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On cobordism maps on periodic Floer homology

Guanheng Chen

Abstract

In this article, we investigate the cobordism maps on periodic Floer ho-

mology (PFH). In the first part of the paper, we define the cobordism maps

on PFH via Seiberg Witten theory as well as the isomorphism between PFH

and Seiberg Witten cohomology. Furthermore, we show that the maps sat-

isfy the holomorphic curve axiom. In the second part of the paper, we

give an alternative definition of these maps by using holomorphic curve

method, provided that the symplectic cobordisms are Lefschetz fibration

satisfying certain nice conditions. Under additionally certain monotonicity

assumptions, we show that these two definitions are equivalent.

1 Introduction

Let (Σ, ωΣ) be a connected closed 2-dimensional symplectic manifold and φ : Σ → Σ

be a symplectomorphism. Let Y be a mapping torus of (Σ, φ), i.e.,

Y = R×Σ/(t+ 2π, x) ∼ (t, φ(x)).

The coordinate vector field ∂t and the volume form ωΣ descend respectively to a vector

field ∂t and a closed 2-form ωφ on Y under the natural projection R × Σ → Y . The

closed integral curves of ∂t are called periodic orbits.

In [8], M. Hutchings proposes an invariant HP∗(Y, ωφ,Γ, J,ΛP ) for the pair (Y, ωφ)

which is so-called periodic Floer homology, abbreviated as PFH. Here J is a generic

almost complex structure, Γ ∈ H1(Y,Z), and ΛP is a local coefficient. Roughly speak-

ing, the chain complex of HP (Y, ωφ,Γ, J,ΛP ) is generated by periodic orbits, and its

differential is defined by counting holomorphic currents with ECH index one. PFH is

shown to be well defined by Hutchings and Taubes in [13] and [14]. PFH is a sister

version of a more well-known invariant called embedded contact homology, abbreviated

as ECH. In Tuabes’ series of papers [28], [29], [30], [31] and [32], he shows that ECH

is isomorphic to Seiberg Witten cohomology. Likewise, periodic Floer homology is also

isomorphic to a version of Seiberg Witten cohomology.

1

Theorem 1.1 (Theorem 6.2 of [18]). Given a fixed homology class Γ ∈ H1(Y,Z), let ΛP

be a (cΓ, [ω])-complete local coefficient for the periodic Floer homology. Let ΛS denote a

corresponding (c1(sΓ), 2πr[ω])-complete local coefficient for Seiberg Witten cohomology

in the sense of [18]. There exists an isomorphism between periodic Floer homology and

a perturbed version of Seiberg Witten cohomology

Tr∗ : HP∗(Y, ωφ,Γ, J,ΛP ) → HM−∗(Y, sΓ,−πr, J,ΛS),

which reverses the relative grading.

In [33], M. Usher defines an invariant HF∗(Y, [ω],Γ,Λ) for a tuple (Y, π, [ω],Γ),

where π : Y → S1 is a surface fibration over circle, [ω] ∈ H2(Y,R), Γ ∈ H1(Y,Z)

and Λ is a local coefficient. Additionally, he shows that HF is a covariant functor

from the fibered cobordism category (FCOB) to the category of modules over univer-

sal Novikov ring. Roughly speaking, the object in FCOB is surface fibration over the

circle and the morphism is Lefschetz fibration. The invariants HF∗(Y, [ω],Γ,Λ) and

HP∗(Y, ωφ,Γ,ΛP ) are respectively three dimensional analogy of Donaldson-Smith’s in-

variant [4] and Taubes’ Gromov invariant [24], thus one might expect that these two

homologies are isomorphism. Motivated by [33], it is also expected to construct a

TQFT structure on PFH which is an analogy to Usher’s on HF .

To this end, we need to define the cobordism maps on PFH induced by Lefschetz

fibration. The natural way to define the cobordism maps is by counting holomorphic

curves with zero ECH index. However, as explained in Section 5.5 of [10], the ECH

index can be negative in general. Consequently, we need to count broken holomorphic

curves with zero ECH index. The moduli space of broken holomorphic curves has many

components of various dimensions, so it is difficult to handle it. To overcome the techni-

cal difficulties, Hutchings and Taubes define the cobordism maps on embedded contact

homology by using the isomorphism “HM=ECH”[28] and Seiberg Witten theory [12].

Since PFH is a cousin of embedded contact homology, it is natural to expect that

the parallel results hold for PFH. In some special cases, the ECH index is still possible

to be nonnegative. In these cases, can we define the cobordism maps on PFH without

Seiberg Witten theory? If we can, it is natural to ask weather we can generalize the

isomorphism in Theorem 1.1 to cobordism level.

The purpose of this article is to explore the cobordism maps on PFH and we answer

the questions above for certain Lefschetz fibrations. Firstly, we follow the idea in [12]

to define the cobordism maps HP sw(X,ΩX , J,ΛX ) on PFH via Seiberg Witten theory

and Theorem 1.1. Since the Seiberg Witten theory works for any 4-manifold, the maps

HP sw(X,ΩX , J,ΛX) are well-defined for general symplectic cobordisms, not limited

to Lefschetz fibration. These maps HP sw(X,ΩX , J,ΛX) inherit the natural properties

of Seiberg Witten cobordism maps. But beyond that, the maps satisfy an additional

2

property called “holomorphic curve axiom”, which relates the Seiberg Witten equations

and holomorphic curves. Secondly, we focus on the case that the symplectic cobordisms

admit Lefschetz fibration structures. Under certain technical assumptions (♠), we give

an alternative definition of the cobordism maps, denoted by HP (X,ΩX , J,ΛX ). These

maps are defined by counting embedded holomorphic curves with zero ECH index, by

using the same techniques in [13] and [14], and without using Seiberg Witten theory.

Furthermore, in most cases, we can deform the symplectic structure on the Lefschetz

fibration such that it meets these technical assumptions (♠). Finally, under certain

monotonicity assumptions, we show that these two definitions are equivalent. In other

words, Theorem 1.1 holds in cobordism level under the assumptions (♠). The proof of

this part relies heavily on Tuabes’ series of papers [28], [29], [30], and [31].

Finally, we mention that some relevant results on ECH side have been achieved

recently. In order to generalize Taubes’ Gromov invariant to nearly symplectic mani-

folds, C.Gerig defines the cobordism maps on ECH for certain symplectic cobordisms

[5]. In addition, he compares the cobordism maps on ECH with the cobordism maps

on Seiberg Witten cohomology [6]. There are various similarities between the results

here and C.Gerig’s. We will compare our results with Gerig’s in Remark 21. Besides,

J. Rooney defines the cobordism maps on ECH by counting holomorphic curves for

general settings [20]. In his definition, the cobordism maps are defined by counting

holomorphic building and the holomorphic curves are allowed to have negative ECH

index.

2 Preliminaries

Before we state the main results, let us introduce some essential definitions in this

section.

2.1 Surface fibration over S1 and periodic orbits

Let π : Y → S1 be a 3-dimensional fibration over S1. Let ω be a fiberwise non-

degenerate closed 2-form on Y , then there is a decomposition TY = TY hor ⊕ TY vert

with respect to ω, where TY vert = ker π∗ and TxYhor = v ∈ TxY : ω(v,w) = 0,∀w ∈

TxYvert. Let ∂t be the coordinate vector field of S1. The horizontal lift of ∂t is the

unique vector field R on Y such that R ∈ TY hor and π∗(R) = ∂t. Such a vector field

R is called Reeb vector field.

Fix a base point 0 ∈ S1, let φt be the flow generated by R starting at π−1(0)

and φω = φ1, then Y can be identified with a mapping torus Σφω via the following

3

diffeomorphism

F : x ∈ π−1(t) → (t, φ−1t (x)) ∈ R× Σ, (1)

where Σ = π−1(0) and the mapping torus Σφω is defined by

Σφω = R× Σ/(t+ 2π, x) ∼ (t, φω(x)).

Moreover, it is easy to check that F∗(R) = ∂t and F ∗ωφω = ω. The closed integral

curves of R are called periodic orbit. Let γ be a periodic orbit of R, then the intersection

pairing of γ with the fiber, namely d = [γ] · [Σ], is called degree or period of γ. The

linearization of the flow along γ defines a symplectic linear map:

Pγ : (ker π∗|γ(0), ω|γ(0)) → (ker π∗|γ(0), ω|γ(0)).

A periodic orbit γ is called non-degenerate if 1 is not eigenvalue of Pγ . For any non-

degenerate periodic orbit γ, Pγ has eigenvalues λ and λ−1 which are either real and

positive, in which case γ is called positive hyperbolic, or real and negative, in which

case γ is called negative hyperbolic, or on the unit circle, in which case γ is called

elliptic.

Definition 2.1. Let π : Y → S1 be a surface fibration over S1. A 2-form ω ∈ Ω2(Y )

is called admissible if dω = 0 and ω is fiberwise nondegenerate. Given Q > 0, an

admissible 2-form ω is called Q-admissible if all the periodic orbits of φω with degree

less than Q are nondegenerate.

Definition 2.2. Fix Γ ∈ H1(Y,Z), an orbit set with homology class Γ is a finite set

of pairs α = (αi,mi), where αii are distinct irreducible simple periodic orbits and

mii are positive integers. In addition, we require that∑imi[αi] = Γ. An orbit set is

called admissible if mi = 1 whenever αi is hyperbolic. The set of admissible orbit sets

with homology class Γ is denoted by P(Y, ω,Γ).

We will need the following definition when we state the main results and define the

“self-intersection number” (Definition 4.4) of holomorphic curves later.

Definition 2.3. (Cf. Definition 4.1 of [11]) Let Q > 0 and γ be a simple elliptic orbit

with degree d ≤ Q.

• γ is called Q-positive elliptic if the rotation number θ ∈ (0, dQ) mod 1.

• γ is called Q-negative elliptic if the rotation number θ ∈ (− dQ , 0) mod 1.

4

2.2 Symplectic cobordism and Lefschetz fibration

In this subsection, we clarify the meaning of symplectic cobordism between fibered

3-manifolds.

Definition 2.4. Let (Y±, π±, ω±) be a surface fibration over circle together with admis-

sible 2-form. A symplectic cobordism from (Y+, π+, ω+) to (Y−, π−, ω−) is a symplectic

manifold (X,ΩX) such that ∂X = Y+ ⊔ (−Y−) and ΩX |Y± = ω±.

Definition 2.5. Let (X,ΩX ) be a symplectic cobordism between fibered 3-manifolds.

Given ΓX ∈ H2(X, ∂X,Z), the pair (ΓX ,ΩX) is called monotone if

c1(TX) + 2PD(ΓX) = τ [ΩX ]

for some τ 6= 0 ∈ R. If [ΩX ] = τc1(TX) for some τ 6= 0 ∈ R, we just say that (X,ΩX)

is monotone.

When we define the cobordism maps on PFH by using holomorphic curve method,

we focus on the symplectic cobordisms with fibration structures. (See Definition 2.9.)

Definition 2.6. (Cf. [22]) Let X be a compact, connected, oriented, smooth 4-manifold

and B be a compact, connected oriented surface possibly with boundary. A Lefschetz

fibration is a map πX : X → B with the following properties:

1. π−1X (∂B) = ∂X.

2. Each critical point of πX lies in the interior of X.

3. For each critical point of πX , we can find a pair of orientation preserving complex

coordinate charts, one on X, centered at the critical point, and one on B, such

that πX(z1, z2) = z21 + z22 on these charts.

Throughout this article, we assume that the regular fiber of πX : X → B is an ori-

ented connected closed surface and B has two boundary components ∂B = S1+

⊔(−S1

−)

so that ∂X = Y+⊔(−Y−). Note that Y+ and Y− are surface fibrations over circles. We

allow the boundary component to be empty. In addition, we assume that πX is injective

on the set of critical points.

The singular fiber of πX : X → B can be classified as separating and non-

separating, depending on whether the corresponding vanishing cycle is separating or

non-separating. Let z be a critical value of πX and g is genus of regular fiber. In the

case that π−1X (z) = Σ is non-separating, then Σ is an immersed surface of g − 1 with a

double point. If the singular fiber π−1X (z) = Σ is separating, then Σ is a transversely

intersecting pair of embedded surfaces with square −1 and genus adding to g. For more

details, please refer to [22].

5

Definition 2.7. (Cf. [22]) A Lefschetz fibration is called relatively minimal if there is

no fiber containing an embedded sphere with −1 self-intersection number. Such sphere

is called exceptional sphere.

The following definition is an analogy to Definition 2.1.

Definition 2.8. Let πX : X → B be a Lefschetz fibration. A 2-form ωX ∈ Ω2(X) is

called admissible if dωX = 0 and ωX is fiberwise nondegenerate. Near critical points

of πX , ωX is Kahler with respective to the coordinate charts (z1, z2) in Definition 2.6.

Given Q > 0, we say that an admissible 2-form ωX is Q-admissible if ωX |Y+and ωX |Y−

are Q-admissible.

Definition 2.9. Let π± : Y± → S1 be a 3-dimensional fibration over S1 together

with a Q-admissible 2-form ω±. A fiberwise symplectic cobordism from (Y+, π+, ω+)

to (Y−, π−, ω−) is a 4-dimensional Lefschetz fibration πX : X → B together with an

admissible 2-form ωX such that ∂X = Y+⊔(−Y−), πX |Y± = π± and ωX |Y± = ω±.

Given a fiberwise symplectic cobordism (X,πX , ωX), we can construct a symplectic

cobordism (X,ΩX) as follows: Firstly, we need the following lemma.

Lemma 2.10. Let (X,πX) be a Lefschetz fibration, Y = ∂X and U be a collar neigh-

borhood of Y in X such that U ≃ (−ε, 0] × Y for some ε > 0. Also, the projec-

tion πX ≃ id × π under above identification. Let η be a closed 2-form of X, then

η|U = η|0×Y + da for some a ∈ Ω1(U). Moreover, a|0×Y = 0.

Proof. Under the identification U ≃ (−ε, 0] × Y , we can write η = ηs + ds ∧ γs, whereηs ∈ Ω2(Y ) and γs ∈ Ω1(Y ) for each s ∈ (−ε, 0]. Since dη = 0, we have

d3ηs + ds ∧ ∂ηs∂s

− ds ∧ d3γs = 0,

where d3 is the exterior derivative of Y . Therefore, d3ηs = 0 and ∂ηs∂s = d3γs for each

s ∈ (−ε, 0]. Integrate both side of the second identity, we have

ηs = η|s=0 +

∫ s

0

∂ητ∂τ

= η|0×Y + d3

(∫ s

0γτ

).

Let a =∫ s0 γτ , then η = η|0×Y + d3a+ ds ∧ ∂a

∂s = η|0×Y + da.

According to Lemma 2.10, we can identify respectively neighborhoods of Y+ and

Y− in (X,πX , ωX) with collars of the form

((−2ε, 0] × Y+, ω+ + dµ+),

((0, 2ε] × Y−, ω− + dµ−).

6

Moreover, πX = id × π± under the identification. Fix a cut off functions φ so that

φ(s) = 0 when s ≤ 12 and φ(s) = 1 when s ≥ 3

4 . Let φ±(s) = φ( s∓2ε). We define a new

admissible 2-form on X by

ωXφ =

ωX , on X \ ((−2ε, 0] × Y+⋃(0, 2ε] × Y−)

ω+ + d(φ+(s+)µ+) on (−2ε, 0] × Y+

ω− + d(φ−(s−)µ−) on (0, 2ε] × Y−,

(2)

where s+ and s− are coordinates of (−2ε, 0] and (0, 2ε] respevtively. It is worth noting

that dωXφ = 0 and ωXφ is monotone whenever ωX is monotone.

Fix a volume form ωB on B. We can define a symplectic form ΩX = ωXφ+Kπ∗XωB

on X, where K is a large positive number so that ΩX ∧ ΩX > 0 everywhere. Fix a

large K, we find collar neighborhoods (−2ε, 0] × S1+ and (0, 2ε] × S1

− of ∂B such that

KωB = ds ∧ dt in these neighborhoods. Then (X,ΩX) is a symplectic cobordism from

(Y+, π+, ω+) to (Y−, π−, ω−).

Remark 1. One can construct the symplectic cobordism from (X,πX , ωX) directly by

taking ΩX = ωX + Kπ∗XωB. The advantage of the above construction is that the

symplectic completion of (X,ΩX) is still a Lefschetz fibration, see the next section.

Notation. To simplify the notation, from now on, we suppress the subscript φ

from the notation and assume that K = 1 all time. Also, given a fiberwise symplectic

cobordism (X,πX , ωX), the symplectic form ΩX over X always refers to the symplectic

form constructed above.

2.3 Symplectic completion

In order to introduce holomorphic curves with asymptotic ends, we define the comple-

tion of the symplectic cobordism or fiberwise symplectic cobordism by adding cylindri-

cal ends.

Suppose that (X,ΩX) is a symplectic cobordism from (Y+, π+, ω+) to (Y−, π−, ω−).

By Moser’s trick (Cf. Proposition 6.4 of [34]), we can identify the neighborhoods of Y+

and Y− in (X,ΩX) symplectically with collars of the form

((−ε, 0] × Y+, ω+ + ds ∧ π∗+dt),((0, ε] × Y−, ω− + ds ∧ π∗−dt).

Granted this identification, define the completion of (X,ΩX) by adding cylindrical ends

X = ((−∞, 0]× Y−) ∪Y− X ∪Y+([0,+∞)× Y+).

7

We call [0,+∞)×Y+ and (−∞, 0]×Y− the ends of X. The symplectic form ΩX extends

to be ω± + ds ∧ π∗±dt over the ends.

It is worth noting that when (X,ΩX ) is constructed from a fiberwise symplectic

cobordism (X,πX , ωX), then the fibration structure automatically extends to πX :

X → B, where B = ((−∞, 0] × S1−) ∪S1

−B ∪S1

+([0,+∞) × S1

+).

Recall that ωB = ds ∧ dt near ∂B. In order to define the almost complex structure

adapted to fibration latter, we fix a ωB-compatible complex structure jB over B such

that jB(∂s) = ∂t over the ends of B throughout.

2.4 Almost complex structures

In order to define the periodic Floer homology and its cobordism maps in different

settings, we need the following several types of almost complex structures.

2.4.1 Almost complex structures on R× Y

Definition 2.11. Let π : Y → S1 be a surface fibration as before and ω be an admis-

sible 2-form on Y . An almost complex structure J on R × Y is called symplectization

admissible if J satisfies the following properties:

1. J is compatible with Ω = ω + ds ∧ π∗dt.

2. J is R-invariant and J(∂s) = R.

3. J maps kerπ∗ to ker π∗ .

Let Jcomp(Y, π, ω) denote the space of all symplectization admissible almost complex

structures. It is contractible and carries a natural C∞-topology. Let Jcomp(Y, π, ω)reg

denote the Baire subset of admissible almost complex structures that satisfy the criteria

[J1], [J2a] and [J2b] in [18].

2.4.2 Almost complex structures on X

Definition 2.12. An almost complex structure J on X is called cobordism admissible

if J satisfies the following properties:

1. There exists J± ∈ Jcomp(Y±, π±, ω±) such that J agrees with J+ and J− respec-

tively on [−ǫ,+∞)× Y+ and (−∞, ǫ]× Y− for some ǫ > 0.

2. J |X is compatible with ΩX .

We use Jcomp(X,ΩX) to denote the space of cobordism admissible almost complex struc-

tures. It is contractible and carries a natural C∞-topology. Fix symplectization admis-

sible almost complex structures J+ and J−, let Jcomp(X,ΩX , J±) denote a subset of

8

Jcomp(X,ΩX) such that J |R±×Y± agrees with J± along R± × γ±, where γ runs over all

periodic orbits with degree less than Q, and R+ = [0,∞) and R− = (−∞, 0].

The following definition only for the case that (X,πX , ωX) is a fiberwise symplectic

cobordism.

Definition 2.13. An almost complex structure J on X is called adapted to the fibration

if J satisfies the following properties:

1. There exists J± ∈ Jcomp(Y±, π±, ω±) such that J agrees with the J+ and J−respectively on [−ǫ,+∞)× Y+ and (−∞, ǫ]× Y− for some ǫ > 0.

2. πX : X → B is complex linear with respect to (J, jB), i.e., jB dπX = dπX J .

3. Away from the critical points of πX , J |ker dπXis compatible with ωX.

4. Near the critical points of πX , under the coordinate charts (z1, z2) in Definition

2.6, J agrees with the standard complex structure.

We use Jtame(X,πX , ωX) to denote the space of almost complex structures adapted to

the fibration, this space is contractible and carries a natural C∞-topology. Fix J± ∈Jcomp(Y±, π±, ω±), define Jtame(X,πX , ωX , J±) similarly as in Definition 2.12.

Remark 2. The symplectic form ΩX gives a decomposition TX = TXhor ⊕ TXvert of

the tangent bundle of X, where TXvert = kerπX∗ and TXhorx = v ∈ TXx|ΩX(v,w) =

0 ∀w ∈ TXvertx . With respect to this splitting, J can be written as J =

[Jhh Jvh

Jhv Jvv

].

Note that the complex linear condition implies that Jvh = 0. In addition, it is easy to

check that J is ΩX-tame, however, J is not compatible with ΩX unless Jhv = 0.

In order to meet the transversality condition, we often require an almost complex

structure J satisfies certain criteria. In the rest of the paper, we refer to such an almost

complex structure J as a generic almost complex structure. The precise meaning will

be explained in Section 4.4.

Let us explain why we need two different kinds of almost complex structures here.

Let (X,πX , ωX) be a fiberwise symplectic cobordism. To define the cobordism maps

HP (X,ΩX , J,ΛX ) by using holomorphic curve method, a genus bound on holomorphic

curves (Lemma 7.1) plays a key role in our proof. To obtain this bound, we require

that πX is complex linear. This is the reason why we need Definition 2.13.

To define the cobordism maps HPsw(X,ΩX ,ΛX), in order to apply the techniques

in [12], we want to perturb the Seribrg Witten equations by symplectic form ΩX . To

this end, we need to find a metric such that ΩX is self-dual. The natural way is to define

the metric by g(·, ·) = ΩX(·, J ·), where J is an almost complex structure compatible

9

with ΩX . So one may consider the subset in Jtame(X,πX , ωX) consisting of almost

complex structures that are compatible with ΩX . We denote this subset by J . Of

course, one also can use the more general almost complex structures in Jcomp(X,ΩX).

When we show that these two definitions of cobordism maps agree, we need the

almost complex structures to be nice enough so that both of HPsw(X,ΩX , J,ΛX) and

HP (X,ΩX , J,ΛX ) are well defined simultaneously. Unfortunately, the potential candi-

date J doesn’t work. Therefore, we need to search the almost complex structure in a

larger space Jcomp(X,ΩX).

2.5 Composition of symplectic cobordism

Let (X+,ΩX+) and (X−,ΩX−) be respectively symplectic cobordisms from (Y+, π+, ω+)

to (Y0, π0, ω0) and from (Y0, π0, ω0) to (Y−, π−, ω−). There is a natural operation on

these two cobordisms by gluing them along their common boundaries. Here we only

focus on the fiberwise symplectic cobordism case, the general case can be formulated

similarly.

Define XR = X+ ∪R×Y0[−R,R]s × Y0 ∪−R×Y0

X−. Note that in a collar neigh-

borhood of Y0, both of ωX+ and ωX− are equal to ω0 by our construction in Section

2.2, so we can glue them together smoothly. The admissible 2-form on XR is defined

by

ωXR=

ωX+ : on X+

ωX− : on X−ω0 : on Y0 × [−R,R]

Observe that if (X+, ωX+) and (X−, ωX−) are monotone with the same coefficient τ ,

then (XR, ωXR) is also monotone. The result (XR, πXR

, ωXR) is called the composition

of (X+, πX+, ωX+

) and (X−, πX− , ωX−).

Let XR be the usual symplectic completion by adding cylindrical ends. Let JX± ∈Jtame(X±, πX± , ωX±) be a generic almost complex structure on X± so that JX+

= J+

on R+ × Y+, JX± = J0 on R∓ × Y0 and JX− = J− on R− × Y−. These almost complex

structures JX+and JX− induces a natural almost complex structure JR over XR. It is

given by

JR =

JX+: on X+

JX− : on X−J+ : on Y+ × [R,∞)

J0 : on Y0 × [−R,R]J− : on Y− × [−R,−∞).

Note that JR ∈ Jtame(XR, πXR, ωXR

).

10

2.6 Local coefficient.

The periodic Floer homology cannot be defined over Z coefficient in general. We need

to introduce a concept called local coefficient in this subsection.

Definition 2.14. Given orbit sets α+ = (α+,i,mi) and α− = (α−,j , nj), define

H2(X,α+, α−) to be the set of relative homology classes of 2-chains in X such that

∂Z =∑

imiα+,i −∑

j njα−,j. Here two 2-chains are equivalent if and only if their

difference is boundary of a 3-chains. If X = [0, 1]×Y , we just denote the set of relative

homology classes by H2(Y, α+, α−).

A local coefficient, ΛP , of R-module for periodic Floer homology is defined as fol-

lows. For each orbit set α, we have a R-module Λα, and for each relative homology class

Z ∈ H2(Y, α, β), we assign an isomorphism ΛZ : Λα → Λβ satisfying the composition

law ΛZ1+Z2= ΛZ2

ΛZ1, where Z1 ∈ H2(Y, α, β) and Z2 ∈ H2(Y, β, γ).

Let ω be an admissible 2-form on (Y, π). Let S ⊂ H2(Y, α, β) be a [ω]-finite set,

that is S⋂Z ∈ H2(Y, α, β) :

∫Z ω ≤ C is finite for any C ∈ R. To ensure that a series

of isomorphisms∑Z∈S

nZΛZ is convergent and the terms of this series can be rearranged,

we required that the local coefficient ΛP is [ω]-complete in the the sense of [17].

Let (X,ΩX) be a symplectic cobordism from (Y+, π+, ω+) to (Y−, π−, ω−) and Λ±P

be [ω±]-complete local coefficient. A X-morphism ΛX between Λ+P and Λ−

P means that

for each relative homology class Z ∈ H2(X,α+, α−), we have an isomorphism ΛX(Z) :

Λα+→ Λα− satisfying the composition law ΛX(Z+ + Z + Z−) = Λ−

Z− ΛX(Z) Λ+

Z+,

where Z± ∈ H2(Y±, α±, β±) and Z ∈ H2(X,β+, α−). For the same reasons mentioned

in last paragraph, we require that ΛX satisfies the following condition:

For any ΩX-finite set S ⊂ H2(X,α+, α−), the set of isomorphism ΛX(Z) ∈Hom(Λα+

,Λα−)|Z ∈ S is equicontinuous. In addition, ΛX(Z)Z∈S converges

to zero as Z runs through S.

It is an analogy of the [ω]-completeness in [17].

The local coefficient, ΛS , of R-module for SeibergWitten cohomology can be defined

similarly. The orbit sets and relative homology class are to be replaced by configurations

and by relative homotopy class respectively. For the definition of configurations and

relative homotopy class, please refer to Section 5.1.

Remark 3. We assume that R = Z2 unless otherwise stated.

2.6.1 Local coefficient and 2-form.

Typical examples of ΛP and ΛX are given as follows.

11

Definition 2.15. Let R be a ring, the universal Novikov ring R over R is defined by

R = ∑iait

λi |i ∈ R, ai ∈ R,∀C > 0,#i|λi < C, ai 6= 0 <∞.

Fix an admissible 2-form ω on Y . We introduce a local coefficient system Rω which

is associated to ω. For the fiber of Rω we take everywhere the universal Novikov ring

R. If Z is a relative homology class from α to β, then Rω(Z) : Rα → Rβ is defined to

be multiplication by t∫Zω. One can check that it is [ω]-complete in the sense of [17].

Let (X,πX , ωX) be a fiberwise symplectic cobordism from (Y+, π+, ω+) to (Y−, π−, ω−).

Then the X-morphism RωX: Rω+

→ Rω− is defined as follows. If Z ∈ H2(X,α+, α−)

is a relative homology class from α+ to α−, then RωX(Z) : Rα+

→ Rα− is defined to

be multiplication by t∫ZωX .

3 Statement of the main results

Let us clarify the notation and conventions we will be using. We fix a number Q > g(Σ)

and a homology class Γ± ∈ H1(Y±,Z) satisfies Γ± · [Σ] ≤ Q throughout. We assume

that the manifolds Y+, Y− and X are connected. In addition, the 2-forms ω+, ω− and

ωX are supposed to be Q-admissible.

The following theorems are the main results of this paper. In the first theorem, we

define the cobordism maps on PFH for general cases, the maps are denoted by HPsw.

The subscript “sw” is used to emphasize that they are defined by Seiberg Witten theory.

Theorem 1. Let (Y±, π±) be a 3-dimensional fibration over S1 with fiber Σ, together

with a Q-admissible 2-form ω±. Fix Γ± ∈ H1(Y±,Z) satisfying g(Σ) < Γ± · [Σ]. Let

(X,ΩX ) be a symplectic cobordism from (Y+, π+, ω+) to (Y−, π−, ω−). Let Λ±P be a

[ω±]-complete local coefficient for the periodic Floer homology and ΛX be a X-morphism

between Λ+P and Λ−

P . Then (X,ΩX) induces a module homomorphism

HP sw(X,ΩX ,ΛX) : HP∗(Y+, ω+,Γ+,Λ+P ) → HP∗(Y−, ω−,Γ−,Λ

−P )

with natural decomposition

HPsw(X,ΩX ,ΛX) =∑

ΓX∈H2(X,∂X,Z),∂Y±ΓX=Γ±

HPsw(X,ΩX ,ΓX ,ΛX) (3)

and satisfying the following properties:

1. (Composition rule) Let (X+,ΩX+) and (X−,ΩX−) be symplectic cobordisms from

(Y+, π+, ω+) to (Y0, π0, ω0) and from (Y0, π0, ω0) to (Y−, π−, ω−) respectively.

Then we have∑

ΓX |X±=ΓX± ,ΓX∈H2(X,∂X,Z)

HPsw(X,ΩX ,ΓX ,ΛX)

= HPsw(X−,ΩX− ,ΓX− ,ΛX−) HPsw(X+,ΩX+,ΓX+

,ΛX+),

12

where (X,ΩX ) is the composition of (X+, ΩX+) and (X−,ΩX−) defined in Section

2.5, and ΛX = ΛX+ ΛX−.

2. (Invariance) Suppose that ΩX1and ΩX2

are symplectic forms on X such that

ΩX1= ΩX2

in collar neighborhoods of Y+ and Y−, then

HPsw(X,ΩX1,ΛX) = HPsw(X,ΩX2

,ΛX).

3. (Commute with U-map)

U+ HPsw(X,ΩX ,ΛX) = HPsw(X,ΩX ,ΛX) U−.

4. (Holomorphic curve axiom) Given a cobordism admissible almost complex struc-

ture J ∈ Jcomp(X,ΩX) such that J± = J |R± × Y± is generic, then there is a

chain map

CPsw(X,ΩX , J,ΛX) : CP∗(Y+, ω+,Γ+, J+,Λ+P ) → CP∗(Y−, ω−,Γ−, J−,Λ

−P )

inducing HPsw(X,ΩX , J,ΛX) with the following properties: If there is no J holo-

morphic current from α+ to α− with zero ECH index, then

< CPsw(X,ΩX , J,ΛX)α+, α− >= 0.

Before we state the second result, let us introduce the assumptions (♠) as follows.

Definition 3.1. Let (X,πX , ωX) be a fiberwise symplectic cobordism from (Y+, π+, ω+)

to (Y−, π−, ω−). Let B be the base manifold of (X,πX ). We say that the fiberwise

symplectic cobordism satisfies assumptions (♠) if any of the following hold:

♠.1 g(B) ≥ 2,

♠.2 When g(B) = 1, each periodic orbit of (Y+, π+, ω+) with degree 1 is either Q-

negative elliptic or hyperbolic, and each periodic orbit of (Y−, π−, ω−) with degree

1 is either Q-positive elliptic or hyperbolic.

♠.3 When g(B) = 0, then we have the following properties:

(a) Y+ 6= ∅ and Y− 6= ∅.

(b) Each periodic orbit of (Y+, π+, ω+) with degree less than Q is either Q-

negative elliptic or hyperbolic.

(c) Each periodic orbit of (Y−, π−, ω−) with degree less than Q is either Q-

positive elliptic or hyperbolic.

13

The assumptions on Q-positive (negative) elliptic orbits are inspired by [11], the

existence of these orbits are used to ensure that the ECH indexes of holomorphic curves

are nonnegative. Under the assumptions (♠), we give a Seiberg Witten free definition

of the cobordism maps in the following theorem.

Theorem 2. Let (Y±, π±) be a 3-dimensional fibration over S1 with fiber Σ, together

with a Q-admissible 2-form ω± on Y±. Fix Γ± ∈ H1(Y±,Z) satisfying g(Σ) < Γ± · [Σ].Let (X,πX , ωX) be a fiberwise symplectic cobordism from (Y+, π+, ω+) to (Y−, π−, ω−)

satisfying (♠). Let Λ±P be a [ω±]-complete local coefficient for the periodic Floer homol-

ogy and ΛX be a X-morphism between Λ+P and Λ−

P . Suppose that (X,πX ) contains no

separating singular fiber, then for generic J ∈ Jtame(X,πX , ωX), (X,πX , ωX) induces

a module homomorphism

HP (X,ΩX , J,ΛX) : HP∗(Y+, ω+,Γ+, J+,Λ+P ) → HP∗(Y−, ω−,Γ−, J−,Λ

−P )

with natural decomposition (3). The map HP (X,ΩX , J,ΛX) is defined by counting

embedded holomorphic curves, without using Seiberg Witten theory. (Recall that the

symplectic form here is given by ΩX = ωX + π∗XωB.) Moreover, the cobordism maps

satisfy the following properties:

1. (Composition rule) Let (X+, πX+, ωX+) and (X−, πX−, ωX−) be respectively fiber-

wise symplectic cobordisms from (Y+, π+, ω+) to (Y0, π0, ω0) and from (Y0, π0, ω0)

to (Y−, π−, ω−). Assume both of them satisfy assumptions (♠). Let (XR, πR, ωXR)

be the composition of (X+, πX+, ωX+

) and (X−, πX− , ωX−), and JR is defined in

Section 2.5. Let ΛXR= ΛX+ ΛX−. Assume that JX± are generic and there

exists R0 ≥ 0 such that JR is generic for any R ≥ R0, then we have

∑

ΓXR|X±=ΓX± ,ΓXR

∈H2(XR,∂XR,Z)

HP (XR,ΩXR,ΓXR

, JR,ΛXR)

= HP (X−,ΩX− ,ΓX− , JX− ,ΛX−) HP (X+,ΩX+,ΓX+

, JX+,ΛX+

),

for any R ≥ R0.

2. (Commute with U-map)

U+ HP (X,ΩX , J,ΛX) = HP (X,ΩX , J,ΛX) U−.

3. (Blowup) Suppose that (ΓX , ωX) is monotone. Take a point x ∈ X in regular fiber

and a generic J ∈ Jtame(X,πX , ωX) such that J is integral near x. Let (X,′ J ′)

be the blowup of (X,J) at x. Then there exists an admissible 2-form ωX′ such

that J ′ ∈ Jtame(X′, πX′ , ωX′). Let E be the homology class of the exceptional

sphere. Regard ΓX ∈ H2(X′, ∂X ′,Z) and assume that ΓX · E = 0. Let ΛX′ be a

14

X ′-morphism such that ΛX(Z) = ΛX′(Z) for any Z with relative class ΓX , then

we have

HP (X,ΩX ,ΓX , J,ΛX) = HP (X ′,ΩX′ ,ΓX , J′,ΛX′).

Remark 4. Here are some remarks about Theorem 2:

• Note that if (X+, πX+, ωX+) and (X−, πX−, ωX−) satisfy assumptions (♠), then

their composition also satisfies the assumptions (♠). Thus the statement about

composition rule in Theorem 2 makes sense.

• The cobordism maps HP (X,ΩX , J,ΛX) in Theorem 2 also can be defined over

local coefficient of Z module. See the discussion in Section 7.6.

• Theorem 2 is also true when Y+ and Y− are disconnected.

• If X has separating singular fibers and each separating singular fiber contains

a exceptional sphere. Let Eii denote the homology class of these exceptional

spheres. Assume that ΓX ∈ H2(X, ∂X,Z) such that ΓX · Ei ≤ 0 for all i. Then

the cobordism maps HP (X,ΩX ,ΓX , J,ΛX) are still well defined. We will discuss

this in Section 7.4.

Theorem 3. Given the same assumptions of Theorem 2, suppose that (X,ωX) or

(ΓX , ωX) is monotone for some ΓX ∈ H2(X, ∂X,Z). Then there exists a nonempty open

subset Vcomp(X,πX , ωX) of Jcomp(X,ΩX) such that for generic J ∈ Vcomp(X,πX , ωX),

both of HP sw(X,ΩX , J,ΓX ,ΛX) and HP (X,ΩX , J,ΓX ,ΛX) are well defined. More-

over, we have

HP (X,ΩX ,ΓX , J,ΛX ) = HP sw(X,ΩX ,ΓX , J,ΛX ).

In particular, the cobordism map HP (X,ΩX ,ΓX , J,ΛX) is independent of the choice

of J ∈ Vcomp(X,πX , ωX).

Remark 5. This remark concerns the monotonicity assumptions in Theorem 3. These

assumptions are only used to guarantee that the existence of Vcomp(X,πX , ωX) and do

not play any role in other argument.

When we define the cobordism maps for cases ♠.1 and ♠.2, the methods in this

paper rely on a genus bound in Lemma 7.1. Roughly speaking, Vcomp(X,πX , ωX) is the

set of cobordism admissible almost complex structures such that the genus bound holds.

Under the monotonicity assumptions, we show that it is nonempty and open.

The author believes that the conclusions in Theorem 3 should be true without the

monotonicity assumptions. Here are some thoughts of the author which support the

15

above opinion. For convenience, the local coefficients are taken to be the one in Subsec-

tion 2.6.1. Then the chain maps CP (X,ΩX ,ΓX , J) and CP sw(X,ΩX ,ΓX , J) can be re-

garded as Taylor series. In fact, Section 7.3 provides a nonempty open subset of almost

complex structures VLcomp(X,πX , ωX , J±) such that for generic J ∈ VL

comp(X,πX , ωX , J±),

the genus bound holds for any curve with energy less than L, and the coefficients of

CP (X,ΩX ,ΓX , J) are well defined up to order L.

If one could define HP (X,ΩX ,ΓX , J) for generic J and show that it is indepen-

dent of J by using any method which beyond this paper, then take a generic JL ∈VLcomp(X,πX , ωX , J±), the methods in this paper still can show that the chain maps of

HP and HPsw are equal up to order L, i.e.,

CPsw(X,ΩX ,ΓX , JL) = CP (X,ΩX ,ΓX , JL) + o(L).

As the cobordism maps are independent of J , then CPsw and CP are chain homotopic to

CPsw(X,ΩX ,ΓX , J) and CP (X,ΩX ,ΓX , J) respectively, where J is an almost complex

structure which is independent of L. Take L → ∞, then we get the same conclusions

of Theorem 3.

Remark 6. In case ♠.3, we do not use the genus bound in Lemma 7.1. The conclusions

in Theorem 3 are true without the monotonicity assumptions. Moreover, the open set

Vcomp(X,πX , ωX) = Jcomp(X,ΩX ).

Remark 7. For the case that Γ± · [Σ] = 0, the map HP (X,ΩX , J,ΛX) is still well-

defined by counting holomorphic curves with zero ECH index. The manifold (X,ΩX)

can be arbitrary symplectic cobordism. Moreover, HP (X,ΩX , J,ΛX) satisfies the fol-

lowing properties:

1. Suppose that Y± = ∅ and b+2 (X) > 1, then

HP (X,ΩX , J,ΛX )(Λ[∅]) =∑

A∈H2(X,Z)

Gr(X,A)ΛX (A) Λ[∅],

where Gr(X,A) is the Gromov invariant defined in [24].

2. Suppose that Y+ 6= ∅ or Y− 6= ∅ and X is relatively minimal Lefschetz fibration,

also, the genus of the fiber is at least two, then HP (X,ΩX , J,ΛX ) is a canonical

isomorphism.

3. In above two cases, HP (X,ΩX , J,ΛX) is independent on the choice of almost

complex structures.

The first statement follows from the definition of Taubes’ Gromov invariant. The sec-

ond statement follows from the observation that the only closed holomorphic curve

in X with zero ECH index is the empty curve. In both cases, HP (X,ΩX ,ΛX) =

HPsw(X,ΩX ,ΛX).

16

Remark 8. In many applications, the tuple (Y, π, ω,Γ) satisfies the monotonicity prop-

erty which is similar to Definition 2.5. In this case, the periodic Floer homology of

(Y, π, ω,Γ) can be defined with Z2 or Z coefficient. It follows from the observation that

the holomorphic curves with the same ECH index have same energy. The compactness

of the moduli space implies that there are only finitely many holomorphic curves that

contribute to the differential.

This observation is true for the cobordism case as well. (See the proof of Corollary

7.16.) If the pair (ΓX ,ΩX) is monotone in the sense of Definition 2.5, then the cobor-

dism maps HPsw(X,ΩX ,ΓX) and HP (X,ΩX ,ΓX) given in Theorems 1 and 2 can be

defined with Z2 or Z coefficients. In Remark 19, we explain more about the case in

Theorem 1, as the cobordism maps are defined by counting solutions to Seiberg Witten

equations rather than holomorphic curves.

Note that the same argument is not available when (X,ΩX) is monotone. One may

use a similar argument as in Lemma 5.11 to control the energy of holomorphic curves

with fixed genus. However, the curves that contribute to the cobordism maps do not

have priori upper bound on the genus.

Besides the main results above, here we state a result which concerns the assump-

tions (♠). The proposition that follows asserts that after suitable perturbation, the

assumptions (♠) holds in most cases, the exceptional cases are that the base manifold

B is a disk. The proposition is an analogy of Theorem 2.5.2 of [3] in PFH setting. It

will be proved in Section 6.

Proposition 3.2. Let (X,πX , ωX) be a fiberwise symplectic cobordism from (Y+, π+, ω+)

to (Y−, π−, ω−). Given Q ≥ 1, we can always find ω′± such that

• Every periodic orbit of ω′+ with degree less than Q is either Q-negative elliptic or

hyperbolic.

• Every periodic orbit of ω′− with degree less than Q is either Q-positive elliptic or

hyperbolic.

• The cohomology class of ω′± is unchanged, i.e., [ω′

±] = [ω±].

• Given a metric g± over Y±, for any δ > 0, there exists a constant c0 depending

on g±, such that we can arrange that |ω′± − ω±|g± ≤ c0δ.

Moreover, we can find an admissible 2-form ω′X such that (X,πX , ω

′X) is a fiberwise

symplectic cobordism from (Y+, π+, ω′+) to (Y−, π−, ω′

−).

We have organized the rest of the paper in the following way: Section 4 is an

overview of the foundation of holomorphic curve and periodic Floer homology. We

17

prove Theorem 1 in Section 5. In Section 6, we prove Proposition 3.2. Section 7

presents some partial results about defining cobordism maps HP (X,ΩX , J,ΛX) on

PFH by using holomorphic curve method. Finally, in Sections 8 and 9, we explore

the relationship of HP sw(X,ΩX , J,ΛX ) and HP (X,ΩX , J,ΛX) under monotonicity

assumptions.

Acknowledgements

The author would like to thank Prof.Yi-Jen Lee for suggesting the problem, as well

as for her helpful comments. He also wants to thank the anonymous referees whose

comments, suggestions, and corrections, have greatly improved this paper. Without

their support, this paper would not have been possible.

4 ECH index and periodic Floer homology

This section supplies the background material that is used subsequently to define the

periodic Floer homology and its cobordism maps. For more details, please refer to [10].

4.1 J-holomorphic curves and currents

Definition 4.1. A J holomorphic curve is a map u : (C, j) → X satisfying du +

J du j = 0, where (C, j) is a Riemann surface possibly with punctures. Two holo-

morphic curves u : (C, j) → X and u′ : (C ′, j′) → X are equivalent if there exists a

biholomorphism ϕ : (C, j) → (C ′, j′) such that u = u′ ϕ.

Definition 4.2. A holomorphic curve u is called somewhere injective if u−1u(z) = z

for some z ∈ C, we also call it simple.

Remark 9. Sometimes we just use C to denote the holomorphic curves, especially in

the case that u is somewhere injective.

Given orbit sets α+ = (α+,i,mi) and α− = (α−,j , nj), the moduli space

MJX(α+, α−) is a collection of equivalent class of holomorphic maps u : C → X with

the following properties: u has positive ends at covers of α+,i with total multiplicity mi,

negative ends at covers of α−,j with total multiplicity nj, and no other ends. Here C is

a compact Riemann surface possibly with punctures and may be disconnected. When

X = R × Y and J ∈ Jcomp(Y, π, ω), we just use MJY (α+, α−) to denote the moduli

space. Given Z ∈ H2(X,α+, α−), we use MJX(α+, α−, Z) to denote the elements in

MJX(α+, α−) with relative homology class Z.

18

Given a curve u ∈ MJX(α+, α−), the ΩX-energy EΩX

(u) of u which is defined in [1]

is given by

EΩX(u) =

∫

C∩R+×Y+

ω+ +

∫

C∩R−×Y−ω− +

∫

C∩u−1(X)u∗ΩX .

By Stokes formula, the ΩX-energy only depends on α+, α− and its relative homology

class. For i ∈ Z and L ∈ R, define

MJ,LX,I=i(α+, α−) =

⊔

I(α+,α−,Z)=i EΩX(Z)<L

MJX(α+, α−, Z),

where I(α+, α−, Z) is the ECH index. We will review the definition of I(α+, α−, Z) in

the upcoming section.

Remark 10. In the case that (X,πX , ωX) is a fiberwise symplectic cobordism, the ΩX-

energy of a holomorphic curve u ∈ MJ(α+, α−) is EΩX(u) =

∫C u

∗ωX+dvol(B), where

d = [Σ] · [α±]. Therefore, a uniform upper bound on∫C u

∗ωX ensures that we can use

the Gromov compactness in [1].

Most of the time, we only care about the underlying current of the holomorphic

curves, rather than the holomorphic maps. Thus it is convenient to introduce the notion

of holomorphic currents.

Definition 4.3. A J-holomorphic current from α+ = (α+,i,mi) to α− = (α−,j , nj)is a finite set of pairs C = (Ca, da), where Caa are distinct, irreducible, some-

where injective J holomorphic curves with EΩX(Ca) < ∞ and daa are positive inte-

gers. If Ca is a J-holomorphic curve whose positive ends are asymptotic to α+(a) =

(α+,i,mia) and negative ends are asymptotic to α−(a) = (α−,j , nja), then mi =∑adamia and nj =

∑adanja. Let MJ

X(α+, α−) denote the moduli space of J holomor-

phic currents from α+ to α−.

Note that for each u : C → X in MJX(α+, α−), we can associate a holomorphic

current C ∈ MJX(α+, α−) in the following way. According to Theorem 6.19 of [34], there

is a factorization u = v ϕ, where v : C → X is a somewhere injective holomorphic

curve and ϕ : C → C is a branched covering. Then the underlying holomorphic current

of u is C = (C, d), where d is degree of ϕ. If C is reducible, then C is a union of

holomorphic currents of all irreducible components.

The following definition generalizes the concept of intersection numbers to punc-

tured holomorphic curves.

Definition 4.4 (Cf. Definition 4.7 of [11]). Let C be a simple irreducible holomorphic

curves from α+ to α−, the self intersection number is defined to be

C ⋆ C =1

2(2g(C) − 2 + indC + h(C) + 2eQ(C) + 4δ(C)) ∈ 1

2Z.

19

where h(C) is the number of ends of C at hyperbolic orbits and δ(C) ≥ 0 is a count

of the singularities of C in X with positive integer weights, and eQ(C) is the total

multiplicity of all elliptic orbits in α+ that are Q-negative, plus the total multiplicity of

all elliptic orbits in α− that are Q-positive.

If C and C ′ are two distinct simple irreducible holomorphic curves, then C ⋆C ′ ∈ Z

are defined to be the algebraic count of their intersection points.

Remark 11. It is worth noting that if C is closed, then C ⋆ C agrees with the usual

self intersection number C · C by adjunction formula.

Definition 4.5. A holomorphic current C = (Ca, da) ∈ MJX(α+, α−) is called em-

bedded if it satisfies the following properties

• For any a, Ca is embedded and da = 1.

• Ca ⋆ Cb = 0 for any a 6= b.

A holomorphic curve u ∈ MJX(α+, α−) is called embedded if its underlying holomorphic

current is embedded.

4.2 ECH index and Fredholm index

In this subsection, we briefly review the ECH index and Fredholm index for holomorphic

curves.

ECH Index. Let C be a holomorphic current from α+ = (α+,i,mi) to α− =

(α−,j , nj). Let τ be a homotopy class of symplectic trivializations τ+i of the restriction

of (ker π+∗, J+) along α+,i and τ−j of the restriction of (ker π−∗, J−) along α−,j. The

ECH index is defined by the following formula:

I(C) = cτ (C) +Qτ (C) +∑

i

mi∑

p=1

µτ (αp+,i)−

∑

j

nj∑

q=1

µτ (αq−,j),

where cτ (C) and Qτ (C) are respectively the relative Chern number and the relative

self-intersection numbers (See Section 4.2 of [9]), and µτ is the Conley-Zehender index.

The ECH index I only depend on orbit sets α+, α− and relative homology class of C.Fredholm Index. Let u : C → X be a J-holomorphic curve from α+ = (α+,i,mi)

to α− = (α−,j , nj). For each i, let ki denote the number of ends of u at α+,i, and

let piakia=1 denote their multiplicities. Likewise, for each j, let lj denote the number

of ends of u at α−,j, and let qjbljb=1 denote their multiplicities. Then the Fredholm

index of u is defined by

indu = −χ(C) + 2cτ (u∗TX) +

∑

i

ki∑

a=1

µτ (αpia+,i)−

∑

j

lj∑

b=1

µτ (αqjb−,j).

20

Sometimes we write indC instead of indu.

The following theorem summarizes two important relationships between Fredholm

index and ECH index for simple holomorphic curve.

Theorem 4.6 ([8], [9]). Let C ∈ MJX(α+, α−) be a simple holomorphic curve, then

• ind(C) ≤ I(C)− 2δ(C), equality holds if and only if C is admissible. (Definition

of admissible will be given in next section.)

• If α+ and α− are admissible, then ind(C) = I(C) mod 2,

where δ(C) is a count of the singularities of C in X with positive integer weights.

The following theorem will be also used frequently:

Theorem 4.7 (Theorem 5.1 [9]). Let C = (Ca, da) be a holomorphic current in X,

then

I(C) ≥∑

a

daI(Ca) +∑

a

da(da − 1)Ca ⋆ Ca + 2∑

a6=b

dadbCa ⋆ Cb. (4)

4.3 ECH partition

Given a simple periodic orbit γ and a holomorphic curve C, suppose that C has positive

ends at covers of γ whose total covering multiplicity is m, the multiplicities of these

covers form a partition of the integer m, which is denoted by p+(γm, C). We called it

a positive partition of C at γ. Similarly, we define p−(γm, C) for negative ends and

called it a negative partition of C at γ.

Definition 4.8. Let u : C → X be a J holomorphic curve from α+ to α− without

R-invariant cylinder. u is called positively admissible if the positive partition at γ is

p+(m,γ), i.e., p+(γm, C) = p+(m,γ). Likewise, u is called negatively admissible if the

negative partition at γ is p−(m,γ). u is called admissible if u are both positively and

negatively admissible. Here p+(m,γ) and p−(m,γ) are defined in [8], [9]. When u is

admissible, we also say that u satisfies the ECH partition condition.

Definition 4.9. A connector u : C → R × Y is a union of branched cover of trivial

cylinders with zero Fredholm index. Here each connected component of C is a punctured

sphere. A connector is trivial if it is a union of unbranched cover of trivial cylinders,

otherwise, it is nontrivial.

Remark 12. For the sake of convenient, we use stronger definition here. Note that the

original definition of connector in [8] doesn’t involve any constrain on Fredholm index.

In this article, we don’t use the precise definition of p+(m,γ) and p−(m,γ), we only

need to know the following two facts.

21

• Let γ be an elliptic orbit, for any partition p(m,γ), there is no nontrivial con-

nector from p+(m,γ) to p(m,γ). Likewise, there is no nontrivial connector from

p(m,γ) to p−(m,γ). Cf. exercise 3.13, 3.14 of [10].

• Suppose that u is a simple holomorphic curve, then u is admissible if and only if

I(u) = ind(u). Cf. [8], [9].

4.4 Meaning of generic almost complex structure

The purpose of this subsection is to explain the precise meaning of “generic almost

complex structure”. To this end, we first need to recall the behavior of holomorphic

curve at the ends. The following descriptions follow Section 2 of [14].

Let γ be a non-degenerate periodic orbit of (Y, π, ω). Let ϕγ : S1 ×D → Y be the

coordinate in Lemma 2.3 of [18]. Granted this coordinate, we can express the ends of

the holomorphic curve as a graph of certain functions. More details are explained as

follows: Suppose that C is a somewhere injective J-holomorphic curve in X and E ⊂ C

is a positive end at γqE . Let ϕγ = Id× ϕγ : R× S1 ×D → R × Y , then there exists a

constant s0 > 0 such that ϕ−1γ (E ∩ ([s0,+∞)× Y )) is the image of a map

[s0,+∞)× R/2πqE → R× S1 ×D

(s, τ) → (s, (τ, ς(τ, s))),(5)

and ς(τ, s) = e−λqEs(ςqE

+ r(τ, s)), where the notation is explained as follows. λqEis the

samllest positive eigenvalue of the set of 2πqE periodic eigenfunctions of operator:

Lγ : C∞(R;C) → C∞(R;C)

η → i

2

dη

dt+ νη + µη, (6)

and ςqE(possibly zero) is a 2πqE -periodic eigenfunction of Lγ . ν and µ are respectively

2π-periodic real and complex functions, they are determined by the pair (ω, J). r is an

error term such that |r| ≤ e−ε|s| for some ε > 0.

Let J1tame(X,πX , ωX , J±) and J1comp(X,ΩX , J±) be subsets of Jtame(X,πX , ωX)

and Jcomp(X,ΩX , J±) respectively with the following properties:

J.0 If J ∈ J1tame(X,πX , ωX , J±) or J1comp(X,ΩX , J±), then all simple J holomor-

phic curves are Fredholm regular in the sense of [34], possibly except for fibers.

Lemma 4.10. J1tame(X,πX , ωX , J±) and J1comp(X,ΩX , J±) are respectively Baire

subset of Jtame(X,πX , ωX , J±) and Jcomp(X,ΩX , J±).

Proof. The lemma can be proved by the standard argument, for example, Chapter 7

of [34] or Section 3.4 of [21].

22

Let C be an embedded holomorphic curve with zero ECH index. Let E ⊂ C

denote an end, reintroduce the notation qE , λE and ςqEas above. Introduce a subset

divE ⊂ 1, . . . , qE as follows. An integer q ∈ divE if either of the following is true:

1. q = qE ,

2. q is a proper divisor of qE , and there is a 2πq-periodic eigenfunction ςq of Lγ with

eigenvalue λq of same sign as λqE, such that |λqE

| ≥ |λq| > 0.

The following conditions are analogs of the criteria [Ja], [Jb] in [18]. We intro-

duce subsets J2tame(X,πX , ωX , J±) ⊂ J1tame(X,πX , ωX , J±) and J2comp(X,ΩX , J±) ⊂J1comp(X,ΩX , J±) as follows: J ∈ J2tame(X,πX , ωX , J±) or J2comp(X,ΩX , J±) if every

embedded J holomorphic curve C with zero ECH index satisfies the following properties:

J.1 For each end E ⊂ C, divE = qE.

J.2 [Non-degenerate] For each end E ⊂ C, ςqE6= 0.

J.3 [Non-overlapping ]Let E and E′ denote two distinct pairs of either negative or

positive ends of C such that γE = γE ′ , qE = qE ′ = q. Fix 2πq periodic functions

ςqE, ςq

E ′ for E and E′-version of Lγ . Then ςqE

(t) 6= ςqE ′ (t+ 2πk) for any k.

Lemma 4.11. J2tame(X,πX , ωX , J±) and J2comp(X,ΩX , J±) are respectively Baire

subsets of Jtame(X,πX , ωX , J±) and Jcomp(X,ΩX , J±).

Proof. The proof follows the same argument in Section 3 of [14]. Propositions 3.2

and 3.9 of [14] show that the symplectization admissible almost complex structures

that satisfy the criterions analogy to J.0, J.2 and J.3 form a Baire subset. Note

that our lemma differs from the analogs in [14] only in the following aspects: The

symplectization R × Y and index 1 holomorphic curves are respectively replaced by

X and index 0 holomorphic curves. Also, the symplectic form and almost complex

structures are changed to the one in our setting accordingly. But observe that our

symplectic form and almost complex structures agree with the one in symplectization

case over the ends of X . We sketch the proof as follows.

Similar to [14], we define the universal moduli space Cn, the essential difference is toreplace index 1 curves by index zero curves. Let Jn be subset of J ∈ Jcomp(X,ΩX , J±)

such that there is no J-holomorphic curve in Cn violates J.0 and J.2. It is easy to check

that Jn is open, so it suffices to show that Jn is dense. Fix a J , let U be a suitable

neighborhood of J . Fix a section ψ : U → Cn, where ψ(J ′) = (J ′, C ′). Recall that our

J ′ is fixed along the periodic orbits. Therefore, fix a positive end E of C, expression (5)

defines a map U → BE , where BE is eigenspace corresponding to the smallest positive

eigenvalue. Equivalently, we can express this map as a function aE: U → R or C. The

holomorphic curve C ′ has degenerate end if and only if J ′ ∈ a−1E

(0).

23

The strategy of [14] is to show that aEis submersion, then by implicit function the-

orem, codima−1E

(0) = dimBE ≥ 1. As a consequence, Jn is dense. Lemmas 3.6 and 3.7

of [14] can be copied to our case with only notation change. To show that aEis submer-

sion, we need to construct a j ∈ TJU such that ∇j aE= (da

E)J dψJ (j) 6= 0. Note that

dψJ (j) = D−1C (jC) in our case, since there is no R-action, where jC ∈ Hom0,1(TC,NC)

is determined by j (Cf. (3.7) of [14]). We can construct f ∈ Hom0,1(TC,NC) as in

[14], because, the construction only takes place in the end E . Since f is supported

away from trivial half cylinder R+ × γ, we can choose a j such that jC = f . Again,

the analysis in pages 50-52 of [14] only takes place in the end E , the argument can be

applied to our case to show that ∇j aE6= 0.

Consider a subset Jn of J ∈ Jn such that there is no J holomorphic curve in Cnviolates J.3. Define B = ⊕EBE . Compare to [14], there is no need to modulo R here,

because of the lack of global R-action. Define p : U → B by using expression (5). In fact,

p is submersion by applying the same argument Lemma 3.10 of [14]. The holomorphic

curve C corresponding J ∈ U has overlapping ends if and only if p(J) ∈ Z, where Z is

a subspace of B with positive codimension. By implicit function theorem, Jn is dense.

Define J2comp(X,ΩX , J±) = ∩nJn, then for J ∈ J2comp(X,ΩX , J±), there is no

simple index zero J-holomorphic curve with degenerate ends and overlapping pairs.

If C is embedded with I(C) = 0, then C satisfies J.1 because of the ECH partition

condition.

The case for J2tame(X,πX , ωX , J±) is similar.

Let J3comp(X,ΩX , J±) be subset of Jcomp(X,ΩX , J±) such that J |R×Y± belongs

to Jcomp(Y±, π±, ω±)reg. It is easy to check that it is a Baire subset. The intersec-

tion of J2comp(X,ΩX , J±) and J3comp(X,ΩX , J±) is denoted by Jcomp(X,ΩX , J±)reg.

Jtame(X,πX , ωX , J±)reg is defined in the same way with only notation change.

In this paper, an almost complex structure J is called generic if J belongs to

Jtame(X,πX , ωX , J±)reg or J ∈ Jcomp(X,ΩX , J±)reg.

4.5 Closed holomorphic curves in X

Let πX : X → B be the completion of a fiberwise symplectic cobordism as before and

J ∈ Jtame(X,πX , ωX). Note that if u : C → X is a somewhere injective closed J-

holomorphic curve, then πX u : C → B is holomorphic by assumption. Hence, πX uis constant by maximum principle. In other words, the closed J holomorphic curves in

X are contained in fibers. Conversely, πX is complex linear implies that all irreducible

components of the fibers are somewhere injective holomorphic curves.

24

4.6 Review of periodic Floer homology

Let π : Y → S1 be a surface fibration over S1 together with a Q-admissible 2-form ω.

Fix Γ ∈ H1(Y,Z) with g(Σ) < d = Γ · [Σ] ≤ Q. Take a generic symplectization admissi-

ble almost complex structure J on R×Y and a [ω]-complete local coefficient system ΛP

in the sense of [17], then one can define periodic Floer homology HP∗(Y, ω,Γ, J,ΛP )

as follows. The chain complex of periodic Floer homology is freely generated by ad-

missible orbit sets with homology class Γ, and the differential is defined by counting

holomorphic currents between generators with ECH index 1. More precisely,

∂ =∑

α∈P(Y,ω,Γ)

∑

β∈P(Y,ω,Γ)

∑

Z∈H2(Y,α,β)

(#2MJ

Y,I=1(α, β, Z)/R)ΛZ .

There is an additional structure on periodic Floer homology which is called U -map.

It is defined as follows: Fix a point y ∈ R×Y , let MJY,I=2(α, β, Z)y be the holomorphic

curves in MJY,I=2(α, β, Z) such that y lies in the image of the curves. Suppose that α

and β are admissible orbit sets, then it is a compact one dimensional manifold. The

U -map is defined by

Uy =∑

α∈P(Y,ω,Γ)

∑

β∈P(Y,ω,Γ)

∑

Z∈H2(Y,α,β)

(#2MJ

Y,I=2(α, β, Z)y/R)ΛZ .

It satisfies ∂ U = U ∂ and it is independent of y in homology level. (Cf. [15])

Remark 13. When d ≤ g(Σ), the periodic Floer homology and U -map are still well-

defined by using a larger class of almost complex structures. (Cf. [18])

It is worth noting that the cases d = Γ · [Σ] = 0 or Y = ∅, in either case, P(Y, ω,Γ)

consists of only one element, the empty set. For any J ∈ Jcomp(Y, π, ω)(not necessarily

generic), the closed holomorphic curve in R × Y has homology class m[Σ] for some

m ≥ 1 and I(m[Σ]) = 0 mod 2. Hence, MJY,I=1(∅, ∅) = ∅ and ∂∅ = 0. In conclusion,

HP (Y, ω,Γ,ΛP ) = ΛP .

5 Cobordism maps on HP via Seiberg Witten

theory

In this section, we follow [12] to define the cobordism maps HP sw(X,ΩX , J,ΛX) via

Seiberg Witten theory and Theorem 1.1. Firstly, let us have a quick review of Seiberg

Witten cohomology and its cobordism maps. In Section 5.2, we show that the isomor-

phism in Theorem 1.1 is canonical. This is a counterpart of Section 3 of [12]. The

amount of work here is relatively fewer, because, ECH admits a filtration structure

25

while PFH does not. Sections 5.3 and 5.4 present a convergence result which is used

to prove the holomorphic curve axiom for Theorem 1. This part is corresponding to

Section 7 of [12]. With all these preparations, we prove Theorem 1.

5.1 Review of Seiberg Witten theory

For our purpose, we only introduce a version of Seiberg Witten cohomology and its

cobordism maps defining by non-exact perturbation. Most of what follows here para-

phrase parts of the accounts in [18] and chapter 29 of [17].

5.1.1 Seiberg Witten coholomogy

Given a closed Riemannian 3-manifold (Y, g), a Spinc structure s on Y is a pair (S, cl),

where S is a rank 2 Hermitian vector bundle over Y and cl : TY → End(S) is a bundle

map such that

cl(u)cl(v) + cl(v)cl(u) = −2g(u, v), (7)

for any u, v ∈ TY and cl(e1)cl(e2)cl(e3) = 1, where e1, e2, e3 is an orthonormal frame

for TY . Let A be a Hermitian connection on detS, it determines a Spinc connection

on S compatible with metric and satisfying

∇A(cl(u)ψ) = cl(∇gu)ψ + cl(u)∇Aψ, (8)

where u ∈ TY , and ψ is a section of S, and ∇g is the Levi-Civita connection of g. The

Dirac operator is defined by DAψ =3∑

i=1cl(ei)∇A,eiψ. The set of isomorphism class of

Spinc structures is an affine space over H2(Y,Z).

Let (Y, π) be a surface fibration over S1 together with an admissible 2-form ω. Let

J ∈ Jcomp(Y, π, ω), now we define a metric g on Y such that

• |R|g = 1 and g(R, v) = 0 for any v ∈ ker π∗.

• g|ker π∗ is given by ω(·, J ·).

It is worth noting that ∗3ω = π∗dt.

With the above choice of metric, given a Spinc structure s = (S, cl), then there is

a decomposition

S = E ⊕ EK−1, (9)

where E is a Hermitian line bundle and K−1 = ker π∗, regards as complex line bundle

via J . Given Γ ∈ H1(Y,Z), we define a Spinc structure sΓ such that c1(E) = PD(Γ).

When E is the trivial line bundle, there is a canonical connection AK−1 such that

26

DAK−1(1, 0) = 0. In general case, any connection A on detS can be written as 2A +

AK−1 for some connection A of E.

Given a connection A of E and a section ψ of S, this pair (A,ψ) is called configu-

ration. A configuration (A,ψ) satisfies the three dimensional Seiberg Witten equations

if the following equations hold:DAψ = 0

∗3FA = r(q3(ψ)− i ∗3 ω)− 12 ∗3 FAK−1

− 12 i ∗3 ℘3,

(10)

where q3(ψ) is a certain quadratic form and r is a large positive constant and ℘3 is a

closed 2-form with cohomology class 2πc1(sΓ). In fact, the solutions to equations (10)

are critical points of the Chern-Simon-Dirac functional:

a(A,ψ) = −1

2

∫

Y(A−A0) ∧ d(A−A0)−

∫

Y(A−A0) ∧ (FA0

+1

2FAK−1

)

− i

2

∫

Y(A−A0) ∧ (2rω + ℘3) + r

∫

Yψ∗DAψ,

(11)

where A0 is a fixed reference connection of E.

Two pairs (A,ψ) and (A′, ψ′) are gauge equivalent if there exists u ∈ Map(Y, S1)

such that A′ = A − u−1du and ψ′ = uψ. The chain complex CM∗(Y,−πr, sΓ,ΛS) is

a free module generated by gauge equivalent classes of solutions to (10). By Lemma

29.12 of [17], there is no reducible solution (ψ = 0) to (10) whenever Γ · [Σ] 6= g(Σ)− 1.

Given c± ∈ CM∗(Y,−πr, sΓ,ΛS), let (A±, ψ±) be a representative of c±, then the

differential < δc+, c− > is defined by counting index one solutions to the following flow

line equations (possibly with an additional choice of abstract perturbation):

∂∂sψ(s) +DA(s)ψ(s) = 0

∂∂sA(s) + ∗3FA = r(q3(ψ)− i ∗3 ω)− 1

2 ∗3 FAK−1

− 12 i ∗3 ℘3

(12)

with condition lims→±∞

(A(s), ψ(s)) = u± · (A±, ψ±), modulo gauge equivalence and R

action. For more details about the abstract perturbation, please refer to [17]. The

homology of (CM∗(Y,−πr, sΓ,ΛS), δ) is denoted by HM∗(Y, sΓ,−πr, J,ΛS).

5.1.2 Cobordism maps on Seiberg Witten cohomology

Let (Y+, g+) and (Y−, g−) be two Riemannian 3-manifolds. Let X be a cobordism

from Y+ to Y−, i.e., ∂X = Y+⊔(−Y−). Given a Riemannian metric g on X such that

g|Y± = g±, a Spinc structure sX on X consists of a rank 4 Hermitian vector bundle

S = S+ ⊕ S− over X, where S+ and S− are rank 2 Hermitian vector bundles, together

with a map clX : TX → End(S) satisfying (7) and

clX(e1)clX(e2)clX(e3)clX(e4) =

(−1 0

0 1

),

27

whenever e1, e2, e3, e4 is an orthonormal frame for TX. The definitions of Spinc

connection and Dirac operator are similar to the three dimensional case.

Let s± = (SY± , clY±) ∈ Spinc(Y±) and sX = (S+ ⊕ S−, clX) ∈ Spinc(X), we say

that sX |Y± = s± if SY± = S+|Y± and clY±(·) = clX(ν±)−1clX(·), where ν+ is outward

unit normal vector of Y+ and ν− is inward unit normal vector of Y−.

Let (X,ΩX) be a symplectic cobordism from (Y+, π+, ω+) to (Y−, π−, ω−) and J ∈Jcomp(X,ΩX) which agrees with symplectization-admissible almost complex structures

J+ and J− on the ends. The metric on X is given by g(·, ·) = ΩX(·, J ·). It is worth

noting that ΩX is self-dual with respect to the metric g and |ΩX |g =√2. (Cf. Lemma

1 of [16]) Moreover, g agrees with ds2+ g± on the ends [0,+∞)×Y+ and (−∞, 0]×Y−respectively. Similar to three dimensional case, there is a decomposition

S+ = E ⊕ EK−1X , (13)

where E is a Hermitian line bundle and K−1X is the canonical bundle of (X,J). Fur-

thermore, the splitting (13) agrees with the splitting (9) on the ends [0,∞) × Y+ and

(−∞, 0] × Y−. Again as three dimensional case, when E is the trivial bundle, there is

a canonical connection AK−1 such that DAK−1(1, 0) = 0.

Fix a 2-form ℘3,± on Y± such that [℘3,±] = 2πc1(s±) and let ℘4 be a closed 2-form

on X which agrees with ℘3,+ and ℘3,− on the ends. For a connection A on E and a

section ψ of S+, the perturbed version of 4-dimensional Seiberg Witten equations is

DAψ = 0

F+A = r

2(q4(ψ)− iΩX)− 12F

+AK−1

− i2℘

+4 ,

(14)

where q4(ψ) is a certain quadratic form and ℘+4 is the self-dual part of ℘4. After a

gauge transformation, we may assume that ∇A = ∂s +∇A(s) (temporal gauge) on the

ends, where A(s) is connection of E|s×Y± . Given c± ∈ CM∗(Y±,−π±,r, s±,Λ±S ) and

their representatives (A±, ψ±), let MrX,ind=0

(c+, c−) be the moduli space of solutions

to (14)(possibly with an additional choice of abstract perturbation) with boundary

conditions

lims→±∞

(A(s), ψ(s)) = (A±, ψ±),

modulo gauge equivalence. The solution (A,ψ) is also called instanton.

Let BX(c+, c−) = (A,Ψ)/ ∽, where (A,Ψ) is a configuration such that (A,Ψ) is

asymptotic to a representative of c± on each end in the sense of last paragraph, modulo

the gauge transformation. Given an element in BX(c+, c−), its relative homotopy class

refers to the element in π0BX(c+, c−). The cobordism maps in chain level is given by

CM(X,ΩX , J, r,ΛX ) =∑

c+,c−

∑

Z∈π0BX(c+,c−)

#MrX,ind=0

(c+, c−,Z)ΛX (Z).

28

This map induces a homomorphism in homology level,

HM(X,ΩX , J, r,ΛX ) : HM∗(Y+,−π+,r, s+, J+,Λ+S ) → HM∗(Y−,−π−,r, s−, J−,Λ

−S ).

The cobordism maps HM(X,ΩX , J, r,ΛX ) satisfies the natural composition rule.(Cf.

Chapter 26 of [17])

Remark 14. In general, to define either HM∗(Y, sΓ,−πr,ΛS) or HM(X,ΩX , r,ΛX ),

one needs to choose a suitable abstract perturbation. But for the same reasons as in [28]

and [12], they play a minor role in the arguments. So we suppress them form notation.

5.2 Canonical isomorphism

Although not explicitly exhibited in [18], the isomorphism in Theorem 1.1 is canonical.

The purpose of this section is to make this point clear.

5.2.1 Invariance of HM and cobordism maps

The cohomology HM∗(Y, sΓ,−πr, J,ΛS) defined in Section 5.1, in fact, is an invariant

in the following sense:

Theorem 5.1 (Theorem 34.4.1 [17]). HM∗(Y, sΓ,−πr, J,ΛS) is independent on J

and r. It depends only on [ω], Y , Γ and ΛS .

To say more about the theorem, given two pairs (ω0, J0) and (ω1, J1) such that

[ω0] = [ω1], then we can take a homotopy ρ = (ωs, Js)s∈[0,1] from (ω0, J0) to (ω1, J1),

where [ωs] = [ω0] and Js ∈ Jcomp(Y, π, ωs) for each s ∈ [0, 1]. Let rss∈[0,1] be a

homotopy between r0 and r1, and fix a generic abstract perturbation p = pss∈[0,1],then the data ρ, rss∈[0,1] and p induce an isomorphism

HM(ρ, rs)p : HM∗(Y, sΓ,−π1r1 , J1,ΛS)p1 → HM∗(Y, sΓ,−π1

r0 , J0,ΛS)p0 .

In fact, HM(ρ, rs)p is induced by the chain map

CM(ρ, rs)p : CM∗(Y, sΓ,−π1r1 , J1,ΛS)p1 → CM∗(Y, sΓ,−π0

r0 , J0,ΛS)p0 ,

where CM(ρ, rs)p is defined by counting index zero solutions to the following equa-

tions:

∂∂sΨ(s) +DA(s)Ψ(s) = Ss

∂∂sA(s) + ∗3FA = (q3(Ψ)− i ∗3 rsωs)− 1

2 i ∗3 ℘3 + Ts,(15)

where the pair (Ss,Ts) comes from the abstract perturbation p.

Lemma 5.2. HM(ρ, rs)p mentioned above satisfies the following properties:

29

1. HM(ρ, rs)p only depends on the homotopy class of ρ and rss∈[0,1].

2. Let (ρ, rss∈[0,1]) and (ρ′, r′ss∈[0,1]) be two homotopies such that (ρ, rs)s=0 =

(ρ′, r′s)s=1, then HM(ρ′ ρ, r′s rs) = HM(ρ′, r′s) HM(ρ, rs).

Note that any two such data ρ, rss∈[0,1] and ρ′, r′ss∈[0,1] lie inside the same

homotopy class. Therefore, HM(ρ, rs) only depends on (ωi, Ji)i=0,1 and rii=0,1.

Let (X,ΩX ) be a symplectic cobordism from (Y+, π+, ω+) to (Y−, π−, ω−), then the

cobordism map HM(X,ΩX ,ΛX) is invariance in the following sense:

Lemma 5.3. Let ρ± = (ω±, J±s)s∈[0,1] and rss∈[0,1] be homotopies, and ΩX0and

ΩX1be two symplectic forms such that ΩX1

= ΩX2in a small collar neighborhood of

Y±, then we have the following commutative diagram.

HM∗(Y+, sΓ+,−π+

r1 , J+1,Λ+S )

HM(X,ΩX1,J1,r1,ΛX)

HM(ρ+,rs)// HM∗(Y+, sΓ,−π+

r0 , J+0,Λ+S )

HM(X,ΩX0,J0,r0,ΛX)

HM∗(Y−, sΓ− ,−π−r1 , J−1,Λ

−S )HM(ρ−,rs)

// HM∗(Y−, sΓ− ,−π−r0 , J−0,Λ

−S )

Proof of Lemma 5.2 and 5.3. The lemmas can be proved by using the general outline

in chapter VII of [17]. It is worth noting that one needs to establish the finiteness

results for the parameterized moduli spaces when rss∈[0,1] is not a constant path.

This can be done by using the same argument in Section 31 of [17].

In conclusion, Lemma 5.2 implies thatHM∗(Y, sΓ,−πr, J,ΛS)p = HM∗(Y, sΓ, ω,ΛS)

is independent of r and almost complex structure J and abstract perturbation. Lemma

5.3 gives us a well-defined cobordismmapHM(X,ΩX ,ΛX) betweenHM∗(Y±, sΓ± , ω±,Λ±S ).

5.2.2 Review of Q-δ flat approximation.

Let ω be a Q-admissible 2-form over π : Y → S1 and J ∈ Jcomp(Y, π, ω). Let

ωΣ = ω|π−1(0) and φω be time one flow of R starting at π−1(0). It satisfies φ∗ωωΣ = ωΣ.

As explained in Section 2.1, Y can be identified with a mapping torus Σφω via dif-

feomorphism (1). The definition of Q-δ flat approximation of (φω, J) is given by the

following lemma:

Lemma 5.4 (Lemma 2.1, Lemma 2.2, 2.4 of [18]). Given Q ≥ 1, 0 < δ ≪ 1 and

J ∈ Jcomp(Y, π, ω), we can modify (φω, J) to (φω′ , J ′) such that

1. (φω′ , J ′) = (φω, J) outside a δ-neighborhood of the periodic orbits with degree less

than Q.

30

2. The set of φω′ periodic orbits with degree less than Q is identical to set of φω

periodic orbits with degree less than Q. Moreover, if γ is elliptic or hyperbolic

as defined using φω, then then it is respectively elliptic or hyperbolic as defined

using φω′ with the same rotation number.

3. Given a periodic orbit γ of φω′ with degree less than Q, then there is an embedding

ϕ : S1τ ×Dz → Σφω′

such that

(a) The Reeb vector field is given by

qϕ−1∗ (R) = ∂τ − 2i(νγz + µγ z + r′)∂z + 2i(νγ z + µγz + r′)∂z.

where

i. If γ is elliptic, then νγ = 12θ and µγ = 0 and θ is an irrational number.

ii. If γ is positive hyperbolic, then νγ = 0 and µγ = − i4π log |λ|, where λ

is the eigenvalue of the linear Poincare return map with |λ| > 1.

iii. If γ is negative hyperbolic, then νγ = 14 , µγ = − i

4π log |λ|eiτ , where λis the same as above.

iv. r′ = 0 in a small neighborhood of z = 0.

(b) ϕ∗J ′(∂z) = i∂z and J ′ satisfies |J − J ′| ≤ c0δ and |∇(J − J ′)| ≤ c0.

(c) ω′|τ×D = i2dz ∧ dz and [ω′] = [ω].

In order to make the proof of Lemma 5.5 that follows clear, we outline the con-

struction of Q-δ flat approximation (ω′, J ′) as follows:

1. Let γ be a periodic orbit with degree q ≤ Q and z = (z1, z2) be symplectic

coordinates centered on the corresponding periodic points of φω. Construct a

tubular neighborhood ϕγ : S1 ×D∗ → Σφω such that

(a) ϕ−1γ∗ (

1

q∂t) = ∂τ − 2i(νz + µz + r)∂z + 2i(νz + µz + r)∂z, (16)

where ν ∈ C∞(S1;R), µ ∈ C∞(S1;C), and |r| ≤ c0|z|2, |∇r| ≤ c0|z|.

(b) Jϕγ∗∂z = iϕγ∗∂z along γ.

2. Fix a homotopy (νλ, µλ)λ∈[0,1] satisfying the following properties:

(a) (ν0, µ0) = (νγ , µγ) and (ν1, µ1) = (ν, µ), where (νγ , µγ) is defined in [18].

31

(b) Let Kλ ∈ SL(2 : R) such that Kλz = η(2π), where η is a solution to the

following equations:

i2dηdτ + νλη + µλη = 0

η(0) = z.

We require that Kλ is conjugated to K0 in SL(2 : R) for any λ ∈ [0, 1].

3. Fix ρ0 ∈ (0, δ16), and let λ∗(z) = χ( log |z|log ρ0), we define

r(1) = (νλ∗ − νγ)z + (µλ

∗ − µγ)z + λ∗r+ e,

where e is a generic small perturbation which is supported in |z| ∈ (ρ7

16

0 , ρ5

16

0 ).

Let D ⊂ D∗ with radius c−10 δ. Define u(1)τ : D → D∗τ∈[0,2π] to be u

(1)τ (z) =

η(τ), where η is a solution to the following equations

i2dηdτ + νγη + µγ η + r(1)(η) = 0

η(0) = z.

4. In general, u(1)τ is not area-preserving. But one can use Morser’s trick to construct

a diffeomorphism ρτ : D∗ → D∗ such that u∗τ (i2dz ∧ dz) = i

2dz ∧ dz, where

uτ = ρτ u(1)τ . Define

φ′ω =

(ϕ−1

γ )∗u 2πkq

on ϕγ(2πkq ×D) for any k ∈ 0, 1, . . . , q − 1

φω otherwise,

then φ′ω induces a Q-admissible 2-form ω′ on Y , it is what we require.

5. The almost complex structure J ′ is defined to be J outside the image of ϕγ for

periodic orbits γ with degree less than Q. Let R′ be the Reeb vector of ω′, we

require that J ′∂s = R′, the only ambiguity concerns the action of ker π∗. In the

image of ϕγ , it is straightforward to construct an almost complex structure J ′ ∈Jcomp(Y, π, ω

′) such that it satisfies ϕγ∗J ′(∂z) = iϕγ∗(∂z) in a small neighborhood

of γ, |J − J ′| ≤ c0δ and |∇(J − J ′)| ≤ c0.

According to Proposition 2.1 of [18], the tautological identification of the respective gen-

erators induces a canonical isomorphism Ψ : HP (Y, ω,Γ, J,ΛP ) → HP (Y, ω′,Γ, J ′,ΛP ).

The following lemma is the counterpart of Proposition B.1 of [28].

Lemma 5.5. Let (ω0, J0) and (ω1, J1) be Q-δi flat approximations of (ω, J). Then there

exists a family of pairs (ωs, Js)s∈[0,1] such that (ωs, Js)s=i = (ωi, Ji) for i = 0, 1 and

(ωs, Js) is Q-δ flat approximation of (ω, J) for each s ∈ [0, 1], where δ = minδ0, δ1.

32

Proof. Given tubular neighborhoods ϕγ : S1 ×D∗ → Σφω described in the first point

of the outline above. Note that the constructions of (ω0, J0) and (ω1, J1) only depend

on the choice of the radius ρ and homotopy (νλ, µλ)λ∈[0,1].Let δ = minδ0, δ1 and ρ∗ ≪ ρ0, ρ1 ∈ (0, δ16). We can take a decreasing family

ρs from ρ0 to ρ∗. For each s, we can follow above outline to construct a Q-δ flat

approximation of (ω, J), and this give a homotopy of Q-δ flat approximation from

(ω0, J0) to the ρ∗ version of (ω0, J0). Similar conclusion can be made for (ω1, J1).

Based on the above understanding, we assume that (ω0, J0) and (ω1, J1) are defined

by using ρ∗ ∈ (0, δ16) but different homotopies (νλ0 , µλ0)λ∈[0,1] and (νλ1 , µλ1 )λ∈[0,1]respectively. In fact, we can insert a family of homotopies (νλs , µλs )s,λ∈[0,1] betweenthem so that it satisfies (a) and (b) in the second point of our outline for each s ∈ [0, 1].

(Cf. Proof of Proposition B.1 of [28]) Therefore, for each s, we can follow the outline

to construct a Q-δ flat approximation (ωs.Js) for (ω, J).

5.2.3 Isomorphism between HP and HM

With the previous understanding, we show that the isomorphism in Theorem 1.1 is

canonical in this subsection.

Definition 5.6. Let F,G : CP (Y+, ω+,Γ+,Λ+P ) → CP (Y−, ω−,Γ−,Λ

−P ) be two homo-

morphisms, where F =∑

α+,α−

∑Z∈H2(X,α+,α−)

fZΛZ and G =∑

α+,α−

∑Z∈H2(X,α+,α−)

gZΛZ .

We say that F equals to G up to order L if fZ = gZ for any Z with∫Z ωX < L, denoted

by F = G+ o(L). The analogy can be defined for the maps between chain complexes of

Seiberg Witten cohomology.

Suppose that (ω, J) is Q-δ flat, by Theorem 6.1 of [18], we have a bijective map

Tr : CP∗(Y, ω,Γ, J,ΛP ) → CM−∗(Y, sΓ,−πr, J,ΛS). (17)

In addition, given L > 0, there exists rL > 0 such that for any r > rL, we have

#MJ,LY,I=1(α, β)/R = #M

J,LY,ind=1(Tr(α), Tr(β))/R. (18)

A priori, the isomorphism Tr∗ is not induced by Tr straightforward in general case,

because the constant rL may go to infinity as L → ∞.

To define Tr, recall that the cohomology HM−∗(Y, sΓ,−πr, J,ΛS) is independent

on r. We can take unbounded generic sequences Ln∞n=1 and rn∞n=1 such that rn >

rLn and a sequences of homotopies rn,s|s ∈ [0, 1]∞n=1 such that rn,s=0 = r and rn,s=1 =

rn, also a sequence of generic abstract perturbations. Define

Tr = limn→∞

CM(rn,s) Trn : CP∗(Y, ω,Γ, J,ΛP ) → CM−∗(Y, sΓ,−πr, J,ΛS). (19)

33

For each n, CM(rn,s) can be factored as in Section 4 of [28]. Using the limit argument

in Lemma 4.6 of [28], one can show that the energy of the trajectory which contributes

to each small piece is positive unless that it is close to the constant trajectory. More-

over, the energy of the broken trajectory which consists of ”constant trajectory” is still

close to zero. Use these properties one can show that energy of the trajectory which

contributes to CM(rn,s) has an n-independent lower bound, provided that r is suffi-

ciently large. Hence, the above limit is well defined. Also, note that Tr may dependent

on the choice of the sequences and homotopies, but Tr∗ does not.

The following lemma is a counterpart of Lemma 3.4 of [12]. To simplify the notation,

for any two Q-δ flat approximations, we suppress the canonical isomorphism between

them in both chain complex level and homology level.

Lemma 5.7. Suppose that ρ = (ωs, Js)s∈[0,1] such that (ωs, Js) is Q-δ flat and Js ∈Jcomp(Y, π, ωs)

reg for each s ∈ [0, 1]. Given L > 0, then there exists cL > 1 such that

if mins∈[0,1]

rs ≥ cL, CM(ρ, rs) is chain homotopy to Tr0 T−1r1 + o(L).

Proof. The argument basically is the same as Lemma 3.4 of [12]. Arguing by contra-

diction, suppose that for each positive integer j, there is a path rj,s|s ∈ [0, 1] such

that the conclusion fails and limj→∞

mins∈[0,1]

rj,s = +∞.

For each j and positive integer k, we choose a family of abstract perturbations

pj,k,s|s ∈ [0, 1] suitable for defining HM(ρ, rj,s) and satisfying

1. |pj,k,s|P < k−1,

2. The pj,k,s instantons between generators of CM∗(Y, sΓ,−πrj,s , Js,ΛS) have non-

negative index.

Now fix j and k, let N be a positive large integer. Taking a partition of [0, 1],

0 = s0 < s1 < . . . sN = 1 and |si − si−1| ≤2

N.

Let Ii : CM∗(Y, sΓ,−πri , Ji,ΛS) → CM∗(Y, sΓ,−πri−1

, Ji−1,ΛS) be the chain

map which is defined by ρ|[si−1,si], rj,s|[si−1,si] and pj,k,s|[si−1,si]. Let I = I1 · · · IN ,

then Lemma 5.2 implies that I is chain homotopy to CM(ρ, rs).

If Ii is canonical bijection of the generators up to order L, i.e., Ii = Trsi−1 T−1

rsi+

o(L), then I = Tr0 T−1r1 + o(L), contradict with our assumption. Therefore, for each

N , we can find iN such that IiN is not canonical bijection of the generators up to order

L.

Let N → ∞, using the compactness argument as the proof of Theorem 34.4.1 [17]

and the second property of our pj,k,s, there exists sj,k ∈ [0, 1] and an index zero, non-R

invariant pj,k,sj,k-instanton dj,k between generators of CM∗(Y, sΓ,−πrsj,k, Jsj,k ,ΛS).

34

Moreover, the total drop in the sj,k-version of Chern-Simon-Dirac functional along dj,k

is uniformly bounded by rj(c0 + L), where rj = maxs∈[0,1]

rj,s.

Passing a subsequence, we can assume that limj→∞

limk→∞

sj,k = s∗. The argument in

Section 5 of [18] and Section 4 of [31] can be repeated almost verbatim to conclude the

following: the instantons dj,k converges to a non-R invariant Js∗- holomorphic currents

(possibly broken) C in R×Y between generators of CP∗(Y, ωs∗,Γ, Js∗ ,ΛP ). In addition,

by Theorem 5.1 of [2], I(C) = 0. However, our Js∗ here is generic, this is impossible.

Corollary 5.8. Suppose that (ω0, J0) and (ω1, J1) are Q-δ flat approximations of (ω, J),

then the following diagram commutes.

HP∗(Y, ω0,Γ, J0,ΛP )

Ψ0,1

Tr0∗// HM−∗(Y, sΓ,−π0

r0 , J0,ΛS)

HM(ρ,rs)

HP∗(Y, ω1,Γ, J1,ΛP )Tr1∗

// HM−∗(Y, sΓ,−π1r1 , J1,ΛS)

The map Ψ0,1 is induced by canonical bijection of the generators.

Proof. By Lemma 5.5, we can find a homotpy ρ = (ωs, Js)s∈[0,1] from (ω0, J0) to

(ω1, J1) so that (ωs, Js) is Q-δ flat pairs for each s ∈ [0, 1].

Taking increasing unbounded sequences Ln∞n=1 and rn∞n=1 such that rn >

maxrLn , cLn. Let ρi = (ωi, Ji)s∈[0,1] and homotopy rin,ss∈[0,1] such that rin,s=0 =

ri and rin,s=1 = rn, i = 0, 1. By Lemmas 5.7 and 5.2, we know that HM(ρ, rs) is

induced by the following map

CM(ρ0, r0n,s) CM(ρ, rn) CM(ρ1, r

1n,s)

−1

= CM(ρ0, r0n,s) (Trn T−1

rn + o(Ln)) CM(ρ1, r1n,s)

−1.

The corollary follows from (19) and taking n→ ∞.

Let Ψ : HP∗(Y, ω,Γ, J,ΛP ) → HP∗(Y, ω′,Γ, J ′,ΛP ) be the canonical isomorphism

induced by Q-δ flat approximation. Let

T ′r∗ : HP∗(Y, ω

′,Γ, J ′,ΛP ) → HM−∗(Y, sΓ,−π′r, J

′,ΛS)

be the isomorphism provided by Theorem 1.1. Take a homotopy ρ = (ωs, Js)s∈[0,1]from (ω, J) to (ω′, J ′), then we have an isomorphism

T∗ : HP∗(Y, ω,Γ, J,ΛP ) → HM−∗(Y, sΓ, [ω],ΛS), (20)

where T∗ is defined by T∗ = ir HM(ρ, r)T ′r∗Ψ and ir : HM

−∗(Y, sΓ,−πr, J,ΛS) →HM−∗(Y, sΓ, [ω],ΛS) is the direct limit isomorphism. By Corollary 5.8 and Lemma 5.2,

T∗ is independent of the choice of the Q-δ flat approximation and r.

35

Corollary 5.9 (Cf. Corollary 6.7 of [18]). HP∗(Y, ω,Γ,ΛP ) is independent on J .

Proof. For any two almost complex structures J0, J1 ∈ Jcomp(Y, π, ω)reg, the isomor-

phism between HP∗(Y, ω,Γ, J0,ΛP ) and HP∗(Y, ω,Γ, J1,ΛP ) is given by T −11∗ T0∗.

5.3 Convergence theorem

Proposition 5.11 that follows asserts that a sequence of Seiberg Witten solutions to

(14) give rise to broken holomorphic curves. It is a counterpart of Proposition 7.1 of

[12]. In this subsection, we assume that (ω±, J±) is Q-δ flat, unless otherwise stated.

Recall that T±r is the bijection in (17) and s : X → R is the coordinate function.

We prove a convergence result for a slightly general setting than we need here. Given

a symplectic form ΩX over X, assume that there exists a positive function K : (−∞, ǫ]∪[−ǫ,∞) → R such that ΩX = ω± + K2ds ∧ π∗±dt over (−∞, ǫ] × Y− ∪ [−ǫ,∞) × Y+.

Additionally, we assume that K satisfies the following properties:

1. |K − 1|C2 ≤ c0e− s

c0 .

2. There exists a positive constant C such that C−1 ≤ K ≤ C.

Take a J ∈ Jcomp(X,ΩX), note that J is still compatible with ΩX over X. Define

a metric g = ΩX(·, J ·). Obviously, g is not product metric on the ends. But ΩX still

is a self-dual harmonic 2-form with respective to g and |ΩX |g =√2. Along the ends of

X , the metric g satisfies the following properties:

1. g = K2ds2 + g3, where g3 is an s-dependence metric on Y±.

2. |R±|g3 = K and g3(v,w) = ω±(v, Jw) for any v,w ∈ ker π±∗.

3. ∗3ω± = Kπ∗±dt.

4. For any a ∈ Ω1(Y ) and b ∈ Ω2(Y ), we have

∗4(ds ∧ a) = K−1 ∗3 a, and ∗4 b = Kds ∧ ∗3b.

Write FA = ds ∧ EA + FA(s) over the ends of X, then (14) becomes

1K∇A,sψ +Dg3

A(s)ψ = 0

1KEA + ∗3FA(s) +

12 ∗3 FAK−1

− r(q3(ψ) − iKdt) + i2 ∗3 ℘3 = 0.

(21)

Note that EA = ∂sA if A is in temporal gauge.

36

Note that the splitting (13) still holds. We decompose ψ = (α, β) with respective

to this splitting. We have the following relation.

clX(ΩX)α = −2iα and clX(ΩX)β = 2iβ

clY (Kdt)α = iα and clY (Kdt)β = −iβ.(22)

The following definition is an analogy of the ΩX-energy of holomorphic curve.

Definition 5.10. Given a configuration d = (A,ψ) ∈ BX(c+, c−), the ΩX energy of v

is defined as follows:

FΩX(d) =

i

2π

∫

R−×Y−FA ∧ ω− +

i

2π

∫

XFA ∧ ΩX +

i

2π

∫

R+×Y+

FA ∧ ω+.

Note that FΩX(d) only depends on c± and the relative homotopy class of d.

The main result of this section is as follows:

Proposition 5.11. Let (A+, ψ+) and (A−, ψ−) be solutions to r-version of (10) and

[(A±, ψ±)] = T±r (α±). Let d = (A,ψ) be a solution to r-version of (14) which is

asymptotic to (A±, ψ±) at the ends. Assume that FΩX(d) ≤ L. Given any δ > 0, there

exists κδ ≥ 1 and c0 ≥ 1 such that the following are true for r ≥ κδ:

1. Each point in X where |α| ≤ 1− δ has distance less than c0r− 1

2 from α−1(0).

2. There exist

(a) a positive integer N ≤ c0 and an open cover of R as⋃

1≤k≤N Ik, each of

length at least 2δ−1, with [−1, 1] ⊂ Ik0 and

(b) a broken holomorphic current Ck1≤k≤N from α+ to α− such that for each

k, we have

i. supz∈C∈Ck

dist(z, α−1(0)) + supz∈α−1(0)

dist

z,

⋃

C∈CkC

< δ.

ii. Let I ⊂ Ik be an interval of length 1 and let ν denote the restriction to

s−1(I) with |ν|∞ = 1 and |∇ν| ≤ δ−1. Then

| i2π

∫

s−1(I)FA ∧ ν −

∫

Cν| ≤ δ,

where A = A− 12(α∇Aα− α∇Aα).

The essential difference between [12] and our case is how to get the uniformly upper

bound on the function M, where M is defined as follows. Under the assumption in

37

Proposition 5.11 and let d = (A,ψ) be a solution to the r-version of (14), we define a

function M : R → R by

M(s) =

∫

[s,s+1]r(1− |α|2).

Firstly, let us summarize the analytic results which we need in the proof of Propo-

sition 5.11.

1. An r-independent upper bound on the function M. The precise statement and

the proof will be given in Lemma 5.17 latter.

2. Provided that the last point is true, we can obtain the counterparts of Lemmas

5.3, 5.6, 5.7, 5.9, 5.10, 5.11 and 5.12 in [18]. Because these lemmas carry over

almost verbatim to our setting, we will not repeat them here. We only remind

that the counterpart of Lemmas 5.12, |β| ≤ c0r−1 and |∇Aβ| ≤ c0r

− 1

2 are true at

the points x ∈ X that has distance c−10 equal or less to R± × γ± and |s(x)| ≥ 1.

But this is enough for our purpose. The proof of these Lemmas can be copied with

the relevant argument in Section 3 of [31]. The proof there relies on the elliptic

estimates for certain differential inequalities. One can deduce these differential

inequalities from the Seiberg Witten equations and Weizenbock formula. Since

elliptic estimates are local, the proof in [31] can be applied to our cases without

essential change. We refer to the lemmas mentioned above as priori estimates for

the Seiberg Witten equations (14).

Remark 15. One can compare the parallel results in ECH setting (Sections 3 of [31])

with PFH setting (our case here or [18]), the only difference is that the assumptions

Av < r2 or fv > −r2 and E(A) ≤ L in ECH setting are replaced by FΩX(d) ≤ L.

These kinds of assumptions are equivalent to give an upper bound on the topological

energy of Seiberg Witten equations. In Remark 17, we will explain a little more about

the difference.

With the above analytic results in hand, we follow Section 4 in [31] to sketch how to

use them to prove the convergence theorem. For the convenience of readers, we restate

some key lemmas (Lemmas 5.12, 5.13) in [31] and [12].

The following proposition is a local version of Proposition 5.11. Roughly speaking,

it asserts that given a sequence of solutions (Arn , ψrn), there exists a subsequence of

(Arn , ψrn) converges to a holomorphic current C in certain sense over any compact

subset of X.

Proposition 5.12. (Cf.Prop 4.1 [31] and Prop 7.8 [12]) Under the same assumptions

in Proposition 5.11 and fixed a δ > 0. Let I be a connected subset of R of length at

least 2δ−1 + 16. Let I ⊂ I be a connected subset of points with distance at least 7 from

38

∂I and |I| ≥ 2δ−1. Then there exists κδ ≥ 1 and c0 ≥ 1 such that the following are true

for r ≥ κδ:

1. Each point in s−1(I) where |α| ≤ 1− δ has distance less than c0r− 1

2 from α−1(0).

2. There exists holomorphic current C defined in a neighborhood of the closure of

s−1(I), such that

(a) supz∈C∈C∩s−1(I)

dist(z, α−1(0)) + supz∈α−1(0)∩s−1(I)

dist

(z,⋃

C∈CC

)< δ.

(b) Let ν be a 2-form with support in s−1(I) such that |ν| ≤ 1 and |∇ν| ≤ δ−1,

then

| i2π

∫

s−1(I)FA ∧ ν −

∫

Cν| ≤ δ,

where A = A− 12(α∇Aα− α∇Aα).

Proof. (Sketch) With the above analytic results in hand, Proposition 4.1 and its proof

in [31] can be carried over almost verbatim to our setting. To see how these work, let

Fr be the current associate to the curvature of Ar, i.e., Fr(ν) =i2π

∫X FAr ∧ ν for any

2-form ν with compact support in a compact set K ⊂ X . By the analytic result above,

there is an r-independent bound on the L1 norm of curvature, i.e., |FAr |L1(K) ≤ CK .

As a result, the current Fr converges weakly to a current F∞. By the technique called

positive cohomology assignment which is introduced in [23], the support of F∞ is a

holomorphic current C. Moreover, it satisfies the properties (a) and (b).

If s−1(I) is non-compact, then one can take a sequence of compact sets to cover

s−1(I). The argument above supplies a holomorphic current in each compact set. The

argument on page 2874-2875 of [31] shows that the holomorphic currents over these

compact sets can be patched together.

Basically, Lemmas 4.6, 4.7 and 4.8 in [31] concern the following phenomenon: If the

energy of holomorphic curve is sufficiently small, then it closes to the trivial cylinder

in certain sense. The statement and the proof there can be adapted to our setting with

only notation change. We omit them here.

The following lemma is an analogy of Lemma 4.9 in [31] and Lemma 7.10 [12] in

our setting.

Lemma 5.13. (Cf. Lemma 4.9 in [31] and Lemma 7.10 [12]) Under the same assump-

tions in Proposition 5.11. Let I ⊂ R − 0 be a connected subset with length at least

16. Given ε > 0, let I = k ∈ Z|∫s−1[k,k+1]FA ∧ ω ≥ ε, where ω = ω+ or ω−. Let

I ′ ⊂ I− (∪k∈I [k, k + 1]) be a connected component. Then i2π

∫s−1(I′) FA ∧ ω ≥ −ε2.

39

Proof. Let I ′ be the portion of I ′ with distance at least 16 + Rε′ from ∂I ′, where Rε′

is defined in Lemma 4.9 of [31]. Let I ′′ be the rest, i.e., I ′′ = I ′ − I ′. It suffices to

estimate i2π

∫s−1(I′) FA ∧ ω and i

2π

∫s−1(I′′) FA ∧ ω.

To obtain the lower bound for i2π

∫s−1(I′′) FA ∧ ω, note that the argument in [31]

only use the priori estimates for Seiberg Witten equations. Since the relevant analytic

results are also true in our setting, we can copy the relevant argument to get lower

bounded for i2π

∫s−1(I′′) FA ∧ ω,.

To estimate∫s−1(I′) FA ∧ω, the proof in [31] use the exactness of da in ECH setting

while our ω is not exact. But we can make the following adjustment: We decompose

ω = ω0 +∑

γ dyγ , where γ runs over simple periodic orbits with degree less than Q,

yγ is a 1-form supports in a 10ε′ neighborhood of γ and ω0 supports away from γ.

The term∫s−1(I′) FA ∧ dyγ =

∫s−1(∂I′) FA ∧ yγ can be estimated as (4-14) in [31]. As

a result, we obtain the lower bound for∫s−1(I′) FA ∧ dyγ . Let C be the holomorphic

current provided by Proposition 5.12, ([31] uses notation ϑ), therefore,

|∫

s−1(I′)FA ∧ ω0 −

∫

C∩s−1(I′)ω0| < δ.

Note that the argument in Lemma 4.9 of [31] show that C|s is contained in a 2ε′

neighborhood of γ for any s ∈ I ′, thus∫C∩s−1(I′) ω0 = 0. Hence,

∫s−1(I′) FA ∧ ω0 >

−δ.

The consequence of Lemma 5.13 is that there are only finitely many elements in I.The solutions (Ar, ψr)r over I − (∪k∈I [k, k + 1]) have small energy. Apply Proposi-

tion 5.12 to (Ar, ψr)r, (Ar, ψr)r supplies a nontrivial holomorphic current in each

connected component of s−1(∪k∈I [k, k + 1])), while in each connected component of

s−1(I − (∪k∈I [k, k + 1])), the holomorphic current is close to the trivial cylinders and

they are corresponding to ends of holomorphic curves. Roughly speaking, the above

construction supplies us with the broken holomorphic current promised in Proposition

5.11. For more details, we refer the reader to Section 4 of [31].

5.4 Energy Bound

In this subsection, we prove the promised bound for M. Firstly, let us introduce some

functionals as follows:

1. S(A,ψ) =(F+

A

2− cl−1

X (ΨΨ∗)0 + iη

4,DAΨ

),

where A = 2A+AK−1 , Ψ =√2rψ and η = 2rΩX + 2℘+

4 .

2. As in [17], we define the topological energy and analytic energy as follows:

40

(a) Eanal(XR)(A,ψ) =1

4

∫

XR

|FA|2 +∫

XR

|∇AΨ|2 +∫

XR

2| i4η − cl−1

XR(ΨΨ∗)0|2

+1

4

∫

XR

Rg|Ψ|2 − i

∫

XR

FA ∧ 1

2∗ ℘4,

where XR = x ∈ X ||s(x)| ≤ R.(b) Etop(XR)(A,ψ) =

1

4

∫

XR

FA ∧ FA −∫

∂XR

< Ψ,DAΨ > +i

∫

XR

FA ∧ (rΩX +1

2℘4)

+

∫

∂XR

H

2|Ψ|2,

where H is mean curvature.

3. We follow [18] to define a functional QF as follows:

QF (A) = i

∫

Y±(A−A0) ∧ ω±,

where A0 is a fixed reference connection of E → Y±. We abuse notation A0 to

denote a reference connection of E → X such that A0 is R invariant on the ends.

4. Let a± be the Chern-Simon-Dirac functional for Y±. Recall that we have defined

them in (11).

Lemma 5.14. Assume that ℘4 is a closed 2-form such that ℘4 = ℘3 when |s| ≥0. Suppose that d = (A,ψ) is a solution to r-version of (14) which is asymptotic to

solutions (A+, ψ+) and (A−, ψ−) to (10) at the ends. Then there exists a constant

c0 > 0 such that for r ≥ c0, we have

1. |α| ≤ 1 + c0r ,

2. |β|2 ≤ c0r−1(1− |α2|) + c2

0

r2.

Proof. The proof is the same as Lemma 7.3 of [12].

The next lemma shows that the Chern-Simon-Dirac functional is almost decreasing.

Since the metric g3 varies along the ends, the Chern-Simon-Dirac functional a(d|s)should be understood as the one defined by metric g3|s×Y .

Lemma 5.15. Assume that ℘4 is a closed 2-form such that ℘4 = ℘3 when |s| ≥ 0.

Suppose that d = (A,ψ) is a Seiberg Witten solution to r-version of (33) which is

asymptotic to solutions (A+, ψ+) and (A−, ψ−) to (10) at the ends. For any 0 < s− ≤s+ or s− ≤ s+ < 0, we have

a(d|s−)− a(d|s+)

=1

2

∫ s+

s−

(∫

Y(K|B(A,ψ)|2 +K−1|EA|2 + 2rK|Dg3

A ψ|2 + 2rK−1|∇A,sψ|2)volg3

)ds

+

∫ s+

s−

(∫

Y

(r < ψ, (∂sD

g3)ψ > +rh(ψ,Dg3A ψ)

)volg3

)ds,

41

where B(A,ψ) = ∗3FA − r(q3(ψ)− iKdt)+ 12 ∗3 FAK−1

+ i2 ∗3 ℘3, h = ∂sg, and ∂s(D

g3)

is derivative of the Dirac operator defined by metric g3 along s direction. Furthermore,

a(d|s−)− a(d|s+) ≥1

4C

∫

s∈[s−,s+](|B(A,ψ)|2 + |EA|2) + 2r(|Dg3

A ψ|2 + |∇A,sψ|2)− c0r.

(23)

Proof. Without loss of generality, assume that (A,ψ) is in temporal gauge. By direct

computation, we have

∂sa(d|s) = −∫

Y∂sA ∧ (FA(s) +

1

2FAK−1

+ irω +i

2℘3) + r

∫

Y

(< ∂sψ,D

g3A ψ > + < ψ,Dg3

A (∂sψ) >)

+ r

∫

Y< ψ, clY (∂sA)ψ > +r

∫

Y< ψ, (∂sD

g3)ψ > +r

∫

Yh(ψ,Dg3

A ψ)

= −∫

Y∂sA ∧ (FA(s) +

1

2FA

K−1− r ∗3 q3(ψ) + irω +

i

2℘3) + r

∫

Y2Re < ∂sψ,D

g3A ψ >

+ r

∫

Y< ψ, (∂sD

g3)ψ > +r

∫

Yh(ψ,Dg3

A ψ).

Then the conclusion follows from the flow line equations (21).

Using Cauchy-Schwarz inequality, the term rh(ψ,Dg3A ψ) can be absorbed by the

term 2rK|Dg3A ψ|2. By the assumption on K, we have |h|+|∂s(Dg3)| ≤ c0e

− sc0 . Combine

this with Lemma 5.14, we get the inequality (23).

Lemma 5.16. Assume that ℘4 is a closed 2-form such that ℘4 = ℘3 when |s| ≥0. Suppose that d = (A,ψ) is a Seiberg Witten solution to r-version of (14) which

is asymptotic to solutions (A+, ψ+) and (A−, ψ−) to (10) at the ends. Assume that

FΩX(d) ≤ L, then there exists a constant c0 > 0 such that for r ≥ c0, we have

1. |a(d|∂X )| ≤ c0(1 + 2πL)r, where a(d|∂X ) = a+(d|0×Y+)− a−(d|0×Y−).

2. Etop(XR)(A,ψ) ≤ c0(1 + 2πL)r + c0Rr.

3. 18

∫X |FA|2 +

∫X |∇AΨ|2 + 2

∫X | i4η − cl−1

X (ΨΨ∗)0|2 ≤ c0(1 + 2πL)r.

4. M(s) =∫[s,s+1] r(1− |α|2) ≤ c0(1 + 2πL)r

1

2 .

Proof. For the sake of simplicity, we assume that Y− = ∅. The proof is divided into

the following four steps.

Step 1 Note that a(A+, ψ+) + rQF (A+) is gauge-invariant. By Lemma 5.1 in [18],

|a(A+, ψ+) + rQF (A+, ψ+)| ≤ c0r. (24)

42

On the other hand, QF (A+, ψ+) = 2π(FΩX(d)−FΩX

(A0)). Therefore,

QF (A+, ψ+) ≤ 2πL+ c0 (25)

provided that FΩ(d) ≤ L. Combine Lemma 5.15, (24) and (25), we have

a(d|∂X) ≥ a(A+, ψ+)− c0r ≥ −(c0 + 2πL)r. (26)

By direct computation, we have

Etop(X)(A,ψ) =1

4

∫

XFA0

∧FA0+i

4

∫

XFA0

∧(rΩX+1

2℘4)−2a(d|∂X )+

∫

∂XrH|ψ|2.

(27)

By our assumption, |H| ≤ c0. Therefore, Etop(X)(A,ψ) ≤ c0(1 + 2πL)r.

Step 2 Replace X by XR and a(d|∂X ) by a(d|∂XR) in (26) and (27), then we complete

the proof of the second assertion in the lemma. Note that the extra term c0Rr

comes from the energy i4

∫XR

FA0∧ rΩX .

Step 3 Follows from Lemma 5.14 and Cauchy-Schwarz inequality, we have

Eanal(X)(A,ψ) ≥ 1

4

∫

X|FA|2 +

∫

X|∇AΨ|2 +

∫

X2| i4η − cl−1

X (ΨΨ∗)0|2

− i

∫

XFA ∧ 1

2∗ ℘4 − c0r

≥ 1

4

∫

X|FA|2 +

∫

X|∇AΨ|2 +

∫

X2| i4η − cl−1

X (ΨΨ∗)0|2 −1

8

∫

X|FA|2

− c0

∫

X|℘4|2 − c0r

≥ 1

8

∫

X|FA|2 +

∫

X|∇AΨ|2 +

∫

X2| i4η − cl−1

X (ΨΨ∗)0|2 − c0r.

By direct computation, we know that

|S(A,ψ)|2L2(XR) = Eanal(XR)(A,ψ) − Etop(XR)(A,ψ)

for any R ≥ 0 (Cf. Section 4.5 of [17]). In particular, Etop(X)(A,ψ) = Eanal(X)(A,ψ)

when (A,ψ) is a solution to (14). Thus we have

1

4

∫

XFA0

∧ FA0+ i

∫

XFA0

∧ (rΩX +1

2℘4)− 2a(d|∂X )

≥ 1

8

∫

X|FA|2 +

∫

X|∇AΨ|2 +

∫

X2| i4η − (ΨΨ∗)0|2 − c0r

=⇒ 1

8

∫

X|FA|2 +

∫

X|∇AΨ|2 +

∫

X2| i4η − (ΨΨ∗)0|2 ≤ −2a(d|∂X ) + c0r.

In particular, a(d|∂X ) ≤ c0r, and hence |a(d|∂X )| ≤ c0(1 + 2πL)r. Moreover,

1

8

∫

X|FA|2 +

∫

X|∇AΨ|2 +

∫

X2| i4η − cl−1

X (ΨΨ∗)0|2 ≤ c0(1 + 2πL)r. (28)

These deduce the first and third bullets of the lemma.

43

Step 4 To prove the last bullet of the lemma. By relation (22) and Lemma 5.14, we have∫

X2| iη

4− cl−1

X (ΨΨ∗)0|2 =∫

Xr2(1− |α|2)2 + e, (29)

where |e| ≤ 11000

∫X r2(1− |α|2)2 + c0r. Combine this with (28), we have

∫

Xr(1− |α|2)2 ≤ c0(1 + 2πL). (30)

By Holder inequality,

∫

Xr(1− |α|2) ≤ c0r

1

2

(∫

Xr(1− |α|2)2

) 1

2

≤ c0(1 + 2πL)1

2 r1

2 . (31)

To estimate∫[s,s+1]×Y+

r(1 − |α|2) for s ≥ 0, the argument is the same as the

proof of Lemma 5.4 of [18]. By the first bullet of the lemma and inequalities (26)

and Lemma 5.15,∫

R+×Y+

(|EA|2 + |B(A,ψ)|2 + 2r(|∇A,sψ|2 + |DA(s)ψ|2)) ≤ c0(1 + 2πL)r, (32)

in particular,∫[s,s+1]×Y+

|EA|2 ≤ c0(1 + 2πL)r.

By the Seiberg Witten equations (14), (22) and Lemma 5.14,

i

∫

[s,s+1]×Y+

K2ds ∧ π∗+dt ∧ (K−1 ∗3 EA) + i

∫

[s,s+1]×Y+

K2ds ∧ π∗+dt ∧ (FA(s) +1

2FA

K−1)

=

∫

[s,s+1]×Y+

r(1− |α|2) + e, (33)

where |e| ≤ r100

∫[s,s+1]×Y+

(1− |α|2) + c0.

The first term of (33)

|i∫

[s,s+1]×Y+

Kds ∧ π∗+dt ∧ ∗3EA| ≤ c0

(∫

[s,s+1]×Y+

|EA|2) 1

2

≤ c0(1 + 2πL)r1

2 ,

and the second term of (33) is c1(sΓ+) · [π∗+(Kdt)], these give the promised bound

on∫[s,s+1]×Y+

r(1− |α|2) in the lemma.

Lemma 5.17. Assume that ℘4 is a closed 2-form such that ℘4 = ℘3 when |s| ≥ 0.

Suppose that d = (A,ψ) is a Seiberg Witten solution to r-version of (14) which is

asymptotic to solutions (A+, ψ+) and (A−, ψ−) to (10) at the ends to (10) at the ends.

Assume that FΩ(d) ≤ L, then there exists a constant c0 > 0 such that for r ≥ c0,

M(s) =

∫

[s,s+1]r(1− |α|2) ≤ c0(1 + 2πL).

44

Proof. Fix R ≥ 2, note that the conclusions in Lemma 5.16 are still true by replacing

X by XR. Let w = 1 − |α|2, then w satisfies the following equation (Cf. the proof of

Lemma 2.2 of [23])

1

2d∗dw + r|α|2w − |∇Aα|2 + ew = 0, (34)

where |ew| ≤ c0(|α|2 + |β|2 + |∇Aβ|2).Let χ be a cut-off function such that χ = 1 on X 1

2R and χ = 0 on X\X 3

4R. Multiply

both sides of the equality (34) by χ, then we have

1

2d∗d(χw) − 1

2w(d∗dχ)− < dχ, dw > +χr|α|2w − χ|∇Aα|2 + χew = 0. (35)

Integrate both sides of (35), then∫

XR

χr|α|2w ≤∫

XR

χ|∇Aα|2 +∫

XR

2|dχ||∇Aα|+∫

XR

χ|ew|+ c0

≤ c0

∫

XR

(|∇Aα|2 + |∇Aβ|2) + c0.

By the third point of Lemma 5.16,∫XR

(|∇Aα|2 + |∇Aβ|2) ≤ c0(1 + 2πL). Hence,∫X 1

2R

r|α|2w ≤ c0(1 + 2πL).

On the other hand, inequality (30) tells us that that∫

X 12R

rw2 =

∫

X 12R

r(1− |α|2)2 ≤ c0(1 + 2πL).

Therefore,∫X 1

2R

r(1 − |α|2) =∫X 1

2R

rw2 +∫X 1

2R

r|α|2w ≤ c0(1 + 2πL). In particular,

M(0) ≤ c0(1 + 2πL).

To bound M(s) =∫[s,s+1] r(1− |α|2) for s > 0, we may assume that M(s) attains

its maximum at some s∗ ∈ (1,∞). Let χ∗ be a cut off function such that χ∗ = 1 on

s∗ − 1 ≤ s ≤ s∗ + 1 and χ∗ = 0 whenever |s− s∗| ≥ 2.

Recall the inequality (32), apply Weizenbock formula for DA(s) and use the fact

that∫s×Y+

i2πFA(s) ∧ π∗+(Kdt) = c1(E) · [π∗+(Kdt)], we have

c0(1 + 2πL)r ≥∫

[0,+∞)×Y+

(|EA|2 + |B(A,ψ)|2 + 2r|∇A(s),sψ|2 + 2r|DA(s)ψ|2)

≥∫

[0,+∞)×Y+

χ∗(|EA|2 + |B(A,ψ)|2 + 2r|∇A(s),sψ|2 + 2r|DA(s)ψ|2)

≥∫

[0,+∞)×Y+

χ∗(|EA|2 + |BA|2 + 2r|∇Aψ|2 + r2w2)− c0r. (36)

Therefore,∫[0,+∞)×Y+

χ∗(rw2+2|∇Aα|2 +2|∇Aβ|2) ≤ c0(1+ 2πL). Replace the cut off

function χ by χ∗ in (35), and repeat the same argument, we obtain∫

s∈[s∗−1,s∗+1]r|α|2w ≤ c0(1 + 2πL).

45

Hence,∫s∈[s∗−1,s∗+1] r(1− |α|2) ≤

∫s∈[s∗−1,s∗+1](r|α|2w + rw2) ≤ c0(1 + 2πL).

Remark 16. Assume that K ≡ 1. Using the same limit argument in the proof of

Proposition 5.2 of [12], the following statement is true. Let (Ar±, ψ

r±)r≥r0 be a family

of solutions to (10) with sufficiently small abstract perturbation p±. Let drr≥r0 =

(Ar, ψr)r≥r0 be a family of solutions to (14) which is perturbed by the small abstract

perturbation p. Assume that dr is asymptotic to (Ar±, ψ

r±) at the ends, where p = p± on

the ends. Suppose that FΩX(dr) ≤ L and (Ar

±, ψr±)r≥r0 converges to orbit set α± in

the current sense, then there is a broken J holomorphic current between α+ and α−.

Remark 17. One could compare the result here with Proposition 5.1 of [31] and Propo-

sition 7.1 of [12], the proof in our setting is easier than the ECH setting. The reasons

are as follows.

• Let d = (A,ψ) be an instanton which is asymptotic to (A±, ψ±). In both PFH

and ECH cases, we need to control the quantity Av = a−(A−, ψ−)− a+(A+, ψ+).

In PFH case, the Chern-Simon-Dirac functional is of the form

a = a0 −r

2QF ,

where a0 is gauge invariant. Choose a suitable gauge, Lemma 5.1 of [18] shows

that |a0| ≤ c0r. Therefore, the quantity Av is completely controlled by QF (A+)−QF (A−), while QF (A+)−QF (A−) is controlled by FΩX

(d).

However, in ECH case, the relevant part is more complicated. At generators, the

Chern-Simon-Dirac functional is of the form

a =1

2(cs− rE).

The quantity Av is not determined by the energy i∫X FA∧dλ = E(A+)−E(A−).

The reason is as follows. If the spin-c structure is torsion, one may follow the

same argument in PFH setting to show that Av is controlled by E(A+) and

E(A−). However, the energy does not give any bound on E(A+) and E(A−).

In fact, E(A−) and i∫X FA ∧ dλ are controlled by E(A+), however, this is non-

trivial and is not obvious from the perspective of Seiberg Witten theory. The

same problems happen when the spin-c structure is non-torsion. Also, note that

the Chern-Simon functional cs is not gauge-invariant in this case, we need to

take into account the grading of the generators. Thus one also needs the spectral

flow estimate.

• Another difference is that the energy i∫Y FA ∧ dt is a topological invariant, while

it is not in the ECH setting. This difference simplifies the analysis in many

46

places. In fact, this already causes differences in the last bullet. The correspond-

ing quantities of E(A±) are constant in the PFH setting. This is why we can get

the uniform bound |a0| ≤ c0r. Another example is that it is very difficult to get

the estimate (36) in ECH setting, because, one needs to make effort to get the

bound on i∫s×Y FA(s) ∧ λ. Lemmas 5.2 and 5.3 of [31] and argument in pages

79-81 are trying to realize the similar bound for∫s×Y FA(s) ∧ λ.

5.5 Proof of Theorem 1

In order to reduce the proof of Theorem 1 to the case that (ΩX |Y± , J |R±×Y±) is Q-δ

flat, we prove the following lemma firstly. It plays the same role as Lemma 6.6 of [12].

Lemma 5.18. Let (X,πX , ωX) be a fiberwise symplectic cobordism from (Y+, π+, ω+) to

(Y−, π−, ω−). Let J ∈ Jtame(X,πX , ωX)reg or J ∈ Jcomp(X,ΩX)reg such that J = J±on R± × Y±. Given sufficiently small δ > 0, then there exist an admissible 2-form ω′

X

and an almost complex structure J ′ ∈ Jtame(X,πX , ω′X)reg or J ′ ∈ Jcomp(X,Ω

′X )reg

accordingly so that (ω′X , J

′)|R±×Y± = (ω′±, J

′±) is a Q-δ flat approximation of (ω±, J±),

where Ω′X = ω′

X+ωB. The almost complex structure J ′ satisfies estimates |J ′−J | ≤ c0δ

and |∇(J ′ − J)| ≤ c0. Moreover, (X,ω′X) is monotone if (X,ωX) is monotone.

Proof. For simplicity, we assume that Y− = ∅. Let R+ be the Reeb vector field of ω+.

By Lemma 5.4, we can find a Q-δ flat approximation of (ω+, J+), the result is denoted

by (ω′+, J

′+).

According to the construction of Q-δ flat approximation, we know that ω′+ = ω+ +

d(h+dt) and |R+ − R′+| ≤ c0δ, where h+ is a function which supports in δ-tubular

neighborhoods of periodic orbits with degree less than Q, R+ and R′+ are Reeb vector

fields of ω+ and ω′+ respectively. It is easy to check that |dvh+| ≤ c0δ, where d

vh+ is

the component of dh+ which doesn’t contain dt. Consequently, |h+| ≤ c0δ2,

Let U+ = [0,−ǫ]×Y+ be a collar neighborhood of Y+ such that ΩX |U+= ω++ds∧

π∗+dt and J |U+= J+. We define a new admissible 2-form ω′

X on X by

ω′X =

ω+ + d(φ(s)h+) ∧ π∗+dt : on U+

ωX : on X \ U+,

where φ(s) is a cut off function such that φ = 1 near s = 0 and φ = 0 near s = −ǫ,and |φ′| ≤ c0ǫ

−1. Note that ω′X has the same cohomology class as ωX , therefore, ω′

X

is monotone whenever ωX is monotone. In addition, ω′X + π∗XωB is still symplectic

whenever δ ≪ ǫ is small enough.

Let ω+s = ω+ + d(φ(s)h+) ∧ dt|s×Y+and Rs be Reeb vector field of (ω+s, π

∗+dt).

Let Jss∈[0,1] be a family of almost complex structures on ker π+ starting from J+|ker π+

47

and ending at J ′+|ker π+

, and Js|ker π+is compatible with ωs|ker π+

for each s. The almost

complex structure J ′ is defined as follows:

1. J ′(∂s) = Rs, J′ maps kerπ+ to itself and J ′|ker π+

= Js|ker π+in U+.

2. J ′ agrees with J in X \ U+.

By construction, we have |J ′ − J | ≤ c0δ and |∇(J ′ − J)| ≤ c0. It is straight for-

ward to check that J ′ ∈ Jtame(X,πX , ω′X) and J ′ ∈ Jcomp(X,Ω

′X) whenever J ∈

Jtame(X,πX , ωX) and J ∈ Jcomp(X,ΩX) respectively.

Finally, we make a small perturbation such that the resulting almost complex struc-

ture J ′ ∈ Jtame(X,πX , ω′X)reg or Jcomp(X,ΩX)reg. In addition, we make that such a

perturbation supports in U+ and the image of ϕγ but disjoints with the periodic or-

bits.

Remark 18. The conclusions in Lemma 5.18 are also true when (X,ΩX) is a sym-

plectic cobordism and J ∈ Jcomp(X,ΩX), because, the construction only takes place in

collar neighborhoods of Y+ and Y− where ΩX = ω± + ds ∧ π∗±dt.

Proof of Theorem 1. The homomorphism HP sw(X,ΩX ,ΛX) is defined by

HP sw(X,ΩX ,ΛX) = T −1− HM(X,ΩX ,ΛX) T+.

Note that we have a natural decomposition

HPsw(X,ΩX ,ΛX) = T −1− HM(X,ΩX ,ΛX) T+

=∑

sX∈Spinc(X),sX |Y±=sΓ±

T −1− HM(X,ΩX , sX ,ΛX) T+

=∑

sX∈Spinc(X),sX |Y±=sΓ±

HP sw(X,ΩX , sX ,ΛX).

It is well known that Spinc(X) is an affine space over H2(X, ∂X,Z). Therefore,

HPsw(X,ΩX ,ΛX) =∑

ΓX∈H2(X,∂X,Z),∂Y±ΓX=Γ±

HPsw(X,ΩX ,ΓX ,ΛX).

The first three assertions of Theorem 2 are direct consequences of the corresponding

properties of cobordism maps on Seiberg Witten Floer cohomology.

To prove the fourth point in the theorem, first of all, we assume that (ω±, J±) is

Q-δ flat. Fix an almost complex structure J ∈ Jcomp(X,ΩX) such that J |R±×Y± = J±is generic, then the chain map CP sw(X,ΩX , J,ΛX) is defined by

CP sw(X,ΩX , J,ΛX) = limr→∞

(T−r )−1 CM(X,ΩX , J, r,ΛX ) T+

r . (37)

48

Suppose that CP sw(X,ΩX , J,ΛX) is non-zero, then the existence of holomorphic cur-

rent C follows immediately from Proposition 5.11. The analogy of Theorem 5.1 of [2]

tells us that C has zero ECH index.

For the case that (ω±, J±) is not Q-δ flat, we can use the argument in Section 6 of

[12] to show that the conclusion still holds. The main points are sketched as follows:

By Lemma 5.18 and Remark 18, we can construct a sequence of pair (ΩXn, Jn) over

X such that each (ΩXn|Y± , Jn|R±×Y±) is a Q- 1n flat approximation of (ω±, J±) and JnC0-converges to J . The pair (ΩXn, Jn) plays the same role as (λ′n, J

′n) in Section 6 of

[12]. Define the chain map by

CP sw(X,ΩX , J,ΛX) = limn→∞

(Ψ−n )

−1 CP sw(X,ΩXn , Jn,ΛX) Ψ+n , (38)

where CPsw(X,ΩXn , Jn,ΛX) is (ΩXn, Jn) version of (37) and Ψ±n is the canonical iso-

morphism induced by Q- 1n flat approximation. The analogy of commutative diagram in

Lemma 6.5 of [12] can be proved similarly, thus CPsw(X,ΩX , J,ΛX) induces the cobor-

dism map HPsw(X,ΩX , J,ΛX ). If the chain map (38) is non-trivial, then Proposition

5.11 supplies us a Jn holomorphic current Cn for sufficiently large n. By Taubes’Gromov

compactness (Lemma 6.8 of [12]), Cn converges to a J holomorphic current C. This

complete the proof of the holomorphic curve axiom.

Note that Lemma 6.8 of [12] is weaker than the usual one (eg. Lemma 9.9 of [8]),

because, Jn only C0-converges to J . The new ingredient of the proof is Lemma 6.13

of [12], it also can be applied to our case. In sum, the analogy of Lemma 6.8 of [12] is

also true in our setting.

Remark 19. The convergence theorem 5.11 implies that a sequence of instantons drrin Mr

X(c+, c−) gives a relative homology class Z, where Z is the relative class of holo-

morphic curves given by the instantons. Note that the relative homology class Z only

depends on the relative homotopy class of drr. Moreover, this correspondence is 1-1.

If the instantons contribute to the cobordism maps, then the ECH index of Z is zero.

(Cf. Theorem 5.1 of [2])

Assume that (ΓX ,ΩX) is monotone, then there are only finitely many relative ho-

mology classes supporting holomorphic curves with zero ECH index.(Cf. Corollary 7.16)

The observations in the last paragraph imply that there are only finitely many relative

homotopy classes such that MrX,ind=0

(c+, c−,Z) is nonempty. In particular, the cobor-

dism map HPsw(X,ΩX ,ΓX) can be defined with Z2 or Z coefficient.

49

6 Proof of Proposition 3.2

Let (Y, π) be a surface fibration over circle together with an admissible 2-form ω, and

J ∈ Jcomp(Y, π, ω). By Lemma 5.4, we can assume that (ω, J) is Q-δ flat. Let γ be an

elliptic orbit with degree q ≤ Q, then there is a small tubular neighborhood of γ such

that it can be identified with the following standard form

(S1τ ×Dz, qR = ∂τ − iθz∂z + iθz∂z, ω =

i

2dz ∧ dz +

(θ

2zdz +

θ

2zdz

)∧ dτ

),

together with the standard almost complex structure J(∂x) = ∂y. Assume that 0 <

θ < 1, otherwise, we change of coordinate (t = τ, w = e−ikτz) for suitable k ∈ Z.

Let us assume that 0 < θ < 12 , the construction that follows also works for 1

2 <

θ < 1. The only difference is that one needs to choose a suitable function C(r). (See

Remark 20) Take a disk Dδ′ ⊂ D with radius 0 < δ′ ≪ 1 and a smooth function

f : D → R such that f is constant outside Dδ′ and f = f(r). Define a new admissible

2-form by

ωf = ω + df ∧ dτ.

By directly computation, the Reeb vector field of ωf over S1 ×Dδ′ is

qRf = ∂τ − (θ +f ′

r)x∂y + (θ +

f ′

r)y∂x.

Let C(r) be a smooth function satisfying the following properties:

1. C(r) = 12 − θ on D δ′′

2

, where 0 < δ′′ ≤ 12δ

′.

2. C(r) ≥ 0 and C(r) = 0 outside Dδ′′ . Also, C′(r) ≤ 0.

Take f ′/r = C(r), then we know that f(r) =∫ r0 sC(s)ds and f ′(0) = 0. Therefore, γ

is still a periodic orbit of Rf with degree q.

The Reeb flow is

dx

dτ= (θ +C(r))y

dy

dτ= −(θ + C(r))x.

When r ≤ δ′′2 , above equations become

dx

dτ=

1

2y

dy

dτ= −1

2x.

The linear Poincare return map is given by sending (x, y) to (−x,−y) when r ≤ δ′′2 .

Remark 20. If 12 < θ < 1, we can take C(r) to be a smooth function such that

50

1. C(r) = 12 − θ on D δ′′

2

, where 0 < δ′′ ≤ 12δ

′.

2. C(r) ≤ 0 and C(r) = 0 outside Dδ′′ . Also, C ′(r) ≥ 0.

Lemma 6.1. The set of φωfperiodic orbits with degree less than q is identical to the

set of φω periodic orbits with degree less than q.

Proof. Let γq′ be a periodic orbit with degree q′ ≤ q. Suppose that there exists τ0 such

that γq′(τ0) ∈ Dδ′′ , otherwise, there is nothing to prove. We assume that τ0 = 0.

First of all, we claim that γq′(τ) ∈ D2δ′′ for any τ . Let τmax = supτ ∈ [0, 2π]|γq′ (τ) ∈Dδ′′, if τmax = 2π, then we have done. If τmax < 2π, then for τ > τmax, γq′(τ) satisfies

the ODE ∂τx = θy and ∂τy = −θx. It cannot escape outside D2δ′′ , because, it is a

rotation. In particular, the degree of γq′ is at least q, and hence q′ = q.

Now the Reeb vector is qRf = ∂τ − (θ + C(r))∂φ in D2δ′′ , where (r, φ) is the polar

coordinate. The Reeb flow equation becomes

dr

dτ= 0

dφ

dτ= −(θ + C(r)).

Therefore, r = r0 and φ(τ) = φ0 − (θ + C(r0))τ . Since 0 < θ ≤ θ + C(r0) ≤ 12 ,

φ(2π) 6= φ(0) mod 2π unless that r0 = 0. In other words, γq = r = 0 = γ.

Define a new coordinate (t′, x′, y′) by

t′ = τ x′ = cos(1

2τ)x− sin(

1

2τ)y y′ = sin(

1

2τ)x+ cos(

1

2τ)y.

Under the coordinate above, (S1τ ×D δ′′

4

, ωf , qRf ) can be identified with

([0, 2π] ×D δ′′

4

/((2π, x, y) ∼ (0,−x,−y)), dx′ ∧ dy′, ∂t′),

and the almost complex structure J satisfies J(∂x′) = ∂y′ .

Take a function h with compact support on D δ′′4

such that h = 12((x

′)2 − (y′)2) on

D δ′′8

. Define

ω′f = dx′ ∧ dy′ + εdh ∧ dt′.

Then γ = x′ = y′ = 0 still is a periodic orbit. Moreover, for sufficiently small ε > 0,

there is no periodic orbit with period q other than γ. The linear Reeb flow is

dx′

dt= −εy′

dy′

dt= −εx′.

Then the linear Poincare return map has two real eigenvalues −eε and −e−ε. Therefore,

γ is negative hyperbolic.

51

Proof of Proposition 3.2. First of all, we can always perturb (ω+, J+) such that it is

Q-δ flat. If there exists a periodic orbit γ of ω+ which is not Q-negative elliptic or hyper-

bolic, then we perform the above construction to make it become negative hyperbolic.

This process may create another elliptic orbit, but they have larger degree. Repeat

this process until all periodic orbits with degree less than Q satisfy the conclusion.

Obviously, the cohomology class of ω+ does not change under the above construction.

Also, from the construction, we know that

|ω′+ − ω+| ≤ c0δ.

We can modify ω− in the same way.

By using cut-off functions, we can always modify ωX such that ωX = ω± in a collar

neighborhood of Y±. Since ω′± = ω±+ dµ± for some µ±, we can define ω′

X by the same

formula (2).

7 Cobordismmaps on HP via holomorphic curve

In principle, the cobordism maps in chain level should be defined by

CP (X,ΩX , J,ΛX ) =∑

α±∈P(Y±,ω±,Γ±)

∑

Z∈H2(X,α+,α−)

#2MJX,I=0(α+, α−, Z)ΛX(Z).

(39)

However, Section 5.5 of [10] points out that (39) doesn’t make sense in general, be-

cause, the moduli space MJX,I=0(α+, α−, Z) is not compact due to the appearance of

holomorphic curves with negative ECH index.

In this section, we show that actually definition (39) works for fiberwise symplectic

cobordism (X,πX , ωX) satisfying the assumptions (♠). Firstly, we show that the ECH

index is nonnegative and investigate the holomorphic currents with small indexes (0 ≤I ≤ 1). Then we construct the cobordism maps separately for the following three

cases: (X,πX ) contains no separating fiber, monotone case and (X,πX ) is not relatively

minimal. Finally, we show that the cobordism maps satisfy certain natural properties.

Remark 21. It is worth mentioning that recently C.Gerig defines the cobordism maps

on ECH for certain symplectic cobordisms [5]. It is worth comparing our results with

C.Gerig’s on ECH. In both cases, we borrow the idea in [11] to use the Q-positive

(negative) orbits to guarantee the nonnegative of the ECH index. Besides, our proof

here is also based on the following two simple observations:

O.1 If C is a holomorphic curve in X with at least one end, then g(C) ≥ g(B). (See

Lemma 7.1 below.)

52

O.2 If Y+ 6= ∅ and Y− 6= ∅, and C is a holomorphic curve in X with at least one end,

then C has at least one positive end and one negative end.

Both of these observations come from the fibration structure, which does not appear

in the ECH setting. In the ECH case, even the ECH index is nonnegative, C.Gerig

needs to deal with certain multiply covered holomorphic planes. Compare to his case,

our situation is even better, the multiply covered holomorphic curves cannot contribute

to the cobordism maps because of the observations above. For example, there is no

holomorphic plane in case ♠.3 due to O.2.

In the remaining of this paper, we focus on the fiberwise symplectic

cobordism case. We always assume that the fiberwise symplectic cobordism

(X,πX , ωX) satisfies the assumptions (♠), unless otherwise stated.

7.1 Index computation

In this subsection, we show that the ECH index of holomorphic current is nonnegative

under assumptions (♠). Moreover, the moduli space MJX,I=i(α+, α−) only consists of

embedded holomorphic curves in the sense of Definition 4.5, where i = 0 or 1.

Lemma 7.1. Let α+ and α− be orbit sets of (Y+, π+, ω+) and (Y−, π−, ω−) respectively,

and J ∈ Jtame(X,πX , ωX). Suppose that C is an irreducible somewhere injective J

holomorphic curve from α+ to α− in X and C is not closed, then g(C) ≥ g(B).

Proof. Let d = [α±] · [Σ], then C ∩ [Σ] = d. The intersection positivity implies that C

intersects a generic fiber with d distinct points. Since πX : X :→ B is complex linear,

πX : C → B is a degree d branched cover.

Suppose that C has k+ positive ends and k− negative ends, 1 ≤ k± ≤ d, then by

Riemann-Hurwitz formula,

2− 2g(C)− k+ − k− = d(−2g(B)) − b

⇒ g(C) = 1 + dg(B) +1

2b− 1

2(k+ + k−) (40)

⇒ g(C) ≥ 1 + dg(B)− d ≥ g(B) + (d− 1)(g(B) − 1) ≥ g(B).

where b ≥ 0 is the sum over all the branch points of the order of multiplicity minus

one.

Recall that C ⋆ C is the generalized self-intersection number in Definition 4.4. The

following lemma tells us that it is nonnegative.

Lemma 7.2. Let α+ and α− be orbit sets of (Y+, π+, ω+) and (Y−, π−, ω−) respec-

tively (not necessarily admissible), and C ∈ MJX(α+, α−) be an irreducible simple

53

holomorphic curve. Assume that C is not closed, then C ⋆ C ≥ i ≥ 0 for generic

J ∈ Jtame(X,πX , ωX), where i = 1 in case ♠.1, i = 12 in case ♠.2 and i = 0 in case

♠.3.

Proof. We divide the proof into three cases, and we prove the result case by case.

• In case ♠.1, the conclusion follows from Lemma 7.1 and Definition 4.4.

• In case ♠.2, C ⋆ C ≥ 0 by Lemma 7.1. It suffices to rule out the possibility that

C ⋆ C = 0. If C ⋆ C = 0, it follows from Definition 4.4 that C satisfies

g(C) = 1 and indC = eQ(C) = δ(C) = h(C) = 0. (41)

By (41) and the assumption ♠.2, all ends of C are asymptotic to elliptic orbits

with degree larger than 1. Substitute g(C) = g(B) = 1 for (40), we deduce that

k± = d, where k+ and k− are the numbers of positive and negative ends of C

respectively, and d is the degree of πX : C → B. Consequently, all the ends of C

are asymptotic to periodic orbits with degree 1, we get a contradiction.

• In case ♠.3, by observation O.2, every holomorphic curve inX which is not closed

has at least one positive end and at least one negative end. Therefore, 2eQ(Ca)+

h(Ca) ≥ 2 under our assumption. Then the conclusion follows immediately from

Definition 4.4.

Corollary 7.3. Let α+ and α− be orbit sets of (Y+, π+, ω+) and (Y−, π−, ω−) re-

spectively (not necessarily admissible), and C ∈ MJX(α+, α−). Assume that C doesn’t

contain closed components. Then for generic J ∈ Jtame(X,πX , ωX), I(C) ≥ 0.

Proof. By Theorem 4.6, I(Ca) ≥ ind(Ca) ≥ 0 for generic J ∈ Jtame(X,πX , ωX). Ac-

cording to Lemma 7.2 and inequality (4), we have I(C) ≥ 0.

The following two lemmas analyze the holomorphic curves with ECH index zero

and one.

Lemma 7.4. Let α+ and α− be orbit sets of (Y+, π+, ω+) and (Y−, π−, ω−) respectively,

and C = (Ca, da) ∈ MJX(α+, α−) without closed components. Assume that at least

one of α+ and α− is admissible. For generic J ∈ Jtame(X,πX , ωX), if I(C) = 0, then

1. For each a, I(Ca) = ind(Ca) = 0.

2. C is embedded in the sense of Definition 4.5.

3. If u is a J holomorphic curve whose associated current is C, then u is admissible.

54

Proof. For cases ♠.1 and ♠.2, the conclusions follows immediately from Lemma 7.2,

Theorem 4.6 and 4.7.

For case ♠.3, if da > 1 for some a, then we must have Ca ⋆ Ca = 0. Recall that

h(Ca) + 2eQ(Ca) ≥ 2 because of observation O.2. The equality Ca ⋆ Ca = 0 forces

g(Ca) = indCa = δ(Ca) = 0 and h(Ca) + 2eQ(Ca) = 2.

If eQ(Ca) = 1 and h(Ca) = 0, then Ca has only one end, which contradicts with O.2.

As a result, the only possibility is that

g(Ca) = indCa = δ(Ca) = eQ(Ca) = 0, and h(Ca) = 2. (42)

Thus Ca has respectively one positive end and one negative end at hyperbolic orbit.

However, α+ or α− is admissible, this forces da = 1.

Lemma 7.5. Let α+ and α− be admissible orbit sets of (Y+, π+, ω+) and (Y−, π−, ω−)

respectively. Let C = (Ca, da) ∈ MJX(α+, α−) without closed components. For generic

J ∈ Jtame(X,πX , ωX), if I(C) = 1, then

1. C is embedded in the sense of Definition 4.5.

2. There exists a unique a0 such that I(Ca0) = ind(Ca0) = 1. Without loss of

generality, we always assume that a0 = 1. For a ≥ 1, I(Ca) = ind(Ca) = 0.

3. If u is a J holomorphic curve whose associated current is C, then u is admissible.

Proof. We divide the proof into three cases as before, and we prove them case by case.

• Fist of all, we consider case ♠.1. The first and third conclusions and uniqueness

of a0 follow from the formula (4). The third conclusion of the lemma follows from

Theorem 4.6. To show the second conclusion, it suffices to rule out the case that

all Ca have I(Ca) = 0 but I(C) = 1.

Arguing by contradiction. Suppose that I(C) = 1 but I(Ca) = 0 for any a. By

(5.5) of [9], we have

I(∑

a

Ca) =∑

a

I(Ca) + 2∑

a6=b

(Ca ⋆ Cb − lτ (Ca, Cb)) + µτ (∑

a

Ca)−∑

a

µτ (Ca),

Suppose that each Ca has ends at γ with total multiplicity ma and m =∑

ama,

where γ is an embedded periodic orbit. If γ is elliptic and its rotation number is

θ, then

m∑

q=1

µτ (γq)−

∑

a

ma∑

q=1

µτ (γq) =

m∑

q=1

−∑

a

ma∑

q=1

2⌊qθ⌋ = even.

55

Since α+ and α− are admissible, the same conclusion automatically holds when

γ is hyperbolic. Hence, we have I(∑aCa) =

∑aI(Ca) mod 2 which contradicts

with I(C) = 1.

Therefore, there exists a0 = 1 such that I(C1) = 1. Since ind(C1) ≤ I(C1) and

I(C1) = ind(C1) mod 2, we must have ind(C1) = 1.

• Now we consider case ♠.2. If da > 1 for some a, then Ca ⋆ Ca = 12 . Recall that

g(Ca) ≥ 1 under the assumptions. Definition 4.4 and equality Ca⋆Ca = 12 deduce

that Ca satisfies any one of the following:

g(Ca)− 1 = h(Ca) = δ(Ca) = eQ(Ca) = 0, and indCa = 1, (43)

g(Ca)− 1 = indCa = δ(Ca) = eQ(Ca) = 0, and h(Ca) = 1,

and Ca has no ends at elliptic orbits.(44)

The reasons for second line in (44) are as follows. If g(Ca) = 1, the Riemann-

Hurwitz formula (40) tells us that all the ends of Ca are asymptotic to periodic

orbits with degree 1. On the other hand, the assumption ♠.2 and eQ(Ca) = 0

imply that Ca has no ends at elliptic orbits with degree 1.

Firstly, the case (43) cannot happen. If h(Ca) = 0, then all the ends of Ca

are asymptotic to elliptic orbits. As a result, indCa = 0 mod 2. We obtain a

contradiction.

In the case (44), all the ends of Ca are asymptotic to hyperbolic orbits. The

assumption that α+ and α− are admissible forces da = 1.

Repeat the same argument in case ♠.1, we obtain the conclusion.

• For case ♠.3, if da > 1 for some a, then Ca⋆Ca = 12 or Ca⋆Ca = 0. If Ca⋆Ca = 0,

follow from the proof of Lemma 7.4, then Ca satisfies (42).

If Ca ⋆ Ca = 12 , by 2eQ(Ca) + h(Ca) ≥ 2, one can check that Ca satisfies any one

of the following:

g(Ca) = δ(Ca) = eQ(Ca) = 0, and indCa = 1, h(Ca) = 2, (45)

or g(Ca) = δ(Ca) = indCa = eQ(Ca) = 0, and h(Ca) = 3, (46)

or g(Ca) = δ(Ca) = indCa = 0, and eQ(Ca) = 1, h(Ca) = 1. (47)

In either of these cases, Ca has at least one end at hyperbolic orbits. Then

da > 1 contradicts with the assumption that α+ and α− are admissible. As a

result, da = 1 for any a. Repeat the same argument in case ♠.1, we get the same

conclusion.

56

Assume that u is a holomorphic curve with 0 ≤ I(u) ≤ 1. If we remove the

admissible assumption on α±, then u may not be embedded. The following lemma

asserts that the Fredholm index of u is still nonnegative for generic J .

Lemma 7.6. Let α+ and α− be orbit sets of (Y+, π+, ω+) and (Y−, π−, ω−) respec-

tively (not necessary to be admissible). For generic J ∈ Jtame(X,πX , ωX), any u ∈MJ

X,I=i(α+, α−) without closed irreducible components has nonnegative Fredholm in-

dex, i.e.,

indu ≥ 0,

where i = 0 or i = 1.

Proof. From the proof of Lemmas 7.4 and 7.5, we know that u has no nodes but u

could be multiple covers of simple curves, as our α± may not be admissible anymore.

Without loss of generality, we may assume that u is an irreducible non-simple

curve. Then u is branched cover of an embedded curve C. According to the discussion

in Lemmas 7.4 and 7.5, C satisfies (42) or (44) or (45) or (46) or (47). The covering

degree is denoted by d. In the cases (42) or (44) or (45) or (46), all the ends of u are

asymptotic to hyperbolic orbits, by Riemann-Hurwitz formula, we have

indu = dindC + b ≥ 0,

where b ≥ 0 is the sum over all the branch points of the order of multiplicity minus

one.

In the case (47), C is a holomorphic cylinder from γm+ to γn−. One of them is elliptic

orbit while the other one is hyperbolic. Without loss of generality, assume that γ+ is

an embedded elliptic orbit and γ− is an embedded hyperbolic orbit. Note that γ− is

negative hyperbolic, otherwise, indC = 1 mod 2. We can take a trivialization τ such

that µτ (γp+) = −1 for any p ≤ Q and µτ (γ−) = 1. Therefore, indC = 0 implies that

2cτ (C) = 1 + n.

By Riemann-Hurwitz formula, we have

indu = b+ 2dcτ (C) +∑

i

µτ (γpi+ )−

∑

j

µτ (γqj− )

= b+ d(1 + n)−∑

i

−∑

j

qj ≥ 0.

The last step in above inequality is a consequence of∑

i ≤ d and∑

j qj = dn.

One can perform the same argument when γ+ is hyperbolic and γ− is elliptic, then

we obtain the same conclusion.

57

7.2 (X, πX) contains no separating singular fiber.

In this subsection, we define the cobordism maps HP (X,ΩX , J,ΛP ) by counting holo-

morphic curves in the case that X contains no separating singular fiber.

7.2.1 Compactness and Transversality

Recall that the fibers of (X,πX ) are holomorphic curves. The following lemma rules

out the fibers provided that the degree Γ± · [Σ] is large enough.

Lemma 7.7. Suppose that (X,πX) contains no separating singular fiber. Let C =

(Ca, da) ∈ MJX(α+, α−). Assume that d = α± · [Σ] > g(Σ) − 1. If C contains closed

components, then I(C) ≥ 2 for generic J ∈ Jtame(X,πX , ωX).

Proof. By our assumptions, the homology class of regular fibers and singular fibers are

[Σ], thus we can rewire C = C′+m[Σ] for some m ≥ 1 and C′ has no closed component.

By definition of ECH index,

I(C)− I(C′) = m < c1(TX), [Σ] > +2mQτ (C, [Σ]) +m2Qτ ([Σ], [Σ]).

According to the definition of Qτ and adjunction formula, we have

Qτ (C, [Σ]) = C ∩ [Σ] = d,Qτ ([Σ], [Σ]) = [Σ] · [Σ] = 0,

< c1(TX), [Σ] >= χ(Σ) + [Σ] · [Σ] = 2− 2g(Σ).

Therefore, I(C) = I(C′) + 2m(d− g(Σ) + 1) ≥ 2.

Lemma 7.8. Let α+ and α− be admissible orbit sets of (Y+, π+, ω+) and (Y−, π−, ω−)

respectively. Assume that d = α± · [Σ] > g(Σ) − 1 and (X,πX ) contains no separating

singular fiber. Then for generic J ∈ Jtame(X,πX , ωX) and any L > 0, the moduli space

MJ,LX,I=0(α+, α−) is compact.

Proof. The proof here is similar to Lemma 7.19 of [13]. Let un∞n=1 ⊂ MJ,LX,I=0(α+, α−)

be a sequence of holomorphic curves and their underlying currents are Cn∞n=1. By

Taubes’ Gromov compactness (Cf. Lemma 6.8 of [1]), Cn∞n=1 converges to a broken

holomorphic current C∞ = C0, . . . , CN in the sense of Section 9 of [8]. Convergent

as a current implies that [Cn] = [C∞] for large n. Then J0(Cn) = J0(C∞), where J0

is a number which only depends on relative homology class. For the precise definition

of J0, please refer to [9]. According to Corollary 6.11 and Proposition 6.14 of [9],

J0(Cn) = J0(C∞) implies that the genus of domains of un∞n=1 have an n independent

upper bound.

58

With the above understanding, we can apply the Gromov compactness in [1] to

un∞n=1. After passing a subsequence, un∞n=1 converges to a broken holomorphic

curve u∞ = u−N− , . . . , u0, . . . , uN+, where u0 ∈ MJX(α0, β0) and ui ∈ MJ+

Y+(αi, βi)

for i > 0 and ui ∈ MJ−Y−

(αi, βi) for i < 0 , αi = βi+1, αN+ = α+ and β−N− = α−.

Moreover,

I(u∞) = I(u−N−) · · ·+ I(u0) + · · ·+ I(uN+) = 0. (48)

By Lemma 7.7, u∞ doesn’t contain closed components.

By Lemma 7.3 and the fact that I(ui) ≥ 0 in R×Y±, (48) forces I(ui) = 0 for all i.

For i 6= 0, I(ui) = 0 implies that ui are branched cover of cylinders with nonnegative

Fredholm index. By Lemma 7.6, ind(u0) ≥ 0. The additivity of the Fredholm index

implies ind(ui) = 0 for i. In particular, ui is a connector for i 6= 0.

Since un is admissible for all n, uN+ is positively admissible. Recall that there

is no nontrivial connector from p+(γ,m) to other partitions of γm. As a result, uN+

is a trivial connector, which is ruled out in the holomorphic building. Therefore, u∞has no positive level. Similarly, u∞ has no negative level. In conclusion, u∞ = u0 ∈MJ,L

X,I=0(α+, α−).

Corollary 7.9. For J ∈ J1tame(X,πX , ωX , J±) and any L > 0, the moduli space

MJ,LX,I=0(α+, α−) is a finite set. In particular, the equation (39) is meaningful.

Proof. The corollary follows from Lemmas 7.8 and 4.10.

7.2.2 Chain map

To show that (39) is a chain map, we carry out the following compactness result. It is

similar to Lemma 7.23 of [14].

Lemma 7.10. Let α+ and α− be admissible orbit sets of (Y+, π+, ω+) and (Y−, π−, ω−)

respectively with degree d > g(Σ) − 1. Suppose that (X,πX) contains no separating

singular fiber. For any L > 0 and generic J ∈ Jtame(X,πX , ωX), let un∞n=1 ⊂MJ,L

X,I=1(α+, α−), then un∞n=1 converges to a broken holomorphic curve u∞, either

u∞ ∈ MJ,LX,I=1(α+, α−) or u∞ = u0, . . . , ui, . . . , uN+ or u∞ = u−N− , . . . ui, . . . u0,

where u0 ∈ MJ,LX,I=0(β+, β−) and it is embedded, uN+ ∈ MJ+,L

Y+,I=1(α+, β+), u−N− ∈

MJ−,LY−,I=1(β−, α−) for some β± and uii are connectors for i 6= 0,±N±.

Proof. Arguing as Lemma 7.8, un∞n=1 ⊂ MJ,LX,I=1(α+, α−) converges to a broken

holomorphic curve u∞ = u−N− , . . . , u0, . . . uN+ in the sense of [1] and

I(u−N−) · · · + I(u0) + · · ·+ I(uN+) = 1.

Here we only consider case ♠.3, the other two cases are similar.

59

First of all, keep in mind that 0 ≤ I(ui) ≤ 1 and ui has nonnegative Fredholm index,

for any N− ≤ i ≤ N+. These follow from Corollary 7.3, Lemma 7.6 and Proposition

7.15 of [13]. Note that u0 could be a multiply covered holomorphic curve.

If I(uN+) = I(u−N−) = 0, then u±N± is a union of branched cover of trivial cylin-

ders. By the facts that ind(u±N±) ≥ 0 and ind(u±N±) = 0 mod 2 whenever α+ and

α− are admissible, we have ind(u±N±) = 0 and u±N± is connector. Since each un is ad-

missible, u∞ is also admissible. As a result, u±N+ and u±N− must be trivial connectors,

which are ruled out in the holomorphic building. Hence, u∞ = u0 ∈ MJ,LX,I=1(α+, α−).

If I(uN+) = 1, then I(ui) = 0 for all i < N+. By Lemma 1.7 of [13], ui is

branched cover of trivial cylinders with ind(ui) ≥ 0 for i 6= 0, N+. By Lemma 9.5 of

[8], uN+ contains a nontrivial component with I = ind = 1. Additivity of Fredholm

index implies that ind(ui) = 0 for all i < N+. Argue as before, u−N− is a trivial

connector, which is ruled out in the holomorphic building. Hence, u∞ has no negative

level. If u0 contains a multiply covered component, then (42) tells us that it is a

branched cover of a holomorphic cylinder with nonempty positive end and nonempty

negative end at hyperbolic orbits. However, this contradicts with our assumption that

α− is admissible. Therefore, u0 is embedded. In conclusion, u0 ∈ MJX,I=0(β+, α−, Z ′),

uN+ ∈ MJ+Y+,I=1(α+, β+) and uii are connectors for 0 < i < N+.

If I(u−N−) = 1, then we can get the similar conclusion.

Lemma 7.11. For generic J ∈ Jtame(X,πX , ωX), CP (X,ΩX , J,ΛX ) is a chain map,

namely,

CP (X,ΩX , J,ΛX) ∂+ = ∂− CP (X,ΩX , J,ΛX),

where ∂± is the differential of CP∗(Y±, ω±,Γ±, J±,Λ±P ).

Proof. We assume that Y− = ∅ for simplicity. To show that CP (X,ΩX , J,ΛX) is a

chain map, it suffices to prove that

∑

Z1+Z2=W

#2MJ+Y+,I=1(α, β, Z1)#2MJ

X,I=0(β,Z2) = 0, (49)

where W ∈ H2(X,α), Z2 ∈ H2(X,β) and Z1 ∈ H2(Y+, α, β).

The usual strategy is to express (49) as a count of ∂MJX,I=1(α,W ). However, the

compactness result in Lemma 7.10 tells us that the broken holomorphic curves from

∂MJX,I=1(α,W ) not only involves elements fromMJ+

Y+,I=1(α, β, Z1) andMJX,I=0(β,Z2),

but also involves connectors.

In fact, the similar phenomenon already happen in the proof of ∂2 = 0. (See Lemma

7.23 of [14].) In the proof of ∂2 = 0, the key point is to glue a pair (u−, u+) consisting

of I = 1 holomorphic curves u− and u+ in R × Y , by inserting connectors in between.

In our case, the pair of holomorphic curves above is substituted by (u−, u+), where u−is a curve in X with I(u−) = 0 and u+ is a curve in R× Y+ with I(u+) = 1.

60

The strategy of gluing two holomorphic curves with I(u±) = 1 in [13] and [14] still

can be applied to our cases without essential change, because, their gluing analysis

mainly depends on what is happening near the neck region. In addition, our curves u+

and u− are admissible. By Theorem 1.13 of [13], when the orbit set β is admissible, the

gluing coefficient is 1, otherwise, it is zero. In other words, there is exactly one way (in

mod 2 sense) to glue (u−, u+) together when β is admissible. The argument in Section

7.3 of [13] leads to (49).

A summary of Hutchings and Taubes’ gluing argument can be found in Section 6.5

of [3], besides, it indicates the minor changes in the cobordism case.

In conclusion, CP (X,ωX , J,ΛX) induces a homomorphism

HP (X,ωX , J,ΛX) : HP∗(Y+, ω+,Γ+, J+,Λ+P ) → HP∗(Y−, ω−,Γ−, J−,Λ

−P )

in homology level.

7.3 Monotone case

As explained in Section 2.4.2, in order to compare the maps HPsw(X,ΩX , J,ΛX) and

HP (X,ΩX , J,ΛX ), we need to find almost complex structures in Jcomp(X,ΩX) such

that these two maps can be defined simultaneously. Under the monotonicity assump-

tions, we show that there is a subset Vcomp(X,πX , ωX)reg which meets our requirement.

Recall that when we define the cobordism maps in cases ♠.1 and ♠.2, we need to

use the genus bound g(C) ≥ g(B) to guarantee that C ⋆ C ≥ 12 . However, this may

not be true if we use J ∈ Jcomp(X,ΩX), because, the projection πX generally is not

complex linear anymore. However, at least intuitively, the bound g(C) ≥ g(B) should

be still true when J is close to Jtame(X,πX , ωX). The following lemma summarizes the

goal of this subsection.

Lemma 7.12. Assume that (X,ωX) is monotone, then there is a nonempty open subset

Vcomp(X,πX , ωX , J±) of Jcomp(X,ΩX , J±) such that it satisfies following properties:

Fix a generic J ∈ Vcomp(X,πX , ωX , J±), let C be a simple irreducible J-holomorphic

curve and it is not a fiber on the end of X. Assume additionally that C is not an

exceptional sphere or torus with zero self intersection number, then C satisfies

2C ⋆ C = 2g(C) − 2 + ind(C) + h(C) + 2eQ(C) + 4δ(C) ≥ i, (50)

where i = 2 in case ♠.1 and i = 1 in case ♠.2.

We postpone the proof for the lemma to Section 7.3.2.

61

This nice property guarantees that we can define HP (X,ωX , J,ΛX ) for generic

J ∈ Vcomp(X,πX , ωX , J±) as before. Moreover, using such an almost complex struc-

ture J , the assumption that (X,πX ) only contains nonseparating fibers can be dropped.

In the case that (ΓX , ωX) is monotone, the similar conclusion can be obtained. We use

Vcomp(X,πX , ωX) to denote ∪J±Vcomp(X,πX , ωX , J±), where J± runs over all symplec-

tization admissible almost complex structures.

Remark 22. Note that we don’t use the property g(C) ≥ g(B) in case ♠.3. So in this

section, we only consider cases ♠.1 and ♠.2, unless otherwise stated. In case ♠.3,

we can define the cobordism maps for generic J ∈ Jcomp(X,ΩX , J±) as before, without

the monotonicity assumptions. The only difference is that the holomorphic torus and

sphere may appear. Note that if they appear, then they lie inside the interior of X,

because, their intersection number with the fibers on the ends is zero.

Firstly, let us consider the case that there is no holomorphic sphere. Then the ECH

index is still nonnegative. But the curves in MJX,I=i(α+, α−) may contain multiple

covers of torus for i = 0, 1. According to [24], the transversality of multiple tori still

can be obtained for generic J . Also note that the multiple tori cannot shift to the

symplectization ends or other levels, or break into curves with asymptotic ends under

Gromov compactness. Under these observations, the cobordism maps can be defined by

the previous argument.

For the cases that the holomorphic sphere with C ⋆ C = −1 appears, the discussion

is the same as when X is not relatively minimal. We discuss this case in Section 7.4.

Without loss of generality, in this section and Section 7.4 subsequently, we assume

that (X,πX) has only one singular fiber. If the singular fiber is separating, then it is a

union of two embedded surfaces Σ1 and Σ2. Sections 7.3.1 and 7.3.2 consider the case

that (X,πX) is relatively minimal. The case that (X,πX ) is not relatively minimal will

be considered in Section 7.4.

Definition 7.13. Given L > 0, we define VLcomp(X,πX , ωX , J±) to be a subset of

Jcomp(X,ΩX , J±) such that J ∈ VLcomp(X,πX , ωX , J±) if the following is true:

S.1 For any irreducible J holomorphic curve C in X which has at least one end and∫C ωX ≤ L, then g(C) ≥ g(B).

S.2 Let C ∈ MJ(α+, α−) with d = [α±]·[Σ] ≥ 1 and∫C ωX ≤ L. If g(C) = g(B) = 1,

then on each end of X, C has exactly d-ends.

S.3 For any irreducible closed holomorphic curve C in X with∫C ωX ≤ L, then C

is homologous to m[Σ] +m1[Σ1] +m2[Σ2] for some nonnegative intergers m, m1

and m2.

62

Lemma 7.14. VLcomp(X,πX , ωX , J±) is a nonempty open subset of Jcomp(X,ΩX , J±).

Proof. Firstly, it is worth noting that

Jcomp(X,ΩX , J±) ∩ Jtame(X,πX , ωX , J±) 6= ∅ ⊂ VLcomp(X,πX , ωX , J±)

for any L > 0. In particular, VLcomp(X,πX , ωX , J±) is nonempty for any L > 0.

For openness, we argue by contradiction. Suppose that VLcomp(X,πX , ωX , J±) is not

open, then there exists a sequence of irreducible holomorphic curves (Cn, Jn)∞n=1

with∫CnωX ≤ L that do not satisfy the conclusions and Jn converges to J∞ ∈

VLcomp(X,πX , ωX , J±) in C∞ topology. By Gromov compactness [1], (Cn, Jn)∞n=1

converges to (u∞, J∞), where u∞ is possibly broken with nodes.

Firstly, we deal with the case that each Cn has at least one end, so does u∞. Let

u0 be an irreducible component in cobordism level of u∞ which has at least one end. If

Cn violates S.1, then the genus of the domain of u0 is strictly less than g(B). However,

it is impossible since J∞ ∈ VLcomp(X,πX , ωX , J±).

If Cn violates S.2, i.e., g(Cn) = g(B) = 1, but Cn has strictly less than d-ends. If

d = 1, then there is nothing to prove, because, we assume that Cn has at least one end.

Let us assume that d ≥ 2. Since J∞ ∈ VLcomp(X,πX , ωX , J±), the cobordism level of

u∞ has exactly one irreducible component u0 with at least one end, otherwise, combine

with the discussion in the last paragraph, the arithmetic genus of the domain of u∞is strictly greater than 1, contradict with g(Cn) = 1. Moreover, the domain of u0 has

genus one and u0 has d-ends. As a result, u∞ is broken. Obviously, the arithmetic

genus of the domain of u∞ is strictly greater than 1, we get a contradiction.

Assume that each Cn is closed and it violates S.3. Then u∞ cannot be broken,

otherwise, the top (bottom) level of u∞ has no positive (negative) ends. Then its

homology class must be m[Σ] +m1[Σ1] +m2[Σ2] for some nonnegative integers m,m1

and m2 because of J∞ ∈ VLcomp(X,πX , ωX , J±). As a result, this is also true for large

n, then we get a contradiction.

Corollary 7.15. For generic J ∈ VLcomp(X,πX , ωX , J±), let C be a simple irreducible

J-holomorphic curve which has at least one and∫C ωX ≤ L. Then 2C ⋆ C ≥ i, where

i = 2 in case ♠.1 and i = 1 in case ♠.2 .

Proof. Follows from Lemma 7.14 and repeat the same argument in Lemma 7.2.

Remark 23. Note that any closed simple holomorphic curve C in X such that C⋂X 6=

∅ must lie inside the interior of X. For generic J ∈ VLcomp(X,πX , ωX , J±), such a curve

C is Fredholm regular.

63

7.3.1 (ΓX , ωX) is monotone

Corollary 7.16. Suppose that (X,πX ) is relatively minimal and (ΓX , ωX) is monotone,

then there exists a nonempty open subset Vcomp(X,πX , ωX , J±) of Jcomp(X,πX , ωX , J±)

with the following significance: For generic J ∈ Vcomp(X,πX , ωX , J±), the cobordism

map HP (X,ΩX ,ΓX , J,ΛX) is well defined.

Proof. Let C1 and C2 be holomorphic currents with the same relative homology class

ΓX . Then by definition of ECH index, we have

I(C1)− I(C2) =< c1(TX) + 2PD(ΓX), C1 − C2 >= τ

(∫

C1ωX −

∫

C2ωX

).

Therefore, the ωX-energy of the elements in⊔

Z∈H2(X,α+,α−),[Z]=ΓX

MJX,I=i(α+, α−, Z)

are constant which only depends on α±. Their maximum is denoted by L∗. Take

Vcomp(X,πX , ωX , J±) = VL∗comp(X,πX , ωX , J±).

Assume that g(Σ) ≥ 2. Since (X,πX) is relatively minimal, g(Σi) ≥ 1. By adjunc-

tion formula, it is easy to check that

I(m[Σ] +m1[Σ1] +m2[Σ2]) < 0. (51)

According to Lemma 7.14 and Lemma 2.4 in [10], for generic J in Vcomp(X,πX , ωX , J±),

the holomorphic current consists of closed holomorphic curves must be of the form

(C, d), where C is a fiber in the ends of X. If g(Σ) ≤ 1, then the singular fiber is

nonseparate, otherwise, (X,πX ) is not relatively minimal.

In both cases, by Corollary 7.15, (50) is true for all irreducible simple holomorphic

curves with energy less than L, except for the close curves with the same homology as

the fibers. Replace Lemma 7.2 by Corollary 7.15, then repeat the previous arguments

in Lemmas 7.3, 7.4, 7.5, 7.6, 7.7, 7.8, 7.10 and 7.11, HP (X,ΩX ,ΓX , J,ΛX ) is well

defined.

7.3.2 (X,ωX) is monotone

Observe that there are only finitely many pairs of orbit sets (α+, α−) with [α±]·[Σ] ≤ Q.

We label these pairs by (αk+, α

k−)

NQ

k=1. For each pair (αk+, α

k−), fix a Jk holomorphic

curve uk : Ck → X which violates the conditions S.1 or S.2 in Definition 7.13, where

Jk ∈ Jcomp(X,ΩX , J±). Denote L0 = maxk∫Cku∗kωX and I0 = maxk|indCk|. Note

that in either case, g(Ck) ≤ g(B).

The consequence of the following lemma is that if C is a simple holomorphic curve

which violates the requirements S.1 or S.2 in Definition 7.13, then the Fredholm index

of C is larger than 3.

64

Lemma 7.17. Suppose that (X,ωX) is monotone, there exists L∗ > 0 with the following

significance: Given L ≥ L∗, for J ∈ VLcomp(X,πX , ωX , J±), given orbit sets α+ and α−

(at least one is nonempty) and an irreducible J holomorphic curve C ∈ MJ(α+, α−)

(not necessarily simple) with 0 ≤ indC ≤ 3, then the following are true:

1. g(C) ≥ g(B),

2. If g(C) = g(B) = 1, then C has exactly d = α± · [Σ]-ends.

Proof. Let J ∈ VLcomp(X,πX , ωX , J±) and C be an irreducible J holomorphic curve from

α+ to α− with 0 ≤ ind(C) ≤ 3 and∫C ωX > L. Assume additionally that g(C) ≤ g(B).

There exists k such that α± = αk±. By definition of the Fredholm index,

ind(C)− ind(Ck) = χ(Ck)− χ(C) + 2 < c1(TX), C − Ck > +µindτ (C)− µindτ (Ck). (52)

Since there are only finitely many orbit sets α± with degree less thanQ and g(C), g(Ck) ≤g(B), we can find a constant cQ > 0 which only dependents on Q and g(B) such that

|µindτ (C)− µindτ (Ck)|+ |χ(Ck)− χ(C)| ≤ cQ. (53)

Combine (52) and (53), we have | < c1(TX), C − Ck > | ≤ c0, where c0 = I0 + cQ + 3.

On the other hands, by the monotonicity assumption,

|∫

CωX −

∫

Ck

ωX | = | < ωX , C − Ck > | = |τ < c1(TX), C − Ck > | ≤ |τ |c0, (54)

In other words, the ωX-energy of holomorphic curve which violates the desired proper-

ties is uniformly bounded.

Take L ≥ L∗ ≥ L0+ |τ |c0. Suppose that there exists J ∈ VLcomp(X,πX , ωX , J±) and

an irreducible J holomorphic curve C violated the conclusions of the lemma. Then C

must satisfy 0 ≤ ind(C) ≤ 3 ,∫C ωX > L and g(C) ≤ g(B). However,

∫

CωX −

∫

Ck

ωX > L∗ −∫

Ck

ωX ≥ |τ |c0

contradicts with (54).

Corollary 7.18. Assume that (X,ωX) is monotone. For J ∈ VL∗comp(X,πX , ωX , J±)reg,

let C be an irreducible simple J holomorphic curve with at least one end, then 2C⋆C ≥ i,

where i = 2 in case ♠.1 and i = 1 in case ♠.2.

Proof. If∫C ωX ≤ L∗, then the conclusion follows from Corollary 7.15.

If∫C ωX > L∗, by Lemma 7.17, either indC < 0 or indC ≥ 4. Since J is generic,

the former case cannot happen. The conclusion follows from the definition of C⋆C.

65

Now we examine (50) for closed simple holomorphic curve C. By Remark 23, we

assume that C lies inside the interior of X, otherwise, C is a fiber over the ends of X.

Lemma 7.19. Assume that (X,ωX) is monotone and (X,πX ) is relatively minimal.

For generic J ∈ VL∗comp(X,πX , ωX , J±), let C be a simple closed J holomorphic curve

in X, then either (50) is still true for C or [C] = m[Σ] for m ≥ 0.

Proof. By Definition 7.13, any such a closed curve satisfies either [C] = m[Σ]+m1[Σ1]+

m2[Σ2] or∫C ωX > L∗. By (51) and Remark 23, the former case cannot happen if

g(Σ) ≥ 2, and [C] = m[Σ] when g(Σ) ≤ 1. Hence, it suffices to consider∫C ωX > L∗.

If g(C) ≥ 2, then (50) is automatically true for C. Suppose that g(C) ≤ 1, by the

monotonicity assumption,

ind(C) = 2 < c1(TX), C > +2g(C)− 2 =2

τ

∫

CωX + 2g(C) − 2.

Since L∗ > 2|τ |, either ind(C) < 0 or ind(C) ≥ 4. The former case cannot appear for

generic J and (50) is true for C in the latter case.

Proof of Lemma 7.12. Define Vcomp(X,πX , ωX , J±) = VL∗comp(X,πX , ωX , J±). The lemma

is a direct consequence of Corollary 7.18 and Lemma 7.19.

Lemma 7.20. Suppose that (X,ωX) is monotone and (X,πX ) is relatively minimal.

Then there exists a nonempty open subset Vcomp(X,πX , ωX , J±) of Jcomp(X,ΩX , J±)

such that for J ∈ Vcomp(X,πX , ωX , J±)reg, HP (X,ΩX , J,ΛX) is well defined.

Proof. Replace Lemma 7.2 by Lemma 7.12, then we repeat the same argument in Lem-

mas 7.3, 7.4, 7.5, 7.7, 7.8, 7.10 and 7.11 to define the cobordism mapsHP (X,ΩX , J,ΛX).

7.4 (X, πX) is not relatively minimal

This section concerns the case that (X,πX) is not relatively minimal. Without loss of

generality, assume that Σ2 is an exceptional sphere. Its homology class is denoted by

E. The discussion here follows [24]. To define the cobordism maps, we need further

constrains on the holomorphic currents.

Definition 7.21. A holomorphic current C = (Ca, da) is called admissible if da = 1

whenever Ca is a holomorphic sphere with Ca · Ca = −1. Given orbit sets α+ and

α−, and a relative homology class Z, let MJ,adX (α+, α−, Z) denote the moduli space of

holomorphic curves whose underlying holomorphic currents are admissible.

Under certain assumptions, we claim that the cobordism maps are still well defined.

The results are summarized in the following lemma.

66

Lemma 7.22. Let ΓX ∈ H2(X, ∂X,Z), we consider any one of the following cases:

C.1 (X,ωX) or (ΓX , ωX) is monotone. Take J ∈ Vcomp(X,πX , ωX)reg.

C.2 The relative class satisfies ΓX ·E ≤ 0. Take J ∈ Jtame(X,πX , ωX)reg.

Replace the moduli space MJX,I=0(α+, α−, Z) by MJ,ad

X,I=0(α+, α−, Z) in equation (39),

then it is still meaningful. Moreover, the map defined by equation (39) is a chain map,

and it induces a well-defined homomorphism HP (X,ΩX ,ΓX , J,ΛX ).

It is worth noting that I(m[Σ] + m1[Σ1] + meE) ≤ 0 and equality holds if and

only if m = m1 = 0 and me = 1 when g(Σ) ≥ 2. In the monotone case C.1, by

the observation above and Lemma 7.12, any holomorphic current can be written as

C +meE, where C doesn’t contain any E component and I(C) ≥ 0. When g(Σ) = 0,

let C be a exceptional sphere with [C] = m[Σ] + m1[Σ1] + m2[Σ]. Without loss of

generality, assume that m2 ≥ m1, hence we can write [C] = m[Σ] + m2[Σ2]. By

adjunction formula, it is easy to check that C is an exceptional sphere if and only if

m = 0 and m2 = 1. Therefore, we still have the decomposition C +meE. For the case

g(Σ) = 1, the homology class of exceptional sphere is not necessarily to be E, but the

discussion that follows still works, we omit this case here.

In the case C.2, for any holomorphic current C with relative class ΓX , we can

write C = C0 +m[Σ] +m1[Σ1] +meE, where C0 doesn’t contain any closed component.

Then C · E = ΓX · E ≤ 0 implies that me ≥ m1. Since E + [Σ1] = [Σ], we write

C = C′ + (me −m1)E, where C′ = C0 + (m+m1)[Σ]. By Corollary 7.3 and Lemma 7.7,

we have I(C′) ≥ 0.

Lemma 7.23. For generic J , let Cn∞n=1 be a sequence of admissible holomorphic

currents in MJ,LX,I=i(α+, α−), where 0 ≤ i ≤ 1. Then Cn∞n=1 converges to a broken

holomorphic currents C∞ in the current sense. Decompose C∞ = C + meE, where Cdoesn’t contain any e component. Assume that I(C) ≥ 0, then me = 1.

Proof. By Taubes’ Gromov compactness, Cn∞n=1 converges to C∞ (possibly broken)

in the sense of Lemma 9.9 of [8]. By definition,

I(C∞) = I(C +meE) = I(C) + I(meE) + 2meC ·E = I(C) +me −m2e + 2meC ·E.

On the other hand, C∞ ·E = C ·E−me = Cn ·E ≥ −1. As a result, I(C∞) ≥ me(me−1).

By our assumption, me = 1.

Proof of Lemma 7.22. Let us consider case C.2. Write C = C′ +meE as before. Ob-

serve that if C is admissible with 0 ≤ I(C) ≤ 1, then by Lemma 7.7, C′ doesn’t con-

tain any closed component and me = 0 or 1. Note that the exceptional sphere is

67

Fredholm regular due to the automatic transversality theorem.(Cf. Page 51 of [10]).

Take a generic J ∈ Jtame(X,πX , ωX), replace the moduli space MJX,I(α+, α−, Z) by

MJ,adX,I (α+, α−, Z). By Lemma 7.23, the compactness results in Lemmas 7.8 and 7.10

still hold, because, there is no holomorphic curve with negative ECH index created after

taking Gromov compactness. Moreover, all the holomorphic curves which contribute to

the cobordism maps are embedded. By using the same argument as before, we define

cobordism maps HP (X,ΩX ,ΓX , J,ΛX), even though X is not relatively minimal. This

leads to the last bullet in Remark 4.

Similarly, by the same reasons as above, the cobordism maps can be defined by

counting embedded holomorphic curves in monotone case C.1.

7.5 Properties of HP (X,ΩX , J,ΛX)

7.5.1 Composition rule

Let (X+, πX+, ωX+

) be a fierwise symplectic cobordism from (Y+, π+, ω+) to (Y0, π0, ω0)

and (X−, πX− , ωX−) be a fiberwise symplectic cobordism from (Y0, π0, ω0) to (Y−, π−, ω−).

Given JX± ∈ Jtame(X±, πX± , ωX±), define (XR, πXR, ωR) and JR as Section 2.5.

Lemma 7.24. Assume that (X+, πX+, ωX+

) and (X−, πX− , ωX−) satisfy the assump-

tions (♠) and they contain no separating singular fiber. Introduce (XR, πXR, ωR) and

JR as Section 2.5. Suppose that JX± is generic, and there exists R0 ≥ 0 such that JR

is generic for any R ≥ R0. Then

∑

ΓXR0|X±=ΓX± ,ΓXR0

∈H2(XR0,∂XR0

,Z)

HP (XR0,ΩXR0

,ΓXR0, JR0

,ΛXR0)

= HP (X−,ΩX− ,ΓX− , JX− ,ΛX−) HP (X+,ΩX+,ΓX+

, JX+,ΛX+

),

where ΛXR= ΛX+ ΛX−.

Proof. To simplify the notation, we assume that R0 = 0. Let α− and α+ be admis-

sible orbit sets. Fix a relative homology class Z ∈ H2(X0, α+, α−) with [Z] = ΓX0.

Let Rn∞n=1 be an increasing unbounded sequence and un ∈ MJRn

XRn ,I=0(α+, α−, Z).

Arguing as Lemma 7.8, un∞n=1 converges to a broken holomorphic curve u∞ =

v−m, . . . u−, τ1 . . . τk, u+, . . . v+l in the sense of SFT [1] with I(u∞) = 0, where u+,

u−, τi, v+i and v−j are respectively holomorphic curves in X+, X−, R× Y0, R× Y+ and

R× Y−.

Corollary 7.3 forces I(u+) = I(u−) = I(τi) = I(v+j ) = I(v−j ) = 0. Furthermore, by

the argument similar to Lemmas 7.8 and 7.10, τij , v+j j and v−j j are connectors.

Since u∞ is admissible, v+j j and v−j j are trivial connectors and they are ruled out

by the holomorphic building. Thus negative ends of u− are asymptotic to α− and

68

positive ends of u+ are asymptotic to α+. By Lemma 7.4, both of u+ and u− are

embedded.

Let M =⊔

R≥0 MJRXR,I=0(α+, α−, Z), then M is a 1-dimensional manifold, because,

JR is generic for R ≥ 0. The boundary of M has two components ∂1M and ∂2M,

where ∂1M = MJXX,I=0(α+, α−Z), and ∂2M is contributed by the broken holomorphic

curves described in the previous paragraph. By the gluing arguments in [13] and [14],

we have

#2MJXX0,I=0(α+, α−, Z)

=∑

α0∈P(Y0,ω0,Γ0)

∑

Z=Z++Z−

#2MJX+

X+,I=0(α+, α0, Z+)#2M

JX−X−,I=0(α0, α−, Z−).

This formula implies the conclusion of the lemma.

7.5.2 Blowup formula

In this subsection, we prove a simple version of blowup formula. The proof is the same

as Theorem 4.2 in [19].

Take a generic almost complex structure J ∈ Jtame(X,πX , ωX) which is integral in

a small neighborhood U of x ∈ X, where x lies in a regular fiber. Then we perform

a complex blowing up at x, the result is denoted by (X ′, J ′), also, J ′ = J outside U .

Then πX′ = πX pr : X ′ → B still is a Lefschetz fibration and πX′ is complex linear

with respect to J ′ and jB , where pr : X ′ → X is the blowing down map. After blowing

up, there is one additional critical point and its critical value is πX(x). One of the

irreducible components of π−1X′ (πX(x)) is an exceptional sphere with self interesting −1.

Let E denote its homology class. There is a standard way to find a symplectic form ΩX′

on X ′ such that ΩX′ tames J ′ and ΩX′ = pr∗ΩX outside a neighborhood of pr−1(x).

(Cf. [21] ) Then we can define an admissible 2-form by ωX′ = ΩX′ −ωB and clearly we

have J ′ ∈ Jtame(X′, πX′ , ωX′). According to Lemma 7.22, HP (X ′,ΩX′ , J ′′,ΓX ,ΛX′) is

well defined for any generic J ′′, provide that ΓX ·E ≤ 0.

Lemma 7.25. Suppose that (ΓX , ωX) is monotone. Introduce (X ′, ωX′ , J ′) and E as

above. Regard ΓX as an element in H2(X′, ∂X ′,Z) and assume that ΓX · E = 0,

then HP (X ′,ΩX′ , J ′,ΓX ,ΛX′) is well defined, where ΛX′ is a X ′-morphism such that

ΛX(Z) = ΛX′(Z) for any Z with relative class ΓX . Furthermore,

HP (X,ΩX , J,ΓX ,ΛX) = HP (X ′,ΩX′ , J ′,ΓX ,ΛX′).

Proof. By the monotonicity assumption, the energy of curves in MJX,I=0(α+, α−, Z) is

fixed, denoted by L, where Z is a relative homology class with [Z] = ΓX . By Corollary

7.9, MJ,LX,I=0(α+, α−) is a finite set. So we can take a sufficiently small neighborhood

U of x such that any holomorphic curve in MJ,LX,I=0(α+, α−) does not interest U .

69

Let e = pr−1(x) be the exceptional sphere representing the class E. e is a J ′

holomorphic curve in X ′. For every u ∈ MJ ′X′,I=0(α+, α−, Z) and [Z] = ΓX , u does

not intersect e due to the assumption ΓX · E = 0. As a result, I(pr u) = 0. Since

pr is biholomorphic outside pr−1(x), pr u is a holomorphic curve in X. That is

pr u ∈ MJX,I=0(α+, α−, Z). Moreover, u is Fredholm regular. As a consequence, there

is a bijection between MJ ′X′,I=0(α+, α−, Z) and MJ

X,I=0(α+, α−, Z) and this lead to the

conclusion.

7.5.3 Commute with U-map

The argument here is similar to Section 2.5 of [15]. We sketch the main points as

follows.

Fix a point x ∈ X, we define the cobordism maps with mark point x as follows: Let

MJX,I=2(α+, α−, x, Z) be the moduli space of holomorphic curves whose image intersect

x. Given u ∈ MJX,I=2(α+, α−, x, Z) and let C be the associated holomorphic current.

Note that the components of C which interest x have ind ≥ 2.

Based on the above understanding, we have I(u) ≥ 2 for u ∈ MJX(α+, α−, x, Z).

Repeat the argument in Lemma 7.8 to show that MJX,I=2(α+, α−, x, Z) is a compact

zero dimensional manifold. We define the cobordism maps with one mark point by

CP (X,ΩX , J,ΛX)x =∑

α±

∑

Z∈H2(X,α+,α−)

#2MJX,I=0(α+, α−, x, Z)ΛX(Z).

It is a chain map due to the same argument in Lemma 7.11. So HP (X,ωX , J,ΛX)x is

well defined.

Let y± ∈ Y± = ∂±X, and γ± be a path in X such that γ±(0) = x and γ±(1) = y±,

and γ±(t) lies in the interior of X for 0 ≤ t < 1. Let MJX,I=i(α+, α−, γ+, Z) be the

moduli space holomorphic curves whose image intersect γ±. When i = 1, using the

same argument as in Lemma 7.8 to show that it is a 0 dimensional compact manifold.

As a result,

K+ =∑

α±∈P(Y±,ω±,Γ±)

∑

Z∈H2(X,α+,α−)

#MJX,I=1(α+, α−, γ+, Z)ΛZ

is well defined.

When i = 2, MJX,I=2(α+, α−, γ±, Z) is a manifold of dimension 1. The broken

holomorphic curve u∞ = u−N− , . . . , u0, . . . , uN+ which comes from the limits of curves

in MJX,I=2(α+, α−, γ±, Z) consists of the following possibilities:

1. uN+ is a holomorphic curve with I = 1 and u0 ∈ MJX,I=1(β, α−, γ+, Z ′), and the

other levels are connectors. Moreover, there is no negative level.

70

2. uN+ is a holomorphic curve with I = 2 and it interest y+. u0 ∈ MJ

X,I=0(β, α−, Z ′),

and the other levels are connectors. Moreover, there is no negative level.

3. u−N− is a holomorphic curve with I = 1 and u0 ∈ MJX,I=1(β, α−, γ+, Z ′), and

the other levels are connectors. Moreover, there is no positive level.

4. u∞ = u0 ∈ MJX,I=2(α+, α−, x, Z).

The gluing analysis in [13] and [14] shows that

CP (X,ΩX , J,ΛX)x − U+,y CP (X,ΩX , J,ΛX) = ∂+ K+ −K+ ∂−.

Therefore, HP (X,ΩX , J,ΛX)x = U+ HP (X,ΩX , J,ΛX). Similarly, we show that

HP (X,ΩX , J,ΛX )x = HP (X,ΩX , J,ΛX ) U−. As a consequence,

U+ HP (X,ΩX , J,ΛX) = HP (X,ΩX , J,ΛX) U−.

7.6 Z-coefficient

So far, the cobordism maps HP (X,ΩX , J,ΛX) are defined over the local coefficient of

Z2-module. In fact, they also can be defined over the local coefficient of Z-module by

endowing a coherent orientation on the moduli space MJX,I=0(α+, α−, Z).

Given an admissible orbit set α± ∈ P(Y±, π±,Γ±), we can define the determinant

line bundle detD → MJ(α+, α−) with fiber detDu = Λmax kerDu ⊗ ΛmaxcokerDu,

where Du is the linearization of ∂Ju. Introduce a 2-elements set Λ(α+, α−) to be the

orientation sheaf of detD. The gluing principle in Section 3.b of [30] tells us that

the collection of Λ(α+, α−)α± is a coherent system of orientations in the following

sense: Given orbit sets α+ and α−, there is a canonically associated Z2 module, which

is denoted by Λ(α±). Moreover, there is a canonical isomorphism from Λ(α+, α−) to

Λ(α+)Λ(α−).

With the above understanding, a set of orientations oPFH(α+, β+) ∈ Λ(α+, α−)is called coherent if there exists oPFH(α±) ∈ Λ(α±)α± such that oPFH(α+, α−) =

oPFH(α+)oPFH(α−).

Fix a choice of coherent orientation, the moduli space MJX,I=0(α+, α−, Z) is a finite

set with signs which are determined by the orientations. Granted this understood, we

can define the cobordism maps HP (X,ΩX , J,ΛX) over the local coefficient of Z-module

through replacing the mod two count ”#2” by integer count ”#”.

7.7 Proof of Theorem 2

Proof. Suppose that (X,πX , ωX) is a fiberwise symplectic cobordism from (Y+, π+, ω+)

to (Y−, π−, ω−) satisfying assumptions (♠). Also, assume that Γ± · [Σ] > g(Σ). Up to

71

now, we have shown that the cobordism maps HP (X,ΩX , J,ΛX) are well defined for

generic J ∈ Jtame(X,πX , ωX) whenever (X,πX) contains no separating singular fiber,

and it is well defined for generic J ∈ Vcomp(X,πX , ωX) whenever (X,ωX) is monotone.

The promised properties of HP (X,ΩX , J,ΛX) follow from Lemmas 7.24, 7.25 and the

discussion in Section 7.5.3.

Up to now, we have completed the proof of Theorem 2.

8 Embedded holomorphic curves and Seiberg-

Witten equations

In this section, we aim to prove the following theorem.

Theorem 8.1. Suppose that there exists R0 ≥ 0 such that ℘4 agrees with ℘3± when

|s| ≥ R0 and J ∈ Vcomp(X,πX , ωX) is generic. Also, assume that (ωX , J)|R±×Y± is Q-δ

flat. Given the same assumptions in Theorem 3 and any L, there exists rL > 0 such

that for any r ≥ rL, there is a bijective map

Ψr : MJ,LX,I=0(α+, α−) → M

J,LX,ind=0(T

+r (α+), T

−r (α−)),

where MJ,LX,ind=0(T

+r (α+), T

−r (α−)) is the moduli space of index zero solutions to (14)

with i2π

∫X FA ∧ ωX ≤ L. Moreover, the image of Ψr is nondegenerate.

The proof of Theorem 8.1 relies heavily on [29], [30], [31] and [18] with only minor

changes. In these papers, they prove a similar result that there is a 1-1 correspondence

between instantons and I = 1 holomorphic curves when X is symplectization of a con-

tact three manifold or a mapping torus. Note that Theorem 8.1 differs from the analogs

in [29], [30], [31] and [18] in the following aspects: The manifold R×Y , symplectization

admissible almost complex structure and symplectic form are respectively replaced by

X , cobordism admissible almost complex structure and ΩX here. Besides, the I = 1

holomorphic curves are replaced by I = 0 holomorphic curves in our case, because, X

does not admit global R action. Note that our holomorphic curves are embedded with

I = 0, they are counterparts of the nontrivial embedded I = 1 holomorphic curves in

symplectization case.

Except for the convergence results in Sections 4 and 5 of [31], the analysis in [29],

[30], [31] or [18] mainly takes place either near a holomorphic curve C or its ends. Since

our symplectic form and cobordism admissible almost complex structure agree with

symplectization case on the ends of X , i.e., ΩX |R±×Y± = ω±+ds∧π∗±dt and J |R±×Y± ∈Jcomp(Y±,, π±, ω±), the relevant analysis on the ends of C can be applied to our situation

without modification. For the relevant analysis on the tubular neighborhood of C, the

72

form of ΩX plays no role there, so their arguments also apply equally well in the current

situation.

Let us briefly outline the proof, it can be divided into the following three parts:

1. Firstly, we construct the map Ψr. The construction is sketched as follows, it is

corresponding to Sections 4-7 of [29]. Given an embedded holomorphic current

C, we star by following Section 5 of [29] to construct a line bundle E → X and a

pair (A∗, ψ∗) which is close to solving (14). Then using the perturbation analysis

to deform (A∗, ψ∗) to be a true solution (AC , ψC) and define Ψr(C) = [(AC , ψC)].

We will review the ideal of the construction later. We point out that the analysis

of this part is local, and whether ΩX is exact or not plays no role to the analysis,

the arguments in [29] also can be applied to our setting without essential change.

When C = ∅, the manifold R× Y has natural R action, so [29] can take (A∅, ψ∅)

simply by the R invariant solution corresponding to empty orbits. Since our

(X,πX ) is not R invariant, we will construct (A∅, ψ∅) in Section 8.1 in detail.

2. Secondly, to verify that the map Ψr is well defined, we need to check that (AC , ψC)

is regular with zero Fredholm index. This corresponding to Section 3.a of [30].

The main points of this part have been summarized in Section 5.1 of [2]. Thus

we skip this part here.

3. From the construction, it is not difficult to check that Ψr is injective. To show

that Ψr is surjective, the proof also is divided into three parts. [18] calls them

estimation, convergence, and perturbation respectively. They are corre-

sponding to Section 3, Sections 4-5 and Sections 6-7 of [31] respectively. We will

clarify their meaning latter. We point out that there is no essential difference in

estimation and perturbation parts. For the convergence part, we already

established it in Section 5.3.

For simplicity, we assume that Y− = ∅ in this section.

8.1 Construct a trivial Seiberg Witten solution.

We begin to construct Ψr in Theorem 8.1 by considering a special case that C = ∅. We

follow the argument in [29] to construct a solution (A∅, ψ∅) to Seiberg Witten equations

(14) which corresponds to the emptyset of X. Also, in Section 8.1.2, we show that this

solution is small.

Deformation operator. Given a configuration (A,ψ), the deformation operator

73

D : Γ(iT ∗X ⊕ S+) → Γ(i(Ω2+(X)⊕ iR)⊕ S−) at (A,ψ) is defined by

d+b− 2−3

2 r1

2 (q(ψ, η) + q(η, ψ))

DAη + (2r)1

2 cl(b)ψ

∗d ∗ b− 2−1

2 r1

2 (η†ψ − ψ†η),

(55)

where (b, η) ∈ Γ(iT ∗X⊕S+). The first two lines come from the linearization of Seiberg

Witten equations and the last line is gauge fixing condition.

Take E to be the trivial line bundle, denoted by IC, then S+ = IC ⊕ K−1X and

S− = ∧0,1T ∗X . Let (AI , ψI) be the trivial approximation solution, i.e., AI is the trivial

connection on IC and ψI = (1C, 0). Using DI to denote the deformation operator at

(AI , ψI). Note that DI is an elliptic operator from Γ(iΩ1⊕S+) to Γ((iΩ0⊕Ω2+)⊕S−).Let h ∈ Γ(iΩ1 ⊕ S+), we define the H-norm as follows,

|h|2H =

∫

X|∇h|2 + r

4|h|2,

where the covariant derivative acts on section of iT ∗X as the Levi-Civita covariant

derivative and on section of S+ as covariant derivative that is defined by Levi-Civita

covariant and 2AI+AK−1 . One can define theH-norm for section of Γ((iΩ0⊕Ω2+)⊕S−)similarly. Keep in mind that |h|L4 ≤ c0|h|H because of the Sobolev inequality.

Weizenbock formula:

DID∗I = ∇∗∇+ 2r +R0 +

√rR1

D∗IDI = ∇∗∇+ 2r + R0 +

√rR1.

(56)

Here Ri and Ri (i = 0, 1) are bounded endomorphism of iR⊕Ω2+⊕S− and iTX∗⊕S+

respectively.

Lemma 8.2. Let h ∈ Γ(iΩ1⊕S+) and f ∈ Γ((iΩ0⊕Ω2+)⊕S−) with compact supports,

then there exists c0 > 0 such that for any r ≥ c0, DI and D∗I satisfy the following

estimates:

1c0|h|2H ≤ |DIh|2L2 ≤ c0|h|2H ,

1c0|f|2H ≤ |D∗

I f|2L2 ≤ c0|f|2H ,

Proof. The lemma follows from the Weizenbock formula (56) and integration by part.

Let H and L be the completion of smooth sections in Γ(iΩ1 ⊕ S+) and Γ((iΩ0 ⊕Ω2+)⊕ S−) with compact supports with respect to H norm and L2 norm respectively.

74

Lemma 8.3. There exists κ > 0 and c0 > 0 such that for r ≥ c0, DI : H → L is

invertiable. Moreover, |D−1I | ≤ κ.

Proof. By Lemma 8.2, DI is injective.

To show that DI is surjective, given g ∈ L, we define a functional Fg : L∩L21((iΩ

0⊕Ω2+)⊕ S−) → R by

Fg(f) =

∫

X|D∗

If|2 − 2 < g, f >,

where L21((iΩ

0⊕Ω2+)⊕S−) is the completion of smooth sections with compact support

with respect to L21-norm. By Holder inequality and Lemma 8.2, we have

Fg(f) =

∫

X|D∗

I f|2 − 2 < g, f >≥ 1

c0|f|2H − c0

r|g|2L2 ≥ −c0

r|g|2L2 > −∞.

Consequently, m = inff∈L∩L2

1

Fg is well-defined. We claim that there exists a minimizer of

Fg. The reasons are given as follows:

Firstly, there exists a sequence fi ⊂ L∩L21((iΩ

0⊕Ω2+)⊕S−) such that limi→∞

Fg(fi) =

m. The limit and Lemma 8.2 imply that fi have a uniform bound on H norm. After

passing a subsequence, fi converges weakly to f in the sense of H norm. By Lemma

8.2, D∗I is a bounded linear operator, thus D∗

Ifi → D∗I f converges weakly in the sense

of L2 norm. Therefore, limi→∞

|D∗I fi|L2 ≥ |D∗

I f|L2 . As a result,

m = limi→∞

Fg(fi) ≥ limi→∞

|D∗I fi|L2 − 2

∫

X< f, g >≥ Fg(f).

Hence, f is a minimizer of Fg. Furthermore, it is easy to check that Fg is convex.

Therefore, the minimum of Fg is unique.

Let f ∈ L∩L21((iΩ

0⊕Ω2+)⊕S−) be the minimizer above. The standard variational

argument show that D∗If is a weak solution to DID

∗I f = g. Elliptic regularity (Cf. [7])

implies that h = D∗If is L

21,loc. Then h = D∗

I f ∈ H follows from Lemma 8.2. Hence, DI

is surjective.

Finally, the bound on |D−1I | follows from Lemma 8.2.

8.1.1 Contraction mapping argument

In this section, let κ denote the bound of |D−1I | in Lemma 8.3. Also, we use µ to denote

−12℘

+4 + i

2F+A

K−1.

Let b0 = (b0, η0) denote the small solution to (3-34) in [29] such that (A∅+, ψ∅+) =

(AI+(2r)1

2 b0, ψI+η0) satisfies equations (10) with an additional gauge fixed condition.

We regard b0 as an R invariant section on R×Y+. By Lemma 3.10 in [29], |b0| ≤ c0r−1

and |∇b0| ≤ c0r−1.

75

Let χ be a cut-off function such that χ = 1 when s ≥ R0, χ = 0 on s ≤ R0−κ−4 and

|∇χ| ≤ c0κ4. Define (A∅, ψ∅) = (AI , ψI) + ((2r)

1

2 (bh + χb0), (χα0 + αh, χβ0 + βh)). We

want to find h∅ = (bh, ηh) ∈ H such that (A∅, ψ∅) satisfies the Seiberg Witten equations

(14) together with the gauge fixing condition in the last line of (55). It is equivalent to

solve the following equation:

DI(χb0 + h∅) + r1

2 (χb0 + h∅) ∗ (χb0 + h∅) = v, (57)

where v = (2r)−1

2 iµ. Equation (57) is equivalent to

v = DIh∅ + r1

2 h∅ ∗ h∅ + χ(DIb0 + r1

2b0 ∗ b0) + r1

2χ(χ− 1)b0 ∗ b0 + 2r1

2h∅ ∗ χb0 +∇χ ∗ b0.

Note that DIb0 + r1

2 b0 ∗ b0 = v+ and v+ = v when s ≥ R0 by assumption. By

Lemma 8.3, we can rewrite equation (57) as follows:

h∅ = D−1I ((v − χv+)− r

1

2h∅ ∗ h∅ + r1

2χ(1− χ)b0 ∗ b0 − 2r1

2 h∅ ∗ χb0 −∇χ ∗ b0).

Define a map T : H → H by

T (h∅) = D−1I ((v− χv+)− r

1

2 h∅ ∗ h∅ + r1

2χ(1− χ)b0 ∗ b0 − 2r1

2h∅ ∗ χb0 −∇χ ∗ b0).

Lemma 8.4. Let κ ≥ 1 be the constant in Lemma 8.3. Suppose that |(o − χo+)|L2 ≤1κ4 r

− 1

2 and r is sufficiently large. Let B be a ball in H centered at the origin with radius1κ2 r

− 1

2 . Then T : B → B is a contraction mapping. In particular, equation (57) has a

unique solution.

Proof. Let h ∈ B. Then we have

|T (h)|H≤ κ

(|(v− χv+)|L2 + r

1

2 |h ∗ h|L2 + r1

2 |χ(1− χ)b0 ∗ b0|L2 + 2r1

2 |h ∗ χb0|L2 + |∇χ ∗ b0|L2

)

≤ 1

κ3r−

1

2 + c0κr1

2 |h|2H + c0κr− 3

2 + c0κr− 1

2 |h|H + c0κr−1 ≤ 1

κ2r−

1

2 .

Therefore, T maps B to itself.

For h1, h2 ∈ B,

|T (h1)− T (h2)|H= |D−1

I (r1

2h1 ∗ h1 − r1

2 h2 ∗ h2 + 2r1

2 (h1 − h2) ∗ χb0)|H≤ κ

(r

1

2 |h1|H + r1

2 |h2|H + |b0|∞)|h1 − h2|H ≤ δ|h1 − h2|H ,

where 0 < δ < 1. Therefore, T : B → B is a contraction mapping.

76

8.1.2 Sup-norm estimate

We show that the sup-norm of h∅ is small in the following sense:

Lemma 8.5. Suppose that |(v − χv+)|L2 ≤ 1κ4 r

− 1

2 and r is sufficiently large enough.

Let h∅ be the solution in Lemma 8.4 and κ be the constant in Lemma 8.3, then there

exists κ0 > 1 such that if κ ≥ κ0, we have |h∅|∞ ≤ c0r−1.

Proof. Recall that h satisfies the equation:

DIh∅ + r1

2 h∅ ∗ h∅ + r1

2 h∅ ∗A = B, (58)

where A = 2χb0 and B = r1

2χ(1 − χ)b0 ∗ b0 −∇χb0 + (v − χv+). By Lemma 3.10 in

[29], it is easy to check that A and B satisfy:

|A| ≤ c0r−1, |∇A| ≤ c0,

|B| ≤ c0r− 1

2 , |∇B| ≤ c0.

Keep in mind that the operators DIh and D∗Ih are both of the form ∇h+ r

1

2O(1) ∗ h.By the Weizenbock formula (56), we obtain

∇∗∇h∅ + 2rh∅ + R0h∅ +√rR1h∅

= D∗IB− r

1

2D∗I(h∅ ∗ h∅)− r

1

2D∗I(h∅ ∗ A)

= I + II + III.

Take inner product of above equation with h∅, then we have

| < III, h∅ > | = |r 1

2 < D∗I(h∅ ∗ A), h∅ > | ≤ c0(r

1

2 |A||∇h∅||h∅|+ r|A||h∅|2 + r1

2 |∇A||h∅|2)≤ c0r

− 1

2 |h∅||∇h∅|+ c0r1

2 |h∅|2.

| < I, h∅ > | = | < D∗IB, h∅ > | ≤ c0|∇B||h∅|+ c0r

1

2 |B||h∅| ≤ c0|h∅|.

By definition,

r1

2h∅ ∗ h∅ =

−2−3

2 (r)1

2 q(ηh, ηh)

(2r)1

2 cl(bh)ηh

0

(59)

then

II = r1

2D∗I(h∅ ∗ h∅)

=(−2d∗(r

1

2 q(ηh, ηh)) + 2(2)1

2 irIm(ψ∗I (cl(bh)ηh)),DAI

((2r)1

2 cl(bh)ηh) + (2)1

2 rcl(q(ηh, ηh))ψI

)

= (A,B) .

77

Hence,

|A| ≤ c0r1

2 |∇ηh||ηh|+ c0r|bh||ηh|,|B| ≤ c0r

1

2 |∇bh||ηh|+ c0r1

2 |bh||∇ηh|+ c0r|ηh||ηh|.

By our construction, (A∅, ψ∅) = (AI , ψI) + ((2r)1

2 (bh + χb0), (χα0 + αh, χβ0 + βh))

satisfies the Seiberg Witten equations. By Lemma 5.14, we have |ψ∅| ≤ 1 + c0r . In

particular, |ηh| ≤ c0. Therefore,

| < II, h∅ > | ≤ c0|A||bh|+ c0|B||ηh|≤ c0r

1

2 |∇ηh||ηh||bh|+ c0r|bh|2|ηh|+ c0r

1

2 |∇bh||ηh||ηh|+ c0r1

2 |bh||∇ηh||ηh|+ c0r|ηh|3

≤ c0r1

2 |∇h∅||h∅|+ c0r|h∅|2.

Here we have used the fact that |ηh| is bounded. On the other hand,

< ∇∗∇h∅ + 2rh∅ + R0h∅ +√rR1h∅, h∅ >

≥ 1

2d∗d|h∅|2 + 2r|h∅|2 + |∇h∅|2 − c0(1 + r

1

2 )|h∅|2.

Hence, we obtain a differential inequality:

d∗d|h∅|2 + r|h∅|2 ≤ c0|h∅|+ c0r|h∅|2.

Let g(x, y) be the Green function of d∗d + r over X . It satisfies the following

properties:

• 0 < g(x, y) ≤ c0dist(x, y)−2e−

√rdist(x,y)/c0 and |∇g(x, y)| ≤ c0dist(x, y)

−1e−√rdist(x,y)/c0 .

•∫X g(x, y)voly ≤ c0r

−1.

The standard elliptic regularity implies that |h∅| → 0 as s → +∞, (Cf. (4-11) of

[27]) thus we may assume that |h∅|(x) = |h∅|∞ for some x ∈ X. Then

|h∅|∞ =|h∅|(x) ≤ c0r

∫

X|h∅|g(x, y)dy + c0r

−1

≤ c0r

(∫

X|h∅|3

) 1

3(∫

Xg

3

2

) 2

3

+ c0r−1

≤ c0r2

3

(∫

X|h∅|3

) 1

3

+ c0r−1

≤ c0r2

3 |h∅|2

3

L4 |h∅|1

3

L2 + c0r−1

≤ c0r1

2 |h∅|H + c0r−1 ≤ c0κ

−2.

(60)

Hence, |h∅| ≤ c0κ−2 for sufficiently large r.

78

On the other hand,

| < II, h∅ > | ≤ c0r1

2 |∇ηh||ηh||bh|+ c0r|bh|2|ηh|+ c0r

1

2 |∇bh||ηh||ηh|+ c0r1

2 |bh||∇ηh||ηh|+ c0r|ηh|3

≤ c0r1

2 |∇h∅||h∅|2 + c0r|h∅|3.

Then we obtain

d∗d|h∅|2 + r|h∅|2 ≤ c0|h∅|+ c0r|h∅|4 + c0r|h∅|3. (61)

Again using the Green function, we have

|h∅|∞ ≤ c0

(∫

Xg(x, y)r|h∅|2dy +

∫

Xg(x, y)r|h∅|3dy + r−1

)

≤ c0(|h∅|∞ + |h∅|2∞)|h∅|∞ + c0r−1.

Therefore, |h∅| ≤ c0r−1 if κ is sufficiently large.

8.2 Iteration

This subsection is not necessary if we choose ℘3 and ℘4 in the following ways: Recall

that [℘3] = 2πc1(sΓ). Since C = ∅, the line bundle E is trivial. We can take ℘3 =

iFAK−1

+ dµ+. Using a suitable cut-off function χ which supports in the ends, we

can construct µ = χµ+ and take ℘4 = iFAK−1

+ dµ. With these choices, we have

|(v − χv+)|L2 ≤ κ−4r−1

2 provided that |µ+| ≪ 1 sufficiently small. Then we can apply

Lemma 8.4 directly to construct a solution to (57).

If we choose ℘3 and ℘4 in an arbitrary way, then the assumption |(v − χv+)|L2 ≤κ−4r−

1

2 may not hold. But we can use the following argument to solve (57). Firstly,

we reintroduce the coordinate function s of X. Let f : X → R be a Morse function

such that f = 4 on ∂X = Y+. Then we define a new coordinate function s on X such

that s = f on X and s = s + 4 on R+ × Y+. In general s is not smooth, but we can

use mollifier to make it smoothly.

Let χn = χ(s+ nκ−4 − 4) and µn = χnµ. Note that |χn+1 − χn| ≤ c0κ−4. First of

all, we solve the following equations:

DA0ψ0 = 0

F+A0

= r2(q(ψ0, ψ0)− iΩX) + iµ0

∗d ∗ b0 − 2−1

2 r1

2 (η∗0ψI − ψ∗Iη0) = 0,

(62)

where A0 = AI+(2r)1

2 b0h+χ(2r)1

2 b0, ψ0 = ψI+η0h+χη0, b0 = b0h+χb0 and η0 = η0h+χη0.

Since |(v − χv+)|L2 = (2r)−1

2 |χ(µ − µ)|L2 ≤ c0κ−4r−

1

2 , by Lemma 8.4, we can find a

unique solution h0 = (b0h, η0h) ∈ H. Moreover, |h0|H ≤ κ−2r−

1

2 and |h0|∞ ≤ c0r−1.

79

Next, we solve the equations

DA1ψ1 = 0

F+A1

= r2(q(ψ1, ψ1)− iΩX) + iµ1

∗d ∗ b1 − 2−1

2 r1

2 (η∗1ψI − ψ∗Iη1) = 0,

(63)

where A1 = AI + (2r)1

2 b0h + (2r)1

2χb0 + (2r)1

2 b1h, ψ1 = ψI + η0h + χη0 + η1h, b1 =

(2r)−1

2 (A1 − AI), η1 = ψ1 − ψI and h1 = (b1h, η1h) ∈ H. Then we need to solve the

following equation

D0h1 + r1

2h1 ∗ h1 = v1,

where D0h1 = DIh1 + r1

2 (χb0 + h0) ∗ h1 and v1 = (2r)−1

2 (χ1 − χ0)µ. Note that

|v1|L2 ≤ c0κ−4r−

1

2 .

As before, define T (h1) = −D−1I (r

1

2 (χb0 + h0) ∗ h1 + r1

2 h1 ∗ h1 − v1). Let B be a

ball in H with center at origin and radius less than 1κ2 r

− 1

2 and h1 ∈ B. Then

|T (h1)|H ≤ κ(r1

2 |(χb0 + h0) ∗ h1|L2 + r1

2 |h1 ∗ h1|L2 + |v1|L2)

≤ κ(|h0|∞|h1|H + |b0|∞|h1|H + c0r1

2 |h1|2H + c0κ−4r−

1

2 )

≤ c0κ−3r−

1

2 + c0κ−4r−

1

2 ≤ κ−2r−1

2 .

Also, it is not hard to check that |T (h1) − T (h′1)|H ≤ δ|h1 − h′1|H and 0 < δ < 1. By

the contraction mapping theorem, we can find the solution (A1, ψ1). One can replace

h by h = h0 + h1 and repeat the argument in Lemma 8.5 to show that |h|∞ ≤ c0r−1.

Note that in (60) we need to use the bound |h|H ≤ κ−2r−1

2 , but now h = h0 + h1.

Fortunately, we can still show that |h|∞ ≤ c0κ−2 by the following inequalities:

|h|∞ =|h|(x) ≤ c0r

∫

X|h|g(x, y)dy + c0r

−1

≤ c0r

∫

X|h1|g(x, y)dy + |h0|∞ + c0r

−1

≤ c0r

∫

X|h1|g(x, y)dy + c0r

−1

≤ c0r1

2 |h1|H + c0r−1 ≤ c0κ

−2.

(64)

Therefore, with the above sightly modification, we still have |h|∞ ≤ c0r−1.

By induction, suppose that we have solved the n-th equation and the solution

satisfies |h0 + · · · + hn| ≤ c0r−1. To solve the (n + 1)-th equation, let An+1 = An +

(2r)1

2 bn+1h , ψn+1 = ψn + ηn+1

h and hn+1 ∈ H. As before, it equivalent to find the fixed

point of

T (hn+1) = −D−1I (r

1

2 (χb0 + h0 + · · ·+ hn) ∗ hn+1 + r1

2hn+1 ∗ hn+1 − vn+1),

80

where vn+1 = i(2r)−1

2 (µn+1−µn). Now the contribution of (χb0+ h0 + · · ·+ hn) ∗hn+1

is

κr1

2 |(b0 + h0 + · · ·+ hn) ∗ hn+1|L2

≤ κ(|b0|∞ + |h0 · · ·+ hn|∞)|hn+1|H ≤ c0κ−1r−

3

2 .

As before, we can show that T is contraction mapping. Moreover, repeat the argument

in Lemma 8.5 with the sightly modification as (64), we still have

|h0 + · · ·+ hn+1|∞ ≤ c0r−1.

Hence, there exists a positive integer N = O(κ4) such that h∅ = (bh, ηh) = h0 +

. . . hN and A∅ = AI+(2r)1

2 bh+χ(2r)1

2 b0, ψ∅ = ψI+ηh+χη0, b = bh+χb0, η = ηh+χη0

satisfies the following equations

DAψ = 0

F+A = r

2 (q(ψ,ψ) − iΩ) + iµ

∗d ∗ b− 2−1

2 r1

2 (η∗ψI − ψ∗Iη) = 0.

(65)

Moreover, according to our construction, (A∅, ψ∅) converges to (A∅+, ψ∅+) as s→ ±∞.

8.3 General case that C 6= ∅Proposition 8.6 (Cf. Prop 7.1 of [29]). Let C = (Ca, 1) be an embedded holomorphic

curve in X. There exists c0 > 1 and a finite dimensional normed vector space V0. Let

B be a radius c−10 ball in K, where K is the Banach space defined in Section 5.b of [29].

Then for any r ≥ c0, there exists a linear map q : K → V0 such that

1. |q(ξ)| ≤ c0|ξ|L2 and q is surjective.

2. For any v ∈ V0, there exists a unique ξv ∈ q−1(v) ∩ B such that b(ξv) = q(ξv) +

h(ξv) satisfies the following equations:

Db+ r

1

2 b ∗ b = v

lims→∞

b = b±.(66)

Moreover, |ξv|K ≤ (r−1

2+16σ + |v|).

3. dim(V0) = I(C).

4. Let b(ξ) = (b, η) be the solution to (66), then (Ar, ψr) = (Aξ+(2r)1

2 b, ψξ+η) is a

solution to Seiberg Witten equations (67). Moreover, (Ar, ψr) is non-degenerate

with Fredholm index I(C).

81

5. limr→∞

i2π

∫X FAr ∧ ωX =

∫C ωX .

Proof. The proof of the first and second bullets of the proposition closely tracks Taubes’

argument in Sections 4-7 of [29]. We summarize the main points in the following five

steps.

Step 1: The first step is to construct a line bundle E → X . Let ΞC be a collection

of the limit of the s-constant slice of C as s → ∞. One can define an open cover

UCC∈C , Uγγ∈ΞC , U0 of X as in [29], where UC is a tubular neighborhood of C, Uγ

is a tubular neighborhood of R≥R0× γ, and U0 is the rest. Let C ∈ C and π : NC → C

be the normal bundle. E is defined by gluing π∗NC |UC, and the trivial line bundles

over Uγγ∈ΞC and U0. The Spin-c bundle is S+ = E ⊕ (E ⊗K−1X ).

Step 2: The next step is to construct an approximation solution (A∗, ψ∗) that is

close to solving (14). We need to introduce vortices firstly. The Vortex equations is an

two dimensional analogy of Seiberg Witten equations. A pair (A,α) is called n-vortices

if they satisfy the following equations

∂Aα = 0

∗FA = −i(1− |α|2)12π

∫C(1− |α|2) = n,

where A ∈ Ω1(C, iR) and α is a complex function over C. The moduli space of n-

vortices is denoted by Cn. It is diffeomorphic to symmetric product Symn(C). For

more details about the vortices, please refer to Section 2 of [29].

The fiber of NC is a complex plane, so one can consider the vortex equations over

there. Assign each fiber of NC a vortex , a collection of these vortices give us a pairs

(AC,r, ψC,r) defined near C, where AC,r and ψC,r are connection and section of E

respectively. Since our C is embedded, there is a canonical choice of vortices over each

fiber which is called symmetric vortices. This vortex is corresponding to the origin

under isomorphism Cn ≃ Symn(C)

Let (γ,mγ) ∈ ΞC , i.e., C is asymptotic to γ with total multiplicities mγ . Thus the

ends of C lie inside Uγ . Again, we assign vortices on each fiber of the normal bundle of

R≥R0× γ. The choice of these vortices is determined by the asymptotic behavior (5).

A collection of them gives us a pair (Aγ,r, ψγ,r) is defined over Uγ .

Let (A0, ψ0 = (1C, 0)) be the trivial solution over U0, where A0 is trivial con-

nection. The approximation solution (A∗, ψ∗) is defined by gluing (AC,r, ψC,r)C∈C ,

(Aγ,r, ψγ,r)γ∈ΞC and (A0, ψ0) together. It is worth noting that the constructions of

(AC,r, ψC,r)C∈C and (Aγ,r, ψγ,r)γ∈ΞC take place near C and the choice of (A0, ψ0)

is canonical, that is why the constructions of [29] can be applied to our case without

essential change.

82

Step 3: Given vortices in Cn, one can perturb the vortices by using sections of

T1,0Cn and exponential map. Since our (A∗, ψ∗) comes from vortices over each fiber of

the normal bundle, we can deform (A∗, ψ∗) in a similar way. Follow Section 5.b of [29],

we can define a Banach space K which is an analogy of T1,0Cn. Given ξ ∈ K, we can

deform (A∗, ψ∗) by using ξ, the result is denoted by (Aξ, ψξ).

Suppose that C is asymptotic to an orbit set α+, according to Section 3 of [29] (also

see [18]), we can construct a configuration (A+, ψ+) from α+ satisfying equations (10)

together with an additional gauge fixed condition. Let b = (b, η) ∈ Γ(iΩ1 ⊕ S+), we

want to find b such that (A,ψ) = (Aξ +(2r)1

2 b, ψξ + η) satisfies the following equations

DAψ=0

F+A = r

2 (q(ψ,ψ) − iΩX)− 12F

+A

K−1− i

2℘+4

∗d ∗ b− 2−1

2 r1

2 (η∗ψξ − ψξ∗η) = 0,

(67)

with asymptotic behavior lims→±∞

(A,ψ) = (A±, ψ±). In particular, [(A,ψ)] ∈ MJ,rX (c+, c−).

The equations (67) can be schematically as

Db+ r

1

2 b ∗ b = v

lims→±∞

b = b±,(68)

where D is the deformation operator of Seiberg Witten equations at (Aξ, ψξ).

Observed by Taubes, the bundle iT ∗X⊕S+ over UC can be identified as VC0⊕VC1,

where VC0 = π∗N ⊕ π∗N and VC1 = π∗T 0,1C ⊕ (π∗N2 ⊗ π∗T 0,1C). Granted this

identification, the deformation operator D can be rewritten as

D =

(∂Hθ ϑ∗Cξ,r

ϑCξ,r ∂Hθ

)+ r, (69)

where r is a small error term, ϑCξ,r is the deformation operator of the vortex equations

and ϑ∗Cξ ,ris its L2 adjoint. The similar identification also holds on U0 and Uγ± . The

operator ϑ∗Cξ,ris not surjective, so does D. One cannot expect to use contraction

mapping principle directly.

As [29], we can define a L2 projection Πξ : L2((iΩ0 ⊕ Ω2+) ⊕ S−) → L2((iΩ0 ⊕Ω2+)⊕S−). Roughly speaking, the Πξ maps elements in VC0 ⊕VC1 to the cokerϑ∗Cξ ,r

.

Using the properties of vortices, identification (69) and the same argument in Lemma

6.1 of [29], we know that (1−Πξ)D is surjective in suitable Sobolev space. Hence, we

want to use the contraction mapping principle to solve the following equation firstly,

(1−Πξ)(Db + r1

2 b ∗ b) = (1−Πξ)v. (70)

83

Step 4: To solve (70), however, the error term (1 − Πξ)v is not small enough to

use the contraction mapping principle. Write b = q+ h, equation (70) is equivalent to

(1−Πξ)(Dq+ r1

2 q ∗ q+ 2r1

2 q ∗ h) = (1−Πξ)(v − vh), (71)

where vh = Dh+ r1

2 h ∗ h and Πξq = 0.

Section 6.d of [29] constructs a h = h(ξ) so that the error term (1 − Πξ)(v − vh)

is small enough. The construction of h in our case is same as Lemma 6.3 of [29] with

the following adjustment: Replace b0 by χb0 + h∅ in proof of Lemma 6.3, where χ

and h∅ is defined in Section 8.1.1. Note that Lemma 8.5 guarantees that the estimates

(6-31) in Lemma 6.3 of [29] hold, and the remaining part of proof takes place near the

holomorphic curves, hence the construction in [29] can be applied to cobordism case

with only notation changes.

Given h(ξ), for any sufficiently small ξ, then we can find q = q(ξ) satisfying equation

(71) by using contraction mapping principle.

Step 5: To deal the remaining part of (68), we need to consider equation

Πξ(Dq+ r1

2 q ∗ q+ 2r1

2 q ∗ h− v+ vh) = 0. (72)

The argument in Step 4 works for any small ξ, thus one hope to vary ξ such that (72)

holds. Let T (ξ) to be r1

2 times the left hand side of (72). We can write

T (ξ) = T0 + T1(ξ) + T2(ξ),

where T0 is the ξ = 0 version of T , T1 is the linerization of T , and T2 is a quadric error

term. The argument in [29] can show that T0 is closed to zero. Under the identification

in Section 7.d of [29], T1 is surjective provided that DCC∈C are surjective, where DC

is the linearization of the holomorphic curve. Therefore, we can use the contraction

mapping argument to find solutions ξ to equation (72).

To prove the last point of the proposition, rewrite (Aξ, ψξ) = (A∗, ψ∗) + tξ, then

(Ar, ψr) = (Aξ , ψξ) + ((2r)1

2 br, ηr) = (A∗, ψ∗) + ((2r)1

2 bξ, ηξ),

where ((2r)1

2 bξ, ηξ) = ((2r)1

2 br, ηr) + tξ. Note that the pair (bξ, ηξ) is asymptotic to

(bξα+, ηξα+

), where (bξα+, ηξα+

) is used to construct (A+, ψ+) from orbit set α+, cor-

responding to the solution to equation (3-6) in [29]. By Lemma 3.10 of [29], dωX = 0

and Stokes’ formula, we have

|∫

X(2r)

1

2dbξ ∧ ωX | = |∫

Y+

(2r)1

2 bξα+∧ ω+| ≤ c0r

1

2 |bξα+|L2 ≤ c0r

− 1

2 .

Hence, it suffices to prove the statement for A∗. Then the last point of the proposition

follows from the following Lemma 8.7.

84

Lemma 8.7. Let (A∗, ψ∗) be the approximation solution which is constructed in Propo-

sition 8.6, then

limr→∞

i

2π

∫

XFA∗ ∧ ωX =

∫

CωX .

Proof. The proof of this lemma relies heavily on the construction of (A∗, ψ∗) in Section

5 of [29] and the properties of vortices. In what follows, we use the notation and

terminology of [29].

Recall that when we construct (A∗, ψ∗), we need to assign vortices to each fiber of

normal bundle of C. In fact, the collection of these vortices form a tuple of vortices

sections (cCC∈C , cγ+γ+∈ΞC ) of certain vector bundles, called vortices bundle. For

the details, please refer to Section 2.e of [29]. In particular, cγ+ here is a map from

[R,+∞)× S1 to Cmγ+.

Given C ∈ C, by the construction and properties of vortices, FA∗ = dVAC,r =

r(1−|αC,r|2)12∇θs∧∇θs on the region where χC = 1. Also, |FA∗ | ≤ c0e−rσ on the part

where χC 6= 1. Thus

∫

X⋂

UC

i

2πFA∗ ∧ ωX

=

∫

X⋂

UC

1

2πr(1− |αC,r|2) i

2∇θs ∧ ∇θ s ∧ ωX +O(e−rσ), (73)

=

∫

C⋂UC

ωX +O(e−rσ).

The last step follows from the fact that∫Nz

12π (1− |αC,r|2) i2∇θs ∧ ∇θ s = 1 along each

fiber of NC .

Given γ+ ∈ ΞC+, define cγ+0: [R,+∞) × S1 → Cmγ+

be a smooth vortex section

so that cγ+0= cγ+ on s ≤ R∗ − R and cγ+0

is vortices when s ≥ R∗, where R∗ is the

constant defined in (4-9) of [29]. Using c0 = (cC )C∈C , (cγ+0)γ∈ΞC+, we can construct

an approximation solution (A∗0, ψ

∗0) as before. Note that

∫X FA∗ ∧ ωX =

∫X FA∗

0∧ ωX

because of Stokes’ theorem.

On Uγ , we can divide it into three parts X1, X2 and X3. For the definition of Xi,

please refer to Section 6 of [29]. By the observation in the last paragraph, we only need

to prove the statement for FA∗0. Since FA∗

0∧ω+ = 0 on s ≥ R∗, there is no contribution

from X1. Now we consider the case on X2. Suppose χγ = 1, then A∗ = θ0+Aγ,r0 +Aγ,r

and FA∗ = dVAγ,r+xγ,r−xγ,r, where xγ,r is defined in (2-37) of [29]. Roughly speaking,

it is derivative of A∗ in the horizontal direction. As before, on the part where χγ 6= 1,

85

the contribution is O(e−rσ) and

∫

X⋂(Uγ∩X2)

i

2πFA∗ ∧ ωX

=

∫

X⋂(Uγ∩X2)

1

2πr(1− |αγ,r|2) i

2dz ∧ dz ∧ ωX +

∫

X⋂(Uγ∩X2)

(xγ − xγ,r) ∧ ωX +O(e−rσ),

= mγ

∫

R×γ⋂

Uγ∩X2

ωX +O(e−rσ) +

∫

X⋂(Uγ∩X2)

(xγ − xγ,r) ∧ ωX (74)

=

∫

X⋂(Uγ∩X2)

(xγ,r − xγ,r) ∧ ωX +O(e−rσ).

By (2-2) of [29], |xγ,r| ≤ c0r1

2

∑z′∈Zγ(w,t) e

−r12 |z−z′|, where Zγ(w, t) is defined in Section

5.d of [29]. Thus we have |∫X

⋂(Uγ∩X2)

xγ,r ∧ ωX | ≤ c0r− 1

2+8σ.

On the intersection of Uγ ∩X3 and near an end E of C, recall that

A∗ = χE ((1− χγ)(θ0 − u−1γ duγ) + χγ(θ0 +Aγ,r

0 +Aγ,r)) + (1− χE )AE ,

where AE , χE and χγ are defined in page 2648 of [29]. Then

FA∗ = dχE (1− χγ)((θ0 − u−1

γ duγ)− Aγ,r)+ χEdχγ

(Aγ,r − (θ0 − u−1

γ duγ))

+ dχE (Aγ,r − A

E ) + (1− χE )dAE + χEχγdA

γ,r

= (I) + (II) + (III) + (IV ) + (V ).

By the exponential decay of vortices and (4-9) of [29], the contribution of (I) and (II)

are c0e−rσ .

By (6-45), (6-46) and (6-48) of [29],

|Aγ,r − AE | ≤ c0r

1

2 |z|2e−√

r|η|c0 + c0e

−√

r|η|c0 .

Hence, the contribution of (III) is c0r−1+8σ.

Arguing as the cases in (73) and (74), also using (6-45), (6-46) and (6-48) of [29],

the contributions of (IV ) + (V ) are∫C∩X3

ωX +O(r−1

2+8σ).

8.4 Proof of Theorem 8.1

Proof. For each C = Ca ∈ MJ,LX,I=0(α+, α−), Proposition 8.6 gives us a unique solu-

tion (Ar, ψr) to Seiberg Witten equations (14). Moreover, (Ar, ψr) is nondegenerate

with zero Fredholm index and limr→∞

i2π

∫X FAr ∧ ωX =

∫C ωX .

So we can define the following meaningful map

Ψr : MJ,LX,I=0(α+, α−) → M

J,LX,ind=0(T

+r (α+), T

−r (α−)),

86

by sending C ∈ MJ,LX,I=0(α+, α−) to [(Ar, ψr)], where M

J,LX,ind=0(T

+r (α+), T

−r (α−)) is the

moduli space of solutions to Seiberg Witten equations (14) with zero index.

Note that Ψr is injective for the following reasons. Let (Ar, ψr) = Ψr(C), then FAr

satisfies the following properties:

1. The L2-norm of FAr over the |s| ≤ R portion U0 is bounded by c0.

2. For each Ca ∈ C, for every |s0| ≤ R−1, the L2-norm of FAr over the [s0−1, s0+1]

portion UC is greater than c−10 r

1

2 .

The two properties above can be checked directly from the construction. Let C, C′ ∈MJ,L

X,I=0(α+, α−) and C 6= C′. There exists c′0 ≥ c100 such that for r ≥ c′0, then there

exists a component C ′a of C′ and |s0| ≤ R− 1 such that the [s0 − 1, s0 + 1] part of UC′

a

lies inside the C version’s U0 part. Above two properties imply that Ψr(C) 6= Ψr(C′).

To show that Ψr is onto for sufficiently large r, as summarized in [18], we can

divide the proof into three parts, which are called estimation, convergence and

perturbation.

1. The estimation part: As remark on page 38, the relevant estimates in [18] can

be carried over almost verbatim to our setting.

2. The convergence part: Suppose that Ψr is not onto for any r, then there exists a

sequence (An, ψn)∞n=1 such that each (An, ψn) ∈ MJ,LX,ind=0(T

+rn(α+), T

−rn(α−))

is not image of Ψrn and rn → ∞. By Proposition 5.11 and Theorem 5.1 of [2],

after passing a subsequence, (An, ψn)∞n=1 converges to a broken holomorphic

current C with I(C) = 0.

According to our assumptions, properties of Vcomp(X,πX , ωX) and the counter-

parts of Lemmas 7.3, 7.4 and 7.7, C is not broken and it is embedded.

3. The perturbation part: Let (An, ψn)∞n=1 and C as above. We have the following

refinement of the convergence (Cf. Lemma 6.2 of [31]): Given any δ > 0, then

there exists a subsequence of (An, ψn)∞n=1 such that

supz∈C∈C

dist(z, α−1n (0)) + sup

z∈α−1n (0)

dist

(z,⋃

C∈CC

)< δr

− 1

2n (75)

for sufficiently large n. To prove (75), the argument is the same as Section 7 of

[31]. We define the special sections ok|p as (7-9) of [31], where p lies in C ∈ Cor half R-invariant cylinders. Note that the proof of Lemmas 4.10, 7.1, 7.2, 7.3

and 7.4 in [31] are local, they are still true in our case. Copy the argument in

Lemmas 7.5 and 7.7 in [31], we can derive (75).

87

With (75), we follow Section 6.b of [31] to construct a gauge transformation

un such that un · (An, ψn) is close to the approximation solution (A∗, ψ∗). The

construction relies on Lemma 6.4 of [29]. The proof only concerns the behavior

of (An, ψn) near C, so it is still true in our case. Write

un · (An, ψn) = (Aξ, ψξ) + ((2rn)1

2 bξ, ηξ),

where (Aξ, ψξ) is defined in Step 3 of Proposition 8.6. By contraction mapping

principle and argument in 6.c of [31], we find a new gauge transformation, still

denoted by un, such that b(ξ) = (bξ, ηξ) satisfies (68). Let q(ξ) = b(ξ) − h(ξ),

where h(ξ) is defined in step 4 of Proposition 8.6.

If Πξq(ξ) = 0 and ξ ∈ B, where B ⊂ K is provided by Proposition 8.6, then the

uniqueness of equations (71), (72) implies that u∗n(An, ψn) = Ψrn(C), this lead

to our goal. The tasks of Sections 6.d and 6.e in [31] are to find such ξ ∈ B such

that Πξq(ξ) = 0. The analysis in 6.d and 6.e of [31] are again local, they can be

applied to our situation without modification.

9 Proof of Theorem 3

Before we prove Theorem 3, we want to reduce the proof to the case that (ωX , J)|R±×Y±

is Q-δ flat firstly. Assume that (X,ωX) or (ΓX , ωX) is monotone, then the existence

of nonempty open subset Vcomp(X,πX , ωX) has been proved in Section 7.3. Take an

almost complex structure J ∈ Vcomp(X,πX , ωX)reg, by Lemma 5.18, we can always

perturb (ωX , J), the result is denoted by (ω′X , J

′), such that (ω′X , J

′)|R±×Y± is Q-δ

flat. In what follows, we show that the cobordism maps are unchanged under the

perturbation.

To simplify the notation, assume that Y− = ∅. Given any L > 0, our goal is to find

a pair (ω′X , J

′) satisfying the conclusions in Lemma 5.18 and there exists a bijection

between MJ,LX,I=0(α+) and MJ ′,L

X,I=0(α+). We follow the strategy in Appendix of [28]

to construct such a pair (ω′X , J

′). For every periodic orbit γ with degree less than Q,

reintroduce the coordinate ϕγ : S1τ ×Dz → Y+, (ν, µ) in (16), and (νγ , µγ) is defined in

Lemma 5.4. We fix a homotopy (ντ , µτ )τ∈[0,1] starting from (ν, µ) to (νγ , µγ) which

is used to define the Q-δ approximation. Furthermore, there is a uniform lower bound

to the absolute value of the eigenvalue of (ντ , µτ )-version of (6).

Take a N ≫ 1, using the data (ν kN, µ k

N)Nk=0, we want to construct a sequence

of pair (ωX,k, Jk)Nk=0 inductively, where ωX,k is an admissible 2-form and Jk is a

cobordism admissible almost complex structure with respective to ωX,k, such that

88

F.1 (ωX,0, J0) = (ωX , J) and (ωX,N , JN ) = (ω′X , J

′). Also, (ω′X , J

′)|[0,∞)×Y+is Q-δ

approximation of (ω+, J+).

F.2 The difference (ωX,k+1−ωX,k, Jk+1−Jk) is supported in [−32ǫ,∞)×Uγ,ρk+1

, where

Uγ,ρk+1is image via the map ϕγ from of the set in S1 ×D where |z| ≤ ρk+1.

F.3 The half cylinder [−2ǫ,∞)×γ is Jk holomorphic for any k and any periodic orbit

γ with degree less than Q.

F.4 We have estimates |ωX,k − ωX,k+1| ≤ c0ρk+1N−1 , |Jk − Jk+1| ≤ c0ρk+1N

−1 and

|∇(Jk − Jk+1)| ≤ c0N−1.

F.5 There exists a bijective map between MJk,LX,I=0(α+) and MJk+1,L

X,I=0 (α+).

F.6 For any k, MJk,LX,I<0(α+) = ∅ and the curves in MJk,L

X,I=0(α+) are embedded and

Fredholm regular.

The sequence 0 < ρN < ρN−1 < · · · < ρ0 ≪ δ will be determined during the construc-

tion. Appendix of [28] (Also see [18].) constructs such sequence of pairs inductively

for symplectization case. In other words, given (ω+, J+), we can define a sequence

(ω+,k, J+,k)Nk=0 with properties F.1, F.2, F.3, F.4, F.5, and F.6. For cobordism

case, we just need to interpolate this sequence with (ωX , J) over a collar neighborhood

of ∂X. Assume that J0, · · · , Jk have been defined. We can define Jk+1 as follows: De-

fine (ωX,k+1, Jk+1) = (ωX , J) over X−[32ǫ, 0]×Y+ and (ωX,k+1, Jk+1) = (ωk+1,+, Jk+1,+)

over [0,∞)×Y+. Over the region [−32ǫ, 0]×Y+, define (ωX,k+1, Jk+1) to be the interpo-

lation between (ωk+1,+, Jk+1,+) and (ω+, J+), which is similar to the proof of Lemma

5.18. Obviously, we can arrange the interpolation such that Jk+1 satisfies the condi-

tions F.1, F.2 and F.3. Since J+,k − J+,k+1 satisfies the condition F.4, we can further

arrange that Jk+1 − Jk satisfies this condition.

To prove F.5 and F.6, firstly observe that we have the following facts. Let C ∈MJk,L

X,I=0(α+) and C be an irreducible component of C, then

1. Jk+1 is 1N -close to Jk in C1 sense. Also, the factor 1

N is very small.

2. If C∩[−32ǫ,∞)×Y+ equals to union of [−3

2ǫ,∞)×γ, then C is a Jk+1 holomorphic

by F.2 and F.3. Otherwise, C is Jk+1 holomorphic except for where it intersects

the region [−32ǫ,∞)×Uγ,ρk+1

. The intersection of C with [−32ǫ,∞)×Uγ,ρk+1

can

only occur in the following two ways. (Cf. (A-14) of [28])

(a) Intersection occurs in a disk of radius R−1 in C centered around each inter-

section points C ∩ (−32ǫ,∞)× γ. (By slightly increasing ǫ, we may assume

that C doesn’t intersect −32ǫ × γ.)

89

(b) The ends of C which are asymptotic to covered of γ.

3. The condition F.4 and the last point implies that C is nearly Jk+1 holomorphic

in L2 sense.

According to the above observations, we can follow the Appendix of [28] to perturb

C to be Jk+1 holomorphic. The main points are sketched as follows. Let NC be the

normal bundle of C and η be a section of NC . The deformation of C along η is Jk+1

holomorphic if and only if η satisfies the following equation:

DCη + p1η + (R1 + p2)∇η +R0(η) + p0 = 0, (76)

where DC is the deformation operator of C with respect to Jk. Here pi3i=0 and

Ri1i=0 are certain operators. They satisfy the same estimates (A-17) in [28]. Lemma

A.2 of [28] shows that DCη + p1η + +p2∇η is invertible as a map between suitable

Banach spaces. To see how this works. Write η = η1 + η2, where η1 has support far

out on the ends of C and η2 is supported in the rest of C. Since we have a relevant

version of (A-14) [28] and the same estimates on pi2i=1, these can be used to make

sure that the contributions of pi2i=1 terms are small. Recall that there is a uniform

lower bound to the absolute value of the eigenvalue of (ντ , µτ )-version of (6). Over the

ends of C, this property implies that ||DCη1 + p1η1 + p2∇η1||1 ≥ c−10 ||η1||2, where c0 is

independent of k and || · ||i, i = 1, 2, are suitable norms. By induction, we have lower

bound ||DCη2||1 ≥ c−1k ||η2||2. Therefore, DCη + p1η + p2∇η is invertible. Combine

these with the relevant version (A-17) in [28] and the relevant estimates on pi3i=0, by

contraction mapping principle, there exists a unique solution to (76). With the above

understanding, we have an injective map

F : MJk,LX,I=0(α+) → MJk+1,L

X,I=0 (α+).

To see why F is onto for sufficiently small ρk+1, by Taubes’Gromov compactness, as

ρk+1 → 0, the holomorphic curves Ck+1 ∈ MJk+1,LX,I=0 (α+) converge to an unbroken curve

in MJk,LX,I=0(α+). Since the curves in MJk,L

X,I=0(α+) is regular, thus Ck+1 is in the image

of F for sufficiently small ρk+1. In sum, Jk+1 satisfies all the properties F.1, F.2, F.3,

F.4, F.5 and F.6. Follow Lemma A.4 in [28] to compare the deformation operator

DF(C) and DC + p1 + p2∇, their difference is controlled by ρk+1. Therefore, DF(C) is

also invertible.

For (ωX,N , JN ), we can further perform a small perturbation such that JN is

generic, Moreover, the moduli space MJN ,LX,I=0(α+) is unchanged, because, the curve

in MJN ,LX,I=0(α+) is Fredholm regular.

We summarize the above discussion into the following lemma.

90

Lemma 9.1. Given L > 0, there exists a Q-δ flat pair (ω′X , J

′) satisfying the con-

clusions in Lemma 5.18. Moreover, there is a bijection between MJ,LX,I=0(α+, α−) and

MJ ′,LX,I=0(α+, α−).

Proof of Main Theorem 3. Let ωX be a Q-admissible 2-form. Let J be a generic almost

complex structure in Vcomp(X,πX , ωX)reg. Taking an unbounded increasing sequence

Ln∞n=1 and take δ = 1n in Lemma 9.1, we can find a sequence of pairs (ωXn, Jn)∞n=1

such that (ωXn, Jn)|R±×Y± is Q- 1n flat approximation of (ωX , J)|R±×Y± . By Lemma

9.1, we have

CP (X,ΩX , J,ΛX) = CP (X,ΩXn, Jn,ΛX) + o(Ln), (77)

where limn→∞

o(Ln) = 0.

For each n, according to Theorem 8.1, there exists a constant rLn > 0 such that

CP (X,ΩXn, Jn,ΛX) = (T−r )−1 CM(X,ΩXn, Jn, r,ΛX) T+

r + o(Ln). (78)

for any r > rLn . Let r → ∞, by (37), we have

CP (X,ΩXn, Jn,ΛX) = CPsw(X,ΩXn, Jn,ΛX) + o(Ln). (79)

Combine above equalities (77) and (79), (38) and by taking n→ ∞, then we obtain

CP (X,ΩX , J,ΛX ) = CP sw(X,ΩX , J,ΛX ).

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(Guanheng Chen) Department of Mathematics, The Chinese University of Hong Kong.

E-mail adress: ghchen@math.cuhk.eduh.hk

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