monopoles and contact structures - harvard...

Monopoles and contact structures

P. B. Kronheimer1 and T. S. Mrowka2

1 Harvard University, C 2 Massachusetts Institute of Technology, C and

California Institute of Technology, P

1. Introduction

(i) Four-manifolds with contact boundary

The monopole invariants, or Seiberg-Witten invariants, introduced by Witten [27]are invariants of a smooth, closed, oriented 4-manifold X . When b+(X ) is greaterthan 1, they can be regarded as defining a map

SW : Spinc (X )→ Z,

where Spinc (X ) denotes the set of isomorphism classes of Spinc-structures on X[17]. The sign of the invariant depends also on a choice of a homology orienta-tion for X , as defined in [4]. In this paper we shall adapt the construction of themonopole invariants to the situation of a 4-manifold with contact boundary.

In more detail, we consider a connected oriented 4-manifold X with non-empty boundary, equipped with an oriented contact structure ξ on ∂X (which weregard as an oriented field of 2-planes). We require ξ to be compatible with theboundary orientation of ∂X . This means that if we choose any 1-form θ on ∂Xwhich annihilates the field of 2-planes ξ , then the 3-form θ ∧ dθ is positive. (Thenon-vanishing of the 3-form is the contact condition.) We also need to specify ahomology orientation of (X , ξ ). The definition of a homology orientation in thissetting is explained in an appendix to this paper.

The contact structure ξ on ∂X determines a preferred Spinc-structure sξ in aneighborhood of the boundary (see section 2(ii)). Write Spinc (X , ξ ) for the setof isomorphism classes of extensions of sξ to the interior. (Thus an element ofSpinc (X , ξ ) is given by a Spinc-structure s on X together with an isomorphismbetween s and sξ at the boundary; the latter will usually be omitted from our

1Partially supported by NSF grant number DMS-95319642Partially supported by NSF/NYI grant number DMS-9357641 and a Sloan Fellowship

2 1. Introduction

notation.) We shall extend the monopole invariants to this setting, defining themas a map

SW : Spinc (X , ξ )→ Z.

No restriction on b+(X ) is necessary for this definition. If ξ denotes the samecontact structure with the opposite orientation for the contact 2-planes, then theSpinc-structure determined by ξ near the boundary is sξ (the complex conjugateSpinc-structure), and for suitable choice of homology orientations there is a relation

SW (X ,ξ )(s) = SW (X ,ξ )(s),

for all s ∈ Spinc (X , ξ ).On a closed manifold X , the monopole invariants are defined, roughly speak-

ing, by counting solutions to the monopole equations (or Seiberg-Witten equations)on X . Motivation for considering these equations in conjunction with contactstructures comes from three directions. First, there is the wealth of topological re-sults related to contact structures and their cousins, foliations of 3-manifolds; seefor example [9, 12]. Second, there is the material in [5], suggesting a study of theYang-Mills equations in a setting similar to ours. Third, and most closely con-nected with the present paper, there is the recent work of Taubes, who investigatedthe properties of the equations on a symplectic 4-manifold [23, 24, 25], using themto answer several important questions in 4-dimensional symplectic topology.

A symplectic form ω on X determines a preferred Spinc-structure s0 (see sec-tion 2) and a preferred homology orientation. Amongst the results of Taubes citedabove is the following theorem [23]: if (X , ω) is a closed symplectic 4-manifoldwith b+(X ) > 1, then SW (s0) and SW (s0) are non-zero; in fact, SW (s0) = 1when X is given the canonical homology orientation. This result has the follow-ing extension to our situation. A symplectic structure on the oriented 4-manifold(X , ∂X ) is said to be compatible with the contact structure ξ on the boundary ifthe symplectic 2-form ω is positive on the oriented contact 2-planes. As in theclosed case, a compatible symplectic structure ω determines a preferred elements0 ∈ Spinc (X , ξ ), as well as a preferred homology orientation, and we have:

Theorem 1.1. If ω is a symplectic form on X which is compatible with the contact structureξ on ∂X, then with the canonical homology orientation,

SW (s0) = 1,

where s0 is the element of Spinc (X , ξ ) determined by ω.

There is a companion theorem to the result of [23], which shows that a sym-plectic structure restricts the set of Spinc-structures on which SW can be non-zero.

(ii) Applications 3

(This set is in any event finite.) For a closed symplectic manifold, the theorem isgiven in [24], and we shall extend it to our setting. To state the result, recall firstthat in the closed case the set Spinc (X ) is a principal homogeneous space for thegroup H2(X ;Z). In our situation, Spinc (X , ξ ) is a principal homogeneous spacefor the relative cohomology H2(X , ∂X ;Z). In either case, we denote the action bys 7→ s+ e, and for any two structures s1, s2 we have a difference element s1 − s2.

Theorem 1.2. If ω is a symplectic form on X which is compatible with the contact structureξ on ∂X, and if SW (s) 6= 0, then

[ω]^ (s− s0) ≥ 0

in H4(X , ∂X ;R), with equality only if s = s0. In particular, if ω is exact, then s0 is theonly element of Spinc (X , ξ ) for which the monopole invariant is non-zero.

The proofs of Theorems 1.1 and 1.2 will turn out to be routine modificationsof the proofs of the original theorems from [23] and [24], once the definition andproperties of our invariants are in place. The analysis needed to justify the defini-tions themselves is more delicate, but again rests on work of Taubes, particularly theestimates for solutions of the monopole equations contained in [25].

(ii) Applications

We shall derive two topological corollaries from the main results of this paper.First we shall derive a finiteness result for the set of homotopy types of semi-fillablecontact structures on a given 3-manifold. A closed, contact 3-manifold (Y, ξ ) issaid to be symplectically fillable [9] if Y is the oriented boundary of a symplecticfour-manifold (X , ω) in such a way that the restriction of ω to the oriented contact2-planes at the boundary is positive (this is the compatibility condition describedabove). This definition does not require Y to be connected. One says that (Y, ξ )is symplectically semi-fillable if it is a union of components of a symplectically fillablecontact 3-manifold. It is known that every oriented 2-plane field on a 3-manifoldis homotopic to a contact structure, but not every 2-plane field is homotopic to afillable or semi-fillable contact structure. This follows from the results of [7], whichcan be seen as constraining the Euler class of the 2-plane bundle. However, on any3-manifold, there are infinitely many homotopy classes of 2-plane fields even withEuler class zero. We shall prove:

Theorem 1.3. For any closed 3-manifold Y , there are only finitely many homotopy classesof 2-plane fields which are realized as semi-fillable contact structures.

4 1. Introduction

This theorem implies a similar statement about taut foliations of 3-manifolds,because of the following result due to Eliashberg and Thurston. Recall that a folia-tion of a 3-manifold by oriented 2-dimensional leaves is taut if for every leaf L thereis a closed curve in the 3-manifold which meets L and is transverse to the leaves.

Theorem 1.4 (Eliashberg-Thurston [10]). Let Y be a closed oriented 3-manifoldwith a smooth foliation ^ by oriented 2-dimensional submanifolds. Suppose this is not thefoliation of S1 × S2 by the leaves θ × S2. In the Grassmann bundle of 2-dimensionaloriented subspaces of TY , let U be any neighborhood of the section determined by the tangentplanes to the leaves of ^. Then there exists an oriented contact structure ξ on Y which iscompatible with the orientation of Y and is contained in the neighborhood U. Furthermore,if ^ is taut, then (Y, ξ ) is symplectically semi-fillable.

In particular, the class of 2-plane fields homotopic to the tangent distribution ofa taut foliation contains a semi-fillable contact structure (except in the case of onefoliation of S1 × S2), and from Theorem 1.3 we deduce:

Corollary 1.5. For any closed 3-manifold Y , the number of homotopy classes of 2-planefields which are realized as the tangent distributions to smooth, taut foliations is finite.

For a second application of our main results, note that an example of an exactsymplectic form arises when (X , ∂X ) is a Stein domain: that is, when X has a com-plex structure J and carries a pluri-subharmonic function φ for which ∂X is a levelset where φ is a maximum. There is a natural contact structure ξ on the boundaryin this case, given by the J-invariant 2-planes in T (∂X ). This contact structure iscompatible, in the above sense, with the Kahler form ω = i∂∂φ. Theorem 1.2 thenhas the following corollary, for which an earlier proof was given by Lisca and Matic:

Corollary 1.6 (Lisca-Matic [18]). Suppose that the 4-manifold (X , ∂X ) has thestructure of a Stein domain for two complex structures J1 and J2. Then if the resultingcontact structures ξ1 and ξ2 on the boundary are isotopic, the Spinc -structures determined byJ1 and J2 must be isomorphic, and in particular one must have c1( J1) = c1( J2).

As pointed out in [18], this corollary allows one to exhibit interesting examples ofcontact structures on a 3-manifold, which are homotopic as oriented 2-plane fieldsbut not homotopic through contact structures. These examples arise from Eliash-berg’s surgery construction for Stein domains [8]. In fact, only the exact symplecticform is needed in Theorem 1.2, not the complex structure; the construction of theformer is rather less delicate (see [26]).

Corollary 1.6 is parallel to one of the first applications of the monopole invari-ants: if X is a compact Kahler manifold which is minimal (in the sense of algebraicgeometry) with c1(X )2 > 0 and b+(X ) > 1, then s0 and its conjugate are the onlybasic Spinc-structures, and it follows, for example, that any orientation-preservingself-diffeomorphism of X must preserve c1(X ) up to sign.

(iii) Further properties 5

(iii) Further properties

An important notion in 3-dimensional contact geometry is that of an overtwistedcontact structure [9]. A contact structure ξ on Y 3 is overtwisted if there is anembedded disk D in Y whose boundary is Legendrian (i.e. everywhere tangent toξ ) while D itself is transverse to ξ along the boundary. It is possible to show that if(X , ξ ) is a 4-manifold with an overtwisted contact structure on its boundary, thenthe invariant SW vanishes identically. In particular, by combining this statementwith Theorem 1.1, one recovers a result first proved by Eliashberg using differentmethods, namely the statement that overtwisted contact structures are not fillable[9]. The vanishing also echoes another result due to Eliashberg [6], which statesthat the classification of overtwisted contact structures is essentially soft: it is thesame as the homotopy classification of 2-plane fields on the 3-manifold.

The proof of the vanishing theorem for overtwisted structures is based on threeingredients: first, the results in [26] and [8], concerning exact symplectic structuresand Legendrian surgeries; second, the fact that the elements of Spinc (X , ξ ) withnon-zero invariant constrain the genus of closed embedded surfaces by an adjunc-tion inequality; and third, an excision property of our invariants. We hope to returnto the excision property and its corollaries in a future paper [16].

Acknowledgements. Much of this work was done during the Fall of 1995, whileEliashberg was visiting Harvard. The authors benefitted greatly from the lecturecourse he gave and from many discussions. In addition, the work of Lisca andMatic [18] provided a major impetus for this research.

2. Preliminaries and definition of the invariants

(i) The Seiberg-Witten equations

In this section we set our conventions for the Seiberg-Witten equations. Let X bean oriented Riemannian 4-manifold with a Spinc-structure s. We regard s as beingdefined by a triple (W+,W −, ρ ), where W ± are Hermitian 2-plane bundles andρ : T ∗X → Hom(W+,W −) satisfies the Clifford relation

ρ∗(e )ρ(e ) = |e|2IdW+ .

The map ρ extends to an injective linear map

ρ : Λ∗(T ∗X )→ End(W+ ⊕W −),

and our notation will not distinguish between these two usages. A connection Ain W + ⊕W − is called a spin connection if the induced connection in End(W + ⊕

6 2. Preliminaries and definition of the invariants

W −) agrees with the Levi-Civita connection on the image of ρ. The set of spinconnections is an affine space for the space of imaginary 1-forms: the action is givenby

(A, a) 7→ A + a IdW+⊕W− .

We will simply write A + a for the right-hand side. For a spin-connection A, letA denote the induced connection in det(W+). Notice that

FA+a = FA + 2 da.

A choice of spin connection A gives rise to Dirac operators

D±A : Γ(W ±)→ Γ(W∓).

With our conventions,

D±A+a = D±A + ρ(a)|W± .

Finally, the Seiberg-Witten equations, or monopole equations, for a pair (A,Φ)consisting of a spin connection A and a section Φ of W +, are the following:

ρ(F+A

)− Φ⊗Φ∗ = 0

D+AΦ = 0.

Here ρ(F+A ) is viewed as an element of isu(W +), as is Φ ⊗ Φ∗, and the bracesdenote the traceless part of the endomorphism.

(ii) Almost complex structures and 2-plane fields

There is a simple relationship between almost complex structures and Spinc-struc-tures in four dimensions.

Lemma 2.1. On an oriented Riemannian 4-manifold (X , g ), there is a one-to-one corre-spondence between

(a) compatible almost complex structures J ;

(b) self-dual 2-forms ω of length√

2; and

(c) isomorphism classes of pairs (s,Φ) consisting of a Spinc -structure and a unit-lengthsection Φ of the associated bundle W +.

(ii) Almost complex structures and 2-plane fields 7

Proof. The correspondence between ω and J in (a) and (b) is given by J θ = − ∗(ω ∧ θ ) where θ is a 1-form. Given a pair (s,Φ) as in (c), one obtains a complexstructure J from the multiplication by i in W − via the isomorphism T ∗X →W −

given by:

θ 7→ ρ(θ )Φ.

The corresponding form ω is the unique self-dual 2-form such that ρ(ω)(Φ) =−2iΦ. The reverse construction, obtaining the Spinc-structure s and section Φfrom J or ω, goes as follows. Put

W + = Λ0,0 ⊕ Λ0,2 and W − = Λ0,1.

Then define Clifford multiplication

ρ : T ∗X ⊗W + →W −

to be the symbol of√

2(∂ + ∂∗) : Ω0,0⊕Ω0,2→ Ω0,1. This defines a Spinc structure.The section Φ corresponds to the section 1 of Λ0,0.

Note that when J or ω is given, the above construction gives an actual Spinc

structure and spinor, rather than just an isomorphism class; we give these a name,and introduce also the preferred spin connection in this context (see [23]):

Definition 2.2. We write s0 and Φ0 for the Spinc-structure and spinor obtainedfrom J or ω by the above construction. We write A0 for the unique spin connectionfor s0 with the property that D+A0

Φ0 = 0. We call s0 and A0 the canonical Spinc-structure and spin connection.

The existence of a unique A0 as specified in this definition is straightforward.

On an oriented Riemannian 3-manifold Y , a Spinc-structure s is a pair (W , ρ )consisting of a Hermitian 2-plane bundle W and a map ρ : T ∗X → End(W ) sat-isfying the Clifford relation. The following lemma can be seen as arising from theprevious one, applied to translationally-invariant structures on the cylinder R×Y :

Lemma 2.3. On an oriented Riemannian 3-manifold Y , there is a one-to-one correspon-dence between

(a) oriented 2-plane fields ξ ;

(b) 1-forms θ of length 1; and

(c) isomorphism classes of pairs (s,Φ) consisting of a Spinc -structure and a unit-lengthspinor Φ.


Proof. Given an oriented 2-plane field ξ , there is a unique unit-length 1-form θ

which annihilates ξ and is positive on the positively-oriented normal field to ξ .This gives a bijection between (a) and (b). If a pair (s,Φ) is given, there is a unique1-form θ such that the +i and −i eigenspaces of ρ(θ ) are CΦ and Φ⊥ respectively.The pair (s,Φ) can be recovered from θ or ξ much as in the previous lemma.

(iii) Definition of the invariants

Let X be a compact, connected oriented 4-manifold with non-empty boundary∂X . The boundary may have more than one component. Let ξ be a contactstructure on ∂X , compatible with the boundary orientation. From the data (X , ξ ),and some additional choices, we can construct a complete Riemannian manifold(X+, g0), with a symplectic structure ω0 defined outside a compact set, as follows.As a manifold, we take X+ to be the union of X with a cylinder [1,∞) × ∂X ,identifying the boundary of X with the boundary of the cylinder 1 × ∂X . Now,if Y is any contact manifold, a standard construction gives [1,∞)×Y (or just R×Y )a symplectic structure. We will spell out our conventions for this construction,applied to ∂X .

We pick a 1-form θ on ∂X whose kernel is the 2-plane field ξ , compatiblewith the orientation of ξ , as in the proof of Lemma 2.3. We must choose also acomplex structure for ξ : an automorphism J of ξ with J 2 = −1, such that for anyvector e in ξ , the pair (e, Je ) is a positively oriented basis. There is now a uniqueRiemannian metric g1 on ∂X with the following three properties: the 1-form θ hasunit length; dθ is equal to 2(∗θ ); and the restriction of the metric to ξ is such thatJ is an isometry. This means that at each point there is a frame ei with dual frameei such that θ is e1, its exterior derivative dθ is 2e2 ∧ e3, and Je2 = e3.

Define a symplectic form ω0 on [1,∞)× ∂X by the formula

ω0 = 12d(t2θ )

= t dt ∧ θ + 12 t2 dθ, (1)

and a metric g0 by

g0 = dt2 + t2g1.

Then g is compatible with ω0, in that ω0 has length√

2 and ∗ω0 = ω0. These twogive X+ a metric and symplectic structure outside a compact set. Pick any smoothextension of g0 to all of X (also called g0).

On X+ \ X , Definition 2.2 provides a canonical Spinc structure s0, a spinor Φ0and a spin connection A0. As in the introduction, we write Spinc (X , ξ ) for the setof isomorphism classes of Spinc structures s on X+ equipped with an isomorphism

(iii) Definition of the invariants 9

s → s0 on X+ \ X . (Our notation will suppress the latter.) Pick an elements = (W +,W−, ρ ) of Spinc (X , ξ ), and for convenience choose an extension to allof X of the spinor Φ0 and the spin connection A0, keeping the same notation forthese extensions.

We will now introduce suitable function spaces on X+ to define a moduli spaceof pairs (A,Φ) which solve a particular deformation of the monopole equations andwhich are asymptotic to (A0,Φ0) on the ends of X+. If V → X+ is a hermitianvector bundle equipped with a unitary connection A, we write L2

k,A(V ), or simplyL2

k , for the completion of the space of smooth, compactly-supported sections of Vwith respect to the norm

‖s‖2L2k ,A=∫

X+

(|∇kAs|2 + · · · + |∇As|2 + |s|2). (2)

On X+, as on any complete Riemannian manifold with injectivity radius boundedbelow and bounded geometry, this coincides with the set of distributional sectionss whose first k covariant derivatives belong to L2 [3]. We shall also use the Sobolevembedding of L2

k in L4k−1. We pick an `, not less than 4 (so that L2

` is contained inC1), and define

# = (A,Φ)∣∣ (A − A0) ∈ L2

` and (Φ−Φ0) ∈ L2`,A0

, (3)

and

& = u : X+ → C∣∣ |u| = 1 and 1− u ∈ L2

`+1. (4)

We consider & acting on W + by multiplication. From the Sobolev multiplicationtheorems it follows that the set & is a Hilbert Lie group acting smoothly on #. (Notein passing that for any A with A − A0 ∈ L2

` , the L2k,A-norm is commensurate with

the L2k,A0

-norm for k ≤ `+1, also as a consequence of the multiplication theorems.)The action of & is free: if (A,Φ) is fixed by u, then u must be constant (to

preserve A) and must be 1 on the end of X+ (to preserve Φ where Φ is non-zero).In the next section we shall show that the space @ = #/& is a Hilbert manifold.Write [A,Φ] for the orbit of (A,Φ) under &.

For any η ∈ L2`−1(isu(W+)), let Mη, or Mη(s), denote the subset of @ consist-

ing of those [A,Φ] which satisfy the following modification of the Seiberg-Wittenequations:

ρ(F+A

)− Φ⊗Φ∗ = ρ(F+A0

)− Φ0 ⊗Φ∗0+ η, (5)

D+AΦ = 0. (6)


We find it convenient to restrict the perturbation term η to a suitable Banach spaceof decaying sections. We fix an ε0 > 0 and an extension t of the function t on[1,∞)× ∂X to all of X+. Then we define

1 = e−ε0 tCr(isu(W+)), (7)

for some fixed r ≥ `, and equip this space with the norm

‖η‖1 = ‖eε0 tη‖Cr .

We then introduce the space (s) defined as

(s) = ([A,Φ], η) ∈ @×1

∣∣ (5) and (6) hold.

Theorem 2.4. The space (s) is a Banach submanifold of @ × 1. The projectionπ2 : (s)→ 1 is a proper Fredholm map of index

d(s) = ⟨e(W+,Φ0), [X , ∂X ]

⟩,

and has orientable index bundle. An orientation of the index bundle can be specified in termsof a choice of homology orientation of (X , ξ ).

The formula for d(s) refers to the Euler class of the bundle W + on X , relative tothe distinguished section Φ0 on the boundary. (Note that on a closed 4-manifoldwith a Spinc structure, one has e(W +) = 1

4 (c1(W +)2 − 2χ − 3σ ), where χ and σ

are the Euler number and signature of the manifold.) The individual moduli spacesMη are the fibers π−1

2 (η) of π2. If η is a regular value of π2, the theorem impliesthat Mη is a smooth, compact manifold of dimension d(s). We can now define theinvariant SW discussed in the introduction. We shall see later that SW (s) does notdepend on the choices involved in constructing the metric g0 on X+.

Definition 2.5. Let (X , ξ ) be as above. The monopole invariants of (X , ξ ) constitutea map

SW : Spinc (X , ξ )→ Z,

defined as follows. If d(s) is non-zero, then we set SW (s) = 0. If d(s) is zero, thenSW (s) is defined to be the degree of the Fredholm map π2 of Theorem 2.4. Inother words, it is the number of points in the moduli space Mη = π−1

2 (η) for regularvalues of η, counted with sign.

Remark. In the case of a closed manifold, one can define an invariant SW (s) forSpinc structures s when the dimension of the moduli space, d(s), is even and pos-itive by the familiar device of evaluating µd/2 on Mη, where µ is a standard 2-dimensional cohomology class. In our case, however, the class µ is zero on @.

11

The proof of Theorem 2.4, with the exception of the orientability statement, iscontained in section 3 and is divided up as follows. The proof that π2 is Fredholmis given in subsection 3(ii), and the proof of properness appears in subsection 3(iii).These proofs are carried out in a slightly more general framework, described insubsection 3(i). Subsection 3(iv) contains the proof that the invariants of (X , ξ )are diffeomorphism invariants: they do not depend on the choices involved inconstructing g0 and ω0. The orientability of π2 and the definition of homologyorientation are treated in an appendix.

3. The moduli space

In this section we prove Theorem 2.4. The theorem is stated for a manifold X+

obtained from a manifold X with boundary by adding an open conical end, with asymplectic structure. The proofs do not depend on the rather special geometry ofthe end of X+, but work on any manifold which has a symplectic structure outsidea compact set and whose geometry is sufficiently mild at infinity. For the presenttherefore, we shall work in a more general context, and in the subsection below weintroduce a precise definition of a class of manifolds for which the arguments canbe carried through. We will have use for the extra generality in later sections.

(i) Symplectic structures outside a compact set

We consider a 4-manifold Z equipped with a symplectic structure defined on theend of Z , specified by giving a non-degenerate 2-form ω on the complement ofa compact set K . The manifold Z will have a complete Riemannian metric g,compatible with the symplectic structure on the end. The geometry of (Z , ω, g ) isrequired to be asymptotically euclidean in the following weak sense. (We use thenotation inj(x ) for the injectivity radius of Z about the point x ∈ Z , and expx forthe exponential map at x.)

Condition 3.1. There exists a proper function σ : Z → R+ with the followingproperties.

• The injectivity radius satisfies inj(x ) > σ (x ) for all x.

• For each x ∈ Z , let ex be the map ex : v 7→ expx(σ (x )v ) and let γx be metricon the unit ball in Tx X defined as e∗x ( g )/σ (x )2. Then these metrics havebounded geometry, in the sense that all covariant derivatives of the curvatureare bounded by constants independent of x.

• For each x ∈ Z \ K , let ox similarly be the symplectic form e∗x (ω)/σ (x )2 onthe unit ball. Then ox similarly approximates the translation-invariant form,along with all its derivatives.

12 3. The moduli space

• For all ε > 0, the function e−εσ is integrable on Z .

The data (ω, g ) satisfying Condition 3.1 defines an asymptotically flat almost Kahlerstructure (or structure) on the end of Z .

The definition of implies that all covariant derivatives of the curvatureapproach zero uniformly on the end of the manifold. The example we have inmind is the manifold X+ with conical end, described above. In this example, σ canbe taken to be the coordinate t. The definition also allows for non-trivial topologyof the end. For example, one may take the complex manifold obtained by blowingup C2 at the set of points (n2, 0)n∈N. A suitable Kahler form on this manifold,for which the areas of the exceptional curves go to infinity with n, defines an

structure.Note that if (Z , ω, g ) is and 8 is any neighborhood of the euclidean

metric on B1 in the C∞ topology, we are free to suppose that the metrics γxdefined above lie within 8 for all x: one only has to replace σ by σ/K and observehow the Ck norms scale.

The definitions made in section 2 for the case of X+ carry over in a straightfor-ward manner to an manifold (Z , ω, g ). The symplectic structure determinesa preferred Spinc-structure s0 outside the set K .

Definition 3.2. We define Spinc (Z , ω) to be the set of equivalences classes of pairsconsisting of a Spinc structure s on Z and an isomorphism between s and s0 onthe complement of some compact set K ′ ⊃ K .

Note that Spinc (Z , ω) is a principal homogeneous space for H2c (Z ;Z).

Let s = (W+,W−, ρ ) be an element of Spinc (Z , ω). Fix a spin connection A0and section Φ0 of W + on Z , agreeing with the standard ones outside K ′ containingK , as in the definition. The definition of # and & can be carried over to thissetting without change (see equations (3) and (4)), as can definitions leading to thestatement of Theorem 2.4 above. The space of perturbations 1 is taken to be

1 = e−ε0σCr(isu(W+)).

We shall prove this theorem in the context. After doing so, we can define themonopole invariants of an manifold (Z , ω, g ) as a map

SW : Spinc (Z , ω)→ Z,

just as in Definition 2.5. In this more general setting, it is not clear to what extentthe invariant depends on the compatible metric g on the end of Z . Proposition 3.26gives a preliminary result in that direction.

(ii) Fredholm theory 13

(ii) Fredholm theory

For any (A,Φ) in # we introduce the operators δ1 = δ(A,Φ)1 and δ2 = δ(A,Φ)

2 , whichdescribe the linearization of the action of & and the linearization of the Seiberg-Witten equations respectively. For k ≤ `, we write

δ1 : L2k+1(iR) −→ L2

k (iΛ1)⊕ L2k,A(W+ )

δ2 : L2k+1(iΛ1)⊕ L2

k+1,A(W+) −→ L2k (isu(W+))⊕ L2

k,A(W−),(8)

where

δ1( f ) = (−df , f Φ)

δ2(a, φ ) = (ρ(d+a)− Φφ∗ + φΦ∗, (D+Aφ + ρ(a)Φ)).

Our convention for the norm on isu(W+) is

|η|2 = 12

tr(η2).

With this convention the formal adjoints of these operators are

δ∗1(a, φ ) = −d∗a + iIm(Φ∗φ )

δ∗2(η, ψ ) = (2d∗ρ−1(η)+ 2iImρ−1(Φ∗ψ ),D−Aψ − ηΦ

).

Consider now the operator $ = δ∗1 + δ2 and its formal adjoint $∗. We regard $ asacting in the topologies

L2k+1(iΛ1)⊕ L2

k+1,A(W +) −→ L2k (iR)⊕ L2

k,A(isu(W+))⊕ L2k,A(W−), (9)

where it is continuous for k ≤ `, and similarly with $∗.

Theorem 3.3. For every (A,Φ) in #, the operators $ and $∗ are Fredholm in the abovetopologies, for k ≤ `. Furthermore

ind($) = −ind($∗) = ⟨e(W +,Φ0), [Z ,Z \ K ′]

⟩. (10)

It is a routine matter to deduce from this theorem that @ is a Hausdorff Hilbertmanifold whose tangent space at [A,Φ] is isomorphic to ker

((δ(A,Φ)

1 )∗), and that

the map π2 : (s)→ 1 of Theorem 2.4 is Fredholm and has the index stated. (Thefact that (s) is a Banach manifold is also standard: one can use the transversalityargument of [17].)

The proof of the Theorem 3.3 is broken up into three parts. First, followingTaubes [21] and Gromov-Lawson [13] we introduce a general class of operators forwhich we can prove the Fredholm property. Then we prove that the operators $and $∗ fall into this class. Finally we use excision to calculate the index. Noneof this uses the fact that the form ω defining the structure is closed. Afterproving this result, we will return to Theorem 2.4.


Fredholm theory for admissible operators

Let Z be a complete Riemannian manifold with injectivity radius bounded below.Let E, F → Z be Riemannian vector bundles equipped with metric-compatibleconnections and let

D : C∞(E )→ L2loc(F )

be a first-order elliptic operator given as

D = ρ ∇A + r,

where ρ : Λ1 ⊗ E → F and r : E → F are uniformly bounded bundle maps and ∇Ais a covariant derivative. We assume that an identity of the form∫

Z|Ds|2 =

∫Z

(|∇As|2 + |s|2 + 〈Rs, s〉) (11)

is satisfied, where R is a section of End(E ).

Definition 3.4. The operator D is called admissible if the following holds.

(a) There is a constant C > 0 so that for every compact set K ,∫K

∣∣〈Rs, s〉∣∣ ≤C‖s‖2L21,A(K ),

for all smooth, compactly-supported sections.

(b) For all ε > 0 there is a compact set K ⊂ Z so that∫Z\K

∣∣〈Rs, s〉∣∣ ≤ ε‖s‖2L21,A, (12)

for all smooth, compactly-supported sections.

If we consider D as an operator mapping compactly supported smooth sectionsto L2 sections, then D extends to three potentially different domains.

• The first is L21,A(E ), the completion of C∞0 (E ) with respect to the L2

1,A-norm.

• The second is the completion of C∞0 (E ) with respect to the norm

‖s‖2D =∫

Z

(|Ds|2 + |s|2).The resulting extension is often called the minimal extension. We will denotethis domain by Dommin or by Dommin(D) when the operator is important.


• The third domain is the set of all s ∈ L2(E ) with Ds ∈ L2(F ) as a distribution.This extension is often called the maximal extension. We will denote thedomain of this extension by Dommax or by Dommax(D) when the operatoris important. In other words, s ∈ L2 is in Dommax if the linear functional onL : C∞0 (F )→ R given by L(t ) = ∫Z (D∗t, s ) is bounded in the L2 topology.Then Ds is defined to be the unique t ′ ∈ L2(F ) so that L(t ) = ∫Z 〈t, t ′〉.Dommax is topologized using the same norm as Dommin.

In the language of [14], as unbounded operators, the adjoint of D : Dommin(D)→L2(F ) is D∗ : Dommax(D∗ )→ L2(E ). It follows that, on these spaces, D∗ is Fred-holm if D is, in which case ind(D∗) = −ind(D) (see [14], p.167).

Lemma 3.5. If D is an admissible operator then there is a compact set K ⊂ Z and aconstant C > 0 so that for all s ∈C∞0 (E ) we have∫

Z(|∇As|2 + |s|2) ≤C

(∫Z|Ds|2 +

∫K

(|∇As|2 + |s|2)).

Proof. Choose a compact set K so that the inequality (12) holds with ε = 12 . Then

by (11) we have

‖s‖2L21,A

=∫

Z|Ds|2 −

∫Z〈Rs, s〉

≤∫

Z|Ds|2 +

∫K

∣∣〈Rs, s〉∣∣+ ∫Z\K

∣∣〈Rs, s〉∣∣≤

∫Z|Ds|2 +C

∫K

(|∇As|2 + |s|2)+ 12‖s‖2L2

1,A.

Rearranging this inequality gives the desired result.

Proposition 3.6. For an admissible operator on a complete Riemannian manifold all threedomains coincide:

L21,A(E ) = Dommin(D) = Dommax(D).

Furthermore, if D∗ is also admissible, then the maps D and D∗ : L21,A→ L2 are Fredholm

and

ind(D) = −ind(D∗ ).

Proof. The proof follows the argument in [13]. From the definitions it follows that

L21,A ⊂ Dommin ⊂ Dommax.


From Lemma 3.5 it follows that L21,A ⊃Dommin. Since Dommin is a closed subset of

Dommax, the equality Dommin = Dommax will follow if we can show that Domminis dense in Dommax.

To do this, first construct a sequence of cut-off functions βm with the followingproperties:

• the sets β−1m (1) are all compact and their union is Z ;

• the derivatives |∇βm| are uniformly bounded by a constant b.

For example, if τ is a lower bound for the injectivity radius, one can take a uniformlylocally-finite collection of geodesic balls Bm of radius τ such that the half-size ballscover Z . Then one can take a standard radial bump-function ψm on each ball Bm,strictly positive on the half-size balls, and so obtain a partition of unity by functions

φn = ψn/(∑

m

ψm)

whose gradients are uniformly bounded. Then the cut-off functions βm can bedefined by

βm =m∑

n=1

φn . (13)

Now fix s ∈ Dommax. By local elliptic regularity, the section sm = βms is in L21

for all m > 0. Thus the sequence sm is contained in Dommin and converges in L2

to s. Now,∫Z|Ds −Dsm|2 =

∫Z|(1− βm )Ds − ρ(dβm )s|2

≤∫

Wm

(|1− βm|2|Ds|2 + 2bλ|Ds||s| + b2

λ2|s|2

),

where λ bounds the norm of ρ pointwise and Wm is the set on which βm 6= 1. Theright hand side converges to zero as m→∞, which finishes the proof of the densityassertion.

The finite-dimensionality of the kernel of D and the closed range propertyfollow from the next lemma.

Lemma 3.7. Let si ⊂ L21,A be a sequence with the following properties:

• ‖si‖L2 ≤ 1;

• limi→∞ ‖Dsi‖L2 = 0.


Then si has a strongly L21,A-convergent subsequence converging to s ∈ ker(D).

Proof. Local elliptic theory implies that we can find an L21,loc section s ∈ ker(D) and

an exhaustion of Z by a sequence Kα of compact sets so that after passing to asubsequence and relabeling, the si converge in the strong L2

1 topology on each Kα

to s. The estimate of Lemma 3.5 implies

‖si − s‖2L21,A≤C

(‖Dsi‖2L2 + ‖s − si‖L21,A(K )

).

The right-hand side of the this inequality approaches zero as i →∞.

It remains to prove the finite-dimensionality of the cokernel. By the closedrange property, an element of the cokernel of D is represented by t ∈ L2 withD∗t = 0 in the weak sense. Thus t ∈Dommax(D∗), by the definition of the maximaldomain. Since D∗ is admissible, we have Dommin(D∗ ) = Dommax(D∗), and thekernel of D∗ on these domains is finite dimensional. This finishes the proof.

The admissibility of $ and $∗

On R4 with its standard metric and symplectic form, the operator $ correspondingto the canonical solution (A0,Φ0) satisfies∫

R4|$(a, φ )|2 =

∫R4

(|∇a|2 + |∇A0φ|2 + |φ|2 + |a|2

)for compactly-supported sections. This operator is therefore admissible. The re-lation above depends on Clifford-algebra identities, the Weitzenbock formula andthe correct choice of norms, to ensure the cancellation of cross-terms. The nextproposition deals with the same calculation in the general case.

Proposition 3.8. For all compactly-supported (a, φ ) ∈ iΩ1 ⊕ Γ(W +) we have∫Z

∣∣$(a, φ )∣∣2 = ∫

Z

(|∇a|2 + |∇Aφ|2 + |Φ|2

(|φ|2 + |a|2)+Ric(a, a)+ s

4|φ|2 + 1

2

⟨ρ(F+

A)φ, φ

⟩− 2⟨a ⊗ φ,∇AΦ

⟩− ⟨φ, ρ(a)D+AΦ⟩).

(14)

Here Ric is the Ricci tensor of the Riemannian metric and s is the scalar curvature.

Proof. We will use the following identities in the calculation.

Lemma 3.9. Let f be an imaginary-valued function, a a section of iΛ1, ω a section ofΛ+, Φ and φ sections of W +, ψ a section of W − and γ a section of Λp. Then:


(a) tr(ρ(iω)2

) = 4|ω|2Λ+,

(b)∣∣Im(Φ∗φ )

∣∣2 + 12 tr(Φφ∗ + φΦ∗2) = |Φ|2|φ|2,

(c)∣∣ρ(a)Φ

∣∣2 = |Φ|2|a|2,

(d)⟨ρ( f )Φ, φ

⟩ = ⟨ f , iIm(Φ∗φ )⟩,

(e)⟨ρ(a)Φ, ψ

⟩ = ⟨a, 2iIm(ρ−1(Φ∗ ⊗ ψ ))⟩,

(f) 2⟨ρ(iω)Φ, φ

⟩= tr(ρ(iω)Φφ∗ + φΦ∗

),

(g) D+A(ρ(γ )Φ

) = ρ(dγ + d∗γ )Φ− 2ιγ∇AΦ+ (−1)pρ(γ )D+AΦ.

The second-to-last term in the last equation is defined via a local orthonormalframe e1, e2, e3, e4 by the equation:

ιγ∇AΦ =4∑

i=1

ρ(ιeiγ )∇Ai Φ ∈W ±

.

The verification of these identities is left to the reader. Given this result, we verify(14). The first equality below comes from the definition of $, the second comesfrom the identities above, and the last comes from integration by parts and theWeitzenbock formulae for the operators dd∗ ⊕ 2d∗d+ on Ω1 and D−AD+A on Γ(W+ ).∫

Z

∣∣$(a, φ )∣∣2

=∫

Z

(|d∗a|2 − 2

⟨d∗a, iIm(Φ∗φ )

⟩+ ∣∣Im(Φ∗φ )∣∣2

+∣∣ρ(d+a)

∣∣2 − 2⟨ρ(d+a), Φφ∗ + φΦ∗

⟩+ ∣∣Φφ∗ + φΦ∗∣∣2

+ |D+Aφ|2 + 2⟨D+Aφ, ρ(a)Φ

⟩+ ∣∣ρ(a)Φ∣∣2)

=∫

Z

(|d∗a|2 + 2|d+a|2 + |D+Aφ|2

+∣∣Im(Φ∗φ )

∣∣2 + ∣∣Φφ∗ + φΦ∗∣∣2 + 2

∣∣ρ(a)Φ∣∣2

− 2⟨ρ(d∗a)Φ, φ

⟩− 2⟨ρ(d+a)Φ, φ

⟩+ 2⟨φ,D+(ρ(a)Φ)

⟩)=∫

Z

(|∇a|2 +Ric(a, a)+ |∇Aφ|2

+ s4|φ|2 + 1

2

⟨ρ(F+A )φ, φ

⟩+ |Φ|2(|φ|2 + |a|2)− 2⟨ρ(d∗a)Φ, φ

⟩− 2⟨ρ(d+a)Φ, φ

⟩+ 2⟨φ, ρ(d+a + d∗a)Φ

⟩+ 2⟨a ⊗ φ,∇AΦ

⟩− ⟨φ, ρ(a)D+AΦ⟩).


The stated equality now follows.

We have a similar proposition for the formal adjoint:

Proposition 3.10. For all compactly supported triples

( f , η, ψ ) ∈ iΩ0 ⊕ Γ(isu(W+))⊕ Γ(W −)

we have∫Z

∣∣$∗( f , η, ψ )∣∣2

=∫

Z

(|d f |2 + 4

∣∣∇(ρ−1(η))∣∣2 + |∇Aψ |2 + |Φ|2

(| f |2 + |η|2 + |ψ |2)+ 4⟨5ρ−1(η), ρ−1(η)

⟩+ s4|ψ |2 + 1

2

⟨ρ(F−A )ψ, ψ

⟩+ 4⟨ψ, ιρ−1(η)∇AΦ

⟩+ 2⟨ψ, f D+AΦ

⟩+ 2⟨ψ, ηD+AΦ

⟩).

(15)

Here 5 is the endomorphism of Λ+ given by −0+ + s/6, where 0 is the Weyl curvature.

Proof. Using the Weitzenbock formulae for d+d∗ on Ω+ and D+AD−A on Γ(W −),we calculate:∫

Z

∣∣$∗( f , η, ψ )∣∣2

=∫

Z

(|d f |2 + 4

∣∣d∗ρ−1(η))∣∣2 + 4

∣∣Im(ρ−1(Φ∗ ⊗ ψ )∣∣2

− 2⟨d f , 2iIm(ρ−1(Φ∗ ⊗ ψ ))

⟩+ 4⟨d∗(ρ−1(η)), 2iIm(ρ−1(Φ∗ ⊗ ψ ))

⟩+ | f |2|Φ|2 + |D−Aψ |2 + |ηΦ|2

+ 2⟨D−Aψ, f Φ

⟩− 2⟨D−Aψ, ηΦ

⟩)=∫

Z

(|d f |2 + 4

∣∣∇ρ−1(η))∣∣2 + 4

⟨5ρ−1(η), ρ−1(η)

⟩+ |Φ|2|ψ |2− 2⟨d f , 2iIm(ρ−1(Φ∗ ⊗ ψ ))

⟩+ 4⟨d∗(ρ−1(η)), 2iIm(ρ−1(Φ∗ ⊗ ψ ))

⟩+ | f |2|Φ|2 + |∇Aψ |2 + s

4|ψ |2 + 1

2

⟨ρ(F−A )ψ, ψ

⟩+ |η|2|Φ|2+ 2⟨ψ, ρ(df )Φ+ f D+AΦ

⟩− 4⟨ψ, ρ(d∗ρ−1(η))Φ

⟩+ 4⟨ψ, ιρ−1(η)∇AΦ

⟩− 2⟨ψ, ηD+Φ

⟩).

The desired equality follows.


The admissibility of $ and $∗ follows from the two propositions. For the caseof (A0,Φ0), the relevant facts are that the pointwise norm of the Riemann curvatureof an manifold tends to zero on the end, as do FA0

and ∇A0Φ0. For a general(A,Φ) ∈ #, one uses in addition the fact that each of FA − FA0

, ∇AΦ−∇A0Φ0 and

1− |Φ|2 are in L2.

Calculation of the index

From the admissibility of $ and $∗ and Proposition 3.6, it follows that $ and$∗ are Fredholm in the topologies of Theorem 3.3 for k = 0. To calculate theindex we argue as follows. We consider the case where the Spinc-structure underconsideration is the canonical Spinc-structure for some almost complex structure onZ . In this case we will find a pair(A,Φ) ∈ # so that $ and $∗ have trivial kernel.The index is therefore zero, in agreement with the formula in Theorem 3.3, sincethe canonical section Φ0 on the end of Z extends to a nowhere-vanishing sectionon all of Z . The deduction of the general case from the almost complex case andthe compact case is a standard application of the excision property, and will beomitted.

Lemma 3.11. If (A,Φ) satisfy the pointwise estimate

|Φ(x )|2 ≥ 12

maxx∈Z

(|Ric(x )| + |5(x )| + 1

4|s(x )| + 1

2|FA(x )|

+ |∇AΦ(x )| + |DAΦ(x )|),

then $ and $∗ have trivial kernel.

Proof. Combining the given inequality with Propositions 3.8 and 3.10, we obtain∫Z|$(a, φ )|2 ≥

∫Z

(|∇a|2 + |∇Aφ|2 + 1

2|Φ|2(|a|2 + |φ|2)).

and ∫Z|$∗( f , η, ψ )|2

≥∫

Z

(|d f |2 + |∇ρ−1(η)|2 + |∇Aψ |2 + 1

2|Φ|2(| f |2 + |η|2 + |ψ |2)).

It follows immediately that ker($) and ker($∗) are trivial.

(iii) Compactness 21

Corollary 3.12. For the Spinc -structure arising from an almost complex structure J onZ, extending the almost complex structure defined on the end of Z by the AFAK structure,we have

ind($A,Φ) = 0

for all (A,Φ) ∈ #.

Proof. Let (A0,Φ0) be the extension to all of Z of the standard spin connection andunit-length spinor, as determined by J . Notice that if we scale the metric on Z bya factor λ, the L2

1 Sobolev spaces remain unchanged, and we obtain a continuousfamily $λ of Fredholm operators. While (A0,Φ0) may not satisfy the inequality ofLemma 3.11 for the original metric, the inequality holds once λ is sufficiently large.The index is therefore zero for (A0,Φ0), and hence for any other (A,Φ), since #is connected.

Proof of Theorem 3.3

The results so far establish Theorem 3.3 for the case k = 0. The cases 0 < k ≤ ` areroutine consequences of the following lemma.

Lemma 3.13. Let (A,Φ) ∈ # and write $ = $(A,Φ). Suppose (a, φ ) belongs toL2(iΛ1)⊕ L2(W+) and

$(a, φ ) ∈ L2k (iR)⊕ L2

k,A(isu(W +))⊕ L2k,A(W −)

for some k < `. Then

(a, φ ) ∈ L2k+1(iΛ1)⊕ L2

k+1,A(W +).

Proof. For smooth compactly supported sections it is straightforward to deduce aninequality of the following form:∫

X|∇k

A$(a, φ )|2 ≥∫

X|∇k+1

A (a, φ )|2 −C‖(a, φ )‖2L2k.

Using cut-off functions βm as in (13), one shows that if the integral on the left-handside is finite, then βm(a, φ ) is Cauchy in L2

k+1.

(iii) Compactness

We now turn to proving that the Fredholm map π2 defined in Theorem 2.4 isproper, so showing in particular that the moduli spaces Mη which we are studying


are compact. We continue to work in the more general context of a manifold Zwith an structure (ω, g ).

The argument has three steps. The first step (Lemma 3.14) is to derive a point-wise estimate on the norm of |Φ| in any solution (A,Φ). The second step is toprove that solutions of the equations decay exponentially on the end of Z , withconstants that may a priori depend on a certain ‘energy’ of the solution (Proposi-tion 3.15 and Corollary 3.16). The final step is to give a uniform bound on theenergy of any exponentially decaying solution (Lemma 3.17).

Lemma 3.14. There are constants κ1 and κ2, depending only on the AFAK manifold,such that for all [A,Φ] in the fiber Mη, we have

supx∈Z|Φ(x )|2 ≤ κ1‖η‖C0 + κ2.

Proof. Since Φ−Φ0 is in L2` and ` > 2, the pointwise norm |Φ−Φ0| tends to zero

on the end of Z . So if |Φ| is ever larger than 1, then the maximum is achieved atsome x ∈ Z . At this point, we have

0 ≤ 12

∆|Φ|2

= 〈∇∗A∇AΦ,Φ〉 − 〈∇AΦ,∇AΦ〉≤ 〈∇∗A∇AΦ,Φ〉= 〈D−AD+AΦ,Φ〉 − 1

2

⟨ρ(F+A )Φ,Φ

⟩− s4〈Φ,Φ〉

= −14|Φ|4 − 1

2

⟨(ρ(F+A0

)− Φ0Φ∗0)Φ,Φ

⟩− 12〈ηΦ,Φ〉 − s

4〈Φ,Φ〉.

Since FA0 , Φ0 and s are uniformly bounded on the manifold, the requiredestimate follows.

Outside a compact subset K ′ ⊃ K , the Spinc-structure s is identified with thecanonical one s0 with W + = Λ0,0 ⊕Λ0,2. Thus we can write a section Φ of W + asa pair (α, β ) ∈ Λ0,0 ⊕ Λ0,2; such a pair is mapped to a spinor by the formulae

α 7→ ρ(α)Φ0

β 7→ 2ρ(β )Φ0.

We write a spin connection A as A0 + a, and regard the 1-form a as a connectionin the trivial line bundle. Outside K ′, the vanishing of DAΦ becomes [23]

∂aα + ∂∗a β = 0. (16)


If we write 2η = ρ(−iη0ω)+ ρ(η1 − η1), where

(η0, η1) ∈ R⊕Λ0,2,

then the equation (5) is equivalent to

2iFωa −

14

(1− |α|2 + |β|2) = η0

2F0,2a − 1

2αβ = η1.

(17)

Here Fωa = 1

2〈Fa, ω〉.For any set U ⊂ Z \ K ′, we introduce the notation

EU (A,Φ) =∫

U

((1− |α|2 − |β|2)2 + |β|2 + |∇aα|2 + |∇aβ|2

),

where (a, α, β ) are the forms corresponding to (A,Φ) on Z \ K ′. Here ∇a isthe unique unitary connection in Λ0,2 whose (1, 0)-part is equal to ∂a under theidentification Λ1,0 ⊗ Λ0,2 = Λ1,2.

Proposition 3.15. For any constants E0 > 0 and e > 0, there exist constants ε and C(depending also on Z), such that if ‖η‖1 ≤ e and [A,Φ] ∈ Mη satisfies EZ\K ′ (A,Φ) ≤E0 then (

1− |α|2 − |β|2)2 + |β|2 + |∇aα|2 + |∇aβ|2 ≤Ce−εσ .

We postpone the proof of this Proposition to the end of the subsection.

Corollary 3.16. For any element of Mη, there exists a gauge representative (A,Φ) in #such that A − A0 and Φ−Φ0 decay exponentially along with their first derivatives.

Proof. Let (A,Φ) be any gauge representative. We can approximate (A,Φ) in theL2` topology by a smooth configuration (A1,Φ1) equal to (A0,Φ0) on the end of

Z . By the Fredholm theory of the previous subsection, we can then apply a gaugetransformation u1 ∈ & after which (A,Φ) satisfies the Coulomb condition

δ∗1(A − A1,Φ−Φ1) = 0,

where δ1 is the linearization of the gauge group action at (A1,Φ1). Because theperturbation term η belongs to e−ε0σCr ⊂ L2

r , the linear theory shows that, in thisgauge, (A − A0,Φ−Φ0) is in L2

`+1. (Recall that r was chosen to be at least `.)With (A,Φ) in this Coulomb gauge, let (a, α, β ) be the corresponding forms on

Z \K ′. We need to find a gauge in which a, 1− α and β decay exponentially, along


with their first derivatives. Proposition 3.15 shows that gauge-invariant quantitiessuch as 1− |α|2, |∇aα| etc. have exponential decay.

As α is non-vanishing outside some compact set K1, we can find a uniqueu : Z \ K1→ S1 such that

uα|α| = 1.

Because of the additional regularity of (A,Φ), this gauge transformation on Z \K1has 1− u ∈ L2

`+1, by Kato’s inequality and the Sobolev multiplication theorems. Inparticular, it is homotopic to 1, and can therefore be extended to all of Z . Thus weobtain u as an element of &, and the gauge-transformed solution (a− u−1du, uα, uβ )has the required decay, because uα = |α| and

a − u−1du = u(∇aα)/|α| − (∇|α|)/|α|.

The next Lemma provides the uniform bound on E, needed to obtain uniformexponential decay.

Lemma 3.17. There are constants κ3, κ4 depending only on the AFAK manifold, suchthat for all [A,Φ] ∈ Mη

EZ\K ′ (A,Φ) ≤ κ3‖η‖21 + κ4.

Proof. For any (A,Φ) ∈ # with Φ supported in Z \ K ′ and (A− A0,Φ−Φ0) decay-ing exponentially along with their first derivatives, we have the following identityfor the corresponding forms (a, α, β ) on Z \ K ′:∫

Z\K ′

(∣∣∂aα + ∂∗a β∣∣2 + 2

∣∣iFωa − 1

8

(1− |α|2 + |β|2)∣∣2 + 2

∣∣F0,2a − 1

4 αβ∣∣2

+ 12 iFω

a − 2|iFωa |2 − 2|F0,2

a |2)

=∫

Z\K ′

(12

(|∇aα|2 + |∇aβ|2 + 〈iFω

∇β, β〉)

+ 132

(1− |α|2 − |β|2)2 + 1

8 |β|2 − 2⟨N ∂aα, β

⟩).

(18)

Here N : Λ1,0 → Λ0,2 is formed from the Nijenhuis tensor of J . In the presentnotation, a proof of this identity can be found in the exposition [15] of the resultsof [25]. In our non-compact setting, the exponential decay is used to justify the


necessary integration by parts: we are using the fact that if ζ is a 3-form on Z andboth ζ and dζ are bounded by a multiple of e−εσ for some ε, then∫

Zdζ = 0. (19)

The proof of (19) uses the integrability of the function e−εσ (the fourth condition inthe definition (3.1) of AFAK) together with the existence of the cut-off functionsβm defined in the proof of Proposition 3.6.

Now choose compact subsets K ′ = K 1 ⊂ K 2 ⊂ K 3 with the property that|N | ≤ 1/32 and |Fω

∇ | ≤ 1/8 on Z \ K 2, so that |N∂aα| ≤ |∇aα|/32 for example.Choose a cut-off function c supported in Z \ K 2 and equal to 1 on Z \ K 3. Let[A,Φ] be in Mη and let (A,Φ) be a representative with exponential decay, as in theconclusion of Corollary 3.16. We can apply the above identity to the pair (A, Φ)where Φ = cΦ. Write (a, α, β ) for the corresponding forms. The bounds on N andFω

∇ allow us to use the Peter-Paul inequality to conclude that the right-hand sideof the identity above is an upper bound for∫

Z\K 3

(14 |∇aα|2 + 1

2 |∇aβ|2 + 132 (1− |α|2 − |β|2)2 + 1

16 |β|2).

On the other hand, since (A,Φ) satisfy the Seiberg-Witten equations (17) and (16),the integrand on the left-hand side is identically equal to

12

(|η0|2 + |η1|2)+ 1

2 iFωa − 2|iFω

a |2 − 2|F0,2a |2

on Z \ K 3 where c is 1. On the pre-compact set K 3 \ K ′, the contribution to theintegral is bounded by a function of ‖η‖C1 by Lemma 3.14 and elliptic regularity.We therefore have∫

Z\K 3

(14 |∇aα|2 + 1

2 |∇aβ|2 + 132 (1− |α|2 − |β|2)2 + 1

16 |β|2)

≤ κ(‖η‖2C1 + ‖η‖2L2)+ κ ′ + ∫

Z\K 3

12 iFω

a .

The integrand in the last term is i4da ∧ ω. Since a = A − A0 decays exponen-

tially, we can integrate by parts, using the fact that ω is closed, to obtain

i4

∫∂K 3

a ∧ ω.

This expression can be controlled using Lemma 3.14. After adjusting the constants,one obtains a bound for EZ\K 3 of the required form, and hence also a bound forEZ\K ′ .


We can now complete the proof that π2 : (s)→ 1 in Theorem 2.4 is proper.Suppose [Ai ,Φi] ∈ Mηi and ηi converges to η in the topology of 1. By Proposi-tion 3.15 and Lemma 3.17, there are constants C and ε, independent of i, such thaton Z \ K ′ we have

(1− |αi |2 − |βi |2)2 + |βi |2 + |∇aiαi |2 + |∇aiβi |2 ≤Ce−εσ . (20)

By local elliptic regularity, similar bounds apply to all higher covariant derivativesof α and β for suitable constants C , ε.

Lemma 3.18. There is a compact set K 4 ⊂ Z and gauge transformations ui ∈ & suchthat after passing to a subsequence, the transformed solutions ui (Ai ,Φi ) converge strongly onZ \ K 4 in the L2

` topology.

Proof. The uniform exponential decay (20) means in particular that αi is non-vanishing for all i outside some compact set K 4. As in the proof of Corollary 3.16,we can apply gauge transformations ui ∈ & so that on Z \ K 4 the correspondingtriples (ai , αi , βi ) satisfy

αi

|αi | = 1 on Z \ K 4.

As before, the restriction of the connection matrices ai to Z \ K 4 can now beexpressed in terms of gauge invariant quantities, and they therefore have uniformexponential decay in this new gauge. A diagonal argument and elliptic regularityallows us to pass to a subsequence which converges in the L2

` topology on compactsubsets of Z \ K4. The uniform exponential decay assures us that such a subse-quence converges strongly in the L2

` topology on Z \ K 4.

Lemma 3.19. For any compact 4-dimensional submanifold K 5 ⊂ Z with boundary, thereexist gauge transformations vi ∈ L2

`+1(K 5) such that after passing to a subsequence, thetransformed solutions vi (Ai ,Φi ) converge strongly on K 5.

Proof. This is proved in [17].

Choose K 4 and K 5 as in these two Lemmas with K 4 properly contained inK 5, and pass to a subsequence so that after applying gauge transformations ui onZ \ K 4 and vi on K 5 the convergence statements of each of the Lemmas holds.After passing to a further subsequence, the ratio wi = uiv−1

i converges strongly to agauge transformation w on K 5 \ K 4. Choose N sufficiently large that |wi − wN |is less than 1/2 for all i ≥ N , and let u′i = uiu

−1N and v′i = viv

−1N . For i large,


wi = wN exp(2π iθi ) for a unique function θi on the overlap satisfying |θi | < 1.Now define gauge transformations si on all of Z by

si =

v′i exp(2π iγ θi ) on K 5

u′i on Z \ K 5,

where γ is a function equal to 0 on K 4 and 1 on Z \ K 5. The sequence si(Ai ,Φi )then converges strongly on all of Z .

Proof of exponential decay

We now return to the proof of Proposition 3.15, which was postponed above. Thefirst two Lemmas establish that, under the hypotheses of the Proposition, we canfind a σ0 such that |α|2 ≥ 1/2 on the subset of Z where σ ≥ σ0.

Lemma 3.20. Let (a, α, β ) be a solution to the equations (17) and (16) on R4 for thestandard metric and symplectic form, with η = 0. Suppose that ER4 (a, α, β ) is finite. Thenthe solution is gauge-equivalent to the trivial solution with α = 1, β = 0 and a the productconnection.

Proof. The assumption that ER4 is finite implies that |α|(x ) tends to 1 and |β|(x )tends to 0 as x tends infinity. Indeed, if there is a divergent sequence of points xiwith, say, |β|(xi ) ≥ κ , then the restrictions of the solution to the balls B1(xi ) has aconvergent subsequence, because of the uniform bound on the energy. The limitsolution on the unit ball has |β|(0) ≥ κ and therefore has EB1(0) non-zero, fromwhich one sees that the the original solution has EU (a, α, β ) = ∞, where U is theunion of the balls B1(xi ).

On flat R4 with η = 0, if one applies the operator ∂∗a to the equation (16), oneobtains the relation

12

∆|β|2 = −14

(1+ |α|2 + |β|2)|β|2 − |∇aβ|2

≤ 0,

which now means that β is identically zero, from the maximum principle, since |β|tends to zero at infinity.

Equation (17) now says that F0,2a = 0, so a defines a holomorphic structure on

the trivial bundle; equation (16) says that α is a holomorphic section. Since |α|tends to 1 at infinity on R4, the section α is nowhere zero, by Hartog’s theorem. Ifwe write |α|2 = eh, then the equations reduce to the following equation for h (seeequation (23) below):

2∆h + eh − 1 = 0.


Since h approaches zero at infinity, the maximum principal shows that h is identi-cally zero. The solution is therefore trivial.

Lemma 3.21. There exists a constant σ0, depending only on Z, ‖η‖1 and E0, such if(A,Φ) is a solution on Z and the corresponding forms (a, α, β ) satisfy EZ\K ′ ≤ E0, then

|α|2(z ) ≥ 1/2

for all z with σ (z ) > σ0.

Proof. Let ηi be a sequence in 1 with bounded norms, and suppose there is a cor-responding sequence of solutions (Ai ,Φi ) and points zi ∈ Z approaching infinity,with |αi |2(zi ) < 1/2. Suppose also that the energy EZ\K ′ (ai , αi , βi ) is boundedby E0. The definition of means that the balls Bσ (zi )(zi ) have metrics andsymplectic forms which approach the standard structure on R4. By restricting thesolutions (Ai ,Φi ) to these balls and taking a suitable subsequence, we obtain a se-quence of configurations on an increasing sequence of balls Bσ (zi )(0) in R4 whichconverge (after gauge transformation) on compact sets to a configuration (A,Φ)which solves the equations with η = 0. This limiting solution has |α|2 ≤ 1/2 at theorigin, and has energy bounded by E0. This contradicts the previous Lemma.

Proposition 3.15 follows from Lemma 3.21, the definition of , and thefollowing Proposition:

Proposition 3.22.

(a) There exist positive constants K , ν with the following properties. Suppose that(a, α, β ) satisfy the equations on the ball Bσ (0) ⊂ R4 of radius σ ≥ 1, with thestandard metric and symplectic form, and η = 0. Suppose also that

|α|2 ≥ 1/2.

Then on the half-sized ball Bσ/2(0), each of∣∣1− |α|2∣∣, |β|2, |∇aα|2 and |∇aβ|2

are bounded above by the constant Ke−νσ .

(b) There is a neighborhood 8 of the standard metric and symplectic form on B1 in theC∞ topology with the following property. Suppose the metric g and symplectic formω on Bσ (0) are not standard, but that the euclidean coordinates are gaussian normalcoordinates and the rescaled forms (S∗σ g )/σ 2 and (S∗σω)/σ 2 on the unit ball B1 liein 8. (Here Sσ : B1 → Bσ is the dilation). Then the same conclusion holds, withan appropriate adjustment of the constants.


(c) Similarly, the same conclusion holds, though with different constants K and ν, if theright-hand side η is non-zero but satisfies ‖η‖Cl (Bσ ) ≤Ce−ε0σ .

Proof. This proposition can be deduced from differential inequalities for solutionsof the equations which are derived in [25], where the purpose was also to deriveexponential decay estimates, though in a slightly different context. We will onlytreat the case that η = 0; it will be clear that a non-zero η satisfying the stated boundwill not affect the argument.

First, in order to compare our calculation more closely with that in [25], wepull back by the dilation map to obtain a solution on a geodesic unit ball B1 of theequations

∂aα + ∂∗a β = 0

2iFωa −

r4

(1− |α|2 + |β|2) = 0

2F0,2a − r

2αβ = 0,

(21)

where r = σ 2.

Lemma 3.23. There exists a universal constant C such that if the metric on B1 is suffi-ciently close to the euclidean metric, any solution (a, α, β ) of the equations (21) on B1 forr ≥ 1 satisfies

|α|2 + |β|2 ≤C

on the smaller ball B0.95.

Proof. Let (a, α, β ) be a solution of the equations (21) on B1. From the equations,one obtains the differential inequality

12

∆(|α|2 + |β|2)+ r

4

(|α|2 + |β|2)(|α|2 + |β|2 − 1− z1/r) ≤ 0 (22)

for some z1 depending only on the metric (equation (2.2) of [25], or [17]). Thuswe may suppose that the function s = |α|2 + |β|2 satisfies

∆s + r2

s(s − 2) ≤ 0.

Suppose now that s1 is a positive function of the radial distance on the interior ofB1 satisfying

∆s1 +r2

s1(s1 − 2) ≥ 0


and s1 → +∞ at the boundary of B1. The existence of such a function is an el-ementary exercise, and there is even a function s1 which satisfies this inequalitysimultaneously for all metrics sufficiently close to euclidean. The maximum prin-cipal shows first that such a function must satisfy s1 ≥ 2 everywhere. A secondapplication shows that s − s1 can have no positive local maxima: at any local maxi-mum, one has s1(s1 − 2) ≥ s(s − 2), and combining this with the inequality s1 ≥ 2shows that s1 ≥ s there.

Thus s ≤ s1 on the interior of the ball, and the constant C can be taken to bethe supremum of s1 over the smaller ball.

The C0 bound on α and β gives bounds on all covariant derivatives of α, β andthe curvature, though the scaling means that these take the form

|Fa| ≤ r · const.

and so on. Further applications of the differential inequalities will be based on thefollowing Lemma:

Lemma 3.24. Let s be a solution of the differential inequality

12

∆s + rV s ≤ h,

s|∂ = t,

on Bρ , a geodesic ball of radius ρ ≤ 1, for some metric g. Suppose that V ≥ v0 > 0 andh ≥ 0. Then for any ρ ′ < ρ we have

supBρ′

s ≤(

supBρ

hrV

)+ (sup |t|)e−ν

√r,

where ν depends only on v0, ρ, ρ ′ and the metric g.

Proof. Let s0 be the solution to the first equation with h = 0 on the right-hand sideand the same boundary conditions. The maximum principle shows that rV (s −s0) ≤ h at the point where (s − s0) is maximum. We must therefore only show that

|s0| ≤ (sup |t|)e−ν√

r

on the ball Bρ ′. To obtain an upper bound, let T = sup t, and let

s1 =Tcosh cucosh cρ

,


where u is the radial distance function on B1 and c = (r v0/k )1/2. Then s1 ≥ s0 onthe boundary, and a short calculation shows that

12

∆s1 + rV s1 ≥ 0,

if k is sufficiently large, depending only on g. By the maximum principal, s0 ≤ s1.On the smaller ball, s1 satisfies a bound of the required form.

Now let (a, α, β ) again be a solution of the equations (21). Because of thedifferential inequality (22), we can apply Lemma 3.24 to the function

s = |α|2 + |β|2 − 1− z1

r

on the ball B0.95. Using the fact that |α|2 + |β|2 ≤C on the boundary, one sees that

|α|2 + |β|2 ≤ 1+ z2

r

on the ball B0.9, for some constant z2 which has absorbed the exponential termfrom Lemma 3.24. Similarly, one can imitate the proof of Proposition 2.3 of [25],using the lemma again in place of the un-adorned maximum principle, to obtainthe inequality

|β|2 ≤ z4

r

(1− |α|2 + 1

r2

)on B0.8, for some z4 depending only on g and ω.

The next step from [25] is to estimate |F−a |. (Note that |F+a | can be estimateddirectly from the equations.) With s = |F−a| and

q0 =r

4√

2

(1+ κ1

r

)(1− |α|2)− κ2r |β|2 + κ3,

the inequality (2.21) from [25] states that

12

∆(s − q0)+ r4|α|2(s − q0) ≤ |5|s,

if the constants κi are suitably chosen (they depend only on g and ω). Here |5| isan expression in the curvature of g. This inequality depends on the inequality for|β|2 which was just shown to hold on B0.8. Since |5| ≤ κ4 for some constant κ4,we have the inequality

12

∆(s − q0)+ r4

(|α|2 − 4κ4

r

)(s − q0) ≤ κ4q0.


The right-hand side is bounded by κ5r , for some κ5 which is independent of r andthe solution, so when r is large we can use Lemma 3.24 again to deduce that on theball B0.75, one has the estimate

|F−a | = s

≤ q0 + κ= r

4√

2

(1+ κ1

r

)(1− |α|2)− κ2r |β|2 + κ6.

(The constant κ above absorbed both terms on the right-hand side of the conclusionof the Lemma). One can now follow the same line as Proposition 2.8 of [25] todeduce estimates on the covariant derivatives of α and β: on the ball B0.7, one has

|∇aα|2 ≤ z5r(

1− |α|2 + 1r

),

|∇aβ|2 ≤ z6

(1− |α|2 + 1

r

).

Consider next the function

h = log |α|2.In the case of a holomorphic section α of a line bundle with unitary connection ahaving F0,2

a = 0, one has the standard identity

12

∆h = −iΛ∂∂ log |α|2

= −iΛFa

= −2iFωa .

In the non-integrable case, when α is not holomorphic, one has

12

∆h = −2iFωa +

2<(α∂∗a ∂aα)

|α|2 − 2<( iΛ(∂aα ∧ ∂aα)

α2

).

We now use the equations (21) to replace ∂aα with −∂∗a β, and use the relation

∂a ∂a = F0,2a + N∂a,

where N : Λ1,0 → Λ0,2 is the Nijenhuis tensor. After using (21) to express thecurvature terms in terms of α and β, the result is

12

∆h = r4

(1− |α|2 − |β|2)− 2<

( α∂∗a N ∗β|α|2

)+ 2<

( iΛ(∂∗a β ∧ ∂aα)α2

). (23)

(iv) Varying the structure 33

Now write Vh = (eh − 1)/h and use the previous estimates to bound |β|2, ∂aα andthe terms involving the derivatives of β. The result is an inequality

12

∆h + r4Vhh ≥ −z7

√r − z8

on B0.7. Using the lemma once more (with the opposite sign), one obtains a lowerbound for h of the form zr−1/2 and hence a bound

1− |α|2 ≤ z9√r

(24)

on B0.6.We now return to the line of [25], where equation (4.15) provides the differen-

tial inequality

12

∆y1 +r4|α|2y1 ≤

(z10r

(1− |α|2)+ r

8

)y1

for the quantity

y1 = |∇aα|2 + r32|∇aβ|2 + r2

z11|β|2.

The inequality (24) and the positivity of y1 means that we can rearrange this toobtain

12

∆y1 +r

32y1 ≤ 0

once r is sufficiently large, and so the lemma gives the required exponential boundy1 ≤ Ke−ν

√r on the ball B0.55. A similar bound for 1 − |α|2, on the ball B0.5,

follows from the formula for ∆h above.

(iv) Varying the AFAK structure

Having now proved Theorem 2.4 in the context, we have an invariant of

manifolds (Z , ω, g ), taking the form of a map SW : Spinc (Z , ω)→ Z. In section2, we showed how to pass from (X , ξ ), a 4-manifold with contact boundary, to an manifold X+. This passage involved a choice of suitable metric, and to justifyDefinition 2.5 as defining an invariant of pairs (X , ξ ), we have to show that theinvariant SW is independent of the choice made.

Rather than prove a result which is specific to the situation of X+, we remainwith the setting, and consider a family (Zt, ωt, gt ) of manifolds, t ∈[0, 1]. We do not aim for the greatest generality (see [16]), but restrict ourselves tothe following situation.


Condition 3.25. We consider a family (Zt, ωt , gt ) with the following properties:

• the underlying manifold Z = Zt is independent of t;

• there exists a proper function σ on Z which is independent of t, such thatthe conditions of (3.1) hold for all t;

• the ratios gs g−1t of the metrics are uniformly bounded on Z ;

• both gt and ωt vary continuously with t in the topology of C∞(Z ), as definedusing the metric g0.

Consider now the canonical Spinc structure on the end of Z which correspondsto ωt , and let s ∈ Spinc (Z , ωt ) be any extension of the canonical Spinc structure tothe whole of Z . Let W ±(t ) be the corresponding spin bundles. The definitionof the canonical Spinc structure exhibits W ±(t ) as a t-dependent subbundle of thebundle of differential forms, Λ∗(Z ). We can fix an isomorphism W ±(t )→W ±(0),by using the parallel transport of the connection defined by g0-orthogonal projec-tion in Λ∗(Z ). The above conditions then ensure that, under this identification,the space L2

`,A0(t )(W+) is independent of t, as are the spaces L2

` (Λ1) and the space

L2`+1(C). The affine spaces

#t =(A0(t ),Φ0(t )

)+ (L2` (Λ

1)⊕ L2`,A0(t )(W

+))

therefore form the fibers of a trivial Hilbert vector-bundle

#∗ =⋃t

#t × t

over the interval [0, 1], and this bundle is acted on by the gauge group &, whichis independent of t. Let @∗ be the quotient space, #∗/&. Fix an exponentiallydecaying form η, and consider the moduli spaces Mη,t on Zt . Their union forms asubset Mη,∗ of @∗.

It is a routine matter to follow through the estimates involved in the Fredholmtheory, to show that under the conditions (3.25), the space @∗ is a Hilbert manifold-with-boundary, and the subset Mη,∗ is a smooth, compact submanifold for genericη giving an oriented cobordism between Mη,0 and Mη,1. We have therefore proved:

Proposition 3.26. If (Z , ωt, gt ) is a family of AFAK manifolds satisfying Conditions3.25, then their monopole invariants,

SW : Spinc (Z , ω)→ Z,

are independent of t.

In particular, for compact oriented 4-manifolds (X , ξ ) with contact boundary,our invariant is a diffeomorphism invariant.

35

4. Symplectic manifolds

(i) Patching symplectic forms

In this section we prove Theorems 1.1 and 1.2. We suppose (X , ξ ) is a 4-manifoldwith contact boundary, and that ω is a symplectic form on X , compatible withξ . Let (X+, ω0, g0) be the complete manifold obtained by adding the cone[1,∞)× ∂X to X , as described in section 2(iii). The manifold X+ now has a sym-plectic form ω on the compact submanifold X , and a form ω0 on the complementof X ; but the compatibility condition between ω and ξ does not ensure that the twoforms match on ∂X . Indeed, ω|∂X may not even be exact, and there is thereforeno reason to suppose that there would be a symplectic form on X+ agreeing withω on X and equal to ω0 outside some larger compact set. We do, however, havethe following:

Lemma 4.1. Let U be a collar-neighborhood of ∂X in X. Then there exists a familyof AFAK structures (X+, ωt, gt ), coinciding with the original structure at t = 0 and havingthe property that ω1 extends to a symplectic form on all of X+, agreeing with ω on X \U.Furthermore, the family can be chosen so as to satisfy the conditions (3.25).

Proof. We start by constructing the symplectic form ω1. Let U1 be a collar neigh-borhood of ∂X in X , and choose an identification of U1 with [0, 1]× ∂X . Writes for the first coordinate on U1, so that ∂X = s−1(1).

Pick a 1-form θ on ∂X whose kernel is ξ , and pull it back to a form on U1,also called θ . The compatibility condition between ω and ξ means that ds ∧ θ ∧ ωis a positive 4-form at ∂X , and by making U1 small enough, we can arrange thatthis form is positive on the collar.

Let X denote the interior of X and let U 1 =U1 ∩X . Let f (s ) be any functionof s ∈ [0, 1) which is increasing, is identically zero on [0, 1/3) and tends to infinityas s tends to 1. The 1-form f (s )θ on U 1 can then be extended by zero to all of X .We set

ω = ω + d( f θ )

on X , and we then have

ω2 = ν1 f f + ν2 f + ν3 f + ν4,

where

ν1 = 2ds ∧ θ ∧ dθ

ν2 = 2ds ∧ θ ∧ ων3 = 2dθ ∧ ων4 = ω2

.

36 4. Symplectic manifolds

Each of the 4-forms above is strictly positive on U1, with the possible exception ofν3. Let a be the maximum value of the function |ν3/ν1|, and let b be the minimumvalue of |ν4/ν3|. The form ω is non-degenerate at points where either f is less thanb or f is greater than a. There is no difficulty in choosing f so that f is greaterthan a at all points where f > b, so that ω is a symplectic form on X .

Let U 2 ⊂ U 1 be the set where s ≥ 1/3, and let U 3 ⊂ U 2 be the set wheref (s ) ≥ 1/2. Let ψ : X → X+ be a diffeomorphism which is the identity onX \U 2 , and maps the collar U 3 to [1,∞)× ∂X by the map

(s, x ) 7→ ((2 f (s ))1/2, x

).

The latter condition means that the pull-back ψ∗(ω0) is equal to d( f θ ) on U 3 .Finally, let ω1 be the push-forward ψ∗(ω) on X+. This is a symplectic form on allof X+ which agrees with the original ω on X \U2 and is asymptotic to ω0 on theend of X+, in that the pointwise norm of the difference,

|ω1 − ω0|g0 = |ψ∗(ω)|g0,

decays like a multiple of t−2, where t is the first coordinate on [1,∞)× ∂X .On the end of X+, we can interpolate linearly between ω0 and ω1 to obtain a

family of forms which will be non-degenerate outside some fixed compact set K .With a suitable choice of compatible Riemannian metrics, these define a family of structures satisfying the conditions of (3.25).

(ii) Proof of Theorems 1.1 and 1.2

We shall now prove a version Theorems 1.1 and 1.2 in the context. Theoriginal theorems follow immediately from this one, in view of the construction ofthe previous subsection and Proposition 3.26.

Theorem 4.2. Let (Z , ω, g ) be an AFAK manifold whose symplectic structure ω extendsto all of Z, let s0 ∈ Spinc (Z , ω) be the canonical Spinc structure determined by ω, and lets = s0 + e be any element. Then if SW (s) 6= 0 then

e^ [ω] ≥ 0

in H4c (Z ), with equality only if s = s0. With the standard homology orientation,

SW (s0) = 1.

Proof. We follow [24]. Let (A0,Φ0) be the canonical spin connection and spinorfor the Spinc structure s0, on all of Z . Let E → Z be a line bundle equipped witha trivialization outside a compact set, having first Chern class e ∈ H2

c (Z ), so that

(ii) Proof of Theorems 1.1 and 1.2 37

the spin bundles for s and s0 are related by W + =W +0 ⊗ E. An element (A,Φ) of

#(Z , s) can be written in terms of a triple (a, α, β ), where a is a connection in E,α ∈ Ω0,0(E ) and β ∈ Ω0,2(E ). The relationship between these two forms is givenby the same formulae as in section 3(iii), and the equations take the form of (16)and (17). We consider the equations with η = 0.

Following [25], we deform the equation (17), introducing a parameter r ≥ 1 asfollows:

2iFωa −

r4

(1− |α|2 + |β|2) = 0

2F0,2a − r

2αβ = 0.

(25)

The invariant SW (s) can be calculated by counting solutions to the deformedequation, for any r . The identity (18) has the following modification, appropriateto the deformed equations:∫

Z

(∣∣∂aα + ∂∗a β∣∣2 + 2

∣∣iFωa − r

8

(1− |α|2 + |β|2)∣∣2 + 2

∣∣F0,2a − r

4 αβ∣∣2

+ r2 iFω

a − 2|iFωa |2 − 2|F0,2

a |2)

=∫

Z

(12

(|∇aα|2 + |∇aβ|2 + 〈iFω

∇β, β〉)

+ r2

32

(1− |α|2 − |β|2)2 + r2

8 |β|2 − 2⟨N ∂aα, β

⟩).

Note that the integral is now over all of Z . For r sufficiently large, we can applythe Peter-Paul inequality to the right-hand side to see that, for a solution of theequations, we have∫

Z

r2 iFω

a ≥∫

Z

(14 |∇aα|2 + 1

2 |∇aβ|2 + r2

32 (1− |α|2 − |β|2)2 + r2

16 |β|2).

As in the proof of Lemma 3.17, a gauge transformation can be chosen so that, inthe trivialization of E on the end of Z , the connection form a decays like e−cσ forsome c > 0. This allows us to conclude that the left-hand side of the inequalityabove computes the pairing πe^ [ω].

We see immediately that there are no solutions if the pairing is negative. If thepairing is zero, then β = 0 and α is a covariant-constant section of length 1, so thebundle E is trivial and e = 0. It then follows that the solution is gauge-equivalentto the standard solution (A0,Φ0). One can show that the standard solution is non-degenerate for r sufficiently large by imitating the proof of Lemma 3.11. Fromthe definition of the canonical homology orientation in the symplectic or almostcomplex case, as given in the Appendix, the standard solution contributes+1 to theinvariant.

38 5. The finiteness theorem

5. The finiteness theorem

In this section we will prove the finiteness result, Theorem 1.3. Some of the con-structions in the proof can be seen as first steps in defining an element of monopoleFloer homology from a contact structure, i.e. recasting our invariants as a map

SW : contact structures on Y → HF(Y ).

The full construction of the monopole Floer homology would be a rather long andtechnical story and is not directly relevant to proving the theorem. We thereforecontent ourselves with developing the minimum of machinery, at the expense ofmaking our arguments seem a bit more ad hoc.

(i) Homotopy classification of two-plane fields

The homotopy classification of oriented 2-plane fields on a 3-manifold goes backto Pontryagin. We recast the result in terms of Spinc-structures. Let Y be a closedoriented 3-manifold and let Ξ denote the space of oriented 2-plane fields on Y .The construction of Lemma 2.3 gives a map

p : π0(Ξ)→ Spinc (Y ),

to the set of Spinc-structures on Y . In addition, it identifies the fiber of p overt = (W , ρ ) with the set of homotopy classes of non-vanishing sections ofW modulothe action of automorphisms of t. (In this section, we use t to denote a typical3-dimensional Spinc-structure, and continue to use s for a Spinc-structure on a 4-manifold.) If Φ0 and Φ1 are two non-vanishing sections of W , there is a differenceelement δ(Φ0,Φ1) ∈ Z given by a relative Euler class:

δ(Φ0,Φ1) = e([0, 1]×W ;Φ0,Φ1

)[[0, 1]×Y, ∂

]. (26)

The difference element determines the homotopy class of Φ1 when Φ0 is given.For an automorphism of t presented as a map u : Y → S1, we have

δ(Φ0, uΦ1) = δ(Φ0,Φ1)+ ([u]^ c1(W ))[Y ],

where [u] denotes the element of H1(Y ) determined by u. It is therefore appropri-ate to introduce the following definition.

Definition 5.1. If c1(W ) is not zero or a torsion element, let div(t) denote thedivisibility of c1(W ) in H2(Y )/torsion. Otherwise define div(t) to be zero.

(ii) The equations on a cylinder 39

Thus δ descends to a difference element

δ(ξ0, ξ1) ∈ Z/div(t)Z

for pairs of elements in p−1(t) ⊂ π0(Ξ). Again, the difference element determinesthe homotopy class of ξ1 when ξ0 is given. We have thus proved:

Proposition 5.2. Given ξ0 ∈ p−1(t), the map which sends ξ ∈ p−1(t) to δ(ξ0, ξ ) iden-tifies this fiber with Z/div(t)Z.

(ii) The equations on a cylinder

Let Y be a closed, oriented 3-manifold with Riemannian metric. Fix t ∈ Spinc (Y )and consider the unperturbed monopole equations on the cylinder R×Y with thecorresponding 4-dimensional spin-structure:

ρ(F+A

)− Φ⊗Φ∗ = 0

D+AΦ = 0.

If A is in temporal gauge and we identifyW + on the 4-manifold with the pull-backof the spin bundle W , the equations can be interpreted [17] as the downward gra-dient flow equations for the following functional (the Chern-Simons-Dirac func-tional):

(B,Ψ) = −12

∫Y

(B − B0) ∧ (FB + FB0 )−∫

Y〈Ψ,DBΨ〉.

This is a function on the space #(Y, t) consisting of pairs (B,Ψ) where B is aspin-connection on Y and Ψ is a section of W . A base-point B0 in the set of spin-connections has been chosen, and DB is the 3-dimensional Dirac operator. If µ isa real 2-form on Y , we introduce also the functional

µ(B,Ψ) = −12

∫Y

(B − B0) ∧ (FB + FB0 + 2iµ)−∫

Y〈Ψ,DBΨ〉.

This function is invariant under the identity component of the gauge group if µ isclosed: in this case, if u is any gauge transformation, one has

µ(B − u−1du, uΨ)− µ(B,Ψ) = [u]^(−4π2c1(W )+ 2π [µ]

). (27)

We will always take µ to be closed. The downward gradient-flow equation for theperturbed functional are equivalent to the following equations on the cylinder:

ρ(F+A+ iµ+)− Φ⊗Φ∗ = 0

D+AΦ = 0.(28)


The set of critical points of µ is invariant under the whole gauge group.Critical points are solutions of the equations

ρ(FB + iµ)− 12

Ψ⊗Ψ∗ = 0

DBΨ = 0.

We write Nµ(Y, t) for the set of gauge-equivalence classes of solutions to theseequations on Y . We call a solution (B,Ψ) reducible if Ψ = 0 and non-degenerate ifthe kernel of the Hessian is equal to the tangent space to the gauge group orbitthrough (B,Ψ). Note that if Y is a rational homology sphere then there is a uniquereducible solution θ for each Spinc-structure t.

Proposition 5.3.

(a) For any closed µ, the space Nµ(Y, t) is compact, and is empty for all but finitelymany t ∈ Spinc (Y ).

(b) If µ0 is a given closed form which does not represent 2π c1(W ), then there is a Baire setof exact Cr forms µ1 such that Nµ0+µ1 (Y, t) consists of finitely many non-degenerate,irreducible critical points.

Proof. The first claim follows from the usual C0 bound on |Ψ| and the resultingC0 bound on the curvature of a solution. The non-degeneracy of the irreduciblesolutions for generic choice of exact perturbation µ1 is Proposition 3 of [11]. Theonly remaining point is to note that the equations have no reducible solutions unlessµ represents 2π c1(W ).

For any two non-degenerate solutions α, β ∈ Nµ, let Mµ(α, β ) denote the mod-uli space of solutions to (28) on R ×Y which are asymptotic to α at −∞ and β at∞. An element of Mµ(α, β ) determines a path γ in #(Y, t)/&. Let γ be a lift ofγ to #(Y, t). When α and β are irreducible, we let i(γ ) denote the spectral flowof the Hessian of µ along γ . Thus i(γ ) is equal to the formal dimension of thecomponent of Mµ(α, β ) containing the given solution.

Now suppose γ1, γ2 are two paths between α and β, and let γ1, γ2 be liftsso that γ1(−∞) = γ2(−∞). There is a unique gauge transformation u such thatuγ1(∞) = γ2(∞), and we have

i(γ1)− i(γ2) = ([u]^ c1(W ))[Y ].

Thus we can define a difference element

i(α, β ) ∈ Z/div(t)Z,

depending only on α and β, independent of the chosen path.

(iii) Manifolds with cylindrical ends 41

(iii) Manifolds with cylindrical ends

Let X be an oriented Riemannian 4-manifold with cylindrical end modeled onR+ ×Y . Fix a Spinc-structure s on X , let t be the induced Spinc-structure onY , and let µ be a closed 2-form on Y for which Nµ(Y, t) is a finite set of non-degenerate, irreducible solutions. (Such a choice is possible if b1(Y ) is nonzero, byProposition 5.3.) For each α ∈ Nµ(Y, t), we define a moduli space of solutions tothe monopole equations as follows. Let Γ = (Aα,Φα ) be a spin connection andspinor on X which agree with the pull-back of some gauge representative of αon the end of X . Let ε be any positive quantity smaller than the least positiveeigenvalue of the Hessian of at α, and define

#(Γ) = (A,Φ)∣∣ (A − Aα ) ∈ e−ετL2

` and (Φ−Φα ) ∈ e−ετL2`,Aα

, (29)

and

& = u : X → C∣∣ |u| = 1 and 1− u ∈ e−ετL2

`+1. (30)

Here τ : X → R is any function agreeing with the coordinate t on the cylinderR+ ×Y . Fix r > ` and choose a section η of isu(W+) such that η+ ρ(icµ) ∈ e−ετCr ,where c : X → R+ is a cutoff function supported in the cylinder and equal to 1 onthe end. Define Mη(Γ) to be the set of &-equivalence classes of pairs (A,Φ) ∈ #(Γ)solving the perturbed monopole equation:

ρ(F+A

)− Φ⊗Φ∗ = η,D+AΦ = 0.

(31)

If Γ1, Γ2 are two configurations as above, there is a gauge transformation u,defined on the end of the manifold and pulled back from Y there, such that u(Γ1) =Γ2 on the end. If u extends to all of X , then Mη(Γ1) = Mη(Γ2) and we sayΓ1 and Γ2 are equivalent; the obstruction to such an extension is an element ofH1(Y )/ i∗(H1(X )). We define Mη(X , α) to be the union of the spaces Mη(Γ) as Γruns through a set of representatives for the equivalence classes.

Let i(Γ) be the index of the linearization of these equations on the given func-tion spaces, with appropriate gauge-fixing. If Γ1, Γ2 differ by a gauge transforma-tion u : Y → S1 on the end of X , then

i(Γ1)− i(Γ2) = ([u]^ c1(W ))[Y ].

We define

i(X , s, α) ∈ Z/div(t)Z

to be the residue class of i(Γ).


Proposition 5.4. If i(Γ) < 0, then there is a Baire set of perturbations η such that themoduli space Mη(Γ) is empty.

Proof. The required Fredholm theory for manifolds with cylindrical ends is stan-dard, see [19, 1, 22]. Given this, the conclusions follow from transversality resultsdeduced exactly as in the compact case, see [17]. Note that there can be no re-ducible solutions because of our choice of µ.

Proposition 5.5. Let (A,Φ) be any solution to the equations (31) with finite variationof µ on the cylindrical end. Then (A,Φ) is gauge equivalent to an element of some#(Γ) and hence represents an element of Mη(X , α), for some α ∈ Nµ(Y, t).

Proof. Given the non-degeneracy of the critical point set of µ, this followsfrom standard exponential decay results for perturbations of gradient flow equa-tions, see [20].

(iv) Index of critical points and two-plane fields

Let Y be as above, t a Spinc-structure, and suppose µ is again chosen so that Nµ(Y, t)consists of non-degenerate, irreducible solutions. By comparing the index theory inthe cylindrical-end case with the case of manifolds with contact boundary, we candefine an index, I (α), for elements α ∈ Nµ(Y, t) taking values in π0(Ξ) as follows.Choose X with boundary Y so that t extends to a Spinc-structure s. Choose a unit-length section Φ0 of W +|Y so that the relative Euler class satisfies the congruence

e(W +;Φ0) ≡ i(X , s, α) (mod div(t)).

Then define I (α) = [ξ ] where ξ is the two-plane field defined by Φ0. It is a straight-forward matter to check that the required X always exists and that the definition isindependent of the choice of X and Φ0 satisfying the given congruence. Noticethat if t is the Spinc structure determined by a contact structure ξ and s ∈ Spinc (X , ξ ),then

dim Mη(X , ξ , s) ≡ i(X , s, α) (mod div(t)).

In other words, I is defined so that the equality I (α) = [ξ ] means that the dimensionof the cylindrical-end moduli space M (X , s, α) is the same as that of the

moduli space M (X , ξ , s), modulo div(t).

(v) Proof of Theorem 1.3 43

(v) Proof of Theorem 1.3

Let (X , ξ ) be a 4-manifold with contact boundary, and assume at present that theboundary Y is connected. Fix s in Spinc (X , ξ ), and let µ be a closed 2-form on Y .Let t again be the Spinc-structure on Y determined by ξ .

Proposition 5.6. Suppose the following three conditions hold:

• the invariant SW (s) is non-zero;

• for some metric gY , the critical point set Nµ(Y, t) consists of finitely many non-degenerate, irreducible solutions;

• the cohomology class [µ] extends to X.

Under the above assumptions we have the following.

(a) The critical-point set Nµ(Y, t) is non-empty.

(b) In addition, if div(t) = 0 so that the difference-element δ on two-plane fields takesvalues in Z, then there exist α and β in Nµ(Y, t) with

δ(I (β ), [ξ ]

) ≥ 0 and δ([ξ ], I (α)

) ≥ 0.

Proof. Let X+ be the manifold with conical end constructed from X . Let g1be a Riemannian metric on X+ which agrees with the structure on the conicalend and which contains an isometric copy of [−1, 1]×Y with the metric dt2 + gY .Choose a perturbing term η1 ∈ 1 which is equal to ρ(−iµ) on this cylinder. Forall R > 1, let gR and ηR be obtained by replacing the cylinder [−1, 1] ×Y by acylinder isometric to [−R,R]×Y . The complement of the interior of the cylinderconsists of two components, which we denote by X in (the compact piece) and Xout(the non-compact piece).

The hypothesis SW (s) 6= 0 means that, for all R, the moduli space MηR ( gR )is non-empty. Let [AR,ΦR] belong to this moduli space, and let γR : [−R,R]→#(Y, t)/&Y be the path determined by the restriction of [AR,ΦR] to the cylinder.

Lemma 5.7. The variation of µ along the path γR is bounded, independent of R.

Proof. Let γ (t ) = (B(t ),Ψ(t )) be a lift of γR to #(Y, t). As in [17], the uniformbounds on |ΦR| mean that there are gauge transformations uin and uout defined onX in and Xout such that µ(uout(γ (R ))) and µ(uin(γ (−R ))) are both boundedby a constant independent of R. Furthermore, uout can be chosen so that 1 −


uout belongs to L2`+1; in particular, its restriction to R ×Y lies in the identity

component, so that

µ(uout(γ (R ))

) = µ(γ (R )

).

Because [µ], [uin] and c1(W ) all extend over X in, we have([uin]^ (−4π2c1(W )+ 2π [µ])

)[Y ] = 0,

so

µ(uin(γ (−R ))

) = µ(γ (−R )

),

from equation (27).

The first statement in the Proposition now follows, just as in [17], for theremust exist a sequence Ri →∞ and cylinders [ni , ni + 1]×Y ⊂ [−Ri ,Ri]×Y , suchthat the restrictions of [ARi ,ΦRi ] to these cylinders converge in a temporal gaugeto a translation-invariant solution.

For the second statement, suppose div(t) = 0. Let X in and Xout be the man-ifolds obtained by attaching cylinders [0,∞) ×Y and (−∞, 0] to X in and Xoutrespectively. Choose ηin on X in such that ηin + ρ(iµ) ∈ e−εtCr on the end, andhaving the generic property that

Mηin (X in, s, α) = ∅

for all α with i(X in, s, α) < 0. Note that, because div(t) is zero, i(X in, s, α) is aninteger, equal to i(Γ) for any choice of Γ. Note also that, because SW (s) is non-zero, the formal dimension of the moduli space for X+ is zero, and by excision thismeans that

i(X in, s, α) = δ([ξ ], I (α)).

Similarly, choose ηout on Xout, asymptotic to ρ(iµ) on the cylindrical end anddecaying exponentially on the conical end. Combining the Fredholm theory for manifolds with the cylindrical-end theory, we have a self-evident constructionfor moduli spaces Mηout (Xout, ξ , β ). The formal dimension of these moduli spacesis given by the difference element δ(I (β ), [ξ ]). We can choose ηout so that thesemoduli spaces are empty whenever this difference element is negative.

Let Ri be any sequence tending to infinity, and let η′i be any sequence of per-turbing terms on (X+, gRi ) which converge on compact sets to ηin and ηout on thecylindrical-end manifolds. The moduli spaces Mη′i are non-empty on X+ for all i

(v) Proof of Theorem 1.3 45

as SW (s) 6= 0, and after passing to a subsequence, we can suppose there are solu-tions which converge on compact sets to configurations (Ain,Φin) and (Aout,Φout)on the cylindrical-end manifolds. These limits solve the equations with perturbingterms ηin and ηout respectively. By the Lemma above, these solutions have finitevariation of the Chern-Simons-Dirac functional on the cylindrical ends, and there-fore define elements of moduli spaces Mηin (X in, α) and Mηout (Xout, ξ , β ) for someα, β in Nµ(Y, t), by Proposition 5.5. The required inequalities follow from thechoice of perturbing terms and the discussion in the previous paragraph.

This proposition has the following adaptation in the case that the boundary Yof (X , ξ ) is disconnected. Suppose Y = Y1 ∪ . . . ∪Yn, let µi be the restriction of µto Yi , and let ti be the Spinc-structure on Yi induced from t.

Proposition 5.8. Suppose the following three conditions hold:

• the invariant SW (s) is non-zero;

• for some metric gY1, the critical point set Nµ1 (Y1, t1) consists of finitely many non-degenerate, irreducible solutions;

• the cohomology class [µ1] extends to X.

Under the above assumptions we have the following.

(a) The critical-point set Nµ1 (Y1, t1) is non-empty.

(b) In addition, if div(t1) = 0 so that the difference-element δ on two-plane fields takesvalues in Z, then there exist α and β in Nµ1 (Y1, t1) with

δ(I (β ), [ξ ]

) ≥ 0 and δ([ξ ], I (α)

) ≥ 0.

Proof. If one stretches along the cylinder corresponding to Y1 then the proof of theprevious proposition goes through under the given hypotheses.

We now prove a weakened version of Theorem 1.3. If λ is a real 2-dimensionalcohomology class on Y , let us say that a contact structure ξ is λ-fillable if there is asymplectic filling (X , ω) such that λ extends to X .

Proposition 5.9. For any connected, oriented 3-manifold Y and any non-zero class λ ∈H2(X ;R), the number of elements of π0(Ξ) realized by λ-fillable contact structures is finite.


Proof. Fix a Riemannian metric gY and choose a closed 2-form µ representing2πελ, having chosen ε so that ελ is not an integer class. The critical point setNµ(Y, t) then contains no reducible solutions, for any t, and by Proposition 5.3 wecan alter µ by an exact form so that all elements of Nµ(Y, t) are non-degenerate.

If ξ is a λ-fillable contact structure and X is the manifold carrying the filling,then by Theorem 1.1 the invariant SW (s0) is non-zero, where s0 is the canonicalSpinc-structure on X . The restriction of s0 to Y is the Spinc-structure t = p([ξ ]),and by Proposition 5.6 above, Nµ(Y, t) is non-empty for this Spinc-structure.

It therefore follows from Proposition 5.3 that there are only finitely many t ∈Spinc (Y ) which arise as p([ξ ]) for λ-fillable contact structures ξ . To prove theproposition, it remains to show that, for each t, only finitely many elements of thefiber p−1(t) arise in this way. If div(t) is non-zero, then p−1(t) itself is finite. Ifdiv(t) is zero, then the fiber is an affine copy of Z. If we regard it thus as an orderedset, the second part of Proposition 5.6 states that the elements of p−1(t) which arisefrom λ-fillable contact structures lie in the finite segment [Imin, Imax], where

Imin = min I (α) | α ∈ Nµ(Y, t)

Imax = max I (α) | α ∈ Nµ(Y, t) .

This completes the proof of the Proposition.

To conclude the proof of Theorem 1.3 for connected Y we will reduce the gen-eral case to the case handled by the above Proposition. The following elementarylemma is a special case of results in [26] or [8].

Lemma 5.10. Let (Y, ξ ) and (Y ′, ξ ′) be fillable contact 3-manifolds with fillings X andX ′. Then the connected sum Y #Y ′ carries a contact structure which is equal to the given onesaway from the connected sum region and is fillable by the boundary connected sum of X andX ′.

Thus there is a map from π0(ΞY ) × π0(ΞY ′ ) to π0(ΞY #Y ′). If we have a fixedξ ′ on Y ′ with zero or torsion first Chern class (so that the divisibility of thecorresponding Spinc-structure is zero) then this map is injective as a function of[ξ ] ∈ π0(ΞY ).

Fix Y ′ and a non-zero class λ ∈ H2(Y ;R) for which there exists a λ-fillablecontact structure ξ ′ with zero first Chern class. For example a small tubular neigh-borhood of the standard Lagrangian torus S1 × S1 ⊂ C × C is a Stein domain car-rying a unique nonzero class λ up to scale and we may take Y ′ to be its boundary.Suppose Y is a connected 3-manifold carrying fillable contact structures in infinitelymany homotopy classes. Then Y #Y ′ carries λ-fillable contact structures in infinitelymany homotopy classes by the above lemma. This contradicts Proposition 5.9 andso proves Theorem 1.3 for Y .

Appendix 47

The case of disconnected Y can be treated in the same way, using Propo-sition 5.8 in place of 5.6 and applying the construction to each component inturn.

Appendix: orientations

Let (X , ξ ) be an oriented 4-manifold with contact boundary. To fix the sign ofthe invariant SW (s), we need to orient the moduli spaces Mη(s), i.e. orient theFredholm map π2 as promised in Theorem 2.4. For us, a homology orientation of(X , ξ ) is simply a choice of orientation for the real determinant line of the operator$ at any configuration (A,Φ) ∈ #, for any choice of s ∈ Spinc (X , ξ ). To use thisdefinition, we need to show that such a choice determines a consistent choice oforientations for the determinant lines at all (A,Φ), for all choices of s.

In general, let Z be a non-compact manifold, and let D1 : Γ(E1) → Γ(F1)and D2 : Γ(E2)→ Γ(F2) be real elliptic operators. Suppose that an isomorphism ι :(E1, F1)→ (E2, F2) is defined outside a compact subset of Z , and that ι intertwinesthe symbols of D1 and D2. Such data determines, by the usual construction, anelement of the real K -theory (with compact supports) of T ∗Z and a well-definedindex [2]. We write Ω(D1,D2, ι) for the orientation bundle for this index element.

On an manifold, and in particular on the manifold X+ which one formsfrom (X , ξ ), one has operators

D1 = −d∗ ⊕ ρ B d+ : Ω1(iR)→ Ω0(iR)⊕ isu(W+)

D2 = D−A : Γ(W −)→ Γ(W+).

The latter depends on a choice of s and a connection A. Outside a compact set,the almost complex structure J gives an isomorphism ι J between these operators atthe symbol level: we have

Λ1 ∼= Λ0,1J∼=W −(s0)

and

Λ0 ⊕ su(W +) ∼= Λ0 ⊕Λ+

∼= Λ0,0J ⊕Λ0,2

J

∼=W +(s0);

the symbol of the two operators both coincide with the symbol of√

2(∂ + ∂∗). Theindex of the Fredholm operator $ is an analytic realization of the topological indexin this situation, and what is involved in specifying an orientation of the modulispace is therefore an orientation of Ω(−d∗ ⊕ ρ B d+,D−A, ι J ).

48 Appendix

Notice first that if A1 and A2 are spin connections for possibly different ele-ments s1, s2 in Spinc (X , ξ ) (or Spinc (Z , ω) in the general setting), then theratio

Ω(−d∗ ⊕ ρ B d+,D−A1, ι J ) ⊗ Ω(−d∗ ⊕ ρ B d+,D−A2

, ι J )−1

is isomorphic to Ω(D−A1,D−A2

, id). (The two spin structures coincide on the endof the manifold). There is a canonical orientation for the latter, however, becausethe Dirac operator is complex. Thus a choice of orientation at one configurationdetermines a choice of orientation at all others. Using the same identification,but applied to a path of connections joining A to a gauge transform of A, we seethat the canonical orientations we define are invariant under gauge transformation.Thus we are led to define:

Definition A.1. A homology orientation for a non-compact, oriented 4-manifoldZ equipped with an almost complex structure J on the complement of a compactset is determined by a choice of orientation O of Ω(−d∗ ⊕ ρ B d+,D−A, ι J ), for somechoice of spin connection A and some choice of Spinc-structure s extending thecanonical Spinc-structure defined by J .

If A′ and s′ are another such pair, and O′ is a choice of orientation for Ω(−d∗ ⊕ρ B d+,D−A′ ), we say that O and O′ determine the same homology orientation iftheir difference in Ω(D−A,D

−A′ ) agrees with the complex orientation.

We have seen that for an manifold, such a homology orientation deter-mines an orientation of the moduli spaces, and hence fixes the sign of the invariantSW . Two remarks can be made about this notion of homology orientation. First,consider the case that the almost complex structure on the end of Z is extended tothe whole manifold. Such an extension gives a trivialization of the index element,and hence a preferred orientation. For example, in the case of a 4-manifold (X , ξ )with contact boundary, a compatible symplectic structure determines a preferredhomology orientation. To realize this preferred orientation analytically, one de-forms the operator $ in a family as in Corollary 3.12, scaling the metric by λ, orequivalently scaling Φ0. For sufficiently large λ, the cokernel and kernel are trivial:there is then a canonical orientation.

The second remark concerns how one can compare a homology orientationfor (X , ξ ) with a homology orientation for (X , ξ ). In the more general contextof a manifold Z with an almost complex structure J outside a compact set, onecan similarly ask to compare a homology orientation for (Z , J ) with a homologyorientation for (Z ,− J ). If s1 is a Spinc-structure extending the canonical Spinc-structure determined by J , then s1 naturally extends the Spinc structure determinedby − J . In this situation, if A1 and A1 are conjugate spin connections, the lines

REFERENCES 49

Ω(−d∗ ⊕ ρ B d+,D−A1, ι) and Ω(−d∗ ⊕ ρ B d+,D−A1

, ι) are canonically isomorphic.(Here ι and ι are the isomorphisms between the symbols, as determined by J and− J .) In this way, for each choice of s1 and each homology orientation O for(Z , J ), we obtain a homology orientation O = O(s1) for (Z ,− J ).

There now remains the question of how O(s1) depends on s1. Let s2 be an-other extension of the canonical Spinc-structure, and let A2 be a correspondingconnection. The real lines Ω(D−A1

,D−A2, id) and Ω(D−A1

,D−A2, id) are canonically

isomorphic, because the real operators which are involved are the same. However,the complex structures of the index elements are opposite, so their canonical orien-tations differ by the complex rank (half the real rank) of the index element obtainedfrom the difference of DA1 and DA2 . We therefore have

O(s2) = (−1)(d(s1)−d(s2))/2O(s1).

For the definition of the invariant SW , we are only concerned with the cases whered(si ) = 0, and in such cases the orientation O is independent of si . This justifiesthe assertion made in the introduction, that for suitable choices of homology ori-entations of (X , ξ ) and (X , ξ ), one has the relation

SW (X ,ξ )(s) = SW (X ,ξ )(s),

for all s ∈ Spinc (X , ξ ).

References

[1] M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry:I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69.

[2] M. F. Atiyah and I. M. Singer, The index of elliptic operators: I, Ann. of Math. 93 (1968),484–530.

[3] T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampere equations, Grundlehren derMathematischen Wissenschaften, no. 252, Springer-Verlag, Berlin-New York, 1982.

[4] S. K. Donaldson, The orientation of Yang-Mills moduli spaces and 4-manifold topology, J.Differential Geom. 26 (1987), no. 3, 397–428.

[5] , Boundary value problems for Yang-Mills fields, J. Geom. Phys 8 (1992), 89–102.

[6] Y. Eliashberg, Classification of overtwisted contact structures on 3-manifolds, Invent. Math.98 (1989), no. 3, 623–637.

[7] , Filling by holomorphic discs and its applications, Geometry of low-dimensionalmanifolds, 2 (Durham, 1989), Cambridge Univ. Press, 1990, pp. 45–67.

[8] , Topological characterization of Stein manifolds of dimension > 2, Internat. J. Math(1990), no. 1, 29–46.

50 REFERENCES

[9] , Contact 3-manifolds twenty years since J. Martinet’s work, Ann. Inst. Fourier(Grenoble) 2 (1992), no. 1–2, 165–192.

[10] Y. Eliashberg and W. P. Thurston, Contact structures and foliations on 3-manifolds, TurkishJ. Math 0 (1996), no. 1, 19–35.

[11] K. A. Frøyshov, The Seiberg-Witten equations and four-manifolds with boundary, Math.Res. Lett. 3 (1996), no. 3, 373–390.

[12] D. Gabai, Foliations and 3-manifolds, Proceedings of the International Congress ofMathematicians (Kyoto 1990), vol. I, Math. Soc. Japan, 1991, pp. 601–609.

[13] M. L. Gromov and H. B. Lawson, Positive scalar curvature and the Dirac operator on completeRiemannian manifolds, Inst. Hautes Etudes Sci. Publ. Math (1984), no. 58, 83–196.

[14] T. Kato, Perturbation theory for linear operators, second ed., Grundlehren der Mathema-tischen Wissenschaften, no. 132, Springer-Verlag, Berlin-New York, 1976.

[15] D. Kotschick, The Seiberg-Witten equations on symplectic four-manifolds [after C. H. Taubes],expose 812, Seminaire N. Bourbaki, 1995–1996.

[16] P. B. Kronheimer and T. S. Mrowka, in preparation.

[17] , The genus of embedded surfaces in the projective plane, Math. Res. Lett. 1 (1994),no. 6, 797–808.

[18] P. Lisca and G. Matic, Tight contact structures and the Seiberg-Witten invariants, preprint,1996.

[19] R. B. Lockhart and R. C. McOwen, Elliptic differential operators on noncompact manifolds,Ann. Scuola Norm. Sup. Pisa Cl. Sci. 12 (1985), no. 3, 409–447.

[20] J. W. Morgan, T. S. Mrowka, and D. Ruberman, The L2-moduli space and a vanishingtheorem for Donaldson polynomial invariants, Monographs in Geometry and Topology,no. II, International Press, Cambridge, MA, 1994.

[21] C. H. Taubes, Stability in Yang-Mills theories, Comm. Math. Phys. 91 (1983), no. 2,235–263.

[22] , Gauge theory on asymptotically periodic 4-manifolds , J. Differential Geom. 5(1987), no. 3, 363–430.

[23] , The Seiberg-Witten invariants and symplectic forms, Math. Res. Lett. 1 (1994),no. 6, 809–822.

[24] , More constraints on symplectic forms from Seiberg-Witten invariants, Math. Res.Lett. 2 (1995), no. 1, 9–13.

[25] , SW ⇒ Gr: from the Seiberg-Witten equations to pseudo-holomorphic curves, J.Amer. Math. Soc. 9 (1996), no. 3, 845–918.

[26] A. Weinstein, Contact surgery and symplectic handlebodies, Hokkaido Math. J. 20 (1991),no. 2, 241–251.

[27] E. Witten, Monopoles and four-manifolds, Math. Res. Lett. 1 (1994), no. 6, 769–796.

This version: August 28, 1996.

monopoles and contact structures - harvard...

Documents