0 , ,
2, we associate the Z /ρ -graded Z / V α = Span Z / { S αp } , where { S αp } is the set of sheets of Λ above e α . (Recall that sheets are components of π − x ( e dα ) ∩ Λnot contained in a cusp edge.) The Z /ρ -grading is specified on generators by the Maslovpotential, i.e. in V α the sheet S αp has grading | S αp | = µ ( S αp ) . The basis sets { S αp } are partially ordered by descending z -coordinates of sheets, and we say thata linear map f : V α → V α is strictly upper triangular if(5.1) (cid:104) f ( S αq ) , S αp (cid:105) (cid:54) = 0 ⇒ z ( S αp ) > z ( S αq ) . For d (cid:48) < d , whenever e d (cid:48) α appears along the boundary of e dβ (via a lifting e d (cid:48) α (cid:44) → D d c β → e dβ where c β is the characteristic map for e dβ ), there is an inclusion of sheets ι : { S αp } (cid:44) → { S βq } where ι ( S αp )is the sheet above e dβ that limits to S αp as the x -coordinate approaches e d (cid:48) α . This extends linearlyto an inclusion ι : V α (cid:44) → V β such that V β = ι ( V α ) ⊕ V cusp where V cusp is spanned by those sheets above e dβ that limit to a cusp edge above e d (cid:48) α .Suppose in addition that V α is equipped with a differential d : V α → V α of degree +1 mod ρ making ( V α , d ) into a Z /ρ -graded cochain complex. Then, there is an associated boundarydifferential (cid:98) d : V β → V β obtained by extending d so that on V cusp we have (cid:98) d ( S βq ) = S βp if thesheets S βp and S βq meet at a cusp above e d (cid:48) α so that S βp (resp. S βq ) is the upper (resp. lower)sheet of the cusp. Suppose next that e d (cid:48) α is a 1-cell and f : ( V α , d − ) → ( V α , d + ) is a chainmap (for some pair of differentials d ± on V α ). Then, there is an associated boundary map (cid:98) f : ( V β , (cid:98) d − ) → ( V β , (cid:98) d + ) obtained by extending f to be the identity on V cusp . Definition 5.1. A ρ -graded chain homotopy diagram (abbrv. CHD ) for (Λ , µ, E ) is atriple D = ( { d α } , { f β } , { K γ } ) satisfying:(1) For each 0-cell, e α , d α : V α → V α , d α = 0is a strictly upper triangular differential of degree +1 mod ρ making ( V α , d α ) into acochain complex.(2) For each 1-cell, e β , f β : ( V β , (cid:98) d − ) → ( V β , (cid:98) d + ) , f β ◦ (cid:98) d − = (cid:98) d + ◦ f β is a chain isomorphism of degree 0 mod ρ such that f β − id is strictly upper triangular.Here, (cid:98) d − and (cid:98) d + are the boundary differentials arising from the initial and terminalvertices of e β .(3) For each 2-cell, e γ , K γ : ( V γ , (cid:98) d ) → ( V γ , (cid:98) d ) , (cid:98) f η j j ◦ · · · ◦ (cid:98) f η − (cid:98) f η m m ◦ · · · ◦ (cid:98) f η j +1 j +1 = (cid:98) d ◦ K γ + K γ ◦ (cid:98) d , UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 29 is a homotopy operator of degree − ρ between the compositions of boundary maps (cid:98) f η j j ◦ · · · ◦ (cid:98) f η and (cid:98) f η m m ◦ · · · ◦ (cid:98) f η j +1 j +1 . Here, f , . . . , f j and f j +1 , . . . , f m are the chain mapsassociated to the sequencess of edges that appear as we travel the two paths aroundthe boundary of e γ from the initial vertex, v , to the terminal vertex, v (as in thedefinition of the differential for A (Λ , E )); the exponents η i are ± v to v ; and the maps (cid:98) d and (cid:98) d are the boundary differentials arising from v and v .We write CHD ρ (Λ , µ, E ) or simply CHD ρ (Λ , E ) for the set of all ρ -graded CHDs for (Λ , µ, E ). Remark 5.2.
When Λ ⊂ J M is a Legendrian surface with swallowtail points, the samedefinition applies except that more care needs to be taken in the construction of boundarydifferentials and boundary maps for cells that border swallowtail points. See [34, Section 3].We will not need the precise details near swallowtail points in this article.Given a CHD D = ( { d α } , { f β } , { K γ } ) for (Λ , E ), we can define an algebra homomorphism (cid:15) : A (Λ , E ) → Z / ∀ e α , (cid:15) ( a αp,q ) = (cid:104) d α ( S αq ) , S αp (cid:105) when z ( S αp ) > z ( S αq ); ∀ e β , (cid:15) ( b βp,q ) = (cid:104) f β ( S βq ) , S βp (cid:105) when z ( S βp ) > z ( S βq ); ∀ e γ , (cid:15) ( c γp,q ) = (cid:104) K γ ( S γq ) , S γp (cid:105) when z ( S γp ) > z ( S γq ).That is, the values of (cid:15) on generators associated to a given cell of E are the above diagonalmatrix coefficients of the map associated to the cell by D . Proposition 5.3.
For any (Λ , µ, E ) consisting of a ρ -graded Legendrian knot or surface (in-cluding cobordisms) equipped with a compatible polygonal decomposition E , the above defines abijection Aug ρ (Λ , E ) ↔ CHD ρ (Λ , E ) between ρ -graded augmentations of the cellular DGA A (Λ , E ) and ρ -graded CHDs.Proof. This is proven in Proposition 3.7 of [34] by checking that (cid:15) ◦ ∂ = 0 holds on the generatorsof a given cell if and only if the corresponding condition from (1)-(3) holds for that cell. Becauseof the local nature of the proof, it applies equally well in the cobordism case. (cid:3) Note that when E is a compatible polygonal decomposition for a Legendrian surface, Σ, and E (cid:48) is a CW sub-complex of E above which Σ restricts to a Legendrian Λ, we get a restrictionmap on CHDs CHD ρ (Σ , E ) → CHD ρ (Λ , E (cid:48) ) , D (cid:55)→ D| Λ by forgetting the data associated to cells not belonging to E (cid:48) . Proposition 5.4.
Let
Σ : Λ − → Λ + be a ρ -graded compact Legendrian cobordism and E bea compatible polygonal decomposition for Σ restricting to decompositions E ± for Λ ± . Then,using the bijection from Proposition 5.3 to identify augmentations and CHDs, the (cellular)augmentation map i ∗− × i ∗ + : Aug ρ (Σ , E ) → Aug ρ (Λ − , E − ) × Aug ρ (Λ + , E + ) associated to theimmersed DGA map F cell (Σ) satisfies ( i ∗− × i ∗ + )( D ) = ( D| Λ − , D| Λ + ) . Proof.
This is clear since the maps i ± : A (Λ ± , E ± ) (cid:44) → A (Σ , E ) are inclusions of sub-DGAs. (cid:3) Morse complex families.
We now review the definition of Morse complex families forLegendrian knots and surfaces, as in [22, 34], allowing for the case of Legendrian cobordisms.Morally, a Morse complex family can be thought of as a CHD with all chain maps and chainhomotopies factored into elementary pieces.
The 1-dimensional case.
Let (Λ , µ ) ⊂ J M be a 1-dimensional ρ -graded Legendrian knot.Recall that we use the respective notations π x : J M → M and π xz : J M → M × R for thebase and front projections. Definition 5.5. A ρ -graded Morse complex family (abbr. MCF ) for (Λ , µ ) is a pair C = ( H, { d ν } ) consisting of the following items:(1) A handleslide set H which is a finite collection of points in M such that each p ∈ H isequipped with lifts u ( p ) , l ( p ) ∈ Λ not belonging to cusp edges and satisfying π x ( u ( p )) = π x ( l ( p )) = p , z ( u ( p )) > z ( l ( p )), and such that µ ( u ( p )) = µ ( l ( p )). We refer to p ∈ H asa (formal) handleslide of C with upper and lower endpoints at u ( p ) and l ( p ).Denote by Λ sing the base projection of the singular set of π xz (Λ) (crossings and cusps), and let { R ν } be the collection of path components of M \ ( H ∪ Λ sing ). Note that above each R ν , thesheets of Λ are totally ordered by their z -coordinates, so they can be enumerated as S ν , . . . , S νn ν with z ( S ν ) > · · · > z ( S νn ν ) . We then have Z /ρ -graded vector spaces V ( R ν ) = Span Z / { S νi } , and we sometimes omit thesuperscript ν in the notation for sheets.(2) The second piece of information defining an MCF is a collection of strictly upper trian-gular (as in (5.1)) differentials { d ν } of degree +1 mod ρ making each ( V ( R ν ) , d ν ) into a Z /ρ -graded cochain complex.Moreover, H and { d ν } are required to satisfy the following Axiom 5.6. Axiom 5.6.
Let R and R be connected components of π x (Λ) \ ( H ∪ Λ sing ) that border eachother at a point p ∈ H ∪ Λ sing .(1) If p ∈ H is a handleslide with upper endpoint u ( p ) ∈ S u and l ( p ) ∈ S l , then thehandleslide map h u,l ( S i ) = (cid:26) S i , i (cid:54) = l,S u + S l , i = l, gives a chain isomorphism h u,l : ( V ( R ) , d ) ∼ = → ( V ( R ) , d ).(2) If p is the base projection of a crossing between sheets S k and S k +1 , then the permutationmap Q ( S k ) = S k +1 , Q ( S k +1 ) = S k , Q ( S i ) = S i , i / ∈ { k, k + 1 } is a chain isomorphism, Q : ( V ( R ) , d ) ∼ = → ( V ( R ) , d ).(3) Suppose that p is a cusp point such that the two sheets of the cusp, S k and S k +1 , existabove R but not above R . Then, identifying sheets of R and R whose closuresintersect at p gives an inclusion ι : V ( R ) (cid:44) → V ( R ) with V ( R ) = ι ( V ( R )) ⊕ V cusp , V cusp = Span Z / { S k , S k +1 } . We require that ι ◦ d = d ◦ ι and d S k +1 = S k , i.e. as a complex ( V ( R ) , d ) is the splitextension of ( V ( R ) , d ) by ( V cusp , d (cid:48) ) where d (cid:48) maps the lower sheet of the cusp to theupper sheet of the cusp. Remark 5.7. (1) The requirement in Axiom 5.6 (3) used here is stronger than in someother versions of the definition, eg. in [22, 23, 24]. In the terminology of [22], withour definition we only consider Morse complex families with “simple births and deaths”.The collection of equivalence classes of MCFs is not affected by making this restriction;see [22, Proposition 3.17].(2) If Λ ⊂ J R , then the collection of differentials { d ν } is uniquely determined by Λ and H , since one can work from left to right and apply Axiom 5.6 to determine the { d ν } inductively. However, it should be noted that not every handleslide set H for Λ willproduce an MCF. For example, consider the case when R and R are seperated by UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 31 a crossing point, and d has already been determined. If (cid:104) d S k +1 , S k (cid:105) (cid:54) = 0, then thedifferential d = Q ◦ d ◦ Q − required by Axiom 5.6 (2) will not be strictly uppertriangular.In figures, we picture handleslides as vertical segments connecting u ( p ) and l ( p ) on the frontprojection of Λ. When convenient, differentials d ν may be indicated above R ν by drawing adotted arrow from S i to S j whenever (cid:104) d ν S j , S i (cid:105) = 1. See for example Figures 5 and 9 below.5.2.2. The 2-dimensional case.
Consider a 2-dimensional ρ -graded compact Legendrian surface,(Σ , µ ) ⊂ J M where dim M = 2. We allow for ∂M (cid:54) = ∅ , but in this case require that Σis properly embedded, eg. Σ could be a compact Legendrian cobordism in J ([0 , × R ) or J ([0 , × S ). We again use the notation Σ sing ⊂ M for the base projection of the singular setof π xz (Σ) (which now includes crossings, cusps, and swallowtail points). Definition 5.8. A ρ -graded MCF for Σ is a triple ( H , H − , { d ν } ) consisting of:(1) A set of super-handleslide points , H − ⊂ Int ( M ) \ Σ sing , equipped with upper andlower endpoint lifts to Σ satisfying µ ( u ( p )) = µ ( l ( p )) − p ∈ H − .(2) A collection of handleslide arcs which is an immersion H : X → M of a compact1-manifold, transverse to ∂M . Aside from the exceptions at boundary points specifiedin Axiom 5.9 below, we require that H is transverse to H ∪ Σ sing and that the onlyself intersections are at transverse double points in Int ( M ). In addition, H should beequipped with upper and lower endpoint lifts defined on the interior of Xu, l : Int ( X ) → Σ , π x ◦ u = π x ◦ l = H , satisfying u ( s ) > l ( s ) and µ ( u ( s )) = µ ( l ( s )) ∀ s ∈ Int ( X ) . Endpoints of handleslide arcs must satisfy Axiom 5.9 below.(3) A collection of differentials { d ν } of degree +1 mod ρ making each ( V ( R ν ) , d ν ) into a Z /ρ -graded cochain complex where { R ν } is the collection of path components of M \ ( H − ∪ H ( X ) ∪ Σ sing ). Note that H − ∪ H ( X ) ∪ Σ sing is a stratified subset of M whose1-dimensional strata are handleslide arcs , crossing arcs , and cusp arcs . For each such1-dimensional stratum, we require that the complexes ( V ( R ) , d ) and ( V ( R ) , d ) thatappear adjacent to the stratum are related as in Axiom 5.6 (1)-(3).Before stating Axiom 5.9 we introduce some convenient terminology. Above M \ Σ sing sheetsof Σ are locally labelled S , . . . , S n with descending z -coordinate. We will refer to a handleslidearc that has its upper and lower lift on sheets S i and S j as an ( i, j ) -handleslide . We cautionthat this terminology only applies locally since the labeling of upper and lower sheets of ahandleslide arc may change when the arc passes through Σ sing . We use a similar terminologyfor super-handleslide points. Axiom 5.9.
Endpoints of handleslide arcs can occur at transverse intersections of H and ∂M .All other endpoints of handleslide arcs occurring in the interior of M are as specified in thefollowing:(1) Suppose that p ∈ M is a transverse double point for H | Int ( X ) where a ( u , l )-handleslidearc crosses a ( u , l )-handleslide arc. If l = u , then at p there is a single endpoint of a( u , l )-handleslide arc.(2) Let p ∈ H be a ( u, l )-super-handleslide point, and fix some region R ν bordering p . • For every i < u such that (cid:104) d ν S u , S i (cid:105) = 1, there is a single ( i, l )-handleslide arc withan endpoint at p . • For every l < j such that (cid:104) d ν S j , S l (cid:105) = 1, there is a single ( u, j )-handleslide arc withan endpoint at p . (3) Let p ∈ Σ sing be the base projection of an upward swallowtail point, i.e. one where inthe front projection the crossing arc appears below the third sheet that that limits to theswallowtail point. The two cusp arcs in Σ sing that meet at p divide a small neighborhood N ⊂ M of p into halfs N − and N + above which Σ has ( n −
2) or n sheets respectively.Let k be such that above N − (resp. above N + ) the sheets that border the swallowtailpoint are numbered S k (resp. S k , S k +1 , S k +2 ) with S k +1 and S k +2 meeting along thecrossing arc, c , that ends at p and is contained in N + . Let ( V ( R ) , d ) be the complexassociated to the region, R , that borders p on the same side as N − . • There are two ( k + 1 , k + 2)-handleslide arcs contained in N + and ending at p , oneon each side of c . • For each i < k such that (cid:104) d S k , S i (cid:105) = 1 there is a ( i, k )-handle slide arc in N + withendpoint at p .When p is a downward swallowtail, the requirement is symmetric. The sheets S k in N − (resp. S k , S k +1 , S k +2 in N + ) correspond now to sheets S l − in N − (resp. S l , S l − , S l − ).There are now two ( l − , l − N + on opposite sides of c with endpointsat p , and for each l − < j with (cid:104) d S j , S l − (cid:105) = 1 there is an ( l, j + 2)-handleslide arc in N + with endpoint at p .The three types of endpoints for handleslide arcs are pictured in Figure 5. Remark 5.10. (1) In Axiom 5.9 (2), if the condition holds for some region R ν bordering p ,then it holds for every region that borders p .(2) In Axiom 5.9 (3), in a neighborhood of a swallowtail point p , assume that a differential d has been assigned to the region R that borders p in N − . Then, as long as handleslidearcs are placed in N + as specified by the axiom, there is always a unique way to assigndifferentials to regions in N + to produce an MCF near p .(3) See [34, Section 6.1] for a further discussion of constructions of MCFs.Note that when Σ and M are 2-dimensional and have boundary, an MCF C for Σ restricts toan MCF for the 1-dimensional Legendrian Σ | ∂M ⊂ J ( ∂M ). Definition 5.11.
Let C and C be two MCFs for a 1-dimensional Legendrian knot Λ ⊂ J M .We say that C and C are equivalent if there exists an MCF C for the product cobordism j (1 [0 , · Λ) ⊂ J ([0 , × M ) such that C| { }× M = C and C| { }× M = C . Remark 5.12.
The statement of Definition 5.11 is slightly different than the definition of equiv-alence in [22, 23, 24]. These earlier works did not consider MCFs for 2-dimensional Legendriansand instead present a collection of elementary moves on 1-dimensional MCFs with two MCFs C and C declared to be equivalent if they can be related by a sequence of elementary moves. Thetwo versions of the definitions are equivalent, since the elementary moves simply describe the bi-furcations that occur along the slices, C| { t }× M , of a generic 2-dimensional MCF C for j (1 [0 , · Λ)as t increases from 0 to 1. For instance, as numbered in [23, Section 4.3], Move 1 correspondsto passing a critical point of t ◦ H ; Moves 2-6 correspond to a transverse self intersection of H with Move 6 demonstrating the handleslide endpoints required by Axiom 5.9 (1); Moves 7, 8,and 10 are transverse intersections of H with the base projection of a crossing arc of j (1 [0 , · Λ);Moves 9 and 11 are transverse intersections of H and the base projection of a cusp arc; andMove 15 corresponds to passing a super-handleslide point. The Moves 12-14 do not occur inour case since we only consider MCFs with “simple births and deaths”, and this does not affectthe resulting equivalence classes. Indeed, when two such MCFs with only “simple births anddeaths” are related by a sequence of the Moves 1-15, possibly passing through some MCFs withnon-simple births or deaths, we can always arrive at an alternate sequence of moves avoidingnon-simple cusps by treating the implicit handleslides discussed in [23] as explicit handleslideslocated near a cusp. See [22, Proposition 3.17]. UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 33 (1) ( m, j ) ( i, j ) ( i, m ) S i S m S j x = ax = b (2) S u S l x = ax = b (3) x x x zx = ax = b Figure 5.
The three possible endpoints for handleslide arcs specified by Axiom5.9 (1)-(3). The left column depicts the base projection (to M ) of H (in red), H − (a green star) and Σ sing , (in blue). The center and right column depict slicesof the front projection, π xz (Σ), at x = a and x = b with dotted black arrowsfrom S i to S j used to indicate when (cid:104) dS j , S i (cid:105) = 1. In (3), a downward swallowtailpoint is pictured.5.3. MCFs and CHDs.
Suppose now that (Λ , µ ) ⊂ J M is a ρ -graded Legendrian knot orsurface equipped with a compatible polygonal decomposition E . Definition 5.13.
An MCF C will be called nice with respect to E if the handleslide sets(including super-handleslides when dim ( M ) = 2) are transverse all vertices and edges of E except at the endpoints of handleslide arcs that occur at swallowtail points as specified byAxiom 5.9 (3). Moreover , in a neighborhood of each upward (resp. downward) swallowtailpoint, s , all of the ( i, k )- (resp. ( l, j + 2)-) handleslide arcs specified by Axiom 5.9 (3) appearon the S side of the crossing arc ending at s . Denote by MCF ρ (Λ , E ) the set of all nice ρ -gradedMCFs for (Λ , µ, E ).Any MCF can be made nice after a perturbation of the handleslide sets. In particular, whendim Λ = 1, for any E , the set of equivalence classes of nice MCFs, MCF ρ (Λ , E ) / ∼ , coincideswith the set of equivalence classes of arbitrary MCFs for Λ.5.3.1. From MCF to CHD: 1-dimensional case.
Assume now that Λ ⊂ J M is 1-dimensionaland that C is a ρ -graded MCF for Λ that is nice with respect to E . We use C to produce a ρ -graded CHD, Φ( C ) = ( { d α } , { f β } ), for (Λ , E ) as follows: This technical requirement will not be relevant to the later discussion in the present article. It is includedin order for Proposition 5.16 to be true. See [34]. • For each 0-cell, e α ∈ E , by identifying sheets whose closures intersect we obtain anisomorphism V ( e α ) ∼ = V ( R ν )where R ν is a region that either contains e α or is adjacent to e α . (If e α is at a cusp of Λ,then R ν must be taken to be the bordering region above which the cusp sheets do notexist.) We define d α : V ( e α ) → V ( e α ) to agree with d ν under this isomorphism. (This isindependent of the choice of R ν by Axiom 5.6 (2).) • For each 1-cell, e β , let p , . . . , p s ∈ H , s ≥
0, be the sequence of handleslide points of C that appear when e β is traversed according to its orientation. Let u i and l i be theindices above e β of the sheets that contain the upper and lower lift of p i respectively,and put(5.2) f β : V ( e β ) → V ( e β ) , f β = h u s ,l s ◦ · · · ◦ h u ,l where the h u i ,l i are the handleslide maps from Axiom 5.6 (1). Proposition 5.14.
The construction defines a ρ -graded CHD, Φ( C ) = ( { d α } , { f β } ) , for (Λ , E ) .Moreover, Φ gives a surjective map from MCF ρ (Λ , E ) to CHD ρ (Λ , E ) .Proof. To verify that f β : ( V ( e β ) , (cid:98) d − ) → ( V ( e β ) , (cid:98) d + ) is a chain map with respect to the boundarydifferentials arising from the vertices at the ends of e β , use Axiom 5.6. Or, see [34, Section 5]. Forsurjectivity, recall that the standard row reduction algorithm shows that any upper triangularmap with 1’s on the main diagonal can be factored into a product of handleslide maps. (cid:3) We will see in Proposition 5.17 below that Φ is bijective on equivalence classes.5.3.2.
MCFs and CHDs in 2-dimensions.
Now, let (Σ , µ, E ) be a Legendrian surface in J M (allowing for Σ and M to have non-empty boundary) with Maslov potential and polygonaldecomposition. The construction from Proposition 5.14 applies more generally to produce froma nice MCF C ∈
MCF ρ (Σ , E ) an assignment of differentials and chain maps to the 0- and 1-cellsof E . • Given e α , choose any bordering region R ν whose sheets are in bijection with those of e α and use V ( e α ) ∼ = V ( R ν ) to define d α via d ν . • Given e β , push e β slightly into the interior of a neighboring 2-cell whose sheets are inbijection with the sheets of e β , and again define f β as a composition of handleslide mapsas in (5.2).Note that Propositions 5.2 and 5.3 of [34] show that the d α and f β are well defined. Definition 5.15.
Let C = ( H − , H , { d ν } ) and D = ( { d α } , { f β } , { K γ } ) be a nice MCF and aCHD for (Σ , E ) where dim Σ = 2. We say that C and D agree on the -skeleton if the { d α } and { f β } are related to C as in the above construction.In the case where ∂ Σ = ∅ , (1) and (2) of the following proposition appear as Proposition 6.4and Proposition 5.5 of [34]. Proposition 5.16.
Let (Σ , µ, E ) be a ρ -graded Legendrian surface possibly with boundary equippedwith a compatible polygonal decomposition. (1) Given any nice ρ -graded MCF C for (Σ , E ) , there exists a ρ -graded CHD D for (Σ , E ) that agrees with C on the -skeleton. (2) Given any ρ -graded CHD D for (Σ , E ) , there exists a nice ρ -graded MCF C for (Σ , E ) that agrees with D on the -skeleton. As a technical requirement, when e β has an endpoint at a swallowtail point, s , the endpoint of the shiftedversion of e β should not be seperated (locally) from e β by any of the handleslide arcs with endpoints at s . See[34]. UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 35
Moreover, in (2) C can be constructed to extend any given nice MCF C X defined above a -dimensional (CW) sub-complex X ⊂ E that agrees with D on X .Proof. There is no issue with allowing ∂ Σ (cid:54) = ∅ in the proofs of (1) and (2) from [34]. To seethis, and to verify the statement about C X , we briefly review the proofs. In proving either (1)or (2), the construction starts by defining the required D or C above the 1-skeleton. [For (2),start with C X above X , and then assign handleslide marks along all remaining edges by arguingas in the proof of surjectivity of Φ in Proposition 5.14.] Then, the proofs in [34] show that (onceadditional handleslide endpoints are placed at swallowtail points as specified by Axiom 5.9 (3)) C can be extended over a 2-cell if and only if D can. (cid:3) Bijection between equivalence classes of MCFs and augmentations.Proposition 5.17.
Given (Λ , µ, E ) with Λ ⊂ J M and dim Λ = 1 , the map Φ :
MCF ρ (Λ , E ) → CHD ρ (Λ , E ) = Aug ρ (Λ , E ) from Proposition 5.14 induces a bijection (cid:98) Φ :
MCF ρ (Λ) / ∼ → CHD ρ (Λ , E ) / ∼ = Aug ρ (Λ , E ) / ∼ . In the statement, the equality
CHD ρ (Λ , E ) = Aug ρ (Λ , E ) is from Proposition 5.3, and underthis equality the equivalence relation on CHD ρ (Λ , E ) is DGA homotopy. Proof.
Let E (cid:48) denote the product polygonal decomposition for j (1 [0 , · Λ) as constructed inSection 4.3.
Lemma 5.18.
Two CHDs D and D for (Λ , E ) correspond to homotopic augmentations if andonly if there exists a CHD D for ( j (1 [0 , · Λ) , E (cid:48) ) such that D| { }× Λ = D and D| { }× Λ = D .Proof. Using the computation of A ( j (1 [0 , · Λ) , E (cid:48) ) in terms of A (Λ , E ) found in Proposition4.5, we see that, in the notations found there, two augmentations (cid:15) and (cid:15) for Λ are DGAhomotopic if and only if there exists a single augmentation α : A ( j (1 [0 , · Λ) , E (cid:48) ) → Z / i ∗ α = (cid:15) and i ∗ α = (cid:15) . The statement of the lemma is simply the translation of thisstatement to CHDs resulting from the bijection from Proposition 5.3. (cid:3) To complete the proof of Proposition 5.17, we need show that for any C , C ∈ MCF ρ (Λ) wehave C ∼ C if and only if (cid:15) (cid:39) (cid:15) where (cid:15) i is the augmentation corresponding to Φ( C ) underthe bijection from Proposition 5.3. Suppose that C ∼ C . Then, there exists an MCF, C , for j (1 [0 , · Λ) that restricts to C i on { i } × M for i = 0 ,
1. Applying Proposition 5.16 (1), we obtaina CHD D on Λ × [0 ,
1] that agrees with C on the 1-skeleton. In particular, D| { i }× Λ = Φ( C i ), soLemma 5.18 implies that Φ( C ) ∼ Φ( C ).Conversely, suppose that Φ( C ) ∼ Φ( C ). Then, there exists a CHD D ∈
CHD ρ ( j (1 [0 , · Λ) , E (cid:48) )restricting to Φ( C i ) on { i } × Λ, i = 0 ,
1. Using the relative statement at the end of Proposition5.16, there is an MCF C for j (1 [0 , · Λ) restricting to C i on { i } × Λ. Thus, C ∼ C . (cid:3) The bijection between equivalence classes of MCFs and augmentations leads to the followingcharacterization of the (cellular) induced augmentation set of a compact Legendrian cobordism(see Section 3.2 for the definition).
Proposition 5.19.
Let Σ ⊂ J ([0 , × M ) be a compact ρ -graded Legendrian cobordism from (Λ , E , µ ) to (Λ , E , µ ) . Using the bijection (cid:98) Φ :
MCF ρ (Λ) / ∼ ∼ = → Aug ρ (Λ , E ) / ∼ , the inducedaugmentation set I cell Σ ⊂ Aug ρ (Λ , E ) / ∼ × Aug ρ (Λ , E ) / ∼ viewed as a subset I MCF Σ ⊂ MCF ρ (Λ , E ) / ∼ × MCF ρ (Λ , E ) / ∼ Crossings: Right cusps:
Figure 6.
The allowed locations for handleslides in an A -form MCF. satisfies I MCF Σ = { ([ C| Λ ] , [ C| Λ ]) | C ∈ MCF ρ (Σ) } . Proof.
From Proposition 5.4, under the identification
CHD ρ (Λ i , E i ) / ∼ = Aug ρ (Λ i , E i ) / ∼ wehave I cell Σ = { ([ D| Λ ] , [ D| Λ ]) | D ∈ CHD (Σ , E ) } . Then, the result follows easily using Proposition 5.16. (cid:3) A -form MCFs. From Proposition 5.19 and Corollary 4.11, we see that I MCF Σ is invariant ofΣ up to Legendrian isotopy rel. ∂ Σ. Moreover, I MCF Σ is equivalent in a canonical way to both thecellular induced augmentation set I cell Σ and, via Corollary 4.12, to the induced augmentation set I Σ of the conical version of Σ, G std (Σ). For computing I MCF Σ in explicit examples, it is convenientto have an efficient characterization of the sets, MCF ρ (Λ) / ∼ , not requiring the cellular DGAwhich often has a large number of generators. For Legendrian links in J R , work of Henry[22, 24], which we now review, gives an explicit bijection MCF ρ (Λ) / ∼ ↔ Aug ρ (Λ res ) / ∼ whereΛ res is the Ng resolution of Λ. Recall that the DGA A (Λ res ) has generators in bijection withthe crossings and right cusps of the front projection of Λ; see [27, Section 2]. Definition 5.20 ([22]) . A ρ -graded MCF C for a Legendrian link Λ ⊂ J R is in A -form if itshandleslides occur only in the following locations:(1) Handleslides connecting the two crossing strands appear immediately to the left of somesubset of the crossings of π xz (Λ).(2) In the case ρ = 1, handleslides connecting the two strands that meet at a right cuspappear immediately to the left of some subset of the right cusps of π xz (Λ).(3) Near a right cusp, any number of additional handleslides may appear connecting theupper (resp. lower) strand of the right cusp to a strand above (resp. below) the rightcusp. These handleslides are located to the right of the handleslide from (2), if it exists.See Figure 6. Remark 5.21.
The definition appears slightly different from that found in [22, 24] as thehandleslides from (3) are implicit handleslides in these references; see Remarks 5.7 (1) and 5.12.Moreover, an A -form MCS is uniquely determined by its handleslides of type (1) and (2), sincethe handleslides of type (3) are determined by the form of the complexes ( R ν , d ν ) near rightcusps.Given an A -form MCF, C , for Λ ⊂ J R we can define an algebra homomorphism (cid:15) C : A (Λ res ) → Z / b i ∈ A (Λ res ) we set (cid:15) C ( b i ) to be 1 (resp. 0) if a handleslideconnecting the two strands of the crossing or cusp appears (resp. does not appear) to the leftof b i . UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 37
Proposition 5.22 ([22, 24]) . For any Λ ⊂ J R , the correspondence C (cid:55)→ (cid:15) C defines a surjectivemap from the set of ρ -graded A -form MCFs for Λ to Aug ρ (Λ res ) . Moreover, this induces abijection MCF ρ (Λ) / ∼ ∼ = → Aug ρ (Λ res ) / ∼ . Proof.
In [22], a map (cid:98)
Ψ :
MCF ρ (Λ) / ∼ → Aug ρ (Λ res ) / ∼ is defined and shown to be surjective.Moreover, this map has the property that when C is an A -form MCF we have (cid:98) Ψ([ C ]) = [ (cid:15) C ], andit is shown that every MCF is equivalent to an A -form MCF. Finally, in [24] it is shown that (cid:98) Ψis injective. (cid:3) Examples
In this section, we apply the methods developed in Section 5 to compute induced augmen-tations and augmentation sets for some particular Legendrian fillings. In Section 6.1, we giveexamples of oriented Legendrian fillings inducing augmentations that cannot be induced by anyoriented embedded Lagrangian fillings. In Section 6.2, we prove Theorem 1.4 by constructing2 n Legendrian fillings of the Legendrian torus knot, T (2 , n + 1), each with a single degree 0Reeb chord, and distinguished by their induced augmentation sets.6.1. Initial examples.
Before launching into examples, we summarize the method that we usefor computing induced augmentations and induced augmentation sets arising from a (compact)Legendrian filling, Σ. For a 1-dimensional Legendrian, Λ, with admissible transverse decompo-sition E (cid:116) , Proposition 5.17 and Corollary 4.12 provide a commutative diagram of bijections andinclusions(6.1) MCF ρ (Λ) / ∼ ∼ = Aug ρ (Λ , E || ) / ∼ ∼ = Aug ρ ( (cid:101) Λ) / ∼∪ ∪ ∪ I MCF Σ ∼ = I cell Σ ∼ = I (cid:101) Σ where, in the top row, from left to right the sets are the equivalence classes of MCFs of Λ;homotopy classes of augmentations of the cellular DGA of (Λ , E || ); and homotopy classes ofaugmentations of the LCH DGA A ( (cid:101) Λ) where (cid:101)
Λ is Legendrian isotopic to Λ. (It is the standardgeometric model from Section 4.5.) The bottom row consists of the (equivalent) versions of theinduced augmentation set in these three different settings (where (cid:101)
Σ = G std (Σ) is Σ with itspositive end extended to become a conical cobordism). The MCFs in I MCF Σ are those that arisefrom restricting a 2-dimensional MCF, C Σ , for Σ to Λ. Thus, elements of I MCF Σ (and so alsoinduced augmentations in I (cid:101) Σ ) may be constructed by specifying a 2-dimensional MCF for Σ,and then observing its restriction to Λ. To keep track of the different MCFs of Λ we use thefurther bijection from Proposition 5.22 with augmentations of the Ng resolution, MCF ρ (Λ) / ∼ ∼ = Aug (Λ res )following Section 5.5. That is, once a 2-dimensional MCF has been restricted to the Legen-drian boundary Λ, if necessary, we transform it by an equivalence into A -form to arrive at anaugmentation of Λ res .For the rest of the section, all of the augmentations we talk about are of Λ res , and we willrefer to Reeb chords of Λ res as crossings and right cusps of the front projection. Example 6.1.
An augmentation that can be induced by a Legendrian filling for a knot thatdoes not have any oriented embedded Lagrangian fillings.The work [30, Section 8.3] gives an infinite family of such knots without explicitly computingthe induced augmentations. One of them is the knot 4 shown in Figure 7 where a conicalLegendrian filling is constructed by a clasp move (see Figure 17 below for a discussion of localmoves for constructing Legendrian cobordisms) followed by Reidemeister I moves and an unknot move; the pictures are slices of the front projection of the corresponding compact filling, Σ ⊂ J ([0 , × M ), with t ∈ [0 ,
1] constant. As shown in [30, Section 8.3], the clasp move canbe realized by a conical Legendrian cobordism with a single Reeb chord, so that a conicalversion of Σ (after an appropriate Legendrian isotopy) is a disk with a single double point inthe Lagrangian projection. Figure 7 indicates the MCF for Σ as a movie of handleslides. Notethat the handleslides that appear after the Reidemeister I moves are as required in Axiom 5.9(3). Using the correspondence between A-form MCFs and augmentations of Λ res , the inducedaugmentation (cid:15) sends the crossings b and b to 1 and all other Reeb chords to 0. Note thata Z -valued Maslov potential for 4 extends (uniquely) over Σ, and the MCF for Σ is 0-graded.Correspondingly, (cid:15) is the unique 0-graded augmentation of Λ res . As discussed in [30, Section8.3], the 4 knot does not have any oriented embedded Lagrangian fillings due to a restrictionon the Thurston-Bennequin number. b b ↓↓∅ Figure 7.
A conical Legendrian filling of Legendrian 4 constructed by a movie.The arrow indicates the negative t direction. Red vertical line segments indicateshandle slides in the MCF.We note that 4 does admit non-orientable embedded Lagrangian fillings. Such a filling,constructed via applying two pinch moves (see Figure 17) and then an unknot move, is picturedin Figure 8. It induces the 1-graded augmentation (cid:15) that sends b , b , a , and c to 1 and otherReeb chords to 0. Question 6.2.
Can the augmentation (cid:15) induced by the Legendrian filling from Figure 7 alsobe induced by a non-orientable embedded Lagrangian filling?We suspect that the answer is no, but we do not know of any obstructions to such a non-orientable filling that are currently available in the literature. a c ∅ Figure 8.
A non-orientable conical Legendrian filling of Legendrian 4 with em-bedded Lagrangian projection. A 1-graded MCF inducing the augmentation (cid:15) ispictured. Example 6.3.
Two augmentations for the same knot, one can be induced by an embeddedLagrangian filling and one cannot.
UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 39
Figure 9 shows a Legendrian of knot type m (8 ). The knot has a genus one embeddedLagrangian filling constructed by doing three pinch moves (see Figure 17) and closing up the twodisjoint unknots (see the first row in Figure 9). The associated MCF induces the augmentation (cid:15) that sends b , b , b to 1 and others to 0. b b b b b b b b b b b b ∅∅ Figure 9.
The first row gives an embedded Lagrangian filling of m (8 ) con-structed by a movie. The second row gives a conical Legendrian filling of m (8 )through a movie. The arrow indicates negative t direction.One the other hand, one can construct a conical Legendrian filling through doing three claspmoves and then close up the unknot as shown in the movie in the second row of Figure 9.This is a Legendrian disk with three Reeb chords. The associated MCF as shown in Figure9 induces the augmentation (cid:15) that sends b , b , b and b to 1 and other Reeb chords to 0.This augmentation cannot be induced by an embedded orientable Lagrangian filling since itslinearized homology has Poincare polynomial t − + 4 + 2 t . By Seidel’s Isomorphism [12, 11], ifan augmentation was induced by an embedded orientable genus g filling, using the Z / Z / Z / t + 2 g .6.2. Proof of Theorem 1.4.
The theorem provides examples of knots with an arbitrary (fi-nite) number of immersed fillings, all having isomorphic DGAs, but distinguished by inducedaugmentation sets.
Theorem 1.4.
For each n ≥ , there exists n distinct conical Legendrian fillings, Σ , . . . , Σ n of the max-tb Legendrian torus knot T (2 , n + 1) such that (i) the Σ i are all orientable with genus n − and have Z -valued Maslov potentials, (ii) each Σ i has a single Reeb chord of degree , (iii) and the induced augmentation sets satisfy I Σ i (cid:54) = I Σ j when i (cid:54) = j . Note that each of the DGAs A (Σ i ) is Z -graded with a single generator in degree 0 and thedifferential is necessarily zero for grading reasons. Thus, they are all isomorphic. Proof.
Consider the Legendrian torus knot T (2 , n + 1) as shown in Figure 10 part ( a ). Thereare 2 n “eye shapes” that one can do a clasp move on to get a (2 , n −
1) Legendrian torus knotas shown in part ( b ). One can then do 2 n − T (2 , n −
1) knot and close theunknot in part ( c ). Let Σ i be the conical Legendrian filling of T (2 , n + 1) that is constructedby performing the first clasp move at the “eye shape” between b i and b i +1 . According to the b b b b n +1 ( a ) ( b )( c ) c c n − Figure 10.
Part ( a ) is a Legendrian (2 , n +1) torus knot and ( b ) is a Legendrian(2 , n −
1) torus knot. Part ( c ) is the unknot got after doing 2 n − T (2 , n − b ) and ( c ), the pinching procedure on T (2 , n −
1) induces anaugmentation of T (2 , n −
1) that only sends c to 1 and all others to 0.The clasp move produces a degree 0 Reeb chord, a , (see Proposition 8.3 of [30] for the degreecomputation) of the resulting conical Legendrian surface Σ i , and the rest of the cobordism canbe constructed without Reeb chords. Thus, A (Σ i ) has two augmentations, a (cid:55)→ a (cid:55)→ I Σ i can have at most two elements. In fact, I Σ i does have two elementsinduced by the two MCFs of the conical Legendrian surface shown in Figure 11. For a subset I of the set { , · · · n + 1 } , denote by (cid:15) I the algebra map from A ( T (2 , n + 1)) to Z / { b i | i ∈ I } to 1 and all other Reeb chords to 0. Then, Figure 11 shows I Σ i = { (cid:15) { } , (cid:15) { } } , if i = 1 { (cid:15) { } , (cid:15) { ,i,i +2 } } , if 1 < i < n { (cid:15) { } , (cid:15) { , n } } , if i = 2 n. (cid:3) Every augmentation is induced by an immersed filling
With the characterization of induced augmentation sets for Legendrian cobordisms in termsof MCFs now in place, the present section establishes Theorem 1.2 from the introduction. Forconvenience, we repeat the statement.
Theorem 1.2.
Let Λ ⊂ J R have the Z /ρ -valued Maslov potential µ where ρ ≥ , and let (cid:15) : A (Λ) → Z / be any ρ -graded augmentation. (1) If ρ (cid:54) = 1 , there exists a conical Legendrian filling Σ of Λ with Z /ρ -valued Maslov potentialextending µ together with a ρ -graded augmentation α : A (Σ) → Z / such that (cid:15) (cid:39) (cid:15) (Σ ,α ) .Moreover, if ρ is even, then Σ is orientable. (2) If ρ = 1 , then there exists a conical Legendrian cobordism Σ : U → Λ where U is thestandard Legendrian unknot with tb ( U ) = − together with an augmentation α : A (Σ) → Z / such that (cid:15) (cid:39) f ∗ Σ α . UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 41 ( a ) ( b ) ( c ) b b b c b b b c b b i b i +1 b i +2 c c i b b i b i +1 b i +2 c c i b b n − b n b n +1 c c n − b b n − b n b n +1 c c n − Figure 11.
Two MCFs for the clasp move constructing Σ i pictured in the baseprojection. These MCFs extend the MCF from Figure 10 (b) and (c) to all ofΣ i . Columns ( a ) , ( b ) , ( c ) are for i = 1, 1 < i < n and i = 2 n , respectively. Theblack dotted lines represent double points (in the front projection) and red linesrepresent handleslides. Proof of Theorem 1.2.
Note that the orientability of Σ when ρ is even follows from the existenceof the Z /ρ -valued Maslov potential on Σ. See Remark 2.9.In view of Corollary 3.8, if Λ and Λ are Legendrian isotopic and the statement holds forΛ , then it also holds for Λ . Therefore, we can replace Λ with its standard geometric model G std (Λ , E (cid:116) ) with respect to some admissible transverse decomposition (see Section 5.3), so thatCorollary 4.12 and Proposition 5.19 apply to reduce Theorem 1.2 to the following proposition. (cid:3) Proposition 7.1.
Let C Λ be any ρ -graded MCF for Λ ⊂ J R with respect to the Maslov potential µ . (1) If ρ (cid:54) = 1 , then there exists a compact Legendrian filling, Σ ⊂ J ([0 , × R ) , of Λ with aMaslov potential extending µ such that C Λ can be extended to a ρ -graded MCF, C Σ , on Σ . (2) If ρ = 1 , then the same statement holds except that Σ is a compact Legendrian cobordismfrom the Legendrian unknot U to Λ . We will prove Proposition 7.1 at the end of this section after establishing some preliminariesabout extending MCFs over various elementary cobordisms.7.1.
Normal rulings and SR -form MCFs. In [22], Henry introduced a class of MCFs (the“ SR -form MCFs”) for 1-dimensional Legendrian links that are in a clear many-to-one corre-spondence with the normal rulings of Λ. Moreover, Henry proved that any MCF is equivalentto an SR -form MCF. It will be convenient to make use of this result in proving Proposition 7.1,so we briefly discuss the relevant definitions.See any of [23, 20, 35] for a detailed definition of normal rulings for Legendrian linksin J R . Here, we recall that a normal ruling, σ , for Λ ∈ J R (where Λ has generic frontand base projections) may be viewed as a continuous family of fixed point free involutions, σ x : Λ ∩ { x = x } → Λ ∩ { x = x } , defined for x ∈ R \ Λ sing , i.e. for all values of x such that Switches:Departures:Returns:
Figure 12.
Near a crossing the two sheets that cross cannot be paired with oneanother by a normal ruling σ . The figure indicates the allowed configurations forthe crossing sheets and their companion sheets. (There may be other sheets of Λin between those pictured.) π xz (Λ) has no crossing or cusp along the vertical line x = x . For each connected component R ν ⊂ R \ Λ sing there is a resulting involution on the sheets of Λ above R ν , notated simply as σ : { S νi } → { S νi } , which partitions { S νi } into a collection of disjoint pairs (because of the fixed point free condition).Moreover, if two regions R i and R i +1 share a border at a cusp or crossing, then the pairings ofsheets above R i and R i +1 are required to be related in a standard way. In particular, sheetsthat meet at a cusp point are paired near the cusp point, and every crossing of π xz (Λ) is eithera departure , a return , or a switch for σ as in Figure 12. The degree of a crossing of π xz (Λ)with respect to a Z /ρ -valued Maslov potential µ is defined to be the difference µ ( u ) − µ ( l ) ∈ Z /ρ where u and l denote the upper and lower strands at the left side of the crossing. We say that σ is ρ -graded with respect to µ if all switches of σ have degree 0 mod ρ . On each region R ν ⊂ R \ Λ sing , we can use σ to define a standard ruling differential (7.1) d σν : V ( R ν ) → V ( R ν ) , d σν S νj = (cid:26) S νi , σ ( S νj ) = S νi and z ( S νi ) > z ( S νj ) , , else . Definition 7.2. A ρ -graded MCF C = ( H, { d ν } ) for a 1-dimensional Legendrian link Λ is in SR -form with respect to a ρ -graded normal ruling σ for Λ if the only handleslides of C are asfollows:(1) At every switch of σ , handleslides are placed as specified in Figure 13.(2) At some subset of the degree 0 (mod ρ ) returns of σ , handleslides are placed as in Figure13.(3) In the 1-graded case, at some subset of the right cusps of Λ, a single handeslide connectsthe two cusp strands near the cusp. Remark 7.3. (1) It can be shown that for any ρ -graded normal ruling σ of Λ and anychosen subset of the ρ -graded returns of Λ (and also of the right cusps of Λ in the1-graded case), there is an SR -form MCF with handleslide set as in Figure 13.(2) Whenever C = ( H, { d ν } ) is an SR-form MCF and R ν ⊂ R \ ( H ∪ Λ sing ) is a region outsideof the collection of handleslides near switches, returns, and cusps, the differential d ν agrees with the standard ruling differential d σν defined in (7.1). UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 43
Switches:Returns:Right cusps:
Figure 13.
The placement of handleslides near switches and some subset ofthe degree 0 returns of σ for an SR-form MCF. When ρ = 1, handleslides are alsoplaced next to some collection of right cusps as indicated.The following proposition is Theorem 6.17 from [22]. Proposition 7.4.
Any ρ -graded MCF on a Legendrian link Λ ⊂ J R is equivalent to a SR-formMCF with respect to some ρ -graded normal ruling of Λ . Construction of MCFs.
The following results from [22] and [34] are useful for construct-ing MCFs.
Proposition 7.5.
Let Λ ⊂ J M be a -dimensional Legendrian, and let C = ( H, { d ν } ) be an ρ -graded MCF for Λ . (1) Suppose that H (cid:48) is a handleslide set for the product cobordism j (1 [0 , · Λ) ⊂ J ([0 , × M ) as in Definition 5.8 (2) (and without super-handleslide points) that agrees with H whenrestricted to { } × Λ and satisfies Axiom 5.9. Then, there is a unique ρ -graded MCF for j (1 [0 , · Λ) that agrees with C on { } × Λ and has handleslide set H (cid:48) . (2) Construct a handleslide set H (cid:48) for j (1 [0 , · Λ) ⊂ J ([0 , × M ) by extending eachhandleslide point p of C to a handleslide arc along [0 , × { p } , and for some region R ν ⊂ M \ (Λ sing ∩ H ) and some i < j such that µ ( S νi ) = µ ( S νj ) − placing a single ( i, j ) -super handleslide point, q , in the interior of [0 , × R ν and adding handleslide arcsin [0 , × R ν as specified by Axiom 5.9 (2) that connect q to { } × R ν with monotonicallyincreasing [0 , component. Then, there is a unique ρ -graded MCF for j (1 [0 , · Λ) thatagrees with C on { } × Λ and has handleslide set H (cid:48) .Proof. In all cases, it needs to be shown that it is possible to produce the required collection ofdifferentials { d ν } extending the given ones and so that Axiom 5.6 is satisfied. (1) and (2) areboth consequences of Proposition 3.8 from [22] which shows that the differentials { d ν } can beextended from [0 , t − (cid:15) ] × M to [0 , t + (cid:15) ] × M when a generic bifurcation of the t -slices of H occurs at t = t . (2) corresponds to Henry’s Move 17 in [22]. Alternatively, see Proposition 6.3of [34]. (cid:3) Next, we make a sequence of observations (B1)-(B5) about extending MCFs along varioustypes of elementary cobordisms that will form the building blocks for the proof of Proposition7.1. In all cases it should be observed that the Maslov potential for the given 1-dimensionalLegendrian extends in a unique way over the elementary cobordism, though we mostly leavethis implicit. Moreover, although we omit the terminology, all MCFs under consideration are ρ -graded. II: III:I:Other:
Figure 14.
Generic bifurcations of the front and base projections during aLegendrian isotopy include the Reidemeister moves I-III, and instances where twofront singularities share the same x -coordinate (marked “Other” in the figure).Vertical and horizontal reflections of all moves are allowed.(B1) If C and C are equivalent MCFs on Λ ⊂ J R , then there exists a MCF C on j (1 [0 , · Λ) ⊂ J ([0 , × R ) that restricts to C i above { i } × R , for i = 0 , Proof.
This is the definition of equivalence. (cid:3) (B2) Let Λ t , 0 ≤ t ≤ ⊂ J ([0 , × R ) such that the { t } × R slices of the front projection of Σ are the frontprojections of the Λ t . Any MCF on Λ extends over Σ. Proof.
It suffices to consider the case where the isotopy contains a single bifurcation of the frontor base projection, i.e. a single Reidemeister move or an instance at which two singularities(crossings or cusps) at different locations in the front diagram share the same x -coordinate. SeeFigure 14.Before applying the move, we apply (B1) together with Henry’s result (Proposition 7.4) toassume that the given MCF C is in SR-form with respect to some normal ruling σ . • For a Type III move or any of the “Other” moves (both directions): Use Proposition 7.5to first move all of the handleslides outside of the x -interval that contains the picturedpart of the diagram where the move occurs. Then, we apply the move, and check thatthere is a unique way to assign differentials to the new region(s) that arise from the moveso that Axiom 5.6 is satisfied.This is a straightforward case-by-case check that we illustrate by considering the TypeIII move in detail. (Other cases are left to the reader.) The t -slices before (resp. after)the move occurs intersect a sequence of 4 regions of π x (Σ) \ (Σ sing ∪ H ), which we label R A , R B , R C , and R D (resp. R A , R B (cid:48) , R C (cid:48) , and R D ); see Figure 15. The differentials on R A , R B , R C , and R D are already specified by C . Suppose that the three sheets thatintersect at the triple point are numbered as S νk , S νk +1 , S νk +2 above these regions. (Recallthat above any particular region we label sheets with decreasing z -coordinate, eg. thesheet labeled S Ak becomes S Bk +1 when it passes through the crossing locus.) Write Q i i +1 for a linear map that interchanges the i and i + 1 sheets. According to Axiom 5.6 (2)we must define the differentials on R B (cid:48) and R C (cid:48) so that the following maps are chainisomorphisms,(7.2) Q k +1 k +2 : ( V ( R A ) , d A ) ∼ = → ( V ( R B (cid:48) ) , d B (cid:48) ) , and Q k k +1 : ( V ( R B (cid:48) ) , d B (cid:48) ) ∼ = → ( V ( R C (cid:48) ) , d C (cid:48) ) . In order for this to define a valid MCF we only need to check that Axiom 5.6 (2) issatisfied along the border between R C (cid:48) and R D , i.e., we need to show that Q k +1 k +2 : ( V ( R C (cid:48) ) , d C (cid:48) ) → ( V ( R D ) , d D ) UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 45 xz xt R A R B R B (cid:48) R C R C (cid:48) R D Figure 15.
Labeling of the regions in the base projection near a triple point.is a chain map. In view of (7.2), this is equivalent to the composition Q k +1 k +2 ◦ Q k k +1 ◦ Q k +1 k +2 : ( V ( R A ) , d A ) → ( V ( R D ) , d D ) being a chain isomorphism. Since we know thatAxiom 5.6 is satisfied by C , when we pass from R A to R D by way of R B and R C we seethat Q k k +1 ◦ Q k +1 k +2 ◦ Q k k +1 : ( V ( R A ) , d A ) → ( V ( R D ) , d D ) is a chain isomorphism, sowe can just note the braid relation Q k k +1 ◦ Q k +1 k +2 ◦ Q k k +1 = Q k +1 k +2 ◦ Q k k +1 ◦ Q k +1 k +2 . • For a Type I move ( ← direction): A crossing, b , a left cusp, c l , and a right cusp, c r ,of Λ all vanish at a swallowtail point, s , during the move. The crossing, b , must bea switch for σ , so under the SR-form assumption, the handleslides appearing near thecrossing are exactly those required to have endpoints at the swallowtail point by Axiom5.9 (3). Extend these to handleslide arcs with endpoints at s . Let ∂ ν l (resp. ∂ ν r ) denotethe differential from C assigned to the region R ν l (resp. R ν r ) that borders c l on the left(resp. c r on the right). After the Type I move, R ν l and R ν r merge to become a singleregion, so it is important to note that ∂ ν l and ∂ ν r agree. This is the case since by Remark7.3 (2), they both agree with the standard ruling differential for σ . • For a Type I move ( → direction): To extend C , we just add additional handleslide arcswith end points at the swallowtail point as required by Axiom 5.9. Differentials for thenew regions can be defined since the handleslide set on the slices after the swallow tailpoint occurs is in SR-form for the ruling σ (cid:48) obtained from σ by making the new crossinginto a switch. Alternatively, see Proposition 6.2 of [34]. • For a Type II move ( ← direction): Once again, the strategy is to use an equivalence tomove all of the handleslides out of the x -inteval, I , where the front diagram is picturedand then perform the move. It is then routine to check that the required differentialscan be defined to complete the extension of C to C . The two pictured crossings of Λ area departure followed by a return, so (since C is in SR-form) the only handleslides thatneed to be moved are those that may appear near the return. Figure 16 illustrates a 2-dimensional MCF on j (1 [0 , · Λ) that will remove all handleslides from I as required. ThisMCF involves a single super-handleslide point, so Proposition 7.5 (1) and (2) produce therequired differentials. (A similar procedure applies for the horizontally and/or verticallyreflected versions of Move II.) • For a Type II move ( → direction): There are no handleslides in the interval wherethe move occurs. We perform the move and then check that the differentials may beextended. (cid:3) Remark 7.6.
An alternate approach to establishing (B2) is made possible by the correspon-dence between MCFs and augmentations. When Φ is a Legendrian isotopy from Λ − to Λ + , as inSection 3.2, there is an invertible conical Legendrian cobordism, Σ Φ , with embedded Lagrangianprojection. Thus, because the induced augmentation set I Σ has the form found in equation (3.4)the calculation of I Σ in terms of MCFs via Proposition 5.19 and Corollary 4.12 implies that anyMCF for Λ − can be extended over (a compact version of) Σ.Along with moves associated to 1-dimensional Legendrian isotopies, the generic front bifur-cations of the t -slices of a Legendrian cobordism include the Clasp Move , the
Pinch Move , xz xt Figure 16.
Removing a handleslide at a return to prepare for a Type II move.The top row illustrates the MCF on a sequence of t -slices with t increasing. Thebottom row depicts the MCF in the base projection. If the return is as in the 2ndor 3rd column of Figure 12, there will be a second handleslide arc with endpointat the super-handleslide point. Neither of its endpoints are on the cusp sheets, soit can be moved away from the pictured part of the front projection. Clasp:Pinch:Unknot: xz xt ∅ Figure 17.
The Clasp, Pinch, and Unknot Moves, pictured as front projectionslices (left) and in the base projection of the corresponding 2-dimensional cobor-dism (right). Shading in the base projection indicates the region where the twocusp sheets exist.
Cusp Tangency:
Figure 18.
The Cusp Tangency Move refers to the interchange of the far leftand far right diagrams. This is accomplished by a Type II Move and a ClaspMove, as pictured.and the
Unknot Move as pictured in Figure 17. These moves correspond to a local maximumor minimum in the t -direction for the crossing locus (in the case of the Clasp Move) or the cusplocus (for the Pinch and Unknot Move). It will also be convenient to consider a (non-generic) Cusp Tangency Move pictured in Figure 18 which can be realized by a combination of theClasp Move with a Legendrian isotopy.(B3) (a) An MCF can be extended along the → direction of the Clasp Move if there is nohandleslide that connects the crossing sheets and has its x -coordinate between thetwo crossings. UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 47
Figure 19.
The requirement on handleslides from (B3) that allows for MCFsto be extended through a clasp move (left) or cusp tangency (right).(b) An MCF can be extended along the Cusp Tangency Move (either direction) providedthere is no handleslide that connects the crossing strands and has its x -coordinateto the left of the crossing.See Figure 19. Proof.
Let Σ be the Legendrian surface corresponding to the Clasp Move ( → ), with Λ andΛ denoting the 1-dimensional slices before and after the move. Let C be an MCF for Λ with no handleslides between the crossings. Then, there is a sequence of three adjacent regions R A , R B , R C for C where R B is between the crossings and R A and R C are to the left and to theright. Since there are no handleslides between the crossings, we can define a MCF for Σ byextending all handleslide points of C along straight line segments in the t -direction. Axiom 5.6(2) for C shows that the differentials on the two regions R A and R B agree, so that there is awell defined differential on the common region for Σ that contains them.The Cusp Tangency Move follows from the first case of (B3) since the hypothesis restrictingthe location of handleslides before the move implies that the move can be realized (in eitherdirection) by a Type II Move followed by a Clasp Move ( → direction) in such a way that thereare no handleslides between the crossings when the Clasp Move is applied. (cid:3) (B4) An SR-form MCF for Λ associated to a ρ -graded normal ruling σ can be extended to a ρ -graded MCF on the elementary cobordism Σ arising from applying a Pinch Move ( → direction) to adjacent sheets of Λ, S νk and S νk +1 , above a region where they are paired by σ .Note that the Z /ρ -valued Maslov potential µ for Λ extends over Σ since the fact that σ is ρ -graded implies that µ ( S νk ) = µ ( S νk +1 ) + 1 mod ρ . Proof.
At the location of the pinch move, the differential for the SR-form MCF, d ν , is thestandard ruling differential for σ . (See Remark 7.3 (2).) Thus, d ν S k +1 = S k , and S k and S k +1 do not appear in the differentials of the other generators. That is, the complex for ( V ( R ν ) , d ν )splits as a direct sum ( V ( R ν ) , d ν ) = ( C , d ) ⊕ ( C , d ) with C spanned by { S k , S k +1 } and C spanned by the rest of the sheets. Therefore, when we extend the MCF over the surface byusing ( C , d ) in the region where S k and S k +1 do not exist, the Axiom 5.6 is satisfied. (cid:3) (B5) An MCF C can be extended along the Unknot Move ( ← direction), provided there isno handleslide connecting the two sheets of the unknot. (Note that such a handleslidecannot exist if C is ρ -graded and ρ (cid:54) = 1.) Proof.
Using (B1) we can assume C is in SR-Form on the slice that precedes the unknot move.It then follows that (i) there are no handleslides with endpoints on the unknot sheets, and (ii)in the region where the unknot exists ( V ( R ν ) , d ν ) = ( C , d ) ⊕ ( C , d ) where C is spanned bythe unknot sheets, S νk and S νk +1 , and d ν S νk +1 = S k . Moreover, by Axiom 5.6 the differentials onthe two regions adjacent to the unknot both agree with d . Thus, C extends in an obvious wayover the surface. (cid:3) n − kk + 1 n Figure 20.
The elementary tangles l n − ,nk , r n,n − k , and σ n,nk . The picture illus-trates l , , r , , and σ , from left to right. X Y
Figure 21.
The front projection corresponding to the word w Λ = Xl k σ k − σ k − · · · σ k − s σ k +1 σ k +2 · · · σ k + t Y when s = 3 and t = 2. Remark 7.7.
Although we will not need to use them in our proof of Theorem 1.2, it is alsoeasy to give necessary and sufficient conditions for an MCF to extend over the remaining moves.(1) Any MCF can be extended over the Pinch Move ( ← direction) or the Unknot Move ( → direction). (This is easy to see directly, but also follows from the fact that there areconical Legendrian cobordisms realizing these moves that have embedded Lagrangianprojections; cf. [15, 6].)(2) An MCF can be extended over the Clasp Move ( ← direction) if and only if (cid:104) d ν S k +1 , S k (cid:105) =0 where R ν is the region where the crossings will appear and S k and S k +1 are the twocrossing sheets.7.3. Construction of the Legendrian filling.
The front projection of a generic LegendrianΛ ⊂ J R can be represented as a word w Λ that is a product (left to right concatenation) ofelementary Legendrian tangles, each one of which contains a single crossing or cusp. We notatethese elementary tangles as l n − ,nk , r n,n − k , and σ n,nk in the case of a tangle with a left cusp,right cusp, or crossing respectively, where n ≥ ≤ k ≤ n −
1. The superscripts indicatethe number of strands of the tangle at its left and right vertical boundaries, while k is thenumbering of the upper of the two strands that is involved with the crossing or cusp when wenumber strands as 1 to n from top to bottom. See Figure 20. In the following, we suppress thesuperscripts from notation. Proof of Proposition 7.1.
Assume C Λ is an ρ -graded MCF for Λ ⊂ J R . We first prove theproposition assuming ρ (cid:54) = 1, and then close by indicating the minor modifications to the proofwhen ρ = 1.Let w Λ be the word representing the front projection of Λ, and note that w Λ can be writtenin the form(7.3) w Λ = Xl k σ k − σ k − · · · σ k − s σ k +1 σ k +2 · · · σ k + t Y where s, t ≥ Y contains no left cusps. (For example, just take the l k term to be the leftcusp of Λ with largest x -coordinate, and put s = t = 0.) See Figure 21. UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 49
We prove the following statement by a nested induction. The outer induction is on c and theinner induction is on n . Inductive Statement: If w Λ has c left cusps and can be written in the form (7.3) such thatthe length of Y is | Y | = n , then there exists a compact Legendrian filling Σ ⊂ J ([0 , × R )with a ρ -graded MCF C such that C| Λ = C Λ .The case c = 0 is trivial since Λ = ∅ . For fixed c ≥
1, the case n = 0 is vacuously true.Assuming n ≥
1, we write Y = zY (cid:48) where z is an elementary tangle that is necessarily a rightcusp or a crossing, z = r j or z = σ j . We consider cases depending on the vertical location ofthis right cusp (in Cases 1-5) or crossing (in Cases 6-10). By symmetry we can assume j ≤ k .In addition, using (B1) and Proposition 7.4 we can assume that C Λ is in SR-form with respectto a ρ -graded normal ruling σ . Case 1: z = r j with j ≤ k − s −
2. Then, z is a right cusp above all the sheets with cuspsor crossings in the product l k σ k − σ k − · · · σ k − s σ k +1 σ k +2 · · · σ k + t . So, a Legendrian isotopy of Λmodifies the front diagram by Xl k σ k − σ k − · · · σ k − s σ k +1 σ k +2 · · · σ k + t ( r j Y (cid:48) ) → ( Xr j ) l k − σ k − σ k − · · · σ k − − s σ k − σ k · · · σ k − t Y (cid:48) . Use (B2) to extend C Λ over the Legendrian isotopy. Then, the inductive hypothesis on n appliesto complete the construction of (Σ , C ). Case 2: z = r k − s − . Using a Legendrian isotopy, we have w Λ → Xl k σ k − σ k − · · · σ k − s r k − s − σ k − σ k · · · σ k − t Y (cid:48) . Note that we must have s ≥
1, since otherwise a “zig-zag” l k r k − would occur in Λ. This is notpossible since MCFs do not exist for stabilized Legendrian links. The idea is to try to move thecusp l k up next to the r k − s − . Consider the first crossing to the right of l k , x = σ k − .Subcase A: x is a switch for σ . Since C is in SR-form for σ , we apply a → Pinch Move to the right of x (extending the MCF using (B4)) followed by a Type I Reidemeister Move to producean MCF on a surface with the slices · · · l k σ k − · · · ( B → · · · l k σ k − r k l k · · · → · · · l k σ k − · · · σ k − s r k − s − σ k − σ k · · · σ k − t Y (cid:48) → · · · σ k − · · · σ k − s r k − s − l k − σ k − σ k · · · σ k − t Y (cid:48) . Then, the inductive hypothesis on n applies.Subcase B: x is not a switch for σ . Then, it is a departure, so since C Λ is in SR-form, it has nohandleslides connecting the crossing strands to the left of x = σ k − . Then, we can apply(B3) to extend past the Cusp Tangency Move: Xl k σ k − σ k − · · · σ k − s r k − s − σ k − σ k · · · σ k − t Y (cid:48) ( B → Xl k − σ k σ k − · · · σ k − s r k − s − σ k − σ k · · · σ k − t Y (cid:48) → Xl k − σ k − · · · σ k − s r k − s − σ k − σ k − σ k · · · σ k − t Y (cid:48) . We then repeat this argument with x = σ k − . Continuing in this manner, we mustencounter the switch case at some point; if not, the left cusp would eventually be movednext to the right cusp, and we would have an MCF for a stabilized link which is impos-sible. (The front would have a “zig-zag”, l k − s r k − s − .) Case 3: s ≥ z = r k − s . Then, the presence of the term σ k − s r k − s shows that Λ is stabilizedwhich is impossible. (There is a standard Legendrian isotopy that will turn the “fish tail” fromthe σ k − s r k − s product into a “zig-zag”.) Case 4: z = r j with k − s < j < k . Then, a type II move can be applied to produce aLegendrian isotopy Xl k σ k − σ k − · · · σ k − s σ k +1 σ k +2 · · · σ k + t ( r j Y (cid:48) ) → ( Xr j − ) l k − σ k − · · · σ k − s σ k − σ k · · · σ k − t Y (cid:48) , where the product σ k − · · · σ k − s just means the identity tangle if s = 2. Again, we can apply(B2) and then the inductive hypothesis on n to produce (Σ , C ). Case 5: z = r k .Subcase A: s = t = 0. Then, we have a standard Legendrian unknot in the middle of the diagram. As long as ρ (cid:54) = 1, there can be no handleslides connecting the strands of this unknot,so we apply (B5) to extend C as we remove the unknot. This decreases c , so that theinductive hypothesis applies.Subcase B: Exactly one of s > t >
0. Then, we can apply a Type I Reidemeister move toreduce the number of cusps. The (Σ , C ) is then constructed via (B2) and the inductivehypothesis on c .Subcase C: s > t >
0. If the crossing σ k − is a switch for the normal ruling σ associated to C Λ ,then we can apply a pinch move, as in Case 2, followed by two Type I moves to decrease c while extending C via (B4) and (B2). If σ k − is not a switch, then we apply a CuspTangency Move as in (B3) to arrive at Xl k − σ k − · · · σ k − s σ k σ k +1 σ k +2 · · · σ k + t r k Y (cid:48) . At this point, Case 4 (reflected vertically) applies.
Case 6: z = σ j with j ≤ k − s −
2. As in Case 1, move z to the left of l k and then apply theinductive hypothesis on n . Case 7: z = σ k − s − . Using a Legendrian isotopy, we can group σ k − s − into the product X ( l k σ k − σ k − · · · σ k − s σ k +1 σ k +2 · · · σ k + t )( σ k − s − Y (cid:48) ) → X ( l k σ k − σ k − · · · σ k − s σ k − s − σ k +1 σ k +2 · · · σ k + t ) Y (cid:48) and apply the inductive hypothesis on n . Case 8: s ≥ z = σ k − s . Using an argument similar to Case 2, we attempt to apply cusptangency moves to move the left cusp directly next to z . Initially, assume k − > k − s . If thecrossing x = σ k − directly to the right of l k is a switch, we can apply a Pinch Move followed bya Type I Reidemeister Move to go from · · · l k σ k − σ k − · · · ( B → · · · l k σ k − r k l k σ k − · · · → · · · l k σ k − · · · σ k − s σ k − s σ k +1 σ k +2 · · · σ k + t Y (cid:48) → · · · σ k − · · · σ k − s σ k − s l k σ k +1 σ k +2 · · · σ k + t Y (cid:48) and then apply induction on n .If x = σ k − is not a switch, then it is a departure. Then, we apply (B3) to extend C duringthe sequence Xl k σ k − σ k − · · · σ k − s σ k +1 σ k +2 · · · σ k + t ( σ k − s Y (cid:48) ) ( B → Xl k − σ k σ k − · · · σ k − s σ k − s σ k +1 σ k +2 · · · σ k + t Y (cid:48) → Xl k − σ k − · · · σ k − s σ k − s σ k σ k +1 σ k +2 · · · σ k + t Y (cid:48) . We repeat this argument until we either find a switch or arrive at a word of the form Xl k − s +1 σ k − s σ k − s σ k − s +2 · · · σ k + t Y (cid:48) If the first σ k − s is a switch, then the usual combination of Pinch Move and Type I ReidemeisterMove produces Xl k − s +1 σ k − s σ k − s +2 · · · σ k + t Y (cid:48) which has the form (7.3) with | Y (cid:48) | = n − σ k − s is a depar-ture, then we can extend C over another Cusp Tangency Move and then apply a ReidemeisterII move: Xl k − s +1 σ k − s σ k − s σ k − s +2 · · · σ k + t Y (cid:48) ( B → Xl k − s σ k − s +1 σ k − s σ k − s +2 · · · σ k + t Y (cid:48) → Xl k − s +1 σ k − s +2 · · · σ k + t Y (cid:48) . Again, we are able to use the inductive hypothesis on n . UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 51
Case 9: z = σ j with k − s < j < k . Then, a Type III Reidemeister move can be applied: Xl k σ k − σ k − · · · σ k − s σ k +1 σ k +2 · · · σ k + t ( σ j Y (cid:48) ) → ( Xσ j − ) l k σ k − σ k − · · · σ k − s σ k +1 σ k +2 · · · σ k + t Y (cid:48) . We apply (B2) and then the inductive hypothesis on n to produce (Σ , C ). Case 10: z = σ k .Subcase A: s = t = 0. This case cannot occur since the appearance of the product l k σ k would showthat Λ is stabilized.Subcase B: Exactly one of s > t >
0. Then, a Type II Move allows us to apply induction on n .For instance, when s > t = 0, Xl k σ k − σ k − · · · σ k − s ( σ k Y (cid:48) ) → Xl k σ k − σ k σ k − · · · σ k − s Y (cid:48) → Xl k − σ k − · · · σ k − s Y (cid:48) . Subcase C: Both s > t >
0. We have Xl k σ k − · · · σ k − s σ k +1 · · · σ k + t ( σ k Y (cid:48) ) → Xl k σ k − σ k +1 σ k ( σ k − · · · σ k − s )( σ k +2 · · · σ k + t ) Y (cid:48) . If the σ k − is a switch, then we apply (B4) to do a Pinch Move followed by a Type I andType II Move: Xl k σ k − σ k +1 σ k · · · ( B → Xl k σ k − r k l k σ k +1 σ k · · ·→ Xl k σ k +1 σ k · · ·→ Xl k +1 ( σ k − · · · σ k − s )( σ k +2 · · · σ k + t ) Y (cid:48) → ( Xσ k − · · · σ k − s )( l k +1 σ k +2 · · · σ k + t ) Y (cid:48) . Finally, if σ k − is not a switch, we use (B3) and apply a Cusp Tangency Move followedby a Type III Move: Xl k σ k − σ k +1 σ k · · · ( B → Xl k − σ k σ k +1 σ k · · ·→ Xl k − σ k +1 σ k σ k +1 ( σ k − · · · σ k − s )( σ k +2 · · · σ k + t ) Y (cid:48) → ( Xσ k +1 ) l k − ( σ k − · · · σ k − s )( σ k σ k +1 σ k +2 · · · σ k + t ) Y (cid:48) . Then, induction on n applies.This completes the proof when ρ (cid:54) = 1. The only place where the hypothesis ρ (cid:54) = 1 was used inthe above induction was Subcase A of Case 5 where (B5) was applied to remove a Legendrianunknot component. When ρ = 1, instead of applying (B5) for this subcase, we can simply applya Legendrian isotopy to move the unknot component to the left of the rest of the front. In thismanner, the inductive argument produces a cobordism Σ from a disjoint union of standardLegendrian unknots (each with 2 cusps and no crossings) to Λ together with an MCF C on Σextending C Λ . Finally, we can apply the ( ← ) direction of the Pinch Move to join all of theunknots into a single unknot via a cobordism. The MCF C extends in a clear way over thiscobordism. (cid:3) Appendix A. Isomorphism between the cellular and immersed LCH functors
This appendix fills in the details of Section 4.5, and together with Appendix B provides aproof of Proposition 4.9.In the case of a closed Legendrian knot or surface, Λ ⊂ J M , [31, 32, 33] constructs a stabletame isomorphism between the cellular DGA and the Legendrian contact homology DGA. Theproof of the isomorphism of functors from Proposition 4.9 extends the approach of [32, 33] tothe case of cobordisms. (1) (2) (9) (8) (12) Figure 22.
A selection of the 14 square types allowed in a transverse squaredecomposition enumerated as in [32] and pictured in the base projection. See [32,Figure 4] for the full list of square types. The dotted curves are projections ofcrossing arcs, and the solid curves are projections of cusp edges with the shadingindicating the portion of the square where the cusp sheets exist. Note that onlysquares of type (1), (2), and (9) will appear in the collar neighborhoods of theboundary of a Legendrian cobordism. Codimension 2 front singularities appearabove the type (8) and (12) squares which contain a single triple point and cusp-sheet intersection point respectively. Swallowtail points appear above type (13)and (14) squares (not pictured). For each of the square types, the number ofsheets of the Legendrian as well as the positioning of the crossing and/or cuspsheets, relative to the other sheets, may be arbitrary (as allowed by the requiredcrossing and cusp sets).A.1.
Review of the isomorphism for closed surfaces.
Let us briefly outline the isomor-phism from [31, 32, 33]. Starting with a Legendrian surface, Σ ⊂ J M with generic base andfront projection, in [31] the cellular DGA is shown to be independent of the choice of com-patible polygonal decomposition, E , (as in Definition 4.1) for Σ; see Proposition 4.7 for thecorresponding result for cobordisms. Thus, it suffices to construct a stable tame isomorphismbetween the LCH DGA and the cellular DGA for one particular choice of E . To do this, [32,Section 3] constructs a transverse square decomposition for Σ, E (cid:116) , that is a CW complexdecomposition of a neighborhood of π x (Σ) into squares together with a choice of orientationsfor 1-cells of E (cid:116) satisfying: • For each 2-cell, e α , a characteristic map for e α is given of the form c α : [ − , × [ − , ∼ = → e α , and the c α satisfy some smoothness requirements discussed in [32, Section 3.1]. • Above each square of E (cid:116) , when viewed using the parametrization given by c α , the orien-tation of edges is from the lower left to upper right corner of [ − , × [ − , • Additional technical requirements listed as (A1)-(A4) in [32, Section 3.3-3.4] are imposed.Given (Σ , E (cid:116) ), after a Legendrian isotopy Σ is replaced with a standard form Legendrian, (cid:101) Σ,whose sheets above squares of E (cid:116) , when viewed in the [ − , × [ − ,
1] coordinates from char-acteristic maps, match an explicit model found in [33, Sections 2-3]; a metric (cid:101) g for M is alsodefined in terms of E (cid:116) in [33, Construction 1.3]. In this article, we will call ( (cid:101) Σ , (cid:101) g ) the standardgeometric model for Σ with respect to E (cid:116) . In addition, an associated compatible polygonaldecomposition, E || , for (cid:101) Σ is constructed from E (cid:116) ; see [32, Section 3.6]. Over the course of [32,Sections 5-7], rigid GFTs of ( (cid:101) Σ , (cid:101) g ) are enumerated to provide a computation of the LCH DGA A ( (cid:101) Σ , (cid:101) g ). Finally, in [32, Section 8] a stable tame isomorphism A ( (cid:101) Σ , (cid:101) g ) ∗ S ∼ = A cell ( (cid:101) Σ , E || ) ∗ S (cid:48) isestablished. UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 53 (PV) (1Cr) (Cu)
Figure 23.
The three edge types that appear above the 1-cells of a 1-dimensionaltransverse decomposition: Plain vanilla (PV), one crossing (1Cr), and left cusp(Cu).A.2.
Transverse decompositions for -dimensional Legendrians and compact cobor-disms. In generalizing this approach to Legendrian cobordisms, we will want to work withtransverse square decompositions for compact Legendrian cobordisms with a standard form ina collar neighborhood of the boundary. As a preliminary, we revisit transverse decompositionsin the 1-dimensional case.
Definition A.1. A transverse decomposition for a 1-dimensional Legendrian, Λ ⊂ J M , isa CW decomposition of a neighborhood of π x (Λ) ⊂ M such that(1) there are no singular points of π xz (Λ) (crossings or cusps) above 0-cells of E , and(2) there is at most one singular point of π xz (Λ) above any 1-cell of E .We also add the technical requirement:(T3) The characteristic maps for 1-cells of E are diffeomorphisms c β : [ − , → e β with the property that characteristic maps for adjacent 1-cells glue together smoothlyat 0-cells, eg. if e α is the terminal vertex for both e β and e β (cid:48) , then( − δ, δ ) → M, t (cid:55)→ (cid:26) c β (1 + t ) , t ≤ ,c β (cid:48) (1 − t ) , t ≥ , is smooth.A standard subdivision of E is a second transverse decomposition, E (cid:48) , for Λ obtained from E by dividing every 1-cell that does not contain a cusp point in half. Moreover, the edges of E (cid:48) should be oriented so that:(1) The two new edges e α (cid:48) , e α (cid:48)(cid:48) ∈ E (cid:48) that arise from subdividing e α ∈ E by adding a newvertex v are both oriented away from v .(2) Edges that contain a cusp point of Λ are oriented in the direction where the number ofsheets increases, i.e. towards the vertex where the two cusp sheets exist.Call a transverse decomposition, E , admissible if(1) it is a standard subdivision of some other transverse decomposition for Λ, and(2) no two edges of E that both contain cusp points of Λ share the same initial vertex. Remark A.2. (1) Clearly, any 1-dimensional Legendrian Λ ⊂ J M with generic front andbase projections has many admissible transverse decompositions.(2) Above each edge, e β ∈ E , of an admissible transvserse decomposition, when viewed usingthe orientation of e β , the front projection π xz (Λ) topologically matches one of the edgetypes (PV), (1Cr), or (Cu) pictured in Figure 23. Definition A.3.
Given an admissible transverse decomposition, E (cid:116) , define the associatedcompatible polygonal decomposition , E || , by the following procedure: For each 1-cell e β ∈E (cid:116) of type (Cu) or (1Cr), we move the location of the initial vertex of e β to the base projection of E EE || J ( S ) = S × R xz (cid:15) − (cid:15) E E E Figure 24. (left) A Legendrian knot in J S pictured with an initial transversedecomposition, E ; its standard subdivision, E ; and the associated compatiblepolygonal decomposition E || for E . Note that E is an admissible transverse de-composition, but it would not have been if the (PV) 1-cell appearing at the farright of E had not been included. [Then, the two (Cu) 1-cells would have sharedthe same initial vertex in E .] (right) A collared transverse square decompositionfor a compact Legendrian cobordism Σ ⊂ J ([0 , × M ).the crossing or cusp point of e β . (Note that the definition of admissible transverse decompositionshows that it never happens that a vertex of E (cid:116) is simultaneously the initial vertex of two differentedges of type (Cu) or (1Cr).) See Figure 24. Definition A.4.
Let Σ ⊂ J ([0 , × M ) be a compact Legendrian cobordism from Λ to Λ with dim Λ = dim Λ = 1. Suppose that E and E are transverse decompositions for Λ andΛ . We say that a transverse square decomposition E for Σ (in particular E is a decompositionof a neighborhood of π x (Σ) ⊂ [0 , × M ) is collared by E and E if, for some (cid:15) > , (cid:15) ] × M (resp. [1 − (cid:15), × M ) Σ has the form j (1 [0 ,(cid:15) ] · Λ ) (resp. j (1 [1 − (cid:15), · Λ ))with notation as in (4.1);(ii) the squares of E in [0 , (cid:15) ] × M (resp. [1 − (cid:15), × M ) have the form [0 , (cid:15) ] × e β (resp.[1 − (cid:15), × e β (cid:48) ) where e β ∈ E (resp. e β (cid:48) ∈ E );(ii) and all edges in [0 , (cid:15) ] × M or [1 − (cid:15), × M that run parallel to the [0 ,
1] are orientedaway from ∂ ([0 , × M ). See Figure 24 (right).We make the further technical requirement that characteristic maps for collar squares havethe form(A.1) [ − , × [ − , → [0 , (cid:15) ] × e β or [ − , × [ − , → [1 − (cid:15), × e β (cid:48) ( t, x ) (cid:55)→ ( l − ( t ) , c β ( x )) ( t, x ) (cid:55)→ ( l + ( t ) , c β (cid:48) ( x ))where l − : [ − , → [0 , (cid:15) ] and l + : [ − , → [1 − (cid:15),
1] are linear with l − (resp. l + ) orientationpreserving (resp. reversing). Lemma A.5.
Let E , E be admissible transverse decompositions for -dimensional Legendrianknots Λ , Λ ⊂ J M . Then, for any compact Legendrian cobordism, Σ ⊂ J ([0 , × M ) , thereexists a transverse square decomposition E (cid:116) for Σ that is collared by E and E , and satisfies theconditions (A1)-(A4) from [32, Section 3] . Note that just as in [32, Section 3], we can construct a compatible polygonal decomposition, E || , for Σ that is associated to E (cid:116) . Along ∂ ([0 , × M ), E || agrees with the associated compatiblepolygonal decompositions for E and E as defined in Definition A.3. Proof.
Since E and E are admissible, they are subdivisions of some other transverse decompo-sitions E (cid:48) and E (cid:48) for the Λ k . Build E (cid:116) by starting with collars of E (cid:48) , E (cid:48) , and then following Steps UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 55 E (cid:48) k , k = 0 ,
1, that do notcontain cusp edges are divided into fourths with all edges oriented away from the center of thesubdivision. Squares in the collars of E (cid:48) k that do have cusp edges are divided in half in the middleof the [0 ,
1] direction, and edges that intersect the cusp locus are oriented in the direction wherethe number of sheets increase. Note that with this subdivision process, the initial collaring by E (cid:48) and E (cid:48) becomes a collaring of E (cid:116) by E and E as required, provided we reverse the orientationof the edges parallel to [0 ,
1] in all squares that border { } × M or { } × M . Finally, (A1)-(A3)are easily verified, and (A4) holds in the collar neighborhood since the definition of admissibletransverse decomposition requires that edges with cusps never share initial vertices. (cid:3) A.3.
A standard geometric model for -dimensional Legendrians. Let E (cid:116) be an ad-missible transverse decomposition for the 1-dimensional Legendrian Λ ⊂ J M . From the pair,(Λ , E (cid:116) ), we produce a standard geometric model for Λ that is an object of Leg ρim (i.e. anordered pair consisting of a ρ -graded Legendrian in J M and a metric on M ) that we will denote G std (Λ , E (cid:116) ) = ( (cid:101) Λ , (cid:101) g ) ∈ Leg ρim . The construction of G std (Λ , E (cid:116) ) is a restriction of the corresponding 2-dimensional construc-tion from [33, Section 2]. Extend Λ ⊂ J M to the product cobordism Σ = j (1 [0 , · Λ) whichhas π xz (Σ) = [0 , × π xz (Λ). The transverse decomposition E (cid:116) gives rise to a transverse squaredecomposition, E Σ (cid:116) , for Σ with squares in bijection with the edges e α ∈ E (cid:116) . The squares of E Σ (cid:116) have characteristic maps of the form[ − , × [ − , → [0 , × e α , ( x , x ) → (( x + 1) / , c α ( x ))and are of Type (1), (2), or (9) (as in Figure 22) when e α is an edge of type (PV), (1Cr),or (Cu) respectively. We follow the convention from [32] of numbering sheets above a squareas S , S , . . . , S n so that above (1 ,
1) (equivalently, above the terminal vertex of e α ) we have z ( S i ) > z ( S i +1 ). We usually denote by S k and S k +1 the pair of sheets that cross or meet at acusp in squares of Type (2) or (9). For Type (1), (2) and (9) squares with all possible valuesof n and 1 ≤ k < k + 1 ≤ n (with n ≤ N for some fixed N which we take to be the maximumnumber of sheets of Λ above any point of M ), [33, Section 2] constructs defining functions F , . . . , F n : [ − , × [ − , → R for sheets S , . . . , S n . (However, in the case S i is a cusp sheet, the domain of F i is [ − , × [ − / , F i (above all squares of E Σ (cid:116) ) fit together to define asmooth Legendrian (cid:101) Σ. Define (cid:101)
Λ to be the restriction of (cid:101)
Σ to J ( { } × M ). We denote thedefining functions for (cid:101) Λ above a 1-cell e α ∈ E (cid:116) as(A.2) g , . . . , g n : [ − , → R , g i ( x ) = F i ( − , x ) . (The domain of g i is actually [ − / ,
1] in the case of a cusp sheet.) In (1Cr) edges, the crossingis located at x = 0, and for (Cu) edges the cusp is located at x = − / (cid:101) Λ , (cid:101) g ) by taking (cid:101) g to be the Euclidean metric in theparametrizations of 1-cells by [ − , M because of (T3) fromDefinition A.1. Moreover, it agrees with the restriction of the metric from [33] that is used forcomputations of gradient vector fields of difference functions of (cid:101) Σ. The defining formulas for the F i are found in the very last paragraph of [33, Section 2]. A.4.
Isomorphism with cellular DGA in -dimensional case. The LCH DGA of a stan-dard geometric model, A ( (cid:101) Λ , (cid:101) g ), can be computed as in [32, Section 5]. There, a list of Properties1-7 appears concerning the location of Reeb chords of (cid:101) Λ above edges and vertices of E (cid:116) , as wellas properties specifying the direction of gradients of local difference functions −∇ ( g i − g j ). Inparticular, the Reeb chords of (cid:101) Λ are notated as a αp,q , b βp,q , (cid:101) b βk +1 ,k , and are as follows:(1) For each 0-cell e α ∈ E (cid:116) , let N ( e α ) be the ball of radius 1 / M with centered at e α .Then, for each pair of sheets S p , S q ∈ π − x ( N ( e α )) ∩ Λ with z ( S p ) > z ( S q ), there is aReeb chord a αp,q located in N ( e α ).(2) For each 1-cell e β ∈ E (cid:116) and each pair of sheets S p , S q above [1 / , / ⊂ [ − , ∼ = e β with z ( S p ) > z ( S q ), there is a Reeb chord b βp,q located in coordinates in (1 / , / ⊂ [ − , ∼ = e β .(3) When e β is a (1Cr) 1-cell such that sheets S k and S k +1 cross at x = 0, with z ( S k ) >z ( S k +1 ) when x > z ( S k ) < z ( S k +1 ) when x <
0, there is one additional Reebchord, (cid:101) b βk +1 ,k , located in ( − / , − / ⊂ [ − , A ( (cid:101) Λ , (cid:101) g ) is associated to either a 0-or 1-cell of E (cid:116) .Let us compare the generators of A ( (cid:101) Λ , (cid:101) g ) with the generators of the cellular DGA, A (Λ , E || ),where E || is the compatible polygonal decomposition associated to E (cid:116) . Notice that the 0- and1-cells of E (cid:116) and E || are in bijective correspondence, and for each e dα ∈ E || there is an injectivemap from the generators of A (Λ , E || ) associated to e dα to the generators of A ( (cid:101) Λ , (cid:101) g ) associated to e dα . The only generators of A ( (cid:101) Λ , (cid:101) g ) not in the image of this map are(1) the generators (cid:101) b βk +1 ,k located in (1Cr) edges; and(2) the generators a αk,k +1 in 0-cells that are the initial vertex of a (1Cr) edge above whichthe sheets S k and S k +1 cross.[The latter generators do not appear in A (Λ , E || ) since in E || the sheets S k and S k +1 cross rightat the location of the 0-cell.] We refer to generators of either of these two types as exceptionalgenerators of A ( (cid:101) Λ , (cid:101) g ), and we denote by Ex ( (cid:101) Λ) the set of all such exceptional generators.The computation of A ( (cid:101) Λ , (cid:101) g ) arising from the combination of [32, Proposition 5.10 and Lemma8.3] is summarized in terms of the cellular DGA as follows. Proposition A.6. (1) In A ( (cid:101) Λ , (cid:101) g ) , the exceptional generators satisfy ∂ (cid:101) b βk +1 ,k = a − k,k +1 where a − k,k +1 is the generator associated to the initial vertex of e β that corresponds to the twocrossing sheets of e β . (2) Let I = I ( Ex ( (cid:101) Λ)) be the 2-sided ideal generated by all exceptional generators. Then, thegenerators of the quotient DGA A ( (cid:101) Λ , (cid:101) g ) /I are in canonical bijection with the generatorsof A (Λ , E || ) , and this bijection extends to a DGA isomorphism A ( (cid:101) Λ , (cid:101) g ) /I ∼ = A (Λ , E || ) . Note that, since A ( (cid:101) Λ , (cid:101) g ) /I is obtained from A ( (cid:101) Λ , (cid:101) g ) by repeated applications of Proposition2.1, the quotient map p : A ( (cid:101) Λ , (cid:101) g ) → A ( (cid:101) Λ , (cid:101) g ) /I is a DGA homotopy equivalence (and underlies In [32] the Properties 1-7 are stated as properties of a standard geometric model for a 2-dimensional Legen-drian surface that hold in a neighborhood of 0-cells and 1-cells. These properties have obvious restatements inthe 1-dimensional case that are valid for ( (cid:101) Λ , (cid:101) g ). UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 57 a stable tame isomorphism). Given (Λ , E (cid:116) ), the composition q Λ : A ( (cid:101) Λ , (cid:101) g ) p → A ( (cid:101) Λ , (cid:101) g ) /I ∼ = → A (Λ , E || )will be refered to as the canonical quotient map . Since q Λ is a homotopy equivalence, it isinvertible in the DGA homotopy category DGA ρ hom , and it follows that the the correspondingimmersed map Q Λ = (cid:0) A ( (cid:101) Λ , (cid:101) g ) q Λ → A (Λ , E || ) id ← (cid:45) A (Λ , E || ) (cid:1) ∈ DGA ρ im is an isomorphism in DGA ρ im .A.5. Comparison of cellular and LCH functors for immersed cobordisms.
Essentially,Proposition 4.9 shows that if we use the isomorphism Q Λ to identify the cellular and LCH DGAsassociated to 1-dimensional Legendrians, then the immersed maps associated to a cobordism Σby the two versions of the immersed LCH functor are equivalent. To give a clean statement, itis convenient to first introduce a third cobordism category.With 1-dimensional M fixed, define the transverse Legendrian category , Leg ρ (cid:116) , to haveobjects (Λ , E (cid:116) ) where Λ ⊂ J M is a ρ -graded Legendrian and E (cid:116) is an admissible transversedecomposition for Λ. Morphisms are compact ρ -graded Legendrian cobordisms up to combina-torial equivalence of front and base projections (as in Section 4.4). In the preceding, we haveconstructed maps on objects G std : Leg ρ (cid:116) → Leg ρ im , and G || : Leg ρ (cid:116) → Leg ρ cell , (Λ , E (cid:116) ) (cid:55)→ ( (cid:101) Λ , (cid:101) g ) (Λ , E (cid:116) ) (cid:55)→ (Λ , E || ) . Both G std and G || are functors with the following actions on morphisms: • The morphism spaces in
Leg ρ (cid:116) and Leg ρ cell only depend on Λ and are the same in bothcategories. Define G || ([Σ]) = [Σ]. • Given a compact cobordism Σ from (Λ , E ) to (Λ , E ), choose any conical Legendriancobordism Σ (cid:48) ⊂ J ( R > × M ) that (i) matches the boundary conditions j ( s · G std (Λ ))(resp. j ( s · G std (Λ ))) when s is sufficiently close to 0 (resp. to + ∞ ), and (ii) hasfront and base projections combinatorially equivalent to Σ. Note that the compactlysupported Legendrian isotopy type of Σ (cid:48) depends only on the compactly supportedLegendrian isotopy type of Σ, so setting G std ([Σ]) = [Σ (cid:48) ] makes G std into a well definedfunctor. Remark A.7.
Later, we will work with a more specific representative of G std ([Σ]) constructedvia a choice of transverse decomposition of Σ that is collared by E and E .We have now more carefully defined all of the items that appear in the statement of Propo-sition 4.9. For convenience, we repeat the statement here. Proposition 4.9.
The canonical quotient map construction, (Λ , E (cid:116) ) ∈ Leg ρ (cid:116) (cid:55)→ Q Λ ∈ Hom
DGA ρ im ( A ( (cid:101) Λ , (cid:101) g ) , A (Λ , E || )) gives an isomorphism (invertible natural transformation) from the functor that is the composition Leg ρ (cid:116) G std → Leg ρim F → DGA ρ im . to the functor Leg ρ (cid:116) G || → Leg ρ cell F cell → DGA ρ im . That is, the Q Λ are invertible, and for any compact Legendrian cobordism Σ : (Λ − , E (cid:116) − ) → (Λ + , E (cid:116) + ) with generic base and front projection we have a commutative diagram in DGA ρ im , (A.3) A ( (cid:101) Λ + , (cid:101) g + ) F ( G std (Σ)) (cid:47) (cid:47) Q Λ+ (cid:15) (cid:15) A ( (cid:101) Λ − , (cid:101) g − ) Q Λ − (cid:15) (cid:15) A cell (Λ + , E || + ) F cell ( G || (Σ)) (cid:47) (cid:47) A cell (Λ − , E −|| ) . Before giving the proof of Proposition 4.9 below, we discuss the geometric model for Legen-drian cobordisms that will be used to compute the immersed map associated to G std (Σ).A.6. Geometric model for Legendrian cobordisms.
Let Σ : (Λ − , E (cid:116) − ) → (Λ + , E (cid:116) + ) be acompact cobordism in Leg ρ (cid:116) , so that the E (cid:116) ± are admissible transverse decompositions for Λ ± .According to Proposition A.5, we can find a transverse square decomposition E (cid:116) Σ that is collaredby the E (cid:116) ± and satisfies the technical requirements (A1)-(A4) of [32]. The standard coordinatemodels for squares of Type (1)-(14) constructed in [33, Sections 2 and 3] piece together toproduce a Legendrian Σ ⊂ J ([0 , × M ) together with a metric g . We will modify thisconstruction in the collar squares in order to associate to (Σ , E (cid:116) Σ ) a standard geometricmodel , ( (cid:101) Σ , (cid:101) g ), which will be a conical Legendrian cobordism (cid:101) Σ ⊂ J ( R > × M ) from (cid:101) Λ − to (cid:101) Λ + equipped with a metric (cid:101) g suitably related to (cid:101) g ± , where G std (Λ ± ) = ( (cid:101) Λ ± , (cid:101) g ± ). The conicalcobordism ( (cid:101) Σ , (cid:101) g ) then defines an immersed DGA map(A.4) A ( (cid:101) Λ + , (cid:101) g + ) f (cid:101) Σ → A ( (cid:101) Σ , (cid:101) g ) ← (cid:45) A ( (cid:101) Λ − , (cid:101) g − )as in Theoreom 2.12 whose immersed homotopy class is F ( G std (Σ)). The construction of ( (cid:101) Σ , (cid:101) g )and the computation of (A.4) is an extension of the approach from [32, 33] and appears in theAppendix B. Here, we just state the result.A computation of the DGA of the standard geometric model, (Σ , g ), of a closed Legendriansurface with transverse square decomposition appears in [32, Sections 5-7] as follows. The Reebchords that are the generators of A (Σ , g ) are located above neighborhoods, denoted N ( e dα ),of the 0-, 1-, and 2-cells of E (cid:116) . Moreover, the Reeb chords located in any particular N ( e dα )form a sub-DGA of A (Σ , g ), and computations of the generators and differentials in these sub-DGAs appear in [32, Propositions 5.9 and 5.10, Theorems 6.5 and 7.4, Lemmas 8.3 and 8.4].Taken together this gives a computation of A (Σ , g ) summarized in [32, Proposition 8.6]. Thisgeneralizes to Legendrian cobordisms as follows. Proposition A.8.
Let Σ ⊂ J ([0 , × M ) be compact Legendrian cobordism with transversesquare decomposition E (cid:116) collared by transverse decompositions E ± (cid:116) for Λ ± . (1) The standard geometric model, ( (cid:101) Σ , (cid:101) g ) , has its DGA, A ( (cid:101) Σ , (cid:101) g ) , determined by E (cid:116) as in [32] . (2) The sub-DGA of A ( (cid:101) Σ , (cid:101) g ) associated to generators corresponding to cells above { } × M (resp. above { } × M ) is identified (via canonical bijection of generators) with A ( (cid:101) Λ − , (cid:101) g − ) (resp. with A ( (cid:101) Λ + , (cid:101) g + ) ). Using j ± : A ( (cid:101) Λ ± , (cid:101) g ± ) (cid:44) → ( (cid:101) Σ , (cid:101) g ) for the corresponding inclusionmaps, the induced DGA map from (A.4) is (A.5) F ( G std (Σ)) = (cid:2) A ( (cid:101) Λ + , (cid:101) g + ) j + → A ( (cid:101) Σ , (cid:101) g ) j − ← (cid:45) A ( (cid:101) Λ − , (cid:101) g − ) (cid:3) . The proof of Proposition A.8 appears at the end of Appendix B after the construction of( (cid:101) Σ , (cid:101) g ). UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 59
Proof of Proposition 4.9.
The q Λ are homotopy equivalences, so they are isomorphisms in DGA ρ hom .Since the Q Λ are the image of q Λ under the functor from Proposition 2.7, they are invertible in DGA ρ im .It remains to verify the commutativity of the diagram (4.5). The map F ( G std (Σ)) at the topof the diagram has been characterized in (A.5) in terms of a transverse decomposition E (cid:116) forΣ collared by E − (cid:116) and E + (cid:116) . There is an associated compatible polygonal decomposition E || of Σconstructed as in [31, Section 3.6] that moreover restricts to E −|| and E + || above { } × M and { } × M respectively. We then have F cell ( G || (Σ)) = (cid:2) A (Λ + , E + || ) i + → A (Σ , E || ) i − ← (cid:45) A (Λ − , E −|| ) (cid:3) . We need to show that the compositions F cell ( G || (Σ)) ◦ Q Λ + and Q Λ − ◦ F ( G std (Σ)) agree. Fromthe definition of composition we have(A.6) F cell ( G || (Σ)) ◦ Q Λ + = (cid:2) A ( (cid:101) Λ + , (cid:101) g + ) p ◦ q Λ+ −→ B p ◦ i − ← (cid:45) A (Λ − , E −|| ) (cid:3) where B = (cid:16) A (Λ + , E + || ) ∗ A (Σ , E || ) (cid:17) / I ( { x − i + ( x ) | x ∈ A (Λ + , E + || ) } )and for i = 1 , p i denotes the quotient map restricted to the i -th factor of the free product.Note that p : A (Σ , E ) → B is an isomorphism and p − ◦ p = i + , so (A.6) becomes(A.7) F cell ( G || (Σ)) ◦ Q Λ + = (cid:2) A ( (cid:101) Λ + , (cid:101) g + ) i + ◦ q Λ+ −→ A (Σ , E || ) i − ← (cid:45) A (Λ − , E −|| ) (cid:3) . The other composition is(A.8) Q Λ − ◦ F ( G std (Σ)) = (cid:2) A ( (cid:101) Λ + , (cid:101) g + ) p ◦ j + −→ B p ← (cid:45) A (Λ − , E −|| ) (cid:3) where B = A ( (cid:101) Σ , (cid:101) g ) ∗ A (Λ − , E −|| ) / I ( { j − ( x ) − q Λ − ( x ) | x ∈ A ( (cid:101) Λ − , (cid:101) g ) } ) . This time p induces an isomorphism B ∼ = A ( (cid:101) Σ , (cid:101) g ) / I ( Ex ( (cid:101) Λ − ))where Ex ( (cid:101) Λ − ) denotes the set of exceptional generators of (cid:101) Λ − (viewed via j − as a subset of theexceptional generators of (cid:101) Σ), and using this isomorphism (A.8) becomes(A.9) Q Λ − ◦ F ( G std (Σ)) = (cid:2) A ( (cid:101) Λ + , (cid:101) g + ) p ◦ j + −→ A ( (cid:101) Σ , (cid:101) g ) / I ( Ex ( (cid:101) Λ − )) h − ← (cid:45) A (Λ − , E −|| ) (cid:3) where p is projection and h − is the composition A (Λ − , E −|| ) ∼ = → A ( (cid:101) Λ − , (cid:101) g − ) / I ( Ex ( (cid:101) Λ − )) (cid:44) →A ( (cid:101) Σ , (cid:101) g ) / I ( Ex ( (cid:101) Λ − )) where the isomorphism is from Proposition A.6; any generator x ∈ A (Λ − , E −|| )is mapped by h − to the equivalence class of the corresponding Reeb chord of (cid:101) Λ − (which alsoappears as a Reeb chord of (cid:101) Σ).To complete the proof we show that the immersed maps from (A.7) and (A.9) are bothimmersed homotopic to a third immersed map M that we now define. Consider the furtherquotient B = A ( (cid:101) Σ , (cid:101) g ) / I ( Ex ( (cid:101) Λ − ) ∪ Ex ( (cid:101) Λ + )), and the immersed map(A.10) M = (cid:0) A ( (cid:101) Λ + , (cid:101) g + ) k + ◦ q Λ+ −→ A ( (cid:101) Σ , (cid:101) g ) / I ( Ex ( (cid:101) Λ − ) ∪ Ex ( (cid:101) Λ + )) k − ← (cid:45) A (Λ − , E −|| ) (cid:1) where k ± is the composition A (Λ ± , E ±|| ) ∼ = → A ( (cid:101) Λ ± , (cid:101) g ± ) / I ( Ex ( (cid:101) Λ ± )) (cid:44) → A ( (cid:101) Σ , (cid:101) g ) / I ( Ex ( (cid:101) Λ − ) ∪ Ex ( (cid:101) Λ + )) . Claim 1: Q Λ − ◦ F ( G std (Σ)) = [ M ]. To verify the claim, note that since the exceptional generators, Ex ( (cid:101) Λ − ) = { a , b , . . . , a m , b m } come in pairs with ∂b i = a i , we can apply Proposition 2.1 m times to get a stable tameisomorphism ϕ : A ( (cid:101) Σ , (cid:101) g ) / I ( Ex ( (cid:101) Λ − ) ∪ Ex ( (cid:101) Λ + )) ∗ S → A ( (cid:101) Σ , (cid:101) g ) / I ( Ex ( (cid:101) Λ − )) . Using the properties of ϕ from Proposition 2.1, it can be checked that ϕ produces an im-mersed DGA homotopy between M and the map from (A.9). [To shorten notation, write (cid:101) A = A ( (cid:101) Σ , (cid:101) g ) / I ( Ex ( (cid:101) Λ − )). There are a sequence of quotient maps q i : (cid:101) A /I i → (cid:101) A /I i +1 , where I i = I ( a , b , . . . , a i , b i ), with a sequence of homotopy inverses g i : (cid:101) A /I i +1 → (cid:101) A /I i satisfying ϕ | A ( (cid:101) Σ , (cid:101) g ) / I ( Ex ( (cid:101) Λ − ) ∪ Ex ( (cid:101) Λ + )) = g ◦ g ◦ · · · ◦ g m − . Since k + ◦ q Λ + = q m − ◦ · · · ◦ q ◦ ( p ◦ j + ) we use that g i and q i are homotopy inverse to compute ϕ ◦ ( k + ◦ q Λ + ) = ( g ◦ · · · ◦ g m − ) ◦ ( q m − ◦ · · · ◦ q ◦ p ◦ j + ) (cid:39) p ◦ j + . The remaining identity, ϕ ◦ k − = h − , follows from (2) of Proposition 2.1.] Claim 2: F cell (Σ) ◦ Q Λ + = [ M ].The quotient A ( (cid:101) Σ , (cid:101) g ) / I ( Ex ( (cid:101) Λ − ) ∪ Ex ( (cid:101) Λ + )) has all exceptional generators canceled from thesub-DGAs A ( (cid:101) Λ ± , (cid:101) g ± ). The remaining generators are in bijection with the generators of thecellular DGAs A (Λ ± , E ±|| ), and moreover using this identification the differentials agree (byProposition A.6). Now, Section 8 of [32] gives a procedure in the case that Σ is a closedLegendrian surface to obtain a stable tame isomorphism, ϕ , between A ( (cid:101) Σ , (cid:101) g ) and A (Σ , E || ). Themap ϕ is obtained from composing (in the sense of [30, Remark 2.2]) stable tame isomorphismsthat arise from the following steps:(1) Cancel all exceptional generators in pairs, to go from A ( (cid:101) Σ , (cid:101) g ) to a quotient A ( (cid:101) Σ , (cid:101) g ) /I .(2) Cancel some generators of A (Σ , E || ) that correspond to certain cells of E || located neartriple points and cusp-sheet intersections of Σ. (These are codimension two singularitiesof the front projection.) Write the resulting quotient as A (Σ , E || ) /J .(3) Apply an explicitly defined tame isomorphism Φ : A ( (cid:101) Σ) /I → A (Σ , E || ) /J .This procedure generalizes to the case where Σ is a cobordism to provide a stable tame iso-morphism, ϕ , between B = A ( (cid:101) Σ , (cid:101) g ) / I ( Ex (Λ − ) ∪ Ex (Λ + )) and A (Σ , E || ). Furthermore, recallingthat within both B and A (Σ , E || ) we have already identified sub-DGAs corresponding to Λ − and Λ + with A (Λ ± , E ±|| ), the map ϕ can be seen to restrict to the identity on these sub-DGAs.[To verify, note that although we now begin the process from (1)-(3) with exceptional generatorsfrom Ex (Λ − ) ∪ Ex (Λ − ) already cancelled, we can still proceed in (1) to cancel the remainingexceptional generators of Σ in pairs by following the Steps 1-3 at the end of the proof of Theo-rem 8.2 from [32]. The cancellations in part (2) all take place near codimension 2 singularitiesin the interior of [0 , × M , so do not involve generators associated to Λ ± . When generatorsare cancelled in pairs during (1) and (2), Proposition 2.1 is applied to produce stable tame iso-morphisms, and each of these isomorphisms individually restricts to the identity on the A (Λ ± )sub-DGAs by Proposition 2.1 (2) because none of the canceled generators belong to these DGAs.Finally, at (3) the map Φ is the identity except on generators associated to cells that borderswallowtail points (and no such cell occurs in Λ ± .)] The DGA maps from (A.7) and (A.10)have their image in the sub-DGAs associated to Λ ± and already agree when these sub-DGAsare identified with A (Λ ± , E ±|| ). Since ϕ restricts to the identity on these sub-DGAs, it followsthat ϕ gives the required immersed homotopy. (cid:3) UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 61
Appendix B. A geometric model for Legendrian cobordisms
In this appendix we construct a standard geometric model for Legendrian cobordisms, andthen prove Proposition A.8.Recall that we are given a compact Legendrian cobordism Σ ⊂ J ([0 , × M ) from Λ − to Λ + as well as a transverse square decomposition, E (cid:116) , for Σ that is collared by admissible transversedecompositions E ± for Λ ± . We need to construct a standard geometric model for Σ that wewill denote G std (Σ , E (cid:116) ) = ( (cid:101) Σ , (cid:101) g )that will be a conical Legendrian cobordism in J ( R > × M ) from G std (Λ − , E − ) = ( (cid:101) Λ − , (cid:101) g − )to G std (Λ + , E + ) = ( (cid:101) Λ + , (cid:101) g + ) such that the metric (cid:101) g has the form g R × (cid:101) g − (resp. g R × (cid:101) g + ) on(0 , /T ] × M (resp. on [ T, + ∞ ) × M ) for T (cid:29) g R is the Euclidean metric on R > .To do this, we will construct from (Σ , E (cid:116) ) a Morse minimum cobordism , Σ min ⊂ J ([0 , × M ),(as in Section 2.5) from (cid:101) Λ − to (cid:101) Λ − with metric g min , and then define ( (cid:101) Σ , (cid:101) g ) to be an associatedconical cobordism, (Σ conic , g conic ), constructed from (Σ min , g min ) as in Construction 2.14. Then,Proposition 2.15 computes the immersed DGA map associated to Σ in terms of Σ min as F ( G std (Σ)) = (cid:2) A ( (cid:101) Λ + ) i + → A (Σ min ) i − ← (cid:45) A ( (cid:101) Λ − ) (cid:3) where the inclusions i ± arise as in Section 2.5 from the Morse minimums. Thus, to completethe proof of Proposition A.8, we will check that this immersed map is determined by E (cid:116) as inthe statement of the proposition. This is done in Proposition B.8 below.B.1. Construction of Σ min . Recall that the squares, e α , of the transverse square decomposi-tion E (cid:116) are equipped with characteristic maps, c α : [ − , × [ − , ∼ = → e α , satisfying properties as discussed in Section A.1 and Definition A.4. By Definition A.4, thereexists (cid:15) > E (cid:116) in [0 , (cid:15) ] × M (resp. [1 − (cid:15), × M ) squares havethe form [0 , (cid:15) ] × e α (resp. [1 − (cid:15), × e α ) with e α ∈ E − (resp. e α ∈ E + ). The construction of(Σ min , g min ) is carried out in the coordinates coming from the above parametrizations c α , andis slightly different in the collar squares than in the other squares of E (cid:116) . Construction of Σ min above [ (cid:15), − (cid:15) ] × M : This follows the construction from [33] precisely. In[33, Sections 2 and 3], defining functions F , . . . , F n : [ − , × [ − , → R are constructed forthe different square types (1)-(14). Above [ (cid:15), − (cid:15) ] × M , (cid:101) Σ is defined to appear as the 1-jets ofthe F , . . . , F n when viewed using the parametrization of squares by [ − , × [ − , Construction of Σ min above [0 , (cid:15) ] × M and [1 − (cid:15), × M : In order to arrange that Σ min hasthe appropriate Morse minimum ends near { } × M and { } × M , we will need to slightlyalter the defining functions F , . . . , F n . In order to still be able to apply the computation ofthe LCH DGA from [32] to these squares it is important that this modification preserves therelevant properties listed in [32, Section 5-8]. In the following, we construct the modified definingfunctions, and record properties of their partial derivatives that will allow for verification of theproperties from [32].Recall that the collar squares that border the left and right boundaries ∂ ([0 , × M ) are inbijection with the 1-cells of the admissible transverse decompositions E − (cid:116) and E + (cid:116) for Λ − andΛ + . For e α ∈ E ± (cid:116) , write e β for the corresponding collar square. As in Definition A.4, in collarsquares the characteristic maps provide parametrizations of the form(B.1) [ − , × [ − , ∼ = → e β , ( t, x ) (cid:55)→ ( l ± ( t ) , c α ( x )) In the case of functions that define sheets that meet a cusp edge above the square the domain is actually aproper subset of [ − , × [ − , − − / − / ts s = α ( t ) − − / ts s = m (1 + t ) Figure 25. (left) The plateau function α ( t ). (right) The red curve is a schematicof the graph of the function h : [ − , − / → R used in the definition of the G i .where l ± is linear as in (A.1). Moreover, with this parametrization, the left edges {− } × [ − , ∂ ([0 , × M ).Let e β be a given collar square. Note that e β must have either Type (1), (2), or (9), (as inFigure 22), and let F , . . . , F n : [ − , × [ − , → R denote the 2-variable defining functionsfrom [33] for the square type of e β . In addition, let g , . . . , g n : [ − , → R , g i ( x ) = F i ( − , x ),denote the 1-variable defining functions for (cid:101) Λ − or (cid:101) Λ + above [ − , ∼ = e α (as defined in (A.2)).We will construct Σ min above [ − , × [ − , ∼ = e β to have defining functions, G , . . . , G n : [ − , × [ − , → R , of the form(B.2) G i ( t, x ) = (1 − α ( t )) h ( t ) g i ( x ) + α ( t ) F i ( t, x )where α is a non-decreasing plateau function satisfying(B.3) α ( t ) = (cid:26) , t ∈ [ − / , , , t ∈ [ − , − / . Moreover, h : [ − , − / → R > satisfies • h ( t ) = m (1 + t ), for t ∈ [ − / , − /
4] where m > • h ( t ) is positive, and increasing with h (cid:48) ( t ) > − , − / • h ( t ) has a unique critical point that is a non-degenerate local minimum at t = − < m (cid:28)
1, (this is done in B.1.1 below), weneed to first observe some properties of the F i .As in [32, 33], we use the notations F i,j := F i − F j and g i,j = g i − g j for the associated difference functions . The F i are notated so that when i < j , F i,j (+1 , +1) >
0. Since we onlyhave squares of type (1), (2), and (9), it follows from the construction in [33] that when i < j , F i,j ( t, x ) > t, x ) in the domain of F i,j , except in two cases:(i) In a Type (2) square when i = k, j = k + 1 are the two sheets that cross. Then,sgn ( F k,k +1 ) = sgn ( x ) and the crossing locus is precisely the subset where x = 0.(ii) In a Type (9) square, the two cusp sheets meet at a cusp edge when x = − /
8, sothe corresponding difference function satisfies F k,k +1 ( t, x ) > x > − /
8, and F k,k +1 ( t, x ) = 0 when x = − / − / , − / × [ − , F i,j are specified in thenext two lemmas. Lemma B.1.
For any ( t, x ) ∈ [ − / , − / × [ − , and any i < j we have ∂ t F i,j ( t, x ) > except for the following two cases: UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 63 (1)
When S k and S k +1 cross along x = 0 (in a Type (2) square), we havesgn ( ∂ t F k,k +1 ( t, x )) = sgn ( x ) . (2) When S k and S k +1 meet at a cusp at x = − / (in a Type (9) square), we have ∂ t F i,j ( t, x ) ≥ with strict inequality when x ≥ .Proof. With ( t, x ) ∈ [ − , × [ − , F i,j are F i,j ( t, x ), the formula forthe F i from [33, Section 2] (at the end of the section) becomes ∀ ( t, x ) ∈ [ − / , − / × [ − , , F i ( t, x ) = φ ( x ) · f Ui ( t ) + (1 − φ ( x )) · f Dσ D ( i ) ( t ) + f Li ( x ); F i,j ( t, x ) = φ ( x ) · f Ui,j ( t ) + (1 − φ ( x )) · f Dσ D ( i ) ,σ D ( j ) ( t ) + f Li,j ( x )where σ D ( i ) is the position of sheet S i above the initial vertex of e α (this is defined to have σ D ( k ) = σ D ( k + 1) = k − . k and k + 1 meet at a cusp.) The f U , f D , f L are 1-dimensional functions from [33, Section 2] that are specified by the edge type of the U (up), D (down), and L (left) edge of the square. The φ function is a non-decreasing plateau functionwith φ ( x ) = (cid:26) , x ∈ [ − , − / , x ∈ [1 / , , satisfying φ (0) = 1 / φ (cid:48) (0) >
0. Thus, for all ( t, x ) ∈ [ − / , − / × [ − , ∂ t F i,j ( t, x ) = φ ( x ) · ( f Ui,j ) (cid:48) ( t ) + (1 − φ ( x )) · ( f Dσ D ( i ) ,σ D ( j ) ) (cid:48) ( t ) . (B.4)In (B.4), ( f Ui,j ) (cid:48) ( t ) >
0, and as long as σ D ( i ) < σ D ( j ) holds, we also have ( f σ D ( i ) ,σ D ( j ) ) (cid:48) ( t ) > U and D edges areType (PV), “plain vanilla”, so when i < j the only critical point of f i,j in ( − ,
1) is a localmax in [1 / , / σ D ( i ) ≥ σ D ( j ) is as in (1) and (2). For (1), we have f Dσ D ( k ) ,σ D ( k +1) = f Dk +1 ,k = − f Uk,k +1 . (The f U and f D functions are identical since both these edgesare plain vanilla with the same number of sheets.) Thus, we have ∂ t F i,j ( t, x ) = (2 φ ( x ) − · ( f Uk,k +1 ) (cid:48) ( t ) , so sgn ( ∂ t F i,j ( t, x )) = sgn (2 φ ( x ) −
1) = sgn ( x ) . For (2), σ D ( k ) = σ D ( k + 1) = k − .
5, so the second term in (B.4) is 0. (cid:3)
Next, note that in the construction from [33, Section 2.1] there is a fixed sequence of values η i,j ∈ (1 / , /
4) for i < j with the property that η i,j < η i (cid:48) ,j (cid:48) when ( i, j ) ≺ ( i (cid:48) , j (cid:48) ) with respectto the lexicographic ordering, ≺ . The η i,j specify the location of (non-exceptional) Reeb chordsabove the subsets of edges of squares that appear in coordinates as (1 / , / ⊂ [ − , g i,j ( x ) = F i,j ( − , x ), i < j each have a singlelocal maximum in (1 / , /
4) located at x = η i,j . These local maxima correspond to the Reebchords b βi,j discussed in Section A.4. In addition, for a (1Cr) edge, (cid:101) η k +1 ,k ∈ ( − / , − / g k +1 ,k in ( − / , − / (cid:101) b βk +1 ,k .There are no other critical points of the g i,j in ( − ,
1) other than the critical point of g k,k +1 at − / S k and S k +1 meet at a cusp in a Type (9) square. Lemma B.2.
For any ( t, x ) ∈ [ − / , − / × ( − , and i < j we havesgn ( ∂ x F i,j ( t, x )) = sgn ( g (cid:48) i,j ( x )) , and in particular ∂ x F i,j ( t, x ) > , if x ∈ [ − / , / . Proof.
Observe that sgn(( f Li,j ) (cid:48) ( x )) = sgn( g (cid:48) i,j ( x )) (these functions both have there critical pointsat the same x -values). Compute ∂ x F i,j ( t, x ) = φ (cid:48) ( x ) · ( f Ui,j ( t ) − f Dσ D ( i ) ,σ D ( j ) ( t )) + ( f Li,j ) (cid:48) ( x ) . When x / ∈ ( − / , / φ (cid:48) ( x ) = 0, and sgn ( ∂ x F i,j ( t, x )) = sgn (( f Li,j ) (cid:48) ( x )) is clear.For x ∈ [ − / , / | f Ui,j ( t ) | + | f Di,j ( t ) | is C -small and, except in the case that ( i, j ) = ( k, k + 1) and L is a (1Cr) edge, ( f Li,j ) (cid:48) ( x ) ≥ / C -small in this case means much smaller than 1 / i, j ) = ( k, k + 1) and x ∈ [ − / , /
4] we have f Uk,k +1 = − f Dσ D ( k ) ,σ D ( k +1) , so we get ∂ x F k,k +1 ( t, x ) = φ (cid:48) ( x ) · (2 f Uk,k +1 ( t )) + ( f Lk,k +1 ) (cid:48) ( x )and both terms are positive. (cid:3) B.1.1.
Definition of the constant m . In [ − / , × [ − , G , . . . , G n take theform(B.5) G i ( t, x ) = (1 − α ( t )) m (1 + t ) g i ( x ) + α ( t ) F i ( t, x ) , where m is a constant whose value we will now specify. Let R = [ − / , − / × [1 / ,
1] and R = [ − / , − / × [ − , − / R , the F i,j are non-vanishing in R , so by compactness there exists C > | F i,j ( t, x ) | ≥ C, ∀ i < j and ( t, x ) ∈ R . Moreover, the only time a difference function is 0 in R is the F k,k +1 function in a Type (9)square which vanishes along the cusp edge. Thus, C > | F i,j ( t, x ) | ≥ C, ∀ i < j and ( t, x ) ∈ R except when F i,j = F k,k +1 in a Type (9) square. We now fix the constant 0 < m (cid:28) m · || g i,j || C ([ − , < C for all difference functions g i,j associated to (cid:101) Λ − and (cid:101) Λ + . Lemma B.3.
Suppose that ( t, x ) ∈ R (cid:96) , (cid:96) = 1 or , and we have defining functions F i and F j , i < j , that are defined above all of R (cid:96) (i.e. neither of the sheets defined by F i and F j meets acusp edge above R (cid:96) .) Then, sgn ( ∂ t G i,j ( t, x )) = sgn ( F i,j ( t, x )) . Proof.
From (B.5) compute, ∂ t G i,j ( t, x ) = α (cid:48) ( t ) ( F i,j ( t, x ) − m (1 + t ) g i,j ( x )) + (1 − α ( t )) mg i,j ( x ) + α ( t ) ∂ t F i,j ( t, x ) . Then, use the definition of m and that we are in R (cid:96) , we havesgn ( F i,j ( t, x ) − m (1 + t ) g i,j ( x )) = sgn( F i,j ( t, x )) = sgn( g i,j ( x )) = sgn( ∂ t F i,j ( t, x ))where the last equality is from Lemma B.1. Since α (cid:48) ( t ) ≥ ≤ α ( t ) ≤
1, the resultfollows. (cid:3)
Lemma B.4.
For any ( t, x ) ∈ [ − / , − / × ( − , and any i < j we havesgn ( ∂ x G i,j ( t, x )) = sgn ( g (cid:48) i,j ( x )) if x ∈ ( − , − / ∪ [1 / , , and ∂ x G i,j ( t, x ) > if x ∈ [ − / , / . Proof.
Compute ∂ x G i,j ( t, x ) = (1 − α ( t )) m (1 + t ) g (cid:48) i,j ( x ) + α ( t ) ∂ x F i,j ( t, x ) . Then, use that sgn ( g (cid:48) i,j ( x )) = sgn ( ∂ x F i,j ( t, x )) and ∂ x F i,j ( t, x ) > x ∈ [ − / , /
4] asestablished in Lemma B.2. (cid:3)
UGMENTATIONS AND IMMERSED LAGRANGIAN FILLINGS 65
Corollary B.5.
The G i,j with i < j have no critical points in ( − , − / × [ − , except forthe critical points of G k,k +1 along the cusp edge in the Type (9) square.Proof. When t ∈ [ − / , − /
4] this follows from Lemmas B.3 and B.4. When t ∈ ( − , − /
8] wecompute from (B.2) ∂ t G i,j = h (cid:48) ( t ) · g i,j ( x ) and ∂ x G i,j = h ( t ) · g (cid:48) i,j ( x ) . Since h (cid:48) ( t ) >
0, the only possible critical points are when g i,j = 0. This occurs only along thecusp edge in the Type (9) square, and along the crossing locus, where x = 0, in the Type (2)square. In the latter case, g (cid:48) i,j ( x ) > (cid:3) Proposition B.6.
The Legendrians defined in local coordinates piece together to form a smoothMorse minimum cobordism Σ min ⊂ J ([0 , × M ) from (cid:101) Λ − to (cid:101) Λ + .Proof. Smoothness of Σ min above vertices and edges of E (cid:116) follows as in [33] from (i) the technicalassumptions about the parametrizations, c α , and (ii) the form of the defining functions F i,j nearthe boundaries of squares. (See Section 4.1 of [33].) Moreover, the definition of the definingfunctions G i above the collar squares shows that Σ agrees with j ( h ◦ l − ± ( s ) · Λ ± ) in neighborhoodsof { } × M and { } × M (here l − : [0 , (cid:15) ] → [ − ,
1] or l + : [1 − (cid:15), → [ − ,
1] is the linear functionfrom (B.1)). Since h ◦ l − − (resp. h ◦ l − ) is positive with a non-degenerate local min at s = 0(resp. at s = 1), it follows that Σ min has the correct Morse minimum form near ∂ ([0 , × M )as in Section 2.5. (cid:3) B.2.
Definition of the metric g min . The metric g min is also defined via the parametrizations c α : [ − , × [ − , → e α . For non-collar squares, e α ⊂ [ (cid:15), − (cid:15) ] × M the metric is definedas in [33, Construction 1.3], i.e. Euclidean except in a neighborhoods of the corners where ithas a standard form that depends only on the number of other squares that have vertices atthat corner. In the collar squares, the metric g min is defined to appear in the local coordinates( t, x ) ∈ [ − , × [ − ,
1] as ( c ( t ) · g R ) × g R where • c ( t ) : [ − , → R > is monotonic non-decreasing, • c ( t ) = (cid:15) / t near −
1, and • c ( t ) = 1, for t ∈ [ − / , (cid:15) / l ± from (B.1). Proposition B.7.
This construction defines a smooth metric g min on [0 , × M that has theform g R × (cid:101) g − near { } × M and g R × (cid:101) g + near { } × M .Proof. For smoothness, note that from the construction of E (cid:116) all vertices where collar squaresborder non-collar squares are 4-valent. As a result, the metric in all neighboring compactsquares is also Euclidean near these corners. (See [33, Construction 1.3].) The statementabout the form of g min near { } × M , follows since in [0 , (cid:15) ] × M , keeping in mind that bydefinition (cid:101) g − appears Euclidean in the local coordinates from 1-dimensional characteristic maps c α : [ − , → e α ⊂ M , we see that g min has the form( l − ± ) ∗ (cid:0) ( (cid:15) / g R (cid:1) × (cid:101) g − = g R × (cid:101) g − . A similar calculation applies near { } × M . (cid:3) B.3.
Computation of F ( G std (Σ)) . With the construction of (Σ min , g min ) complete we turn tothe evaluation of the immersed map(B.6) A ( (cid:101) Λ + ) i + → A (Σ min ) i − ← (cid:45) A ( (cid:101) Λ − )whose immersed homotopy class agrees with F ( G std (Σ)), as discussed at the start of this section.We need to verify that (B.6) is determined by E (cid:116) as in statements (1) and (2) from PropositionA.8. In [32] Reeb chords of a closed Legendrian with transverse square decomposition are identifiedand associated to the different cells of E (cid:116) . The precise number of Reeb chords above a 0-, 1-, or2- cell is determined by the appearance of Σ above e dα , i.e. the number of sheets above a 0-cell,or the edge or square type of Σ above a 1- or 2-cell. Proposition A.8, stated more precisely,becomes the following: Proposition B.8. (1)
The Reeb chords of Σ min all appear in collections corresponding to the -, -, and -cells of the e α squares precisely as in [32] with their differentials satisfyingthe same formulas from [32, Propositions 5.9 and 5.10, Theorems 6.5 and 7.4, Lemmas8.3 and 8.4] . (2) The Reeb chords corresponding to - and -cells contained in { } × M and { } × M areprecisely the Reeb chords of Σ min that are the generators if the sub-DGAs i ± ( A (Λ ± )) ⊂A (Σ min ) .Proof. The computation of the DGA, A (Σ min ), proceeds in the same manner as in the case of aclosed Legendrian surface from [32, Sections 5-8]. There a list of Properties 1-19 satisfied by thestandard geometric model ( (cid:101) Σ , (cid:101) g ) (in the closed case) can be found, and in [33, Section 5] it isshown that the properties are maintained when ( (cid:101) Σ , (cid:101) g ) is perturbed to achieve regularity. Theseproperties describe enough of the behavior of defining functions of (cid:101) Σ and their gradient vectorfields to locate all Reeb chords and enumerate rigid GFTs. In particular, each d -cell e dα ∈ E (cid:116) hasa neighborhood with the property that any GFT starting in N ( e dα ) must have its entire imagein N ( e dα ). This allows the differentials of Reeb chords in each N ( e dα ) to be computed in a localmanner.In the following, we indicate how the same computation can be applied to the Morse minimumcobordism Σ min by considering the collar and non-collar squares of E (cid:116) separately. Non-collar squares, e α ⊂ [ (cid:15), − (cid:15) ] × M : Here, the coordinate model used for Σ min is exactly thesame as in [32, 33], so the computations there apply directly. Collar squares, e α , in [0 , (cid:15) ] × M or [1 − (cid:15), × M : These squares are parametrized by ( t, x ) ∈ [ − , × [ − ,
1] with the t coordinate corresponding to the [0 , (cid:15) ] or [1 − (cid:15),
1] factor and t = − e α on ∂ ([0 , × M ). In these coordinates, above [ − / , × [ − , − , − / × [ − , min appear as(B.7) G i,j ( t, x ) = h ( t ) · g i,j ( x ) , where g i,j is the corresponding difference function for (cid:101) Λ ± , and the only critical points of any G i,j are along the cusp edge in a Type (9) square when G i and G j define the two cusp sheets. [In allother cases, we use that h ( t ) > h (cid:48) ( t ) >
0, and g (cid:48) i,j ( x ) (cid:54) = 0 when g i,j ( x ) = 0 to see that ∂ t G i,j or ∂ x G i,j is non-zero.] In [ − / , − / × [ − , min has noReeb chords in this region. Finally, since h (cid:48) ( −
1) = 0 the Reeb chords above {− } × [ − ,
1] arein bijection with critical points of the g i,j ( x ). By construction, see (A.2), g i,j ( x ) = F i,j ( − , x )where the F i : [ − , × [ − ,
1] are the usual defining functions (from [33]) matching the squaretype of e α , so the Reeb chords above [ − , − / × [ − ,
1] indeed match the Reeb chords ofthe corresponding square from E (cid:116) above [ − , − / × [ − ,
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Massachusetts Institute of Technology
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