aa r X i v : . [ m a t h . G T ] J u l 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 bal-anced sutured manifolds, in the spirit of [12]. Applications include a new proof ofProperty P for knots.
Contents
Floer homology for sutured manifolds is an invariant
SFH ( M, γ ) of “bal-anced sutured 3-manifolds” (
M, γ ), introduced by Juh´asz in [12, 13]. Itincorporates the knot Floer homology of Ozsv´ath-Szab´o and Rasmussen[25, 27] as a special case, and it provides a framework in which to adapt thearguments of Ghiggini and Ni [11, 23, 22] to reprove, for example, that knotFloer homology detects fibered knots.The construction that forms the basis of Juh´asz’s invariant is an adap-tation of Ozsv´ath and Szab´o’s Heegaard Floer homology for 3-manifolds.The purpose of the present paper is to show how something very similarcan be done using either monopole Floer homology [18] or instanton Floer
The work of the first author was supported by the National Science Foundationthrough NSF grant number DMS-0405271. The work of the second author was supportedby NSF grants DMS-0206485, DMS-0244663 and DMS-0805841. homology [4] in place of the Heegaard version. We will define an invariantof balanced sutured manifolds by gluing them up, with some extra pieces, toform a closed manifold and then applying ordinary Floer homology, of eithervariety, to this closed manifold. Many of the theorems and constructions ofGhiggini, Ni and Juh´asz can be repeated in this context. In particular, ourconstruction leads to candidates for “monopole knot homology” and “in-stanton knot homology”: the monopole and instanton counterparts of theHeegaard knot homology groups. Adapting the arguments of [11] and [23],we shall also prove that fibered knots can be characterized using either ofthese invariants.The definition of instanton knot homology which arises in this way, mo-tivated by Juh´asz’s sutured manifold framework, is not new. It turns outto be exactly the same as an earlier instanton homology for knots, definedby Floer twenty years ago [8]. We conjecture that, over a field of character-istic zero, the knot homology groups of Ozsv´ath-Szabo and Rasmussen areisomorphic to Floer’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.The same definition could be applied with Heegaard Floer homology: it isnot clear to the authors whether the resulting invariant of sutured manifoldswould be the same as Juh´asz’s invariant, but we would conjecture that thisis the case. It seems, at least, that the construction recaptures Heegaardknot homology [26]. Some things are missing however. Our constructionleads to knot homology groups which lack (a priori) the Z grading as wellas the additional structures that 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 argu-ment presented here is independent of the work of Feehan and Leness [6]concerning Witten’s conjecture, and does not require any tools from contactor symplectic topology. As a related matter, we show that instanton homol-ogy captures the Thurston norm on an irreducible 3-manifold, answering aquestion raised in [15]. Acknowledgments.
The authors would like to thank Andr´as Juh´asz and JakeRasmussen for helpful comments and corrections to an earlier version of thispaper.
We follow the notation of [18] for monopole Floer homology. Thus, to aclosed, connected, oriented 3-manifold Y equipped with a spin c structure s , we associate three varieties of Floer homology groups with integer coeffi-cients, d HM • ( Y, s ) , d HM • ( 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 d HM • and d HM ∗ does not arise, but wepreserve the former notation as a visual clue. Unless c ( s ) is torsion, thesegroups are not Z -graded, but they always have a canonical Z / · · · → HM • ( Y, s ) i → d HM • ( Y, s ) j → d HM • ( Y, s ) p → HM • ( Y, s ) → · · · . If c ( s ) is not torsion, then HM • ( Y, s ) is zero and d HM • ( Y, s ) and d HM • ( Y, s )are canonically isomorphic, via j . In this case, we simply write HM • ( Y, s )for either d HM • ( Y, s ) or d HM • ( Y, s ). All these groups can be non-zero onlyfor finitely many spin c structures on a given Y : we write d HM • ( Y ) = M s d HM • ( Y, s )for the total Floer homology, taking the sum over all isomorphism classes ofspin c structure, with similar notation for the d HM and HM cases. 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”,a group homomorphism exp : R → R × . (1)We will use polynomial notation for the exponential map, writing t = exp(1)and so writing exp( n ) as t n . Let B ( Y, s ) denote the Seiberg-Witten con-figuration space for a spin c structure s on Y ; that is, B ( Y, s ) is the spaceof gauge equivalences classes [ A, Φ] consisting of a spin c connection A anda section Φ of the spin bundle. Given a smooth 1-cycle η in Y with realcoefficients, we can associate to each path z : [0 , → B ( Y, s ) a real number r ( z ) by r ( z ) = i π Z [0 , × η tr F A z , where A z is the 4-dimensional connection on [0 , × Y arising from the path z . Now define a local system Γ η 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 t r ( z ) . Following [18, section 22], we obtain Floer homologygroups with coefficients in Γ η ; they will be R -modules denoted d HM • ( Y ; Γ η ) , d HM • ( Y ; Γ η ) , HM • ( Y ; Γ η ) . These still admit a direct sum decomposition by isomorphism classes of spin c structures. The following is essentially Proposition 32.3.1 of [18]: Proposition 2.1.
If there is an integer cohomology class that evaluates as on [ η ] , and if t − t − is invertible in R , then HM • ( Y ; Γ η ) is zero; thus weagain have an isomorphism j between d HM • ( Y ; Γ η ) and d HM • ( Y ; Γ η ) . In the situation of the proposition, we once more drop the decorationsand simply write HM • ( Y ; Γ η ) = M s HM • ( Y, s ; Γ η )for this R -module. Cobordisms between 3-manifolds give rise to maps between their Floer ho-mology groups. More precisely, if W is a compact, oriented cobordism from Y to Y , equipped with a homology-orientation in the sense of [18], then W gives rise to a map d HM ( W ) : d HM • ( Y ) → d HM • ( Y )with similar maps on d HM • and HM • . If η and η are 1-cycles in Y and Y respectively, then to obtain a maps between the Floer groups with localcoefficients, we need an additional piece of data: a 2-chain ν in W with ∂ν = η − η . In this case, we obtain a map which we denote by d HM ( W ; Γ ν ) : d HM • ( Y ; Γ η ) → d HM • ( Y ; η ) . The map d HM ( W ) and its relatives are defined by taking a sum over allspin c structures on W . In the case of d HM ( W ; Γ ν ), the spin c contributionsare weighted according to the pairing of the curvature of the connection withthe cycle ν . There is a corresponding invariant for a closed 4-manifold X with b + ≥ ν . In [18], this invariant of ( X, ν )is denoted by m ( X, ν ) (or m ( X, [ ν ]), because only the homology class of ν matters); it is an element of R defined by m ( X, [ ν ]) = X s m ( X, s ) t h c ( s ) , [ ν ] i , (2)where m ( X, s ) denotes the ordinary Seiberg-Witten invariant for a spin c structure s . Monopole Floer homology detects the Thurston norm of a 3-manifold Y .We recall from [18] what lies behind this slogan. Let F ⊂ Y be a closed,oriented, connected surface in our closed, oriented 3-manifold Y . We shallsuppose F is not sphere. Then we have a vanishing theorem [18, Corollary40.1.2], which states that HM • ( Y, s ) = 0for all spin c structures s satisfying h c ( s ) , [ F ] i > F ) − . (Note that this condition implies that c ( 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 thiscase, the non-vanishing theorem asserts that if F is genus-minimizing in itshomology class, then there exists a spin c structure s c with HM • ( Y, s c ) = 0and h c ( s c ) , [ F ] i = 2 genus( F ) − . 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 aspin c structure on Y . The non-vanishing result holds for any spin c structure s c 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 explainin the context of instanton homology in section 7.8 below.It is convenient to introduce the following shorthand. We denote the setof isomorphism classes of spin c structures on a closed oriented manifold Y by S ( Y ). If F ⊂ Y is a closed, connected oriented surface of genus g ≥ S ( Y | F ) for the set of isomorphisms classes of spin c structures s on Y satisfying the constraint h c ( s ) , [ F ] i = 2 genus( F ) − , (3)and we write HM • ( Y | F ) ⊂ HM • ( Y )for the subgroup HM • ( Y | F ) = M s ∈S ( Y | F ) HM • ( Y, s ) . (4)Note again that all the spin c structures in S ( Y | F ) have non-torsion firstChern class. When a local system Γ η is given, we define HM • ( Y | F ; Γ η )similarly. If F is a surface with more than one component, we define S ( Y | F ) = \ F i ⊂ F S ( Y | F i )where the F i are the components, and we define HM • ( Y | F ) accordingly.As a special case, we have Lemma 2.2.
Let F be a closed, connected, oriented surface of genus at least , and let Y = F × S . Regard F as a surface F × { p } in Y . Then we haveHM • ( Y | F ) = Z . Indeed, if F is given a metric of constant negative curvature and Y is giventhe product metric, then the complex that computes HM • ( Y | F ) has a singlegenerator, corresponding to a single, non-degenerate solution of the Seiberg-Witten equations.Proof. This is standard. The spin c that contributes is the product spin c structure, which corresponds to the 2-plane field tangent to the fibers ofthe map Y → S . The unique gauge-equivalence class of solutions to theequations is a pair [ A, Φ] with Φ covariantly constant.
Corollary 2.3.
Let Y be the product F × S , as in the previous lemma.Then for any local coefficient system Γ η , we haveHM • ( Y | F ; Γ η ) = R , where R is the coefficient ring. So far, following [18], we have discussed connected 3-manifolds and con-nected 4-dimensional cobordisms between them. Because of the special roleplayed by reducible connections, one must be careful when generalizing; butthere are simple situations where the discussion can be carried over withoutdifficulty to the case of 3-manifolds with several components. The analysisof the Seiberg-Witten equations on a manifold with cylindrical ends is car-ried out in [18] for an arbitrary number of ends, and our task here is just topackage the resulting information.Let W be cobordism from Y to Y , and suppose that each of these hascomponents Y = Y , ∪ · · · ∪ Y ,r Y = Y , ∪ · · · ∪ Y ,s . Although we label them this way, no ordering of the components need bechosen at this point. We may allow either r or s (or both) to be zero, andwe do not require W to be connected. If W has any closed components, weinsist that each such component has b + ≥ F ⊂ Y and F ⊂ Y . We will suppose that each componentof Y contains a component of F and that all components of F have genus2 or more. Thus we have non-empty surfaces F ,i = F ∩ Y ,i ⊂ Y ,i . We make a similar hypothesis for F . We can regard the union F ∪ F asa subset of W , and we suppose that we are given a surface F W ⊂ W whichcontains F ∪ F in addition perhaps to other components. The notation wepreviously used for spin c structures with constraints on c can be extendedto this case: we write S ( W | F W ) for the set of spin c structures s on W suchthat (3) holds for every component of F W .We can define HM • ( Y | F ) by taking a product over the components of Y . That is, we should define the configuration space B ( Y ) as the product ofthe B ( Y ,i ), and we should construct HM • ( Y | F ) as the Floer homology ofthe Chern-Simons-Dirac functional on the components of this product spacewhich belong to the appropriate spin c structures. The only slight twist hereis in understanding the orientations of moduli spaces that are needed to fixthe signs.We therefore digress to consider orientations. For a cobordism suchas W above, perhaps with several components, we define a 2-element setΛ( W ) of homology orientations of W as follows. Attach cylindrical ends tothe incoming and outgoing ends to get a complete manifold W + , and let t be function which agrees with the cylindrical coordinate on the ends. Thefunction t tends to + ∞ on the outgoing ends and −∞ on the incoming ends.Consider the linearized anti-self-duality operator δ = d ∗ ⊕ d + acting on theweighted Sobolev spaces δ : L ,ǫ ( i Λ ) → L ǫ ( i Λ ⊕ i Λ + ) , where L k,ǫ = e − ǫt L k . Fix a spin c structure on W and let D + A be the Diracoperator, for a spin c connection A that is constant on the ends. We consider D A acting on weighted Sobolev spaces of the same sort, and we write P = δ + D + A . These are the linearized Seiberg-Witten equations on W , with Coulombgauge fixing, at a configuration where the spinor is zero. There exists ǫ > P is Fredholm for all ǫ in the interval (0 , ǫ ). Wedefine Λ( W ) to be the set of orientations of the determinant line of P , for any ǫ in this range. Using weighted Sobolev spaces here is equivalent to usingordinary Sobolev spaces and replacing P by a zeroth-order perturbationwhich on the ends has the form P − ǫ Θwhere Θ is obtained from applying the symbol of P to the vector field ∂/∂t along the cylinder. The Dirac operator is irrelevant at this point because itis complex and its real determinant is therefore canonically oriented; so wecould use the operator δ instead.Now suppose that α and α are gauge-equivalence classes correspondingto non-degenerate critical points of the Chern-Simons-Dirac functional on Y and Y respectively. Let γ = ( A, Φ) be any configuration on W + thatis asymptotic to these gauge-equivalence classes on the ends. Let P γ bethe corresponding operator (acting on the Sobolev spaces without weights).Define Λ( W ; α , α ) to be the set of orientations of the determinant of P A .This is independent of the choice of γ , in a canonical manner. If Λ and Λ are two 2-element sets, we use the notation Λ Λ to denote the 2-elementset formed by the obvious “multiplication” (the set of bijections from Λ toΛ ). With this in mind, we defineΛ( α , α ) = Λ( W )Λ( W ; α , α ) . An excision argument makes this independent of W . Given now a 3-manifold Y (with several components) and a non-degenerate critical point α , wechoose cobordism X from the empty set to Y and we defineΛ( α ) = Λ( ∅ , α ) . We then have Λ( α , α ) = Λ( α )Λ( α ) . What this last equality means in practice is this. If we are given acobordism W with a choice of homology orientation in Λ( W ) and a modulispace M = M ( W ; α , α ), then a choice of orientation of M is the sameas a choice of bijection from Λ( α ) to Λ( α ). In the case that Y = Y ,the cylindrical cobordism has a canonical homology-orientation because theoperator P is invertible; so in this Λ( α )Λ( α ) orients the moduli spaces.The appropriate definition of HM • ( Y , s ) for spin c structures s that are non-torsion on each component is therefore to take the complex to be C • ( Y , s ) = M α Z Λ( α )and to define the differential using the corresponding orientation of the mod-uli spaces. In this way, we construct HM • ( Y | F ) and HM • ( Y | F ). If wesupply W with a homology orientation in the above sense, then W definesa map HM ( W | F W ) : HM • ( Y | F ) → HM • ( Y | F ) . (5)The notation HM ( W | F W ) is meant to imply that we use only the spin c structures from S ( W | F W ).The complex C • ( Y , s ) just defined can be considered as a tensor productover the connected components of Y : C • ( Y , s ) = O i C • ( Y ,i , s i ) , (6)0but there are some choices involved. Let us pick an ordering of the compo-nents. For each i , let X i be a cobordism from the empty set to Y ,i . Usingthe standard convention for the orientation of a direct sum, we can thenidentify Λ( X ) = Λ( X )Λ( X ) · · · Λ( X r ) , and similarly with Λ( X, α ). In this way, we can specify an isomorphism Z Λ( α ) → Z Λ( α , ) ⊗ · · · ⊗ Z Λ( α ,r ) . This allows us to identify the complexes on the left and right in (6) asgroups. Ordering issues mean that there will be the expected alternatingsigns appearing when we compare the differentials on the left and right. Asusual with products in homology, what results from this is a split short exactsequence 0 → O i HM • ( Y ,i | F ,i ) → HM • ( Y | F ) → T → 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 Y with non-torsion spin c structure s . Let Z be the 4-manifold S × Y . Wecan pull back the spin c structure to Z , and we still call it s . For clarity,suppose that b + ( Z ) is bigger than 1, so that m ( Z, s ) is defined. To fix thesign of m ( Z, s ), we need a homology orientation of Z ; but a product suchas Z has a preferred homology orientation. To define it, we must specify anorientation for the determinant of P on Z . The operator P − ǫ Θ is invertiblefor small ǫ , and we use this to to orient the determinant. Now let α be non-degenerate critical point for the (possibly perturbed) Chern-Simons-Diracfunctional on ( Y, s ). This pulls back to an isolated, non-degenerate solutionon Z to the 4-dimensional Seiberg-Witten equations, say ˆ α . This solutioncontributes either +1 or − m ( Z, s ). We have the followinglemma. Lemma 2.4.
The solution ˆ α contributes +1 or − to the invariant m ( Z, s ) according as the critical point α has odd or even grading in C • ( Y, s ) , for thecanonical Z/ grading.Proof. We have two operators differing by zeroth-order terms P = P − ǫ Θ P = P γ . P s be a homotopy between them. We have a determinant line for thisfamily of operators over the interval [0 , P and P at the two ends gives the determinant line a canonical orientation at thetwo ends. The sign with which ˆ α contributes is, by definition, +1 or − s = 0 , P s as ddt + L s on S × Y , where L s is a self-adjoint elliptic operator perturbed by a boundedterm, and the canonical mod 2 grading of α is determined, by definition, bythe parity of the spectral flow of the family of operators L s from s = 0 to s = 1.So we must see that the parity of the spectral flow of the operators L s determines whether the invertible operators P and P provide the sameorientation. This is a general fact about families of self-adjoint Fredholmoperators. What we have here are two non-trivial homomorphisms H ( S ) → Z / , 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 ( ∞ ) /O ( ∞ ),at which 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 othersituations of the following sort. Consider again a cobordism W from Y to Y with surfaces F W , F and F as before. Suppose that one of the incomingboundary components is the same as one of the outgoing ones: say Y ,r = Y ,s . We may form a new W ∗ from W by identifying these boundary components,so W ∗ has r − s − Y ,r and Y ,s may belong either to the same or to different compo-nents of W , but we treat these cases together. The surface F W gives rise toa homeomorphic surface F W ∗ in W ∗ . (We push F ,r and F ,s away from theboundary of W before gluing to Y ,r to Y ,r , to keep these surfaces disjoint,if necessary.) If is possible that this process has created a W ∗ which hasone more closed component than W . This new closed component of W ∗ will2have b + at least 1; but we shall suppose that, if there is such a component, ithas b + at least 2. (The case of b + = 1 will be discussed in a slightly differentcontext in the next subsection.)Under this hypothesis on b + for the closed components, we now have anew map HM ( W ∗ | F W ∗ ) : HM • ( Y ∗ | F ∗ ) → HM • ( Y ∗ | F ∗ ) , (8)where Y ∗ is Y \ Y ,r and Y ∗ is defined similarly. The analysis from [18]provides a “gluing theorem” which tells us that the map HM ( W ∗ | F W ∗ ) isobtained from HM ( W | F W ) by a contraction. More precisely, at the chainlevel, ( W, F W ) defines a chain map O i C • ( Y ,i | F ,i ) → O j C • ( Y ,j | F ,j ) . This map can be contracted by taking an alternating trace over C • ( Y ,r | F ,r ) = C • ( Y ,s | F ,s ) , and the result of this contraction is a chain map which is chain-homotopicto the chain map defined by ( W ∗ , F W ∗ ).The cobordism W from Y to Y can also be regarded as a cobordism ˜ W from ˜ Y to ˜ Y , where ˜ Y = Y ∪ ( − Y ,s )and ˜ Y = Y \ Y ,s . (That is, we regard the last outgoing component as an incoming componentwith the opposite orientation.) The relation between the maps defined by W and ˜ W can be put in the same context as the above gluing theorem. Wefirst add an extra component Z to W , where Z is the cylinder [0 , × Y ,s ,regarded as a cobordism from Y ,s ∪ ( − Y ,s ) to the empty set. The mapdefined by W ∪ Z is a tensor product, at the chain level, and the cobordism˜ W can be obtained by gluing an outgoing component of W to an incomingcomponent of Z . All that is left is to understand the map defined by Z .Discounting torsion, this last map is the Poincar´e duality pairing HM • ( − Y ,s | F ,s ) ⊗ HM • ( Y ,s | F ,s ) → Z . As in [18], this pairing depends on a homology orientation of Y ,s , whichreappears as the need to choose a homology orientation for the extra com-ponent Z .3Let us pursue a simple application of this formalism. Let W be again acobordism from Y to Y and let F and F be surfaces in these boundary3-manifolds as above. Suppose that W contains in its interior a product3-manifold Z = G × S where G is connected of genus at least 2. Regard G = G × { p } also asa submanifold of W . Form a new cobordism W † from Y to Y by thefollowing process. Cut W open along Z to obtain a manifold W ′ with twoextra boundary components G × S , then attach a copy of G × D to eachof these boundary components to obtain W † . Set F W = ( F ∪ F ∪ G ) ⊂ WF W † = ( F ∪ F ∪ G ) ⊂ W † . Then we have
Proposition 2.5.
The maps HM ( W | F W ) and HM ( W † | F W † ) are equal, upto sign, as maps HM ( Y | F ) → HM ( Y | F ) . Proof.
Consider the manifold W ′ obtained from W by cutting open along Z . This is a cobordism from Y ∪ Z to Y ∪ Z . The manifold W or W † can be obtained from W ′ by gluing with [0 , × Z or with ( D ∐ D ) × G respectively. We can regard [0 , × Z and ( D ∐ D ) × G as two differentcobordisms from Z to Z , and they both induce maps HM • ( Z | G ) → HM • ( Z | G ) . The result follows from the glueing formalism as long as we know that thesetwo maps on HM • ( Z | G ) are the same. Lemma 2.2 tells us that HM • ( Z | G )is simply Z . The product [0 , × Z of course induces the identity map onthis copy of Z . So it only remains to show that the invariant of manifold D × G in HM • ( Z | G ) is ±
1. This can be seen directly by examining thesolutions of the Seiberg-Witten equations; or one can see indirectly that thismust be so, on the grounds that there exist closed 4-manifolds containing( Z | G ) for which an appropriate Seiberg-Witten invariant is 1. In the previous subsection we discussed gluing results in a context wherethe boundary components of the cobordisms carried spin c structures that4had non-torsion first Chern classes. The non-torsion condition ensures thatreducible solutions on the 3-manifolds play no role. A situation that isalgebraically similar is when the boundary components Y carry 1-cycles η and we use local coefficients for which the vanishing theorem Proposition 2.1applies. We can think of HM • ( Y ; Γ η ) as measuring the contribution ofthe reducible solutions; so in a situation where this group is zero, as inthe Proposition, we can expect simple gluing results. This expectation isconfirmed in the case of connected 3-manifolds by the results of [18, section32]. We will deal here with the simplest situation, in which the boundarycomponents are 3-tori and local coefficients are used.Let W be a compact oriented 4-manifold with boundary, and supposethe oriented boundary consists of a collection of 3-tori, ∂W = T ∪ · · · ∪ T r . We do not need to suppose that W is connected, but we do require thatevery closed component of W has b + at least 2. Let ν ⊂ W be a 2-chainwith ∂ν = η + · · · + η r . We suppose that each η i is a 1-cycle in T i satisfying the hypotheses of Propo-sition 2.1 and that our coefficient ring R has t − t − invertible. We may takeit that each η i is a standard circle. For each i , the map j : d HM • ( T i ; Γ η i ) → d HM • ( T i ; Γ η i )is an isomorphism according to the proposition, so we again just write HM • ( T i ; Γ i )for this group, using j to identify the two. According to [18, section 37],this group is a free R -module of rank 1, HM • ( T i ; Γ η i ) ∼ = R . (The proof in [18] was done in the case that R = R , but only the invertibilityof t − t − is needed.) After choosing a basis element in HM • ( T i ; Γ η i ), weshould expect W to have an invariant living in O i HM • ( T i ; Γ η i ) = R . However, there is a short-cut to defining an R -valued invariant of W , usedin [7] and [18, section 38]. We now describe this short-cut. In the remainder5of this subsection, we will leave aside the question of choosing homology-orientations to fix the sign of the invariants that arise. So a 4-manifold or acobordism will have an invariant that is ambiguous in its overall sign.Let E (1) be a rational elliptic surface and let [ E (1) be the complementof the neighborhood of a regular fiber, so that ∂ [ E (1) = T . Let ν be a2-cycle in E (1) arising from a section meeting the neighborhood of the fibertransversely in a disk, and let ˆ ν be the corresponding 2-chain in [ E (1). Let¯ W be the closed 4-manifold obtained by attaching r copies of [ E (1) to W ,making the attachments in such a way that the 1-cycles in the boundarytori match up: thus the manifold¯ W = W ∪ T [ E (1) · · · ∪ T r [ E (1)contains a 2-cycle ¯ ν = ν ∪ η ˆ ν · · · ∪ η r ˆ ν . We can now compute a Seiberg-Witten invariant of the closed pair ( ¯
W , ¯ ν ),and the result depends only on ( W, ν ), not on the choice of gluing. Thus wemay make a definition:
Definition 2.6.
Let W have boundary a collection of 3-tori, as above, let ν be a 2-chain in W , and let ( ¯ W , ¯ ν ) be the closed manifold obtained byattaching copies of [ E (1). Suppose that every component of ¯ W has b + ≥ m ( W, ν ) ∈ R for the invariant m ( ¯ W , ¯ ν ) of the closed manifold, as defined at (2). ♦ There is a formal device that can be used to extend this definition toinclude the case that ¯ W has closed components with b + = 1. Let E ( n )denote the elliptic surface without multiple fibers and having Euler number n , and let [ E ( n ) be the complement of a fiber. There is a 2-chain ν n just asin the case n = 1. Instead of attaching [ E (1) to each T i to form ¯ W , we cansimilarly attach \ E ( n i ) to T i , for any n i ≥
1. We still refer to the resultingclosed manifold as ¯ W . It contains a 2-cycle ¯ ν as before. By choosing n i larger than 1 when needed, we can ensure that all components of ¯ W have b + least 2. We then define m ( W, ν ) by m ( W, ν ) = ( t − t − ) − P ( n i − m ( ¯ W , ¯ ν ) . (9)By the results of [18, section 38], the quantity on the right is independentof the choice of the n i .6 Suppose next that W contains in its interior another 3-torus T whichintersects ν transversely in a single circle η representing a primitive elementof H ( T ). We can then cut W open along T to obtain W ′ , a manifold whoseboundary consists of ( r + 2) tori. We can denote the two new boundarycomponents by T r +1 and T r +2 . By cutting ν also, we obtain a 2-chain ν ′ in W ′ whose boundary has two new circles η r +1 and η r +2 in the new boundarycomponents. We have the following glueing theorem. (The hypothesis that t − t − is invertible in R remains in place.) Proposition 2.7.
In the above situation, the invariants of ( W, ν ) and ( W ′ , ν ′ ) are equal: thus m ( W, ν ) = m ( W ′ , ν ′ ) 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 invari-ants of suitable closed manifolds. Restating it in such terms, and throwingout the components that do not contain T , we arrive at the following. Let X be a closed, connected 4-manifold with b + ≥
2, and let T ⊂ X be anon-separating 3-torus. Let ν be a 2-cycle in X meeting T transversely ina standard circle η with multiplicity 1. Let X ′ be cobordism from T to T obtained by cutting X open, and let ν ′ be the resulting 2-chain in X ′ . Be-cause of what we already know about the separating case, the propositionis equivalent to the following lemma, which we shall prove. Lemma 2.8.
In the above situation, the map induced by the cobordism, d HM • ( X ′ ; Γ ν ′ ) : d HM • ( T ; Γ η ) → d HM • ( T ; Γ η ) is given by multiplication by the element m ( X, ν ) ∈ R .Proof. It is convenient to arrange first that X ′ has b + at least 1. We can dothis by choosing a standard 2-torus F near T intersecting ν 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 multipliesboth d HM • ( X ′ ; Γ ν ′ ) and m ( X, ν ) by ( t − t − ) n − .We now perturb the Chern-Simons-Dirac functional on T , as in [18,section 37], so that there are only reducible critical points, and we stretch X at T , inserting a cylinder [ − R, R ] × T and letting R increase to infinity7as usual. We consider what happens to the zero-dimensional moduli spaceson X in the limit. Because b + ( X ′ ) is at least 1, we obtain in the limitonly irreducible solutions on the cylindrical-end manifold obtained from X ′ .Furthermore, these irreducible solutions run from boundary-unstable criticalpoints at the incoming end to boundary-stable critical points at the outgoingend. The weighted count of such solutions defines the map −−→ HM ( X ′ ; Γ ν ′ ) : d HM • ( T ; Γ η ) → d HM • ( T ; Γ η )in the notation of [18, subsection 3.5]. We must also obtain in the limitsome (possibly broken) trajectories on the cylindrical part, running fromboundary-stable critical points to boundary-unstable critical points. Fordimension-counting reasons, these trajectories must actually be unbrokenand must be boundary-obstructed. The weighted count of such trajectoriesdefines the map j : d HM • ( T ; Γ η ) → d HM • ( T ; Γ η ) . Thus m ( X, ν ) is equal to the contraction by the Kronecker pairing of twochain maps which on homology define the composite j ◦ −−→ HM • ( X ; Γ ν ′ ) : d HM • ( T ; Γ η ) → d HM • ( T ; Γ η ) . It follows that m ( X, ν ) is the trace of this composite map. The composite isequal to d HM • ( X ′ ; Γ ν ′ ), 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 casethat W has some additional boundary components which are not 3-toribut contain surfaces F of genus 2 or more, as in the previous subsection.That is, we suppose that the boundary of W is a union of 3-tori T , . . . , T r together with a pair of 3-manifolds − Y and Y , each of which may haveseveral components. We suppose also that Y and Y contain surfaces F and F all of whose components have genus 2 or more. We also ask thateach component of Y i contains a component of F i . We shall suppose thatthere is a 2-chain ν in W whose boundary we write as ∂ν = − ζ + ζ + η + · · · + η r . The η i are to be standard circles, one in each torus T i as before. The 1-cycles ζ and ζ will be in Y and Y , but we can allow these to be arbitrary(zero for example). We take F W to be any closed surface in W consistingof F ∪ F together perhaps with additional components. We again suppose8that any closed component of W has b + ≥
2. Then W should give rise to amap HM • ( W | F W ; Γ ν ) : HM • ( Y | F ; Γ ζ ) → HM • ( Y | F ; Γ ζ ) . (10)To define this map, we can again attach ( [ E (1) , ˆ ν ) to each of the 3-tori,to obtain ( ¯ W , ¯ ν ) a cobordism from Y to Y containing a 2-chain ¯ ν and asurface F W . The boundary of ¯ ν is just − ζ + ζ . As in Definition 2.6, wetake HM • ( W : F W ; Γ ν ) to be defined by the map given by the cobordism¯ W . In the event that ¯ W has any closed components with b + = 1, we modifythe construction by using elliptic surfaces E ( n i ) as in (9). Proposition 2.7then has the following variant. Proposition 2.9.
Let W be as above, and let T ⊂ W be a -torus meeting ν transversely in a standard circle with multiplicity . Let W ′ and ν ′ beobtained from W and ν by cutting along T . Suppose that F W is disjointfrom T , so that it becomes also a surface F W ′ in W ′ . Assume as always that t − t − is invertible in R . Then the mapsHM • ( W | F W ; Γ ν ) : HM • ( Y | F ; Γ ζ ) → HM • ( Y | F ; Γ ζ ) HM • ( W ′ | F W ′ ; Γ ν ′ ) : HM • ( Y | F ; Γ ζ ) → HM • ( Y | F ; Γ ζ ) (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 Y to Y containing a -chain η with boundary − ζ ∪ ζ . Let F , F and F W be surfaces as above. Let T ⊂ W be a -torus disjoint from F W and cutting ν in a standard circle η ⊂ T . Form W † by cutting W along T and attaching two copies of D × T in such a way that ∂D × { p } is glued to η in both copies. Let η † be the -chain in W † obtained by attaching -disks D × { p } . Then, as maps fromHM • ( Y | F ; Γ ζ ) to HM • ( Y | F ; Γ ζ ) , we haveHM ( W | F W ; Γ ν ) = ( t − t − ) HM ( W † | F W † ; Γ ν † ) , 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 cobordismsfrom T to T : first, the product cobordism, and second the (disconnected)cobordism formed from two copies of D × T . In each case, there is an9obvious 2-chain whose boundary is the difference of the two copies of η .Each of these cobordisms has an invariant which lives in R , according toDefinition 2.6, or more accurately its correction at (9). In this sense, theproduct cobordism has invariant 1 ∈ R . The invariant of the other cobor-dism is ( t − t − ) − , as can be deduced from the invariants of the ellipticsurfaces. We shall need to understand how monopole Floer homology behaves un-der certain cutting and gluing operations on the underlying 3-manifold. Aformula of the type that we need was first proved by Floer in the contextof instanton homology. Floer’s “excision formula”, as he called it, appliedonly to cutting along tori; but in the monopole homology context one canequally well cut along surfaces of higher genus, as long as one restricts tospin c structures that are of top degree on the surface where the cut is made.We give the proof in the monopole Floer homology context in this section:it is almost identical to Floer’s argument, as presented in [2]. Similar formu-lae have been proved in Heegaard Floer theory, by Ghiggini, Ni and Juh´asz[11, 23, 22, 12, 13].The setup is the following. Let Y be a closed, oriented 3-manifold, ofeither one or two components. In the case of two components, we call thecomponents Y and Y . Let Σ and Σ be closed oriented surfaces in Y ,both of them connected and of equal genus. If Y has two components, thenwe suppose that Σ i is a non-separating surface in Y i for i = 1 ,
2. If Y isconnected, then we suppose that Σ and Σ represent independent homologyclasses. In either case, we write Σ for Σ ∪ Σ . Fix an orientation-preservingdiffeomorphism h : Σ → Σ . From this data, we construct a new manifold˜ Y as follows. Cut each Y along Σ to obtain a manifold Y ′ with four boundarycomponents: with orientations, we can write ∂Y ′ = Σ ∪ ( − Σ ) ∪ Σ ∪ ( − Σ )If Y has two components, then so does Y ′ , and we can write Y ′ = Y ′ ∪ Y ′ . Now form ˜ Y by gluing the boundary component Σ to the boundarycomponent − Σ and gluing Σ to − Σ , using the chosen diffeomorphismof h both times. See Figure 1 for a picture in the case that Y has twocomponents. In either case, ˜ Y is connected. We write ˜Σ for the image of0 Figure 1: Forming a manifold ˜ Y from Y and Y , for the excision theorem. Σ = − Σ in ˜ Y and ˜Σ for the image of Σ = − Σ . So ˜ Y contains a surface˜Σ = ˜Σ ∪ ˜Σ .If we wish to use local coefficients in Floer homology, we will need toaugment this excision picture with 1-cycles η . Specifically, we take a 1-cycle η in Y that intersects each Σ i transversely in a single point p i ( i = 1 , Y has two components, then we may write η = η + η for its two parts. We suppose that the diffeomorphism h ischosen so that h ( p ) = p . When this is done, the 1-cycle gives to a 1-cycle˜ η in the new manifold ˜ Y , as shown, by cutting and gluing.We begin with a statement of the excision theorem with integer coeffi-cients, when the genus of Σ is two or more. Theorem 3.1. If ˜ Y is obtained from Y as above and the genus of Σ and Σ is at least two, then there is an isomorphism of Floer groups with integercoefficients, HM • ( Y | Σ) → HM • ( ˜ Y | ˜Σ) . Remark.
In the case that Y has two components, the left-hand side is thehomology of a tensor product of complexes. In this case, the statement ofthe theorem implies that there is a split short exact sequence HM • ( Y | Σ ) ⊗ HM • ( Y | Σ ) → HM • ( ˜ Y | ˜Σ) → Tor (cid:0) HM • ( Y | Σ ) , HM • ( Y | Σ ) (cid:1) . (12)1Floer’s version of this theorem has Σ and Σ of genus 1, with Y = Y ∪ Y . It uses instanton Floer homology associated to an SO (3) bundlewith non-zero Stiefel-Whitney class on Σ. To obtain a version in monopoleFloer homology when Σ has genus 1, we need to use local coefficients. Wepresent a version that is tailored to our later needs. We recall that Γ η denotes a system of local coefficients with fiber R , a commutative ring as insection 2.1. We suppose, as just discussed, that η meets Σ and Σ each ina single point so that we may form ˜ η as shown. Under these hypotheses, weexpect there to be an isomorphism HM • ( Y ; Γ η ) → HM • ( ˜ Y ; Γ ˜ η ) . We shall not endeavor to prove this variant of Floer’s excision theorem here,because it involves considering reducible solutions on multiple boundarycomponents. Instead, as in section 2.6, we introduce some auxiliary sur-faces F and corresponding constraints on the spin c structures, just to avoidreducibles.Thus we suppose in addition that Y contains an oriented surface F meeting Σ = Σ ∪ Σ transversely, and that the diffeomorphism h : Σ → Σ carries the oriented intersection Σ ∩ F to Σ ∩ F . In this case, we canform an oriented surface ˜ F in the new 3-manifold ˜ Y , by cutting F andregluing. We suppose that neither F nor ˜ F contains a 2-sphere, and thatevery component of Y contains a component of F whose genus is at least 2. Theorem 3.2.
Suppose ˜ Y and ˜ F are obtained from Y and F as above, with Σ and Σ both of genus . Let ˜ η be the -cycle in ˜ Y formed from the cycle η in Y as shown in Figure 1. Assume as usual that t − t − is invertible inthe ring R . Then there is an isomorphism:HM • ( Y | F ; Γ η ) → HM • ( ˜ Y | ˜ F ; Γ ˜ η ) . Remark.
Note again that if Y has two components and R is a field, thenthe left-hand-side is the tensor product HM • ( Y | F ; Γ η ) ⊗ R HM • ( Y | F ; Γ η ) . There is also a simpler way in which local coefficients can enter intothe excision theorem, when the cycle η does not intersect Σ. We state anadaptation of Theorem 3.1 of this sort. Theorem 3.3.
Let ˜ Y be obtained from Y as in Theorem 3.1, with Σ ofgenus at least two. Let η be a 1-cycle in Y , disjoint from Σ . This becomes a cycle also in ˜ Y , which we denote by ˜ η . Then we have an isomorphism of R -modules: HM • ( Y | Σ; Γ η ) → HM • ( ˜ Y | ˜Σ; Γ ˜ η ) . In Theorem 3.3, consider the case that Y = Y ∪ Y and η is contained in Y . In this case, the chain complex that computes the group HM • ( Y | Σ; Γ η )on the left is C • ( Y | Σ ; Γ η ) ⊗ Z C • ( Y | Σ ) , (the tensor product of a complex of free R -modules and a complex of freeabelian groups, both finitely generated). By the K¨unneth theorem, if R has no Z -torsion and HM • ( Y | Σ ; Γ η ) is a free R -module, then the theoremprovides an isomorphism HM • ( Y | Σ ; Γ η ) ⊗ HM • ( Y | Σ ) → HM • ( ˜ Y | ˜Σ; Γ ˜ η ) . (13)As a particular application of this result, we have: Corollary 3.4.
Let Σ ⊂ Y be a closed, oriented surface whose componentshave genus at least and let η be a -cycle in Y whose support lies in Σ .Suppose that R has no Z -torsion. ThenHM • ( Y | Σ; Γ η ) ∼ = HM • ( Y | Σ) ⊗ R . Proof.
Apply the isomorphism of (13) with ( Y , Σ ) = ( Y, Σ) and ( Y , Σ ) =(Σ × S , Σ × { p } ). Take η in Σ × S to be the cycle corresponding to η . ByProposition 2.3 we have HM • ( Y | Σ ; Γ η ) = R . The manifold ˜ Y is another copy of the original Y and ˜Σ is two parallel copiesof Σ. The cycle η becomes now the original 1-cycle η , so HM • ( ˜ Y | ˜Σ; Γ ˜ η ) = HM • ( Y | Σ; Γ η ) . Thus (13) gives an isomorphism
R ⊗ HM • ( Y | Σ) → HM • ( Y | Σ; Γ η ) . Figure 2: A cobordism W from ˜ Y to Y = Y ∪ Y . The proof of Theorem 3.1 is very much the same as Floer’s proof of his origi-nal excision theorem, as described in [2]. The first step (which is common toboth Theorem 3.1 and Theorem 3.2) is to construct a cobordism W from ˜ Y to Y . In the case that Y is disjoint union Y ∪ Y , the cobordism W admitsa map π : W → P , where P is a 2-dimensional pair-of-pants cobordism.This is shown schematically in Figure 2. The 4-dimensional cobordism isthe union of two pieces. The first piece is the product [0 , × Y ′ , where Y ′ asbefore is obtained from Y by cutting open along Σ and Σ . (In the Figure,this appears as the union of two pieces, corresponding to the decompositionof Y ′ as Y ′ ∪ Y ′ .) The second piece is the product of the closed surfaceΣ with a 2-manifold U with corners: U corresponds to the gray-shadedarea in the figure. The two pieces are fitted together as shown, using thediffeomorphism h . If Y is connected, then the picture looks just the same inthe neighborhood of the shaded region, but the product region [0 , × Y ′ isconnected; the cobordism W in this case does not admit a map to the pairof pants.There is a very similar cobordism ¯ W which goes the other way: The-orem 3.1 arises because the cobordisms W and ¯ W give rise to mutuallyinverse maps (in the case of genus at least 2) HM ( W ) : HM • ( ˜ Y | ˜Σ) → HM • ( Y | Σ) HM ( ¯ W ) : HM • ( Y | Σ) → HM • ( ˜ Y | ˜Σ) . when the coefficients are a field.To show that the cobordisms induce mutually inverse maps, let X be the4 Figure 3: The composite cobordism X from ˜ Y to ˜ Y . cobordism from ˜ Y to ˜ Y formed as the union of W and ¯ W . We must showthat X gives rise to the identity map on HM • ( ˜ Y | ˜Σ). This will show that HM ( ¯ W ) ◦ HM ( W ) = 1, and there will be a similar argument for the othercomposite. Note that Σ and ˜Σ are homologous in X , so the map inducedby X really does factor through HM • ( Y | Σ), not just HM • ( Y ).The manifold X is shown schematically in Figure 3 for the case that Y has two components, Y ∪ Y , in which case it admits a map π to thetwice-punctured genus-1 surface, as drawn. Over the shaded region V it isa product, π − ( V ) = Σ × V = Σ × V. If Y is connected, the picture is essentially the same in the neighborhoodof π − ( V ). Let k be the closed curve in V that is shown, and let K be theinverse image K = π − ( k )= Σ × k. (We continue to identify Σ with Σ via h in what follows.) Let X ′ bethe manifold-with-boundary formed by cutting along K . Its boundary istwo copies of K . Let X ∗ be the new cobordism from ˜ Y to ˜ Y obtained byattaching two copies of Σ × D , with ∂D being identified with k : X ∗ = X ′ ∪ (Σ × D ) ∪ (Σ × D ) . Floer’s proof hinges on the fact that the manifold X ∗ is just the productcobordism from ˜ Y to ˜ Y . This means that we only need show that X ∗ gives5 Figure 4: The composite cobordism in the opposite order, from Y to Y , in the casethat Y has two components. rise 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 ( ¯ W ) ◦ HM ( W ) = 1. The picture for the composite ofthe two cobordisms in the other order is shown in Figure 4. The proof thatthis composite gives the identity is essentially the same: the relationshipbetween Y and ˜ Y is a symmetric one, except that we have allowed only Y to have two components. Figure 4 shows the corresponding curve ˜ k in thiscase, along which one must cut, just as we cut along k in the previous case.This completes the proof of Theorem 3.1.The proof of Theorem 3.2 is very similar. The same cobordisms W and¯ W are used. In the cobordism W , there is a 2-chain ν W whose boundaryis η − ˜ η . It consists of the product chain [0 , × η ′ in part of W obtainedfrom [0 , × Y ; while over the shaded region U in Figure 2, the cycle ν W isa section { p } × U of Σ × U . There is a similar 2-chain ν ¯ W in ¯ W , and thesefit together to give a 2-chain ν X in the composite cobordism X (Figure 3).The 3-manifold K ⊂ X lying over the curve k is now a 3-torus, and K meets ν X transversely in a standard circle. The proof now proceeds as before, butusing Corollary 2.10 in place of Proposition 2.5. We learn that the compositecobordism X gives a map which is ( t − t − ) times the map arising from thetrivial product cobordism X † . That is, HM ( ¯ W | F ¯ W ; Γ ν ¯ W ) ◦ HM ( W | F W ; Γ ν W ) = ( t − t − ) . The same holds for the composite in the opposite order. Since t − t − is aunit in R , this means that HM ( W | F W ; Γ ν W ) is an isomorphism, as required.6 In this section, we give the definition of the monopole homology groupsfor balanced sutured manifolds, which are the main object of study in thispaper.
We recall Juh´asz’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, γ ) is a compact, oriented3-manifold M with boundary, equipped with the following data:(a) a closed, oriented 1-manifold s ( γ ) in ∂M , i.e. a collection of disjointoriented circles in the boundary, called the sutures;(b) a union A ( γ ) of annuli, which comprise a tubular neighborhood of s ( γ )in ∂M ; the closure of ∂M \ A ( γ ) is called R ( γ ).These are required to satisfy the following conditions:(a) M has no closed components;(b) if the components of ∂A ( γ ) are oriented in the same sense as thesutures, then it should be possible to orient R ( γ ) so that its orientedboundary coincides with this given orientation of A ( γ );(c) R ( γ ) has no closed components (which implies that the orientation inthe previous item is unique); we call it the canonical orientation ;(d) if we define R + ( γ ) (and R − ( γ ) also) as the subset of R ( γ ) where thecanonical orientation coincides with the boundary orientation (or itsopposite, respectively), then χ ( R + ( γ )) = χ ( R − ( γ )). ♦ It is often helpful to consider sutured manifolds as manifolds with cor-ners: the corners run along the circles ∂A ( γ ) and separate the flat annulifrom the rest of the boundary. Note that M need not be connected. Amodel example is a product sutured manifold ([ − , × T, δ ) . Here T is an oriented surface with non-empty boundary and no closed com-ponents, and the sutures are s ( δ ) = { } × ∂T A ( δ ) are [ − , × ∂T , and wehave R + ( δ ) = { } × TR − ( δ ) = {− } × T. Given a balanced sutured manifold (
M, γ ), we form a closed, orientedmanifold Y = Y ( M, γ ) as follows. The closed manifold is dependent onsome choices, as we shall see. First, we choose an oriented connected surface T whose boundary components are in one-to-one correspondence with thecomponents of s ( γ ). We call T the auxiliary surface . From T we form theproduct sutured manifold ([ − , × T, δ ) as just described. We then gluethe annuli A ( δ ) to the annuli A ( γ ): this is done by a map A ( δ ) → A ( γ )which is orientation-reversing with respect to the boundary orientations andwhich maps ∂R + ( δ ) to ∂R + ( γ ). The result of this step is a 3-manifold withexactly two boundary components, ¯ R + and ¯ R − , which are closed orientablesurfaces of equal genus: ¯ R + = R + ( γ ) ∪ { } × T ¯ R − = R − ( γ ) ∪ {− } × T We require T to be of sufficiently large genus (genus zero may suffice, andgenus two always will) so that two conditions hold:(C1) the genus of ¯ R ± is at least two;(C2) the surface T contains a simple closed curve c such that { } × c and {− } × c are non-separating curves in ¯ R + and ¯ R − respectively.Finally, form Y ( M, γ ) by identifying ¯ R + with ¯ R − using any diffeomorphismwhich reverses the boundary orientations (i.e. preserves the canonical orien-tations), h : ¯ R + → ¯ R − . Inside Y is a closed, connected, non-separating surface ¯ R , obtained fromthe identification of ¯ R + with ¯ R − . We can orient ¯ R using the canonicalorientation of R + ( γ ). As an oriented pair, ( Y, ¯ R ) depends only on twothings, beyond ( M, γ ) itself: first, the choice of genus for T , and second thechoice of diffeomorphism h used in the last step. Definition 4.2.
We call ( Y, ¯ R ) a closure of the balanced sutured manifold( M, γ ) if it is obtained in this way, by attaching to (
M, γ ) a product region[ − , × T satisfying the above conditions and then attaching ¯ R + to ¯ R − bysome h . ♦ Let Y = Y ( M, γ ) be formed from a sutured manifold (
M, γ ) as describedin the previous subsection. Recall that Y contains a connected, orientedclosed surface ¯ R , by construction, whose genus is at least two. We make thefollowing definition: Definition 4.3.
We define the monopole Floer homology of the suturedmanifold (
M, γ ) to be the finitely-generated abelian group
SHM ( M, γ ) := HM • ( Y | ¯ R ) , where Y = Y ( M, γ ) is a closure of (
M, γ ) 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 diffeomorphism h . In section 4.3 we shall prove: Theorem 4.4.
The group SHM ( M, γ ) defined in 4.3 depends only on ( M, γ ) , not on the choice of genus g for the auxiliary surface T or the dif-feomorphism h . There is a version of
SHM with local coefficients that we shall use atsome points along the way. Recall that T is required to contain a curve c that yields non-separating curves {± } × c on ¯ R ± . Let us choose thediffeomorphism h so that h maps { } × c to {− } × c , preserving orientation.Thus the surface ¯ R in Y ( M, γ ) now contains a closed curve ¯ c , the image of {± } × c . Let c ′ be any dual curve on ¯ R : a curve c ′ with ¯ c · c ′ = 1 on ¯ R . Definition 4.5.
We define the monopole Floer homology of the suturedmanifold (
M, γ ) with local coefficients to be the R -module SHM ( M, γ ; Γ η ) := HM • ( Y | ¯ R ; Γ η ) , where the closure Y = Y ( M, γ ) is constructed using a diffeomorphism h satisfying the constraint just described, and η is the 1-cycle in Y carried bythe curve c ′ dual to ¯ c as above. ♦ We shall see that this is independent of the choice of η . When usinglocal coefficients in this way, we can relax the requirement that ¯ R has genus2 or more (condition (C1) above) and allow closures in which ¯ R has genus1:9 Proposition 4.6.
As long as t − t − is invertible in R , the R -moduleSHM ( M, γ ; Γ η ) defined in 4.5 depends only on ( M, γ ) and R , not on theremaining choices. Furthermore, subject to the same condition on R , onecan relax the condition (C1) above and allow ¯ R to have genus when usinglocal coefficients. In the case that ¯ R does have genus 2 or more, we shall also see that wecan take η to be any non-separating curve on ¯ R , rather than a curve dualto ¯ c . We now prove Theorem 4.4: our definition of the monopole Floer homologyof a balanced sutured manifold (
M, γ ) 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 → S be a fibered -manifold whose fiber R is a closedsurface of genus at least . Then HM ( Y | R ) ∼ = Z .Proof. In the case of the product fibration, we have already seen this inthe previous section. If Y h denotes the mapping torus of a diffeomorphism h : R → R , then the excision theorem, Theorem 3.1, in the guise of (12),gives us an injective map HM ( Y h | R ) ⊗ HM ( Y g | R ) → HM ( Y gh | R )with cokernel the Tor term. When g = h − , the mapping tori Y h and Y g areorientation-reversing diffeomorphic; and HM ( Y g | R ) is therefore isomorphicto HM ( Y h | R ) as an abelian group. (This is for the same reason that thehomology and cohomology of a finitely-generated complex of free Z -modulesare isomorphic, as abelian groups.) So we obtain an injective map HM ( Y h | R ) ⊗ HM ( Y h | R ) → Z whose cokernel is torsion. This forces HM ( Y h | R ) to be Z . Corollary 4.8.
Let Y be a closed oriented -manifold containing a non-separating oriented surface ¯ R of genus two or more. Let ˜ Y be obtained from Y by cutting along ¯ R and re-gluing by an orientation-preserving diffeomor-phism h . Then HM ( Y | ¯ R ) and HM ( ˜ Y | ¯ R ) are isomorphic. Proof.
Apply the excision theorem, Theorem 3.1, with Y = Y ∪ Y , taking Y to be the mapping torus of h and Σ = Σ = ¯ R . Lemma 4.7 tells us that HM ( Y | ¯ R ) ∼ = Z , so HM ( Y | ¯ R ) ∼ = HM ( ˜ Y | ¯ R ) by the excision theorem.Consider now the situation of Theorem 4.4. We have a closed 3-manifold Y = Y ( M, γ ) whose construction depends on a choice of genus g for T anda choice of diffeomorphism h . We are always supposing that Y has beenconstructed using an auxiliary surface T subject to the conditions (C1) and(C2). The above corollary tells us that HM ( Y | ¯ R ) is independent of thechoice of h . So the group SHM ( M, γ ), as we have defined it, depends onlyon the choice of g . Let us temporarily write it as SHM g ( M, γ ) . (14)We can apply the same arguments with local coefficients: Theorem 3.3 canbe used in place of Theorem 3.1 to see that SHM g ( M, γ ; Γ η ) (15)(as defined in Definition 4.5) depends at most on the choice of g , not on h (as long 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 haveSHM g ( M, γ ; Γ η ) = SHM g ( M, γ ) ⊗ R . Proof.
In the definition of the local system Γ η , the 1-cycle η is parallel toa curve lying on ¯ R . The result therefore follows from the definitions andCorollary 3.4.Because we already know that SHM g ( M, γ ) is independent of h , theabove lemma establishes that SHM g ( M, γ ; Γ η ) is also independent of h , andthat it is also independent of the choice of η . Next we prove: Proposition 4.10. If t − t − is invertible in the coefficient ring R and R hasno Z -torsion, then the Floer group with local coefficients, SHM g ( M, γ ; Γ η ) ,is independent of g .Proof. Fix g and let T be a surface of genus g . Let Y be the resultingclosure of ( M, γ ), and write ¯ R for the surface it contains. Recall that werequired T to contain a simple closed curve c such that { } × c and {− } × c are non-separating in ¯ R ± . We can form a surface ˜ T of genus g + 1 by1 Figure 5: Increasing the genus of T by 1. the following process. We take a closed surface S of genus 2 containing anon-separating closed curve d . We then cut T along c and cut S along d ,and we reglue to form ˜ T as shown in Figure 5. The figure also shows curves c ′ and d ′ dual to c and d . The curve c ′ is supposed to be extended (out ofthe picture) to become a simple closed curve dual to c in the larger surface¯ R + = R + ( γ ) ∪ { } × T .In forming the closure Y using T , we can arrange that the diffeomor-phism h : ¯ R + → ¯ R − carries { } × c to {− } × c by the identity map on c . This is because any non-separating curve is equivalent to any other inan oriented surface. This implies that Y can be identified with the product S × T over some neighborhood of c in T . So Y contains a torus, S × c . Thedual curve c ′ on ¯ R + becomes a curve (also called c ′ ) in Y which intersectsthe torus S × c once.We will apply the second version of the excision theorem, Theorem 3.2,as follows. We take Y = Y ∪ Y with Y as given, and Y = S × S . Wetake Σ to be the torus S × c inside the product region of Y and Σ tobe S × d . We take η in Y to be η + η , where the cycle η is c ′ and η is { point } × d ′ . These 1-cycles intersect the respective tori once each; and η isof the sort required for the definition of SHM g ( M, γ ; Γ η ) in Definition 4.5.To play the role of the surface F = F ∪ F in Theorem 3.2 we take ¯ R ∪ ¯ R ,where ¯ R is the genus-2 surface { point } × S .The manifold ˜ Y obtained from Y = Y ∪ Y in the excision theorem isanother closure of the original ( M, γ ), using the auxiliary surface ˜ T of genus2one larger than T and a diffeomorphism ˜ h obtained by extending h triviallyover the extra handle. It contains a closed surface ˜ R whose genus is onelarger than the genus of ¯ R . This is the surface obtained from ¯ R and ¯ R by cutting and gluing. So to prove the proposition, we must prove HM ( Y | ¯ R ; Γ η ) ∼ = HM ( ˜ Y | ˜ R ; Γ ˜ η ) . (16)Theorem 3.2, provides an isomorphism HM (( Y ∪ Y ) | ( ¯ R ∪ ¯ R ); Γ η ) → HM ( ˜ Y | ˜ R ; Γ ˜ η ) . But HM ( Y | ¯ R ; Γ η ) is just R by Corollary 2.3, because this manifold isa product, so (16) follows from the K¨unneth theorem. This completes theproof of the proposition. Remark.
Although Figure 5 is drawn so as to make clear that the excisiontheorem is applicable, the topology can be described more simply. Let G denotes the genus-one surface with one boundary component, obtained bycutting S open along d and then removing a neighborhood of d ′ . Then theoperation of forming ˜ T as shown is the same as removing a neighborhoodof the point x = c ∩ c ′ and attaching G to the boundary so created: aconnected sum in other words. The 3-manifold picture is obtained from thisconnected-sum picture by multiplying with by S . That is, we drill out aneighborhood of S × { x } and glue in S × G .Now we can complete the proof of the theorem: Proof of Theorem 4.4.
We have seen that there is no dependence on thechoice of diffeomorphism h , and we have been considering the dependenceon the genus g : we wish to show that SHM g ( M, γ ) is independent of g .From Lemma 4.9 and Proposition 4.10, we learn that the R -module SHM g ( M, γ ) ⊗ R is independent of g whenever R has no Z -torsion and t − t − is invertible.But if A and B are finitely-generated abelian groups and A ⊗ R ∼ = B ⊗ R as R -modules for all such R , then we must have A ∼ = B . For this one can takea universal example for R , namely the ring obtained by inverting t − t − inthe Z [ R ], the group ring of R .Finally, we turn to Proposition 4.6. Up until this point we have beenassuming that ¯ R has genus 2 or more. But the proof of Proposition 4.10works just as well in the genus 1 case. Thus if Y is a closure formed with3¯ R of genus 1 and ( ˜ Y , ˜ R ) is formed as in the proof of Proposition 4.10 with¯ R of genus 2, then HM • ( Y | ¯ R ; Γ η ) ∼ = HM • ( ˜ Y | ˜ Rl Γ ˜ η ) . The group on the right is something we already know to be independent ofother choices: we have therefore HM • ( Y | ¯ R ; Γ η ) = SHM ( M, γ ) ⊗ R . (17)This verifies Proposition 4.6. Juh´asz showed in [12] that knot homology could be obtained as a specialcase of his (Heegaard) Floer homology of a sutured manifold. Specifically,given a knot K in a closed 3-manifold Z , one can form a sutured manifold( M, γ ) by taking M to be the knot complement (with a torus boundary) andtaking the sutures to be two oppositely-oriented meridians. In the monopolecase we have at present no a priori notion of knot homology; but we are freeto take Juh´asz’s prescription as a definition of knot homology and pursuethe consequences. Thus: Definition 5.1.
For a knot K in a closed, oriented 3-manifold Z , we definethe monopole knot homology KHM ( Z, K ) to be the monopole homology ofthe sutured manifold (
M, γ ) associated to (
Z, K ) by Juh´asz’s construction.That is,
KHM ( Z, K ) :=
SHM ( M, γ )where M = Z \ N ◦ ( K ) is the knot complement and s ( γ ) consists of twooppositely-oriented meridians. ♦ To understand what this definition leads to, we must construct a suitableclosure of the sutured manifold.
So let K be a knot in a closed manifold Z , and let ( M, γ ) be the knot com-plement, with two sutures as just described. We can describe a particularlysimple closure of (
M, γ ) as follows, if we temporarily relax the rules andallow the auxiliary surface T to be an annulus. (The reason this is not avalid closure of ( M, γ ) for our purposes is that the resulting surfaces ¯ R ± will4 Figure 6: The part of L lying inside the larger tubular neighborhood N . have genus 1. We will correct this shortly, replacing the annulus by a surfaceof genus 1.) Let N be a closed tubular neighborhood of K , and let N ′ ⊂ N be a smaller one. Let m be a meridian of K , lying outside N ′ but inside N .We will consider M to be Z \ N ′ , and we take two meridional sutures s ( γ )on the boundary of N ′ . If we take T to be an annulus and attach [ − , × T by gluing the two annuli [ − , × ∂T to the sutures A ( γ ), then what resultsis a 3-manifold L with two tori as boundary components: we can identify itwith the complement of a tubular neighborhood of m in M .Figure 6 shows the part of L that lies inside the tubular neighborhood N of K . (The top and bottom are identified.) The figure shows a verticalsolid torus N with a smaller vertical solid torus N ′ 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 ′ )and the boundary of the horizontal solid torus (the boundary of U ). Theseboundary components are ¯ R + and ¯ R − .If we choose a framing of K , then we obtain a fibration of L ∩ N bypunctured annuli E (one of which is shown gray in the figure). We nowform the closure Y = Y ( M, γ ) using T as the auxiliary surface by gluing¯ R + to ¯ R − : on each punctured annulus E , we glue the circle E ∩ ¯ R + to E ∩ ¯ R − . This turns each annulus E into a genus-1 surface F with one5 Figure 7: The closed surface F obtained by gluing two boundary components of E : a genus-one surface with one boundary component. The curve α is the gluinglocus. boundary component. (The remaining boundary component of F lies onthe outer torus, ∂N .) Thus we have seen: Lemma 5.2.
Using an annulus T as the auxiliary surface, a closure of thesutured manifold ( M, γ ) associated to a knot K in Z can be described bytaking a surface F of genus one, with one boundary component, and gluing F × S to the knot complement Z \ N . The gluing is done so that { p } × S is is attached to the meridian of K on ∂N and ∂F × { q } is glued to anychosen 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”in the 3-torus, using any chosen framing of the former and the standardframing of the latter, attaching longitudes to meridians and meridians tolongitudes. We give a name to this closed manifold: Definition 5.3.
We write Y ( Z, K ) for the closed 3-manifold obtained fromthe framed 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, γ ) that is illegitimate, because T is anannulus and ¯ R has genus one. We now described how Y ( Z, K ) gets modifiedif we use a surface ˜ T of genus one (still with two boundary components)instead of T . Figure 7 shows the surface F . The curve α on F is theintersection of F with the torus ¯ R ⊂ Y ( Z, K ) where ¯ R + and ¯ R − are glued.6Thus ¯ R is the torus ¯ R = α × S ⊂ F × S ⊂ Y ( Z, K )The image of [ − , × T in Y ( Z, K ) is a copy of S × T and can be identifiedwith the neighborhood of β × S : S × T = nbd( β ) × S ⊂ F × S , where nbd( β ) ⊂ F is an annular neighborhood of β . The identification ofthe various factors is as indicated: the S factor in S × T becomes the β factor on the right, and the core of the annulus T becomes the S factor onthe right. Recalling the remark made at the end of section 4.3, we see thatto effectively increase the genus of the auxiliary surface by 1, we should:(a) drill out a tubular neighborhood β × D of the circle β × { q } ⊂ F × S ;(b) attach S × G , where G is a genus-one surface with one boundarycomponent, by a diffeomorphism S × ∂G → β × ∂D which preserves the order of the factors.(In the second step, a framing of { q } × β is needed, but we have a preferredone because β lies on { q } × F .) Definition 5.4.
We write ˜ Y ( Z, K ) for the manifold obtained from Y ( Z, K )by the two steps just described. It is a closure of the sutured manifold (
M, γ )associated to the knot K in Z obtained using a genus-one auxiliary surface˜ T ; and it depends only on a choice of framing for K . ♦ While the closure Y ( Z, K ) has a genus-one surface ¯ R , the closure˜ Y ( Z, K ) has a genus-two surface ˜ R . The latter is obtained from ¯ R = α × S by removing a neighborhood of the point ( x, q ) in α × S and adding thegenus-one surface { x } × G . To summarize this discussion, we have the fol-lowing, essentially by definition now: Corollary 5.5.
The monopole knot homology KHM ( Z, K ) can be computedas the ordinary monopole homology HM ( ˜ Y | ˜ R ) , where ˜ Y = ˜ Y ( Z, K ) is asabove. Any framing of K can be used in the construction of ˜ Y . Remark.
Both Y ( Z, K ) and ˜ Y ( Z, K ) can be described alternatively as fol-lows. Let S be a closed surface of genus l , let c be a non-separating simpleclosed curve on S , and let ˆ c be the curve { p } × c in the 3-manifold S × S .Let N (ˆ c ) be a tubular neighborhood. Let Y l be the result of gluing thecomplement of ˆ c to the complement of K : Y l = ( S × S ) \ N ◦ (ˆ c ) ∪ φ Z \ N ◦ ( K ) (18)where φ identifies the meridian curves of ˆ c to the longitudes of K and viceversa. (We give ˆ c the obvious framing, and we recall that a framing of K has been chosen.) Then the manifold Y is Y ( Z, K ), and Y is ˜ Y ( Z, K ).We can also use the simpler manifold Y to compute monopole knothomology, as long as we switch to local coefficients. This is the content ofthe next lemma. Lemma 5.6. If t − t − is invertible in the coefficient ring R and R hasno Z -torsion, then the knot homology KHM ( Z, K ) ⊗ R can be computed asHM • ( Y ; Γ ˆ α ) , where Y is the manifold described in Definition 5.3 and ˆ α isthe curve α × { p } in F × S ⊂ Y , regarded as a -cycle.Proof. According to Proposition 4.6, we can use the closure Y to compute SHM ( M, γ ; Γ η ). Together with Lemma 4.9, this tells us that HM • ( Y ; Γ ˆ α ) ∼ = SHM ( M, γ ) ⊗ R , where ( M, γ ) is the sutured manifold obtained from the knot complementby Juh´asz’s prescription.
Suppose that the knot K ⊂ Z is null-homologous, and let Σ be a Seifertsurface for K : an oriented embedded surface in Z \ N ◦ with boundary asimple closed curve on ∂N . We can frame the knot K so that N is identifiedwith K × D and ∂ Σ is K × { q ′ } for some q ′ ∈ S . We can also regard Σas a surface in the manifold Y ( Z, K ) (Definition 5.3). The union of Σ and F × { q ′ } in Y ( Z, K ) is a closed oriented surface¯Σ = Σ ∪ ( F × { q ′ } ) ⊂ Y ( Z, K ) . Its genus is one more than the genus of Σ. The surface F × { q ′ } ⊂ Y ( Z, K )remains intact in the manifold ˜ Y ( Z, K ) for q ′ = q (Definition 5.4), so we canregard ¯Σ also as a closed surface in ˜ Y = ˜ Y ( Z, K ). Using the surface ¯Σ, we8can decompose
KHM ( Z, K ) according to the first Chern class of the spin c structure. We write KHM ( Z, K ) = M i ∈ Z KHM ( Z, K, i )where
KHM ( Z, K, i ) = M s ∈S ( ˜ Y | ˜ R ) h c ( s ) , [ ¯Σ] i =2 i HM • ( ˜ Y , s ) . If Z is not a homology sphere, then the decomposition by spin c structuresmay depend on the choice of the relative homology class for the Seifertsurface Σ, in which case one should write KHM ( Z, K, [Σ] , i ) (19)for the summands.Some familiar properties of the (Heegaard) knot homology of Ozsv´ath-Szab´o and Rasmussen carry over to this monopole version.
Lemma 5.7.
The groups KHM ( Z, K, i ) and KHM ( Z, K, − i ) are isomor-phic.Proof. The isomorphism arises from the isomorphism between HM • ( Y, s )and HM • ( Y, ¯ s ), where ¯ s is the conjugate spin c structure. Lemma 5.8.
The group KHM ( Z, K, i ) is zero for | i | larger than the genusof Σ .Proof. The adjunction inequality tells us that HM • ( Y, s ) is zero for spin c structures s with c ( s )[ ¯Σ] greater than 2 g ( ¯Σ) −
2. The genus of ¯Σ is onelarger than the genus of Σ.
Lemma 5.9.
For a classical knot K in S of genus g , the monopole knothomology group KHM ( S , K, g ) is non-zero.Proof. We use the description of ˜ Y = ˜ Y ( K ) as the the manifold Y , where Y l is the manifold described by (18). Let S be the genus-2 surface used there,let c be the closed curve on S , and let c ′ be a dual curve on S meeting c once.According to Gabai’s results [9, 10], a Seifert surface Σ of K of genus g arisesas a compact leaf of a taut foliation F K of S \ N ◦ ( K ), and we can ask thatthe leaves of F K meet ∂N ( K ) in parallel circles. On the other hand, S × S has a taut foliation F S which is transverse to the curve ˆ c = { p } × c . This9foliation is obtained from the trivial product foliation by cutting alont thetorus S × c ′ and regluing with a small rotation of the S factor. Together,the foliations F K and F S define a foliation F of ˜ Y = Y . The surface ¯Σsits inside Y as the union of the Seifert surface Σ and the punctured torus( S × c ′ ) \ D . The spin c structure s c determined by F has first Chern classof degree 2 g on ¯Σ and degree 2 on the genus-2 surface, so HM • ( ˜ Y , s c ) is asummand of KHM ( S , K, g ) by definition. The non-vanishing theorem fromsection 2.4 tells us that this group is non-zero. Lemma 5.10.
Let K be a classical knot and let χ ( K, i ) denote the Eulercharacteristic of KHM ( S , K, i ) , computed using the canonical Z / gradingon monopole Floer homology [18]. Then the finite Laurent series X i χ ( K, i ) T i is the symmetrized Alexander polynomial, ∆ K ( T ) , for the knot K , up to anoverall sign.Proof. In different guise, this is essentially the same result as that ofFintushel-Stern [7] and Meng-Taubes [20]. Let ˜ Y = ˜ Y ( S , K ) be the usualclosure of the sutured manifold associated to ( S , K ) as in Definition 5.4,let ˜ R be the genus-2 surface in ˜ Y and let ¯Σ ⊂ ˜ Y be the surface of genus g + 1 formed from a Seifert surface Σ for K and the genus-1 surface F .The Euler characteristic can be computed from the Seiberg-Witten in-variants of the manifold S × ˜ Y . Specifically, regard both ¯Σ and ˜ R as surfacesin X K = S × ˜ Y .
Take ν to be the 2-cycle in X K defined by ¯Σ and consider the generatingfunction m ( X K , [ ν ]) as in (2), but modified to use only spin c structures thatare of top degree on ˜ R . We introduce the notation m ′ ( X K , [ ν ]) = X s ∈S ( X K | ˜ R ) m ( X K , s ) t h c ( s ) , [ ν ] i . We then have X i χ ( K, i ) t i = m ′ ( X K , [ ν ]) . Let X be the same type of 4-manifold as X K , but formed using theunknot in place of K . The corresponding 3-manifold ˜ Y is S × S , where S has genus 2; so X is T × S . The remark following Corollary 5.5 explains0that ˜ Y is formed from ˜ Y by drilling out a neighborhood of a curve ˆ c andgluing in the knot complement M K = S \ N ◦ ( K ). If follows that X K isformed from X by a “knot surgery” in the sense of [7]. This, one drills outa neighborhood of the torus S × ˆ c and glues in S × M K . In the formalismof section 2.6, we can therefore compute the ratio m ′ ( X K , [ ν ]) / m ′ ( X , [ ν ])as the ratio of the invariants associated to ( S × M K , ν K ) and ( S × M , ν ).Here M is the knot complement for the unknot, and ν K and ν are the2-chains defined by Seifert surfaces for K and the unknot respectively. Thisratio is precisely what is calculated in [7] (see also [18, section 42.5]), and itis equal to ∆ K ( t ). The lemma follows.Given a null-homologous knot K in a 3-manifold Z , there is a rathermore straightforward way to arrive at a sutured manifold than the one thatleads to knot homology. We can simply choose a Seifert surface Σ for K andcut the knot complement Z \ N ◦ ( K ) open along Σ. The result is a suturedmanifold ( M Σ , δ ) with a single suture and having R + ( δ ) = R − ( δ ) = Σ. Themonopole Floer homology of this sutured manifold captures the top-degreepart of the monopole knot homology: Proposition 5.11.
In the above situation, let g be the genus of the Seifertsurface Σ , and suppose g = 0 . Then SHM ( M Σ , δ ) is isomorphic toKHM ( Z, K, [Σ] , g ) .Proof. It is sufficient to prove that
SHM ( M Σ , δ ) ⊗ R ∼ = KHM ( Z, K, [Σ] , g ) ⊗ R when the coefficient ring R has t − t − invertible and no Z -torsion. Lemma 5.6tells us that we can compute the right-hand side using the manifold Y as KHM ( Z, K, [Σ] , g ) ⊗ R = HM • ( Y | ¯Σ; Γ ˆ α )where ¯Σ is the surface of genus g + 1 in Y . On the other hand, the samemanifold Y arises as a closure of ( M Σ , δ ) in the sense of section 4.1, so wealso have SHM ( M Σ , δ ) ⊗ R = SHM ( M Σ , δ ; Γ η )= HM • ( Y | ¯Σ; Γ ˆ α ) . This proves the proposition.1
In this section, we adapt the material from [23] to show that the monopoleversion of knot homology detects fibered knots. For the most part, thearguments of [23] carry over with little modification.A balanced sutured manifold (
M, γ ) is a homology product if the in-clusions R + ( γ ) → M and R − ( γ ) → M are both isomorphisms on integerhomology groups. The main target is the following theorem. Theorem 6.1.
Suppose that the balanced sutured manifold ( M, γ ) is tautand a homology product. Then ( M, γ ) is a product sutured manifold if andonly if SHM ( M, γ ) = Z . The application to fibered knots is a corollary:
Corollary 6.2. If K ⊂ S is a knot of genus g , then K is fibered if andonly if KHM ( S , K, g ) = Z .Proof of the corollary. The “only if” direction is a straightforward matter:it follows from Lemma 4.7 and Proposition 5.11. The interesting directionis the “if” direction, and this can be deduced from Theorem 6.1 as follows.Suppose that
KHM ( S , K, g ) = Z . From Lemma 5.10 we learn that theAlexander polynomial of K is monic and that its degree is g . Let Σ be aSeifert surface for K of genus g , and let ( M Σ , δ ) be the balanced suturedmanifold obtained by cutting open the knot complement along Σ. As Niobserves in [23, section 3], the fact that the Alexander polynomial is monictells us that ( M Σ , δ ) is a homology product. The group SHM ( M Σ , δ ) isisomorphic to KHM ( S , K, g ) by Proposition 5.11, so SHM ( M Σ , δ ) = Z .Theorem 6.1 implies that ( M Σ , δ ) is a product sutured manifold, from whichit follows that the knot complement is fibered.We will prove Theorem 6.1 after some preliminary material on furtherproperties of SHM . c structures The following definition of relative spin c structures on sutured manifoldscoincides with that of Juh´asz [13], in slightly different notation. If we regard( M, γ ) as a manifold with corners, then it carries a preferred 2-plane field ξ ∂ on its boundary: on R + ( γ ) and R − ( γ ), we take ξ ∂ to be the tangent planesto the boundary, with the canonical orientation; and on each component of2 A ( γ ) we take ξ ∂ to have, as oriented basis, first the outward normal to M and second the direction parallel to the oriented suture. On a 3-manifold anoriented 2-plane field defines a spin c structure; so ξ ∂ gives a spin c structurein a neighborhood of the boundary. We define S ( M, γ ) to be the set ofextensions of s ∂ to a spin c structure on all of M , up to isomorphisms whichare 1 on ∂M . We refer to elements of S ( M, γ ) as relative spin c structures.Consider the process of forming the closure Y = Y ( M, γ ). When weattach [ − , × T to the annuli in ∂M , the 2-plane field ξ ∂ extends in theobvious way, as the tangents to { p } × T . When we the attach ¯ R + to ¯ R − using h , we obtain a 2-plane field on all of Y ( M, γ ) except the interior of theoriginal M . On the surface ¯ R ⊂ Y , this 2-plane field is the tangent planefield. So we obtain a natural map ǫ : S ( M, γ ) → S ( Y | ¯ R ) . (20) Lemma 6.3.
Let s , s ∈ S ( M, γ ) be relative spin c structures whose dif-ference element in H ( M, ∂M ) is not torsion. Then we can choose T andthe diffeomorphism h so that ǫ ( s ) and ǫ ( s ) are spin c structures in S ( Y | ¯ R ) whose difference is still non-torsion.Proof. The statement only concerns the difference elements. The dual of H ( M, ∂M ; Q ) is H ( M ; Q ), and what we must show is that given a non-zero element α ∈ H ( M ), we can choose T and h so that α is in the imageof the map H ( Y ) → H ( M ) . To do this, consider as an intermediate step the manifold Y ′ with boundary¯ R + ∪ ¯ R − formed from M by attaching [ − , × T . The map H ( Y ′ ) → H ( M ) is surjective. Let β be a class in H ( Y ′ ) which restricts to α . Repre-sent the dual of β by a closed surface ( B, ∂B ) in ( Y ′ , ∂Y ′ ). By adding to B an annulus contained in the product region [ − , × T if necessary, we can beassured that ∂B intersects both ¯ R + and ¯ R − in a collection of curves repre-senting a primitive, non-zero homology class. We can then modify B withoutchanging its class so that ∂B consists of two circles: a non-separating curvein each of ¯ R + and ¯ R − . Finally, we choose the diffeomorphism h : ¯ R + → ¯ R − so as to match up these curves. In this way we obtain a closed surface ¯ B in Y whose dual class in H ( Y ) maps to α in H ( M ).The following corollary is the tool used by Ghiggini [11] in his proof ofthe original version of Corollary 6.2 for genus-1 knots.3 Corollary 6.4.
Suppose that ( M, γ ) admits two taut foliations F and F such that the corresponding spin c structures s and s have non-torsion dif-ference element in H ( M, ∂M ) . Then SHM ( M, γ ) has rank at least .Proof. Choose the closure Y = Y ( M, γ ) so that ǫ ( s ) and ǫ ( s ) are differentspin c structures on Y , as Lemma 6.3 allows. The foliations F and F extendin an obvious way to foliations of Y belonging to the spin c structures ǫ ( s )and ǫ ( s ). By the non-vanishing theorem described in section 2.4, the Floergroups HM • ( Y, ǫ ( s )) and HM • ( Y, ǫ ( s )) both have non-zero rank. Bothof these Floer groups contribute to HM • ( Y | ¯ R ) = SHM ( M, γ ), because thespin c structures ǫ ( s i ) belong to S ( Y | ¯ R ). So SHM ( M, γ ) has rank at least2.
The excision theorems, in addition to their role in showing that
SHM ( M, γ )is well-defined, can be used in a straightforward way to establish some de-composition which related the Floer homology of a sutured manifold (
M, γ )to that of ( M ′ , γ ′ ), obtained from ( M, γ ) by cutting along a surface. Werecord a few types of such decomposition theorem here. To avoid variouscircumlocutions involving tensor products and the K¨unneth theorem, weshall work over Q instead of Z here; and when using local coefficients weshall take R to be a field of characteristic zero: either R with the usualexponential map, or the field of fractions of the group ring Q [ R ]. Proposition 6.5.
Suppose ( M, γ ) is a disjoint union ( M , γ ) ∪ ( M , γ ) and that both pieces are balanced. ThenSHM ( M, γ ; Q ) ∼ = SHM ( M , γ ; Q ) ⊗ SHM ( M , γ ; Q ) . Proof.
It will be sufficient to prove this for the local coefficient versions,
SHM ( M, γ ; Γ η ), because of Lemma 4.9. Form the closures ( Y , ¯ R ) and( Y , ¯ R ) of ( M , γ ) and ( M , γ ) by attaching product regions [ − , × T and [ − , × T respectively. Let c and c be non-separating curves on T and T . When forming the closures Y and Y , choose the diffeomorphisms h and h so that h i maps { } × c i to {− } × c i , as in the proof of Proposi-tion 4.10. Let ˜ T be the connected closed surface obtained from T and T bycutting open along c and c and reattaching, similarly to Figure 5. Let ˜ h bethe diffeomorphism of ˜ T that arises from h and h , and let ˜ Y be the closureof ( M, γ ) that is obtained by attaching ˜ T to ( M, γ ) and gluing up using ˜ h .We now have a connected closure ˜ Y that is related to Y = Y ∪ Y by cut-ting and gluing along 2-tori S × c i . So the excision theorem, Theorem 3.2,4provides an isomorphism HM • ( Y | ¯ R ; Γ η ) → HM • ( ˜ Y | ˜ R ; Γ ˜ η ) , and hence and isomorphism HM • ( Y | ¯ R ; R ) → HM • ( ˜ Y | ˜ R ; R ) . Since R is a field and Y is a disjoint union, the left-hand side is a tensorproduct, and the proposition follows.Next we prove a version of Ni’s “horizontal decomposition” formula. A horizontal surface in ( M, γ ) is a surface S with χ ( S ) = χ ( R + ( γ )) such that ∂S consists of one circle in each of the annuli comprising A ( γ ); it is requiredto represent the same relative homology class as R ± ( γ ) in H ( M, A ( γ )) andshould have [ ∂S ] = [ s ( γ )] in H ( A ( γ )). Cutting along a horizontal surfacecreates a new sutured manifold( M ′ , γ ′ ) = ( M , γ ) ∪ ( M , γ ) . Proposition 6.6 ([23, Proposition 4.1]). If ( M ′ , γ ′ ) is obtained from ( M, γ ) by cutting along a horizontal surface, thenSHM ( M, γ ; Q ) = SHM ( M ′ , γ ′ ; Q ) . Proof.
This follows directly from Theorem 3.1 and Proposition 6.5.We shall also need to decompose sutured manifolds by cutting alongvertical surfaces. We prove a result along the lines of [23] and [13]. A productannulus in (
M, γ ) is an embedded annulus A = [ − , × d in ( M, γ ) suchthat the circle d + = { } × d lies in the interior of R + ( γ ) and d − = {− } × d lies in the interior of R − ( γ ). Proposition 6.7.
Let ( M ′ , γ ′ ) be obtained from ( M, γ ) by cutting along aproduct annulus A . ThenSHM ( M, γ ; Q ) = SHM ( M ′ , γ ′ ; Q ) if we are in either of the following two situations: (a) the curves d + and d − represent non-zero classes in the first homologyof R + ( γ ) and R − ( γ ) respectively; or the curves d + and d − represent the zero class in H ( R + ( γ )) and H ( R − ( γ )) respectively, at least one of them does not bound a disk,and the annulus A separates M into two parts, M ∪ M , one of whichis disjoint from the annuli A ( γ ) .Proof. We begin with case (a) of the proposition. We shall construct closures( Y, ¯ R ) and ( ˜ Y , ˜ R ) for ( M, γ ) and ( M ′ , γ ′ ) which are related to each other asdescribed in the excision theorem, Theorem 3.2, and the result will follow.When we attach the product [ − , × T to ( M, γ ), the curves d + and d − remain non-separating in the closed surfaces ¯ R ± , because T is connected.By taking T to have non-zero genus, we can also ensure that there is acurve c in the interior of T which is non-separating in T . So after attachingthe product region, we have two product annuli [ − , × d and [ − , × c ,with independent non-separating curves d + , c + in ¯ R + in d − , c − in ¯ R − . Wecan close up the manifold using a diffeomorphism h : ¯ R + → ¯ R − such that h ( d + ) = d − and h ( c + ) = c − . The closure ( Y, ¯ R ) of ( M, γ ) that we arrive atin this way contains two tori, Σ = S × c Σ = S × d There is a 1-cycle η lying on ¯ R that is transverse to both of these tori,so Theorem 3.2 is applicable. (This is an instance of that theorem wherethe manifold Y ′ obtained by cutting along Σ and Σ is connected.) Themanifold ( ˜ Y , ˜ R ) obtained from ( Y, ¯ R ) by cutting along Σ ∪ Σ and regluingis a closure of the ( M ′ , γ ′ ), so we are done with case (a).We turn to case (b). Without loss of generality, we suppose that M doesnot meet A ( γ ) and d + does not bound a disk. Let R + , denote R + ( γ ) ∩ M and let R − , denote R − ( γ ) ∩ M . The surface R + , has genus at least 1 and itsonly boundary component is d + . In [23], Ni uses the following observation.The union R + , ∪ A ∪ R − , is isotopic to a horizontal surface in ( M, γ ) to which Proposition 6.6 applies.By cutting along this horizontal surface, the pieces we get from (
M, γ ) are(up to diffeomorphism) (cid:0) [ − , × R + , (cid:1) ∪ [ − , × d M and M ∪ [ − , × d (cid:0) [ − , × R − , (cid:1) . M or M is a product.If M is a product, [ − , × R − , , then the result is entirely straightfor-ward: the surface R − , contains all the annuli A ( γ ). A closure Y of ( M, γ )using an auxiliary surface T can also be regarded as a closure of ( M , γ )using the auxiliary surface R − , ∪ T . So we have SHM ( M, γ ) =
SHM ( M , γ ) . On the other hand, because M is a product, we have SHM ( M , γ ) = SHM ( M ′ , γ ′ ) by Proposition 6.5. Finally, if M is a product, then we cancut M open along a non-separating annulus because R + , has positive genus,and this does not change SHM , by part (a) of the proposition. After cuttingopen M in this way, we arrive at a situation in which proposition (a) appliesagain, and the proof is complete. 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 ho-mology, so the proof carries through with little change. We outline theargument, adapted from [23]. Let (
M, γ ) be a balanced sutured manifoldsatisfying the hypotheses of the theorem, and suppose (
M, γ ) is not a prod-uct sutured manifold. We shall show that
SHM ( M, γ ) has rank at least2. Because of Proposition 6.5, it is sufficient to treat the case that M isconnected. Similarly, because of Proposition 6.6, we may assume that ( M, γ )is “vertically prime”: that is, every horizontal surface in (
M, γ ) is a parallelcopy of either R + ( γ ) or R − ( γ ). By attaching product regions to ( M, γ ) andappealing to Proposition 6.7, we are also free to suppose that (
M, γ ) has onlyone suture. We now consider a maximal product pair i : [ − , × E ֒ → ( M, γ )as in [23, 24] and the induced map i ∗ : H ([ − , × E ) → H ( M ) . There are two cases.
Case 1: i ∗ is not surjective. In this case, Ni establishes that (
M, γ ) admitstwo taut foliations F and F whose difference element is non-torsion in H ( M, ∂M ). It then follows from Corollary 6.4 that
SHM ( M, γ ) has rank2 or more, as required.7
Case 2: i ∗ is surjective. In this case, let ( M ′ , γ ′ ) be the complement ofthe maximal product pair. This is non-empty, because ( M, γ ) is not aproduct sutured manifold. Proposition 6.7 tells us that
SHM ( M, γ ) and
SHM ( M ′ , γ ′ ) have the same rank. Ni observes that the vertically-primecondition on ( M, γ ) implies that M ′ is connected. Furthermore, ( M ′ , γ ′ ) isa homology product, and its top and bottom surfaces R ± ( γ ′ ) are planar,because of the surjectivity of i ∗ . The (connected) surfaces R ± ( γ ′ ) are notdisks, so ( M ′ , γ ′ ) has at least two sutures. Let r ≥ M ′ , γ ′ ). Let S be a planar surface with r +1 boundary components,so that the product sutured manifold [ − , × S has r + 1 sutures. Form anew sutured manifold ( ˜ M , ˜ γ ) by gluing r of the annuli from [ − , × S tothe annuli of ( M ′ , γ ′ ). The resulting sutured manifold ( ˜ M , ˜ γ ) hasrank SHM ( ˜
M , ˜ γ ) = rank SHM ( M ′ , γ ′ )by Proposition 6.7. Furthermore ( ˜ M , ˜ γ ) is a homology product, and itsmaximal product pair is [ − , × S up to isotopy. The construction hasbeen made so that the inclusion of the maximal product pair in ( ˜ M , ˜ γ ) isnot surjective on H , so we now have a situation which falls into Case 1above. It follows that SHM ( ˜
M , ˜ γ ) has rank at least 2; and so too thereforedoes SHM ( M, γ ). This completes Ni’s proof.
In [13], rather general sutured manifold decompositions are considered, andresults of the following sort are obtained. Let (
M, γ ) be a balanced suturedmanifold, and let S ⊂ M be a decomposing surface in the sense of [9]. Thereis a sutured manifold decomposition,( M, γ ) S ( M ′ , γ ′ ) , and we shall suppose that ( M ′ , γ ′ ) is also balanced (which implies that S hasno closed components). Under some mild restrictions on S , Juh´asz provesin [13] that SFH ( M ′ , γ ′ ) is a direct summand of SFH ( M, γ ). An entirelysimilar theorem can be proved in the context of monopole Floer homology,using
SHM ( M, γ ) in place of
SFH ( M, γ ). The following is a restatement ofTheorem 1.3 of [13], though with less specific information about the spin c structures that are involved behind the scenes. In the statement of the theo-rem, an oriented simple closed curve C in R ( γ ) is called boundary coherent ifit either represents a non-zero class in H ( R ( γ )) or it is the oriented bound-ary ∂R of a compact subsurface R ⊂ R ( γ ) with its canonical orientation.8 Figure 8: Adding product 1-handles to a sutured manifold containing a decompos-ing surface S . Theorem 6.8 ([13, Theorem 1.3]).
Let ( M, γ ) be a balanced suturedmanifold and ( M, γ ) S ( M ′ , γ ′ ) a sutured manifold decomposition. Suppose that the decomposing surface S has no closed components, and that for every component V of R ( γ ) , the setof closed components of S ∩ V consists of parallel oriented boundary-coherentsimple closed curves. Then the Heegaard Floer homology SFH ( M ′ , γ ′ ) is adirect summand of SFH ( M, γ ) . We have the following result.
Proposition 6.9.
Theorem 6.8 continues to hold with monopole Floer ho-mology in place of Heegaard Floer homology. That is, with the same hy-potheses, SHM ( M ′ , γ ′ ) is a direct summand of SHM ( M, γ ) .Proof. By Lemma 4.5 of [13], Juh´asz reduces this to the special case of a“good” decomposing surface S , by which is meant a surface S such thatevery component of ∂S intersects both R + ( γ ) and R − ( γ ).Starting from a good decomposing surface S , we can pass to anotherspecial case as follows. Let C be a component of ∂S . By the definition ofa good decomposing surface, C intersects the annuli A ( γ ) in vertical arcs.The number of these arcs counted with sign is zero. Pair up these arcs ac-cordingly; and for each pair attach a product 1-handle as shown in Figure 8.Repeat this with every other boundary component of ∂S . The result ofthis process is a new balanced sutured manifold ( M , γ ) containing a newdecomposing surface S . We have SHM ( M , γ ) ∼ = SHM ( M, γ ), because9adding product handles has no effect. (The inverse operation to adding aproduct handle can also be described as removing a larger product region,by cutting along annuli parallel to the annuli where the handle is attached;so this operation is a special case of one we have seen before.) Furthermore,if ( M ′ , γ ′ ) is what we obtain from ( M , γ ) by sutured manifold decompo-sition along S , then ( M ′ , γ ′ ) is also related to ( M ′ , γ ′ ) by adding addingproduct 1-handles. It therefore suffices to prove that SHM ( M ′ , γ ′ ) is adirect summand of SHM ( M , γ ).Looking at ( M , γ ), 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, γ ) be a balanced sutured manifold and let ( M, γ ) S ( M ′ , γ ′ ) 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 curves C +1 , . . . , C + n in R + ( γ ) and n simple closed curves C − , . . . , C − n in R − ( γ ) .Suppose further that the homology classes of C +1 , . . . , C + n are a collectionof independent classes in H ( R + ( γ )) , and make a similar assumption for R − ( γ ) . Then SHM ( M ′ , γ ′ ) is a direct summand of SHM ( M, γ ) .Proof of the lemma. Form the closure Y = Y ( M, γ ) by attaching a productregion [ − , × T as usual and then choosing the diffeomorphism h in sucha way that h ( C + i ) = C − i (with the opposite orientation) for all i . The resultof this is that Y contains two closed surfaces: first the usual surface ¯ R , andsecond a surface ¯ S obtained from S by identifying C + i with C − i for all i . Theintersection ¯ S ∩ ¯ R consists of n circles, C , . . . , C n . Let F be the orientedsurface obtained from ¯ S ∪ ¯ R by smoothing out the circles of double points,respecting orientations.The same surface F ⊂ Y can be arrived at from a different direction.Start with ( M ′ , γ ′ ). We can write A ( γ ′ ) as a union of components A ( γ ′ ) = A ( γ ) ∪ A , where A ( γ ) are the annuli of the original sutured manifold ( M, γ ) and A are the new annuli. The new annuli can be written as [ − , × D ± i , wherethe collection of curves D ± i are in natural correspondence with the curves C ± i . We now form a closure Y ′ of ( M ′ , γ ′ ) as follows. We attach a productregion [ − , × T ′ to ( M ′ , γ ′ ), where T ′ is a (disconnected) surface T ′ = T ∪ T . T is the surface used to close Y and T is a collection of n annuli T = T , ∪ · · · ∪ T ,n . Although T ′ breaks the rules by being disconnected, we can still effectivelyuse T ′ in constructing SHM ( M ′ , γ ′ ) because of the arguments of section 6.3.In attaching [ − , × T ′ to ( M ′ , γ ′ ) we glue [ − , × ∂T to the annuli A ( γ ) ⊂ A ( γ ′ ) as we did when closing ( M, γ ), and we glue the two components[ − , × T ,i to the two annuli [ − , × D ± i belonging to A .At this point, we have a manifold( M ′ , γ ′ ) ∪ [ − , × ( T ∪ T )with two boundary components ¯ R ′± . The top surface ¯ R ′ + can be describedas a union ¯ R ′ + = ¯ R † + ∪ S + ∪ { +1 } × T . Here ¯ R † + is the surface with boundary obtained by cutting open ¯ R + alongthe circles C + i , and the annuli { +1 } × T are collars of half of the boundarycomponents of ¯ R † + . The surface S + is a copy of S . Up to diffeomorphism,we can forget these annular regions and write¯ R ′ + = ¯ R † + ∪ S + ¯ R ′− = ¯ R †− ∪ S − . That is, ¯ R ′ + is obtained from ¯ R + by cutting open along the circles C + i andinserting a copy of S . Finally, form the closure Y ′ by using a diffeomorphism h ′ : ¯ R ′ + → ¯ R ′− which is equal to h on ¯ R † + and equal to the identity on S .The resulting closure Y ′ of ( M ′ , γ ′ ) is diffeomorphic to Y ; and under thisdiffeomorphism, the surface ¯ R ′ ⊂ Y ′ obtained from ¯ R ′± becomes the surface F . (See Figure 9.) It follows that we can calculate SHM ( M ′ , γ ′ ) as SHM ( M ′ , γ ′ ) = HM • ( Y | F ) . The homology class of F is the sum of the classes of ¯ R and ¯ S . Furthermore, χ ( F ) = χ ( ¯ R ) + χ ( ¯ S ). It follows from the adjunction inequality that theonly spin c structures in S ( Y | F ) which can have non-zero Floer homologyare those in the intersection S ( Y | ¯ R ) ∩ S ( Y | ¯ S ). So we have SHM ( M ′ , γ ′ ) = M s ∈S ( ¯ R ) ∩S ( ¯ S ) HM • ( Y, s ) , Figure 9: Decomposing M along S and then closing up to get F . The collars of ∂M and ∂M ′ are marked with hatching near A ( γ ) and A ( γ ′ ). The product part[ − , × T is not shown in the figure, which is otherwise a faithful representationafter multiplying by S . SHM ( M, γ ) = M s ∈S ( ¯ R ) HM • ( Y, s ) . This shows that
SHM ( M ′ , γ ′ ) is a direct summand of SHM ( M, γ ), as thelemma asserts.As is pointed out in [13], one can use Proposition 6.9 to give an alter-native proof of the non-vanishing of
SHM ( M, γ ) when (
M, γ ) is taut. Oneuses a sutured manifold hierarchy, starting at (
M, γ ) and ending at a prod-uct sutured manifold, whose (monopole) Floer homology we know to be Z ,so showing that Z is a summand of SHM ( M, γ ). 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 instan-ton homology, we refer to [4].
When looking at the monopole Floer homology groups HM • ( Y, s ) of a 3-manifold Y , we could avoid difficulties arising from reducible solutions byconsidering situations where only non-torsion spin c structures s played arole. In instanton homology, reducibles can be avoided by using SO (3)bundles with non-zero w . We proceed as follows.Fix a hermitian line bundle w → Y such that c ( w ) has odd pairingwith some integer homology class. Let E → Y be a U (2) bundle with anisomorphism θ : Λ E → w . Let C be the space of SO (3) connections inad( E ) and let G be the group of determinant-1 gauge transformations of E (the automorphisms of E that respect θ ). The Chern-Simons functional onthe space B = C / G leads to a well-defined instanton homology group whichwe write as I ∗ ( Y ) w [4]. It is also possible to use a slightly larger gauge groupthan G . Fix a surface R ⊂ Y that has odd pairing with c ( w ). Let ξ = ξ R be a real line bundle with w ( ξ ) dual to R . The map E E ⊗ ξ gives riseto a map on the space of connections, ι R : B → B , without fixed points, and there is a quotient B /ι R . This is the same as thequotient of C by a gauge group which has G as an index-2 subgroup. Let us3temporarily write I ∗ ( Y ) w,R for the resulting instanton homology group: itis the fixed space of an induced involution on I ∗ ( Y ). As an example, in thecase Y = T , we have I ∗ ( T ) w = Z ⊕ Z . The involution interchanges thetwo copies of Z , and I ∗ ( T ) w,R = Z whenever w · [ R ] is non-zero. In general, I ∗ ( Y ) w is ( Z / I ∗ ( Y ) w,R is ( Z/ Z coefficients, it will be conve-nient to work with a field of characteristic zero; and in what follows we willtake that field to be C . Thus we will take it that I ∗ ( T ) w,R = C . The monopole Floer homology detects the Thurston norm of a 3-manifold(see section 2.4); but the formulation of this statement requires the decom-position of the monopole Floer homology according to the different spin c structures. In order to relate instanton homology to the Thurston norm,one needs a decomposition of the instanton homology. As suggested in [15],such a decomposition arises from the eigenspaces of natural operators onthe Floer groups.Let Y be again a closed 3-manifold and w a line bundle as above. Givenan oriented closed surface R in Y , there is a 2-dimensional cohomology class µ ( R ) in B (for which our conventions follow [5]) and hence an operationof degree − I ∗ ( Y ) w and I ∗ ( Y ) w,R . There is also the class µ ( y ),for y a point in y , which acts with degree 4. The operators µ ( R ) and µ ( y )commute, so one can look for simultaneous eigenvalues. In the special casethat Y = S × Σ, with Σ a surface of positive genus, the eigenvalues of µ (Σ)and µ ( y ) were computed by Mu˜noz in [21]: Proposition 7.1 ([21, Proposition 20]).
Let w → S × Σ be the line bun-dle whose first Chern class is dual to the S factor. Then the simultaneouseigenvalues of the action of µ (Σ) and µ ( y ) on I ∗ ( S × Σ) w are the pairs ofcomplex numbers ( i r (2 k ) , ( − r for all the integers k in the range ≤ k ≤ g − and all r = 0 , , , . Here i denotes √− .Remark. In [21], the 2-dimensional class called α corresponds to 2 µ (Σ) here,and the class β corresponds to − µ ( y ). Also, the group HF ∗ ( S × Σ) thatappears in [21] is our I ∗ ( S × Σ) w, Σ . Munoz computes the spectrum in the4case of I ∗ ( S × Σ) w, Σ , but the case of I ∗ ( S × Σ) w follows in a straightforwardmanner. Observe, in particular, that because µ (Σ) is an operator of degree2 on a ( Z / λ and iλ will always be isomorphic.As a corollary of this proposition, a similar result holds for a general3-manifold Y . Corollary 7.2.
Let R ⊂ Y be closed connected surface of positive genus,and let w have odd pairing with R . Then the eigenvalues of the action of thepair of operators µ ( R ) and µ ( y ) on I ∗ ( Y ) w are a subset of the eigenvaluesthat occur in the case of the product manifold S × R . That is, they are pairscomplex numbers ( i r (2 k ) , ( − r for integers k in the range ≤ k ≤ g − .Proof. Let R be a copy of R in the interior of the product cobordism W = [ − , × Y . The action of µ ( R ) on I ∗ ( Y ) w can be regarded as beingdefined by this copy of R in the 4-dimensional cobordism. Let W ′ be thecobordism from the disjoint union S × R and Y at the incoming end to Y at the outgoing end, obtained by removing an open tubular neighborhoodof R from W . We have a map defined by W ′ , ψ W ′ : I ∗ ( S × R ) w ⊗ I ∗ ( Y ) w → I ∗ ( Y ) w . The map is surjective, because one obtains the product cobordism by closingoff the boundary component S × Σ. Furthermore, because R is homologousto surfaces in each of the three boundary components, we have, for example ψ W ′ ( µ ( R ) a ⊗ b ) = µ ( R ) ψ W ′ ( a ⊗ b ) . From this relation and the surjectivity of ψ W ′ , it follows that the eigenvaluesof µ ( R ) on the outgoing end Y are a subset of the eigenvalues of the actionof µ ( R ) on S × R . We obtain the result of the corollary by applying asimilar argument to µ ( y ) and to µ ( R ) + µ ( y ).We can now give a definition in instanton homology of something thatwill play the role that HM • ( Y | R ) played in the monopole theory. Definition 7.3.
Let Y be a closed, oriented 3-manifold, w a hermitian linebundle on Y and R ⊂ Y a closed, connected, oriented surface on which5 c ( w ) is odd. Let g be the genus or R , which we require to be positive. Wedefine I ∗ ( Y | R ) w to be the simultaneous eigenspace for the operators µ ( R ), µ ( y ) for the pairof eigenvalues (2 g − , ♦ Remark.
Except in the case that the genus is 1, we could define this moresimply as just the (2 g − µ ( R ), as can be seen from Corol-lary 7.2Although Mu˜noz does not calculate the dimensions of the eigenspaces ingeneral for S × Σ, one can readily read off from the proof of [21, Propo-sition 20] that the dimension of the eigenspace belonging to the largesteigenvalue is 1. That is,
Proposition 7.4.
Let Y = S × R with Σ of positive genus, and let w bethe line bundle dual to the S factor. Then I ∗ ( Y | R ) w = C . There is a simple extension of the above definition to the case that R has more than one component, as long as w is odd on each component. Ifthe components are R m , then the corresponding operators µ ( R m ) commute,and we may take the appropriate simultaneous eigenspace. In general, theaction of µ ( R ) on I ∗ ( Y ) w is not diagonalizable; but one can read off from[21] that the eigenspace of µ ( R ) belonging to the top eigenvalue 2 g − µ ( y ) −
2. That is,ker( µ ( y ) − ∩ ker( µ ( R ) − (2 g − N = ker( µ ( y ) − ∩ ker( µ ( R ) − (2 g − N ≥ Proposition 7.5.
Given any Y , R and w for which I ∗ ( Y | R ) w is defined,and given any other surface Σ ⊂ Y of positive genus, the action of µ (Σ) on I ∗ ( Y | R ) w has eigenvalues belong to the set of even integers in the rangefrom − (2 g − to g − , where g is the genus of Σ .Proof. The action of µ (Σ) on I ∗ ( Y ) w commutes with µ ( R ), so the actionof µ (Σ) does preserve the subspace I ∗ ( Y | R ) w ⊂ I ∗ ( Y ) w . If w is odd on Σ,then the proposition follows from Corollary 7.2 together with the fact that µ ( y ) − w is even on Σ, then one can considera surface in the homology class of R + n Σ and use the additivity of µ .6 Because the actions of µ (Σ ) and µ (Σ ) commute for any pair of classesΣ and Σ , we have a decomposition of I ∗ ( Y | R ) w by cohomology classes (asoutlined in [15]): Corollary 7.6.
There is a direct sum decomposition into generalized eigenspaces I ∗ ( Y | R ) w = M s I ∗ ( Y | R, s ) w where the sum is over all homomorphisms s : H ( Y ; Z ) → Z subject to the constraints (cid:12)(cid:12) s ([ S ]) (cid:12)(cid:12) ≤ S ) − for all connected surfaces S with positive genus and s ([ R ]) = 2 genus( R ) − .The summand I ∗ ( Y | R, s ) w is the simultaneous generalized eigenspace I ∗ ( Y | R, s ) w = \ σ ∈ H ( Y ) [ N ≥ ker (cid:16) µ ( σ ) − s ( σ ) (cid:17) N . It will be convenient at a later point to have a notation for the sort ofhomomorphisms s that arise here. Choosing a notation reminiscent of ournotation for spin c structures, we write H ( Y ) = Hom (cid:0) H ( Y ) , Z (cid:1) and for an embedded surface R ⊂ Y of genus g we write H ( Y | R ) = { s ∈ H ( Y ) | s ([ R ]) = 2 g − } . (21) Let Y be closed, oriented 3-manifold equipped with a line bundle w , andsuppose Σ = Σ ∪ Σ is an oriented embedded surface with two connectedcomponents of equal genus, which we require to be positive. Suppose alsothat c ( w )[Σ ] and c ( w )[Σ ] are equal and odd. We allow that Y has eitherone or two components. In the latter case, we require one of the Σ i to be ineach component. In the former case, when Y is connected, we assume thatΣ and Σ are not homologous. Choose a diffeomorphism h : Σ → Σ , and7lift it to a bundle-isomorphism ˆ h on the restrictions of the line bundle w .From this data, we form ˜ Y by cutting along the Σ i and gluing up using h asbefore. The lift ˆ h can be used to glue up the bundle also, giving us a bundle˜ w → ˜ Y . As before, we write ˜Σ = ˜Σ ∪ ˜Σ for the surfaces in ˜ Y . Theorem 7.7. If ( ˜ Y , ˜Σ) is obtained from ( Y, Σ) as above, then there is anisomorphism I ∗ ( Y | Σ) w ∼ = I ∗ ( ˜ Y | ˜Σ) ˜ w . We interpret the left-hand side as a tensor product in the case that Y hastwo components.Proof. In the case that Σ has genus 1, this result is due to Floer [8, 2]. InFloer’s statement of the result, Y had two components, but the proof doesnot require it. It should also be said that statement the of Floer’s theoremin [2] involves I ∗ ( Y ) w rather than I ∗ ( Y | Σ) w , which leads to an extra factorof two in the dimensions when Y has two components.The case of genus 2 or more is essentially the same, once one knows that I ∗ ( S × Σ i | Σ i ) w has rank 1.In the case of genus 1, note that passing from I ∗ ( Y ) w to I ∗ ( Y | Σ) w canalso be achieved by taking the +2 eigenspace of µ ( y ), for one point y in eachcomponent of Y .Here are two particular applications of the excision theorem. They areboth variants of Proposition 7.4, but involve different line bundles. Proposition 7.8.
Let Y be the product S × Σ , with Σ a surface of genus or more, and let w again be the line bundle dual to the S factor. Let u → Y be a line bundle whose first Chern class is dual to a curve γ lying on { point } × Σ , and write the tensor product line bundle as uw . Then we have I ∗ ( Y | Σ) uw = C . Proof.
Write B for the vector space I ∗ ( Y | Σ) uw and A for the vector space I ∗ ( Y | Σ) 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 man-ifold is a single copy of Y with the line bundle u w . The latter gives thesame Floer homology as the for line bundle w , so we learn that B ⊗ B ∼ = A. We already know that A is one-dimensional, and it follows that B is alsoone-dimensional.8 For the second application, we can dispense with w : Proposition 7.9.
In the situation of Proposition 7.8, the eigenspace of thepair of operators ( µ (Σ) , µ ( y )) on I ∗ ( Y ) u for the eigenvalues (2 g − , isalso one-dimensional.Proof. We can see more generally, that for any λ the eigenspace for ( λ,
2) on I ∗ ( 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 Σ sothat the torus S × c intersects u once. Let Y be S × T , and let u , w and c be similar there to u , w and c . Apply the excision theorem with incomingmanifold Y ∪ Y with the line bundles u w and uw respectively, cuttingalong the tori S × c and S × c . The outgoing manifold is diffeomorphicto Y , with the line bundle uw , which gives the same homology as u . Theexcision theorem gives an isomorphism between the +2 eigenspaces of µ ( y ),which we denote φ : I ∗ ( Y ) (2) uw → I ∗ ( Y ) (2) u . The map that gives rise to the isomorphism in the excision theorem inter-twines (in this instance) the maps µ (Σ) on the outgoing end with µ ( T ) ⊗ ⊗ µ (Σ)on the incoming end. Since µ ( T ) is zero on I ∗ ( S × T ) u w , the map φ actually commutes with µ (Σ). Remark.
The Floer homology group I ∗ ( S × Σ) u is something that appears tobe rather simpler than the more familiar I ∗ ( S × Σ) w . In particular, excisionshows that the it behaves “multiplicatively” in g −
1. The representationvariety that is involved here is easy to identify: the critical point set ofthe Chern-Simons functional is two copies of a torus T g − . The involutioninterchanges ι Σ interchanges the two copies. It seems likely that the Floergroup I ∗ ( S × Σ) w, Σ can be identified with the homology of this torus. Let (
M, γ ) be a balanced sutured manifold. Just as we did in the monopolecase, we attach a connected product sutured manifold [ − , × T to ( M, γ )to obtain a manifold Y ′ with boundary ¯ R + ∪ ¯ R − , a pair of diffeomorphicconnected closed surfaces. As before, we require that there be a closedcurve c in T such that {− } × c and { } × c are both non-separating in theirrespective boundary components. We also pick a marked point, t ∈ T ,9which we did not need before. Now we glue ¯ R + to ¯ R − by a diffeomorphism.We require that h ( t ) = t , so that the resulting closed manifold Y = Y ( M, γ ) contains a standard circle running through t . This circle intersectsonce the closed surface ¯ R obtained by identifying ¯ R ± . We no longer requirethat ¯ R has genus 2 or more: in the instanton case, genus 1 will suffice. Definition 7.10.
The instanton homology of the sutured manifold (
M, γ )is the vector space
SHI ( M, γ ) := I ∗ ( Y | ¯ R ) w , where ( Y, ¯ R ) is obtained from ( M, γ ) by closing as just described, and w isthe line bundle whose first Chern class is dual to the standard circle through t . ♦ 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, γ ) is independent of the choice of genus for T and the choice of diffeomorphism h can be carried over almost verbatimfrom the monopole case, using the excision theorem. It is even somewhateasier to manage, because the case of genus 1 is no longer special. Whenshowing that SHI is independent of the choice of genus, we used twisted co-efficients HM • ( Y | ¯ R ; Γ η ) as an intermediate step in the monopole case. Thecounterpart of twisted coefficients in the proof for the instanton case is theintroduction of the auxiliary line bundle u that appears in Propositions 7.8and 7.9 above. One applies excision along tori, following the same schemeas shown in Figure 5, to increase the genus by 1. On the components S × S ,with S of genus 2 as shown, one should take the line bundle u , where u isthe line bundle whose first Chern class is dual to the dotted curve d ′ . Thisargument shows that I ∗ ( Y | ¯ R ) uw = I ∗ ( ˜ Y | ˜ R ) ˜ u ˜ w where Y and ˜ Y are closures of ( M, γ ) obtained using auxiliary surfaces T and ˜ T of genus g and g + 1. Another application of excision (cutting alongcopies of ¯ R and using Proposition 7.8) shows that I ∗ ( Y | ¯ R ) uw ∼ = I ∗ ( Y | ¯ R ) w = SHI g ( M, γ ) . The proofs of the decomposition results of sections 6.3 and 6.5 carry overwithout change to the instanton setting also. In particular, Proposition 6.9holds in the instanton case:0
Proposition 7.11.
Let ( M, γ ) be a balanced sutured manifold and ( M, γ ) S ( M ′ , γ ′ ) a sutured manifold decomposition satisfying the hypotheses of Theorem 6.8.Then SHI ( M ′ , γ ′ ) is a direct summand of SHI ( M, γ ) .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 spin c 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, γ ) is taut, thenSHI ( M, γ ) is non-zero. The only alternative route known to the authors for proving a non-vanishing theorem for instanton homology is the strategy in [17], whichdraws on results from symplectic and contact topology, as well as on the par-tial proof of Witten’s conjecture relating Donaldson invariants and Seiberg-Witten invariants of closed 4-manifolds [6]. We shall return to non-vanishingtheorems for instanton homology in section 7.8.
Just as we did for the monopole case in section 5, we can take Juh´asz’sprescription as a definition of knot homology. Let K ⊂ Z be again a knot ina closed, oriented 3-manifold. Let ( M, γ ) be the sutured manifold obtainedby taking M to be the knot complement Z \ N ◦ ( K ) and s ( γ ) a pair ofoppositely oriented meridians on ∂K . In the instanton case, there is noneed for ¯ R to have genus 2 or more, so we may use the closure Y ( M, γ )described in Definition 5.3. This is the closure of (
M, γ ) obtained using[ − , × T , where T is an annulus. It is also described in Lemma 5.2 asobtained from Z \ N ◦ ( K ) by attaching F × S , where F has genus one: thegluing is done so that { p }× S is attached to a meridian of K . We summarizethe construction of this instanton knot homology in the following definition.The definition is not new: it is the same “instanton homology for knots” thatFloer defined in [8]. For the purposes of this paper, we call it KHI ( Z, K ):1
Definition 7.13.
The instanton knot homology
KHI ( Z, K ) of a knot K in Z is defined to be the instanton homology of the sutured manifold ( M, γ )above; or equivalently, the instanton homology group I ∗ ( Y | ¯ R ) w . Here Y isobtained from the knot complement by attaching F × S as described, thesurface ¯ R is the torus α × S as shown in Figure 7, and w is the line bundlewith c ( w ) dual to β × { p } ⊂ F × S . ♦ The only difference between this and Floer’s original definition is that wehave used I ∗ ( Y | ¯ R ) w in place of I ∗ ( Y ) w . Since ¯ R has genus one, the formergroup can be characterized as the +2 eigenspace of µ ( y ) acting on the lattergroup. The latter group is the sum of two subspaces of equal dimension, theeigenspaces for the eigenvalues 2 and − K in S , we shall simply write KHI ( K ) for theinstanton knot homology. To get a feel for what this invariant is, let usexamine the set of critical points of the Chern-Simons functional on B , or inother words the space of flat connections in the appropriate SO (3) bundle,modulo the determinant-1 gauge transformations. To do this, we start bylooking at F × S , where F is the genus-1 surface with one boundary com-ponent, and the line-bundle w with c ( w ) dual to β × { p } . The appropriaterepresentation variety can also be viewed as the space of flat SU (2) connec-tions on the complement of the curve β × { p } with the property that theholonomy around a small circle linking β × { p } is the central element − A and let J and J be the holonomies of A around respectively the curves α × { q } and a × S in F × S , where a is apoint on α \ β . The torus α × S intersects the circle β × { p } once, so wehave [ J , J ] = − SU (2). Up to a gauge transformation, we must have J = (cid:18) −
11 0 (cid:19) , J = (cid:18) i − i (cid:19) . Let J be the holonomy around β ′ × { q } , where β ′ is a parallel copy of β .The elements J and J must commute, so J = (cid:18) e iθ e − iθ (cid:19) for some θ in [0 , π ). The angle θ is now determined without ambiguityfrom the gauge-equivalence class of the connection A ; and the matrices J , J and J determine A entirely. We have proved:2 Lemma 7.14.
The representation variety of flat SO (3) connections on F × S for the given w , modulo the determinant-1 gauge group, is diffeomorphicto a circle S , via J as above. Let us examine the restriction of these representations to the boundaryof F × S . On this torus ∂F × S , the flat connections can be regardedas SU (2) connections. The holonomy around the S factor is J , which wehave described above. The holonomy around the ∂F factor is given by thecommutator [ J , J ] = (cid:18) e iθ e − iθ (cid:19) . So for the representation variety described in the lemma, the restriction tothe boundary is a two-to-one map whose image is the space of connectionshaving holonomy around the S factor given by i = (cid:18) i − i (cid:19) . Finally, we can attach F × S to the knot complement S \ N ◦ ( K ), and weobtain the following description of the representation variety. Lemma 7.15.
Let K ⊂ S be a knot and let Y and w be as described inDefinition 7.13. Then the representation variety given by the critical pointsof the Chern-Simons functional in the corresponding space of connections B can be identified with a double cover of the space R ( K, i ) = { ρ : π ( S \ K ) → SU (2) | ρ ( m ) = i } , where m is a chosen meridian. 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 actson R ( K, i ) by conjugation. There is always exactly one point of R ( K, i )which is fixed by the action of this circle, namely the homomorphism ρ which factors through the abelianization H ( S \ K ) = Z . All other orbitsare irreducible: they have stabilizer ±
1, so they are circles. In a genericcase, R ( K, i ) consists of one isolated point corresponding to the abelian(reducible) representation, and finitely many circles, one for each conjugacyclass of irreducible representations. In such a case, the representation varietydescribed in the lemma above is a trivial double-cover of R ( K, i ). It thereforehas two isolated points corresponding to the reducible, and two circles foreach irreducible conjugacy class.3Because it comprises only the +2 eigenspace of µ ( y ), the knot Floerhomology KHI ( K ) has just half the dimension of I ∗ ( Y ) w in Definition 7.13.Heuristically, we can think of each irreducible conjugacy class in R ( K, i )as contributing the homology of the circle, H ∗ ( S ; C ), to the complex thatcomputes KHI ( K ), while the reducible contributes a single C . In any event,if there are only n conjugacy classes of irreducibles and the correspondingcircles of critical points are non-degenerate in the Morse-Bott sense, then itwill follow that the dimension of KHI ( K ) is bounded above by 2 n + 1.For a knot K ⊂ Z supplied with a Seifert surface Σ, there is a decom-position of the instanton knot homology KHI ( Z, K ) as
KHI ( Z, K ) = genus(Σ) M i = − genus(Σ) KHI ( Z, K, [Σ] , i ) . The definition is the same as in the monopole case (19), but uses thegeneralized-eigenspace decomposition of Corollary 7.6 in place of the de-composition by spin c structures. In particular, for a classical knot K ⊂ S ,we can write KHI ( K ) = g M i = − 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 homology
SHM ( M, γ ), where (
M, γ ) is the sutured manifold obtained by cutting openthe knot complement along a Seifert surface of genus g . (See Proposi-tion 5.11.) From the non-vanishing theorem, Theorem 7.12, we thereforededuce a non-vanishing theorem for KHI . Proposition 7.16.
Let K be a classical knot of genus g . Then the instantonknot homology group KHI ( K, g ) is non-zero. In particular, instanton knothomology detects 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 ⊂ S is non-trivial knot, then there exists an irre-ducible homomorphism ρ : π ( S \ K ) which maps a chosen meridian m tothe element i ∈ 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 then4consists of two irreducible critical points, so the rank of I ∗ ( Y ) 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 ). Instanton knot homology detects fibered knots, just as the other versionsdo. We state and prove this here. We need, however, an extra hypothesison the Alexander polynomial. For Heegaard knot homology, and also in themonopole case, we know the Alexander polynomial is determined by theknot homology, and the extra hypothesis is not needed. It seems likely thatthe same holds 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, γ ) is tautand a homology product. Then ( M, γ ) is a product sutured manifold if andonly if SHI ( M, γ ) = 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 oneslight change (a change which is in the spirit of [13]). The key point occursin Case 1 in the proof of Theorem 6.1 (section 6.4), where it is alreadyassumed that (
M, γ ) has just one suture. We described this step using spin c structures, but we can argue using homology instead.Let N be obtained from ( M, γ ) by adding a product region [ − , × T to the single suture. The boundary of N is ¯ R + ∪ ¯ R − . In [23], Ni shows thatif E × I does not carry all the homology of ( M, γ ), then one can find twodecomposing surfaces S and S in N with the following properties. First,the boundaries of S and S are the same and consist of a pair of circles ω + and ω − which represent non-zero homology classes in ¯ R + and ¯ R − . Second,the sutured manifolds ( M ′ , γ ′ ) and ( M ′ , γ ′ ) obtained by decomposition of N along S and S respectively are both taut. Third, if Y is obtained from N by gluing ¯ R + to ¯ R − by a diffeomorphism h with h ( ω + ) = ω − , then theresulting closed surface ¯ S , ¯ S and ¯ R in Y satisfy the following conditions,for some m > S with χ ( ¯ S ) non-zero,[ ¯ S ] = m [ ¯ R ] + [ ¯ S ][ ¯ S ] = m [ ¯ R ] − [ ¯ S ]5and χ ( ¯ S ) = χ ( ¯ S )= mχ ( ¯ R ) + χ ( ¯ S ) . These last conditions imply that H ( Y | ¯ R ) ∩H ( Y | ¯ S ) is disjoint from H ( Y | ¯ R ) ∩H ( Y | ¯ S ). (The notation H is introduced at (21).)For i = 1 ,
2, let F i be the surface in Y obtained by smoothing outthe intersection of ¯ R and ¯ S i (a single circle in both cases). The proof ofLemma 6.10 shows that SHI ( M ′ i , γ ′ i ) = M s ∈H ( Y | F i ) I ∗ ( Y | ¯ R, s ) w = M s ∈H ( Y | ¯ R ) ∩H ( Y | ¯ S i ) I ∗ ( Y | ¯ R, s ) w ⊂ I ∗ ( Y | ¯ R ) w = SHI ( M, γ ) . The disjointness of the two indexing sets for s means that we have SHI ( M ′ , γ ′ ) ⊕ SHI ( M ′ , γ ′ ) ⊂ SHI ( M, γ ) . Finally, both summands on the right are non-zero because these suturedmanifolds are taut.
Corollary 7.19.
Let K be a non-trivial knot in S . Suppose that the sym-metrized Alexander polynomial ∆ K ( T ) is monic and that its degree (by whichwe mean the highest power of T that appears) is g . Then K is fibered if andonly if KHI ( K, g ) is one-dimensional.Proof. The proof given for Corollary 6.2 (the monopole case) needs no al-teration, except that the hypothesis on the Alexander polynomial has beenexplicitly included, rather than being deduced from Lemma 5.10.
Corollary 7.20.
Let K ⊂ S be a knot whose Alexander polynomial ismonic of degree equal to the genus of the knot. Consider the irreduciblehomomorphisms ρ : π ( S \ K ) → SU (2) which map a chosen meridian m to the element i ∈ SU (2) . If there is only one conjugacy class of suchhomomorphisms, and if these homomorphisms are non-degenerate, then K is fibered. Theorem 7.12 asserts the non-vanishing of instanton Floer homology forbalanced sutured manifolds; but the theorem does not say anything directlyabout closed 3-manifolds Y . Nevertheless, with a little extra input, weobtain the following result as a corollary. Theorem 7.21.
Let Y be a closed irreducible -manifold containing aclosed, connected, oriented surface ¯ R representing a non-zero class in secondhomology. Let w be a hermitian line bundle whose first Chern class has oddevaluation on [ R ] . Then I ∗ ( Y | ¯ R ) w is non-zero.Proof. Let M be the manifold obtained by cutting Y open along R , andwrite the boundary of M as R + ∪ R − . We regard M as a sutured manifold,with an empty set of sutures. (The absence of sutures means that M fails tobe balanced.) Let N be the double of M . We can regard R = R + ∪ 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,it will be sufficient to show that I ∗ ( N | R ) w is non-zero. Since R − and R + are homologous in N and of equal genus, we have I ∗ ( N | R ) w = I ∗ ( N | R + ) w , so we could equally well deal with I ∗ ( N | R + ) 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-empty set of circles, and we let T ′ be the surface obtained from T and R bysmoothing these circles of double points. This T ′ has the property that bycutting N open along T and then decomposing further along a non-emptycollection of annuli J , we arrive at a taut, sutured manifold ( N ′′ , δ ′′ ).If T intersects both R + and R − , then ( N ′′ , δ ′′ ) is balanced. If T inter-sects only R + , say, then ( N ′′ , δ ′′ ) fails to be balanced, because its boundarycontains two copies of R − : these are components of ∂N ′′ which fail to meet A ( δ ′′ ), contrary to the definition of balanced. If this is what happens, were-attach these two copies of R − . We rename the resulting manifold as ournew N ′′ and proceed. At this point, ( N ′′ , δ ′′ ) is a balanced sutured manifold.By Theorem 7.12, we know that SHM ( N ′′ , δ ′′ ) is non-zero. We canregard the manifold N as a closure of ( N ′′ , δ ′′ ), but with an auxiliary surfacethat fails to be connected: the auxiliary surface is the collection of annuli J . But as we argued in the proof of Lemma 6.10, a disconnected auxiliarysurface is as good as a connected one here. We can therefore compute7 SHM ( N ′′ , δ ′′ ) as I ∗ ( N | F ) w , where F is the surface in N formed from R ± ( δ ′′ )when making the closure. Thus I ∗ ( N | F ) w = 0 . This surface F can be identified with T ′ in the case that T meets both R + and R − . In the case that T meets only R + , then F is T ′ \ R − . In otherwords, F is obtained by smoothing the circles of double points of either T ∪ R or T ∪ R + . As in the proof Lemma 6.10, the Floer homology I ∗ ( N | F ) w isa direct summand of I ∗ ( N | R + ) w . So the latter is non-zero, and we aredone. Corollary 7.22. If Y is obtained from zero-surgery on a non-trivial knot K ⊂ S , 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 Y is obtained as +1 surgery on K ⊂ S , then π ( Y ) admits a non-trivial homomorphism to SU (2) . In particular, Y is not ahomotopy sphere. This provides a proof of the Property P conjecture that is independentof the work of Feehan and Leness in [6] and independent also of Perelman’sproof of the Poincar´e conjecture.
There are various questions and conjectures which naturally arise. The mostobvious of these is:
Conjecture 7.24.
For balanced sutured manifolds ( M, γ ) , the monopole andHeegaard groups SHM ( M, γ ) and SFH ( M, γ ) are isomorphic. When ten-sored with C , they are both isomorphic to the instanton version, SHI ( M, γ ) . As a special case, we have:
Conjecture 7.25.
With complex coefficients, the knot homologies defined byOzsv´ath-Szab´o and Rasmussen are isomorphic to Floer’s instanton homologyfor knots, KHI ( K ) , as defined here and in [8]. KHI ( K, i ); but it is natural to conjecture that thisis so, just as in the monopole and Heegaard theories. This may be only amatter of repeating [7] in the instanton context:
Conjecture 7.26.
The Euler characteristics of the instanton knot homologygroups KHI ( K, i ) , for i = − g, . . . , g , are the coefficients of the symmetrizedAlexander polynomial of K . If this conjecture is proved, then the hypothesis on the Alexander poly-nomial could be dropped from Corollary 7.19.A loose end in our development of
SHM ( M, γ ) is the lack of a completeaccounting of spin c structures. The material of section 6.2 is a step in theright direction. In [13], Juh´asz proves that his Heegaard Floer homology ofsutured manifolds can be decomposed as a direct sum indexed by the set ofrelative spin c structures S ( M, γ ), and it would be desirable to have a similarstatement for the monopole and instanton cases.Juh´asz [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 ⊂ S be a knot, and consider theirreducible homomorphisms ρ : π ( S \ K ) → SU (2) which map a chosenmeridian m to the element i ∈ SU (2) . Suppose that these homomorphismsare non-degenerate and that the number of conjugacy classes of such homo-morphisms is less then k +1 . Then the knot complement S \ N ◦ ( K ) admitsa foliation of depth at most k , transverse to the torus boundary. The fact that I ∗ ( Y | Σ) w is of rank 1 in the case that Y is a surfacebundle of S with fiber Σ is something that has other applications. Forexample, combined with Donaldson’s theorem on the existence of Lefschetzpencils [3], it yields a fairly direct proof that symplectic 4-manifolds havenon-zero Donaldson invariants. Essentially the same strategy was used byOzsv´ath and Szab´o in the Heegaard context. What the argument showsspecifically is that if X → S is a symplectic Lefschetz fibration whose fiber F has genus 2 or more, and if w is the line bundle dual to a section, thenthe Donaldson invariant D w ( F n ) is non-zero for all large enough n in theappropriate residue class mod 4.Another matter is whether one can relate either the monopole or in-stanton knot homologies to the corresponding Floer homologies of the 3-9manifolds obtained by surgery on the knot, particularly for large integersurgeries. This is 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 differentfrom the definition of KHI ( K ) given in this paper, because instantons withsingularities in codimension-2 were involved. Nevertheless, both theoriesinvolve the same representation variety R ( K, i ). Various versions are definedin [19], but the one most closely related to KHI ( K ) is the “reduced” variant,called RI ∗ ( K ) in [19]. Like KHI ( K ), the group RI ∗ ( K ) is a Floer homologygroup, constructed from a Chern-Simons functional whose set of criticalpoints can be identified with R ( K, i ). The paper [19] develops its theoryfor the gauge group SU ( N ), not just SU (2), and it would be interesting topursue a similar direction with SHI ( M, γ ) and
KHI ( K ).The “hat” version of Heegaard Floer homology, for a closed 3-manifold Y , can also be recovered as a special case of Juh´asz’s SFH , as shown in[12]. The appropriate manifold M is the complement of a ball in Y , andone takes a single annular suture on the result 2-sphere boundary. One cantake this as a definition of a “hat” version of monopole Floer homology. Inthe instanton case, this leads to essentially the same construction that wasused in [19] to avoid reducibles: one replaces Y by Y T and takes w to bea line bundle that is trivial on Y and of degree 1 on a T in the T .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, γ ), as defined in [12], can also be recovered as the Heegaard Floer ho-mology of a closed manifold Y = Y ( M, γ ), of the sort that we have usedhere. If so, it would be interesting to know whether the existing proofs ofthe decomposition theorems in [23] and [13], for example, can be adaptedto prove Floer’s excision theorem in the context of Heegaard Floer theory.
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