aa r X i v : . [ m a t h . S G ] D ec COBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY
JACOB ROONEY
Abstract.
Given an exact symplectic cobordism (
X, λ ) between contact 3-manifolds( Y + , λ + ) and ( Y − , λ − ) with no elliptic Reeb orbits up to a certain action, we de-fine a chain map from the embedded contact homology (ECH) chain complex of( Y + , λ + ) to that of ( Y − , λ − ), both taken with coefficients in Z / Z . The map isdefined by counting punctured holomorphic curves with ECH index 0 in the com-pletion of the cobordism and new objects that we call ECH buildings, answeringa question of Hutchings. Contents
1. Introduction 12. Background 83. The Evaluation Map 134. Index Calculations 165. Degenerations in Cobordisms 226. Obstruction Bundle Gluing 257. Gluing Models and Evaluation Map Calculations 378. The Cobordism Map 46Appendix A. Existence of Models 55Appendix B. Determinant Calculations 59References 63 Introduction
In this paper, we answer a question of Hutchings on the foundations of ECH:given contact 3-manifolds ( Y ± , λ ± ) and an exact symplectic cobordism ( X, λ ) from( Y + , λ + ) to ( Y − , λ − ), how can we define a chain map from the ECH chain complexof Y + to that of Y − by counting J -holomorphic curves? We answer this questionwhen ( Y ± , λ ± ) have no elliptic orbits up to a certain action L . Namely, given theabove setup and assuming that ( Y ± , λ ± ) have no elliptic Reeb orbits up to an action Date : This version: December 3, 2020. L , we define a chain mapΦ X,λ,J, c : ECC L ( Y + , λ + , J + ) → ECC L ( Y − , λ − , J − )by counting J -holomorphic curves in the completion b X and new objects that wecall index ECH buildings . Here, J ± is a generic almost complex structure onthe symplectization R × Y ± , J is a generic almost complex structure on b X thatis compatible with J + at the positive end and with J − at the negative end, and c is a choice of auxiliary data that is explained in Definition 1.4.3. We show inTheorem 1.5.3 that Φ X,λ,J, c is a chain map and is independent of the choice of c .The definition of Φ X,λ,J, c relies on some new developments for holomorphic curvesin the L -supersimple setting of Bao-Honda [BH1, BH2] and Colin-Ghiggini-Honda[CGH1, CGH2, CGH3], and we restrict our attention to that setting throughout thepaper.ECH is isomorphic to both Heegaard Floer homology and Seiberg-Witten Floer(co)homology (see [KLT1, KLT2, KLT3, KLT4, KLT5, CGH1, CGH2, CGH3]), andthe latter isomorphism was used by Hutchings-Taubes in [HT3] to define mapsinduced by exact symplectic cobordisms between contact 3-manifolds. However,a definition of such maps that involves counting J -holomorphic curves has provedelusive. Chris Gerig has given a construction in a specific case [Ge], and Hutchingshas given an example where one must take into account multi-level SFT buildings[Hu3, Section 5].In Sections 2 and 3, we give appropriate background information for ECH andthe evaluation map defined by Bao-Honda. In Sections 4 to 7, we discuss the de-tails of these new developments. In Section 8, we prove the main result of thispaper, namely, that Φ X,λ,J, c is a chain map. The remainder of this section is anoutline of the paper, culminating in the definition of Φ X,λ,J, c ; see Theorem 1.5.3 andDefinition 1.5.2, which depend on some auxiliary definitions in this section.This paper is a heavily revised version of the author’s doctoral thesis [Ro], fromwhich portions of this work have been excerpted.1.1. The L -supersimple setting and filtered ECH. We begin with a discussionof the L -supersimple setting. Recall that the action of a Reeb orbit α on the contactmanifold ( Y, λ ) is the integral A ( α ) = R α λ , while the total action of an orbit set α is the sum A ( α ) = P α ∈ α R α λ . Definition 1.1.1.
A contact form λ on a smooth 3-manifold Y is L -supersimple ifevery Reeb orbit with action less than L is non-degenerate, hyperbolic, and satisfiesthe conclusions of Theorem 2.5.1.Our chain map is defined on the level of filtered ECH , defined as follows. Let( Y, λ ) be a non-degenerate contact 3-manifold, and let J by a generic, compati-ble almost complex structure on R × Y . Let L >
OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 3
ECC L ( Y, λ, J ) ⊂ ECC ( Y, λ, J ) generated by orbit sets α with total action lessthan L . Every non-degenerate contact form can be made into an L -supersimpleform by a small perturbation. That is, for any L > ε >
0, there is a positivesmooth function f on Y that is C -close to 1 such that f λ is L -supersimple. Fur-thermore, if f i λ is L i -supersimple for i = 1 , L < L , we can ensure that theset of Reeb orbits of f λ with action less than L coincides with the correspondingset of Reeb orbits for f λ , i.e., that there is a natural inclusion map ECC L ( Y, f λ, J ) ֒ → ECC L ( Y, f λ, J ) . See [BH1, Theorem 2.0.2] and [CGH1, Theorem 2.5.2] for details.We can reconstruct
ECH ( Y, λ, J ) from these filtered groups in the following way,as described in [CGH0, Theorem 3.2.1]. Let { f i } ∞ i =1 be a sequence of positive smoothfunctions on Y with 1 ≥ f ≥ f ≥ · · · and such that f i λ is L i -supersimple forsome sequence { L i } ∞ i =1 of positive real numbers with lim i →∞ L i = ∞ . Then there is acanonical isomorphism ECH ( Y, λ, J ) ≃ lim i →∞ ECH L i ( Y, f i λ, J ) . Thus, it suffices to define the chain map Φ
X,λ,J, c on each level of the filtration ECC L i ( Y, f i λ, J ), where there are no elliptic Reeb orbits. We do not lose anygenerality in assuming that the contact forms on Y ± are L -supersimple aside fromthe need to assume invariance results of Hutchings-Taubes [HT3].1.2. The ECH index inequality.
The first of our developments is an improve-ment to the ECH index inequality in the L -supersimple setting. On one-dimensionalmoduli spaces, the inequality is in fact an equality and gives information about thetopology of punctured J -holomorphic curves that violate the ECH partition condi-tions. One can also show that the improved equality is an equality for generic curveswith higher Fredholm index using the evaluation map from Section 3. The inequal-ity is implicit in the work of Hutchings [Hu2]. Gardiner-Hind-McDuff give a similarimprovement in [CGHD], and Gardiner-Hutchings-Zhang recently showed that theimproved inequality is an equality for generic curves [CGHZ]. The advantages ofthe L -supersimple setting are that (1) the extra term in the improved inequality isgiven by a simple formula that involves only the multiplicities of the ends of thecurve, and (2) the analysis required to prove generic equality is greatly simplified.The starting point for our improved inequality is Hutchings’ ECH index in-equality from [Hu2]: If u is a somewhere injective J -holomorphic curve in a sym-plectization R × Y , then(1.2.1) I ( u ) ≥ ind( u ) + 2 δ ( u ) , where δ ( u ) is a non-negative count of singularities of u . JACOB ROONEY
Definition 1.2.1.
Let u : ˙Σ → R × Y be a punctured J -holomorphic curve asymp-totic to an orbit set α at the positive ends and to an orbit set β at the negativeends. We say that α is the positive orbit set of u , that β is the negative orbitset of u , and that u goes from α to β . Definition 1.2.2.
Let Γ + ( u ) denote the set of embedded Reeb orbits in the positiveorbit set of u (i.e., forgetting their multiplicities), and let Γ − ( u ) denote the set ofembedded Reeb orbits in the negative orbit set of u . Definition 1.2.3.
The
ECH deficit of u at an orbit γ ∈ Γ + ( u ) is definedas follows. If γ is negative hyperbolic, suppose u has ends at (covers of) γ ofmultiplicities q , . . . , q n , ordered so that the first k ends hove odd multiplicity andthe last n − k ends have even multiplicity. Then∆( u, γ ) = k X i =1 (cid:18) q i −
12 + i − (cid:19) + n X i = k +1 (cid:16) q i − (cid:17) If γ is positive hyperbolic and u has ends at (covers of) γ of multiplicities q , . . . , q n ,then ∆( u, γ ) = n X i =1 ( q i − . The ECH deficit ∆( u, γ ) for γ ∈ Γ − ( u ) is defined similarly. Definition 1.2.4.
The
ECH deficit of u is∆( u ) = X γ ∈ Γ + ( u ) ∆( u, γ ) + X γ ∈ Γ − ( u ) ∆( u, γ ) . Theorem 1.2.5. If J is generic and u is a somewhere injective J -holomorphiccurve in a symplectization, then (1.2.2) I ( u ) ≥ ind( u ) + 2 δ ( u ) + ∆( u ) . Equality holds if A ( α ) < L and ind( u ) = 1 . Degenerations of one-dimensional families in cobordisms.
The nextdevelopment is an analysis of possible degenerations of one-dimensional families ofpunctured holomorphic curves in exact symplectic cobordisms, which we discuss inSection 5.Let ( Y ± , λ ± ) be L -supersimple contact 3-manifolds and let ( X, λ ) be an exactsymplectic cobordism from ( Y + , λ + ) to ( Y − , λ − ). Let J be a generic, L -simple,admissible almost complex structure on the completion ( b X, b λ ) that restricts to L -simple, admissible almost complex structures J + and J − on the ends [0 , ∞ ) × Y + and ( −∞ , × Y − , respectively, of b X . OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 5
Notation 1.3.1.
Let α and β be orbits sets in a contact manifold ( Y, λ ). We denotethe moduli space of J -holomorphic curves u from α to β in R × Y ± with ind( u ) = p and I ( u ) = q by M p,q R × Y ± ( α , β ).Let ( X, λ ) be an exact symplectic cobordism from ( Y + , λ + ) to ( Y − , λ − ), let α bean orbit set in ( Y + , λ + ), and let β be an orbit set in ( Y − , λ − ). We denote the modulispace of J -holomorphic curves u from α to β in the completion b X with ind( u ) = p and I ( u ) = q by M p,qX ( α , β ).Let α and β be generators of ECC L ( Y + , λ + , J + ) and ECC L ( Y − , λ − , J − ), re-spectively. Consider the moduli space M , X X ( α , β ) and let M , X ( α , β ) denote itsSFT compactification as described in [BEHWZ]. We denote an SFT building in ∂ M , X ( α , β ) by [ u − a ] ∪ · · · [ u − ] ∪ u ∪ [ u ] ∪ · · · ∪ [ u b ], where a and b are positiveintegers, the levels go from bottom to top as we read from left to right, the levelswith negative indices are in ( R × Y − ) / R , the level u is in b X , and the levels withpositive indices are in ( R × Y − ) / R . Theorem 1.3.2.
The points in ∂ M , X ( α , β ) are two-level buildings of the form [ u − ] ∪ u or u ∪ [ u ] , where ind( u ) = 0 and ind( u ± ) = 1 . Let γ denotethe negative orbit set of u + . When γ is a generator of the ECH chain complex ECC L ( Y ± , λ ± , J ± ) , we have I ( u ) = 0 and I ( u ± ) = 1 , and both levels are some-where injective. When γ is not a generator of ECC L ( Y ± , λ ± , J ± ) , the buildingsoccur in pairs unless they are of the form u ∪ [ u ] and the following conditions hold: (1) u is somewhere injective; (2) u is multiply covered; (3) I ( u ) > and I ( u ) < ; (4) each Reeb orbit in γ has multiplicity except for finitely many negativehyperbolic orbits γ , γ , . . . , γ k with multiplicities n , n , . . . , n k , respectively; (5) u has n i negative ends at γ i , each with multiplicity ; (6) for each i = 1 , . . . , k , u contains an unbranched, disconnected, n i -fold mul-tiple cover of an embedded holomorphic plane with its positive end at γ i , andeach multiply covered component of u is of this form. The prototypical gluing problem.
The last development is an obstructionbundle gluing calculation for certain branched covers of trivial cylinders with highFredholm index, which we discuss in Section 6. Here, a trivial cylinder is a cylinder R × β ⊂ R × Y + , where β is an embedded Reeb orbit in Y . We use the notationof Hutchings-Taubes from [HT1] for moduli spaces of such branched covers. Definition 1.4.1.
Let β be a Reeb orbit in ( Y + , λ + ). Let M ( a , a , . . . , a k | a − , a − , . . . , a − ℓ ) JACOB ROONEY denote the moduli space of genus 0 branched covers ˙Σ → R × β with ends labeledand asymptotically marked and such that the i th end is asymptotic to an a i -foldcover of β . Definition 1.4.2.
For each n ≥
3, let M n = M (1 , , . . . , | , , . . . , , n positive ends of multiplicity 1, n − u : ˙Σ → R × Y + be an embedded J -holomorphic curve with ind( u ) = 1 such that(1) the positive ends of u are asymptotic to an ECH generator α with totalaction less than L ;(2) the negative ends of u are asymptotic to an orbit set β in which each Reeborbit has multiplicity 1 except for a single negative hyperbolic orbit β ;(3) the curve u has n negative ends at β , each with multiplicity 1;(4) I ( u ) = 1 + (cid:0) n (cid:1) .We wish to glue branched covers in M n to the curve u .The main source of trouble in the above gluing problem is that the moduli spaces M n are not transversely cut out. However, by standard techniques, there should bean obstruction bundle O → [ R, ∞ ) × ( M n / R ) , for R ≫ O ( T,u ) = Hom (cid:0) Coker D Nu , R (cid:1) , where D Nu is the normal part of the linearized ∂ -operator for u .In analogy with [HT2, Definition 5.9], there should also be an obstruction sec-tion s for O whose zero set is the set of branched covers that glue to u + . Suchglued curves lie in the moduli space M n − , R × Y + ( α , β ). In Section 6.4, we describe aperturbation of the asymptotic operator for β that allows us to replace elementsof Coker D Nu with anti-meromorphic 1-forms on ˙Σ, which we use to write down thezero set of s explicitly. Definition 1.4.3.
Let u : ˙Σ → R × Y be a punctured J -holomorphic curve in M n − , R × Y + ( α , β ) with n − β and one negativeend of multiplicity 3 at β . Label the negative ends of u at (covers of) β withby elements of I − = {− , . . . , − n } , where the multiplicity 3 end is labeled 1. Thecurve u satisfies the asymptotic restrictions c ∈ C n − if ev I − ( u ) = c , where theevaluation map ev I − maps u to the leading complex coefficient in the asymptoticexpansion of u at the negative ends labeled by I − . See Definition 3.2.3 for the fulldefinition of the evaluation map. Definition 1.4.4.
We say that c ∈ ( C ∗ ) n − is an admissible asymptotic re-striction if it is not in the big diagonal of ( C ∗ ) n . OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 7
Theorem 1.4.5.
In the prototypical gluing problem, s − (0) is non-empty. If c ∈ C n − is a generic choice of admissible asymptotic restriction and T ≥ R , the mod count of curves in s − (0) with gluing parameter T that satisfy the asymptoticrestriction is . Definition of the chain map.
As described above, there are two contribu-tions to the curve count in the definition of Φ
X,λ,J, c . Suppose that we have ECHgenerators α ∈ ECC ( Y + , λ + , J + ) and β ∈ ECC ( Y − , λ − , J − ), that we writeΦ X,λ,J, c ( α ) = X A ( β ) < A ( α ) h Φ X,λ,J, c ( α ) , β i · β , and that we want to define the coefficient h Φ X,λ,J, c ( α ) , β i . The first contributionis the mod 2 count M , X ( α , β ). The second contribution is the mod 2 count ofnew objects that we call ECH buildings satisfying certain admissible asymptoticrestrictions.
Definition 1.5.1.
Assume the setup described above. An index ECH building from α to β satisfying the admissible asymptotic restriction c is a pair ( u , [ u ])satisfying the following conditions:(1) [ u ] is in ( R × Y + ) / R and u is in b X ;(2) u has positive orbit set α and u has negative orbit set β ;(3) the negative orbit set γ of u coincides with the positive orbit set of u ;(4) the partition of the negative ends of u coincides with the partition of thepositive ends of u , except possibly for some negative hyperbolic Reeb or-bits γ , . . . , γ ℓ in γ of multiplicities m , . . . , m ℓ where the partition for thenegative ends of u at each γ i is (3 , , . . . ,
1) and the partition for the positiveends of u at each γ i is (1 , , . . . , u ) = 0 and I ( u ) = − P ℓj =1 (cid:0) m j (cid:1) ;(6) ind( u ) = P ℓj =1 (2 m j −
4) and I ( u ) = − I ( u ); and(7) [ u ] has a (necessarily unique) representative u that satisfies the asymptoticrestriction c , where we use all of the negative ends at the orbits γ , . . . , γ ℓ for the evaluation map.We denote the set of index 0 ECH buildings from α to β satisfying the admissibleasymptotic restriction c by B ( α , β ; c ). Definition 1.5.2.
Let ( Y ± , λ ± ) be L -supersimple contact 3-manifolds and let ( X, λ )be an exact symplectic cobordism from ( Y + , λ + ) to ( Y − , λ − ). Let J be a generic, L -simple, admissible almost complex structure on the completion ( b X, b λ ) that restrictsto L -simple, admissible almost complex structures J + and J − on the ends [0 , ∞ ) × Y + and ( −∞ , × Y − , respectively, of b X . Let c be a generic choice of admissibleasymptotic restriction. The mapΦ X,λ,J, c : ECC L ( Y + , λ + , J + ) → ECC L ( Y − , λ − , J − ) JACOB ROONEY induced by (
X, λ ) is defined byΦ
X,λ,J, c ( α ) = X A ( β ) < A ( α ) h M , X ( α , β ) + B ( α , β ; c ) i · β . The following theorem is the main result of this paper. Its proof is given inSection 8. The set of generic asymptotic restrictions is described in Definition 8.1.1.
Theorem 1.5.3.
The map Φ X,λ,J, c in Definition 1.5.2 is a chain map and is inde-pendent of the choice of generic, admissible asymptotic restriction c .Acknowledgements. First and foremost, the author thanks Ko Honda for his gen-erous support and endless patience. The author also thanks Michael Hutchings,Katrin Wehrheim, and Erkao Bao for helpful conversations during the developmentof the ideas in this paper. The work for this article was completed while the authorwas at UCLA. 2.
Background
In this section, we establish some notation, briefly review the definition of embed-ded contact homology, and recall some basic facts about the L -supersimple settingof Bao-Honda.2.1. Basic definitions.
Let Y be a smooth 3-manifold, let λ be a non-degeneratecontact form on Y , let ξ = Ker( λ ) be the associated contact structure, and let R λ be the Reeb vector field of λ , defined as the unique vector field on Y satisfying λ ( R λ ) = 1 and dλ ( R λ , · ) = 0. Definition 2.1.1.
An almost complex structure J on R × Y is admissible if itsatisfies the following properties:(1) J is invariant under R -translation;(2) J ( ∂ s ) = R λ , where s is the R -coordinate of R × Y ;(3) J restricts to an orientation-preserving isomorphism of ξ .Let α be a Reeb orbit in ( Y, λ ) and let τ be a trivialization of ξ over α . Wedenote the Conley-Zehnder index of α in the trivialization τ by µ τ ( α ). We recallhere some simple expressions for the Conley-Zehnder index in dimension 3. If α iselliptic, then there is some irrational number θ ∈ (0 ,
1) such that µ τ ( α k ) = 2 ⌊ kθ ⌋ +1.If α is hyperbolic, then µ τ ( α k ) = kn for some integer n . In the latter case, we saythat α is positive hyperbolic if n is even and negative hyperbolic if n is odd. Definition 2.1.2. An orbit set is a tuple of ordered pairs α = (cid:0) ( α , m ) , ( α , m ) , . . . , ( α k , m k ) (cid:1) such that each α i is an embedded Reeb orbit in Y and each m i is a positive integer. OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 9
Definition 2.1.3. If α = (( α , m ) , . . . , ( α k , m k )) is an orbit set, we define µ τ ( α ) = k X i =1 µ τ ( α m k k ) and µ Iτ ( α ) = k X i =1 m k X j =1 µ τ ( α j ) . If β is another orbit set, we define µ τ ( α , β ) = µ τ ( α ) − µ τ ( β ) and µ Iτ ( α , β ) = µ Iτ ( α ) − µ Iτ ( β ) . Punctured holomorphic curves.
Let (Σ , j ) be a closed Riemann surfacewith complex structure j . Let P ⊂ Σ be a finite set of points, called punctures ,which are partitioned into subsets P + and P − of positive and negative punc-tures , respectively. Define ˙Σ = Σ \ P ; we refer to ˙Σ as a punctured Riemannsurface . If J is an admissible almost complex structure on R × Y , a puncturedholomorphic curve is a smooth map u : ˙Σ → R × Y such that du + J ◦ du ◦ j = 0 . A J -holomorphic curve u : ˙Σ → R × Y is said to be multiply covered if it factorsthrough a (possibly branched) cover φ : ˙Σ ′ → ˙Σ for some punctured Riemann surface˙Σ ′ . A curve is said to be simply covered if it is not multiply covered. We alsorefer to such curves as simple .2.3. Moduli spaces.
We distinguish between two types of moduli spaces of J -holomorphic curves, marked and unmarked, and make use of both types. Markedmoduli spaces are used in Section 6 for obstruction bundle gluing problems, andECH is defined using unmarked moduli spaces.Let u : ˙Σ → R × Y be J -holomorphic, and assume that u is asymptotic to Reeborbits α , α , . . . , α n at the positive punctures and to β , β , . . . , β m at the negativepunctures. For each such Reeb orbit, let ( α i ) e denote the underlying embedded Reeborbit for α i , choose a point ζ i on each ( α i ) e , and for each z i ∈ P + , choose an element r i ∈ ( T z i Σ \ { } ) / R + that maps to ζ i under the map α i → ( α i ) e . Similarly, let ( β j ) e denote the underlying Reeb orbit for β j , choose a point η j on each ( β j ) e , and foreach w j ∈ P − , choose an element r j ∈ ( T w j Σ \ { } ) / R + that maps to η j under themap β j → ( β j ) e . We refer to each such choice as an asymptotic marker at therelevant puncture; we refer to markers at positive punctures as positive markers and to markers at negative punctures as negative markers . Let r denote the setof markers that we have chosen.Given orbit sets α and β , the moduli space of marked, punctured holomorphiccurves from α to β in R × Y is the space of pairs ( u, r ), where u is asymptotic to α at the positive punctures and to β at the negative punctures, and r is a set ofasymptotic markers for u , modulo biholomorphisms of domains that send positive punctures to positive punctures, negative punctures to negative punctures, positivemarkers to positive markers, and negative markers to negative markers. Modulispaces of marked curves can be compactified using SFT buildings; see [BEHWZ] fordetails.Unmarked moduli spaces are defined similarly to marked moduli spaces, except wedo not choose asymptotic markers at each puncture. Consequently, we identify twosuch maps if they are related by a biholomorphism of the domains that maps positivepunctures to positive punctures and negative punctures to negative punctures. ECHuses unmarked moduli spaces and identifies two maps if they represent the samecurrent in R × Y .A curve u ∈ M J ( α , β ) has a Fredholm index given byind( u ) = − χ ( ˙Σ) + 2 c ( u ∗ ξ, τ ) + µ τ ( α , β ) , where c ( u ∗ ξ, τ ) is the relative first Chern class of ξ over u in the trivialization τ .See [Hu2, Section 2] for the definition of the relative first Chern class. If M J ( α , β ) istransversely cut out, then the (real) dimension of a neighborhood of u ∈ M J ( α , β )is precisely ind( u ) by results of Dragnev [Dr].2.4. The ECH chain complex.
We now define the ECH chain complex with Z / Z coefficients. (It is possible to define ECH with Z coefficients, but we do nottreat that case here.) Let Γ ∈ H ( Y ) and let J be a generic, admissible almostcomplex structure on R × Y . The groups ECC ( Y, λ, Γ , J ) are generated by orbitssets α = (( α , m ) , ( α , m ) , . . . , ( α k , m k )) such that m i = 1 if α i is hyperbolic andsuch that k X i =1 m i [ α i ] = Γ . Hutchings defines an
ECH index I for J -holomorphic currents C in R × Y . Morespecifically, he defines a relative self-intersection number Q τ ( C ) and sets I ( C ) = c ( ξ | C , τ ) + Q τ ( C ) + µ Iτ ( α , β ) . The differential ∂ counts punctured J -holomorphic currents with ECH index 1 in R × Y going from α to β . More precisely, consider the moduli space M I =1 J ( α , β )of J -holomorphic currents C with I ( C ) = 1 that are asymptotic to α at the positiveends and to β at the negative ends. There is an R -action on M I =1 J ( α , β ) inducedby translation in the R -direction of R × Y . Lemma 2.4.1. If M I =1 J ( α , β ) is non-empty, then A ( β ) < A ( α ) .Proof. See [Hu3, Section 5]. (cid:3)
Lemma 2.4.2. [Hu3, Lemma 5.10] If J is generic and admissible and α and β are orbit sets, then M I =1 J ( α , β ) / R is finite. OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 11
The differential ∂ on the chain complex ECC ( Y, λ, Γ , J ) is defined by ∂ ( α ) = X A ( β ) < A ( α ) (cid:0) M I =1 J ( α , β ) / R (cid:1) · β . Currents counted by the differential ∂ satisfy a rigid requirement on the multiplic-ities of their positive and negative ends. This requirement is crucial in [HT1, HT2]to show that ∂ = 0 and is leveraged extensively in this paper. Definition 2.4.3.
Let α be an embedded hyperbolic Reeb orbit in Y . Let C be a J -holomorphic current in R × Y with positive ends of multiplicities m , m , . . . , m k and negative ends of multiplicities n , n , . . . , n l at covers of α . Set m = P ki =1 m i and n = P lj =1 n j . We say that C satisfies the ECH partition conditions at itspositive ends at α if the multiplicities m i are as in Table 1. Similarly, we say that C m even m odd α positive hyperbolic (1 , . . . ,
1) (1 , . . . , α negative hyperbolic (2 , . . . ,
2) (2 , . . . , , Table 1.
The partition conditions for hyperbolic Reeb orbits.satisfies the partition conditions at its negative ends if the multiplicities n j are as inTable 1 with m replaced by n . We say that C satisfies the ECH partition conditionsif it satisfies the partition conditions at all of its positive and negative ends. Remark . We do not concern ourselves with the partition conditions for ellipticReeb orbits in this paper, as we work completely in the L -supersimple setting.Interested readers can consult [Hu3] for details. Remark . A J -holomorphic curve u : ˙Σ → R × Y gives rise to a J -holomorphiccurrent C = u ( ˙Σ).2.5. The L -supersimple setting. We now review the relevant background for the L -supersimple setting of Bao-Honda. As stated in Section 1, every non-degeneratecontact form can be made into an L -supersimple form by a small perturbation. Theprecise statement of this result, which we take from [BH1], is as follows. Theorem 2.5.1. [BH1, Theorem 2.0.2]
Let λ be a non-degenerate contact form for ( Y, ξ ) . Then, for any L > and ǫ > , there exists a smooth function φ : Y → R + such that (1) φ is ǫ -close to with respect to a fixed C -norm; (2) all the orbits of R φλ of φλ -action less than L are hyperbolic.Moreover, we may assume that (3) each positive hyperbolic orbit α has a neighborhood ( R / Z ) × D δ with coor-dinates ( t, x, y ) such that (a) D δ = (cid:8) x + y ≤ δ (cid:9) , where δ > is small; (b) φλ = H dt + η ; (c) H = c ( α ) − ǫxy , with c ( α ) , ǫ > and c ( α ) ≫ ǫ ; (d) η = 2 x dy + y dx ; (e) α = { x = y = 0 } . (4) each negative hyperbolic orbit α has a neighborhood ([0 , × D δ ) / ∼ withcoordinates ( t, x, y ) , where ∼ identifies (1 , x, y ) ∼ (0 , − x, − y ) and the con-ditions (a) through (e) above hold. One major advantage of working in the L -supersimple setting is that the Fredholmindex is well-behaved under taking multiple covers. Lemma 2.5.2. [BH1, Lemma 3.3.2]
Let ( Y, λ ) be a contact -manifold and let α and β be orbit sets where every orbit is hyperbolic. If v is a J -holomorphic curvefrom α to β in R × Y and u is a degree k branched cover of u with total branchingorder b , then ind( u ) = k ind( v ) + b. In particular, ind( u ) ≥ for all J -holomorphic curves u from α to β in R × Y . Another major advantage of the L -supersimple setting is that, by choosing thealmost complex structure J appropriately, we can ensure that the ∂ -equation islinear for curves that are close to and graphical over trivial cylinders. The set of J for which this assertion is true is described in the following definitions. Definition 2.5.3. [BH2, Definition 3.1.2] Let λ be a contact form on Y . An almostcomplex structure J on R × Y is λ -tame if the following three conditions hold:(1) J is R -invariant;(2) J ( ∂ s ) = gR λ for some positive function g on Y ; and(3) there exists a 2-plane field ξ ′ on Y such that J preserves ξ ′ , dλ is a symplecticform on ξ ′ , and J restricts to an orientation-preserving isomorphism on ξ ′ . Definition 2.5.4. [BH2, Definition 3.1.3] Let
L >
0, let λ be an L -supersimplecontact form, and let α be an embedded Reeb orbit of λ . A λ -tame almost complexstructure J is L -simple for λ if, inside the neighborhood of α given by Theo-rem 2.5.1, the following conditions hold:(1) ξ ′ = Span ( ∂ x , ∂ y );(2) J ( ∂ x ) = ∂ y ; and(3) the function g in Definition 2.5.3 satisfies gR λ = ∂ t + X H , where X H is theHamiltonian vector field of the function H from Theorem 2.5.1 with respectto the symplectic form dx ∧ dy .We can now state the second advantage precisely. OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 13
Proposition 2.5.5. [BH2]
Let λ be an L -supersimple contact form on Y and let J be an L -simple almost complex structure for λ . If u : [ R, ∞ ) × S → R × Y is a J -holomorphic half-cylinder asymptotic to a Reeb orbit α , and if we write u ( s, t ) =( s, t, e u ( s, t )) , then the function e u satisfies ∂ s e u + j ∂ t e u + S e u = 0 , where j is the standard complex structure on R and S = (cid:18) ǫǫ (cid:19) . Proof.
See [BH2] between Definition 3.1.3 and Convention 3.1.4. (cid:3) The Evaluation Map
In this section, we review the Bao-Honda evaluation map from [BH1, BH2]. It isused in Section 7 to cut out 1-dimensional families of holomorphic curves in high-dimensional moduli spaces.Throughout this section, let Y be a smooth 3-manifold, let λ be a non-degenerate, L -supersimple contact form on Y , and let R λ be the Reeb vector field of λ on Y . AllReeb orbits and orbit sets under consideration in this section are tacitly assumedto have (total) action less than L .3.1. The asymptotic operator.
Let γ be a Reeb orbit of λ with period 2 πa ,where a ∈ Z + . Recall from [BH1] that there is an asymptotic operator A γ : W , ( R / πa Z , R ) → L ( R / πa Z , R )defined by A γ = − j ∂∂t − S ( t ) . Here, j is the standard complex structure on R and S ( t ) is a loop of 2 × A γ have dimension at most 2.If γ is negative hyperbolic, then every eigenspace of A γ has real dimension 2, and ifwe label the eigenvalues so that · · · ≤ λ − ≤ λ − < < λ ≤ λ ≤ · · · , then we can choose the corresponding eigenfunctions { f i ( t ) } i ∈ Z \{ } so that theyform an orthonormal basis for L ( R / πa Z , R ). If γ is positive hyperbolic, then theeigenvalues can be labeled so that · · · ≤ λ − ≤ λ − < λ − < < λ < λ ≤ λ ≤ · · · , the eigenspaces for λ ± have real dimension 1, and all other eigenspaces have realdimension 2. The corresponding eigenfunctions can again be chosen to be an or-thonormal basis for L ( R / πa Z , R ). Next, we recall from [BH1, Section 6] some properties of the above-mentionedeigenfunctions. Let u be a punctured holomorphic curve in R × Y and suppose that u has a negative end at γ . If we choose a trivialization τ of the contact structure ξ = Ker( λ ) over γ , then the negative end of u in question can be written in cylindricalcoordinates ( s, t ) ∈ ( −∞ , − R ] × ( R / πa Z ), R ≫
0, as the graph of a function e u ( s, t ),i.e., we have u ( s, t ) = ( s, t, e u ( s, t )) . In the L -supersimple setting, the function η admits a Fourier-type expansion e u ( s, t ) = ∞ X i =1 c i e λ i s f i ( t ) , where the c i are real constants. Similarly, a positive end of u asymptotic to γ canbe written as the graph of a function that has a Fourier-type expansion in negative-indexed eigenfunctions of A γ .Let ρ τ ( f i ) denote the winding number of the eigenfunction f i of A γ . Recall thefollowing facts from [Hu1, Lemma 6.4]. Fact 3.1.1. (1) If i ≤ j , then ρ τ ( f i ) ≤ ρ τ ( f j ) . (2) We have ρ τ ( f ) = (cid:24) CZ τ ( γ )2 (cid:25) and ρ τ ( f − ) = (cid:22) CZ τ ( γ )2 (cid:23) . Definition 3.1.2. [HT2, Definition 3.2] A J -holomorphic curve u has non-degen-erate ends if at each negative (resp. positive) end, the coefficient c (resp. c − ) inthe Fourier-type expansion of u is non-zero. Definition 3.1.3. [HT2, Definition 3.8] A J -holomorphic curve u has non-overla-pping ends if it has non-degenerate ends and and following holds. For every pairof negative (reps. positive) ends asymptotic to covers γ a and γ a of the same Reeborbit γ where the smallest positive (reps. largest negative) eigenvalues of A γ a and A γ a coincide, the leading coefficients in the Fourier-type expansions of u do notdiffer by a factor of a d th root of unity, where d = gcd( a , a ).3.2. The evaluation map.
We now recall the definition of the evaluation map in[BH1] and review some of the map’s properties. Throughout, we use M J ( α , β ) todenote a transversely cut out moduli space of J -holomorphic curves in R × Y withpositive orbit set α and negative orbit set β . Definition 3.2.1.
Let u : ( −∞ , − R ] × ( R / πa Z ) → R × Y be a J -holomorphic half-cylinder, and assume that u is asymptotic to a Reeb orbit γ at the negative end. OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 15
Write u in cylindrical coordinates as the graph of a function e u ( s, t ), and write theFourier-type expansion of e u as e u ( s, t ) = ∞ X i =1 c i e λ i s f i ( t ) , where the c i are real constants. Then the evaluation map on u is defined asev( u ) = ( c , c ) . Definition 3.2.2.
Let u ∈ M J ( α , β ) and label the negative ends of u by 1 , , . . . , m .The evaluation map at the i th negative end of u is defined asev i : M J ( α , β ) → R u ( c , c ) , where we have identified u with a half-cylinder near the i th end and used the eval-uation map from Definition 3.2.1. Definition 3.2.3.
Assume the setup in Definition 3.2.2 and let I = { i , . . . , i p } bea subset of { , . . . , m } . At the l th negative end of u , write the Fourier-type series as X i> c l,i e λ i s f i ( t ) . The evaluation map at the ends specified by I is defined asev I : M J ( α , β ) → Y i ∈ I R u (ev i ( u )) i ∈ I . The total order of the map ev I is defined to be 2 | I | .If all of the ends labeled by I have odd multiplicity, the asymptotic eigenspacesat those ends all have multiplicity 2. If the relevant asymptotic operators are alsocomplex-linear, we can view the eigenspaces as complex vector spaces and takecomplex coefficients with the evaluation map. This modification is used extensivelyin Section 7. We can also define higher-order evaluation maps ev ki : M J ( α , β ) → R k ,where i ∈ I and k >
2, by ev ki ( u ) = ( c , . . . , c k ), where we have identified u with ahalf-cylinder near the i th end. This modification will be used in Section 8. Remark . If ev ki ( u ) = ( c , . . . , c k ) and v is the curve obtained by translating u by a in the R -direction, then ev ki ( v ) = ( e − λ a c , . . . , e − λ k a c k ). Fact 3.2.5.
The above evaluation maps are all smooth.
Transversality for the evaluation map.
One of the key advantages of the L -supersimple setting exploited in [BH1, BH2] is the abundance of transversalityfor the evaluation map at ends of punctured holomorphic curves. We now brieflyjustify why similar transversality results hold for evaluation maps on multiple ends,beginning with a mild generalization of [BH1, Theorem 6.0.4]. Theorem 3.3.1.
Let J be generic, and let M J ( α , β ) be a transversely cut out mod-uli space of curves in R × Y with Fredholm index k . Let K ⊂ M J ( α , β ) be compactand let Z ⊂ R k − be a submanifold. Then there exists a generic J ′ , arbitrarily closeto J , and a compact subset K ′ ⊂ M J ′ ( α , β ) , arbitrarily close to K , such that theevaluation map ev I on K ′ is transverse to Z .Proof. Let u ∈ M J ( α , β ). The perturbation constructed in the proof of [BH1,Theorem 6.0.4] is supported over a single end, so we can repeat the constructionover the relevant ends of u separately. (cid:3) Proposition 3.3.2.
Let J be generic, let M be a transversely cut out moduli spaceof punctured holomorphic curves in R × Y , and consider the evaluation map ev I on M . The set of u ∈ M such that ev I ( u ) intersects a coordinate hyperplane { x i = 0 } in R | I | has codimension in M .Proof. By Theorem 3.3.1, we can make ev I transverse to the coordinate plane { x i =0 } if J is generic. (cid:3) Remark . Analogues of Theorem 3.3.1 and Proposition 3.3.2 hold for higher-order evaluation maps ev ki : M → R k .4. Index Calculations
We prove Theorem 1.2.5 in this section. We use it in Section 5 to classify de-generations of 1-dimensional families of J -holomorphic curves in the L -supersimplesetting. The proof involves strengthening the various inequalities involved in Hutch-ings’ proof of the inequality (1.2.1). The relationship between ∆( u ) and the ECHpartition conditions is partially expressed in the following result, whose proof followseasily from the derivation of the formulas for ∆( u, γ ) in Section 4.3. The subsequentcorollary is a crucial ingredient to our arguments in Section 8. Proposition 4.0.1. If J is generic, u is a somewhere injective J -holomorphic curvein a symplectization, and ∆( u ) = 0 , then u satisfies the ECH partition conditions. Corollary 4.0.2. If J is generic, u is a somewhere injective J -holomorphic curvein a symplectization, and I ( u ) = ind( u ) , then u is embedded and satisfies the ECHpartition conditions. OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 17
Ingredients in the proof of Hutchings’ inequality.
We begin by fixingnotation and collating the results used in Hutchings’ proof of (1.2.1). Our notationclosely, but not exactly, matches that used in [Hu1]. We only analyze negative endsin this discussion; the analysis for positive ends is similar.
Notation 4.1.1.
Let u be a somewhere injective J -holomorphic curve in a sym-plectization. Let β be the negative orbit set of u and fix an embedded Reeb orbit β ∈ Γ − ( u ). Let m be the multiplicity of β in the orbit set β , let n be the numberof negative ends of u that are asymptotic to (covers of) β , and let q , q , . . . , q n bethe multiplicities of these negative ends. Let ζ , ζ , . . . , ζ n be the braids determinedby these negative ends, and let ζ denote the union of the braids ζ , . . . , ζ n . Let τ denote a trivialization of ξ over β . Let µ τ ( β k ) denote the Conley-Zehnder index ofthe k -fold cover of β . For each braid ζ i , let ρ τ ( ζ i ) denote the winding number of ζ i around β it the trivialization τ , let w τ ( ζ i ) the asymptotic writhe of ζ i with respectto τ , and let ℓ τ ( ζ i , ζ j ) denote the linking number of the braids ζ i and ζ j with respectto τ .With the above notation, the five ingredients in the proof of Hutchings’ inequalityare the following.(4.1.1) ρ τ ( ζ i ) ≥ (cid:24) µ τ ( β q i )2 (cid:25) (4.1.2) w τ ( ζ i ) ≥ ( q i − ρ τ ( ζ i )(4.1.3) ℓ τ ( ζ i , ζ j ) ≥ min( q i ρ τ ( ζ j ) , q j τ ( ζ i ))(4.1.4) w τ ( ζ ) ≥ n X i =1 ρ τ ( ζ i )( q i −
1) + X i = j min( q i ρ τ ( ζ j ) , q j ρ τ ( ζ i ))(4.1.5) n X i =1 ρ τ ( ζ i )( q i −
1) + X i = j min( q i ρ τ ( ζ j ) , q j ρ τ ( ζ i )) ≥ m X k =1 µ τ ( β k ) − n X i =1 µ τ ( β q i )4.2. The writhe bound.
We first use Proposition 3.3.2 to improve (4.1.1) slightly.Our proof of the next Lemma closely follows the one given for [Hu1, Lemma 6.6]
Lemma 4.2.1.
Let u be a somewhere injective curve in R × Y . Assume that thecontact form on Y is L -supersimple and that β is an orbit in the negative orbit set β of u . If J is generic, then (4.2.1) ρ τ ( ζ i ) ≥ (cid:24) µ τ ( β q i )2 (cid:25) . Equality holds if A ( β ) < L and ind( u ) = 1 .Proof. The inequality is proved in [Hu1, Lemma 6.6]. So assume that A ( β ) < L and ind( u ) = 1. Let ( s, t ) be cylindrical coordinates over the relevant negative endof u and take an asymptotic expansion u ( s, t ) = s, t, ∞ X i =1 c i e − λ i s f i ( t ) ! of u for s ≪
0, as in Section 3. Since u has Fredholm index 1 and J is generic,Proposition 3.3.2 implies that c = 0. Thus, ρ τ ( ζ i ) equals the winding number of f around β in the trivialization τ . By computations in [HWZ, Section 3], said windingnumber is precisely (cid:24) µ τ ( γ q i )2 (cid:25) . (cid:3) We now turn our attention to (4.1.2). Our proof of the next Lemma closely followsthe one given for [Hu1, Lemma 6.7].
Lemma 4.2.2.
Let u be a somewhere injective curve in R × Y . Assume that thecontact form on Y is L -supersimple and that β is an orbit in the negative orbit set β of u . If J is generic, then (4.2.2) w τ ( ζ i ) ≥ ρ τ ( ζ i )( q i −
1) + ( d i − , where d i = gcd( q i , ρ τ ( ζ i )) . Equality holds if A ( β ) < L and ind( u ) = 1 .Proof. Direct calculation shows that equality holds when ρ τ ( ζ i ) = q i . We proceedby complete induction on ρ τ ( ζ i ). First assume that d i = 1. (This is true when ρ τ ( ζ i ) = 1, but the more general result is useful in the inductive step.) The proofof [Hu1, Lemma 6.7] shows that w τ ( ζ i ) = ρ τ ( ζ i )( q i − ρ τ ( ζ i ) > d i >
1. The same proof shows that ζ i is the cabling of a braid ζ ′ i with q i /d i strands and winding number ρ τ ( ζ i ) /d i by a braid ζ ′′ i with d i strands and windingnumber ρ τ ( ζ ′′ i ) ≥ ρ τ ( ζ i ). Write ρ τ ( ζ ′′ i ) = ρ τ ( ζ i ) + k and d ′ i = gcd( ρ τ ( ζ ′′ i ) , d i ). Weknow inductively that w τ ( ζ ′ i ) = ρ τ ( ζ i ) d i (cid:18) q i d i − (cid:19) and w τ ( ζ ′′ i ) ≥ ( ρ τ ( ζ i ) + k )( d i −
1) + ( d ′ i − , and thus w τ ( ζ i ) = d i w τ ( ζ ′ i ) + w τ ( ζ ′′ i ) ≥ ρ τ ( ζ i )( q i − d i ) + ( ρ τ ( ζ i ) + k )( d i −
1) + ( d ′ i − ρ τ ( ζ i )( q i −
1) + k ( d i −
1) + ( d ′ i − . If k = 0, then d ′ i = d i and w τ ( ζ i ) = ρ τ ( ζ i )( q i −
1) + ( d i − . OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 19 If k >
0, then w τ ( ζ i ) = ρ τ ( ζ i )( q i −
1) + ( d i −
1) + ( k − d i −
1) + ( d ′ i − ≥ ρ τ ( ζ i )( q i −
1) + ( d i − . Now assume that A ( β ) < L and ind( u ) = 1. Equality is also proved by inductionon ρ τ ( ζ i ), and the case ρ τ ( ζ i ) = 1 is handled in the same way (i.e., by provingthe result when d i = 1). So assume ρ τ ( ζ i ) > d i >
1. By Proposition 3.3.2,either ρ τ ( ζ ′′ i ) = ρ τ ( ζ i ) or ρ τ ( ζ ′′ i ) = ρ τ ( ζ i ) + 1. The former is the case k = 0 above,where d ′ i = d i . Here, we know inductively that w τ ( ζ ′′ i ) = ρ τ ( ζ i )( d i −
1) + ( d i −
1) =( ρ τ ( ζ i ) + 1)( d i − w τ ( ζ i ) = d i w τ ( ζ ′ i ) + w τ ( ζ ′′ i )= ρ τ ( ζ i )( q i − d i ) + ( ρ τ ( ζ i ) + 1)( d i − ρ τ ( ζ i )( q i −
1) + ( d i − . The latter is the case k = 1 above, where d ′ i = 1. Here, we again know inductivelythat w τ ( ζ ′′ i ) = ( ρ τ ( ζ i ) + 1)( d i − (cid:3) Linking numbers.
Now we turn out attention to (4.1.3). If J is a genericalmost complex structure on R × Y , any Fredholm index 1, simple curve u in R × Y has non-degenerate and non-overlapping ends. In particular, the proof of [Hu1,Lemma 6.9] implies the following strengthened result. Lemma 4.3.1.
Let u be a somewhere injective curve in R × Y . Assume that thecontact form on Y is L -supersimple and that β is an orbit in the negative orbit set β of u . If J is generic, then (4.3.1) ℓ τ ( ζ i , ζ j ) ≥ min( q i ρ τ ( ζ j ) , q j ρ τ ( ζ i )) . Equality holds if A ( β ) < L and ind( u ) = 1 . Now we put the preceding lemmas together to derive stronger versions of (4.1.4)and (4.1.5) in the L -supersimple setting; these new inequalities are implicit in workof Hutchings [Hu2]. Lemma 4.3.2.
Let u be a somewhere injective curve in R × Y . Assume that thecontact form on Y is L -supersimple and that β is a negative hyperbolic orbit in thenegative orbit set β of u . As in Notation 4.1.1, suppose that u has negative ends ofmultiplicity q , . . . , q n at β . In addition, order the ends of u at β so that q , . . . , q k are the ends with odd multiplicity, ordered so that q ≥ q ≥ · · · ≥ q k and so that q k +1 , q k +2 , . . . q n are the ends with even multiplicity. Then (4.3.2) w τ ( ζ ) ≥ m X i =1 µ τ ( β i ) − n X i =1 µ τ ( β q i ) + k X i =1 (cid:18) q i −
12 + i − (cid:19) + n X i = k +1 (cid:16) q i − (cid:17) . Equality holds if A ( β ) < L and ind( u ) = 1 . Proof.
Choose the trivialization τ so that µ τ ( β ) = 1 and set ρ i = (cid:24) µ τ ( β q i )2 (cid:25) . Set d i = gcd( q i , ρ τ ( ζ i )) and note that by (4.2.1), ρ τ ( ζ i )( q i −
1) + ( d i − ≥ ( ρ i ( q i − , i = 1 , , . . . , kρ i ( q i −
1) + (cid:0) q i − (cid:1) , i = k + 1 , k + 2 , . . . , n . The above inequality, combined with (4.2.1), (4.2.2), and (4.3.1), implies that w τ ( ζ ) = n X i =1 w τ ( ζ i ) + X i = j ℓ τ ( ζ i , ζ j ) ≥ n X i =1 [ ρ τ ( ζ i )( q i −
1) + ( d i − X i = j min( q i ρ τ ( ζ j ) , q j ρ τ ( ζ i )) ≥ n X i =1 ρ i ( q i −
1) + X i = j min( q i ρ j , q j ρ i ) + n X i = k +1 (cid:16) q i − (cid:17) . By a computation in the proof of [Hu2, Lemma 4.19], we have n X i =1 ρ i ( q i −
1) + X i = j min( q i ρ j , q j ρ i ) = m X i =1 µ τ ( β i ) − n X i =1 µ τ ( β q i ) + k X i =1 (cid:18) q i −
12 + i − (cid:19) , and the result follows. (cid:3) Lemma 4.3.3.
Let u be a somewhere injective curve in R × Y . Assume that thecontact form on Y is L -supersimple and that β is a positive hyperbolic orbit in thenegative orbit set β of u . As in Notation 4.1.1, suppose that u has negative ends ofmultiplicity q , . . . , q n at β . Then (4.3.3) w τ ( ζ ) ≥ m X i =1 µ τ ( β i ) − n X i =1 µ τ ( β q i ) + n X i =1 ( q i − . Equality holds if A ( β ) < L and ind( u ) = 1 .Proof. Choose the trivialization τ so that µ τ ( β ) = 0 and set d i = gcd( q i , ρ τ ( ζ i )). By(4.2.1), we have ρ τ ( ζ i ) ≥
0. There are two cases, ρ τ ( ζ i ) = 0 and ρ τ ( ζ i ) >
0, and inboth cases we have w τ ( ζ i ) ≥ q i −
1. Hence, with this choice of τ , the inequalities(4.2.2) and (4.3.1) imply that w τ ( ζ ) = n X i =1 w τ ( ζ i ) + X i = j ℓ τ ( ζ i , ζ j ) ≥ n X i =1 ( q i −
1) + X i = j min( q i ρ τ ( ζ j ) , q j ρ τ ( ζ i )) OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 21 ≥ n X i =1 ( q i − τ , we have m X i =1 µ τ ( β i ) − n X i =1 µ τ ( β q i ) = 0 , and the result follows. (cid:3) Proof of the inequality.
The proofs of Lemmas 4.3.2 and 4.3.3 show thatwe have w τ ( ζ ) ≥ m X i =1 µ τ ( β i ) − n X i =1 µ τ ( β q i ) + ∆( u, β )and that equality holds if A ( β ) < L and ind( u ) = 1. If u has a positive end at α ,computations similar to those in Lemmas 4.3.2 and 4.3.3 show that w τ ( ζ ) ≤ m X i =1 µ τ ( α i ) − n X i =1 µ τ ( α q i ) − ∆( u, α )and that equality holds if A ( α ) < L and ind( u ) = 1. Thus, if we set w τ ( u ) = X positive ends w τ ( ζ ) − X negative ends w τ ( ζ ) , we have w τ ( u ) ≤ µ Iτ ( α , β ) − µ τ ( α , β ) − ∆( u ) , and equality holds if A ( α ) , A ( β ) < L and ind( u ) = 1. Thus, ∆( u ) measures howmuch the curve u violates the ECH partition conditions at its ends. Proof of Theorem 1.2.5.
Recall the relative adjunction formula for somewhere in-jective curves:(4.4.1) c ( u ∗ ξ, τ ) = χ ( ˙Σ) + Q τ ( u ) + w τ ( u ) − δ ( u ) . By the above formula for the asymptotic writhe, we have I ( u ) = c ( u ∗ ξ, τ ) + Q τ ( u ) + µ Iτ ( α , β )= − χ ( ˙Σ) + 2 c ( u ∗ ξ, τ ) − w τ ( u ) + 2 δ ( u ) + µ Iτ ( α , β ) ≥ − χ ( ˙Σ) + 2 c ( u ∗ ξ, τ ) + µ τ ( α ) − µ τ ( β ) + 2 δ ( u ) + ∆( u )= ind( u ) + 2 δ ( u ) + ∆( u ) . Equality clearly holds if A ( α ) < L and ind( u ) = 1. (cid:3) Remark . The inequality (1.2.2) also holds for curves in exact symplectic cobor-disms; equality holds if A ( α ) , A ( β ) < L and ind( u ) = 0. In general, equality holdsif A ( α ) , A ( β ) < L and u has non-degenerate, non-overlapping ends. Degenerations in Cobordisms
In this section, we prove Theorem 1.3.2. We first recall its setup. Let ( Y ± , λ ± ) be L -supersimple contact 3-manifolds and let ( X, λ ) be an exact symplectic cobordismfrom ( Y + , λ + ) to ( Y − , λ − ). Let J be a generic, L -simple, admissible almost complexstructure on the completion ( b X, b λ ) that restricts to L -simple, admissible almostcomplex structures J + and J − on the ends [0 , ∞ ) × Y + and ( −∞ , × Y − , respec-tively, of b X . Let α be a generator of ECC L ( Y + , λ + , J + ) and let β be a generatorof ECC L ( Y − , λ − , J − ). Consider the moduli space M , X ( α , β ) and let M , X ( α , β )denote its SFT compactification as described in [BEHWZ]. As before, we denotean SFT building in ∂ M , X ( α , β ) by [ u − a ] ∪ · · · ∪ [ u − ] ∪ u ∪ [ u ] ∪ · · · [ u b ], where a and b are positive integers, the levels go from bottom to top as we read from left toright, the levels with negative indices are in ( R × Y − ) / R , the level u is in b X , andthe levels with positive indices are in ( R × Y + ) / R .Let [ u − a ] ∪ · · · ∪ [ u − ] ∪ u ∪ [ u ] ∪ · · · [ u b ] be a building in ∂ M , X ( α , β ). ByLemma 2.5.2, each level of the building has non-negative Fredholm index, and thesymplectization levels have positive Fredholm index. Since ind is additive and thetotal Fredholm index of the building is 1, there must be only one symplectizationlevel, which has Fredholm index 1, and the cobordism level u must have Fredholmindex 0.5.1. Multiply covered curves.
We begin with a classification of multiply coveredcurves in b X with non-positive ECH index in the L -supersimple setting. Lemma 5.1.1.
Let u be a J -holomorphic curve in b X with ind( u ) = 0 and connectedimage. The curve u has negative ECH index if and only if it is an unbranched,disconnected cover of a J -holomorphic plane in with ECH and Fredholm index . Inthat case, I ( u ) = − (cid:18) d (cid:19) , where d is the degree of the covering.Proof. Suppose that I ( u ) <
0. By the ECH index inequality (1.2.1), somewhereinjective curves in cobordisms have non-negative ECH index, so u must be a d -foldmultiple cover of a somewhere injective curve v : ˙Σ ′ → b X with ind( v ) ≥ d ≥ I ( u ) ≥ d · I ( v ) + (cid:18) d (cid:19) (2 g ( ˙Σ ′ ) − v ) + h ( v ))from [Hu2], where h ( v ) is the number of ends of v at hyperbolic orbits. Since h ( v ) ≥ v ) ≥
0, the only way for I ( u ) to be negative is if g ( ˙Σ ′ ) = 0 andind( v ) + h ( v ) = 1. Since ind( u ) = 0, Lemma 2.5.2 implies that u is an unbranchedcover of v and ind( v ) = 0. Hence h ( v ) = 1. It follows that u is an unbranched, OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 23 disconnected cover of a plane v . Let γ be the orbit at the positive end of v . If wechoose the trivialization τ of γ ∗ ξ such that c ( v ∗ ξ, τ ) = 0 we see that 0 = ind( v ) = µ τ ( γ ) −
1, so µ τ ( γ ) = 1. Thus, I ( v ) = Q τ ( v ) + 1. If I ( v ) ≥
1, then Q τ ( v ) ≥
0, andan easy computation shows that I ( u ) >
0. Thus, I ( v ) = 0.Suppose there is a component ˙Σ of the domain of u such that ˙Σ → ˙Σ ′ is an m -fold (unbranched) covering with m ≥
2. Then m = χ ( ˙Σ) = 2 − g ( ˙Σ) − m , so g ( ˙Σ) = 1 − m <
0, which is impossible. It follows that every component of thedomain of u maps diffeomorphically onto ˙Σ ′ .Conversely, suppose that u : ˙Σ → b X is such a cover of a plane v : ˙Σ ′ → b X witha positive end at a hyperbolic orbit γ and such that ind( v ) = I ( v ) = 0. As above,we choose the trivialization τ of γ ∗ ξ such that c ( v ∗ ξ, τ ) = 0. Then 0 = ind( v ) = µ τ ( γ ) −
1, so µ τ ( γ ) = 1. Thus, 0 = I ( v ) = Q τ ( v ) + 1, so Q τ ( v ) = −
1. Therelative self-intersection number Q τ is quadratic under taking multiple covers (seethe discussion in [Hu2, Section 3.5]), so Q τ ( u ) = − d and I ( u ) = − d + d X i =1 i = − (cid:18) d (cid:19) , as desired. (cid:3) Definition 5.1.2.
We refer to an unbranched, disconnected, negative-index coverof a plane as in Lemma 5.1.1 as a degenerate cover of said plane.
Lemma 5.1.3.
Let γ be an orbit set with A ( β ) < A ( γ ) . If u ∈ M , X ( γ , β ) ismultiply covered, then u is an immersion and the underlying somewhere injectivecurve is a J -holomorphic cylinder with ECH index and no negative ends.Proof. Assume first that u is a d -fold cover, d ≥
2, of a somewhere injectivecurve v : ˙Σ ′ → b X . Since ind( u ) = 0, Lemma 2.5.2 implies that u is necessarilyan unbranched cover of v . Since β is an ECH generator, it follows immediatelythat u has no negative ends. Since I ( u ) = 0, the inequality (5.1.1) implies that2 g ( ˙Σ ′ ) − h ( v ) ≤
0. Thus, h ( v ) = 1 or 2. If h ( v ) = 1, then v is a plane and, by thearguments in the proof of Lemma 5.1.1, either I ( u ) > I ( u ) <
0. It follows that h ( v ) = 2, v is a cylinder, and I ( v ) = 0. Thus, v is embedded, so u is an immersion.Clearly v has no negative ends. (cid:3) Canceling degenerations.
Now we prove a sequence of lemmas that elimi-nates various cases in our analysis of ∂ M , X ( α , β ) by showing that certain types ofbuildings occur in canceling pairs. Given a two-level building in the boundary, wesay that the negative orbit set of the top level is the intermediate orbit set ofthe building. Lemma 5.2.1.
The symplectization level of a building in ∂ M , X ( α , β ) is somewhereinjective. Proof.
Without loss of generality, assume that the building is of the form u ∪ [ u ].If [ u ] is multiply covered, Lemma 2.5.2 implies that it is a branched cover of atrivial cylinder in ( R × Y + ) / R , contradicting the assumption that its positive orbitset α is a generator of the ECH chain complex for ( Y + , λ + ). (cid:3) Lemma 5.2.2.
The top level of a building in ∂ M , X ( α , β ) has non-negative ECHindex.Proof. First assume that the building is of the form u ∪ [ u ], so that ind( u ) = 1.Then [ u ] is somewhere injective by Lemma 5.2.1, so I ( u ) ≥ u − ] ∪ u , so that ind( u ) = 0. If I ( u ) <
0, then u must contain a degenerate cover of a plane by Lemma 5.1.1.The underlying embedded plane cannot have a negative end since b X is exact; seethe proof of [BH1, Lemma 3.4.2]. Hence, α must contain a Reeb orbit with mul-tiplicity greater than 1, which contradicts the assumption that it is a generator of ECC ( Y + , λ + , J + ). (cid:3) Lemma 5.2.3.
The count of buildings in ∂ M , X ( α , β ) where the bottom level hasnon-negative ECH index and such that the intermediate orbit set γ has at least oneorbit of multiplicity greater than is even.Proof. First assume that the building is of the form u ∪ [ u ]. Then [ u ] is some-where injective by Lemma 5.2.1 and I ( u ) ≥ I ( u ) ≥ I ( u ) + I ( u ) = 1, we see that in fact I ( u ) = 1 and I ( u ) = 0. ByCorollary 4.0.2, [ u ] satisfies the ECH partition conditions, and hence so does u since its negative orbit set β is a generator of ECC ( Y − , λ − , J − ). But then the mul-tiply covered components of u are unbranched covers of cylinders with no negativeends by Lemma 5.1.3, and the count of such buildings is even.Now assume that the building is of the form [ u − ] ∪ u . Then [ u − ] is somewhereinjective by Lemma 5.2.1, so by (1.2.1), I ( u − ) ≥ ind( u − ) = 1. Since I ( u ) ≥ I ( u − ) = ind( u − ) = 1and I ( u ) = 0. By Corollary 4.0.2, [ u − ] satisfies the ECH partition conditions. If u is multiply covered, then its multiply covered components are unbranched coversof cylinders with no negative ends by Lemma 5.1.3. But then α cannot be a gen-erator of ECC ( Y + , λ + , J + ), and we have reached a contradiction. Hence u is alsosomewhere injective. Since γ contains a hyperbolic orbit with multiplicity greaterthan 1, u must either have multiple negative ends with multiplicity 1 asymptoticto the same positive hyperbolic orbit or at least one negative end with multiplicity2 asymptotic to a double cover of a negative hyperbolic orbit. In either case, thecount of such buildings is even. (cid:3) Proof of Theorem 1.3.2.
Note that the building must be of the form u ∪ [ u ]. Let γ denote the intermediate orbit set, and let n γ denote the multiplicity of the orbit OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 25 γ in γ . By Lemma 5.1.1, u must contain a multiply covered component that isa degenerate cover of a plane. Let ^ Γ + ( u ) denote the set of orbits γ in γ suchthat u contains a degenerate cover of a plane whose positive end is at γ . For each γ ∈ ^ Γ + ( u ), let m γ denote the multiplicity of the covering of the plane with itspositive end at γ . By Theorem 1.2.5,(5.2.1) I ( u ) ≥ X γ ∈ ^ Γ + ( u ) (cid:18) m γ (cid:19) and(5.2.2) I ( u ) ≥ − X γ ∈ ^ Γ + ( u ) (cid:18) m γ (cid:19) . Since I ( u ) + I ( u ) = 1, both inequalities must in fact be equalities. Thus, u satisfies the ECH partition conditions except for degenerate covers of planes atorbits in ^ Γ + ( u ).We claim that buildings where u contains other multiply covered componentsoccur in canceling pairs. Any multiple covers besides the degenerate ones have non-negative ECH index. Covers with ECH index 0 are unbranched covers of cylinderswith no negative ends satisfying the partition conditions, and buildings containingsuch curves occur in canceling pairs. There are no multiply covered components of u with positive ECH index, as then the inequality (5.2.2) is strict and I ( u )+ I ( u ) > γ ∈ ^ Γ + ( u ) with m γ < n γ occur in canceling pairs. So assume that such a γ exists. If u has a non-planarcomponent with a positive end asymptotic to γ k for k odd or k ≥ I ( u ) + I ( u ) >
1. The buildings where u hasa non-planar component with a positive end asymptotic to γ occur in cancelingpairs.Finally, we claim that every Reeb orbit γ in Γ + ( u ) \ ^ Γ + ( u ) has multiplicity 1or else the building is part of a canceling pair. So let u ∪ [ u ] be a building suchthat some γ in Γ + ( u ) \ ^ Γ + ( u ) has multiplicity greater than 1. By Theorem 1.2.5,the negative ends of [ u ] at covers of γ satisfy the ECH partition conditions. If γ ispositive hyperbolic, there are at least two negative ends of [ u ] of multiplicity 1 at γ , and such buildings occur in canceling pairs. If γ is negative hyperbolic, there isat least one negative end of [ u ] at γ , and such buildings again occur in cancelingpairs. (cid:3) Obstruction Bundle Gluing
In this section, we set up the gluing machinery in preparation for the proof ofTheorem 1.4.5 in Section 7. We first review the prototypical gluing problem from
Section 1. Recall that ( Y + , λ + ) is a smooth 3-manifold with an L -supersimple con-tact form and u : ˙Σ → R × Y + is an embedded J -holomorphic curve with Fredholmindex 1 such that(1) the positive orbit set of u is an ECH generator α with A ( α ) < L ;(2) the negative ends of u are asymptotic to an orbit set β in which each Reeborbit has multiplicity 1 except for a single negative hyperbolic orbit β ;(3) u has n negative ends at β , each with multiplicity 1;(4) I ( u ) = 1 + (cid:0) n (cid:1) .Recall that, for each n ≥
3, we set M n = M (1 , , . . . , | , , . . . , , n positive ends of multiplicity 1, n − M n to the curve u above. Note that each branched cover in M n has total branching index 2 n − Proposition 6.0.1. [Ro, Proposition 5.2.2]
Let ( Y, λ ) be a non-degenerate contact -manifold and let J be a generic R -invariant almost complex structure on R × Y .Let α be a negative hyperbolic Reeb orbit of λ and let u be a branched cover of thetrivial cylinder R × α in R × Y . If u has k branch points, counted with multiplicity,then ind( u ) = k and dim Coker D Nu = k . In particular, the obstruction bundle O → [ R, ∞ ) × ( M n / R ) has rank n − .Proof. The computation of ind( u ) follows immediately from Lemma 2.5.2. From[We, Theorem 3], we know that dim Ker D Nu = dim Ker D∂ J − k = 0. From thecomputation immediately preceding that theorem, we also know that ind( D Nu ) =ind( u ) − k = − k , so dim Coker D Nu = k . (cid:3) Notation 6.0.2.
For any two subsets { p , . . . , p n } and { q , . . . , q n − } of C , wherethe p i and q j are pairwise distinct, we set A ( z ) = n Y i =1 ( z − p i ) , A ( z ) = n − Y i =1 ( z − p i ) , B ( z ) = n − Y i =2 ( z − q i ) ,A k ( z ) = n Y i =1 i = k ( z − p i ) , A k ( z ) = n − Y i =1 i = k ( z − p i ) , B k ( z ) = n − Y i =2 i = k ( z − q i ) . Note that we suppress the dependence on n for the functions considered above.6.1. Parametrization of the moduli space.
We parametrize the reduced mod-uli space M n / R by choosing a smooth section of the bundle M n → M n / R inthe following way. Curves in M n have genus 0, so the domain for each map isa punctured Riemann sphere b C \ ( P + ∪ P − ), where P + = { p , p , . . . , p n } and P − = { q , q , . . . , q n − } are the (disjoint) sets of positive and negative puncturesand q is the multiplicity 3 negative puncture. View b C as C ∪ {∞} , fix the positivepunctures p n − and p n in C , and fix the negative puncture q to be the point at OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 27 infinity. The other punctures p , . . . , p n − , q , . . . , q n − are free to move in C . Thenthe data consisting of the punctures p , . . . , p n − , q , . . . , q n − ∈ C and θ ∈ R / π Z are sent to the map u : C \ (cid:0) P + ∪ P − (cid:1) → C ∗ z e iθ B ( z ) A ( z ) . Roughly speaking, changing the parameter θ simultaneously rotates the branchpoints of u in the S -factor of the image cylinder.The asymptotic marker τ i ∈ S at each puncture is determined as follows. Foreach positive puncture p i , there is an ǫ > f ( t ),0 < t < ǫ , such that lim t → + f ( t ) = 0 and u ( p i + f ( t )) = e /t . Then(6.1.1) τ i = lim t → + f ( t ) | f ( t ) | = e iθ B ( p i ) A i ( p i ) (cid:12)(cid:12)(cid:12)(cid:12) B ( p i ) A i ( p i ) (cid:12)(cid:12)(cid:12)(cid:12) − . For each negative puncture q j , j = 2 , . . . , n −
2, there is an ǫ > f ( t ), 0 < t < ǫ , such that f ( t ) → t → + and u ( q j + f ( t )) = e − /t . Then τ − j = lim t → + f ( t ) | f ( t ) | = e − iθ A ( q j ) B j ( q j ) (cid:12)(cid:12)(cid:12)(cid:12) A ( q j ) B j ( q j ) (cid:12)(cid:12)(cid:12)(cid:12) − . For q , there is an ǫ > f ( t ), 0 < t < ǫ , such that f ( t ) → t → + and u ( f ( t ) − ) = e − /t . Then τ − = lim t → + f ( t ) | f ( t ) | = e − iθ . The obstruction sections.
Recall that Hutchings-Taubes define a linearizedobstruction section that is homotopic to the full obstruction section and whosezero set is much easier to compute [HT1, HT2]. We define a Z + -indexed family ofsections s m , all homotopic to each other and to s , such that s is the analogue ofthe Hutchings-Taubes linearized section in our setting. In Section 7, we show thatthe count of zeros of s and s are the same.Te define s m , let m ∈ Z + and assume that the positive ends of u and the negativeends of u + are labeled so that the i th positive end of [ u ] ∈ M n / R matches up withthe i th negative end of u . We first restrict our attention to the i th positive endof u . Consider the asymptotic expansion of u over its i th negative end, writtenin cylindrical coordinates, and let Π i,m u denote its projection onto the m leadingeigenspaces of the asymptotic operator A β from Section 3.1. Let σ ∈ Coker( D Nu ) Hutchings-Taubes use s to denote the linearized obstruction section. However, since we use a Z + -indexed family of sections, it makes more sense for us to denote the linearized section by s . and let σ i denote the restriction of σ to the i th positive end of u , written in cylindricalcoordinates. Then set s m ( T, u )( σ ) = n X i =1 h Π i,m u , σ i ( T, · ) i . Notation 6.2.1.
We denote the zero set s − m (0) by Z m . We denote the zero set s − (0) of the full obstruction section by Z .6.3. A basis for the cokernel.
We now choose a convenient basis for the spaceCoker( D Nu ), which we identify with Ker( D Nu ) ∗ . If σ ∈ Coker( D Nu ) and τ is a triv-ialization of ξ over β , let ρ τ ( σ i ) denote the asymptotic winding number of σ restricted to the i th positive end of u in the trivialization τ , defined as follows. Onthe i th positive end, write σ = σ i ⊗ ( ds − idt ) in cylindrical coordinates. Then ρ τ ( σ i )is defined as the winding number of the leading asymptotic eigenfunction in theseries expansion of σ i . Recall from [HWZ, Section 3] that, for each positive end of u , we have 2 ρ τ ( σ i ) ≥ µ τ ( β ) and for each negative end, we have 2 ρ τ ( σ i ) ≤ µ τ ( β ). Lemma 6.3.1. If u ∈ M n , where n ≥ , and σ ∈ Coker( D Nu ) , then (cid:12)(cid:12) σ − (0) (cid:12)(cid:12) ≤ n − , where the zeros of σ are counted with multiplicities.Proof. Note that every zero of σ has negative multiplicity and that χ ( ˙Σ) = 4 − n. On the ends of u , write ρ τ ( σ i ) = (cid:24) µ τ ( β )2 (cid:25) + k i for i > ρ τ ( σ j ) = (cid:22) µ τ ( β )2 (cid:23) − k j for j = − , . . . , − ( n − ρ τ ( σ − ) = (cid:22) µ τ ( β )2 (cid:23) − k − . Then, choosing τ so that µ τ ( β ) = 1, we have0 ≥ σ − (0)= χ ( ˙Σ) + n X i =1 ρ τ ( σ i ) − n − X j =1 ρ τ ( σ − j )= 3 − n + n X i =1 k i + n − X j =1 k − j ≥ − n, as claimed. (cid:3) OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 29
Remark . The proof of Lemma 6.3.1 also shows that a cokernel element σ cannot be too degenerate at the ends. More precisely, we have(6.3.1) 0 ≤ (cid:12)(cid:12) σ − (0) (cid:12)(cid:12) + n X i =1 k i + n − X j =1 k − j ≤ n − . Proposition 6.3.3.
There exists a basis σ , σ , . . . , σ n − , σ n − for Ker( D Nu ) ∗ such that, for each i = 1 , , . . . , n − , the projection of { σ i − , σ i } to the lead-ing eigenspace on the j th positive end of u is a basis for that eigenspace if j = i , n − , or n and vanishes otherwise.Proof. Let σ , σ , . . . , σ n − , σ n − be a basis for Ker( D Nu ) ∗ . We give an algorithmto converting this basis into one with the desired properties.First, note that there must be a pair of basis elements whose projections to theleading eigenspace on the positive end labeled 1 are linearly independent. For if not,then row reduction yields a cokernel element σ with k ≥ n −
2, which contradictsRemark 6.3.2. After possibly relabeling the elements of the basis, we may assumethat σ and σ are the above basis elements. By subtracting appropriate multiplesof σ and σ from the other basis elements, we may assume that k ≥ σ i with i = 1 , σ , σ , . . . , σ ℓ − , σ ℓ are such that, for i = 1 , , . . . , ℓ ,the projection of { σ i − , σ i } to the leading eigenspace on the j th positive endof u , is a basis for that eigenspace if j = i and vanishes if 1 ≤ j ≤ ℓ and j = i . Assume also that, for each σ i with i = 2 ℓ + 1 , ℓ + 2 , . . . , n − , n − k j ≥ j = 1 , , . . . , ℓ . There must be a pair of vectors among σ ℓ +1 , σ ℓ +2 , . . . , σ n − , σ n − whose projections to the leading eigenspace on thepositive end labeled ℓ + 1 are linearly independent. For if not, row reduction yieldsa cokernel element σ with P ℓ +1 j =1 k j ≥ n −
2, which contradicts Remark 6.3.2. Afterpossibly relabeling σ ℓ +1 , σ ℓ +2 , . . . , σ n − , σ n − , we may assume that σ ℓ +1 and σ ℓ +2 are the above basis elements. By subtracting appropriate multiples of σ ℓ +1 and σ ℓ +2 from σ ℓ +3 , σ ℓ +4 , . . . , σ n − , σ n − , we may assume that k ℓ +1 ≥ σ i with i = 2 ℓ + 1 , ℓ + 2.After step n − (cid:3) Deformation of the asymptotic operator.
To make our calculations easier,we now replace the elements of Coker( D nu ), which we identify with Ker( D Nu ) ∗ , withmeromorphic (0 , A β for the orbit β . First, define a homotopy of the asymptotic operator by A β ,ν = − j ∂∂t − (1 − ν ) (cid:18) ǫǫ (cid:19) , where ν ∈ [0 , β wherewe identify (2 π, x, y ) ∼ (0 , x, y ), then A β , is equivalent to the operator − i ∂∂t − . This operator is non-degenerate, as is its pullback to odd covers of β ; however,its pullback to an even cover of β is degenerate. More precisely, let λ + ,ν denotethe smallest positive eigenvalue of the pullback A β k ,ν and λ − ,ν the largest negativeeigenvalue. Then both λ + ,ν and λ − ,ν monotonically converge to 0 as ν → β by putting asymptoticweights δ ν = ( δ ν , . . . , δ ν ) on our Sobolev spaces for ( D Nu ) ∗ , where δ ν = (1 − ν ) λ − , + νδ and δ is a sufficiently small positive real number that depends on n . When ν = 1,the operator ( D Nu ) ∗ is complex-linear, and the elements of Ker( D Nu ) ∗ can be writtenas σ i ( s, t ) ⊗ ( ds − idt ) in cylindrical coordinates over the i th positive end, where σ i ( s, t ) satisfies the equation ( σ i ) s − i ( σ i ) t + 12 σ i = 0 . If we set η i ( s, t ) = e − s/ σ i ( s, t ) over such an end, we see that η i is anti-meromorphicin the usual sense. Finally, we single out the real 1-dimensional subspace of the 0-eigenspace of A β k , that corresponds to the λ + , -eigenspace of A β k , by requiring thatthe leading eigenfunction in the asymptotic expansion of η near an even-multiplicityend be a real scalar multiple of the vector in C representing the stable direction of β k for all t . We say that the leading eigenfunction follows the stable direction of β k . Definition 6.4.1.
A meromorphic (0 , η is a replacement for σ ∈ Ker( D Nu ) ∗ if, in cylindrical coordinates ( s, t ) near each puncture, we have η ( s, t ) = e − s/ σ ( s, t ). Remark . The point of using replacements instead of using elements of Ker( D Nu ) ∗ directly is that we can write down explicit expressions for replacements and henceexplicit equations for the zero sets Z m and Z .6.5. The gluing problem.
We now write down a collection of meromorphic (0 , D Nu ) ∗ from Proposition 6.3.3. Notation 6.5.1.
Set Q k ( z ) = A k ( z ) B ( z )for k = 1 , . . . , n − r i = B ( p i ) A i ( p i )for i = 1 , . . . , n , r − = 1 , and r − j = A ( q j ) B j ( q j ) OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 31 for j = 2 , . . . , n −
2. Note that τ i | r i | = e iθ r i for i > τ − j | r − j | = e − iθ r − j for j ≥ τ − | r − | = e − iθ/ r − . Proposition 6.5.2.
The meromorphic (0 , -forms η k ( z ) = Q k ( z ) d ¯ z are replacements for a basis of Ker( D Nu ) ∗ as constructed in Proposition 6.3.3.Proof. Near p i , we have z = p i + τ i e − ˜ s − it , where (˜ s, t ) ∈ [ R, ∞ ) × ( R / π Z ). Hence u ( z ) = e iθ e ˜ s + it B ( p i + e − ˜ s − it ) A i ( p i + e − ˜ s − it ) , so log | u ( z ) | = ˜ s + log (cid:12)(cid:12)(cid:12)(cid:12) B ( p i + e − ˜ s − it ) A i ( p i + e − ˜ s − it ) (cid:12)(cid:12)(cid:12)(cid:12) . Recall that we require u ( s, t ) = ( s, t, e u ( s, t )) in cylindrical coordinates. Thus, wemust change our ˜ s -coordinate to s = ˜ s + log (cid:12)(cid:12)(cid:12)(cid:12) B ( p i + e − ˜ s − it ) A i ( p i + e − ˜ s − it ) (cid:12)(cid:12)(cid:12)(cid:12) . If ˜ s ≫
0, we have s ≈ ˜ s + log (cid:12)(cid:12)(cid:12)(cid:12) B ( p i ) A i ( p i ) (cid:12)(cid:12)(cid:12)(cid:12) = ˜ s + log | r i | , and consequently z ≈ p i + τ i | r i | e − s − it = p i + r i e − s − it near p i . A similar change mustbe made in cylindrical coordinates around the negative punctures q j , j = 2 , . . . , n − k . We claim that each η k has winding number 1 at p k , p n − ,and p n , has winding number 2 at all other p i , has winding number 1 at q , and haswinding number 0 at all other q j .If we change to cylindrical coordinates around p k , we can write z = p k + τ k e − s − it ,( s, t ) ∈ [ R, ∞ ) × ( R / π Z ). Then the first term in the asymptotic expansion of η k inthe coordinates (˜ s, t ) is approximately − e − iθ r k Q k ( p k ) e − ˜ s + it ⊗ ( d ˜ s − idt ) , which has winding number 1. Similarly, the winding number of η k it cylindricalcoordinates on the positive ends at p n − and p n is also 1. The first term in theasymptotic expansion of η k vanishes in cylindrical coordinates around p i , i = k, n − , n , and the winding number at each of those ends is 2.If we change to cylindrical coordinates around q j , j = 2 , . . . , n −
2, we can write z = q j + τ − j e s + it , ( s, t ) ∈ ( −∞ , − R ] × ( R / π Z ). Then the first term in the asymptoticexpansion of η k in the coordinates (˜ s, t ) is approximately A k ( q j ) B j ( q j ) ⊗ ( d ˜ s − idt ) , which has winding number 0. For q , if we change coordinates to ζ = z − , we seethat η k = − ζ A k ( ζ ) B ( ζ ) d ¯ ζ. Hence, if we change to cylindrical coordinates around ζ = 0, we can write z = τ − e ( s + it ) / , ( s, t ) ∈ ( −∞ , − R ] × ( R / π Z ). Then the first term in the asymptoticexpansion of η k in the coordinates (˜ s, t ) is approximately − e − iθ/ e ( − ˜ s + it ) / ⊗ ( d ˜ s − idt ) , which has winding number 1. (cid:3) Notation 6.5.3.
For each ℓ ≥
1, set(6.5.2) B ℓ = 1( ℓ − d ℓ − Q dz ℓ − ( p ) · · · d ℓ − Q dz ℓ − ( p n )... . . . ... d ℓ − Q n − dz ℓ − ( p ) · · · d ℓ − Q n − dz ℓ − ( p n ) and v ℓ = e iℓθ e − ℓT r ℓ α ,ℓ ... r ℓn α n,ℓ . Corollary 6.5.4. If R ≫ , the section s m on [ R, ∞ ) × ( M n / R ) is close to a sectionwhose zero set is defined by (6.5.3) m X ℓ =1 B ℓ v ℓ = 0 . Proof.
We make the same change to the s -coordinate near a positive puncture as inProposition 6.5.2. Near p i , we have η k ( z ) = " ∞ X ℓ =1 ℓ − d ℓ − Q k dz ℓ − ( p i )( z − p i ) ℓ − dz = − " ∞ X ℓ =1 ℓ − d ℓ − Q k dz ℓ − ( p i ) τ ℓi e − ℓ ( s + it ) ⊗ ( ds + idt ) ≈ − " ∞ X ℓ =1 ℓ − d ℓ − Q k dz ℓ − ( p i ) τ ℓi e − ℓ (˜ s − log | r i | + it ) ⊗ ( d ˜ s + idt )= − " ∞ X ℓ =1 ℓ − d ℓ − Q k dz ℓ − ( p i ) τ ℓi | r i | ℓ e − ℓ (˜ s + it ) ⊗ ( d ˜ s + idt )= − " ∞ X ℓ =1 ℓ − d ℓ − Q k dz ℓ − ( p i ) e iℓθ r ℓi e − ℓ (˜ s + it ) ⊗ ( d ˜ s + idt )Since the obstruction bundle O is complex in this case, we can, following Hutchings-Taubes [HT1] identify the section s m with a section s C m defined by s C m ( T, u )( σ ) = s m ( T, u )( σ ) + i s m ( T, u )( − iσ ) . OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 33
As noted in [HT1], the definition of s C m is equivalent to the replacing the real innerproducts in the original definition with complex inner products. We identify s m with its complexification and compute s m ( T, [ u ])( η k ) = n X j =1 * m X ℓ =1 α j,ℓ e iℓt , − ∞ X ℓ =1 ℓ − d ℓ − Q k dz ℓ − ( p j ) r ℓj e − iℓθ e − ℓT e ℓit + = − n X j =1 m X ℓ =1 ℓ − d ℓ − Q k dz ℓ − ( p j ) r ℓj e iℓθ e − ℓT α j,ℓ , which is the k th component of − P mℓ =1 B ℓ v ℓ . (cid:3) Remark . It will be useful in Appendix A to note that the partial fractiondecomposition of Q i is(6.5.4) Q i ( z ) = 1 − n − X k =2 A i ( q k ) B k ( q k ) 1 q k − z . Corollary 6.5.6.
The equations (6.5.5) ( p n − − p n ) α k, − ( p k − p n ) α n − , + ( p k − p n − ) α n, = 0 ,k = 1 , , . . . , n − , determine Z . Moreover, if ( T, [ u ]) ∈ Z , then p , p , . . . , p n − are determined by p n − , p n , and the coefficients α , , α , , . . . , α n, .Proof. We use the notation r i from Notation 6.5.1. Note that Q k ( p i ) = 0 when i = k , n −
1, or n . Note also that, when m = 1, (6.5.3) has an overall factor of e − T e iθ . Thus, (6.5.3) reduces to0 = Q k ( p k ) r k α k, + Q k ( p n − ) r n − α n − , + Q k ( p n ) r n α n, = α k, ( p k − p n − )( p k − p n ) + α n − , ( p n − − p k )( p n − − p n ) + α n, ( p n − p k )( p n − p n − ) ,k = 1 , . . . , n −
2, which is equivalent to (6.5.5). (cid:3)
The auxiliary gluing problem.
There is one case in Theorem 1.3.2 thatis not addressed by the prototypical gluing problem: the case where the curve in b X has a double cover of a plane, where we must glue a branched cover with twomultiplicity 1 positive ends and one multiplicity 2 negative end. Accordingly, wenow calculate the zero set of the obstruction section for the moduli space M (1 , | M denote the moduli space M (1 , | u ∈ M has onesimple branch point, and we can and do make the identification M = R / π Z . Notethat ind( u ) = dim Coker( D Nu ) = 1.The obstruction bundle O has rank 1. Choose a trivialization τ of ξ over β sothat µ τ ( β ) = 1, as before. Any element σ ∈ Ker( D Nu ) ∗ satisfies0 ≥ σ − (0) = χ ( ˙Σ) + ρ τ ( σ ) + ρ τ ( σ ) − ρ τ ( σ − ) ≥ . Thus, every non-zero element of Ker( D Nu ) ∗ is non-vanishing.We parametrize M in the following way. Fix the positive puncture p = 1, set p = − p − −
1, and let the negative puncture lie at infinity. Then send θ ∈ R / π Z to u θ : C \ {± p, } → C ∗ z e iθ z − . The markers at the positive ends are given by τ = e iθ and τ = − e iθ , while themarker at the negative end is determined by τ − = e − iθ . The meromorphic (0 , η θ ( z ) = e − iθ/ d ¯ z is a replacement for a spanning element of Ker( D Nu θ ) ∗ . In particular, it follows thestable direction at the negative end.To compute the zero set Z , note that, up to a real scalar multiple, we have s ( u θ )( η θ ) = D α , − e iθ/ E + D α , e iθ/ E = D α − α , e − iθ/ E . Thus, there are two values of θ ∈ R / π Z that such that s ( u θ )( η θ ) = 0.The branched covers corresponding to these two values of θ differ only in thechoice of asymptotic marker at the negative end. Thus, the two curves we obtainby gluing also differ only in the choice of asymptotic marker at the multiplicity 2negative end in question. Moduli spaces for ECH consist of holomorphic currents that are not asymptotically marked, so we have over-counted by a factor of 2.6.7. Non-Gluing Results.
We now show that the linearized obstruction sectionfor certain branched covers of R × β never has zeros; these results are used in theproof of Lemma 8.2.1. We assume that the coefficients α j, used in the definition ofthe linearized section are all distinct and non-zero. Notation 6.7.1.
Let n ≥ a , . . . , a k be any positive integers that sumto n . Let M g,n, ( a , . . . , a k ) be the moduli space of genus g ≥ R × β with k positive ends with multiplicities a , . . . , a k and n negative endsall with multiplicity 1. Let M g,n, ( a , . . . , a k ) be the moduli space of genus g ≥ R × β with k positive ends with multiplicities a , . . . , a k , onenegative end with multiplicity 3, and n − ℓ be the number of a j that are odd and order the positive endsso that a , . . . , a ℓ are odd and a ℓ +1 , . . . , a k are even. Proposition 6.7.2.
The linearized obstruction sections over M g,n, ( a , . . . , a k ) and M g,n, ( a , . . . , a k ) have no zeros for any n ≥ , any g ≥ , and any positive integers a , . . . , a k that sum to n and such that ℓ < n . OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 35
We prove Proposition 6.7.2 by exhibiting, for each relevant branched cover u , anelement σ ∈ Ker( D Nu ) ∗ such that s ( u )( σ ) = 0. We begin by describing the space ofreplacements for such σ . Lemma 6.7.3.
Let u : ˙Σ → R × Y + be a branched cover in M g,n, ( a , . . . , a k ) with ℓ < k , where ℓ is as in Notation 6.7.1. Every element σ ∈ Ker( D Nu ) ∗ has a replace-ment in the space V g,n, of meromorphic (0 , -forms on Σ with a pole of order atmost at each point corresponding to a negative puncture and, for all j = 1 , . . . , k , azero of order at least ⌈ a j ⌉ − at the point corresponding to the j th positive puncture.The map Ker( D Nu ) ∗ → Λ , T ∗ Σ that sends σ ∈ Ker( D Nu ) ∗ to its replacement is anisomorphism onto a real subspace of V g,n, with real codimension k − ℓ .Proof. Let π R : R × Y + → R be the projection onto the R -factor and define s = π R ◦ u : ˙Σ → R . After perturbing the asymptotic operator as in Section 6.4, multiply-ing an element σ ∈ Ker( D Nu ) ∗ by e − s/ yields a (0 , s . Continuity then implies that e − s/ σ is globally anti-meromorphic on ˙Σ, and we define our map as σ e − s/ σ . The proof of Proposi-tion 6.5.2 shows that, in cylindrical coordinates, the form d ¯ z has winding number 1around a point corresponding to a positive puncture and winding number − D Nu ) ∗ → Λ , T ∗ Σ is real-linear and injective. The space V g,n, isisomorphic to L ( K − D ), where K is a canonical divisor of Σ, D = k X j =1 (cid:16)l a j m − (cid:17) p j − n X j =1 q j , and L ( K − D ) is the space of meromorphic functions f on Σ with ( f ) ≥ D − K .Since ℓ < n , deg( D ) = − k − n − ℓ < , so by the Riemann-Roch theorem,dim C V g,n, = dim C L ( K − D ) = n − ℓ k + g − . Branched covers u ∈ M g,n, ( a , . . . , a k ) have exactly 2 g + k + n − R Ker D Nu =0 and ind( D Nu ) = ind( u ) − k + n +2 g −
2) = 2 − g − k − n . Hence, dim R Ker( D Nu ) ∗ = n + k + 2 g −
2. The second statement in the lemma now follows. (cid:3)
A small modification of the proof of Lemma 6.7.3 proves the following lemma.
Lemma 6.7.4.
Let u : ˙Σ → R × Y + be a branched cover in M g,n, ( a , . . . , a k ) with ℓ < k , where ℓ is as in Notation 6.7.1. Every element σ ∈ Ker( D Nu ) ∗ has a replace-ment in the space V g,n, of meromorphic (0 , -forms on Σ with a pole of order at most at the point corresponding to the multiplicity negative puncture, a pole oforder at most at each other point corresponding to a negative puncture and, for all j = 1 , . . . , k , a zero of order at least ⌈ a j ⌉ − at the point corresponding to the j th positive puncture. The map Ker( D Nu ) ∗ → Λ , T ∗ Σ that sends σ ∈ Ker( D Nu ) ∗ to itsreplacement is an isomorphism onto a real subspace of V g,n, with real codimension k − ℓ .Proof of Proposition 6.7.2. First, we consider the linearized section over the modulispace M g,n, ( a , . . . , a k ). Given u ∈ M g,n, ( a , . . . , a k ) with domain ˙Σ, it sufficesto exhibit an element σ ∈ Ker( D Nu ) ∗ with winding number ⌈ a k ⌉ at the k th positiveend and with winding number greater than ⌈ a j ⌉ at the j th positive end for all j = 1 , . . . , k −
1. Recall the divisor D from the proof of Lemma 6.7.3 and considerthe divisors D = D + k − X j =1 p j and D = D + k X j =1 p j . Since ℓ < n , both D and D have negative degree, and the Riemann-Roch theoremimplies that dim C ( L ( K − D ) /L ( K − D )) = 1 , where K is a canonical divisor on Σ. Let η ′ be a meromorphic 1-form on Σ corre-sponding to an element of L ( K − D ) that remains non-zero in the quotient. If a k is even, we can multiply η ′ by a complex number to get a form η such that η is inthe image of the map Ker( D Nu ) ∗ → Λ , T ∗ Σ from Lemma 6.7.3. We then take σ tobe the element in Ker( D Nu ) ∗ corresponding to η . The proof for M g,n, ( a , . . . , a k )is a small modification of the previous argument. (cid:3) Proposition 6.7.5.
The linearized obstruction section over M ,n, (1 , . . . , , wherethe branched covers have n positive ends all with multiplicity , has no zeros for any n ≥ .Proof. Let u ∈ M ,n, (1 , . . . , , . . . , n and denote thecorresponding positive punctures by p , . . . , p n . Denote the negative punctures by q , . . . , q n . Note that dim R Ker( D Nu ) ∗ = 2 n −
2. Using the row-reduction strategyfrom the proof of Proposition 6.3.3, we can find a basis { σ , σ , . . . , σ n − σ n − } forKer( D Nu ) ∗ such that for k = 0 , . . . , n −
1, the projections of σ k +1 and σ k +2 to theleading eigenspace on the j th positive end are linearly independent for j = k and j = n and vanish otherwise. Viewing the obstruction bundle as a complex vectorbundle, we see that the meromorphic (0 , η = (¯ z − ¯ p ) · · · (¯ z − ¯ p n )(¯ z − ¯ q ) · · · (¯ z − ¯ q n ) d ¯ z OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 37 is a replacement for an element of Ker( D Nu ) ∗ , and thus s ( u )( η ) = α , − α n, p − p n = 0 , as desired. (cid:3) Gluing Models and Evaluation Map Calculations
In this section, we construct models for the curves obtained after performing thegluing procedure from Section 6. We then use those models to compute the degreeof certain evaluation maps and prove Theorem 1.4.5. Throughout this section, weidentify [ u ] ∈ M n / R with the representative determined by the parametrizationfrom Section 6.7.1. Gluing models.
Assume that the point ( T, [ u ]) ∈ [ R, ∞ ) × ( M n / R ) glues to u . Denote by u u the curve obtained by gluing. Part of the domain of u u can be identified with the Riemann surface Σ obtained from the domain ˙Σ of thebranched cover u by truncating the positive ends at height T . Over Σ , we can write u u as the graph of a section ν of the pullback of the normal bundle of the trivialcylinder R × β in R × Y + . Since the normal bundle is a trivial complex holomorphicline bundle, we can view ν as a complex-valued function on Σ . By the argumentused to prove Lemma 6.7.3, we may assume that ν is genuinely holomorphic awayfrom the punctures. We view Σ as the extended complex plane b C with a finitenumber of disks removed. Using the analysis in the proof of [BH1, Proposition8.7.2], we can write ν in cylindrical coordinates in an annulus around a positivepuncture p i as ν ( s, t ) = s, t, ∞ X ℓ =1 h ( α i,ℓ e − ℓT + d i,ℓ ) e λ i s f i ( t ) i + (lower-order terms) ! , where the d i,ℓ are constants, depending on T , coming from the perturbation in theobstruction bundle gluing construction and satisfy | α i,ℓ | ≫ | e ℓT d i,ℓ | . Definition 7.1.1. A full model associated to ( T, [ u ]) ∈ [ R, ∞ ) × ( M n / R ) iscomplex-valued function g on the domain of u that is holomorphic except for es-sential singularities at p , . . . , p n , that has a zero of order 2 at infinity, that hassimple zeros at q , . . . , q n − , and such that the principal part of g at p i in cylindricalcoordinates ( s, t ) ∈ [ T, ∞ ) × ( R / π Z ), where z = p i + e − ( s + it ) , is ∞ X ℓ =1 ( α i,ℓ e − ℓT + d i,ℓ ) r ℓi e ℓ ( s + it ) , where the r i are as in Notation 6.5.1. Note that a full model associated to ( T, [ u ]), if one exists, is unique and is givenby(7.1.1) g ( z ) = n X i =1 ∞ X ℓ =1 e iℓθ r ℓi ( α i,ℓ e − ℓT + d i,ℓ )( z − p i ) ℓ , Thus, whether or not ( T, [ u ]) has a full model is determined by the locations of thezeros of g . Definition 7.1.2. An order m model associated to ( T, [ u ]) ∈ [ R, ∞ ) × ( M n / R ) isa meromorphic function g on the domain of u with a pole of order m at each positivepuncture p i , a zero of order 2 at infinity, and simple zeros at q , . . . , q n − , such thatthe principal part of g at p i in cylindrical coordinates ( s, t ) ∈ [ T, ∞ ) × ( R / π Z ),where z = p i + e − ( s + it ) , is m X ℓ =1 α i,ℓ e − ℓT r ℓi e ℓ ( s + it ) . Note that an order m model for ( T, [ u ]) ∈ [ R, ∞ ) × M n , if one exists, is uniqueand is given by g ( z ) = n X i =1 m X ℓ =1 e iℓθ e − ℓT r ℓi α i,ℓ ( z − p i ) ℓ . As before, whether or not ( T, [ u ]) has an order m model is determined by thelocations of the zeros of g . As the name suggests, the models defined above arerelated to curves obtained by gluing branched covers of trivial cylinders. An order m model associated to u is the approximation to the function ν on Σ obtained bytruncating the principal part at each singularity to the leading m terms.The following theorem describes when a branched cover has an associated order m model. The proof is given in Appendix A. Theorem 7.1.3.
A point ( T, [ u ]) ∈ [ R, ∞ ) × M n / R has an associated order m model if and only if ( T, [ u ]) ∈ Z m . An examination of the proof of Theorem 7.1.3 yields the following corollary.
Corollary 7.1.4.
A point ( T, [ u ]) ∈ [ R, ∞ ) × M n / R has an associated full model ifand only if ( T, [ u ]) ∈ Z . Evaluation maps in the models.
We compute the evaluation map for thecurves obtained by gluing branched covers of trivial cylinders as in Section 6. Wealso define an evaluation map that is suitable for use in an order 1 model, allowing fortranslations of the glued curve in the R -direction of R × Y + , and compute the countof gluings. Throughout these calculations, s denotes the R -coordinate of R × Y + .The curves obtained by gluing branched covers in M n / R to u live in the modulispace M = M n − , n ( n − / R × Y + ( α , β ). Let E denote the end of M consisting of glued OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 39 curves. The gluing procedure gives a diffeomorphism [ R, ∞ ) × Z ∼ = E / R , where the R -action on E is given by translating curves in the R -direction of R × Y + , and thefull models from Definition 7.1.1 give a section E / R → E . We obtain all curves in E by translating curves in the image of this section in the R -direction of R × Y + , andthus we have an identification [ R, ∞ ) × Z × R ∼ = E . We first compute expressionsfor the components of the evaluation map on curves in the image of this section,i.e., on [ R, ∞ ) × Z × { } . We will abuse notation and let ev j denote the restrictionto E of the evaluation map on M at the negative puncture q j . Proposition 7.2.1.
For any ( T, [ u ]) ∈ [ R, ∞ ) × Z , we have ev ( T, [ u ] ,
0) = e iθ/ n X i =1 p i B ( p i ) A i ( p i ) ( α i, e − T + d i, ) + e iθ/ n X i =1 B ( p i ) A i ( p i ) ( α i, e − T + d i, ) and, for k = 2 , . . . , n − , ev k ( T, [ u ] ,
0) = 1 B k ( q k ) n X i =1 ∞ X ℓ =1 ( − ℓ − ℓ A i ( q k ) B k ( p i ) ℓ A i ( p i ) ℓ ( α i,ℓ e − ℓT + d i,ℓ ) e ( ℓ − iθ Proof.
The result follows directly from (7.1.1) after making the same change to the s -coordinate as in Proposition 6.5.2. (cid:3) To obtain expressions for the components of the evaluation map on all of E , weneed only multiply the above expressions by appropriate exponential functions toaccount for the effect on the evaluation map of translating a curve in the R -directionof R × Y + . The following proposition follows immediately. Proposition 7.2.2.
Let ev j denote the restriction to E ∼ = [ R, ∞ ) × Z × R of theevaluation map on M at the negative puncture q j . Then ev ( T, [ u ] , s ) = e − s/ ev ( T, [ u ] , and, for k = 2 , . . . , n − , ev k ( T, [ u ] , s ) = e − s ev k ( T, [ u ] , Definition 7.2.3.
The model evaluation map on Z × R is defined as follows.Let ( T, [ u ] , s ) ∈ Z × R and let g be the model of order 1 associated to ( T, [ u ]). Wedefine the model evaluation map at the negative puncture q j , j = 2 , , . . . , n −
2, tobe the leading complex asymptotic coefficient in the Fourier-type expansion of g ( z )in cylindrical coordinates around the puncture q j , multiplied by e − s :(7.2.1) mev j ( T, [ u ] , s ) = e − s e − iθ r − j g ′ ( q j ) . Let h ( ζ ) = g ( ζ − ). We define the model evaluation map at the negative puncture q to be the leading complex asymptotic coefficient in the Fourier-type expansion of h ( ζ ) in cylindrical coordinates around ζ = 0, which corresponds to the puncture q ,multiplied by e − s/ :(7.2.2) mev ( T, [ u ] , s ) = e − s/ e − iθ/ h ′′ (0)2 . If I is a subset of { , , . . . , n − } , we define the model evaluation map at thenegative punctures q j , j ∈ I , bymev I ( T, [ u ] , s ) = (mev j ( T, [ u ] , s )) j ∈ I . We now compute the degree of mev I on ( Z ∩ ( { T } × ( M n / R ))) × R when T isfixed and sufficiently large. For simplicity of notation, set(7.2.3) H i = B ( p i ) A i ( p i ) α i, , so that g ( z ) = e iθ e − T n X i =1 B ( p i ) A i ( p i ) α i, z − p i = e iθ e − T n X i =1 H i z − p i . Lemma 7.2.4. If h ( ζ ) = g ( ζ − ) , we have h ′′ (0) = 2 e iθ/ e − T n X i =1 p i H i . Proof.
We want to compute the leading coefficient in the Taylor expansion of e − iθ/ h ( ζ )at ζ = 0. Since h ′ (0) = 0, we have(7.2.4) n X i =1 H i = 0 , and hence e − iθ/ h ( ζ ) = e iθ/ e − T ζ n X i =1 H i − p i ζ = e iθ/ e − T ζ n X i =1 (cid:20) H i − p i ζ − H i (cid:21) = e iθ/ e − T ζ n X i =1 p i H i − p i ζ . The result follows. (cid:3)
Lemma 7.2.5. If ( T, [ u ]) ∈ Z , we have n X i =1 H i ( q k − p i ) · · · ( q k j − p i ) = 0 , where j ∈ { , . . . , n − } and k , . . . , k j are distinct elements of { , . . . , n − } . OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 41
Proof.
The proof is by induction on j ≤ n −
3. Since g ( q k ) = 0 for k = 2 , , . . . , n − n X i =1 H i q k − p i for k = 2 , , . . . , n −
2, which establishes the case j = 1.Now assume the result for some j ≤ n −
4. If k , . . . , k j +1 are distinct elementsof { , . . . , n − } , we see that0 = n X i =1 H i ( q k − p i ) · · · ( q k j − p i ) − n X i =1 H i ( q k − p i ) · · · ( q k j − − p i )( q k j +1 − p i )= n X i =1 H i ( q k − p i ) · · · ( q k j − − p i ) (cid:20) q k j − p i − q k j +1 − p i (cid:21) = ( q k j +1 − q k j ) n X i =1 H i ( q k − p i ) · · · ( q k j +1 − p i ) . Since q k j +1 − q k j = 0, the inductive step follows. (cid:3) Lemma 7.2.6. If ( T, [ u ]) ∈ Z , we can write g ( z ) = e iθ e − T B ( z ) n X i =1 α i, A i ( p i ) 1 z − p i . Proof.
We show inductively that g ( z ) = e iθ e − T m Y k =2 ( z − q k ) n X j =1 H j Q mk =2 ( p j − q k ) 1 z − p j for m = 2 , . . . , n −
2. The lemma then follows by taking m = n − m = 2, use Lemma 7.2.5 to write g ( z ) = e iθ e − T n X i =1 H i z − p i = e iθ e − T n X i =1 (cid:20) H i z − p i − H i q − p i (cid:21) = e iθ e − T ( q − z ) n X i =1 H i ( q − p i )( z − p i )= e iθ e − T ( z − q ) n X i =1 H i ( p i − q ) 1 z − p i . Now assume that the lemma is true for some m with 2 ≤ m ≤ n −
3. ByLemma 7.2.5, we have g ( z ) = e iθ e − T m Y k =2 ( z − q k ) n X j =1 H j Q mk =2 ( p j − q k ) 1 z − p j = e iθ e − T m Y k =2 ( z − q k ) n X j =1 (cid:20) H j Q mk =2 ( p j − q k ) 1 z − p j − H j ( q m +1 − p j ) Q mk =2 ( p j − q k ) (cid:21) = e iθ e − T m +1 Y k =2 ( z − q k ) n X j =1 H j Q m +1 k =2 ( p j − q k ) 1 z − p j , and we are done. (cid:3) Lemma 7.2.7.
For k = 2 , . . . , n − , we have e − iθ r − k g ′ ( q k ) = ( − n e − T ( α n − , − α n, ) q k − ( p n α n − , − p n − α n, ) p n − − p n . Proof.
Set ∆ = Y ≤ i We have ( − n det M ℓ ∆ = p n α n − , − p n − α n, p n − − p n ℓ = n − α n − , − α n, p n − − p n ℓ = n − ℓ < n − . The lemma follows from Claim 7.2.8. The proof of the claim is an exercise in carefulrow-reduction and is given in Appendix B. (cid:3) Proposition 7.2.9. Let I = { , , . . . , n − } and suppose that R ≫ is sufficientlylarge in the prototypical gluing problem. For any fixed T ≥ R and any admissibleasymptotic restriction c ∈ ( C ∗ ) n − , the degree of the restriction of mev I to ( Z ∩ ( { T } × ( M n / R ))) × R is .Proof. Choose an admissible asymptotic restriction c = ( c , . . . , c n − ) ∈ ( C ∗ ) n − .We must count solutions of the equations e − s/ e iθ/ e − T n X i =1 p i H i = c , ( − n e − s e − T ( α n − , − α n, ) q k − ( p n α n − , − p n − α n, ) p n − − p n = c k ,k = 2 , . . . , n − 2, where T is fixed by our assumptions, the p i are fixed, distinctpoints in C by Corollary 6.5.6, q , . . . , q n − are allowed to vary in C \ { p , . . . , p n } , s is allowed to vary in R , and θ is allowed to vary in ( R / π Z ). The last n − q k = ( − n e s e T ( p n − − p n ) c k + ( p n α n − , − p n − α n, ) α n − , − α n, ,k = 2 , . . . , n − 2. Now substitute back into the first equation and consider the normof the left-hand side. If s is very large and positive, the norm is larger than | c | ,while if s is very large and negative, the norm is smaller than | c | . Thus, thereare an odd number of values of s for which the norms of both sides of the firstequation are equal. Since θ ∈ R / π Z , there is a unique choice of θ that solves thefirst equation. (cid:3) Reduction to the first-order model. We now show that the full evaluationmap and the evaluation map in the order 1 model have the same degree. Proposition 7.3.1. Let I = { , , . . . , n − } and suppose that R ≫ is sufficientlylarge in the prototypical gluing problem. For any fixed T ≥ R and any admissibleasymptotic restriction c ∈ ( C ∗ ) n − , the mod degree of the restriction of mev I to ( Z ∩ ( { T } × ( M n / R ))) × R is equal to the mod degree of the restriction of ev I to ( Z ∩ ( { T } × ( M n / R ))) × R .Proof of Proposition 7.3.1. Let c be an admissible asymptotic restriction and let T ≫ R . Define a map F : ( M n / R ) × R → C n − × C n − by F ([ u ] , s ) = ( s ( T, [ u ])( η ) , . . . , s ( T, [ u ])( η n − ) , ev I ( T, [ u ] , s )) . Thus, the set of all ( T, [ u ]) ∈ { T } × ( M n / R ) that glue to u + and satisfy theadmissible asymptotic restrictions c when translated in the R -direction by s is F − ( { } × { c } ).Define a homotopy F ν of F in the following way. For ν ∈ [0 , ], we can define ahomotopy s ν of the obstruction section s such that s = s and s is the linear portionof s ; see, e.g., [BH1, Section 8.7] for a similar construction. Let ev k ( T, [ u ] , s ) denotethe evaluation map at the puncture q k , k = 1 , , . . . , n − 2. Define a homotopy ofeach ev k for ν ∈ [0 , ] by replacing each α i,ℓ e − ℓT + d i,ℓ with α i,ℓ e − ℓT + (1 − ν ) d i,ℓ ,and set ev I,ν = (ev k,ν ) n − k =1 . Then for ν ∈ [0 , ], we define F ν ([ u ] , s ) = ( s ν ( T, [ u ])( η ) , . . . , s ν ( T, [ u ])( η n − ) , ev I,ν ( T, [ u ] , s )) . For ν ∈ [ , η k,ν denote the linear interpolation from η k to Π η k , the projec-tion of η k onto the leading eigenspace at each positive end. Let ev k,ν , ν ∈ [ , ℓ ≥ I,ν = (ev k,ν ) n − j =1 .Then for ν ∈ [ , F ν ([ u ] , s ) = ( s ( T, [ u ])( η ,ν ) , . . . , s ( T, [ u ])( η n − ,ν ) , ev I,ν ( T, [ u ] , s )) . Note that F = F and F ([ u ] , s ) = ( s ( T, [ u ])( η ) , . . . , s ( T, [ u ])( η n − ) , mev I ( T, [ u ] , s )) . Claim 7.3.2. Let K ⊂ ( M n / R ) × R be a compact set such that F − ( { , c } ) is contained in the interior of { T } × K . If T ≫ is sufficiently large, then F − ν ( { , c } ) ∩ ( { T } × ∂K ) = ∅ for all ν ∈ [0 , .Proof of Claim 7.3.2. Suppose that the claim is false. Then there is a sequence { T k , ν k } with T k → ∞ such that F − ν k ( { , c } ) ∩ ( { T k } × ∂K ) = ∅ for all k . We willarrive at a contradiction by showing that the homotopy F ν is very small on { T } × K if T is sufficiently large.Since K is compact, there is a large positive constant C such that | p i − p j | > C − and | q i − q j | > C − for all i = j . In addition, we may assume that C − < s < C . We OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 45 may also assume that each of the punctures p , . . . , p n , q , . . . , q n − , except possiblyfor one of the positive punctures p i , is contained in the disk of diameter C centeredat the origin in C . If all of the punctures in question are contained in the disk, then | p i − q j | < C for all i = j . In this case, we have, for ν ∈ [ , 1] and k = 2 , . . . , n − | ev k,ν ( T, [ u ] , s ) − mev k ( T, [ u ] , s ) | ≤ e C | B k ( q k ) | n X i =1 ∞ X ℓ =2 ℓ (cid:12)(cid:12)(cid:12)(cid:12) A i ( q k ) B k ( p i ) ℓ α i,ℓ A i ( p i ) ℓ (cid:12)(cid:12)(cid:12)(cid:12) e − ℓT ≤ e C n X i =1 ∞ X ℓ =2 ℓC ( ℓ +1)(2 n − | α i,ℓ | e − ℓT , and the right-hand side can be made as small as we like by taking T to be sufficientlylarge.Now assume that one of the positive punctures, say p j , is outside of the above-mentioned disk. There is a constant D > C , depending only on n and C , such thatif | p j | > D , then (cid:12)(cid:12)(cid:12)(cid:12) B k ( p j ) ℓ A j ( p j ) ℓ (cid:12)(cid:12)(cid:12)(cid:12) < (cid:12)(cid:12)(cid:12)(cid:12) A i ( q k ) A i ( p i ) ℓ (cid:12)(cid:12)(cid:12)(cid:12) < i = j and ℓ ≥ 2. Moreover, when C ≤ | p j | ≤ D , we have (cid:12)(cid:12)(cid:12)(cid:12) B k ( p j ) ℓ A j ( p j ) ℓ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( n − ℓ ( C + D ) ( n − ℓ and (cid:12)(cid:12)(cid:12)(cid:12) A i ( q k ) A i ( p i ) ℓ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( n − ℓ + n − ( C + D )for all i = j . It follows that, in this case, we can make | ev k,ν ( T, [ u ] , s ) − mev k ( T, [ u ] , s ) | as small as we like by taking T to be sufficiently large.Since | d i,ℓ | ≪ | α i,ℓ | e − ℓT , a similar estimate shows that, for ν ∈ [0 , ], we can make | ev j,ν ( T, [ u ] , s ) − ev j, ( T, [ u ] , s ) | as small as we like by taking T to be sufficientlylarge. Another similar estimate shows the same result for the leading coefficient atthe puncture q .Now we prove a similar result for the homotopy of the obstruction section. By(6.5.4), we have 1( ℓ − d ℓ − Q i dz ℓ − ( z ) = − n − X k =2 A i ( q k ) B k ( q k ) 1( q k − z ) ℓ for i = 1 , . . . , n and ℓ ≥ 1. If each of p , . . . , p n , q , . . . , q n − is in the disk of diameter C centered at the origin, then, for ν ∈ [ , | s ( T, [ u ])( η k,ν ) − s ( T, [ u ])( η k ) | ≤ n X i =1 ∞ X ℓ =2 n − X k =2 e − ℓT | α i,ℓ | (cid:12)(cid:12)(cid:12)(cid:12) A i ( q k ) B k ( p i ) ℓ B k ( q k ) A i ( p i ) ℓ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C n − n X i =1 ∞ X ℓ =2 n − X k =2 e − ℓT | α i,ℓ | C (2 n − ℓ = n ( n − C n − ∞ X ℓ =2 e − ℓT | α i,ℓ | C (2 n − ℓ , which can be made as small as we like by taking T to be sufficiently large. Whensome positive puncture, say p j , is outside of the disk of diameter C , the sameargument used for the evaluation map shows that we can make | s ( T, [ u ])( η k,ν ) − s ( T, [ u ])( η k ) | as small as we like by taking T to be sufficiently large. A similar estimate showsthat, for ν ∈ [0 , ], | s ν ( T, [ u ])( η k ) − s ( T, [ u ])( η k ) | can be made as small as we like by taking T to be sufficiently large.We now finish the proof of the claim. Since K is compact and F ( { T } × ∂K )does not intersect { , c } , the distance between F ( { T } × ∂K ) and { , c } is boundedbelow by a positive constant. By our above estimates, the homotopy F ν can bemade as small as we on { T } × K like by taking T to be sufficiently large. Thus, thedistance between F ν ( { T } × ∂K ) and { , c } is bounded below by a (possibly smaller)positive constant for all ν ∈ [0 , T k → ∞ in the sequence { T k , ν k } . (cid:3) The claim implies Proposition 7.3.1: if the number of points in F − ( { , c } ) iseven, there is a large compact subset K ⊂ ( M n / R ) × R containing F − ( { , c } ) inits interior such that F − ν ( { , c } ) ∩ ( { T } × ∂K ) = ∅ for some ν ∈ (0 , (cid:3) Proof of Theorem 1.4.5. Combine Proposition 7.2.9 and Proposition 7.3.1. (cid:3) The Cobordism Map In this section, we complete the proof of Theorem 1.5.3 using the evaluationmap discussed in Section 3 together with the degree calculations in Section 7. Weadapt the truncation procedure used in [HT1, Sections 1.3 and 7.3]; see [Hu3,Section 5.4] for an overview. Throughout this section, M denotes the subset of M n − , n ( n − / R × Y + ( α , β ) consisting of curves with with n − β and one negative end of multiplicity 3 at β , and M + denotes thesubset of M , n ( n − / R × Y + ( α , β ) consisting of curves with n negative ends of multiplic-ity 1 at β . Note that u ∈ M + and that the curves obtained by gluing branchedcovers in M n / R to u live in M and that all curves in M are somewhere injective.For each curve in M , label the negative ends asymptotic to β by the elements of I = { , , . . . , n − } , where the multiplicity 3 negative end is given the label 1.There is an evaluation map ev I : M → C n − . OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 47 Setup for truncation. We first collect all of the necessary definitions andauxiliary results for the truncation procedure. Definition 8.1.1. For n ≥ 3, we define the set R n of generic asymptotic re-strictions on curves in M recursively as follows. Let R ′ n denote the set of regularvalues of ev I : M → C n − . For any orbit sets γ ± with A ( γ − ) < A ( γ + ) ≤ A ( α )such that β is an orbit in γ − with multiplicity m ≤ n and for all k = 1 , . . . , n − V m,k ( γ + , γ − ) be the set of somewhere injective J -holomorphic curves from γ + to γ − with Fredholm index k , with ECH index at most (cid:0) n (cid:1) , and such that at most onenegative end asymptotic to β has multiplicity 3 and the remaining negative endsasymptotic to β all have multiplicity 1. Label the negative ends of such curves as-ymptotic to β by the elements of I m = { , . . . , m } ; if there is a negative end at β with multiplicity 3, we require that it be labeled by 1. Let e R m,k ( γ + , γ − ) be the setof regular values of ev I m : V m,k ( γ + , γ − ) → C m . For each subset G = { j , · · · , j m } ⊂ I , let π G : C n − → C m be the projection ( c , . . . , c n − ) ( c j , . . . , c j m ). Each e R m,k ( γ + , γ − ) is a countable intersection of dense, open sets, and hence the same istrue for R Gm,k ( γ + , γ − ) = π − G ( e R m,k ( γ + , γ − )). Finally, define R n = \ m,G, γ ± ,k R Gm,k ( γ + , γ − ) ∩ R ′ n , where the intersection is taken over all m ≤ n , all subsets G ⊂ I m , all pairs oforbit sets γ ± with A ( γ − ) < A ( γ + ) ≤ A ( α ) such that β is an orbit in γ − withmultiplicity m , and all k = 1 , . . . , n − 4. Let e V m,k ( γ + , γ − ) ⊂ V m,k ( γ + , γ − ) denotethe subset of curves with ECH index (cid:0) n (cid:1) . Remark . The pre-image of any c ∈ R n under ev I : V n,k ( γ + , γ − ) → C n − isempty for any k ≤ n − 5. If ( c , . . . , c n − ) ∈ R n , then ( c j , . . . , c j m ) ∈ R m +2 for allsubsets { j , . . . , j m } ⊂ I , and all m = 1 , . . . , n − Notation 8.1.3. If c ∈ C n − is a generic, admissible asymptotic restriction, let K c denote the pre-image of c under ev I : M → C n − . Remark . Since c is generic, K c is a real 1-dimensional submanifold of M . Definition 8.1.5. A pair ( w − , w + ) of J -holomorphic curves is an asymptoti-cally restricted gluing pair with asymptotic restriction c ∈ C n − if w − ∈ e V n, n − ( γ , β ) for some orbit set γ with A ( β ) < A ( γ ) < A ( α ), ev I ( w − ) = c , and w + ∈ M , R × Y + ( α , γ ). We denote the set of all asymptotically restricted gluing pairswith asymptotic restriction c by S c . Definition 8.1.6. A pair ( w − , w + ) of J -holomorphic curves is an inverted asymp-totically restricted gluing pair with asymptotic restriction c ∈ C n − if w + ∈ e V n, n − ( α , γ ) for some orbit set γ with A ( β ) < A ( γ ) < A ( α ), ev I ( w + ) = c , and w − ∈ M , R × Y + ( γ , β ) is a curve that contains n − R × β and an unbranchedcover of R × β of multiplicity 3. We denote the set of all asymptotically restrictedgluing pairs with asymptotic restriction c by S inv c . Definition 8.1.7. For R ≫ 0, let G R be the intersection of K c with the end of M corresponding to Z where the gluing parameter T > R . Remark . By Proposition 7.2.9 and Proposition 7.3.1, the number of compo-nents of G R is finite and odd.The next two definitions are analogues of [HT1, Definition 1.10]. Definition 8.1.9. Let c ∈ C n − be a generic, admissible asymptotic restriction, let( w − , w + ) ∈ S c , let δ > 0, and choose a product metric on R × Y + . Let C c ,δ ( w − , w + )be the set of J -holomorphic curves in K c whose images can be decomposed into twosurfaces with boundary C − ∪ C + such that the following hold.(1) There is a real number R + and a section ψ + of the normal bundle of w + with | ψ + | < δ and such that C + is obtained by translating the portion of theimage of exp w + ( ψ + ) with s ≥ − /δ by R + in the s -direction. Here, exp w + is the exponential map on w + in the normal direction.(2) There is a real number R − and a section ψ − of the normal bundle of w − with | ψ − | < δ and such that C − is obtained by translating the portion ofthe image of exp w − ( ψ − ) with s ≤ /δ by R − in the s -direction. Here, exp w − is the exponential map on w − in the normal direction.(3) We have R + − R − > /δ .(4) The positive boundary circles of C − agree with the negative boundary circlesof C + .Let G c ,δ ( w − , w + ) be the set of curves in C c ,δ ( w − , w + ) that have Fredholm index2 n − (cid:0) n (cid:1) . Definition 8.1.10. Let c ∈ C n − be a generic, admissible asymptotic restriction, let( w − , w + ) ∈ S inv c , let δ > 0, and choose a product metric on R × Y + . Let C inv c ,δ ( w − , w + )be the set of J -holomorphic curves in K c whose images can be decomposed into twosurfaces with boundary C − ∪ C + such that the following hold.(1) There is a real number R + and a section ψ + of the normal bundle of w + with | ψ + | < δ and such that C + is obtained by translating the portion of theimage of exp w + ( ψ + ) with s ≥ − /δ by R + in the s -direction. Here, exp w + is the exponential map on w + in the normal direction.(2) There is a real number R − and a section ψ − of the normal bundle of w − with | ψ − | < δ and such that C − is obtained by translating the portion ofthe image of exp w − ( ψ − ) with s ≤ /δ by R − in the s -direction. Here, exp w − is the exponential map on w − in the normal direction.(3) We have R + − R − > /δ . OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 49 (4) The positive boundary circles of C − agree with the negative boundary circlesof C + .Let G inv c ,δ ( w − , w + ) be the set of curves in C inv c ,δ ( w − , w + ) that have Fredholm index2 n − (cid:0) n (cid:1) .Now we give an analogue of [HT1, Lemma 1.11]. Lemma 8.1.11. Let c ∈ C n − be a generic, admissible asymptotic restriction andlet ( w − , w + ) ∈ S c (resp. S inv c ). There exists a δ > such that for any δ ∈ (0 , δ ) and any sequence { v k } in G c ,δ ( w − , w + ) (resp. G inv c ,δ ( w − , w + ) ), the sequence { [ v k ] } hasa subsequence that converges either to a curve in K c / R or to the building [ w − ] ∪ [ w + ] .There exists an R ≫ such that for any R > R and any sequence { v k } in G R ,the sequence { [ v k ] } has a subsequence that converges either to a curve in K c / R orto a building [ w − ] ∪ [ w + ] ∈ ( M n / R ) × ( M + / R ) .Proof. The lemma is immediate for sequences in G R since G R is a finite union ofopen subsets of K c by Remark 8.1.8. So assume that ( w − , w + ) ∈ S c ; the proof when( w − , w + ) ∈ S inv c is similar. By the compactness results in [BEHWZ], a subsequenceof { [ v k ] } converges to an SFT building [ w ] ∪ · · · ∪ [ w ℓ ], where the levels go frombottom to top as we read from left to right.If ℓ > 1, then [ w ] must contain [ w − ] and [ w ℓ ] must contain [ w + ]. There are noother levels by Lemma 2.5.2, so ℓ = 2. There are no other components of [ w ] or[ w ] by Lemma 2.5.2 since the negative orbit set of [ w − ] is β and the positive orbitset of [ w + ] is α . Thus, [ w ] = [ w − ], and [ w ] = [ w + ].If ℓ = 1, then [ w ] contains either one component with Fredholm index 2 n − n − w ] by Lemma 2.5.2 since the negative orbit setof [ w − ] is β and the positive orbit set of [ w + ] is α . By continuity, [ w ] ∈ K c / R . (cid:3) Finally, we give an analogue of [HT1, Definition 1.12]. Definition 8.1.12. Given an R > R , by Lemma 8.1.11 there is an open subset U ⊂ K c / R such that G R ′ ⊂ U ⊂ G R for some R ′ > R and whose closure U in K c / R has finitely many endpoints.Let c ∈ C n − be a generic, admissible asymptotic restriction, let ( w − , w + ) ∈S c (resp. S inv c ), and let a δ ∈ (0 , δ ). By Lemma 8.1.11, there is an open set U c ( w − , w + ) ⊂ K c / R (resp. U inv c ( w − , w + )) such that G c ,δ ′ ( w − , w + ) ⊂ U c ( w − , w + ) ⊂G c ,δ ( w − , w + ) (resp. G inv c ,δ ′ ( w − , w + ) ⊂ U inv c ( w − , w + ) ⊂ G inv c ,δ ( w − , w + )) for some δ ′ ∈ (0 , δ ) and whose closure U c ( w − , w + ) (resp. U inv c ( w − , w + )) in K c has finitely manyendpoints.8.2. Truncation and the cobordism map. We now truncate K c to obtain acompact 1-manifold with boundary, which we use to prove Theorem 1.5.3. We begin with an analogue of [HT1, Lemma 7.23]. In the following proof, we call coversof trivial cylinders connectors ; an unbranched cover of a trivial cylinder is called a trivial connector , while a branched cover is called a non-trivial connector . Acover of R × β is called a connector over β . A J -holomorphic curve that is not acover of a trivial cylinder is called a non-connector . Lemma 8.2.1. Let c ∈ C n − be a generic, admissible asymptotic restriction. Anysequence { v k } in K c has a subsequence that converges to a curve in K c / R or to a -level building [ w − ] ∪ [ w + ] in ( M n / R ) × ( M + / R ) , S c / R , or S inv c / R .Proof. By [Hu3, Lemma 5.11], we may pass to a subsequence such that every curve isin the same relative homology class after projecting to Y + . By [Hu2, Corollary 6.10],there is an upper bound on the genus of a somewhere injective curve that dependsonly on its relative homology class. Hence, we may pass to a further subsequencesuch that every curve has the same genus. By the compactness results in [BEHWZ],a further subsequence, which we also denote by { [ v k ] } , converges to an SFT building[ w ] ∪ · · · ∪ [ w ℓ ], where the levels go from bottom to top as we read from left to right.If ℓ = 1, then by continuity [ w ] ∈ K c / R . So assume that ℓ > w ] at β the labels and asymptotic markers inducedfrom the sequence { [ v k ] } . Let w be a representative of the class [ w ]. For each i ∈ I , there are two possibilities: ev i ( w ) = 0 ∈ C or ev i ( w ) = 0.If ev i ( w ) = 0, then there is a sequence of translates v ′ k of the curves v k bydistances a k in the R -direction such that ev i ( v ′ k ) → 0. Since ev i ( v k ) = c i for all k ,Remark 3.2.4 implies that a k → ∞ . Then for all d > 2, we have ev di ( w ) = (0 , . . . , w containing the i th negative end is somewhere injective, thenit is a trivial cylinder by Remark 3.3.3; if the component is multiply covered, thenit must be a trivial connector.If ev i ( w ) = 0, then we claim that some translate w ′ of w satisfies ev i ( w ′ ) = c i .If not, then there is some constant C > { t ev i ( w ) | t > } is a distance at least C away from c i in the standard Euclidean metric on C . When j is sufficiently large, we can glue appropriate representatives w ,j , . . . , w ℓ,j of theclasses [ w ] , . . . , [ w ℓ ] to get a curve w j that represents the class [ v k ] and such thatev i ( w j ) = c i . It follows that for any δ > 0, we have | ev i ( w ,j ) − c i | < δ when j issufficiently large, and we have a contradiction.Now we claim that ℓ = 2 and that one of (1) [ w ] ∈ M n / R , (2) ([ w ] , [ w ]) ∈ S c / R ,or (3) ([ w ] , [ w ]) ∈ S inv c / R is true.First, assume that ev I ( w ) = . A small modification of the argument aboveshows that there is some translate w ′ of w such that ev I ( w ′ ) = c . Since c isadmissible, it follows that the components of w − asymptotic to β are somewhereinjective. Since c is generic, Remark 8.1.2 implies that ind( w ) ≥ n − 4, and hence(1.2.2) implies that I ( w ) ≥ (cid:0) n (cid:1) . Thus, ind( w ) ≥ I ( w ) ≥ 1, ind( w ) ≥ n − I ( w ) ≥ (cid:0) n (cid:1) , and by additivity of the Fredholm and ECH indices, we know OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 51 that ind( w ) + ind( w ) = 2 n − I ( w ) + I ( w ) = 1 + (cid:0) n (cid:1) . It follows thatind( w ) = I ( w ) = 1, ind( w ) = 2 n − 4, and I ( w ) = (cid:0) n (cid:1) , so ( w , w ) ∈ S c and weare in case (2).Next, assume that ev I ( w ) = and that all connector components in the buildingare trivial. Thus, w contains n − R × β and one connected, unbranched,3-fold cover of R × β . The component of w containing the negative end labeled i is paired with some negative end of w , which we also label i . If the componentof w containing the negative end labeled i is a trivial connector, it is paired withsome negative end of w , which we also label i . Proceeding in this way, we reacha level w ν i and an end, which we label i , that is paired with a trivial connector in w ν i − and such that ev i ( w ν i ) = 0. Such a level w ν i exists because none of the curves v k contains a component mapping to R × β .We claim that ν i = 2 for all i . If not, then either ν i = m > i or ν i < ν r for some i and r . In the first case, ind( w m ) < n − w ′ ν , . . . , w ′ ν n − such that ev i ( w ν i ) = c i for all i . Then the disjoint union w = w ′ ν ⊔ · · · ⊔ w ′ ν n − is a somewhere injective curve with ind( w ) < n − I ( w ) = c , contradicting Remark 8.1.2. In the second case, for any δ > v ′ k of v k by distances a k in the R -direction when k is sufficiently large sothat(8.2.1) | ev i ( v ′ k ) − ev i ( w ν i ) | < δ and | ev r ( v ′ k ) | < δ. Recall that if λ is the smallest positive asymptotic eigenvalue of β and e λ is thesmallest positive asymptotic eigenvalue of β , then ev i ( v ′ k ) = e − e λ a i / c i if i = 1 andev i ( v k ) = e − λ a i c i if i > 1. Thus, when δ is sufficiently small, the conditions in(8.2.1) contradict the assumption that ev I ( v k ) = c .By a previous argument, some translate w ′ of w satisfies ev I ( w ′ ) = c . It followsthat w is somewhere injective, ind( w ) ≥ n − 4, and I ( w ) ≥ (cid:0) n (cid:1) . Since ind( w ) ≥ I ( w ) ≥ 1, we see that ind( w ) = I ( w ) = 1, ind( w ) = 2 n − 4, and I ( w ) = (cid:0) n (cid:1) . Since w contains n − R × β and one unbranched cover of R × β of multiplicity 3, it follows that ([ w ] , [ w ]) ∈ S inv c , and we are in case (3).Now assume that ev I ( w ) = and that the building contains at least one non-trivial connector. Let v , , . . . , v ,m denote the components of w that are non-trivial connectors. Let e I ⊂ I be the labels of the negative ends of w that arecontained in trivial connectors over β . As before, for each j ∈ e I , the componentcontaining the negative end labeled j is paired with some negative end of w , whichwe also label j . Proceeding as above, we reach a level w ν j and an end, which welabel j , that is paired with a trivial connector over β in w ν j − and such that eitherev j ( w ν j ) = 0 or the end is contained in a non-trivial connector. Let w k , . . . , w k r be the levels containing non-connector components reached by the above procedure, and let w ℓ , . . . , w ℓ e be the levels containing non-trivial connector components com-ponents reached by the above procedure. By an argument from case (2), we musthave r = 1. Let w denote the union of the relevant non-connector components in w k . The curve w is somewhere injective by an argument from case (2). For each i = 1 , . . . , e , let v i, , . . . , v i,d i be the relevant components of the non-trivial connec-tors in w ℓ i . Let I ′ ⊂ e I be the subset of indices where we reach a non-connector inthe above procedure.Recall the definitions of M g,k, (1 , . . . , 1) and M g,k, (1 , . . . , 1) from Notation 6.7.1. Claim 8.2.2. In the above setup, for each i = 1 , . . . , e and each j = 1 , . . . , d i ,there is some k i,j ≥ such that v i,j is in M g i,j ,k i,j , (1 , . . . , for some g i,j > or M g i,j ,k i,j , (1 , . . . , for some g i,j ≥ .Proof of Claim 8.2.2. Let [ u ] be a J -holomorphic curve obtained by gluing [ w ] ∪· · · ∪ [ w ℓ ], where we take the gluing parameters to be large. Over any cylindricalportion of the domain of u where u is close to and graphical over R × β , u can bewritten in cylindrical coordinates ( s, t ) as u ( s, t ) = ( s, t, e u ( s, t )), where e u ( s, t ) has aFourier-type expansion e u ( s, t ) = X i =0 c i e − λ i s f i ( t ) . Perturb J as in [BH2, Lemma 3.4.3] so that c = 0 over each such cylindrical portionand such that the coefficients of f are distinct on distinct cylindrical pieces.Pre-glue the building [ w ] ∪ · · · ∪ [ w ℓ ], with the exception of v i,j and the trivialconnectors below it, to a curve v + . There is a section ψ − of the normal bundle of v i,j defined on the portion of the domain of v i,j obtained by truncating the positive endssuch that the perturbation of v i,j by ψ − (using an appropriate exponential map)coincides with u . There is also a section ψ + of the normal bundle of (a translationof) v + on the portion of the domain of v + obtained by truncating the negative endsasymptotic to β such that the perturbation of v + by ψ + coincides with u .By the proof of [BH1, Claim 8.8.3], we can extend ψ − and ψ + to sections e ψ − and e ψ + defined over the whole domain of v i,j and v + , respectively, that formally satisfythe necessary equation for the obstruction section over the moduli space of branchedcovers containing v i,j has a zero. Our perturbation of J above ensures that the e ψ ± also formally satisfy the necessary equation for the linearized obstruction section s over said moduli space to have a zero. The claim now follows from Proposition 6.7.2and Proposition 6.7.5. (cid:3) By a previous argument, ev I ′ ( w ) = ( c j ) j ∈ I ′ ; hence, w is somewhere injective,and ind( w ) ≥ | I ′ | by Remark 8.1.2. If no v i,j has a negative end with multiplicity3, then | I ′ | + e X i =1 d i X j =1 k i,j = n − , OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 53 and 2 n − ≥ ind( w ) + e X i =1 d i X j =1 ind( v i,j ) ≥ | I ′ | + 2 e X i =1 d i X j =1 ( k i,j − g i,j )= 2 n − e X i =1 d i X j =1 ( g i,j − . Hence, g i,j = 1 for all i and j , and either ind( w ) = 2 | I ′ | + 1 or there exists anadditional component w with ind( w ) = 1 in some level of the building. In thelatter case, w is somewhere injective. In either case, we can glue every somewhereinjective curve in the building [ w ] ∪ · · · ∪ [ w ℓ ] and produce a somewhere injective J -holomorphic curve u in R × Y + with I ( u ) = 1 + (cid:0) n (cid:1) , ∆( u ) = 1 + (cid:0) n − (cid:1) , andind( u ) < n − 4. As in the proof of Claim 8.2.2, we can perturb J to a new, generic J ′ to ensure that the negative ends of u asymptotic to β all non-degenerate andnon-overlapping. Then equality must hold in (1.2.2), and we have a contradiction.Now assume that some, and hence exactly one, v a,b has a negative end withmultiplicity 3. Then | I ′ | + e X i =1 d i X j =1 k i,j = n, and 2 n − ≥ ind( w ) + e X i =1 d i X j =1 ind( v i,j ) ≥ | I ′ | + +2( k a,b − g a,b ) + 2 X i = a,j = b ( k i,j − g i,j )= 2 n − g a,b + 2 X i = a,j = b ( g i,j − , so g a,b = 0 and g i,j = 1 if ( i, j ) = ( a, b ). As before, either ind( w ) = 2 | I ′ | + 1 or thereexists an additional component w with ind( w ) = 1 in some level of the building,and in the latter case, w is somewhere injective. We claim that in either case, i = 1, d = 1, and k , = n . It follows that the only branched cover in the building is acurve in M n and that we are in case (1).To prove the above assertion, glue every curve in the building [ w ] ∪ · · · ∪ [ w ℓ ]except for v a,b to produce a somewhere injective J -holomorphic curve u in R × Y + with I ( u ) = 1+ (cid:0) n (cid:1) and ∆( u ) = (cid:0) n (cid:1) . If i > 1, it follows that ind( u ) > 1, contradicting(1.2.2). If d > 1, we have the same contradiction. If k , < n , then | I ′ | > 0, andhence ind( u ) > 1, again yielding a contradiction. (cid:3) Definition 8.2.3. Let c ∈ C n − be a generic, admissible asymptotic restriction.Choose an open set U as in Definition 8.1.12. For each pair ( w − , w + ) ∈ S c (resp. S inv c ( w − , w + )), choose a δ > U c ( w − , w + ) (resp. U inv c ( w − , w + )) asin Definition 8.1.12. The truncation of K c is the set K ′ c = K c \ U ⊔ G ( w − ,w + ) ∈S c U c ( w − , w + ) ⊔ G ( w − ,w + ) ∈S inv c U inv c ( w − , w + ) . By Lemma 8.2.1, K ′ c is compact. Definition 8.2.4. Let c ∈ C n − be a generic, admissible asymptotic restriction.Let e ∂K ′ c be the set of points of ∂K ′ c that lie in G R . The truncation map is themap B c : ∂K ′ c / R → ( S c / R ) ⊔ ( S inv c / R ) ⊔ ( e ∂K ′ c / R )that sends a curve in e ∂K ′ c / R to itself and every other curve in ∂K ′ c to the 2-levelbuilding into which it is close to breaking. Lemma 8.2.5. Let c ∈ C n − be a generic, admissible asymptotic restriction. If ([ w − ] , [ w + ]) is an element of S c / R or S inv c , then the mod count of points in B − c ([ w − ] , [ w + ]) is if the intermediate orbit set γ is not a generator of the ECHchain complex for ( Y + , λ + ) and is if γ is a generator.Proof. We use the quotient evaluation map from [BH1, Section 6]. Let λ be thesmallest positive asymptotic eigenvalue of β , and let e λ be the smallest positiveasymptotic eigenvalue of β If w ∈ M , ev I ( w ) = ( a , . . . , a n − ), and w ′ is obtainedby translating w a distance s in the R -direction, thenev I ( w ′ ) = ( a e − e λ s/ , a e − λ s , . . . , a n − e − λ s ) . The proof of Lemma 4.3.2 shows that there are no curves w ∈ M with ev I ( w ) = ,since such curves would necessarily have I ( w ) > (cid:0) n (cid:1) . Thus, the evaluation mapdescends to a smooth map on the quotient ev I : M / R → ( C n − \ { } ) / R + ∼ = S n − .Given a pair ([ w − ] , [ w + ]) ∈ S c / R and a neighborhood U − of [ w − ] in e V n, γ , β , n − , wecan identify an open set in M / R with U = [ R, ∞ ) × U − such that as the parameter T ∈ [ R, ∞ ) goes to infinity, the curve breaks into a two-level building in U − ×{ [ w + ] } .The map ev I extends smoothly to the broken curves on the boundary of U . If U − issufficiently small, ev I is a submersion on U − and U . Hence, for every gluing of [ w − ]with [ w + ], there is a unique end of K c that is compactified by adding the building[ w − ] ∪ [ w + ]. In particular, if we choose δ to be sufficiently small when truncating K c and U − is sufficiently small, there is a unique endpoint of K ′ c in U . The proofwhen ([ w − ] , [ w + ]) ∈ S inv c is similar. (cid:3) Proof of Theorem 1.5.3. The mod 2 count of points in e ∂K ′ c is equal to the mod 2count of non-canceling buildings described in Theorem 1.3.2 such that the b X -level OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 55 has an n -fold degenerate cover of a plane at β . Note that in this proof, we denotethe negative orbit set of the symplectization level by β and the negative orbit set ofthe cobordism level by γ . By Lemma 8.2.5, this count is equal to the count of gluingsof pairs ([ w − ] , [ w + ]) in S c / R or S inv c / R whose intermediate orbit set is a generatorof the ECH complex. Pairs in S c / R contribute to the count for Φ X,λ,J, c ◦ ∂ .We claim that pairs ([ w − ] , [ w + ]) ∈ S inv c / R correspond to ECH buildings thatcontribute to the count for ∂ ◦ Φ X,λ,J, c . We can glue w − to the curve u in b X fromTheorem 1.3.2 since the non-trivial component of w − has no negative ends at β .Consider the subset N of the moduli space M , − n ( n +1) / X ( β , γ ) consisting of curvesthat contain a degenerate n -fold cover of a plane with a positive puncture at β . Themultiplicity of any orbit in γ and β besides β is 1, so any component of a curve in N that is not the degenerate cover is somewhere injective. The degenerate covers arecut out transversely by Wendl’s automatic transversality criterion [BH1, Theorem4.2.1]. Thus, N is a transversely cut out 1-manifold. The boundary points of its SFTcompactification are two-level buildings of the form [ v − ] ∪ v or v ∪ [ v ], where v isin b X and v ± are in R × Y − , such that ind( v ) = 0, ind( v ± ) = 1, I ( v ) = − (cid:0) n (cid:1) , and I ( v ± ) = 1. There are finitely many boundary points. Buildings of the form v ∪ [ v ]correspond to pairs in S inv c / R , while buildings of the form [ v − ] ∪ v contribute to thecount for ∂ ◦ Φ X,λ,J, c when v is paired with [ w + ]. The map Φ X,λ,J, c is independentof the choice of c by Remark 8.1.8. The proof of Theorem 1.5.3 is now completewhen the gluing problem is in the case n ≥ n = 2. When we glue branched covers in M (1 , | u , the result lives in the moduli space M , R × Y + ( α , β ). Then M , R × Y + ( α , β ) / R has dimension 1, and all endpoints must be two-level buildings [ w − ] ∪ [ w + ] withind( w − ) = ind( w + ) = 1 and I ( w − ) = I ( w + ) = 1. There is no evaluation map inthis case. (cid:3) Appendix A. Existence of Models In this appendix, we prove Theorem 7.1.3. Recall that our branched covers havemultiplicity 1 positive punctures p , . . . , p n , a multiplicity 3 negative puncture q ,and multiplicity 1 negative punctures q , . . . , q n − . An order m model, if it exists,is necessarily given by g ( z ) = n X i =1 m X ℓ =1 e iℓθ e − ℓT r ℓi α i,ℓ ( z − p i ) ℓ , As before, set h ( ζ ) = g ( ζ − ) and note that h has a removable singularity at ζ = 0and vanishes there. The function g is an order m model associated to a branchedcover u if and only if(A.1) h ′ (0) = 0 and g ( q ) = · · · = g ( q n − ) = 0 , By (7.2.3), (7.2.4), (6.1.1), and (6.5.1), the equations (A.1) are equivalent to(A.2) n X i =1 e iθ e − T r i α i, = 0 and n X i =1 m X ℓ =1 e iℓθ e − ℓT r ℓi α i,ℓ ( q k − p i ) ℓ = 0for k = 2 , . . . , n − Notation A.1. Define ( n − × n matrices A = · · · q − p ) − · · · ( q − p n ) − ... . . . ...( q n − − p ) − · · · ( q n − − p n ) − and, for ℓ > A ℓ = · · · q − p ) − ℓ · · · ( q − p n ) − ℓ ... . . . ...( q n − − p ) − ℓ · · · ( q n − − p n ) − ℓ . Define the ( n − × nm block matrix A = (cid:0) A A · · · A m (cid:1) . Finally, define the n × α ℓ = e iℓθ e − ℓT (cid:16) r ℓ α ,ℓ , · · · , r ℓn α n,ℓ (cid:17) t for all ℓ ≥ mn × α = (cid:0) α t , α t , · · · , α tm (cid:1) t , where a superscript t indicates the transpose of a matrix. Proof of Theorem 7.1.3. The equations (A.2) hold if and only if α ℓ is in the nullspaceof A ℓ for all ℓ = 1 , . . . , m , i.e., if and only if α is in the nullspace of the blockmatrix A . We relate (A.2) to the equations (6.5.3) defining Z m by performing rowoperations on the matrix A to put it in echelon form, at which point the block A ℓ hasbeen converted to the matrix B ℓ from (6.5.2). We first describe the row-reductionalgorithm as a sequence of steps. Fix ℓ and write our starting matrix A ℓ as a matrixof row vectors: A ℓ = — A —...— A n − — . At every step of the row-reduction process, we will refer to the matrix obtained atthat step by e A and the rows of e A by e A , . . . , e A n − . The rows of the original matrix A ℓ will always be denoted A , . . . , A n − . OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 57 The j th step of our row-reduction algorithm, j = 1 , . . . , n − 2, is as follows. If j > 1, then for i = 1 , . . . , j − 1, replace the row e A i with e A i + q j − p j p j − p i e A j and multiply the resulting row by p j − p i q j − p i . If j < n − 2, then for i = j + 1 , . . . , n − e A i with e A i − q j − p j q i − p j e A j and multiply the resulting row by q i − p j q j − q i . Finally, multiply rows 1 , . . . , j by − Notation A.2. If I ⊂ { , . . . , n − } is a set of indices, define A I ( z ) = n − Y i =1 i I ( z − p i ) and B I ( z ) = n − Y i =2 i I ( z − q i ) , where an empty product is defined to be 1. Note that A { i } ( z ) = A i ( z ) and B { i } ( z ) = B i ( z )in the notation from Notation 6.0.2. For any j ∈ { , . . . , n − } , define the set I j = { j, . . . , n − } . Claim A.3. After step j of the row reduction, the i th row e A i of the resulting matrix e A is as follows. For ≤ i ≤ j , ( − j " A − j X k =2 A { i }∪ I j +1 ( q k ) B { k }∪ I j +1 ( q k ) A k . For j + 1 ≤ i ≤ n − , ( − j " A − A I j +1 ( q i ) B I j +1 ( q i ) A i − j X k =2 A I j +1 ( q k )( q k − q i ) B { k }∪ I j +1 ( q k ) A k . In both cases, an empty sum is defined to be the zero row vector.Proof of Claim A.3. We proceed by induction on j . Note that, after step 1, thematrix e A is given by e A = − A ( q − p ) A − A ...( q n − − p ) A n − − A . Now assume that the claim holds at step j . Then the ( j + 1) st row of e A is alreadyin the correct form for step j + 1, and after that step, the other rows of e A are asfollows. For i ≤ j , the i th row is( − j +1 " A − j X k =2 A { i }∪ I j +1 ( q k ) B { k }∪ I j +1 ( q k ) A k + q j +1 − p j +1 p j +1 − p i A − j +1 X k =2 A I j +1 ( q k ) B { k }∪ I j +2 ( q k ) A k ! p j +1 − p i q j +1 − p i = ( − j +1 " A − A { i }∪ I j +2 ( q j +1 ) B I j +1 ( q j +1 ) A j +1 − j X k =2 " A { i }∪ I j +1 ( q k )( q j +1 − p i ) B { k }∪ I j +2 ( q k ) · (cid:16) ( p j +1 − p i )( q k − q j +1 ) + ( q j +1 − p j +1 )( q k − p i ) (cid:17) A k i = ( − j +1 " A − A { i }∪ I j +2 ( q j +1 ) B I j +1 ( q j +1 ) A j +1 − j X k =2 " A { i }∪ I j +1 ( q k )( q j +1 − p i ) B { k }∪ I j +2 ( q k ) ( q j +1 − p i )( q k − p j +1 ) A k = ( − j +1 " A − j +1 X k =2 A { i }∪ I j +2 ( q k +1 ) B { k +1 }∪ I j +3 ( q k +1 ) A k . For j + 2 ≤ i ≤ n − 2, the i th row is( − j " A − A I j +1 ( q i ) B I j +1 ( q i ) A i − j X k =2 A I j +1 ( q k )( q k − q i ) B { k }∪ I j +1 ( q k ) A k ! − q j +1 − p j +1 q i − p j +1 A − j +1 X k =2 A I j +1 ( q k ) B { k }∪ I j +2 ( q k ) A k ! q i − p j +1 q j +1 − q i = ( − j " − A + A I j +2 ( q i ) B I j +2 ( q i ) A i + A I j +2 ( q j +1 )( q j +1 − q i ) B I j +1 ( q j +1 ) A j +1 − j X k =1 " A I j +1 ( q k )( q j +1 − q i )( q k − q i ) B { k }∪ I j +2 ( q k ) · (cid:16) ( q j +1 − p j +1 )( q k − q i ) − ( q i − p j +1 )( q k − q j +1 ) (cid:17) A k i OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 59 = ( − j +1 " A − A I j +2 ( q i ) B I j +2 ( q i ) A i − A I j +2 ( q j +1 )( q j +1 − q i ) B I j +1 ( q j +1 ) A j +1 − j X k =1 A I j +1 ( q k )( q j +1 − q i )( q k − q i ) B { k }∪ I j +2 ( q k ) ( q j +1 − q i )( q k − p j +1 ) A k = ( − j +1 " A − A I j +2 ( q i ) B I j +2 ( q i ) A i − A I j +2 ( q k )( q k − p i ) B { k }∪ I j +2 ( q k ) A k . The claim follows by induction. (cid:3) After step n − e A by ( − n − . ByClaim A.3, the i th row of the resulting matrix is A − n − X k =2 A i ( q k ) B k ( q k ) A k . The j th entry of this row is1 − n − X k =2 A i ( q k ) B k ( q k ) 1 q k − p j when ℓ = 1 and − n − X k =2 A i ( q k ) B k ( q k ) 1( q k − p j ) ℓ when ℓ > ℓ − d ℓ − Q i dz ℓ − ( p j )by Remark 6.5.5, and we are done. (cid:3) Appendix B. Determinant Calculations In this appendix, we prove Claim 7.2.8. For notational simplicity, we abbreviate E ℓ,i = E ℓ ( p , . . . , b p i , . . . , p n ) and E ℓ, ( i,j ) = E ℓ ( p , . . . , b p i , . . . , b p j , . . . , p n ). For com-pactness of notation, we also abbreviate α i = α i, . To reduce bookkeeping withsigns, we will instead compute the determinant (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · · p · · · p n p · · · p n ... . . . ... p n − · · · p n − n α E ℓ, · · · α n E ℓ,n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , which differs from our desired determinant by a factor of ( − n ). Lemma B.1. If n ≥ and ≤ ℓ ≤ n − , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · · p · · · p n p · · · p n ... . . . ... p n − · · · p n − n E ℓ, · · · E ℓ,n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ( , ≤ ℓ ≤ n − − n − )∆ , ℓ = n − . Proof. The proof is by induction on n . Note that E ℓ,k − E ℓ, = ( p − p k ) E ℓ − , (1 ,k ) for k ≥ 2, so that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · · p p · · · p n p p · · · p n ... ... . . . ... p n − p n − · · · p n − n E ℓ, E ℓ, · · · E ℓ,n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · · p − p · · · p n − p p ( p − p ) · · · p n ( p n − p )... ... . . . ...0 p n − ( p − p ) · · · p n − n ( p n − p )0 E ℓ, − E ℓ, · · · E ℓ,n − E ℓ, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ( − n A ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · · p · · · p n ... . . . ... p n − · · · p n − n E ℓ − , (1 , · · · E ℓ − , (1 ,n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Our base case is n = 4. When ℓ = 0 or 1, the result is immediate, as E ,i = E , (1 ,i ) = 1 for all i . When ℓ = 2, we also see that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p p p E ℓ − , (1 , E ℓ − , (1 , E ℓ − , (1 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p p p p + p p + p p + p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . Finally, when ℓ = 3, we see that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p p p E ℓ − , (1 , E ℓ − , (1 , E ℓ − , (1 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p p p p p p p p p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = − ( p − p )( p − p )( p − p ) , and the result follows. OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 61 Now assume the result for n − ℓ = 0 , . . . , n − 2. We prove it for n andall ℓ = 0 , . . . , n − 1. We have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · · p p · · · p n p p · · · p n ... ... . . . ... p n − p n − · · · p n − n E ℓ, E ℓ, · · · E ℓ,n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ( − n A ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · · p · · · p n ... . . . ... p n − · · · p n − n E ℓ − , (1 , · · · E ℓ − , (1 ,n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ( , ≤ ℓ ≤ n − , ( − n A ( p )( − n − )∆ , ℓ = n − ( , ≤ ℓ ≤ n − , ( − n − )∆ , ℓ = n − (cid:3) Proof of Claim 7.2.8. First, we prove the claim when 0 ≤ ℓ ≤ n − 2. We proceed byinduction on ℓ , starting with ℓ = 0 and ℓ = 1. The case ℓ = n − ℓ = 0, note that (6.5.5) implies that α k − α = ( p k − p ) α n − − α n p n − − p n , for k = 1 , . . . , n , which then implies that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · · p p · · · p n p p · · · p n ... ... . . . ... p n − p n − · · · p n − n α α · · · α n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · · p − p · · · p n − p p ( p − p ) · · · p n ( p n − p )... ... . . . ...0 p n − ( p − p ) · · · p n − n ( p n − p )0 α − α · · · α n − α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0since, after expanding along the first column, the last row is a multiple of the first.Now we treat the inductive step. Assume the result is true for ℓ − ≤ n − 3. Weprove it for ℓ . First, note that p α k − p k α = ( p k − p ) p n α n − − p n − α n p n − − p n by (6.5.5), so that E ℓ,k α k − E ℓ, α = E ℓ − , (1 ,k ) ( p α k − p k α ) + E ℓ, (1 , ( α k − α )= ( p k − p ) (cid:20) E ℓ − , (1 ,k ) p n α n − − p n − α n p n − − p n + E ℓ, (1 ,k ) α n − − α n p n − − p n (cid:21) . Hence, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · · p p · · · p n p p · · · p n ... ... . . . ... p n − p n − · · · p n − n E ℓ, α E ℓ, α · · · E ℓ,n α n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · · p − p · · · p n − p p ( p − p ) · · · p n ( p n − p )... ... . . . ...0 p n − ( p − p ) · · · p n − n ( p n − p )0 E ℓ, α − E ℓ, α · · · E ℓ,n α n − E ℓ, α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ( − n − A ( p ) p n α n − − p n − α n p n − − p n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · · p · · · p n ... . . . ... p n − · · · p n − n E ℓ − , (1 , · · · E ℓ − , (1 ,n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ( − n − A ( p ) α n − − α n p n − − p n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · · p · · · p n ... . . . ... p n − · · · p n − n E ℓ, (1 , · · · E ℓ, (1 ,n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) When 0 ≤ ℓ ≤ n − 3, both determinants vanish by Lemma B.1. When ℓ = n − − n − ) − α n − − α n p n − − p n ∆ . The result follows in this case.We now prove the claim when ℓ = n − 1. In this case, we have E n − ,k α k − E n − , α = ( p k − p ) E n − , (1 ,k ) p n α n − − p n − α n p n − − p n , so, by Lemma B.1, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · · p p · · · p n p p · · · p n ... ... . . . ... p n − p n − · · · p n − n E ℓ, α E ℓ, α · · · E ℓ,n α n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · · p − p · · · p n − p p ( p − p ) · · · p n ( p n − p )... ... . . . ...0 p n − ( p − p ) · · · p n − n ( p n − p )0 E ℓ, α − E ℓ, α · · · E ℓ,n α n − E ℓ, α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) OBORDISM MAPS IN EMBEDDED CONTACT HOMOLOGY 63 = ( − n − A ( p ) p n α n − − p n − α n p n − − p n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · · p · · · p n ... . . . ... p n − · · · p n − n E n − , (1 , · · · E n − , (1 ,n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ( − n − A ( p ) p n α n − − p n − α n p n − − p n ( − n − )∆ = ( − n − ) − p n α n − − p n − α n p n − − p n ∆ . 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