Knots, sutures and excision

P. B. Kronheimer and T. S. Mrowka

Harvard University, Cambridge MA 02138Massachusetts Institute of Technology, Cambridge MA 02139

Abstract. We develop monopole and instanton Floer homology groups for balancedsutured manifolds, in the spirit of [12]. Applications include a new proof of PropertyP for knots.

Contents

1 Introduction 12 Background on monopole Floer homology 33 Floer’s excision theorem 194 Monopole Floer homology for sutured manifolds 255 Knot homology 336 Fibered knots 407 Instantons 52

1 Introduction

Floer homology for sutured manifolds is an invariant SFH.M; / of “balancedsutured 3-manifolds” .M; /, introduced by Juhasz in [12, 13]. It incorporatesthe knot Floer homology of Ozsvath-Szabo and Rasmussen [25, 27] as a specialcase, and it provides a framework in which to adapt the arguments of Ghigginiand Ni [11, 23, 22] to reprove, for example, that knot Floer homology detectsfibered knots.

The construction that forms the basis of Juhasz’s invariant is an adapta-tion of Ozsvath and Szabo’s Heegaard Floer homology for 3-manifolds. Thepurpose of the present paper is to show how something very similar can be

The work of the first author was supported by the National Science Foundation throughNSF grant number DMS-0405271. The work of the second author was supported by NSFgrants DMS-0206485, DMS-0244663 and DMS-0805841.

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done using either monopole Floer homology [18] or instanton Floer homol-ogy [4] in place of the Heegaard version. We will define an invariant of bal-anced sutured manifolds by gluing them up, with some extra pieces, to form aclosed manifold and then applying ordinary Floer homology, of either variety,to this closed manifold. Many of the theorems and constructions of Ghiggini,Ni and Juhasz can be repeated in this context. In particular, our construc-tion leads to candidates for “monopole knot homology” and “instanton knothomology”: the monopole and instanton counterparts of the Heegaard knothomology groups. Adapting the arguments of [11] and [23], we shall also provethat fibered knots can be characterized using either of these invariants.

The definition of instanton knot homology which arises in this way, moti-vated by Juhasz’s sutured manifold framework, is not new. It turns out to beexactly the same as an earlier instanton homology for knots, defined by Floertwenty years ago [8]. We conjecture that, over a field of characteristic zero, theknot homology groups of Ozsvath-Szabo and Rasmussen are isomorphic toFloer’s instanton knot homology.

Monopole Floer homology for balanced sutured manifolds is defined insection 4, and the definition is adapted to the instanton case in section 7. Thesame definition could be applied with Heegaard Floer homology: it is not clearto the authors whether the resulting invariant of sutured manifolds would bethe same as Juhasz’s invariant, but we would conjecture that this is the case. Itseems, at least, that the construction recaptures Heegaard knot homology [26].Some things are missing however. Our construction leads to knot homologygroups which lack (a priori) the Z grading as well as the additional structuresthat are present in the theory developed in [25] and [27].

In the setting of instanton homology, we obtain new non-vanishing theo-rems. Among other applications, the non-vanishing theorems lead to a newproof of Property P for knots. In contrast to the proof in [17], the argumentpresented here is independent of the work of Feehan and Leness [6] concerningWitten’s conjecture, and does not require any tools from contact or symplectictopology. As a related matter, we show that instanton homology captures theThurston norm on an irreducible 3-manifold, answering a question raised in[15].

Acknowledgments. The authors would like to thank Andras Juhasz and JakeRasmussen for helpful comments and corrections to an earlier version of thispaper.

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2 Background on monopole Floer homology

2.1 Monopole Floer homology recalled

We follow the notation of [18] for monopole Floer homology. Thus, to a closed,connected, oriented 3-manifold Y equipped with a spinc structure s, we asso-ciate three varieties of Floer homology groups with integer coefficients,

zHM�.Y; s/; bHM�.Y; s/; HM�.Y; s/:

The notation using � in place of the more familiar � was introduced in [18] todenote that, in general, there is a completion involved in the definition. In allthat follows, the distinction between bHM� and bHM� does not arise, but wepreserve the former notation as a visual clue. Unless c1.s/ is torsion, thesegroups are not Z-graded, but they always have a canonical Z=2 grading.

The three varieties are related by a long exact sequence

� � � ! HM�.Y; s/i! zHM�.Y; s/

j! bHM�.Y; s/

p! HM�.Y; s/! � � � :

If c1.s/ is not torsion, then HM�.Y; s/ is zero and zHM�.Y; s/ and bHM�.Y; s/are canonically isomorphic, via j . In this case, we simply write HM�.Y; s/ foreither zHM�.Y; s/ or bHM�.Y; s/. All these groups can be non-zero only forfinitely many spinc structures on a given Y : we write

zHM�.Y / DM

s

zHM�.Y; s/

for the total Floer homology, taking the sum over all isomorphism classes ofspinc structure, with similar notation for the bHM and HM cases.

2.2 Local coefficients

We can also define a version of Floer homology with a local system of coef-ficients. The following definition is adapted from [18, section 22.6]. Let R

denote any commutative ring with 1 supplied with an “exponential map”, agroup homomorphism

exp W R! R�: (1)

We will use polynomial notation for the exponential map, writing

t D exp.1/

4

and so writing exp.n/ as tn. Let B.Y; s/ denote the Seiberg-Witten configura-tion space for a spinc structure s on Y ; that is, B.Y; s/ is the space of gaugeequivalences classes ŒA;ˆ� consisting of a spinc connection A and a section ˆof the spin bundle. Given a smooth 1-cycle � in Y with real coefficients, we canassociate to each path z W Œ0; 1�! B.Y; s/ a real number r.z/ by

r.z/ Di

2�

ZŒ0;1���

trFAz;

where Az is the 4-dimensional connection on Œ0; 1� � Y arising from the pathz. Now define a local system �� on B.Y; s/ by declaring its fiber at everypoint to be R and declaring the map R ! R corresponding to a path z to bemultiplication by tr.z/. Following [18, section 22], we obtain Floer homologygroups with coefficients in ��; they will be R-modules denoted

zHM�.Y I��/; bHM�.Y I��/; HM�.Y I��/:

These still admit a direct sum decomposition by isomorphism classes of spinc

structures. The following is essentially Proposition 32.3.1 of [18]:

Proposition 2.1. If there is an integer cohomology class that evaluates as 1 on Œ��,and if t � t�1 is invertible in R, then HM�.Y I��/ is zero; thus we again have anisomorphism j between zHM�.Y I��/ and bHM�.Y I��/.

In the situation of the proposition, we once more drop the decorations andsimply write

HM�.Y I��/ DM

s

HM�.Y; sI��/

for this R-module.

2.3 Cobordisms

Cobordisms between 3-manifolds give rise to maps between their Floer homol-ogy groups. More precisely, if W is a compact, oriented cobordism from Y1 toY2, equipped with a homology-orientation in the sense of [18], then W givesrise to a map

zHM.W / W zHM�.Y1/! zHM�.Y2/

with similar maps on bHM� and HM�. If �1 and �2 are 1-cycles in Y1 andY2 respectively, then to obtain a maps between the Floer groups with local

5

coefficients, we need an additional piece of data: a 2-chain � in W with @� D�2 � �1. In this case, we obtain a map which we denote by

zHM.W I��/ W zHM�.Y1I��1/! zHM�.Y2I �2/:

The map zHM.W / and its relatives are defined by taking a sum over allspinc structures on W . In the case of zHM.W I��/, the spinc contributionsare weighted according to the pairing of the curvature of the connection withthe cycle �. There is a corresponding invariant for a closed 4-manifold X withbC � 2 containing a closed 2-cycle �. In [18], this invariant of .X; �/ is denotedby m.X; �/ (or m.X; Œ��/, because only the homology class of � matters); it isan element of R defined by

m.X; Œ��/ DX

s

m.X; s/t hc1.s/;Œ��i; (2)

where m.X; s/ denotes the ordinary Seiberg-Witten invariant for a spinc struc-ture s.

2.4 Adjunction inequalities and non-vanishing theorems

Monopole Floer homology detects the Thurston norm of a 3-manifold Y . Werecall from [18] what lies behind this slogan. Let F � Y be a closed, oriented,connected surface in our closed, oriented 3-manifold Y . We shall suppose Fis not sphere. Then we have a vanishing theorem [18, Corollary 40.1.2], whichstates that

HM�.Y; s/ D 0

for all spinc structures s satisfying

hc1.s/; ŒF �i > 2 genus.F / � 2:

(Note that this condition implies that c1.s/ is not torsion.) This vanishingtheorem is usually referred to as the “adjunction inequality”. Accompanyingthis result is a rather deeper non-vanishing theorem, which we state (for thesake of simplicity) in the case that the genus of F is at least 2. In this case, thenon-vanishing theorem asserts that if F is genus-minimizing in its homologyclass, then there exists a spinc structure sc with

HM�.Y; sc/ ¤ 0

andhc1.sc/; ŒF �i D 2 genus.F / � 2:

6

Slightly more specifically, Gabai’s theorem from [9] tells us that Y admits ataut foliation having F as a compact leaf. A foliation in turn determines aspinc structure on Y . The non-vanishing result holds for any spinc structuresc arising in this way. This result appears as Corollary 41.4.2 in [18]. Thetechniques of this paper provide an alternative proof, which we will explain inthe context of instanton homology in section 7.8 below.

It is convenient to introduce the following shorthand. We denote the setof isomorphism classes of spinc structures on a closed oriented manifold Y byS.Y /. If F � Y is a closed, connected oriented surface of genus g � 2, thenwe write S.Y jF / for the set of isomorphisms classes of spinc structures s on Ysatisfying the constraint

hc1.s/; ŒF �i D 2 genus.F / � 2; (3)

and we writeHM�.Y jF / � HM�.Y /

for the subgroup

HM�.Y jF / DM

s2S.Y jF /

HM�.Y; s/: (4)

Note again that all the spinc structures in S.Y jF / have non-torsion first Chernclass. When a local system �� is given, we define HM�.Y jF I��/ similarly. IfF is a surface with more than one component, we define

S.Y jF / D\Fi�F

S.Y jFi /

where the Fi are the components, and we define HM�.Y jF / accordingly.As a special case, we have

Lemma 2.2. Let F be a closed, connected, oriented surface of genus at least 2,and let Y D F � S1. Regard F as a surface F � fpg in Y . Then we have

HM�.Y jF / D Z:

Indeed, if F is given a metric of constant negative curvature and Y is given theproduct metric, then the complex that computes HM�.Y jF / has a single gen-erator, corresponding to a single, non-degenerate solution of the Seiberg-Wittenequations.

7

Proof. This is standard. The spinc that contributes is the product spinc struc-ture, which corresponds to the 2-plane field tangent to the fibers of the mapY ! S1. The unique gauge-equivalence class of solutions to the equations isa pair ŒA;ˆ� with ˆ covariantly constant.

Corollary 2.3. Let Y be the product F � S1, as in the previous lemma. Then forany local coefficient system ��, we have

HM�.Y jF I��/ D R;

where R is the coefficient ring.

2.5 Disconnected 3-manifolds, part I

So far, following [18], we have discussed connected 3-manifolds and connected4-dimensional cobordisms between them. Because of the special role playedby reducible connections, one must be careful when generalizing; but there aresimple situations where the discussion can be carried over without difficulty tothe case of 3-manifolds with several components. The analysis of the Seiberg-Witten equations on a manifold with cylindrical ends is carried out in [18] foran arbitrary number of ends, and our task here is just to package the resultinginformation.

Let W be cobordism from Y1 to Y2, and suppose that each of these hascomponents

Y1 D Y1;1 [ � � � [ Y1;r

Y2 D Y2;1 [ � � � [ Y2;s:

Although we label them this way, no ordering of the components need be cho-sen at this point. We may allow either r or s (or both) to be zero, and we donot requireW to be connected. IfW has any closed components, we insist thateach such component has bC � 2.

As a simple way to avoid reducible connections, let us give a closed, ori-ented surface F1 � Y1 and F2 � Y2. We will suppose that each component ofY1 contains a component of F1 and that all components of F1 have genus 2 ormore. Thus we have non-empty surfaces

F1;i D F1 \ Y1;i

� Y1;i :

We make a similar hypothesis for F2. We can regard the union F1 [ F2 asa subset of W , and we suppose that we are given a surface FW � W which

8

contains F1 [ F2 in addition perhaps to other components. The notation wepreviously used for spinc structures with constraints on c1 can be extended tothis case: we write S.W jFW / for the set of spinc structures s on W such that(3) holds for every component of FW .

We can define HM�.Y1jF1/ by taking a product over the components ofY1. That is, we should define the configuration space B.Y1/ as the product ofthe B.Y1;i /, and we should construct HM�.Y1jF1/ as the Floer homology ofthe Chern-Simons-Dirac functional on the components of this product spacewhich belong to the appropriate spinc structures. The only slight twist here isin understanding the orientations of moduli spaces that are needed to fix thesigns.

We therefore digress to consider orientations. For a cobordism such as Wabove, perhaps with several components, we define a 2-element set ƒ.W / ofhomology orientations of W as follows. Attach cylindrical ends to the incom-ing and outgoing ends to get a complete manifold W C, and let t be functionwhich agrees with the cylindrical coordinate on the ends. The function t tendstoC1 on the outgoing ends and �1 on the incoming ends. Consider the lin-earized anti-self-duality operator ı D d�˚dC acting on the weighted Sobolevspaces

ı W L21;�.iƒ1/! L2� .iƒ

0˚ iƒC/;

where L2k;�D e��tL2

k. Fix a spinc structure on W and let DCA0

be the Diracoperator, for a spinc connection A0 that is constant on the ends. We considerDA acting on weighted Sobolev spaces of the same sort, and we write

P D ı CDCA0:

These are the linearized Seiberg-Witten equations onW , with Coulomb gaugefixing, at a configuration where the spinor is zero. There exists �0 > 0 such thatthe operator P is Fredholm for all � in the interval .0; �0/. We define ƒ.W / tobe the set of orientations of the determinant line of P , for any � in this range.Using weighted Sobolev spaces here is equivalent to using ordinary Sobolevspaces and replacing P by a zeroth-order perturbation which on the ends hasthe form

P � �‚

where ‚ is obtained from applying the symbol of P to the vector field @=@talong the cylinder. The Dirac operator is irrelevant at this point because it iscomplex and its real determinant is therefore canonically oriented; so we coulduse the operator ı instead.

9

Now suppose that ˛1 and ˛2 are gauge-equivalence classes correspond-ing to non-degenerate critical points of the Chern-Simons-Dirac functional onY1 and Y2 respectively. Let D .A;ˆ/ be any configuration on W C that isasymptotic to these gauge-equivalence classes on the ends. Let P be the cor-responding operator (acting on the Sobolev spaces without weights). Defineƒ.W I˛1; ˛2/ to be the set of orientations of the determinant of PA. This isindependent of the choice of , in a canonical manner. If ƒ1 and ƒ2 are two2-element sets, we use the notation ƒ1ƒ2 to denote the 2-element set formedby the obvious “multiplication” (the set of bijections from ƒ1 to ƒ2). Withthis in mind, we define

ƒ.˛1; ˛2/ D ƒ.W /ƒ.W I˛1; ˛2/:

An excision argument makes this independent of W . Given now a 3-manifoldY (with several components) and a non-degenerate critical point ˛, we choosecobordism X from the empty set to Y and we define

ƒ.˛/ D ƒ.¿; ˛/:

We then haveƒ.˛1; ˛2/ D ƒ.˛1/ƒ.˛2/:

What this last equality means in practice is this. If we are given a cobor-dism W with a choice of homology orientation in ƒ.W / and a moduli spaceM D M.W I˛1; ˛2/, then a choice of orientation of M is the same as a choiceof bijection from ƒ.˛1/ to ƒ.˛2/. In the case that Y1 D Y2, the cylindricalcobordism has a canonical homology-orientation because the operator P isinvertible; so in this ƒ.˛1/ƒ.˛2/ orients the moduli spaces. The appropriatedefinition of HM�.Y1; s/ for spinc structures s that are non-torsion on eachcomponent is therefore to take the complex to be

C�.Y1; s/ DM˛1

Zƒ.˛1/

and to define the differential using the corresponding orientation of the modulispaces. In this way, we construct HM�.Y1jF1/ and HM�.Y2jF2/. If we supplyW with a homology orientation in the above sense, then W defines a map

HM.W jFW / W HM�.Y1jF1/! HM�.Y2jF2/: (5)

The notation HM.W jFW / is meant to imply that we use only the spinc struc-tures from S.W jFW /.

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The complex C�.Y1; s/ just defined can be considered as a tensor productover the connected components of Y1:

C�.Y1; s/ DOi

C�.Y1;i ; si /; (6)

but there are some choices involved. Let us pick an ordering of the compo-nents. For each i , let Xi be a cobordism from the empty set to Y1;i . Using thestandard convention for the orientation of a direct sum, we can then identify

ƒ.X/ D ƒ.X1/ƒ.X2/ � � �ƒ.Xr/;

and similarly with ƒ.X; ˛1/. In this way, we can specify an isomorphism

Zƒ.˛1/! Zƒ.˛1;1/˝ � � � ˝ Zƒ.˛1;r/:

This allows us to identify the complexes on the left and right in (6) as groups.Ordering issues mean that there will be the expected alternating signs appear-ing when we compare the differentials on the left and right. As usual withproducts in homology, what results from this is a split short exact sequence

0!Oi

HM�.Y1;i jF1;i /! HM�.Y1jF1/! T ! 0 (7)

where T is a torsion group. If W is closed and has more than one component,the invariant is a product of the contributions from each component.

There is another sign issue to discuss. Consider the case of a 3-manifold Ywith non-torsion spinc structure s. Let Z be the 4-manifold S1 � Y . We canpull back the spinc structure to Z, and we still call it s. For clarity, supposethat bC.Z/ is bigger than 1, so that m.Z; s/ is defined. To fix the sign ofm.Z; s/, we need a homology orientation of Z; but a product such as Z hasa preferred homology orientation. To define it, we must specify an orientationfor the determinant of P on Z. The operator P � �‚ is invertible for small�, and we use this to to orient the determinant. Now let ˛ be non-degeneratecritical point for the (possibly perturbed) Chern-Simons-Dirac functional on.Y; s/. This pulls back to an isolated, non-degenerate solution on Z to the 4-dimensional Seiberg-Witten equations, say O . This solution contributes eitherC1 or �1 to the invariant m.Z; s/. We have the following lemma.

Lemma 2.4. The solution O contributes C1 or �1 to the invariant m.Z; s/ ac-cording as the critical point ˛ has odd or even grading in C�.Y; s/, for the canon-ical Z=2 grading.

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Proof. We have two operators differing by zeroth-order terms

P0 D P � �‚

P1 D P :

Let Ps be a homotopy between them. We have a determinant line for thisfamily of operators over the interval Œ0; 1�, and the invertibility of P0 and P1at the two ends gives the determinant line a canonical orientation at the twoends. The sign with which O contributes is, by definition, C1 or �1 accordingas these two orientations at s D 0; 1 are homotopic.

On the other hand, we can write Ps as

d

dtC Ls

on S1 � Y , where Ls is a self-adjoint elliptic operator perturbed by a boundedterm, and the canonical mod 2 grading of ˛ is determined, by definition, by theparity of the spectral flow of the family of operators Ls from s D 0 to s D 1.

So we must see that the parity of the spectral flow of the operators Lsdetermines whether the invertible operators P0 and P1 provide the same orien-tation. This is a general fact about families of self-adjoint Fredholm operators.What we have here are two non-trivial homomorphisms

H1.S/! Z=2;

where S is a suitable space of self-adjoint operators. One can argue as in [18],following [1], that one may take S to have the homotopy of U.1/=O.1/, atwhich point it is clear that these two are the same.

A consequence of the lemma is that the invariant m.Z; s/ is the equal tothe Euler characteristic of HM�.Y; s/, computed using the canonical mod 2grading. From the lemma and excision, we obtain similar results in other situ-ations of the following sort. Consider again a cobordismW from Y1 to Y2 withsurfaces FW , F1 and F2 as before. Suppose that one of the incoming boundarycomponents is the same as one of the outgoing ones: say

Y1;r D Y2;s:

We may form a new W � from W by identifying these boundary components,so W � has r � 1 incoming and s � 1 outgoing boundary components. Themanifolds Y1;r and Y2;s may belong either to the same or to different compo-nents of W , but we treat these cases together. The surface FW gives rise to a

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homeomorphic surface FW � in W �. (We push F1;r and F2;s away from theboundary of W before gluing to Y1;r to Y2;r , to keep these surfaces disjoint,if necessary.) If is possible that this process has created a W � which has onemore closed component than W . This new closed component of W � will havebC at least 1; but we shall suppose that, if there is such a component, it has bCat least 2. (The case of bC D 1 will be discussed in a slightly different contextin the next subsection.)

Under this hypothesis on bC for the closed components, we now have anew map

HM.W �jFW �/ W HM�.Y �1 jF�1 /! HM�.Y �2 jF

�2 /; (8)

where Y �1 is Y1 n Y1;r and Y �2 is defined similarly. The analysis from [18] pro-vides a “gluing theorem” which tells us that the map HM.W �jFW �/ is ob-tained from HM.W jFW / by a contraction. More precisely, at the chain level,.W; FW / defines a chain mapO

i

C�.Y1;i jF1;i /!Oj

C�.Y2;j jF2;j /:

This map can be contracted by taking an alternating trace over

C�.Y1;r jF1;r/ D C�.Y2;sjF2;s/;

and the result of this contraction is a chain map which is chain-homotopic tothe chain map defined by .W �; FW �/.

The cobordism W from Y1 to Y2 can also be regarded as a cobordism QW

from QY1 to QY2, whereQY1 D Y1 [ .�Y2;s/

andQY2 D Y2 n Y2;s:

(That is, we regard the last outgoing component as an incoming componentwith the opposite orientation.) The relation between the maps defined by Wand QW can be put in the same context as the above gluing theorem. We firstadd an extra componentZ toW , whereZ is the cylinder Œ0; 1��Y2;s, regardedas a cobordism from Y2;s[.�Y2;s/ to the empty set. The map defined byW [Zis a tensor product, at the chain level, and the cobordism QW can be obtainedby gluing an outgoing component of W to an incoming component of Z. Allthat is left is to understand the map defined by Z. Discounting torsion, thislast map is the Poincare duality pairing

HM�.�Y2;sjF2;s/˝HM�.Y2;sjF2;s/! Z:

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As in [18], this pairing depends on a homology orientation of Y2;s, which reap-pears as the need to choose a homology orientation for the extra componentZ.

Let us pursue a simple application of this formalism. Let W be again acobordism from Y1 to Y2 and let F1 and F2 be surfaces in these boundary3-manifolds as above. Suppose that W contains in its interior a product 3-manifold

Z D G � S1

where G is connected of genus at least 2. Regard G D G � fpg also as asubmanifold of W . Form a new cobordism W � from Y1 to Y2 by the follow-ing process. Cut W open along Z to obtain a manifold W 0 with two extraboundary components G � S1, then attach a copy of G �D2 to each of theseboundary components to obtain W �. Set

FW D .F1 [ F2 [G/ � W

FW � D .F1 [ F2 [G/ � W�:

Then we have

Proposition 2.5. The maps HM.W jFW / and HM.W �jFW �/ are equal, up tosign, as maps

HM.Y1jF1/! HM.Y2jF2/:

Proof. Consider the manifold W 0 obtained from W by cutting open along Z.This is a cobordism from Y1 [ Z to Y2 [ Z. The manifold W or W � can beobtained fromW 0 by gluing with Œ0; 1��Z or with .D2qD2/�G respectively.We can regard Œ0; 1��Z and .D2qD2/�G as two different cobordisms fromZ to Z, and they both induce maps

HM�.ZjG/! HM�.ZjG/:

The result follows from the glueing formalism as long as we know that thesetwo maps on HM�.ZjG/ are the same. Lemma 2.2 tells us that HM�.ZjG/is simply Z. The product Œ0; 1� � Z of course induces the identity map on thiscopy of Z. So it only remains to show that the invariant of manifold D2 � Gin HM�.ZjG/ is ˙1. This can be seen directly by examining the solutions ofthe Seiberg-Witten equations; or one can see indirectly that this must be so, onthe grounds that there exist closed 4-manifolds containing .ZjG/ for which anappropriate Seiberg-Witten invariant is 1.

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2.6 Disconnected 3-manifolds, part II

In the previous subsection we discussed gluing results in a context where theboundary components of the cobordisms carried spinc structures that hadnon-torsion first Chern classes. The non-torsion condition ensures that re-ducible solutions on the 3-manifolds play no role. A situation that is alge-braically similar is when the boundary components Y carry 1-cycles � and weuse local coefficients for which the vanishing theorem Proposition 2.1 applies.We can think of HM�.Y I��/ as measuring the contribution of the reduciblesolutions; so in a situation where this group is zero, as in the Proposition, wecan expect simple gluing results. This expectation is confirmed in the case ofconnected 3-manifolds by the results of [18, section 32]. We will deal here withthe simplest situation, in which the boundary components are 3-tori and localcoefficients are used.

Let W be a compact oriented 4-manifold with boundary, and suppose theoriented boundary consists of a collection of 3-tori,

@W D T1 [ � � � [ Tr :

We do not need to suppose that W is connected, but we do require that everyclosed component of W has bC at least 2. Let � � W be a 2-chain with

@� D �1 C � � � C �r :

We suppose that each �i is a 1-cycle in Ti satisfying the hypotheses of Proposi-tion 2.1 and that our coefficient ring R has t � t�1 invertible. We may take itthat each �i is a standard circle. For each i , the map

j W zHM�.Ti I��i/! bHM�.Ti I��i

/

is an isomorphism according to the proposition, so we again just write

HM�.Ti I�i /

for this group, using j to identify the two. According to [18, section 37], thisgroup is a free R-module of rank 1,

HM�.Ti I��i/ Š R:

(The proof in [18] was done in the case that R D R, but only the invertibility oft � t�1 is needed.) After choosing a basis element in HM�.Ti I��i

/, we shouldexpect W to have an invariant living inO

i

HM�.Ti I��i/ D R:

15

However, there is a short-cut to defining an R-valued invariant of W , used in[7] and [18, section 38]. We now describe this short-cut. In the remainder of thissubsection, we will leave aside the question of choosing homology-orientationsto fix the sign of the invariants that arise. So a 4-manifold or a cobordism willhave an invariant that is ambiguous in its overall sign.

Let E.1/ be a rational elliptic surface and let bE.1/ be the complement ofthe neighborhood of a regular fiber, so that @bE.1/ D T 3. Let �1 be a 2-cycle inE.1/ arising from a section meeting the neighborhood of the fiber transverselyin a disk, and let O�1 be the corresponding 2-chain in bE.1/. Let NW be the closed4-manifold obtained by attaching r copies of bE.1/ to W , making the attach-ments in such a way that the 1-cycles in the boundary tori match up: thus themanifold

NW D W [T1bE.1/ � � � [Tr

bE.1/contains a 2-cycle

N� D � [�1O�1 � � � [�r

O�1:

We can now compute a Seiberg-Witten invariant of the closed pair . NW ; N�/, andthe result depends only on .W; �/, not on the choice of gluing. Thus we maymake a definition:

Definition 2.6. Let W have boundary a collection of 3-tori, as above, let � bea 2-chain in W , and let . NW ; N�/ be the closed manifold obtained by attachingcopies of bE.1/. Suppose that every component of NW has bC � 2. Then wewrite

m.W; �/ 2 R

for the invariant m. NW ; N�/ of the closed manifold, as defined at (2). ˙

There is a formal device that can be used to extend this definition to includethe case that NW has closed components with bC D 1. Let E.n/ denote theelliptic surface without multiple fibers and having Euler number n, and let1E.n/ be the complement of a fiber. There is a 2-chain �n just as in the casen D 1. Instead of attaching bE.1/ to each Ti to form NW , we can similarly attach1E.ni / to Ti , for any ni � 1. We still refer to the resulting closed manifold as NW .It contains a 2-cycle N� as before. By choosing ni larger than 1 when needed, wecan ensure that all components of NW have bC least 2. We then define m.W; �/

bym.W; �/ D .t � t�1/�

P.ni�1/m. NW ; N�/: (9)

By the results of [18, section 38], the quantity on the right is independent ofthe choice of the ni .

16

Suppose next that W contains in its interior another 3-torus T which in-tersects � transversely in a single circle � representing a primitive element ofH1.T /. We can then cut W open along T to obtain W 0, a manifold whoseboundary consists of .r C 2/ tori. We can denote the two new boundary com-ponents by TrC1 and TrC2. By cutting � also, we obtain a 2-chain �0 in W 0

whose boundary has two new circles �rC1 and �rC2 in the new boundary com-ponents. We have the following glueing theorem. (The hypothesis that t � t�1

is invertible in R remains in place.)

Proposition 2.7. In the above situation, the invariants of .W; �/ and .W 0; �0/ areequal: thus

m.W; �/ D m.W 0; �0/

in the ring R.

Proof. There are two cases, according as T is separating or not. The separat-ing case is treated in [18, section 38]. We deal here with the non-separatingcase. The definitions mean that both sides are to be interpreted as invariantsof suitable closed manifolds. Restating it in such terms, and throwing out thecomponents that do not contain T , we arrive at the following. Let X be aclosed, connected 4-manifold with bC � 2, and let T � X be a non-separating3-torus. Let � be a 2-cycle in X meeting T transversely in a standard circle �with multiplicity 1. Let X 0 be cobordism from T to T obtained by cutting Xopen, and let �0 be the resulting 2-chain in X 0. Because of what we alreadyknow about the separating case, the proposition is equivalent to the followinglemma, which we shall prove.

Lemma 2.8. In the above situation, the map induced by the cobordism,

bHM�.X 0I��0/ W bHM�.T I��/! bHM�.T I��/

is given by multiplication by the element m.X; �/ 2 R.

Proof. It is convenient to arrange first that X 0 has bC at least 1. We can dothis by choosing a standard 2-torus F near T intersecting � transversely andforming a fiber sum at F with an elliptic surface E.n/. From what we knowabout separating 3-tori, we can conclude that this modification multiplies bothbHM�.X 0I��0/ and m.X; �/ by .t � t�1/n�1.

We now perturb the Chern-Simons-Dirac functional on T , as in [18, sec-tion 37], so that there are only reducible critical points, and we stretch X atT , inserting a cylinder Œ�R;R� � T and letting R increase to infinity as usual.

17

We consider what happens to the zero-dimensional moduli spaces on X in thelimit. Because bC.X 0/ is at least 1, we obtain in the limit only irreducible so-lutions on the cylindrical-end manifold obtained from X 0. Furthermore, theseirreducible solutions run from boundary-unstable critical points at the incom-ing end to boundary-stable critical points at the outgoing end. The weightedcount of such solutions defines the map

��!HM.X 0I��0/ W bHM�.T I��/! zHM�.T I��/

in the notation of [18, subsection 3.5]. We must also obtain in the limit some(possibly broken) trajectories on the cylindrical part, running from boundary-stable critical points to boundary-unstable critical points. For dimension-counting reasons, these trajectories must actually be unbroken and must beboundary-obstructed. The weighted count of such trajectories defines the map

j W zHM�.T I��/! bHM�.T I��/:

Thus m.X; �/ is equal to the contraction by the Kronecker pairing of two chainmaps which on homology define the composite

j ı��!HM�.X I��0/ W bHM�.T I��/! bHM�.T I��/:

It follows that m.X; �/ is the trace of this composite map. The composite isequal to bHM�.X 0I��0/, and the Floer group here is a free R-module of rank1, so the result follows.

There is a straightforward modification of the above results in the case thatW has some additional boundary components which are not 3-tori but containsurfaces F of genus 2 or more, as in the previous subsection. That is, we sup-pose that the boundary ofW is a union of 3-tori T1; : : : ; Tr together with a pairof 3-manifolds �Y1 and Y2, each of which may have several components. Wesuppose also that Y1 and Y2 contain surfaces F1 and F2 all of whose compo-nents have genus 2 or more. We also ask that each component of Yi containsa component of Fi . We shall suppose that there is a 2-chain � in W whoseboundary we write as

@� D ��1 C �2 C �1 C � � � C �r :

The �i are to be standard circles, one in each torus Ti as before. The 1-cycles�1 and �2 will be in Y1 and Y2, but we can allow these to be arbitrary (zerofor example). We take FW to be any closed surface in W consisting of F1 [

18

F2 together perhaps with additional components. We again suppose that anyclosed component of W has bC � 2. Then W should give rise to a map

HM�.W jFW I��/ W HM�.Y1jF1I��1/! HM�.Y2jF2I��2

/: (10)

To define this map, we can again attach .bE.1/; O�1/ to each of the 3-tori, toobtain . NW ; N�/ a cobordism from Y1 to Y2 containing a 2-chain N� and a surfaceFW . The boundary of N� is just��1C�2. As in Definition 2.6, we take HM�.W WFW I��/ to be defined by the map given by the cobordism NW . In the event thatNW has any closed components with bC D 1, we modify the construction by

using elliptic surfaces E.ni / as in (9). Proposition 2.7 then has the followingvariant.

Proposition 2.9. Let W be as above, and let T � W be a 3-torus meeting �transversely in a standard circle with multiplicity 1. Let W 0 and �0 be obtainedfrom W and � by cutting along T . Suppose that FW is disjoint from T , so that itbecomes also a surface FW 0 in W 0. Assume as always that t � t�1 is invertible inR. Then the maps

HM�.W jFW I��/ W HM�.Y1jF1I��1/! HM�.Y2jF2I��2

/

HM�.W 0jFW 0 I��0/ W HM�.Y1jF1I��1/! HM�.Y2jF2I��2

/(11)

are equal up to sign.

A particular application of this setup will be used in the sequel, a versionof Proposition 2.5. We formulate the result as the following corollary:

Corollary 2.10. Let W be a cobordism from Y1 to Y2 containing a 2-chain �with boundary ��1 [ �2. Let F1, F2 and FW be surfaces as above. Let T �W be a 3-torus disjoint from FW and cutting � in a standard circle � � T .Form W � by cutting W along T and attaching two copies of D2 � T 2 in sucha way that @D2 � fpg is glued to � in both copies. Let �� be the 2-chain in W �

obtained by attaching 2-disks D2 � fpg. Then, as maps from HM�.Y1jF1I��1/

to HM�.Y2jF2I��2/, we have

HM.W jFW I��/ D .t � t�1/HM.W �

jFW � I���/;

to within an overall sign.

Proof. Using Proposition 2.9, this can be proved with the same strategy thatwe applied to Proposition 2.5. That is, we consider two different cobordisms

19

from T to T : first, the product cobordism, and second the (disconnected)cobordism formed from two copies of D2 � T 2. In each case, there is an ob-vious 2-chain whose boundary is the difference of the two copies of �. Eachof these cobordisms has an invariant which lives in R, according to Defini-tion 2.6, or more accurately its correction at (9). In this sense, the productcobordism has invariant 1 2 R. The invariant of the other cobordism is.t � t�1/�1, as can be deduced from the invariants of the elliptic surfaces.

3 Floer’s excision theorem

3.1 The setup

We shall need to understand how monopole Floer homology behaves undercertain cutting and gluing operations on the underlying 3-manifold. A formulaof the type that we need was first proved by Floer in the context of instantonhomology. Floer’s “excision formula”, as he called it, applied only to cuttingalong tori; but in the monopole homology context one can equally well cutalong surfaces of higher genus, as long as one restricts to spinc structures thatare of top degree on the surface where the cut is made. We give the proof inthe monopole Floer homology context in this section: it is almost identical toFloer’s argument, as presented in [2]. Similar formulae have been proved inHeegaard Floer theory, by Ghiggini, Ni and Juhasz [11, 23, 22, 12, 13].

The setup is the following. Let Y be a closed, oriented 3-manifold, of eitherone or two components. In the case of two components, we call the compo-nents Y1 and Y2. Let†1 and†2 be closed oriented surfaces in Y , both of themconnected and of equal genus. If Y has two components, then we suppose that†i is a non-separating surface in Yi for i D 1; 2. If Y is connected, then wesuppose that †1 and †2 represent independent homology classes. In eithercase, we write † for †1 [ †2. Fix an orientation-preserving diffeomorphismh W †1 ! †2. From this data, we construct a new manifold QY as follows. Cuteach Y along† to obtain a manifold Y 0 with four boundary components: withorientations, we can write

@Y 0 D †1 [ .�†1/ [†2 [ .�†2/

If Y has two components, then so does Y 0, and we can write Y 0 D Y 01[Y02. Now

form QY by gluing the boundary component †1 to the boundary component�†2 and gluing †2 to �†1, using the chosen diffeomorphism of h both times.See Figure 1 for a picture in the case that Y has two components. In either

20

Figure 1: Forming a manifold QY from Y1 and Y2, for the excision theorem.

case, QY is connected. We write Q†1 for the image of †1 D �†2 in QY and Q†2 forthe image of †2 D �†1. So QY contains a surface Q† D Q†1 [ Q†2.

If we wish to use local coefficients in Floer homology, we will need to aug-ment this excision picture with 1-cycles �. Specifically, we take a 1-cycle � in Ythat intersects each †i transversely in a single point pi (i D 1; 2) with positiveorientation. If Y has two components, then we may write � D �1 C �2 for itstwo parts. We suppose that the diffeomorphism h is chosen so that h.p1/ D p2.When this is done, the 1-cycle gives to a 1-cycle Q� in the new manifold QY , asshown, by cutting and gluing.

We begin with a statement of the excision theorem with integer coefficients,when the genus of † is two or more.

Theorem 3.1. If QY is obtained from Y as above and the genus of †1 and †2 is atleast two, then there is an isomorphism of Floer groups with integer coefficients,

HM�.Y j†/! HM�. QY j Q†/:

Remark. In the case that Y has two components, the left-hand side is the ho-mology of a tensor product of complexes. In this case, the statement of thetheorem implies that there is a split short exact sequence

HM�.Y1j†1/˝HM�.Y2j†2/!

HM�. QY j Q†/! Tor�HM�.Y1j†1/;HM�.Y2j†2/

�: (12)

21

Floer’s version of this theorem has†1 and†2 of genus 1, with Y D Y1[Y2.It uses instanton Floer homology associated to an SO.3/ bundle with non-zeroStiefel-Whitney class on †. To obtain a version in monopole Floer homologywhen† has genus 1, we need to use local coefficients. We present a version thatis tailored to our later needs. We recall that �� denotes a system of local coef-ficients with fiber R, a commutative ring as in section 2.1. We suppose, as justdiscussed, that � meets †1 and †2 each in a single point so that we may formQ� as shown. Under these hypotheses, we expect there to be an isomorphism

HM�.Y I��/! HM�. QY I� Q�/:

We shall not endeavor to prove this variant of Floer’s excision theorem here,because it involves considering reducible solutions on multiple boundary com-ponents. Instead, as in section 2.6, we introduce some auxiliary surfaces F andcorresponding constraints on the spinc structures, just to avoid reducibles.

Thus we suppose in addition that Y contains an oriented surface F meeting† D †1 [ †2 transversely, and that the diffeomorphism h W †1 ! †2 carriesthe oriented intersection †1 \ F to †2 \ F . In this case, we can form anoriented surface QF in the new 3-manifold QY , by cutting F and regluing. Wesuppose that neither F nor QF contains a 2-sphere, and that every componentof Y contains a component of F whose genus is at least 2.

Theorem 3.2. Suppose QY and QF are obtained from Y and F as above, with †1and†2 both of genus 1. Let Q� be the 1-cycle in QY formed from the cycle � in Y asshown in Figure 1. Assume as usual that t � t�1 is invertible in the ring R. Thenthere is an isomorphism:

HM�.Y jF I��/! HM�. QY j QF I� Q�/:

Remark. Note again that if Y has two components and R is a field, then theleft-hand-side is the tensor product

HM�.Y1jF1I��1/˝R HM�.Y2jF2I��2

/:

There is also a simpler way in which local coefficients can enter into theexcision theorem, when the cycle � does not intersect†. We state an adaptationof Theorem 3.1 of this sort.

Theorem 3.3. Let QY be obtained from Y as in Theorem 3.1, with † of genus atleast two. Let �0 be a 1-cycle in Y , disjoint from †. This becomes a cycle also inQY , which we denote by Q�0. Then we have an isomorphism of R-modules:

HM�.Y j†I��0/! HM�. QY j Q†I� Q�0

/:

22

In Theorem 3.3, consider the case that Y D Y1 [ Y2 and �0 is contained inY1. In this case, the chain complex that computes the group HM�.Y j†I��0

/

on the left isC�.Y1j†1I��0

/˝Z C�.Y2j†2/;

(the tensor product of a complex of free R-modules and a complex of freeabelian groups, both finitely generated). By the Kunneth theorem, if R hasno Z-torsion and HM�.Y1j†1I��0

/ is a free R-module, then the theorem pro-vides an isomorphism

HM�.Y1j†1I��0/˝HM�.Y2j†2/! HM�. QY j Q†I� Q�0

/: (13)

As a particular application of this result, we have:

Corollary 3.4. Let † � Y be a closed, oriented surface whose components havegenus at least 2 and let � be a 1-cycle in Y whose support lies in †. Suppose thatR has no Z-torsion. Then

HM�.Y j†I��/ Š HM�.Y j†/˝R:

Proof. Apply the isomorphism of (13) with .Y2; †2/ D .Y;†/ and .Y1; †1/ D.† � S1; † � fpg/. Take �0 in † � S1 to be the cycle corresponding to �. ByProposition 2.3 we have

HM�.Y1j†1I��0/ D R:

The manifold QY is another copy of the original Y and Q† is two parallel copiesof †. The cycle �0 becomes now the original 1-cycle �, so

HM�. QY j Q†I� Q�0/ D HM�.Y j†I��/:

Thus (13) gives an isomorphism

R˝HM�.Y j†/! HM�.Y j†I��/:

3.2 Proof of the excision theorems

The proof of Theorem 3.1 is very much the same as Floer’s proof of his originalexcision theorem, as described in [2]. The first step (which is common to bothTheorem 3.1 and Theorem 3.2) is to construct a cobordism W from QY to Y .In the case that Y is disjoint union Y1 [ Y2, the cobordism W admits a map

23

Figure 2: A cobordism W from QY to Y D Y1 [ Y2.

� W W ! P , where P is a 2-dimensional pair-of-pants cobordism. This isshown schematically in Figure 2. The 4-dimensional cobordism is the unionof two pieces. The first piece is the product Œ0; 1� � Y 0, where Y 0 as beforeis obtained from Y by cutting open along †1 and †2. (In the Figure, thisappears as the union of two pieces, corresponding to the decomposition of Y 0

as Y 01 [ Y02.) The second piece is the product of the closed surface †1 with a 2-

manifold U with corners: U corresponds to the gray-shaded area in the figure.The two pieces are fitted together as shown, using the diffeomorphism h. If Yis connected, then the picture looks just the same in the neighborhood of theshaded region, but the product region Œ0; 1� � Y 0 is connected; the cobordismW in this case does not admit a map to the pair of pants.

There is a very similar cobordism NW which goes the other way: Theo-rem 3.1 arises because the cobordisms W and NW give rise to mutually inversemaps (in the case of genus at least 2)

HM.W / W HM�. QY j Q†/! HM�.Y j†/

HM. NW / W HM�.Y j†/! HM�. QY j Q†/:

when the coefficients are a field.To show that the cobordisms induce mutually inverse maps, let X be the

cobordism from QY to QY formed as the union of W and NW . We must show thatX gives rise to the identity map on HM�. QY j Q†/. This will show that HM. NW / ı

HM.W / D 1, and there will be a similar argument for the other composite.Note that† and Q† are homologous in X , so the map induced by X really doesfactor through HM�.Y j†/, not just HM�.Y /.

The manifold X is shown schematically in Figure 3 for the case that Y

24

Figure 3: The composite cobordism X from QY to QY .

has two components, Y1 [ Y2, in which case it admits a map � to the twice-punctured genus-1 surface, as drawn. Over the shaded region V it is a product,

��1.V / D †1 � V

D †2 � V:

If Y is connected, the picture is essentially the same in the neighborhood of��1.V /. Let k be the closed curve in V that is shown, and let K be the inverseimage

K D ��1.k/

D †1 � k:

(We continue to identify †1 with †2 via h in what follows.) Let X 0 be themanifold-with-boundary formed by cutting along K. Its boundary is twocopies of K. Let X� be the new cobordism from QY to QY obtained by attachingtwo copies of †1 �D2, with @D2 being identified with k:

X� D X 0 [ .†1 �D2/ [ .†1 �D

2/:

Floer’s proof hinges on the fact that the manifold X� is just the productcobordism from QY to QY . This means that we only need show that X� givesrise to the same map as X . This desired equality can be deduced from theformalism of section 2.5, for it is precisely Proposition 2.5. This concludesthe proof that HM. NW / ı HM.W / D 1. The picture for the composite of thetwo cobordisms in the other order is shown in Figure 4. The proof that this

25

Figure 4: The composite cobordism in the opposite order, from Y to Y , in the casethat Y has two components.

composite gives the identity is essentially the same: the relationship betweenY and QY is a symmetric one, except that we have allowed only Y to have twocomponents. Figure 4 shows the corresponding curve Qk in this case, alongwhich one must cut, just as we cut along k in the previous case. This completesthe proof of Theorem 3.1.

The proof of Theorem 3.2 is very similar. The same cobordisms W andNW are used. In the cobordism W , there is a 2-chain �W whose boundary is� � Q�. It consists of the product chain Œ0; 1� � �0 in part of W obtained fromŒ0; 1�� Y ; while over the shaded region U in Figure 2, the cycle �W is a sectionfpg � U of †1 � U . There is a similar 2-chain � NW in NW , and these fit togetherto give a 2-chain �X in the composite cobordism X (Figure 3). The 3-manifoldK � X lying over the curve k is now a 3-torus, and K meets �X transversely ina standard circle. The proof now proceeds as before, but using Corollary 2.10in place of Proposition 2.5. We learn that the composite cobordism X gives amap which is .t�t�1/ times the map arising from the trivial product cobordismX�. That is,

HM. NW jF NW I�� NW / ıHM.W jFW I��W/ D .t � t�1/:

The same holds for the composite in the opposite order. Since t � t�1 is a unitin R, this means that HM.W jFW I��W

/ is an isomorphism, as required.

4 Monopole Floer homology for sutured manifolds

In this section, we give the definition of the monopole homology groups forbalanced sutured manifolds, which are the main object of study in this paper.

26

4.1 Closing up sutured manifolds

We recall Juhasz’s definition of a balanced sutured manifold [12], a restrictedversion of Gabai’s notion of a sutured manifold [9]:

Definition 4.1. A balanced sutured manifold .M; / is a compact, oriented 3-manifold M with boundary, equipped with the following data:

(a) a closed, oriented 1-manifold s. / in @M , i.e. a collection of disjoint ori-ented circles in the boundary, called the sutures;

(b) a union A. / of annuli, which comprise a tubular neighborhood of s. /in @M ; the closure of @M n A. / is called R. /.

These are required to satisfy the following conditions:

(a) M has no closed components;

(b) if the components of @A. / are oriented in the same sense as the sutures,then it should be possible to orient R. / so that its oriented boundarycoincides with this given orientation of A. /;

(c) R. / has no closed components (which implies that the orientation inthe previous item is unique); we call it the canonical orientation;

(d) if we define RC. / (and R�. / also) as the subset of R. / where thecanonical orientation coincides with the boundary orientation (or its op-posite, respectively), then �.RC. // D �.R�. //. ˙

It is often helpful to consider sutured manifolds as manifolds with corners:the corners run along the circles @A. / and separate the flat annuli from therest of the boundary. Note that M need not be connected. A model exampleis a product sutured manifold

.Œ�1; 1� � T; ı/:

Here T is an oriented surface with non-empty boundary and no closed com-ponents, and the sutures are

s.ı/ D f0g � @T

with the boundary orientation. The annuli A.ı/ are Œ�1; 1� � @T , and we have

RC.ı/ D f1g � T

R�.ı/ D f�1g � T:

27

Given a balanced sutured manifold .M; /, we form a closed, orientedmanifold Y D Y.M; / as follows. The closed manifold is dependent on somechoices, as we shall see. First, we choose an oriented connected surface Twhose boundary components are in one-to-one correspondence with the com-ponents of s. /. We call T the auxiliary surface. From T we form the productsutured manifold .Œ�1; 1��T; ı/ as just described. We then glue the annuliA.ı/to the annuli A. /: this is done by a map

A.ı/! A. /

which is orientation-reversing with respect to the boundary orientations andwhich maps @RC.ı/ to @RC. /. The result of this step is a 3-manifold withexactly two boundary components, NRC and NR�, which are closed orientablesurfaces of equal genus:

NRC D RC. / [ f1g � T

NR� D R�. / [ f�1g � T

We require T to be of sufficiently large genus (genus zero may suffice, and genustwo always will) so that two conditions hold:

(C1) the genus of NR˙ is at least two;

(C2) the surface T contains a simple closed curve c such that f1g � c andf�1g � c are non-separating curves in NRC and NR� respectively.

Finally, form Y.M; / by identifying NRC with NR� using any diffeomorphismwhich reverses the boundary orientations (i.e. preserves the canonical orienta-tions),

h W NRC ! NR�:

Inside Y is a closed, connected, non-separating surface NR, obtained fromthe identification of NRC with NR�. We can orient NR using the canonical ori-entation of RC. /. As an oriented pair, .Y; NR/ depends only on two things,beyond .M; / itself: first, the choice of genus for T , and second the choice ofdiffeomorphism h used in the last step.

Definition 4.2. We call .Y; NR/ a closure of the balanced sutured manifold .M; /if it is obtained in this way, by attaching to .M; / a product region Œ�1; 1�� Tsatisfying the above conditions and then attaching NRC to NR� by some h. ˙

28

4.2 The definition

Let Y D Y.M; / be formed from a sutured manifold .M; / as described inthe previous subsection. Recall that Y contains a connected, oriented closedsurface NR, by construction, whose genus is at least two. We make the followingdefinition:

Definition 4.3. We define the monopole Floer homology of the sutured mani-fold .M; / to be the finitely-generated abelian group

SHM.M; / WD HM�.Y j NR/;

where Y D Y.M; / is a closure of .M; / as described in Definition 4.2, andthe notation on the right follows (4). ˙

As it stands, this definition appears to depend on the choice of genus, g,for the auxiliary surface T , as well as on the choice of gluing diffeomorphismh. In section 4.3 we shall prove:

Theorem 4.4. The group SHM.M; / defined in 4.3 depends only on .M; /, noton the choice of genus g for the auxiliary surface T or the diffeomorphism h.

There is a version of SHM with local coefficients that we shall use at somepoints along the way. Recall that T is required to contain a curve c that yieldsnon-separating curves f˙1g � c on NR˙. Let us choose the diffeomorphism h

so that h maps f1g � c to f�1g � c, preserving orientation. Thus the surface NRin Y.M; / now contains a closed curve Nc, the image of f˙1g � c. Let c0 be anydual curve on NR: a curve c0 with Nc � c0 D 1 on NR.

Definition 4.5. We define the monopole Floer homology of the sutured mani-fold .M; / with local coefficients to be the R-module

SHM.M; I��/ WD HM�.Y j NRI��/;

where the closure Y D Y.M; / is constructed using a diffeomorphism h sat-isfying the constraint just described, and � is the 1-cycle in Y carried by thecurve c0 dual to Nc as above. ˙

We shall see that this is independent of the choice of �. When using localcoefficients in this way, we can relax the requirement that NR has genus 2 ormore (condition (C1) above) and allow closures in which NR has genus 1:

29

Proposition 4.6. As long as t�t�1 is invertible in R, the R-module SHM.M; I��/

defined in 4.5 depends only on .M; / and R, not on the remaining choices. Fur-thermore, subject to the same condition on R, one can relax the condition (C1)above and allow NR to have genus 1 when using local coefficients.

In the case that NR does have genus 2 or more, we shall also see that we cantake � to be any non-separating curve on NR, rather than a curve dual to Nc.

4.3 Proof of independence

We now prove Theorem 4.4: our definition of the monopole Floer homologyof a balanced sutured manifold .M; / is independent of the choices made inits definition. The proof consists of several applications of Floer’s excisiontheorem. We begin with an observation about mapping tori:

Lemma 4.7. Let Y ! S1 be a fibered 3-manifold whose fiber R is a closedsurface of genus at least 2. Then HM.Y jR/ Š Z.

Proof. In the case of the product fibration, we have already seen this in theprevious section. If Yh denotes the mapping torus of a diffeomorphism h W

R ! R, then the excision theorem, Theorem 3.1, in the guise of (12), gives usan injective map

HM.YhjR/˝HM.Yg jR/! HM.YghjR/

with cokernel the Tor term. When g D h�1, the mapping tori Yh and Yg areorientation-reversing diffeomorphic; and HM.Yg jR/ is therefore isomorphicto HM.YhjR/ as an abelian group. (This is for the same reason that the ho-mology and cohomology of a finitely-generated complex of free Z-modules areisomorphic, as abelian groups.) So we obtain an injective map

HM.YhjR/˝HM.YhjR/! Z

whose cokernel is torsion. This forces HM.YhjR/ to be Z.

Corollary 4.8. Let Y1 be a closed oriented 3-manifold containing a non-separat-ing oriented surface NR of genus two or more. Let QY be obtained from Y1 bycutting along NR and re-gluing by an orientation-preserving diffeomorphism h.Then HM.Y1j NR/ and HM. QY j NR/ are isomorphic.

Proof. Apply the excision theorem, Theorem 3.1, with Y D Y1 [ Y2, takingY2 to be the mapping torus of h and †1 D †2 D NR. Lemma 4.7 tells us thatHM.Y2j NR/ Š Z, so HM.Y1j NR/ Š HM. QY j NR/ by the excision theorem.

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Consider now the situation of Theorem 4.4. We have a closed 3-manifoldY D Y.M; / whose construction depends on a choice of genus g for T anda choice of diffeomorphism h. We are always supposing that Y has been con-structed using an auxiliary surface T subject to the conditions (C1) and (C2).The above corollary tells us that HM.Y j NR/ is independent of the choice of h.So the group SHM.M; /, as we have defined it, depends only on the choice ofg. Let us temporarily write it as

SHMg.M; /: (14)

We can apply the same arguments with local coefficients: Theorem 3.3 can beused in place of Theorem 3.1 to see that

SHMg.M; I��/ (15)

(as defined in Definition 4.5) depends at most on the choice of g, not on h (aslong as conditions (C1) and (C2) hold). However, we can also relate (14) to(15) directly:

Lemma 4.9. If the coefficient ring R has no Z-torsion, then we have

SHMg.M; I��/ D SHMg.M; /˝R:

Proof. In the definition of the local system ��, the 1-cycle � is parallel to acurve lying on NR. The result therefore follows from the definitions and Corol-lary 3.4.

Because we already know that SHMg.M; / is independent of h, the abovelemma establishes that SHMg.M; I��/ is also independent of h, and that itis also independent of the choice of �. Next we prove:

Proposition 4.10. If t � t�1 is invertible in the coefficient ring R and R hasno Z-torsion, then the Floer group with local coefficients, SHMg.M; I��/, isindependent of g.

Proof. Fix g1 and let T be a surface of genus g1. Let Y1 be the resulting closureof .M; /, and write NR1 for the surface it contains. Recall that we required Tto contain a simple closed curve c such that f1g � c and f�1g � c are non-separating in NR˙. We can form a surface QT of genus g1 C 1 by the followingprocess. We take a closed surface S of genus 2 containing a non-separatingclosed curve d . We then cut T along c and cut S along d , and we reglue toform QT as shown in Figure 5. The figure also shows curves c0 and d 0 dual to c

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Figure 5: Increasing the genus of T by 1.

and d . The curve c0 is supposed to be extended (out of the picture) to becomea simple closed curve dual to c in the larger surface NRC D RC. / [ f1g � T .

In forming the closure Y1 using T , we can arrange that the diffeomorphismh W NRC ! NR� carries f1g � c to f�1g � c by the identity map on c. Thisis because any non-separating curve is equivalent to any other in an orientedsurface. This implies that Y1 can be identified with the product S1 � T oversome neighborhood of c in T . So Y1 contains a torus, S1�c. The dual curve c0

on NRC becomes a curve (also called c0) in Y1 which intersects the torus S1 � conce.

We will apply the second version of the excision theorem, Theorem 3.2, asfollows. We take Y D Y1 [ Y2 with Y1 as given, and Y2 D S1 � S . We take †1to be the torus S1 � c inside the product region of Y1 and †2 to be S1 � d . Wetake � in Y to be �1C �2, where the cycle �1 is c0 and �2 is fpointg � d 0. These1-cycles intersect the respective tori once each; and �1 is of the sort requiredfor the definition of SHMg1.M; I��1

/ in Definition 4.5. To play the role ofthe surface F D F1 [ F2 in Theorem 3.2 we take NR1 [ NR2, where NR2 is thegenus-2 surface fpointg � S .

The manifold QY obtained from Y D Y1 [ Y2 in the excision theorem isanother closure of the original .M; /, using the auxiliary surface QT of genusone larger than T and a diffeomorphism Qh obtained by extending h triviallyover the extra handle. It contains a closed surface QR whose genus is one largerthan the genus of NR1. This is the surface obtained from NR1 and NR2 by cutting

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and gluing. So to prove the proposition, we must prove

HM.Y1j NR1I��1/ Š HM. QY j QRI� Q�/: (16)

Theorem 3.2, provides an isomorphism

HM..Y1 [ Y2/j. NR1 [ NR2/I��/! HM. QY j QRI� Q�/:

But HM.Y2j NR2I��2/ is just R by Corollary 2.3, because this manifold is a

product, so (16) follows from the Kunneth theorem. This completes the proofof the proposition.

Remark. Although Figure 5 is drawn so as to make clear that the excision the-orem is applicable, the topology can be described more simply. Let G denotesthe genus-one surface with one boundary component, obtained by cutting Sopen along d and then removing a neighborhood of d 0. Then the operationof forming QT as shown is the same as removing a neighborhood of the pointx D c \ c0 and attaching G to the boundary so created: a connected sum inother words. The 3-manifold picture is obtained from this connected-sum pic-ture by multiplying with by S1. That is, we drill out a neighborhood of S1�fxgand glue in S1 �G.

Now we can complete the proof of the theorem:

Proof of Theorem 4.4. We have seen that there is no dependence on the choiceof diffeomorphism h, and we have been considering the dependence on thegenus g: we wish to show that SHMg.M; / is independent of g. FromLemma 4.9 and Proposition 4.10, we learn that the R-module

SHMg.M; /˝R

is independent of g whenever R has no Z-torsion and t � t�1 is invertible. Butif A and B are finitely-generated abelian groups and A ˝R Š B ˝R as R-modules for all such R, then we must have A Š B. For this one can take auniversal example for R, namely the ring obtained by inverting t � t�1 in theZŒR�, the group ring of R.

Finally, we turn to Proposition 4.6. Up until this point we have been as-suming that NR has genus 2 or more. But the proof of Proposition 4.10 worksjust as well in the genus 1 case. Thus if Y1 is a closure formed with NR1 of genus1 and . QY ; QR/ is formed as in the proof of Proposition 4.10 with NR of genus 2,then

HM�.Y1j NR1I��1/ Š HM�. QY j QRl� Q�/:

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The group on the right is something we already know to be independent ofother choices: we have therefore

HM�.Y1j NR1I��1/ D SHM.M; /˝R: (17)

This verifies Proposition 4.6.

5 Knot homology

Juhasz showed in [12] that knot homology could be obtained as a special caseof his (Heegaard) Floer homology of a sutured manifold. Specifically, given aknot K in a closed 3-manifold Z, one can form a sutured manifold .M; / bytaking M to be the knot complement (with a torus boundary) and taking thesutures to be two oppositely-oriented meridians. In the monopole case we haveat present no a priori notion of knot homology; but we are free to take Juhasz’sprescription as a definition of knot homology and pursue the consequences.Thus:

Definition 5.1. For a knot K in a closed, oriented 3-manifold Z, we define themonopole knot homology KHM.Z;K/ to be the monopole homology of thesutured manifold .M; / associated to .Z;K/ by Juhasz’s construction. Thatis,

KHM.Z;K/ WD SHM.M; /

where M D Z n N ı.K/ is the knot complement and s. / consists of twooppositely-oriented meridians. ˙

To understand what this definition leads to, we must construct a suitableclosure of the sutured manifold.

5.1 Closures of knot complements

So let K be a knot in a closed manifold Z, and let .M; / be the knot comple-ment, with two sutures as just described. We can describe a particularly simpleclosure of .M; / as follows, if we temporarily relax the rules and allow theauxiliary surface T to be an annulus. (The reason this is not a valid closure of.M; / for our purposes is that the resulting surfaces NR˙ will have genus 1. Wewill correct this shortly, replacing the annulus by a surface of genus 1.) Let Nbe a closed tubular neighborhood of K, and let N 0 � N be a smaller one. Letm be a meridian of K, lying outside N 0 but inside N . We will consider M tobe Z nN 0, and we take two meridional sutures s. / on the boundary of N 0.

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Figure 6: The part of L lying inside the larger tubular neighborhood N .

If we take T to be an annulus and attach Œ�1; 1� � T by gluing the two annuliŒ�1; 1� � @T to the sutures A. /, then what results is a 3-manifold L with twotori as boundary components: we can identify it with the complement of atubular neighborhood of m in M .

Figure 6 shows the part of L that lies inside the tubular neighborhoodN of K. (The top and bottom are identified.) The figure shows a verticalsolid torus N with a smaller vertical solid torus N 0 drilled out of it, as wellas a neighborhood U of the meridian m, which has also been removed. Theboundary of L consists of the inner vertical boundary (the boundary of N 0)and the boundary of the horizontal solid torus (the boundary of U ). Theseboundary components are NRC and NR�.

If we choose a framing of K, then we obtain a fibration of L\N by punc-tured annuli E (one of which is shown gray in the figure). We now form theclosure Y1 D Y.M; / using T as the auxiliary surface by gluing NRC to NR�: oneach punctured annulus E, we glue the circle E \ NRC to E \ NR�. This turnseach annulus E into a genus-1 surface F with one boundary component. (Theremaining boundary component of F lies on the outer torus, @N .) Thus wehave seen:

Lemma 5.2. Using an annulus T as the auxiliary surface, a closure of the sutured

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Figure 7: The closed surface F obtained by gluing two boundary components of E:a genus-one surface with one boundary component. The curve ˛ is the gluing locus.

manifold .M; / associated to a knotK inZ can be described by taking a surfaceF of genus one, with one boundary component, and gluing F � S1 to the knotcomplement Z nN . The gluing is done so that fpg � S1 is is attached to themeridian of K on @N and @F � fqg is glued to any chosen longitude of K on@N .

A shorter way to say what we have done is to that we have glued togethertwo knot complements: for the knot K in Z and the standard circle “knot” inthe 3-torus, using any chosen framing of the former and the standard framingof the latter, attaching longitudes to meridians and meridians to longitudes.We give a name to this closed manifold:

Definition 5.3. We write Y1.Z;K/ for the closed 3-manifold obtained from theframed knot K in Z by the construction just described. ˙

As we pointed out at the beginning of this subsection, we have describeda closure of the sutured manifold .M; / that is illegitimate, because T is anannulus and NR has genus one. We now described how Y1.Z;K/ gets modified ifwe use a surface QT of genus one (still with two boundary components) insteadof T . Figure 7 shows the surface F . The curve ˛ on F is the intersection of Fwith the torus NR � Y.Z;K/ where NRC and NR� are glued. Thus NR is the torus

NR D ˛ � S1

� F � S1

� Y.Z;K/

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The image of Œ�1; 1� � T in Y.Z;K/ is a copy of S1 � T and can be identifiedwith the neighborhood of ˇ � S1:

S1 � T D nbd.ˇ/ � S1 � F � S1;

where nbd.ˇ/ � F is an annular neighborhood of ˇ. The identification of thevarious factors is as indicated: the S1 factor in S1 � T becomes the ˇ factoron the right, and the core of the annulus T becomes the S1 factor on the right.Recalling the remark made at the end of section 4.3, we see that to effectivelyincrease the genus of the auxiliary surface by 1, we should:

(a) drill out a tubular neighborhood ˇ �D2 of the circle ˇ � fqg � F � S1;

(b) attach S1 � G, where G is a genus-one surface with one boundary com-ponent, by a diffeomorphism

S1 � @G ! ˇ � @D2

which preserves the order of the factors.

(In the second step, a framing of fqg�ˇ is needed, but we have a preferred onebecause ˇ lies on fqg � F .)

Definition 5.4. We write QY .Z;K/ for the manifold obtained from Y1.Z;K/ bythe two steps just described. It is a closure of the sutured manifold .M; /associated to the knot K in Z obtained using a genus-one auxiliary surface QT ;and it depends only on a choice of framing for K. ˙

While the closure Y1.Z;K/ has a genus-one surface NR, the closure QY .Z;K/has a genus-two surface QR. The latter is obtained from NR D ˛�S1 by removinga neighborhood of the point .x; q/ in ˛�S1 and adding the genus-one surfacefxg � G. To summarize this discussion, we have the following, essentially bydefinition now:

Corollary 5.5. The monopole knot homology KHM.Z;K/ can be computed asthe ordinary monopole homology HM. QY j QR/, where QY D QY .Z;K/ is as above.Any framing of K can be used in the construction of QY .

Remark. Both Y1.Z;K/ and QY .Z;K/ can be described alternatively as follows.Let S be a closed surface of genus l , let c be a non-separating simple closedcurve on S , and let Oc be the curve fpg � c in the 3-manifold S1 � S . Let N. Oc/

37

be a tubular neighborhood. Let Yl be the result of gluing the complement of Octo the complement of K:

Yl D .S1� S/ nN ı. Oc/ [� Z nN

ı.K/ (18)

where � identifies the meridian curves of Oc to the longitudes of K and viceversa. (We give Oc the obvious framing, and we recall that a framing of K hasbeen chosen.) Then the manifold Y1 is Y1.Z;K/, and Y2 is QY .Z;K/.

We can also use the simpler manifold Y1 to compute monopole knot ho-mology, as long as we switch to local coefficients. This is the content of thenext lemma.

Lemma 5.6. If t � t�1 is invertible in the coefficient ring R and R has noZ-torsion, then the knot homology KHM.Z;K/ ˝ R can be computed asHM�.Y1I� O /, where Y1 is the manifold described in Definition 5.3 and O is thecurve ˛ � fpg in F � S1 � Y1, regarded as a 1-cycle.

Proof. According to Proposition 4.6, we can use the closure Y1 to computeSHM.M; I��/. Together with Lemma 4.9, this tells us that

HM�.Y1I� O / Š SHM.M; /˝R;

where .M; / is the sutured manifold obtained from the knot complement byJuhasz’s prescription.

5.2 Properties of monopole knot homology

Suppose that the knot K � Z is null-homologous, and let † be a Seifert sur-face for K: an oriented embedded surface in Z n N ı with boundary a simpleclosed curve on @N . We can frame the knot K so that N is identified withK�D2 and @† isK�fq0g for some q0 2 S1. We can also regard† as a surfacein the manifold Y1.Z;K/ (Definition 5.3). The union of † and F � fq0g inY1.Z;K/ is a closed oriented surface

N† D † [ .F � fq0g/ � Y.Z;K/:

Its genus is one more than the genus of †. The surface F � fq0g � Y1.Z;K/remains intact in the manifold QY .Z;K/ for q0 ¤ q (Definition 5.4), so we canregard N† also as a closed surface in QY D QY .Z;K/. Using the surface N†, wecan decompose KHM.Z;K/ according to the first Chern class of the spinc

structure. We write

KHM.Z;K/ DMi2Z

KHM.Z;K; i/

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whereKHM.Z;K; i/ D

Ms2S. QY j QR/

hc1.s/;Œ N†�iD2i

HM�. QY ; s/:

IfZ is not a homology sphere, then the decomposition by spinc structures maydepend on the choice of the relative homology class for the Seifert surface †,in which case one should write

KHM.Z;K; Œ†�; i/ (19)

for the summands.Some familiar properties of the (Heegaard) knot homology of Ozsvath-

Szabo and Rasmussen carry over to this monopole version.

Lemma 5.7. The groups KHM.Z;K; i/ and KHM.Z;K;�i/ are isomorphic.

Proof. The isomorphism arises from the isomorphism between HM�.Y; s/and HM�.Y; Ns/, where Ns is the conjugate spinc structure.

Lemma 5.8. The group KHM.Z;K; i/ is zero for ji j larger than the genus of †.

Proof. The adjunction inequality tells us that HM�.Y; s/ is zero for spinc

structures s with c1.s/Œ N†� greater than 2g. N†/� 2. The genus of N† is one largerthan the genus of †.

Lemma 5.9. For a classical knot K in S3 of genus g, the monopole knot homol-ogy group KHM.S3; K; g/ is non-zero.

Proof. We use the description of QY D QY .K/ as the the manifold Y2, where Ylis the manifold described by (18). Let S be the genus-2 surface used there, letc be the closed curve on S , and let c0 be a dual curve on S meeting c once.According to Gabai’s results [9, 10], a Seifert surface † of K of genus g arisesas a compact leaf of a taut foliation FK of S3 n N ı.K/, and we can ask thatthe leaves of FK meet @N.K/ in parallel circles. On the other hand, S1 �S hasa taut foliation FS which is transverse to the curve Oc D fpg � c. This foliationis obtained from the trivial product foliation by cutting alont the torus S1 � c0

and regluing with a small rotation of the S1 factor. Together, the foliationsFK and FS define a foliation F of QY D Y2. The surface N† sits inside Y2 asthe union of the Seifert surface † and the punctured torus .S1 � c0/ n D2.The spinc structure sc determined by F has first Chern class of degree 2gon N† and degree 2 on the genus-2 surface, so HM�. QY ; sc/ is a summand ofKHM.S3; K; g/ by definition. The non-vanishing theorem from section 2.4tells us that this group is non-zero.

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Lemma 5.10. Let K be a classical knot and let �.K; i/ denote the Euler char-acteristic of KHM.S3; K; i/, computed using the canonical Z=2 grading onmonopole Floer homology [18]. Then the finite Laurent seriesX

i

�.K; i/T i

is the symmetrized Alexander polynomial,�K.T /, for the knotK, up to an over-all sign.

Proof. In different guise, this is essentially the same result as that of Fintushel-Stern [7] and Meng-Taubes [20]. Let QY D QY .S3; K/ be the usual closure ofthe sutured manifold associated to .S3; K/ as in Definition 5.4, let QR be thegenus-2 surface in QY and let N† � QY be the surface of genus gC 1 formed froma Seifert surface † for K and the genus-1 surface F .

The Euler characteristic can be computed from the Seiberg-Witten invari-ants of the manifold S1 � QY . Specifically, regard both N† and QR as surfacesin

XK D S1� QY :

Take � to be the 2-cycle in XK defined by N† and consider the generating func-tion m.XK ; Œ��/ as in (2), but modified to use only spinc structures that are oftop degree on QR. We introduce the notation

m0.XK ; Œ��/ DX

s2S.XK j QR/

m.XK ; s/thc1.s/;Œ��i:

We then have Xi

�.K; i/t2i D m0.XK ; Œ��/:

LetX0 be the same type of 4-manifold asXK , but formed using the unknotin place of K. The corresponding 3-manifold QY0 is S1 � S , where S has genus2; so X0 is T 2 � S . The remark following Corollary 5.5 explains that QY isformed from QY0 by drilling out a neighborhood of a curve Oc and gluing in theknot complement MK D S2 n N ı.K/. If follows that XK is formed from X0by a “knot surgery” in the sense of [7]. This, one drills out a neighborhood ofthe torus S1 � Oc and glues in S1 �MK . In the formalism of section 2.6, we cantherefore compute the ratio

m0.XK ; Œ��/=m0.X0; Œ��/

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as the ratio of the invariants associated to .S1 �MK ; �K/ and .S1 �M0; �0/.Here M0 is the knot complement for the unknot, and �K and �0 are the 2-chains defined by Seifert surfaces forK and the unknot respectively. This ratiois precisely what is calculated in [7] (see also [18, section 42.5]), and it is equalto �K.t2/. The lemma follows.

Given a null-homologous knot K in a 3-manifold Z, there is a rather morestraightforward way to arrive at a sutured manifold than the one that leads toknot homology. We can simply choose a Seifert surface † for K and cut theknot complement Z n N ı.K/ open along †. The result is a sutured manifold.M†; ı/ with a single suture and having RC.ı/ D R�.ı/ D †. The monopoleFloer homology of this sutured manifold captures the top-degree part of themonopole knot homology:

Proposition 5.11. In the above situation, let g be the genus of the Seifert surface†, and suppose g ¤ 0. Then SHM.M†; ı/ is isomorphic to KHM.Z;K; Œ†�; g/.

Proof. It is sufficient to prove that

SHM.M†; ı/˝R Š KHM.Z;K; Œ†�; g/˝R

when the coefficient ring R has t� t�1 invertible and no Z-torsion. Lemma 5.6tells us that we can compute the right-hand side using the manifold Y1 as

KHM.Z;K; Œ†�; g/˝R D HM�.Y1j N†I� O /

where N† is the surface of genus g C 1 in Y1. On the other hand, the samemanifold Y1 arises as a closure of .M†; ı/ in the sense of section 4.1, so we alsohave

SHM.M†; ı/˝R D SHM.M†; ıI��/

D HM�.Y1j N†I� O /:

This proves the proposition.

6 Fibered knots

6.1 Statement of the result

In this section, we adapt the material from [23] to show that the monopole ver-sion of knot homology detects fibered knots. For the most part, the argumentsof [23] carry over with little modification.

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A balanced sutured manifold .M; / is a homology product if the inclusionsRC. / ! M and R�. / ! M are both isomorphisms on integer homologygroups. The main target is the following theorem.

Theorem 6.1. Suppose that the balanced sutured manifold .M; / is taut anda homology product. Then .M; / is a product sutured manifold if and only ifSHM.M; / D Z.

The application to fibered knots is a corollary:

Corollary 6.2. If K � S3 is a knot of genus g, then K is fibered if and only ifKHM.S3; K; g/ D Z.

Proof of the corollary. The “only if” direction is a straightforward matter: itfollows from Lemma 4.7 and Proposition 5.11. The interesting direction is the“if” direction, and this can be deduced from Theorem 6.1 as follows.

Suppose that KHM.S3; K; g/ D Z. From Lemma 5.10 we learn that theAlexander polynomial ofK is monic and that its degree is g. Let† be a Seifertsurface for K of genus g, and let .M†; ı/ be the balanced sutured manifoldobtained by cutting open the knot complement along †. As Ni observes in[23, section 3], the fact that the Alexander polynomial is monic tells us that.M†; ı/ is a homology product. The group SHM.M†; ı/ is isomorphic toKHM.S3; K; g/ by Proposition 5.11, so SHM.M†; ı/ D Z. Theorem 6.1 im-plies that .M†; ı/ is a product sutured manifold, from which it follows that theknot complement is fibered.

We will prove Theorem 6.1 after some preliminary material on furtherproperties of SHM.

6.2 Spinc structures

The following definition of relative spinc structures on sutured manifolds coin-cides with that of Juhasz [13], in slightly different notation. If we regard .M; /as a manifold with corners, then it carries a preferred 2-plane field �@ on itsboundary: on RC. / and R�. /, we take �@ to be the tangent planes to theboundary, with the canonical orientation; and on each component of A. / wetake �@ to have, as oriented basis, first the outward normal toM and second thedirection parallel to the oriented suture. On a 3-manifold an oriented 2-planefield defines a spinc structure; so �@ gives a spinc structure in a neighborhoodof the boundary. We define S.M; / to be the set of extensions of s@ to a spinc

42

structure on all of M , up to isomorphisms which are 1 on @M . We refer toelements of S.M; / as relative spinc structures.

Consider the process of forming the closure Y D Y.M; /. When we attachŒ�1; 1��T to the annuli in @M , the 2-plane field �@ extends in the obvious way,as the tangents to fpg � T . When we the attach NRC to NR� using h, we obtaina 2-plane field on all of Y.M; / except the interior of the original M . On thesurface NR � Y , this 2-plane field is the tangent plane field. So we obtain anatural map

� W S.M; /! S.Y j NR/: (20)

Lemma 6.3. Let s1; s2 2 S.M; / be relative spinc structures whose differenceelement in H 2.M; @M/ is not torsion. Then we can choose T and the diffeomor-phism h so that �.s1/ and �.s2/ are spinc structures in S.Y j NR/ whose differenceis still non-torsion.

Proof. The statement only concerns the difference elements. The dual ofH 2.M; @M IQ/ is H 1.M IQ/, and what we must show is that given a non-zeroelement ˛ 2 H 1.M/, we can choose T and h so that ˛ is in the image of themap

H 1.Y /! H 1.M/:

To do this, consider as an intermediate step the manifold Y 0 with boundaryNRC[ NR� formed fromM by attaching Œ�1; 1��T . The mapH 1.Y 0/! H 1.M/

is surjective. Let ˇ be a class in H 1.Y 0/ which restricts to ˛. Represent thedual of ˇ by a closed surface .B; @B/ in .Y 0; @Y 0/. By adding to B an annuluscontained in the product region Œ�1; 1� � T if necessary, we can be assuredthat @B intersects both NRC and NR� in a collection of curves representing aprimitive, non-zero homology class. We can then modify B without changingits class so that @B consists of two circles: a non-separating curve in each ofNRC and NR�. Finally, we choose the diffeomorphism h W NRC ! NR� so as to

match up these curves. In this way we obtain a closed surface NB in Y whosedual class in H 1.Y / maps to ˛ in H 1.M/.

The following corollary is the tool used by Ghiggini [11] in his proof of theoriginal version of Corollary 6.2 for genus-1 knots.

Corollary 6.4. Suppose that .M; / admits two taut foliations F1 and F2 suchthat the corresponding spinc structures s1 and s2 have non-torsion differenceelement in H 2.M; @M/. Then SHM.M; / has rank at least 2.

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Proof. Choose the closure Y D Y.M; / so that �.s1/ and �.s2/ are differentspinc structures on Y , as Lemma 6.3 allows. The foliations F1 and F2 extendin an obvious way to foliations of Y belonging to the spincstructures �.s1/and �.s2/. By the non-vanishing theorem described in section 2.4, the Floergroups HM�.Y; �.s1// and HM�.Y; �.s2// both have non-zero rank. Both ofthese Floer groups contribute to HM�.Y j NR/ D SHM.M; /, because the spinc

structures �.si / belong to S.Y j NR/. So SHM.M; / has rank at least 2.

6.3 Decomposition theorems

The excision theorems, in addition to their role in showing that SHM.M; / iswell-defined, can be used in a straightforward way to establish some decompo-sition which related the Floer homology of a sutured manifold .M; / to thatof .M 0; 0/, obtained from .M; / by cutting along a surface. We record a fewtypes of such decomposition theorem here. To avoid various circumlocutionsinvolving tensor products and the Kunneth theorem, we shall work over Q in-stead of Z here; and when using local coefficients we shall take R to be a fieldof characteristic zero: either R with the usual exponential map, or the field offractions of the group ring QŒR�.

Proposition 6.5. Suppose .M; / is a disjoint union .M1; 1/[ .M2; 2/ and thatboth pieces are balanced. Then

SHM.M; IQ/ Š SHM.M1; 1IQ/˝ SHM.M2; 2IQ/:

Proof. It will be sufficient to prove this for the local coefficient versions,SHM.M; I��/, because of Lemma 4.9. Form the closures .Y1; NR1/ and.Y2; NR2/ of .M1; 1/ and .M2; 2/ by attaching product regions Œ�1; 1� � T1and Œ�1; 1��T2 respectively. Let c1 and c2 be non-separating curves on T1 andT2. When forming the closures Y1 and Y2, choose the diffeomorphisms h1 andh2 so that hi maps f1g�ci to f�1g�ci , as in the proof of Proposition 4.10. LetQT be the connected closed surface obtained from T1 and T2 by cutting open

along c1 and c2 and reattaching, similarly to Figure 5. Let Qh be the diffeomor-phism of QT that arises from h1 and h2, and let QY be the closure of .M; / thatis obtained by attaching QT to .M; / and gluing up using Qh. We now have aconnected closure QY that is related to Y D Y1[Y2 by cutting and gluing along2-tori S1�ci . So the excision theorem, Theorem 3.2, provides an isomorphism

HM�.Y j NRI��/! HM�. QY j QRI� Q�/;

44

and hence and isomorphism

HM�.Y j NRIR/! HM�. QY j QRIR/:

Since R is a field and Y is a disjoint union, the left-hand side is a tensor prod-uct, and the proposition follows.

Next we prove a version of Ni’s “horizontal decomposition” formula. Ahorizontal surface in .M; / is a surface S with �.S/ D �.RC. // such that@S consists of one circle in each of the annuli comprising A. /; it is requiredto represent the same relative homology class as R˙. / in H2.M;A. // andshould have Œ@S� D Œs. /� in H1.A. //. Cutting along a horizontal surfacecreates a new sutured manifold

.M 0; 0/ D .M1; 1/ [ .M2; 2/:

Proposition 6.6 ([23, Proposition 4.1]). If .M 0; 0/ is obtained from .M; / bycutting along a horizontal surface, then

SHM.M; IQ/ D SHM.M 0; 0IQ/:

Proof. This follows directly from Theorem 3.1 and Proposition 6.5.

We shall also need to decompose sutured manifolds by cutting along ver-tical surfaces. We prove a result along the lines of [23] and [13]. A productannulus in .M; / is an embedded annulus A D Œ�1; 1� � d in .M; / such thatthe circle dC D f1g � d lies in the interior of RC. / and d� D f�1g � d lies inthe interior of R�. /.

Proposition 6.7. Let .M 0; 0/ be obtained from .M; / by cutting along a productannulus A. Then

SHM.M; IQ/ D SHM.M 0; 0IQ/

if we are in either of the following two situations:

(a) the curves dC and d� represent non-zero classes in the first homology ofRC. / and R�. / respectively; or

(b) the curves dC and d� represent the zero class inH1.RC. // andH1.R�. //respectively, at least one of them does not bound a disk, and the annulus Aseparates M into two parts, M1 [ M2, one of which is disjoint from theannuli A. /.

45

Proof. We begin with case (a) of the proposition. We shall construct closures.Y; NR/ and . QY ; QR/ for .M; / and .M 0; 0/ which are related to each other asdescribed in the excision theorem, Theorem 3.2, and the result will follow.

When we attach the product Œ�1; 1� � T to .M; /, the curves dC and d�remain non-separating in the closed surfaces NR˙, because T is connected. Bytaking T to have non-zero genus, we can also ensure that there is a curve c inthe interior of T which is non-separating in T . So after attaching the productregion, we have two product annuli Œ�1; 1��d and Œ�1; 1��c, with independentnon-separating curves dC; cC in NRC in d�; c� in NR�. We can close up themanifold using a diffeomorphism h W NRC ! NR� such that h.dC/ D d� andh.cC/ D c�. The closure .Y; NR/ of .M; / that we arrive at in this way containstwo tori,

†1 D S1� c

†2 D S1� d

There is a 1-cycle � lying on NR that is transverse to both of these tori, so The-orem 3.2 is applicable. (This is an instance of that theorem where the manifoldY 0 obtained by cutting along †1 and †2 is connected.) The manifold . QY ; QR/obtained from .Y; NR/ by cutting along †1 [†2 and regluing is a closure of the.M 0; 0/, so we are done with case (a).

We turn to case (b). Without loss of generality, we suppose that M1 doesnot meet A. / and dC does not bound a disk. Let RC;1 denote RC. / \M1

and let R�;2 denote R�. / \M2. The surface RC;1 has genus at least 1 andits only boundary component is dC. In [23], Ni uses the following observation.The union

RC;1 [ A [R�;2

is isotopic to a horizontal surface in .M; / to which Proposition 6.6 applies.By cutting along this horizontal surface, the pieces we get from .M; / are (upto diffeomorphism) �

Œ�1; 1� �RC;1�[Œ�1;1��d M2

and

M1 [Œ�1;1��d

�Œ�1; 1� �R�;2

�:

In this way, case (b) is reduced to the case that either M1 or M2 is a product.IfM2 is a product, Œ�1; 1��R�;2, then the result is entirely straightforward:

the surface R�;2 contains all the annuli A. /. A closure Y of .M; / using an

46

auxiliary surface T can also be regarded as a closure of .M1; 1/ using theauxiliary surface R�;2 [ T . So we have

SHM.M; / D SHM.M1; 1/:

On the other hand, because M2 is a product, we have SHM.M1; 1/ D

SHM.M 0; 0/ by Proposition 6.5. Finally, if M1 is a product, then we cancut M1 open along a non-separating annulus because RC;1 has positive genus,and this does not change SHM, by part (a) of the proposition. After cuttingopen M1 in this way, we arrive at a situation in which proposition (a) appliesagain, and the proof is complete.

6.4 Proof of Theorem 6.1

Those ingredients of Ni’s proof from [23] which involve Heegaard Floer ho-mology have all been replicated here in the context of monopole Floer homol-ogy, so the proof carries through with little change. We outline the argument,adapted from [23]. Let .M; / be a balanced sutured manifold satisfying thehypotheses of the theorem, and suppose .M; / is not a product sutured mani-fold. We shall show that SHM.M; / has rank at least 2.

Because of Proposition 6.5, it is sufficient to treat the case that M is con-nected. Similarly, because of Proposition 6.6, we may assume that .M; / is“vertically prime”: that is, every horizontal surface in .M; / is a parallel copyof either RC. / or R�. /. By attaching product regions to .M; / and appeal-ing to Proposition 6.7, we are also free to suppose that .M; / has only onesuture. We now consider a maximal product pair i W Œ�1; 1� � E ,! .M; / asin [23, 24] and the induced map

i� W H1.Œ�1; 1� �E/! H1.M/:

There are two cases.

Case 1: i� is not surjective. In this case, Ni establishes that .M; / admitstwo taut foliations F1 and F2 whose difference element is non-torsion inH 2.M; @M/. It then follows from Corollary 6.4 that SHM.M; / has rank2 or more, as required.

Case 2: i� is surjective. In this case, let .M 0; 0/ be the complement of themaximal product pair. This is non-empty, because .M; / is not a productsutured manifold. Proposition 6.7 tells us that SHM.M; / and SHM.M 0; 0/

47

have the same rank. Ni observes that the vertically-prime condition on .M; /implies that M 0 is connected. Furthermore, .M 0; 0/ is a homology product,and its top and bottom surfaces R˙. 0/ are planar, because of the surjectivityof i�. The (connected) surfaces R˙. 0/ are not disks, so .M 0; 0/ has at leasttwo sutures. Let r � 2 be the number of sutures in .M 0; 0/. Let S be a planarsurface with rC1 boundary components, so that the product sutured manifoldŒ�1; 1�� S has r C 1 sutures. Form a new sutured manifold . QM; Q / by gluing rof the annuli from Œ�1; 1� � S to the annuli of .M 0; 0/. The resulting suturedmanifold . QM; Q / has

rank SHM. QM; Q / D rank SHM.M 0; 0/

by Proposition 6.7. Furthermore . QM; Q / is a homology product, and its maxi-mal product pair is Œ�1; 1� � S up to isotopy. The construction has been madeso that the inclusion of the maximal product pair in . QM; Q / is not surjective onH1, so we now have a situation which falls into Case 1 above. It follows thatSHM. QM; Q / has rank at least 2; and so too therefore does SHM.M; /. Thiscompletes Ni’s proof.

6.5 More decomposition theorems

In [13], rather general sutured manifold decompositions are considered, andresults of the following sort are obtained. Let .M; / be a balanced suturedmanifold, and let S � M be a decomposing surface in the sense of [9]. Thereis a sutured manifold decomposition,

.M; /SÝ .M 0; 0/;

and we shall suppose that .M 0; 0/ is also balanced (which implies that S hasno closed components). Under some mild restrictions on S , Juhasz proves in[13] that SFH.M 0; 0/ is a direct summand of SFH.M; /. An entirely simi-lar theorem can be proved in the context of monopole Floer homology, usingSHM.M; / in place of SFH.M; /. The following is a restatement of Theo-rem 1.3 of [13], though with less specific information about the spinc structuresthat are involved behind the scenes. In the statement of the theorem, an ori-ented simple closed curve C in R. / is called boundary coherent if it eitherrepresents a non-zero class in H1.R. // or it is the oriented boundary @R1 ofa compact subsurface R1 � R. / with its canonical orientation.

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Figure 8: Adding product 1-handles to a sutured manifold containing a decomposingsurface S .

Theorem 6.8 ([13, Theorem 1.3]). Let .M; / be a balanced sutured manifold and

.M; /SÝ .M 0; 0/

a sutured manifold decomposition. Suppose that the decomposing surface Shas no closed components, and that for every component V of R. /, the set ofclosed components of S \V consists of parallel oriented boundary-coherent sim-ple closed curves. Then the Heegaard Floer homology SFH.M 0; 0/ is a directsummand of SFH.M; /.

We have the following result.

Proposition 6.9. Theorem 6.8 continues to hold with monopole Floer homol-ogy in place of Heegaard Floer homology. That is, with the same hypotheses,SHM.M 0; 0/ is a direct summand of SHM.M; /.

Proof. By Lemma 4.5 of [13], Juhasz reduces this to the special case of a“good” decomposing surface S , by which is meant a surface S such that everycomponent of @S intersects both RC. / and R�. /.

Starting from a good decomposing surface S , we can pass to another spe-cial case as follows. Let C be a component of @S . By the definition of a gooddecomposing surface, C intersects the annuli A. / in vertical arcs. The num-ber of these arcs counted with sign is zero. Pair up these arcs accordingly; andfor each pair attach a product 1-handle as shown in Figure 8. Repeat this withevery other boundary component of @S . The result of this process is a newbalanced sutured manifold .M1; 1/ containing a new decomposing surface

49

S1. We have SHM.M1; 1/ Š SHM.M; /, because adding product handleshas no effect. (The inverse operation to adding a product handle can also bedescribed as removing a larger product region, by cutting along annuli parallelto the annuli where the handle is attached; so this operation is a special caseof one we have seen before.) Furthermore, if .M 01;

01/ is what we obtain from

.M1; 1/ by sutured manifold decomposition along S1, then .M 01; 01/ is also

related to .M 0; 0/ by adding adding product 1-handles. It therefore suffices toprove that SHM.M 01;

01/ is a direct summand of SHM.M1; 1/.

Looking at .M1; 1/, we now see that it is sufficient to prove the followinglemma, which is a priori a special case of the proposition.

Lemma 6.10. Let .M; / be a balanced sutured manifold and let

.M; /SÝ .M 0; 0/

be a sutured manifold decomposition. Suppose that S has no closed compo-nents and that the oriented boundary of @S consists of n simple closed curvesCC1 ; : : : ; C

Cn in RC. / and n simple closed curves C�1 ; : : : ; C

�n in R�. /. Sup-

pose further that the homology classes of CC1 ; : : : ; CCn are a collection of inde-

pendent classes in H1.RC. //, and make a similar assumption for R�. /. ThenSHM.M 0; 0/ is a direct summand of SHM.M; /.

Proof of the lemma. Form the closure Y D Y.M; / by attaching a productregion Œ�1; 1� � T as usual and then choosing the diffeomorphism h in sucha way that h.CCi / D C�i (with the opposite orientation) for all i . The resultof this is that Y contains two closed surfaces: first the usual surface NR, andsecond a surface NS obtained from S by identifying CCi with C�i for all i . Theintersection NS \ NR consists of n circles, C1; : : : ; Cn. Let F be the orientedsurface obtained from NS [ NR by smoothing out the circles of double points,respecting orientations.

The same surface F � Y can be arrived at from a different direction. Startwith .M 0; 0/. We can write A. 0/ as a union of components

A. 0/ D A. / [ A1;

where A. / are the annuli of the original sutured manifold .M; / and A1 arethe new annuli. The new annuli can be written as Œ�1; 1� � D˙i , where thecollection of curves D˙i are in natural correspondence with the curves C˙i .We now form a closure Y 0 of .M 0; 0/ as follows. We attach a product regionŒ�1; 1� � T 0 to .M 0; 0/, where T 0 is a (disconnected) surface

T 0 D T [ T1:

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Here T is the surface used to close Y and T1 is a collection of n annuli

T1 D T1;1 [ � � � [ T1;n:

Although T 0 breaks the rules by being disconnected, we can still effectively useT 0 in constructing SHM.M 0; 0/ because of the arguments of section 6.3. Inattaching Œ�1; 1��T 0 to .M 0; 0/we glue Œ�1; 1��@T to the annuliA. / � A. 0/as we did when closing .M; /, and we glue the two components Œ�1; 1� � T1;ito the two annuli Œ�1; 1� �D˙i belonging to A1.

At this point, we have a manifold

.M 0; 0/ [ Œ�1; 1� � .T [ T1/

with two boundary components NR0˙

. The top surface NR0C can be described asa union

NR0C DNR�C[ SC [ fC1g � T1:

Here NR�C

is the surface with boundary obtained by cutting open NRC alongthe circles CCi , and the annuli fC1g � T1 are collars of half of the boundary

components of NR�C

. The surface SC is a copy of S . Up to diffeomorphism, wecan forget these annular regions and write

NR0C DNR�C[ SC

NR0� DNR�� [ S�:

That is, NR0C

is obtained from NRC by cutting open along the circles CCi andinserting a copy of S . Finally, form the closure Y 0 by using a diffeomorphism

h0 W NR0C !NR0�

which is equal to h on NR�C

and equal to the identity on S .The resulting closure Y 0 of .M 0; 0/ is diffeomorphic to Y ; and under this

diffeomorphism, the surface NR0 � Y 0 obtained from NR0˙

becomes the surfaceF . (See Figure 9.) It follows that we can calculate SHM.M 0; 0/ as

SHM.M 0; 0/ D HM�.Y jF /:

The homology class of F is the sum of the classes of NR and NS . Furthermore,�.F / D �. NR/C �. NS/. It follows from the adjunction inequality that the only

51

Figure 9: Decomposing M along S and then closing up to get F . The collars of @Mand @M 0 are marked with hatching near A. / and A. 0/. The product part Œ�1; 1��Tis not shown in the figure, which is otherwise a faithful representation after multiplyingby S1.

52

spinc structures in S.Y jF / which can have non-zero Floer homology are thosein the intersection S.Y j NR/ \ S.Y j NS/. So we have

SHM.M 0; 0/ DM

s2S. NR/\S. NS/

HM�.Y; s/;

whileSHM.M; / D

Ms2S. NR/

HM�.Y; s/:

This shows that SHM.M 0; 0/ is a direct summand of SHM.M; /, as thelemma asserts.

As is pointed out in [13], one can use Proposition 6.9 to give an alternativeproof of the non-vanishing of SHM.M; / when .M; / is taut. One uses asutured manifold hierarchy, starting at .M; / and ending at a product suturedmanifold, whose (monopole) Floer homology we know to be Z, so showingthat Z is a summand of SHM.M; /.

7 Instantons

Much of the contents of this paper can be adapted to the case of (Yang-Mills)instanton homology, instead of (Seiberg-Witten) monopole Floer homology.We present some of this material in this section. For background on instantonhomology, we refer to [4].

7.1 Instanton Floer homology

When looking at the monopole Floer homology groups HM�.Y; s/ of a 3-manifold Y , we could avoid difficulties arising from reducible solutions by con-sidering situations where only non-torsion spinc structures s played a role. Ininstanton homology, reducibles can be avoided by using SO.3/ bundles withnon-zero w2. We proceed as follows.

Fix a hermitian line bundle w ! Y such that c1.w/ has odd pairing withsome integer homology class. Let E ! Y be a U.2/ bundle with an isomor-phism � W ƒ2E ! w. Let C be the space of SO.3/ connections in ad.E/and let G be the group of determinant-1 gauge transformations of E (the au-tomorphisms of E that respect � ). The Chern-Simons functional on the spaceB D C=G leads to a well-defined instanton homology group which we writeas I�.Y /w [4]. It is also possible to use a slightly larger gauge group than G .

53

Fix a surface R � Y that has odd pairing with c1.w/. Let � D �R be a real linebundle with w1.�/ dual to R. The map E 7! E ˝ � gives rise to a map on thespace of connections,

�R W B ! B;

without fixed points, and there is a quotient B=�R. This is the same as thequotient of C by a gauge group which has G as an index-2 subgroup. Letus temporarily write I�.Y /w;R for the resulting instanton homology group:it is the fixed space of an induced involution on I�.Y /. As an example, inthe case Y D T 3, we have I�.T 3/w D Z ˚ Z. The involution interchangesthe two copies of Z, and I�.T 3/w;R D Z whenever w � ŒR� is non-zero. Ingeneral, I�.Y /w is .Z=8-graded. The involution acts with degree 4, and thegroup I�.Y /w;R is .Z=4/-graded.

Although these groups are defined with Z coefficients, it will be convenientto work with a field of characteristic zero; and in what follows we will take thatfield to be C. Thus we will take it that

I�.T3/w;R D C:

7.2 The eigenspace decomposition

The monopole Floer homology detects the Thurston norm of a 3-manifold (seesection 2.4); but the formulation of this statement requires the decompositionof the monopole Floer homology according to the different spinc structures.In order to relate instanton homology to the Thurston norm, one needs a de-composition of the instanton homology. As suggested in [15], such a decom-position arises from the eigenspaces of natural operators on the Floer groups.

Let Y be again a closed 3-manifold and w a line bundle as above. Given anoriented closed surfaceR in Y , there is a 2-dimensional cohomology class�.R/in B (for which our conventions follow [5]) and hence an operation of degree�2 on both I�.Y /w and I�.Y /w;R. There is also the class �.y/, for y a pointin y, which acts with degree 4. The operators �.R/ and �.y/ commute, so onecan look for simultaneous eigenvalues. In the special case that Y D S1 � †,with † a surface of positive genus, the eigenvalues of �.†/ and �.y/ werecomputed by Munoz in [21]:

Proposition 7.1 ([21, Proposition 20]). Let w ! S1�† be the line bundle whosefirst Chern class is dual to the S1 factor. Then the simultaneous eigenvalues ofthe action of �.†/ and �.y/ on I�.S1 �†/w are the pairs of complex numbers

.ir.2k/; .�1/r2/

54

for all the integers k in the range 0 � k � g � 1 and all r D 0; 1; 2; 3. Here idenotes

p�1.

Remark. In [21], the 2-dimensional class called ˛ corresponds to 2�.†/ here,and the class ˇ corresponds to �4�.y/. Also, the group HF�.S1 � †/ thatappears in [21] is our I�.S1 � †/w;†. Munoz computes the spectrum in thecase of I�.S1�†/w;†, but the case of I�.S1�†/w follows in a straightforwardmanner. Observe, in particular, that because �.†/ is an operator of degree 2on a .Z=8/-graded vector space, the eigenspaces of eigenvalues � and i� willalways be isomorphic.

As a corollary of this proposition, a similar result holds for a general 3-manifold Y .

Corollary 7.2. Let R � Y be closed connected surface of positive genus, andlet w have odd pairing with R. Then the eigenvalues of the action of the pair ofoperators �.R/ and �.y/ on I�.Y /w are a subset of the eigenvalues that occur inthe case of the product manifold S1�R. That is, they are pairs complex numbers

.ir.2k/; .�1/r2/

for integers k in the range 0 � k � g � 1.

Proof. Let R0 be a copy of R in the interior of the product cobordism W D

Œ�; 1� � Y . The action of �.R/ on I�.Y /w can be regarded as being defined bythis copy of R in the 4-dimensional cobordism. LetW 0 be the cobordism fromthe disjoint union S1 �R and Y at the incoming end to Y at the outgoing end,obtained by removing an open tubular neighborhood of R0 from W . We havea map defined by W 0,

W 0 W I�.S1�R/w ˝ I�.Y /w ! I�.Y /w :

The map is surjective, because one obtains the product cobordism by closingoff the boundary component S1�†. Furthermore, becauseR0 is homologousto surfaces in each of the three boundary components, we have, for example

W 0.�.R/a˝ b/ D �.R/ W 0.a˝ b/:

From this relation and the surjectivity of W 0 , it follows that the eigenvaluesof �.R/ on the outgoing end Y are a subset of the eigenvalues of the action of�.R/ on S1 � R. We obtain the result of the corollary by applying a similarargument to �.y/ and to �.R/2 C �.y/.

55

We can now give a definition in instanton homology of something that willplay the role that HM�.Y jR/ played in the monopole theory.

Definition 7.3. Let Y be a closed, oriented 3-manifold, w a hermitian line bun-dle on Y and R � Y a closed, connected, oriented surface on which c1.w/ isodd. Let g be the genus or R, which we require to be positive. We define

I�.Y jR/w

to be the simultaneous eigenspace for the operators �.R/, �.y/ for the pair ofeigenvalues .2g � 2; 2/. ˙

Remark. Except in the case that the genus is 1, we could define this more simplyas just the .2g � 2/-eigenspace of �.R/, as can be seen from Corollary 7.2

Although Munoz does not calculate the dimensions of the eigenspaces ingeneral for S1 � †, one can readily read off from the proof of [21, Proposi-tion 20] that the dimension of the eigenspace belonging to the largest eigen-value is 1. That is,

Proposition 7.4. Let Y D S1 � R with † of positive genus, and let w be the linebundle dual to the S1 factor. Then

I�.Y jR/w D C:

There is a simple extension of the above definition to the case that R hasmore than one component, as long as w is odd on each component. If thecomponents are Rm, then the corresponding operators �.Rm/ commute, andwe may take the appropriate simultaneous eigenspace. In general, the actionof �.R/ on I�.Y /w is not diagonalizable; but one can read off from [21] thatthe eigenspace of �.R/ belonging to the top eigenvalue 2g � 2 is simple whenone restricts to the kernel of �.y/ � 2. That is,

ker.�.y/�2/\ker.�.R/� .2g�2//N D ker.�.y/�2/\ker.�.R/� .2g�2//

for all N � 1.

Proposition 7.5. Given any Y , R and w for which I�.Y jR/w is defined, and givenany other surface † � Y of positive genus, the action of �.†/ on I�.Y jR/whas eigenvalues belong to the set of even integers in the range from �.2g � 2/ to2g � 2, where g is the genus of †.

56

Proof. The action of �.†/ on I�.Y /w commutes with �.R/, so the action of�.†/ does preserve the subspace I�.Y jR/w � I�.Y /w . If w is odd on †, thenthe proposition follows from Corollary 7.2 together with the fact that �.y/� 2is zero on this subspace. If w is even on †, then one can consider a surface inthe homology class of RC n† and use the additivity of �.

Because the actions of �.†1/ and �.†2/ commute for any pair of classes†1 and †2, we have a decomposition of I�.Y jR/w by cohomology classes (asoutlined in [15]):

Corollary 7.6. There is a direct sum decomposition into generalized eigenspaces

I�.Y jR/w DMs

I�.Y jR; s/w

where the sum is over all homomorphisms

s W H2.Y IZ/! 2Z

subject to the constraints ˇs.ŒS�/

ˇ� 2 genus.S/ � 2

for all connected surfaces S with positive genus and s.ŒR�/ D 2 genus.R/ � 2.The summand I�.Y jR; s/w is the simultaneous generalized eigenspace

I�.Y jR; s/w D\

�2H2.Y /

[N�0

ker��.�/ � s.�/

�N:

It will be convenient at a later point to have a notation for the sort of homo-morphisms s that arise here. Choosing a notation reminiscent of our notationfor spinc structures, we write

H .Y / D Hom�H2.Y /; 2Z

�and for an embedded surface R � Y of genus g we write

H .Y jR/ D f s 2 H .Y / j s.ŒR�/ D 2g � 2 g: (21)

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7.3 Excision for instanton homology

Let Y be closed, oriented 3-manifold equipped with a line bundle w, and sup-pose † D †1 [†2 is an oriented embedded surface with two connected com-ponents of equal genus, which we require to be positive. Suppose also thatc1.w/Œ†1� and c1.w/Œ†2� are equal and odd. We allow that Y has either oneor two components. In the latter case, we require one of the †i to be in eachcomponent. In the former case, when Y is connected, we assume that †1 and†2 are not homologous. Choose a diffeomorphism h W †1 ! †2, and lift itto a bundle-isomorphism Oh on the restrictions of the line bundle w. From thisdata, we form QY by cutting along the †i and gluing up using h as before. Thelift Oh can be used to glue up the bundle also, giving us a bundle Qw ! QY . Asbefore, we write Q† D Q†1 [ Q†2 for the surfaces in QY .

Theorem 7.7. If . QY ; Q†/ is obtained from .Y;†/ as above, then there is an isomor-phism

I�.Y j†/w Š I�. QY j Q†/ Qw :

We interpret the left-hand side as a tensor product in the case that Y has twocomponents.

Proof. In the case that † has genus 1, this result is due to Floer [8, 2]. InFloer’s statement of the result, Y had two components, but the proof does notrequire it. It should also be said that statement the of Floer’s theorem in [2]involves I�.Y /w rather than I�.Y j†/w , which leads to an extra factor of twoin the dimensions when Y has two components.

The case of genus 2 or more is essentially the same, once one knows thatI�.S

1 �†i j†i /w has rank 1.

In the case of genus 1, note that passing from I�.Y /w to I�.Y j†/w canalso be achieved by taking the C2 eigenspace of �.y/, for one point y in eachcomponent of Y .

Here are two particular applications of the excision theorem. They are bothvariants of Proposition 7.4, but involve different line bundles.

Proposition 7.8. Let Y be the product S1 � †, with † a surface of genus 1 ormore, and let w again be the line bundle dual to the S1 factor. Let u ! Y be aline bundle whose first Chern class is dual to a curve lying on fpointg �†, andwrite the tensor product line bundle as uw. Then we have

I�.Y j†/uw D C:

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Proof. Write B for the vector space I�.Y j†/uw and A for the vector spaceI�.Y j†/w . We apply the excision theorem in a setting where the incomingmanifold is two copies of Y with the line bundle uw and the outgoing manifoldis a single copy of Y with the line bundle u2w. The latter gives the same Floerhomology as the for line bundle w, so we learn that

B ˝ B Š A:

We already know that A is one-dimensional, and it follows that B is also one-dimensional.

For the second application, we can dispense with w:

Proposition 7.9. In the situation of Proposition 7.8, the eigenspace of the pairof operators .�.†/; �.y// on I�.Y /u for the eigenvalues .2g � 2; 2/ is also one-dimensional.

Proof. We can see more generally, that for any � the eigenspace for .�; 2/ onI�.Y /u is the same as the corresponding eigenspace in I�.Y /wu. For this onecan apply the excision theorem as follows. Let c be a closed curve on † sothat the torus S1 � c intersects u once. Let Y1 be S1 � T 2, and let u1, w1 andc1 be similar there to u, w and c. Apply the excision theorem with incomingmanifold Y1[Y with the line bundles u1w1 and uw respectively, cutting alongthe tori S1 � c1 and S1 � c. The outgoing manifold is diffeomorphic to Y ,with the line bundle uw2, which gives the same homology as u. The excisiontheorem gives an isomorphism between the C2 eigenspaces of �.y/, which wedenote

� W I�.Y /.2/uw ! I�.Y /

.2/u :

The map that gives rise to the isomorphism in the excision theorem intertwines(in this instance) the maps �.†/ on the outgoing end with

�.T 2/˝ 1C 1˝ �.†/

on the incoming end. Since �.T 2/ is zero on I�.S1 � T 2/u1w1, the map �

actually commutes with �.†/.

Remark. The Floer homology group I�.S1�†/u is something that appears tobe rather simpler than the more familiar I�.S1 � †/w . In particular, excisionshows that the it behaves “multiplicatively” in g�1. The representation varietythat is involved here is easy to identify: the critical point set of the Chern-Simons functional is two copies of a torus T 2g�2. The involution interchanges�† interchanges the two copies. It seems likely that the Floer group I�.S1 �†/w;† can be identified with the homology of this torus.

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7.4 Instanton Floer homology for sutured manifolds

Let .M; / be a balanced sutured manifold. Just as we did in the monopolecase, we attach a connected product sutured manifold Œ�1; 1� � T to .M; /to obtain a manifold Y 0 with boundary NRC [ NR�, a pair of diffeomorphicconnected closed surfaces. As before, we require that there be a closed curve cin T such that f�1g � c and f1g � c are both non-separating in their respectiveboundary components. We also pick a marked point, t0 2 T , which we didnot need before. Now we glue NRC to NR� by a diffeomorphism. We requirethat h.t0/ D t0, so that the resulting closed manifold Y D Y.M; / containsa standard circle running through t0. This circle intersects once the closedsurface NR obtained by identifying NR˙. We no longer require that NR has genus2 or more: in the instanton case, genus 1 will suffice.

Definition 7.10. The instanton homology of the sutured manifold .M; / is thevector space

SHI.M; / WD I�.Y j NR/w ;

where .Y; NR/ is obtained from .M; / by closing as just described, and w is theline bundle whose first Chern class is dual to the standard circle through t0. ˙

Remark. As an example, it follows from Proposition 7.4 that the instantonhomology of a product sutured manifold is C.

The proof that SHI.M; / is independent of the choice of genus for T andthe choice of diffeomorphism h can be carried over almost verbatim from themonopole case, using the excision theorem. It is even somewhat easier to man-age, because the case of genus 1 is no longer special. When showing that SHI isindependent of the choice of genus, we used twisted coefficients HM�.Y j NRI��/as an intermediate step in the monopole case. The counterpart of twisted co-efficients in the proof for the instanton case is the introduction of the auxiliaryline bundle u that appears in Propositions 7.8 and 7.9 above. One applies ex-cision along tori, following the same scheme as shown in Figure 5, to increasethe genus by 1. On the components S1 � S , with S of genus 2 as shown, oneshould take the line bundle u, where u is the line bundle whose first Chern classis dual to the dotted curve d 0. This argument shows that

I�.Y j NR/uw D I�. QY j QR/ Qu Qw

where Y and QY are closures of .M; / obtained using auxiliary surfaces T andQT of genus g and g C 1. Another application of excision (cutting along copies

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of NR and using Proposition 7.8) shows that

I�.Y j NR/uw Š I�.Y j NR/w

D SHIg.M; /:

7.5 Decompositions of sutured manifolds and non-vanishing

The proofs of the decomposition results of sections 6.3 and 6.5 carry over with-out change to the instanton setting also. In particular, Proposition 6.9 holds inthe instanton case:

Proposition 7.11. Let .M; / be a balanced sutured manifold and

.M; /SÝ .M 0; 0/

a sutured manifold decomposition satisfying the hypotheses of Theorem 6.8.Then SHI.M 0; 0/ is a direct summand of SHI.M; /.

Proof. The proof is the same as the proof of Proposition 6.9; but at the laststep in Lemma 6.10, instead of using the decomposition into spinc structures,one uses the generalized-eigenspace decomposition of Corollary 7.6.

As shown in [13] and mentioned above at the end of section 6.5, a resultsuch as Proposition 7.11 gives a non-vanishing theorem for the case of tautsutured manifolds. We therefore have:

Theorem 7.12. If the balanced sutured manifold .M; / is taut, then SHI.M; /is non-zero.

The only alternative route known to the authors for proving a non-vanishingtheorem for instanton homology is the strategy in [17], which draws on resultsfrom symplectic and contact topology, as well as on the partial proof of Wit-ten’s conjecture relating Donaldson invariants and Seiberg-Witten invariantsof closed 4-manifolds [6]. We shall return to non-vanishing theorems for in-stanton homology in section 7.8.

7.6 Floer’s instanton homology for knots

Just as we did for the monopole case in section 5, we can take Juhasz’s pre-scription as a definition of knot homology. Let K � Z be again a knot in aclosed, oriented 3-manifold. Let .M; / be the sutured manifold obtained by

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taking M to be the knot complement Z nN ı.K/ and s. / a pair of oppositelyoriented meridians on @K. In the instanton case, there is no need for NR tohave genus 2 or more, so we may use the closure Y1.M; / described in Defini-tion 5.3. This is the closure of .M; / obtained using Œ�1; 1� � T , where T isan annulus. It is also described in Lemma 5.2 as obtained from Z nN ı.K/ byattaching F � S1, where F has genus one: the gluing is done so that fpg � S1

is attached to a meridian of K. We summarize the construction of this instan-ton knot homology in the following definition. The definition is not new: itis the same “instanton homology for knots” that Floer defined in [8]. For thepurposes of this paper, we call it KHI.Z;K/:

Definition 7.13. The instanton knot homology KHI.Z;K/ of a knot K in Z isdefined to be the instanton homology of the sutured manifold .M; / above; orequivalently, the instanton homology group I�.Y1j NR/w . Here Y1 is obtainedfrom the knot complement by attaching F � S1 as described, the surface NR isthe torus ˛�S1 as shown in Figure 7, and w is the line bundle with c1.w/ dualto ˇ � fpg � F � S1. ˙

The only difference between this and Floer’s original definition is that wehave used I�.Y1j NR/w in place of I�.Y1/w . Since NR has genus one, the formergroup can be characterized as the C2 eigenspace of �.y/ acting on the lattergroup. The latter group is the sum of two subspaces of equal dimension, theeigenspaces for the eigenvalues 2 and �2.

For a classical knot K in S3, we shall simply write KHI.K/ for the in-stanton knot homology. To get a feel for what this invariant is, let us examinethe set of critical points of the Chern-Simons functional on B, or in otherwords the space of flat connections in the appropriate SO.3/ bundle, modulothe determinant-1 gauge transformations. To do this, we start by looking atF � S1, where F is the genus-1 surface with one boundary component, andthe line-bundle w with c1.w/ dual to ˇ � fpg. The appropriate representationvariety can also be viewed as the space of flat SU.2/ connections on the com-plement of the curve ˇ � fpg with the property that the holonomy around asmall circle linking ˇ�fpg is the central element �1. Consider such a flat con-nection A and let J1 and J2 be the holonomies of A around respectively thecurves ˛ � fqg and a � S1 in F � S1, where a is a point on ˛ n ˇ. The torus˛ � S1 intersects the circle ˇ � fpg once, so we have

ŒJ1; J2� D �1

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in SU.2/. Up to a gauge transformation, we must have

J1 D

�0 �1

1 0

�; J2 D

�i 0

0 �i

�:

Let J3 be the holonomy around ˇ0 � fqg, where ˇ0 is a parallel copy of ˇ. Theelements J1 and J3 must commute, so

J3 D

�ei� 0

0 e�i�

�for some � in Œ0; 2�/. The angle � is now determined without ambiguity fromthe gauge-equivalence class of the connection A; and the matrices J1, J2 andJ3 determine A entirely. We have proved:

Lemma 7.14. The representation variety of flat SO.3/ connections on F �S1 forthe given w, modulo the determinant-1 gauge group, is diffeomorphic to a circleS1, via J3 as above.

Let us examine the restriction of these representations to the boundaryof F � S1. On this torus @F � S1, the flat connections can be regarded asSU.2/ connections. The holonomy around the S1 factor is J2, which we havedescribed above. The holonomy around the @F factor is given by the commu-tator

ŒJ3; J1� D

�e2i� 0

0 e�2i�

�:

So for the representation variety described in the lemma, the restriction to theboundary is a two-to-one map whose image is the space of connections havingholonomy around the S1 factor given by

i D�i 0

0 �i

�:

Finally, we can attach F � S1 to the knot complement S3 n N ı.K/, and weobtain the following description of the representation variety.

Lemma 7.15. Let K � S3 be a knot and let Y1 and w be as described in Def-inition 7.13. Then the representation variety given by the critical points of theChern-Simons functional in the corresponding space of connections B can beidentified with a double cover of the space

R.K; i/ D f � W �1.S3 nK/! SU.2/ j �.m/ D i g;

where m is a chosen meridian.

63

Note that R.K; i/ is a space of homomorphisms, not a space of conjugacyclasses of homomorphisms. The centralizer of i (a circle subgroup) still acts onR.K; i/ by conjugation. There is always exactly one point of R.K; i/ which isfixed by the action of this circle, namely the homomorphism � which factorsthrough the abelianization H1.S3 n K/ D Z. All other orbits are irreducible:they have stabilizer ˙1, so they are circles. In a generic case, R.K; i/ consistsof one isolated point corresponding to the abelian (reducible) representation,and finitely many circles, one for each conjugacy class of irreducible repre-sentations. In such a case, the representation variety described in the lemmaabove is a trivial double-cover of R.K; i/. It therefore has two isolated pointscorresponding to the reducible, and two circles for each irreducible conjugacyclass.

Because it comprises only the C2 eigenspace of �.y/, the knot Floer ho-mology KHI.K/ has just half the dimension of I�.Y1/w in Definition 7.13.Heuristically, we can think of each irreducible conjugacy class in R.K; i/ ascontributing the homology of the circle, H�.S1IC/, to the complex that com-putes KHI.K/, while the reducible contributes a single C. In any event, ifthere are only n conjugacy classes of irreducibles and the corresponding cir-cles of critical points are non-degenerate in the Morse-Bott sense, then it willfollow that the dimension of KHI.K/ is bounded above by 2nC 1.

For a knot K � Z supplied with a Seifert surface †, there is a decomposi-tion of the instanton knot homology KHI.Z;K/ as

KHI.Z;K/ Dgenus.†/M

iD�genus.†/

KHI.Z;K; Œ†�; i/:

The definition is the same as in the monopole case (19), but uses the generalized-eigenspace decomposition of Corollary 7.6 in place of the decomposition byspinc structures. In particular, for a classical knot K � S3, we can write

KHI.K/ DgM

iD�g

KHI.K; i/;

where g is the genus of the knot. Just as in the monopole case, the topsummand KHI.K; g/ can be identified with the instanton Floer homologySHM.M; /, where .M; / is the sutured manifold obtained by cutting openthe knot complement along a Seifert surface of genus g. (See Proposition 5.11.)From the non-vanishing theorem, Theorem 7.12, we therefore deduce a non-vanishing theorem for KHI .

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Proposition 7.16. Let K be a classical knot of genus g. Then the instanton knothomology group KHI.K; g/ is non-zero. In particular, instanton knot homologydetects the genus of a knot.

This proposition provides an alternative proof for results from [17] and[16]. In particular, we have the following corollary:

Corollary 7.17. If K � S3 is non-trivial knot, then there exists an irreduciblehomomorphism � W �1.S

3 nK/ which maps a chosen meridian m to the elementi 2 SU.2/.

Proof. If there is no such homomorphism, then R.K; i/ consists only of thereducible, which is always non-degenerate. The critical point set in B thenconsists of two irreducible critical points, so the rank of I�.Y1/w is at most 2,and the rank of KHI.K/ is therefore at most 1. This is inconsistent with non-vanishing of KHM.K; g/, since KHM.K; g/ is isomorphic to KHM.K;�g/.

7.7 Instanton homology and fibered knots

Instanton knot homology detects fibered knots, just as the other versionsdo. We state and prove this here. We need, however, an extra hypothesis onthe Alexander polynomial. For Heegaard knot homology, and also in themonopole case, we know the Alexander polynomial is determined by the knothomology, and the extra hypothesis is not needed. It seems likely that the sameholds in the instanton case, but we have not proved it.

We begin with a version of Theorem 6.1 for the instanton case.

Theorem 7.18. Suppose that the balanced sutured manifold .M; / is taut anda homology product. Then .M; / is a product sutured manifold if and only ifSHI.M; / D C.

Proof. Ni’s argument, as presented for monopole knot homology in the proofof Theorem 6.1, works just as well for SHI as it does for SHM, with one slightchange (a change which is in the spirit of [13]). The key point occurs in Case1 in the proof of Theorem 6.1 (section 6.4), where it is already assumed that.M; / has just one suture. We described this step using spinc structures, butwe can argue using homology instead.

Let N be obtained from .M; / by adding a product region Œ�1; 1� � T tothe single suture. The boundary of N is NRC [ NR�. In [23], Ni shows thatif E � I does not carry all the homology of .M; /, then one can find two

65

decomposing surfaces S1 and S2 in N with the following properties. First,the boundaries of S1 and S2 are the same and consist of a pair of circles !Cand !� which represent non-zero homology classes in NRC and NR�. Second,the sutured manifolds .M 01;

01/ and .M 02;

02/ obtained by decomposition of N

along S1 and S2 respectively are both taut. Third, if Y is obtained from N bygluing NRC to NR� by a diffeomorphism h with h.!C/ D !�, then the resultingclosed surface NS1, NS2 and NR in Y satisfy the following conditions, for somem > 0 and some closed surface NS0 with �. NS0/ non-zero,

Œ NS1� D mŒ NR�C Œ NS0�

Œ NS2� D mŒ NR� � Œ NS0�

and

�. NS1/ D �. NS2/

D m�. NR/C �. NS0/:

These last conditions imply that H .Y j NR/\H .Y j NS1/ is disjoint from H .Y j NR/\

H .Y j NS2/. (The notation H is introduced at (21).)For i D 1; 2, let Fi be the surface in Y obtained by smoothing out the in-

tersection of NR and NSi (a single circle in both cases). The proof of Lemma 6.10shows that

SHI.M 0i ; 0i / D

Ms2H.Y jFi /

I�.Y j NR; s/w

D

Ms2H.Y j NR/\H.Y j NSi /

I�.Y j NR; s/w

� I�.Y j NR/w

D SHI.M; /:

The disjointness of the two indexing sets for s means that we have

SHI.M 01; 01/˚ SHI.M 02;

02/ � SHI.M; /:

Finally, both summands on the right are non-zero because these sutured man-ifolds are taut.

Corollary 7.19. LetK be a non-trivial knot in S3. Suppose that the symmetrizedAlexander polynomial�K.T / is monic and that its degree (by which we mean thehighest power of T that appears) is g. ThenK is fibered if and only if KHI.K; g/is one-dimensional.

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Proof. The proof given for Corollary 6.2 (the monopole case) needs no alter-ation, except that the hypothesis on the Alexander polynomial has been explic-itly included, rather than being deduced from Lemma 5.10.

Corollary 7.20. Let K � S3 be a knot whose Alexander polynomial is monic ofdegree equal to the genus of the knot. Consider the irreducible homomorphisms� W �1.S

3 n K/ ! SU.2/ which map a chosen meridian m to the element i 2SU.2/. If there is only one conjugacy class of such homomorphisms, and if thesehomomorphisms are non-degenerate, then K is fibered.

7.8 Non-vanishing theorems in the closed case

Theorem 7.12 asserts the non-vanishing of instanton Floer homology for bal-anced sutured manifolds; but the theorem does not say anything directly aboutclosed 3-manifolds Y . Nevertheless, with a little extra input, we obtain thefollowing result as a corollary.

Theorem 7.21. Let Y be a closed irreducible 3-manifold containing a closed, con-nected, oriented surface NR representing a non-zero class in second homology. Letw be a hermitian line bundle whose first Chern class has odd evaluation on ŒR�.Then I�.Y j NR/w is non-zero.

Proof. Let M be the manifold obtained by cutting Y open along R, and writethe boundary of M as RC [ R�. We regard M as a sutured manifold, withan empty set of sutures. (The absence of sutures means that M fails to bebalanced.) Let N be the double of M . We can regard R D RC [ R� as asurface in the closed manifold N . We can “double” the line bundle also; sowe have a line bundle, also denoted by w, on N . By the excision theorem, itwill be sufficient to show that I�.N jR/w is non-zero. Since R� and RC arehomologous in N and of equal genus, we have

I�.N jR/w D I�.N jRC/w ;

so we could equally well deal with I�.N jRC/w instead.From the proof of Theorem 3.13 of [9], we have a closed, oriented surface

T � N with the following properties. The surface T meets R in a non-emptyset of circles, and we let T 0 be the surface obtained from T andR by smoothingthese circles of double points. This T 0 has the property that by cutting N openalong T and then decomposing further along a non-empty collection of annuliJ , we arrive at a taut, sutured manifold .N 00; ı00/.

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If T intersects both RC and R�, then .N 00; ı00/ is balanced. If T intersectsonlyRC, say, then .N 00; ı00/ fails to be balanced, because its boundary containstwo copies of R�: these are components of @N 00 which fail to meet A.ı00/,contrary to the definition of balanced. If this is what happens, we re-attachthese two copies of R�. We rename the resulting manifold as our new N 00 andproceed. At this point, .N 00; ı00/ is a balanced sutured manifold.

By Theorem 7.12, we know that SHM.N 00; ı00/ is non-zero. We can regardthe manifold N as a closure of .N 00; ı00/, but with an auxiliary surface thatfails to be connected: the auxiliary surface is the collection of annuli J . Butas we argued in the proof of Lemma 6.10, a disconnected auxiliary surface isas good as a connected one here. We can therefore compute SHM.N 00; ı00/ asI�.N jF /w , where F is the surface in N formed from R˙.ı

00/ when making theclosure. Thus

I�.N jF /w ¤ 0:

This surface F can be identified with T 0 in the case that T meets both RC andR�. In the case that T meets only RC, then F is T 0 n R�. In other words,F is obtained by smoothing the circles of double points of either T [ R orT [RC. As in the proof Lemma 6.10, the Floer homology I�.N jF /w is a directsummand of I�.N jRC/w . So the latter is non-zero, and we are done.

Corollary 7.22. If Y is obtained from zero-surgery on a non-trivial knotK � S3,then I�.Y /w is non-zero for an odd line bundle w.

Essentially the same theorem and corollary are proved in [17]. But thepresent proof requires considerably less geometry and analysis. From Floer’ssurgery exact triangle, one obtains, as in [17],

Corollary 7.23. If Y1 is obtained asC1 surgery onK � S3, then �1.Y1/ admits anon-trivial homomorphism to SU.2/. In particular, Y1 is not a homotopy sphere.

This provides a proof of the Property P conjecture that is independent ofthe work of Feehan and Leness in [6] and independent also of Perelman’s proofof the Poincare conjecture.

7.9 Questions and conjectures

There are various questions and conjectures which naturally arise. The mostobvious of these is:

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Conjecture 7.24. For balanced sutured manifolds .M; /, the monopole and Hee-gaard groups SHM.M; / and SFH.M; / are isomorphic. When tensored withC, they are both isomorphic to the instanton version, SHI.M; /.

As a special case, we have:

Conjecture 7.25. With complex coefficients, the knot homologies defined byOzsvath-Szabo and Rasmussen are isomorphic to Floer’s instanton homology forknots, KHI.K/, as defined here and in [8].

There are various more modest questions one should ask. We have notshown that the Alexander polynomial can be recovered from the instantonknot homology groups KHI.K; i/; but it is natural to conjecture that this is so,just as in the monopole and Heegaard theories. This may be only a matter ofrepeating [7] in the instanton context:

Conjecture 7.26. The Euler characteristics of the instanton knot homology groupsKHI.K; i/, for i D �g; : : : ; g, are the coefficients of the symmetrized Alexanderpolynomial of K.

If this conjecture is proved, then the hypothesis on the Alexander polyno-mial could be dropped from Corollary 7.19.

A loose end in our development of SHM.M; / is the lack of a completeaccounting of spinc structures. The material of section 6.2 is a step in the rightdirection. In [13], Juhasz proves that his Heegaard Floer homology of suturedmanifolds can be decomposed as a direct sum indexed by the set of relativespinc structures S.M; /, and it would be desirable to have a similar statementfor the monopole and instanton cases.

Juhasz [14] has considered an extension of the fibering theorem, whichprompts naturally a conjecture in the instanton context. Motivated by this,we have:

Conjecture 7.27 (cf. [14]). Let K � S3 be a knot, and consider the irreduciblehomomorphisms � W �1.S3 nK/! SU.2/ which map a chosen meridianm to theelement i 2 SU.2/. Suppose that these homomorphisms are non-degenerate andthat the number of conjugacy classes of such homomorphisms is less then 2kC1.Then the knot complement S3 n N ı.K/ admits a foliation of depth at most 2k,transverse to the torus boundary.

The fact that I�.Y j†/w is of rank 1 in the case that Y is a surface bundleof S1 with fiber† is something that has other applications. For example, com-bined with Donaldson’s theorem on the existence of Lefschetz pencils [3], it

69

yields a fairly direct proof that symplectic 4-manifolds have non-zero Donald-son invariants. Essentially the same strategy was used by Ozsvath and Szabo inthe Heegaard context. What the argument shows specifically is that if X ! S2

is a symplectic Lefschetz fibration whose fiber F has genus 2 or more, and if wis the line bundle dual to a section, then the Donaldson invariant Dw.F n/ isnon-zero for all large enough n in the appropriate residue class mod 4.

Another matter is whether one can relate either the monopole or instantonknot homologies to the corresponding Floer homologies of the 3-manifoldsobtained by surgery on the knot, particularly for large integer surgeries. Thisis how Heegaard knot homology arose in [27].

In a previous paper [19], the authors described another knot-homologyconstructed using instantons. The definition there is distinctly different fromthe definition of KHI.K/ given in this paper, because instantons with singu-larities in codimension-2 were involved. Nevertheless, both theories involvethe same representation variety R.K; i/. Various versions are defined in [19],but the one most closely related to KHI.K/ is the “reduced” variant, calledRI�.K/ in [19]. Like KHI.K/, the group RI�.K/ is a Floer homology group,constructed from a Chern-Simons functional whose set of critical points canbe identified with R.K; i/. The paper [19] develops its theory for the gaugegroup SU.N /, not just SU.2/, and it would be interesting to pursue a similardirection with SHI.M; / and KHI.K/.

The “hat” version of Heegaard Floer homology, for a closed 3-manifoldY , can also be recovered as a special case of Juhasz’s SFH, as shown in [12].The appropriate manifold M is the complement of a ball in Y , and one takesa single annular suture on the result 2-sphere boundary. One can take this asa definition of a “hat” version of monopole Floer homology. In the instantoncase, this leads to essentially the same construction that was used in [19] toavoid reducibles: one replaces Y by Y#T 3 and takes w to be a line bundle thatis trivial on Y and of degree 1 on a T 2 in the T 3.

Finally, as we mentioned in the introduction, it is worth asking whether,in the Heegaard theory, the Floer homology of a balanced sutured manifold.M; /, as defined in [12], can also be recovered as the Heegaard Floer homol-ogy of a closed manifold Y D Y.M; /, of the sort that we have used here. Ifso, it would be interesting to know whether the existing proofs of the decom-position theorems in [23] and [13], for example, can be adapted to prove Floer’sexcision theorem in the context of Heegaard Floer theory.

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