MONOPOLES AND LANDAU-GINZBURG MODELS III: A GLUING THEOREM

DONGHAO WANG

Abstract. This is the third paper of this series. In [Wan20b], we defined the mono- pole Floer homology for any pair pY, ωq, where Y is an oriented 3-manifold with toroidal boundary and ω is a suitable closed 2-form. In this paper, we establish a gluing theorem for this Floer homology when two such 3-manifolds are glued suitably along their common boundary, assuming that BY is disconnected and ω is non-vanishing on BY . As applications, we construct the monopole Floer 2-functor and the generalized cobor- dism maps. Using results of Kronheimer-Mrowka and Ni, we prove that for any such 3-manifold Y that is irreducible, this Floer homology detects the Thurston norm on H2pY, BY ; Rq and the fiberness of Y . Finally, we show that our construction recovers the monopole link Floer homology for any link inside a closed 3-manifold.

Contents

Part 1. Introduction 2

Part 2. The Gluing Theorem 8 2. The Toroidal Bi-category 8 3. Proof of the Gluing Theorem 13 4. Generalized Cobordism Maps 26 5. Invariance of Boundary Metrics 28

Part 3. Monopoles and Thurston Norms 30 6. Closed 3-Manifolds Revisited 30 7. Thurston Norm Detection 38 8. Relations with Sutured Floer Homology 42 9. Fiberness Detection 46 10. Connected Sum Formulae 49 arXiv:2010.04318v1 [math.GT] 9 Oct 2020 References 51

Date: October 12, 2020. 1 2 DONGHAO WANG

Part 1. Introduction 1.1. An Overview. The Seiberg-Witten Floer homology of a closed oriented 3-manifold is defined by Kronheimer-Mrowka [KM07] and has greatly influenced the study of 3-manifold topology. The underlying idea is an infinite dimensional Morse theory: the monopole equations on Rt times a closed 3-manifold is related to the downward gradient flow of the Chern-Simons-Dirac functional. By further exploring this idea, the author extended their construction in [Wan20b] and defined the monopole Floer homology for any pair pY, ωq, where Y is a compact oriented 3-manifold with toroidal boundary and ω is a suitable closed 2-form on Y . This Floer homology categorifies the Milnor-Turaev torsion invariant of Y by the work of Meng-Taubes [MT96] and Turaev [Tur98], and is an instance of (3+1) topological quantum field theories (TQFT) [Wan20b, Theorem 1.5]. However, in [Wan20b], we have concentrated on the analytic foundation of this Floer theory; very little was discussed about its topological properties. The goal of this current paper is to understand the properties of this Floer homology in the special case when

(‹) BY is disconnected, and ω|BY is small and yet non-vanishing. Under this assumption, we will prove that the monopole Floer homology of pY, ωq is a topological invariant: it depends only on the 3-manifold Y , the cohomology class rωs P 2 1 H pY ; iRq and an additional class in H pBY ; iRq. Moreover, when Y is irreducible, this invariant detects the Thurston norm on H2pY, BY ; Rq and the fiberness of Y , generalizing the classical results [KM97, KM07, Ni08] for closed 3-manifolds. ´ In the context of Heegaard Floer homology, the knot Floer homology HFK˚ and HFK˚ were introduced by Oszv´ath-Sz´abo [OS04] and independently Rasmussen [Ras03]. Moti- vated by the sutured manifold technique developed by Juh´asz[Juh06, Juh08],z Kronheimer- Mrowka [KM10] defined the monopole knot Floer homology KHM as the analogue of the hat-version HFK˚. One motivation of this project is to provide an alternative definition of KHM so that the (3+1) TQFTz property will follow easily. This goal will be accomplished in this paper: for any knot K inside a closed 3-manifold Z, we will verify that the monopole Floer homology of the link complement ZzNpK Y mq is isomorphic to KHM pZ,Kq, where m Ă ZzK is a meridian. This confirms a longstanding speculation [Man16] that the knot Floer homology is related to the monopole equations on Rt times the link complement ZzNpK Y mq. Most topological implications of this paper follow immediately from a gluing theorem that computes the monopole Floer homology when two such 3-manifolds pY1, ω1q and pY2, ω2q are glued suitably along their common boundary; see Theorem 2.7. The gluing map

α : HM ˚pY1, ω1q bR HM ˚pY2, ω2q Ñ HM ˚pY1 ˝h Y2, ω1 ˝h ω2q. is in fact an isomorphism and is functorial when considering both 3-manifold and 4-manifold cobordisms. As a result, the (3+1) TQFT structure can be upgraded into a (2+1+1) TQFT: we can construct the monopole Floer 2-functor

HM ˚ : T Ñ R, MONOPOLES AND LANDAU-GINZBURG MODELS III: A GLUING THEOREM 3 from a suitable cobordism bi-category T to the strict 2-category R of finitely generated R-modules. In this paper, we will always work with the mod 2 Novikov ring R to avoid orientation issues. With that said, the Gluing Theorem 2.7 follows immediately from the proof of Floer’s excision theorem [BD95, KM10], and the construction is not considered to be original. However, this is not a bad sign: the monopole Floer homology of pY, ωq is expected to have intimate relations with the existing theory for closed 3-manifolds and for balanced sutured manifolds. This paper is providing the first a few results towards this direction. 1.2. Summary of Results. Let us explain the consequences of the Gluing Theorem 2.7 in more details. Let Y be a connected, compact, oriented 3-manifold whose boundary 2 BY – Σ :“ 1ďjďn Tj is a union of 2-tori. The monopole Floer homology constructed in [Wan20b] relies on some auxiliary data on Σ. In this paper, we focus on the special case when Σ is disconnectedš and choose the following data in order:

‚ a flat metric gΣ of Σ; 1 ‚ an imaginary-valued harmonic 1-form λ P Ω pΣ, i q such that λ| 2 ‰ 0, 1 ď j ď n; h R Tj 2 2 ‚ an imaginary-valued harmonic 2-form µ P ΩhpΣ, iRq such that |xµ, rTj sy| ă 2π and ‰ 0, for any 1 ď j ď n.

Such a quadruple Σ “ pΣ, gΣ, λ, µq is called a T -surface. We choose a closed 2-form 2 ω P Ω pY, iRq on Y such that ω “ µ ` ds ^ λ in a collar neighborhood p´1, 0ss ˆ Σ Ă Y . Denote such a pair pY, ωq, together with other auxiliary data in the construction, by a thickened letter Y.

In [Wan20b], the monopole Floer homology group HM ˚pYq of Y is defined by studying the monopole equations on the completion Y :“ Y Σ r0, `8qs ˆΣ using ω as a perturbation. By [Wan20b, Theorem 1.4], HM ˚pYq is a finitely generated module over R. However, Ť the results from [Wan20b] do not guaranteedp that HM ˚pYq is a topological invariant; see Section 5 for more details. In this papar, we establish this invariance result in the special case described above: Theorem 1.1 (Theorem 5.2). Under above assumptions, the monopole Floer homology 2 group HM ˚pYq depends only on the 3-manifold Y , the class rωs P H pY ; iRq and r˚Σλs P 1 H pΣ; iRq, and is independent of the rest of the data used in the construction. c The group HM ˚pYq admits a decomposition with respect to relative spin structures on Y : HM ˚pYq “ HM ˚pY, sq. sPSpinc pY q àR For any closed irreducible 3-manifold Z, this grading information [KM97, KM07] will p p determine the Thurston norm xp¨q on H2pZ; Rq. The Gluing Theorem 2.7 then allows us to relate HM ˚pYq with the monopole Floer homology of the double Y Y p´Y q; so a similar detection result is obtained for any irreducible 3-manifold with disconnected toroidal boundary. However, our statement below is slightly different from the one in [KM97, KM07], since the author was unable to verify the adjunction inequality. 4 DONGHAO WANG

Theorem 1.2 (Theorem 7.2). Consider the set of monopole classes 2 MpYq :“ tc1psq : HM ˚pY, sq ‰ t0uu Ă H pY, BY ; Zq. and the function ϕYpκq :“ maxaPMpYqxa, κy, κ P H2pY, BY ; Rq. We set ϕY ” ´8 if MpYq “ H. Then p p 1 pϕ pκq ` ϕ p´κqq ď xpκq, @κ P H pY, BY ; q. 2 Y Y 2 R If in addition Y is irreducible, then MpYq is non-empty and the equality holds for any κ. Remark 1.3. Ideally, one would expect that MpYq is symmetric about the origin and so ϕYpκq “ ϕYp´κq whenever MpYq is non-empty; but the author is unable to prove this symmetry directly, due to the presence of the 2-form ω. However, the ideal adjunction inequality ϕYpκq ď xpκq can be verified in some special cases, e.g. when κ is integral and rBκs P H1pBY ; Zq is proportional to the Poincar´edual of r˚Σλs; see Corollary 8.3. ♦ This Thurston norm detection theorem is accompanied with a fiberness detection result. In the context of Heegaard Floer homology, such a result was first established by Ghig- gini [Ghi08] for genus-1 fibered knots and by Ni for any knots [Ni07] in general and any closed 3-manifolds [Ni09a]. In the Seiberg-Witten theory, this was proved by Kronheimer- Mrowka [KM10] for the monopole knot Floer homology KHM , and by Ni [Ni08] for closed 3-manifolds. Due to Remark 1.3, our statement below is slightly weaker than the ideal version that one may hope to prove in the first place:

Theorem 1.4 (Theorem 7.3). For any integral class κ P H2pY, BY ; Zq, consider the sub- group HM ˚pY|κq :“ HM ˚pY, sq. xc1psq,κy“ϕ pκq à Y 1 If Y is irreducible and rankR HM ˚pY|κq “ rankp R HM ˚pY| ´ κpq “ 1, then Y fibers over S . The proof of Theorem 7.3 relies on the fiberness detection result [KM10, Theorem 6.1] for sutured Floer homology. In the original definition of Gabai [Gab83], any 3-manifold with toroidal boundary is a sutured manifold, but it is not balanced, in the sense of Juh´asz[Juh06, Definition 2.2]. Thus the sutured Floer homology, either SFH or SHM , is not previously defined for this class of sutured manifolds. One may regard our construction as a natural extension of SHM , and ask if a decomposition theorem, like [Juh08, Theorem 1.3] and [KM10, Theorem 6.8], continues to hold at this generality. Using the Gluing Theorem 2.7, we establish a preliminary result, Theorem 8.1, towards this direction. Theorem 1.4 then follows from Theorem 8.1 and the works of Kronheimer-Mrowka [KM10] and Ni [Ni09a]. As an immediate corollary, Theorem 1.2 and 1.4 provides a characterization of the product 2 manifold r´1, 1ss ˆ T : Proposition 1.5 (Corollary 9.3). Let Y be any oriented 3-manifold with disconnected toroidal boundary. If Y is connected and irreducible, then rankR HM ˚pYq ě 1. If the 2 equality holds, then Y “ r´1, 1ss ˆ T is a product. Proposition 1.5 was first suggested to the author by Chris Gerig. MONOPOLES AND LANDAU-GINZBURG MODELS III: A GLUING THEOREM 5

1.3. Connected Sum Formulae. The Gluing Theorem 2.7 also allows us to drive the connected sum formulae for reducible 3-manifolds. For any Y1 and Y2, take their connected sum to be Y1#Y2 “ pY1#Y2, ω1#ω2, ¨ ¨ ¨ q. 2 2 The class rω1#ω2s P H pY1#Y2; iRq is canonically determined by ωj P Ω pYj, iRq, j “ 1, 2 2 2 and by requiring that xrω1#ω2s,S y “ 0, where S Ă Y1#Y2 is the 2-sphere separating Y1 and Y2. Proposition 1.6 (Proposition 10.1). Under above assumptions, we have

HM ˚pY1#Y2q “ HM ˚pY1q bR HM ˚pY2q bR V, where V is a 2 dimensional vector space over R. Proposition 1.7 (Proposition 10.3). For any closed 3-manifold Z, one can form the con- nected sum Y#Z in a similar way. Then

HM ˚pY#Zq – HM ˚pYq bR HM ˚pZq where HM pZq is defined as the sutured Floer homology of pZp1q, δq; see Definition 10.2. ˚ Ą Proposition 1.6 and 1.7 are consistent with the connected sum formulae in [Juh06, Propo- Ą sition 9.15] for sutured Heegaard Floer homology SFH, while the same formula for its Seiberg-Witten analogue SHM were obtained in [Li18, Theorem 1.5]. Proposition 1.6 is accompanied with a vanishing result, which concerns the case when 2 the 2-form ω on Y1#Y2 has non-zero pairing with the separating 2-sphere S : Proposition 1.8. Suppose Y is any 3-manifold with toroidal boundary that contains an 2 2 2 embedded 2-sphere S Ă Y . If in addition xc1psq, rS sy “ 0 and xω, rS sy ‰ 0, then the group HM ˚pY, sq vanishes. Proof. This follows from the neck-stretching argumentp in [KM07, Proposition 40.1.3] and p the energy equation in [Wan20b, Proposition 5.4].  1.4. Relations with Link Floer Homology. One motivation of this project is to provide an alternative definition for KHM and the monopole link Floer homology LHM . This goal is accomplished in this paper using the Decomposition Theorem 8.1. n Theorem 1.9 (Corollary 8.5). For any link L “ tLiui“1 in a closed 3-manifold Z, we pick a meridian mi for each component Li, and consider the link complement

Y pZ,Lq “ ZzNpL Y m1 Y ¨ ¨ ¨ Y mnq.

Then for a suitable 2-form ω on Y “ Y pZ,Lq, we have HM ˚pYq – LHM pZ,Lq. By the work of Osv´ath-Sz´abo[OS08] and Ni [Ni09b], the link Floer homology detects the Thurston norm of the link complement for any links in rational homology spheres. The assumption on homology is later removed by Juh´asz[Juh08, Remark 8.5]. The same detection result using monopole link Floer homology LHM was obtained by Ghosh-Li [GL19, Theorem 1.17]. By Theorem 1.9, Theorem1.2 recovers the previous detection results, when the link complement is irreducible. 6 DONGHAO WANG

On the other hand, any 3-manifold Y with toroidal boundary is the link complement of a link L inside a closed 3-manifold Z. Then the link Floer homology of pZ,Lq detects the Thurston norm of Y . However, the statements in [OS08, Ni09a, Juh08] and [GL19] have to reflect the choice of pZ,Lq, and so is different from the one in Theorem 1.2.

1.5. Generalized (3+1) TQFT. In [Wan20b], we only established the (3+1) TQFT structure for a restricted class of cobordisms, called strict cobordisms. For any 3-manifolds pYi, Σiq with toroidal boundary, i “ 1, 2, a cobordism

pX,W q : pY1, Σ1q Ñ pY2, Σ2q is a 4-manifold with corners, with W being a cobordism from Σ1 to Σ2. It is called strict, if Σ1 “ Σ2 and W “ r´1, 1st ˆ Σ1 is a product. To remove this constraint, we use the Gluing Theorem 2.7 to construct generalized cobordism maps:

Theorem 1.10 (Corollary 4.4). We define a category T2 as follows: each object of T2 is a 3-manifold Y “ pY, ω, ¨ ¨ ¨ q with toroidal boundary, decorated by a closed 2-form ω. Each morphism is a triple pX, W, aq where ‚p X,W q : pY1, Σ1q Ñ pY2, Σ2q is a 4-manifold with corners; ‚ a P HM ˚pWq is an element in the monopole Floer homology of W “ pW, ωW , ¨ ¨ ¨ q. Then there is a functor HM ˚ : T2 Ñ R-Mod from T2 to the category of finitely generated R-modules, which assigns each Y to its monopole Floer homology HM ˚pYq. Remark 1.11. To better package the information on 2-forms, the actual construction of T2 in Corollary 4.4 is slightly different. This generalized (3+1) TQFT structure contains strictly less information than the monopole Floer 2-functor in Theorem 2.7, since the func- toriality of a P HM ˚pWq is completely ignored here. ♦ 1.6. A Remark on the Gluing Theorem. Most results of this paper are based on the monopole Floer 2-functor in the Gluing Theorem 2.7. In this subsection, we provide a conceptual reason to explain why the gluing formula has such a particularly simple form and how it fits into a broader context. In the first paper [Wan20a], the author constructed an infinite dimensional gauged Landau-Ginzburg model for any T -surface Σ “ pΣ, gΣ, λ, µq:

pMpΣq,Wλ, GpΣqq.

It is more appropriate to think of HM ˚pYq as an invariant of Y relative to this Landau- Ginzburg model. To develop a gluing theory in general, one has to assign an A8 category A to pMpΣq,Wλ, GpΣqq and upgrade the Floer homology of Y as an A8-module over A. By the work of Seidel [Sei08, Corollary 18.27], we wish to construct a spectral sequence abutting to HM ˚pY1 ˝h Y2q with E1 page equal to

HM ˚pY1q bR HM ˚pY2q. As remarked in [Wan20a, Subsection 2.2], in the special case (‹) that we have discussed so far, the A8 category A is supposed to be trivial: thimbles of different critical points of Wλ are not supposed to intersect at all. This suggests that the spectral sequence collapses at E1-page and the gluing formula is simply a tensor product. MONOPOLES AND LANDAU-GINZBURG MODELS III: A GLUING THEOREM 7

The next interesting special case is when Σ is connected and µ “ 0. This may allow us to develop a gluing theorem for knot complements and recover the knot Floer homology ´ HFK˚ from our construction. Readers are referred to [Wan20b, Subsection 1.5] for more heuristics on this direction. Remark 1.12. On the other hand, the Gluing Theorem 2.7 is subject to certain limitations. When pY1, ω1q and pY2, ω2q are glued along their common boundaries, the 2-forms ω1 and ω2 have to match within a neighborhood of the boundary. This imposes a non-trivial homological constraint on the way they are glued together. ♦ 1.7. Organization. The rest of this paper is organized as follows: In Section 2, we review the basic definition of bi-categories and state the Gluing Theorem 2.7. Section 3 is devoted to its proof. We will follow closely the proof of Floer’s excision theorem in [BD95, KM10]. In Section 4, we construct the generalized cobordism maps and define the functor in Theorem 1.10. In Section 5, we prove the Invariance Theorem 1.1. In Section 6, we review the theory for closed 3-manifolds following the book [KM07] by Kronheimer-Mrowka and adapt their non-vanishing results to the case of non-exact perturbations. Section 7 is devoted to the Thurston norm detection result: Theorem 1.2. After a digression into sutured Floer homology in Section 8, the proof of the fiberness detection result, Theorem 1.4, is supplied in Section 9. The connected sum formulae are derived in Section 10. Acknowledgments. The author is extremely grateful to his advisor, Tom Mrowka, for his patient help and constant encouragement throughout this project. The author would like to thank Chris Gerig for suggesting Proposition 1.5. The author would also like to thank Zhenkun Li for many helpful discussions. This material is based upon work supported by the National Science Foundation under Grant No.1808794. 8 DONGHAO WANG

Part 2. The Gluing Theorem The primary goal of this part is to construct the monopole Floer 2-functor

HM ˚ : T Ñ R where T is the toroidal bi-category constructed in Section 2 and R is the strict 2-category of finitely generated R-modules. Throughout this paper, the base ring R is always the mod 2 Novikov ring. Our main result is the Gluing Theorem 2.7.

2. The Toroidal Bi-category 2.1. 2-categories. In this section, we review the definition of 2-categories following [Ben67]. The Hom-sets of a 2-category form categories themselves. Definition 2.1. A strict 2-category K consists of the following data: (Y1) a collections of objects A, B, C ¨ ¨ ¨ . They are also called 0-cells; (Y2) for any objects A, B P Ob K, there is a category KpA, Bq, whose objects are called 1-cells and morphisms are called 2-cells. The identity morphism of an 1-cell f is denoted by Idf : f Ñ f. The compositions of 2-cells in KpA, Bq are called vertical compositions, denoted by ˝v. (Y3) for any objects A, B, C P Ob K, there is a functor

˝h : KpA, Bq ˆ KpB,Cq Ñ KpA, Bq, called the horizontal composition. (Y4) The horizontal composition ˝h is associative, i.e. for any four 0-cells A, B, C, D, the two different ways of composing ˝h:

´ ˝h p´ ˝h ´q and p´ ˝h ´q ˝h ´ give rise to the same functor from KpA, Bq ˆ KpB,Cq ˆ KpC,Dq to KpA, Dq. (Y5) let 1 be the category with one object and one morphism; for any object A P Ob K, there is a functor 1A : 1 Ñ KpA, Aq picking out the identity 1-cell eA : A Ñ A

and its identity 2-cell IdeA : eA Ñ eA. The functor 1A is the unit of the horizontal composition ˝h. ♦

Example 2.2. Let R be the Novikov field defined over F2, the field of 2-elements:

´ ni R :“ F2rRs “ t aiq : ai P F2, ni P R, lim ni “ ´8u. iÑ8 iě0 ÿ We define R to be the strict 2-category with a single object ‹ such that Rp‹q :“ Rp‹, ‹q is the category of finitely generated R-modules. The horizontal composition is defined using the tensor product of R-modules. The identity 1-cell e P Rp‹q is R itself. ♦ Example 2.3. Let CAT be the strict 2-category consisting of all categories as 0-cells. For any categories A, B, 1-cells in CATpA, Bq are functors from A to B, while 2-cells are given by natural transformations. ♦ For technical reasons, the toroidal bi-category T defined in the next subsection is not strictly associative. However, T is still unital, and the associativity of ˝h still holds up to 2-isomorphisms; so it is an example of bi-categories: MONOPOLES AND LANDAU-GINZBURG MODELS III: A GLUING THEOREM 9

Definition 2.4. A (unital) bi-category K satisfies (Y1)(Y2)(Y3)(Y5) and (Y4’) for any 0-cells A, B, C, D, there is a natural isomorphism apA, B, C, Dq between the two functors below:

´ ˝h p´ ˝h ´q

KpA, Bq ˆ KpB,Cq ˆ KpC,Dq KpA, Dq,

p´ ˝h ´q ˝h ´ which satisfies an associativity coherence condition [Ben67, P.5]. For any triple pf, g, kq, we use apf, g, kq to denote the 2-cell isomorphism:

apf, g, kq : f ˝h pg ˝h kq Ñ pf ˝h gq ˝h k. ♦ 2.2. The Toroidal Bi-category. The primary goal of this subsection is to define the toroidal bi-category T, over which the monopole Floer 2-functor HM ˚ is defined. Each object of T is a quadruple

Σ “ pΣ, gΣ, λ, µq, called a T -surface, where n piq (T1)Σ “ i“1 Σ ‰H is an oriented surface consisting of finitely many 2-tori; we insist here that the surface Σ is non-empty; š (T2) gΣ is a flat metric of Σ; 1 2 (T3) λ P ΩhpΣ, iRq is a harmonic 1-form and µ P ΩhpΣ, iRq is a harmonic 2-form; when restricted to each connected component Σpiq, λ and µ are both non-zero; (T4) for any 1 ď i ď n, |xµ, rΣpiqsy| ă 2π.

For any T -surfaces Σ1 and Σ2, TpΣ1, Σ2q is a full subcategory of the strict cobordism category Cobs defined in [Wan20b, Section 3]. We recall the definition below for the sake of completeness. An object of TpΣ1, Σ2q is a quintuple Y “ pY, ψ, gY , ω, tquq satisfying the following properties:

(P1) Y is a compact oriented 3-manifold with toroidal boundary and ψ : BY Ñ p´Σ1qYΣ2 is an orientation preserving diffeomorphism. Σ1 and Σ2 are regarded as the incoming and outgoing boundaries of Y respectively. When it is clear from the context, the identification map ψ might be dropped from our notations. (P2) Each component of Y intersects non-trivially with both Σ1 and Σ2. (P3) The metric gY of Y is cylindrical, i.e. gY is the product metric 2 ˚ ds ` ψ pgΣ1 , gΣ2 q

within a collar neighborhood r´1, 1qs ˆ Σ1 p´1, 1ss ˆ Σ2 of BY . 2 (P4) ω P Ω pY, iRq is an imaginary valued closed 2-form on Y such that Ť ω “ µ1 ` ds ^ λ1 on r´1, 1qs ˆ Σ1,

ω “ µ2 ` ds ^ λ2 on p´1, 1ss ˆ Σ2. 2 2 In particular, pµ1, µ2q lies in the image ImpH pY ; iRq Ñ H pΣ; iRqq. 10 DONGHAO WANG

1 1 (P5) The cohomology class of p˚1λ1, ˚2λ2q lies in the image ImpH pY ; iRq Ñ H pBY ; iRqq. In particular, there exists a co-closed 2 form ωh such that

ωh “ ds ^ λ1 on r´1, 1qs ˆ Σ1;

ωh “ ds ^ λ2 on p´1, 1ss ˆ Σ2; (P6) tqu is a collection of admissible perturbations (in the sense of [Wan20b, Definition c 13.3]) of the Chern-Simons-Dirac functional Lω, one for each relative spin structure of Y . Remark 2.5. The reason to include (P2) is to rule out closed components of Y when considering horizontal compositions. This is not a serious problem; see Subsection 2.4. ♦

Having defined the objects (1-cells) of TpΣ1, Σ2q, let us now describe the set mor- phisms (2-cells). Since each 3-manifold with toroidal boundary is now decorated by a closed 2-form ω, morphisms will take these forms into account. Given two objects Yi “ pYi, ψi, gi, ωi, tqiuq, i “ 1, 2 in TpΣ1, Σ2q, a morphism

X : Y1 Ñ Y2 is a triple X “ pX, rψX s, rωX scptq satisfying the following properties.

Σ1 Y1 Σ2 Y1 Σ1 Σ2

r´1, 1st ˆ Σ1 X r´1, 1st ˆ Σ2 ù X

Σ1 Σ2. Σ1 Y2 Σ2 Y2

Figure 1. A 2-cell morphism X.

(Q1) X is a 4-manifold with corners, i.e. X is a space stratified by manifolds

X Ą X´1 Ą X´2 Ą X´3 “H

such that the co-dimensional 1 stratum X´1 consists of three parts

X´1 “ p´Y1q Y pY2q Y WX .

where WX is an oriented 3-manifold with boundary BWX “BY1 XBY2. Moreover, BYi “ Yi X WX and X´2 “BY1 YBY2. (Q2) ψX : WX Ñ r´1, 1st ˆ pp´Σ1q Y Σ2q is an orientation preserving diffeomorphism compatible with ψ1 and ψ2. More precisely, we require that

ψX |BY1 “ ψ1 : BY1 Ñ t´1u ˆ pp´Σ1q Y Σ2q,

ψX |BY2 “ ψ2 : BY2 Ñ t1u ˆ pp´Σ1q Y Σ2q,

which hold also in a collar neighborhood of BWX . When no chance of confusion is possible, ψX might be dropped from our notations. Such a pair pX, ψX q is called a strict cobordism from pY1, ψ1q to pY2, ψ2q. rψX s denotes the isotopy class of such a diffeomorphism. 2 (Q3) There exists a closed 2-form ωX P Ω pX, iRq on X satisfying the following properties. MONOPOLES AND LANDAU-GINZBURG MODELS III: A GLUING THEOREM 11

‚ ωX “ ωi (see (P4)) within a collar neighborhood of Yi Ă X´1 for i “ 1, 2; ‚ within a collar neighborhood of WX Ă X´1,

ωX “ µ1 ` ds ^ λ1 on r´1, 1st ˆ r´1, 1qs ˆ Σ1,

ωX “ µ2 ` ds ^ λ2 on r´1, 1st ˆ p´1, 1ss ˆ Σ2.

The existence of such a 2-form ωX is guaranteed by a cohomological condition on 2 1 rωX s P H pX; iRq; see [Wan20b, Section 3 (Q4)]. Any two such forms ωX , ωX are 1 called relative cohomologous if ωX ´ ωX “ da for a compactly supported smooth 1 1-form a P Ωc pX, iRq. We fix the relative cohomology class of ωX , denoted by rωX scpt.

Remark 2.6. It is necessary to record the isotopy class of ψX here, because the diffeomor- 2 2 phism group Diff`pT q is not simply connected. By [EE67, Theorem 1(b)], Diff`pT q has the 1 1 2 same homotopy type of its linear subgroup S ˆS ˆSLp2, Zq, so π1pDiff`pT qq – Z‘Z. ♦

The vertical composition of TpΣ1, Σ2q is defined by composing strict cobordisms. Since 2 we have recorded the relative cohomology class rωX scpt (instead of just rωX s P H pX; iRq), these classes can be concatenated accordingly on the composed manifold. For any 1-cell Y P TpΣ1, Σ2q, the identity 2-cell is given by the product strict cobordism pr´1, 1st ˆ

Y, Idr´1,1st ˆψq, with rωscpt being the class of the pull-back 2-form ω. A metric of X is not encoded in the definition of X. This category is topological, although auxiliary data are specified for its objects (1-cells). The horizontal composition TpΣ1, Σ2qˆTpΣ2, Σ3q Ñ TpΣ1, Σ3q is defined using the dif- feomorphisms ψ. On the level of 1-cells, given any pair pY12, Y23q P TpΣ1, Σ2qˆTpΣ2, Σ3q, the underlying 3-manifold of Y12 ˝h Y23 is formed by gluing ´1 ´1 Y12zr0, 1ss ˆ ψ12 pΣ2q and Y23zr´1, 0ss ˆ ψ23 pΣ2q along the common boundary t0u ˆ Σ2, using the composition

´1 ´1 ψ12 ψ23 ´1 ψ12 pΣ2q ÝÝÑ Σ2 ÝÝÑ ψ23 pΣ2q, which is orientation reversing. The condition (P5) is certified by concatenating the co-closed 2-form pωhq12 with pωhq23. We shall formally write:

Y12 ˝h Y23 “ pY12 ˝h Y23, ω12 ˝h ω23, ¨ ¨ ¨ q. The problem arises from (P6): it is not guaranteed that two admissible perturbations on Y12 and Y23 can be concatenated in a canonical way to form an admissible perturbation on Y12 ˝h Y23. Instead, we will pick a random collection of admissible peturbations for Y12 ˝h Y23 making ˝h not strictly associative. As for 2-cells, their horizontal compositions are formed similarly using ψX instead.

Let us now construct the natural isomorphism apΣ1, Σ2, Σ3, Σ4q in (Y4’): for any triple pY12, Y23, Y34q, define

apY12, Y23, Y34q : Y12 ˝h pY23 ˝h Y34q Ñ pY12 ˝h Y23q ˝h Y34. to be the identity 2-cell. Indeed, as 1-cells in TpΣ1, Σ4q, they have the same underlying metrics and closed 2-forms; only the admissible perturbations may differ from one another. 12 DONGHAO WANG

Finally, for any 0-cell Σ, we define its identity 1-cell eΣ as 2 pY “ r´1, 1ss ˆ Σ, ψ “ Id, gY “ d s ` gΣ, ω “ µ ` ds ^ λ, tq “ 0uq.

For any 1-cell Y12 P TpΣ1, Σ2q, one may set Y12 ˝h eΣ2 and eΣ1 ˝h Y12 to be just Y12, as they already have the same underlying metrics and closed 2-forms by our conventions of horizontal compositions. In this way, the toroidal bi-category T becomes strictly unital.

2.3. The Monopole Floer 2-Functor. The primary goal of this paper is to define the monopole Floer 2-functor: HM : T Ñ R. We expand on the requirement for a 2-functor in the theorem below: Theorem 2.7 (The Gluing Theorem). There exists a 2-functor HM from the toroidal bi- category T to the strict 2-category R of finitely generated R-modules satisfying the following properties:

(G1) for any T-surfaces Σ1, Σ2, the functor

HM ˚ : TpΣ1, Σ2q Ñ Rp‹q is defined as in [Wan20b, Theorem 1.5 & Remark 1.7], which assigns each 1-cell Y to its monopole Floer homology group HM ˚pYq; (G2) for any Σ1, Σ2, Σ3, there is a natural isomorphism αpΣ1, Σ2, Σ3q between the two compositions in the digram below:

HM ˚ ˆ HM ˚ ˝h TpΣ1, Σ2q ˆ TpΣ2, Σ3q Rp‹q ˆ Rp‹q Rp‹q

˝h HM ˚ TpΣ1, Σ3q.

In other words, for any composing pair pY12, Y23q, there is an isomorphism of R- modules:

α : HM ˚pY12q bR HM ˚pY23q Ñ HM ˚pY12 ˝h Y23q,

that is natural with respect to the 2-cell morphisms in TpΣ1, Σ2q and TpΣ2, Σ3q; (G3) α is associative meaning that the digram

Id bα HM ˚pY12q b HM ˚pY23q b HM ˚pY34q HM ˚pY12q b HM ˚pY23 ˝h Y34q

αbId α (2.1) HM ˚pY12 ˝ Y23q b HM ˚pY34q HM ˚pY12 ˝h pY23 ˝h Y34qq α – HM ˚papY12,Y23,Y34qq

HM ˚ppY12 ˝h Y23q ˝h Y34q

is commutative for any triples pY12, Y23, Y34q P TpΣ1, Σ2qˆTpΣ2, Σ3qˆTpΣ3, Σ4q. MONOPOLES AND LANDAU-GINZBURG MODELS III: A GLUING THEOREM 13

(G4) for any T-surface Σ and its identity 1-cell eΣ, there is a canonical isomorphism

ιΣ : HM ˚peΣq Ñ R, such that the gluing map

α : HM ˚pY12q b HM ˚peΣ2 q Ñ HM ˚pY12 ˝h eΣ2 q “ HM ˚pY12q,

is simply Id bιΣ. A similar property holds also for eΣ1 ˝h Y12. 2.4. A convention for TpH, Σq. If we allow the empty surface H to be a 0-cell of the toroidal bi-category T, then we must allow the underlying 3-manifold of an 1-cell Y “ pY, ω, ¨ ¨ ¨ q to have closed components. This is not a serious problem, as long as on each closed component, the 2-form ω is never balanced or negatively monotone with respect to any spinc structures, in the sense of [KM07, Definition 29.1.1]. This will allow us to apply the adjunction inequality in Proposition 6.2. Instead of setting up the theory at this generality, we will simply define TpH, Σq, TpΣ, Hq, as categories in their own rights, and do not regard them as part of the bi-category T. We can still define the horizontal composition

˝h : TpH, Σ1q ˆ TpΣ1, Σ2q Ñ TpH, Σ2q in the usual way and make the assignment Σ ÞÑ TpH, Σq into a 2-functor (in a suitable sense) from T to CAT. However, the latter point of view is not needed for the purpose of this paper. The category TpH, Hq consists of closed 3-manifolds equipped with imaginary valued closed 2-forms that are never balanced or negatively monotone on each component.

3. Proof of the Gluing Theorem In this section, we present the proof of the Gluing Theorem 2.7. The construction of the gluing map α in Theorem 2.7 (G2) is based upon Floer’s Excision Theorem [BD95], which has been adapted to the monopole Floer homology by Kronheimer-Mrowka [KM10]. We will follow the setup of [KM10, Theorem 3.1 & 3.2] closely. This section starts with an overview of the monopole Floer homology defined in [Wan20b], which yields the monopole Floer functor HM ˚ in Theorem 2.7 (G1). The gluing map α is then constructed in Subsection 3.6.

c 3.1. Review. Recall that a spin structure sX on an oriented Riemannian 4-manifold X is ` ´ ˚ a pair pSX , ρ4q where SX “ S ‘ S is the spin bundle, and the bundle map ρ4 : T X Ñ EndpSX q defines the Clifford multiplication. A configuration γ “ pA, Φq P CpX, sq consists of a smooth spinc connection A and a smooth section Φ of S`. Use At to denote the 2 ` 2 ` induced connection of A on S . Let ωX P Ω pX, iRq be a closed 2-form on X and ωX Ź 14 DONGHAO WANG denote its self-dual part. The Seiberg-Witten equations perturbed by ωX are defined on the configuration space CpX, sq by the formula: 1 ρ pF ` q ´ pΦΦ˚q “ ρ pω` q, (3.1) 2 4 At 0 4 X D`Φ “ 0, " A ` ` ´ ˚ ˚ 1 2 where DA :ΓpS q Ñ ΓpS q is the Dirac operator and pΦΦ q0 “ ΦΦ ´ 2 |Φ| b IdS` is the traceless part of the endomorphism ΦΦ˚ : S` Ñ S`. When it comes to an oriented Riemannian 3-manifold Y , the dimensional reduction of (3.1) is obtained by looking at (3.1) on the product manifold Rt ˆ Y and by asking the c configuration pA, Φq to be Rt-invariant. A spin structure s on Y is again a pair pS, ρ3q ` ˚ where the spin bundle S “ S has complex rank 2 and ρ3 : T Y Ñ EndpSq defines the Clifford multiplication. The 3-dimensional Seiberg-Witten equations now read: 1 ˚ ρ pF t q ´ pΨΨ q “ ρ pωq, (3.2) 2 3 B 0 3 DBΨ “ 0. " c 2 where B is a spin connection and Ψ P ΓpY,Sq. Here ω P Ω pY, iRq is a closed 2-form and DB :ΓpY,Sq Ñ ΓpY,Sq denotes the Dirac operator on Y .

3.2. Results from the Second Paper. In this subsection, we review the construction of the monopole Floer homology from [Wan20b], which defines the functor in Theorem 2.7 (G1). For any T -surface Σ, define ´Σ :“ p´Σ, gΣ, ´λ, µq to be the orientation reversal of Σ. Since the category TpΣ1, Σ2q is more or less equivalent to TpH, p´Σ1q Y Σ2q (only the property (P2) may be different), we focus on the case when Σ1 “H. Given any 1-cell Y “ pY, ψ, gY , ω, tquq P TpH, Σq, we first attach a cylindrical end r1, 8qs ˆ Σ to Y to obtain a complete Riemannian manifold:

Y “ Y Yψ r1, 8qs ˆ Σ. 2 The metric on the end is given by d s ` gΣ. The closed 2-form ω is extended constantly as p 2 µ ` ds ^ λ on r1, 8qs ˆ Σ, denoted also by ω P Ω pY , iRq. Let sstd “ pSstd, ρstd,3q be the c 2 standard spin structure on Rs ˆ Σ with c1psstdq “ 0 P H pΣ, iRq. The spin bundle Sstd can be constructed explicitly as p 0,1 Sstd “ C ‘ Λ Σ. See [Wan20b, Section 2] for more details. A relative spinc structure s on Y is a pair ps, ϕq c where s “ pS, ρ3q is a spin structure on Y and ˚ p ϕ : pS, ρ3q|BY Ñ ψ sstd|BY is an isomorphism of spinc structures near the boundary, compatible with ψ : BY Ñ Σ. The set of isomorphism classes of relative spinc structures on Y c SpinRpY q 2 c is a torsor over H pY, BY ; Zq. There is a natural forgetful map from SpinRpY q to the set of isomorphism classes of spinc structures: c c SpinRpY q Ñ Spin pY q, s “ ps, ϕq ÞÑ s,

p MONOPOLES AND LANDAU-GINZBURG MODELS III: A GLUING THEOREM 15

1 1 whose fiber is acted on freely and transitively by H pΣ, Zq{ ImpH pY, Zqq reflecting the c c change of boundary trivializations. Any s P SpinRpY q extends to a relative spin structure on Y , denoted also by s. p The key observation is that on Rs ˆ Σ, the 3-dimensional Seiberg-Witten equations (3.2) p perturbed by ω “ µ ` dsp ^ λ have a canonical Rs-invariant solution, denoted by

pB˚, Ψ˚q, which is also the unique finite energy solution on Rs ˆΣ, up to gauge, by our assumptions on pλ, µq. This result is due to Taubes [Tau01, Proposition 4.4 & 4.7]. See [Wan20b, Theorem 2.6] for the precise statement that we use here. When it comes to Y , each configuration is required to approximate this special solution pB˚, Ψ˚q as s Ñ 8. Take pB0, Ψ0q to be a smooth configuration on Y which agrees with pB˚, Φ˚q on the cylindricalp end r1, 8qs ˆ Σ and consider the configuration space for any 1 k ą 2 : p 2 ˚ CkpY, sq “ tpB, Ψq : pb, ψq “ pB, Ψq ´ pB0, Ψ0q P LkpY , iT Y ‘ Squ. which is acted on freely by the gauge group p p 1 2 p p Gk`1pY q “ tu : Y Ñ S Ă C : u ´ 1 P Lk`1pY, Cqu, using the formula: p upB,pΨq “ pB ´ u´1du, uΨq. p

The perturbed Chern-Simons-Dirac functional on CkpY, sq is then defined as

1 t t 1 1 t t (3.3) LωpB, Ψq “ ´ pB ´ B0q ^ pFBt ` FBt q ` xDBΨ, Ψy ` pB ´ B0q ^ ω. 8 0 2 p p 2 żY żY żY For any 1-cell P TpH, Tq and any relative spinc structure s P Spinc pY q, the monopole Y p p R p Floer homology HM ˚pY, sq is defined as the Morse homology of Lω on the quotient configu- ration space CkpY, sq{Gk`1pY q. However, it is not guaranteed thatp Lω descends to a Morse function on CkpY, sq{Gk`p1pY q, so an admissible perturbation q of Lω, which is encoded already in (P6),p isp neededp to make critical points Morse and moduli spaces of flowlines regular. The mainp p result ofp [Wan20b] says that the monopole Floer homology HM ˚pY, sq is well-defined as a finite generated module over the mod 2 Novikov field R. Since we have assumed in (T3) that for any T -surface Σ,p both λ and µ are non-vanishing on any component Σpiq Ă Σ, we have a stronger statement:

Theorem 3.1 ([Wan20b, Theorem 1.4]). For any 1-cell Y P TpΣ1, Σ2q, the direct sum

HM ˚pYq :“ HM ˚pY, sq, sPSpinc pY q àR is finitely generated over R. In particular,p the group HMp˚pY, sq is non-trivial for only finitely many relative spinc structures. As explained in [Wan20b, Section 17], this Floer homology isp further enhanced into a functor: 16 DONGHAO WANG

Theorem 3.2 ([Wan20b, Theorem 1.6 & Remark 1.7]). For any T -surfaces Σ1, Σ2, there is a functor from TpΣ1, Σ2q to the category of finitely generated R-modules Rp‹q:

HM ˚ : TpΣ1, Σ2q Ñ Rp‹q, which assigns to each 1-cell Y to its monopole Floer homology group HM ˚pYq. Remark 3.3. In the second paper [Wan20b], we focused on connected 3-manifolds with toroidal boundary, but the results generalize to the disconnected case with no difficulty. ♦

For any 2-cell morphism X : Y1 Ñ Y2, the cobordism map

HM ˚pXq : HM ˚pY1q Ñ HM ˚pY2q is constructed as follows. We focus on the case when Σ1 “H. For the underlying strict cobordism X : Y1 Ñ Y2, pick a diffeomorphism ψX : WX Ñ r´1, 1st ˆ Σ and a closed 2- 2 form ωX P Ω pX, iRq belonging to the class rψX s and rωX scpt respectively, as in (Q2)(Q3). Choose a metric gX of X compatible with its corner structures. We attach an end in the spatial direction to obtain a cobordism from Y1 to Y2: X :“ X r´1, 1s ˆ r0, 8q ˆ Σ, p t p s ψ ďX and attach cylindrical ends inp the time direction to form a complete Riemannian manifold:

X :“ p´8, ´1st ˆ Y1 X r1, `8qt ˆ Y2. The closed 2-form ω extends to a 2-form onďX byď setting X p p p ω1 on p´8, ´1st ˆ Y1, ωX “ ω on r1, `8q ˆ Y , $ 2 t 2 & µ ` ds ^ λ on Rt ˆ r0, 8qs ˆpΣ. p The cobordism map HM ˚pX%q is then defined by counting 0-dimensional solutions (modulo gauge) to the Seiberg-Witten equations (3.1) with ωX defined above. Some additional perturbations are required here to make moduli spaces regular; see [Wan20b, Section 16] for more details. The cobordism map HM ˚pXq depends only on the isotopy class of ψX and the relative cohomology class of ωX , and is independent of the planar metric gX of X.

3.3. The Canonical Grading. By [KM07, Lemma 28.1.1], the standard spinor Ψ˚ on Rs ˆ Σ determines a canonical oriented 2-plane field ξ˚ that is Rs-invariant. For any 3-manifold Y with BY – Σ, an oriented 2-plane field ξ is called relative if ξ “ ξ˚ near the boundary. Any homotopy of oriented relative 2-plane fields is supposed to preserve ξ˚ near the boundary BY . Inspired by the construction in [KM07, Section 28], the author introduced a canonical grading on the group HM ˚pY, sq in [Wan20b, Section 18]. The grading set π Ξ pY, sq p is identified with the homotopy classes of oriented relative 2-plane fields that belongs to the relative spinc structure s. p

p MONOPOLES AND LANDAU-GINZBURG MODELS III: A GLUING THEOREM 17

3.4. Euler Characteristics. The 3-manifold Y is homology oriented, if we pick an orien- 3 tation of i“0 HipY ; Rq. Any homology orientation of Y induces a canonical mod 2 grading on HM ˚pY, sq (cf. [MT96][Wan20b, Subsection 18.2]). Then the graded Euler Characteris- À tics of HM ˚pY, sq is well-defined and recovers a classical algebraic invariant for 3-manifolds with toroidalp boundary. For future reference, we record the statement below: p Theorem 3.4 ([MT96, Tur98, Tau01]). The graded Euler Characteristic of HM ˚pY, sq: c SWpY q : SpinRpY q Ñ Z p s ÞÑ χpHM ˚pY, sqq, is independent of the auxiliary data pgY , gΣ; ω, λ, µq and is equal to the Minor-Turaev torsion T pY q up to an overall sign ambiguity. Moreover,p the functionp SWpY q is invariant under the conjugacy symmetry: s Ø s˚. 3.5. Identity 1-cells. Before we proceed to the construction of the gluing map, let us first define the canonical isomorphismp p

ιΣ : HM ˚peΣq Ñ R in Theorem 2.7 (G4). By the definition of the identity 1-cell eΣ, HM ˚peΣq is computed using the product manifold Rs ˆ Σ with ω “ µ ` ds ^ λ. As noted earlier, the 3-dimensional Seiberg-Witten equations (3.2) have a unique finite energy solution pB˚, Ψ˚q up to gauge c for the standard relative spin structure sstd. Moreover, pB˚, Ψ˚q is irreducible and non- degenerate as the critical point of Lω in the quotient configuration space; see the proof of [Wan20b, Proposition 12.1]. By [Wan20b,p Theorem 2.4], any downward gradient flowline of Lω connecting pB˚, Ψ˚q to itself is necessary a constant path. As a result, the monopole Floer chain complex of peΣ, sstdq is generated by this special solution with trivial differential. The canonical isomorphism ιΣ is then defined by sending this generator to 1 P R. When s ‰ sstd, the 3-dimensionalp equations (3.2) have no solutions at all, so HM ˚peΣ, sq “ t0u. p3.6. pThe Gluing Map. Having defined the functor HM ˚ in Theorem 2.7 (G1)p and the canonical isomorphism ιΣ in (G4), let us construct the gluing map α in (G2) in this subsec- tion. The idea is borrowed from the proof of Floer’s excision theorem [BD95] and [KM10, Theorem 3.2]. We focus on the special case when Σ1 “ Σ3 “H and construct the map

α : HM ˚pY1q bR HM ˚pY2q Ñ HM ˚pY1 ˝h Y2q, for any 1-cells Y1 P TpH, Σq and Y2 P TpΣ, Hq. The general case is not really different. Let Yi be the underlying 3-manifold of Yi and Y1 ˝h Y2 be the closed 3-manifold obtained by gluing Y1 and Y2 along t0u ˆ Σ. In what follows, we also work with the truncated 3-manifolds

Y1,´ :“ ts ď ´1u Ă Y1, Y2,´ :“ ts ě 1u Ă Y2. The gluing map α is induced from an explicit strict cobordismp between p X : Y1 Y2 Ñ r´1, 1ss ˆ Σ Y1 ˝h Y2, ž ž 18 DONGHAO WANG as we describe now. Let Ω be an octagon with a prescribed metric such that the boundary of Ω consists of geodesic segments of length 2, and the internal angles are always π{2. Moreover, this metric is hyperbolic somewhere in the interior and flat near the boundary.

1 ´1 γ4 γ3 h “ 1 ´1 Ω ´1 1 γ1 γ2 ´1 1 Figure 2. The surface Ω with corners

The product Ω ˆ Σ is now a 4-manifold with corners. The desired strict cobordism X is obtained then by attaching r´1, 1st ˆ Y1,´ to γ1 ˆ Σ and r´1, 1st ˆ Y2,´ to γ2 ˆ Σ. Arrows in Figure 2 indicate the positive direction of the time coordinate t. To define the closed 2-form ωX , let h :Ω Ñ R be a function that equals to 1 on γ2 Yγ4 and to ´1 on γ1 Y γ3. Also, h is required to be the linear function on the other four boundary segments of Ω. Set

(3.4) ωX “ µ ` dh ^ λ on Ω ˆ Σ. The 4-manifold cobordism

X : Y1 Y2 Ñ pRs ˆ Σq pY1 ˝h Y2q can be now schematically shownž as follows: ž p p p Rs ˆ Σ r´1, 1st ˆ r1, 8qs ˆ Σ ù ø r´1, 1st ˆ p´8, ´1ss ˆ Σ

Y1 ñ ð Y2

p p r´1, 1st ˆ Y1 ù ø r´1, 1st ˆ Y2 Y1 ˝h Y2 Figure 3

The arrows “ Ñ ” in Figure 3 indicate the positive direction of the spatial coordinate s. As a result, the complete Riemannian manifold X obtained by attaching cylindrical ends to X has two planar ends: Rt ˆ r1, `8qs ˆ Σ and Rt ˆ p´8, ´1ss ˆ Σ. Remark 3.5. In order to make ω into a smooth form on X , the function h :Ω ˆ Σ Ñ p X R must extend to a smooth function on X such that h ” s on any planar end. One may think of h as an extension of the spatial coordinate s over the region Ω ˆ Σ. ♦ MONOPOLES AND LANDAU-GINZBURG MODELS III: A GLUING THEOREM 19

On the other hand, one can draw the cobordism X vertically and regard Ω as the part of the pair-of-pants cobordism that contains the saddle point: p Y1 Ω ˆ Σ Y2 1, 1 1, Σ 1, 1 , 1 Σ r´ st ˆ r 8qspˆ ù ø r´ pst ˆ p´8 ´ ss ˆ

Rs ˆ Σ

r´1, 1st ˆ Y1 ù ø r´1, 1st ˆ Y2 Y1 ˝h Y2 Figure 4. Draw X vertically.

Since the base ring R is also a field, we use thep K¨unneth formula to identify

HM ˚pY1 Y2q – HM ˚pY1q bR HM ˚pY2q.

The gluing map α is then definedž as the cobordism map induced from X “ pX, ωX q and multiplied by a normalizing constant η P R:

HM ˚pXq α : HM ˚pY1q bR HM ˚pY2q ÝÝÝÝÝÑ HM ˚peΣq bR HM ˚pY1 ˝h Y2q ιΣbId ÝÝÝÝÑ R bR HM ˚pY1 ˝h Y2q ηˆ ÝÝÑ HM ˚pY1 ˝h Y2q, where n 1 |2πixµ,rΣpiqsy| η “ ´1 P R with ti “ q , i“1 ti ´ ti piq ź n piq and Σ is the i-th component of Σ “ i“1 Σ . The inverse β of α is induced from the 1 1 1 opposite cobordism of X, denoted by X “ pX , ωX q, and is normalized by the same constant η: š 1 η b ´ HM ˚pX q β : HM ˚pY1 ˝h Y2q ÝÝÝÝÝÑ HM ˚peΣq bR HM ˚pY1 ˝h Y2q ÝÝÝÝÝÝÑ HM ˚pY1q bR HM ˚pY2q

Rs ˆ Σ

Y1 ˝h Y2

Y1 Y2

1 p Figure 5. The opposite cobordismp X .

The choice of the normalizing constant η is justified by the followingp theorem, which says that the gluing map α is indeed an isomorphism with inverse β. 20 DONGHAO WANG

Theorem 3.6 (Floer’s Excision Theorem). The gluing map α and β constructed above are mutual inverses to each other, i.e., α β Id , β α Id . ˝ “ HM ˚pY1˝hY2q ˝ “ HM ˚pY1qbHM ˚pY2q The gluing map α preserves the canonical grading on HM ˚pYiq. There are natural concatenation maps: c c c ´ ˝h ´ : SpinRpY1q ˆ SpinRpY2q Ñ SpinRpY1 ˝h Y2q, π π π ´ ˝h ´ :Ξ pY1, s1q ˆ Ξ pY2, s2q Ñ Ξ pY1 ˝h Y2, s1 ˝h s2q. c Indeed, any two relative spin structures si “ psi, ϕiq, i “ 1, 2 can be composed using the map p p p p ϕ´1 ϕ1 p 2 pS1, ρ3,1q|BY1 ÝÑ sstd|Σ ÝÝÑpS2, ρ3,2q|BY2 , c to produce a spin structure on Y1 ˝h Y2. Meanwhile, any oriented relative 2-plane fields pξ1, ξ2q can be composed, since they agree with the canonical 2-plane field ξ˚ near Σ. In the special case that we have considered so far, Y1 ˝h Y2 is a closed 3-manifold, so c c π SpinRpY1 ˝h Y2q “ Spin pY1 ˝h Y2q.Ξ pY1 ˝h Y2, sq is the subset of π0pΞpY1 ˝h Y2qq, the homotopy classes of oriented 2-plane fields on Y1 ˝h Y2, that belongs to s; see [KM07, P. 585] for the precise definition of π0pΞpY1 ˝h Y2qq.

Theorem 3.7. The gluing map α : HM ˚pY1qbR HM ˚pY2q Ñ HM ˚pY1 ˝h Y2q preserves the relative spinc grading and the canonical grading by the homotopy classes of oriented relative 2-plane fields, meaning that α restricts to a map

α : HM ˚pY1, s1q b HM ˚pY2, s2q Ñ HM ˚pY1 ˝h Y2, sq, s1˝hs2“s à which is an isomorphism by Theoremp 3.6. Moreover,p if an element px1, xp2q belongs to the p pπ π grading prξ1s, rξ2sq P Ξ pY1, s1q ˆ Ξ pY2, s2q, then αpx1 b x2q is in the grading rξ1s ˝h rξ2s. The rest of Subsection 3.6 is devoted to the proof of Theorem 3.6 and 3.7. p p Proof of Theorem 3.6. The argument in [KM10, Theorem 3.2] carries over to our case with little change. We focus on the second identity β α Id to explain the ˝ “ HM ˚pY1qbHM ˚pY2q choice of the normalizing constant η. The map β ˝ α is identical to the one induced by 1 X ˝v X:

k ˆ Σ

Y1 Y2 Figure 6 p p MONOPOLES AND LANDAU-GINZBURG MODELS III: A GLUING THEOREM 21

1 To compare X ˝v X with the product cobordism from Y1 Y2 to itself, we stretch the neck along a union of 3-tori k ˆ Σ, where k is the red circle in Figure 6. To specify the š closed 2-form ω4 in the Seiberg-Witten equations (3.1) as we vary the metrics, we regard k 1 as a curve in Ω ˝v Ω: k 1 ´1

Y1 ñ ð Y2 ´1 1 p p

1 Y1 ð Ω ñ Y2

p 1 ´1 p

Figure 7

Here Ω1 is the opposite cobordism of Ω, regarded as part of a pair-of-pants cobordism. In 1 Figure 7, the top horizontal edge is identified with the bottom edge. On Ω ˝v Ω, one may 1 homotope the function h :Ω ˝v Ω Ñ R rel boundary such that h ” 1{2 on r´1{2, 1{2ss ˆ k, the tubular neighborhood of k colored orange in Figure 7. As we stretch the neck 1 r´1{2, 1{2ss ˆ k ˆ Σ, the 2-form ω4 is set to be µ ` dh ^ λ on pΩ ˝v Ωq ˆ Σ. In particular, ω ” µ on the neck r´1{2, 1{2ss ˆ k ˆ Σ. 1 This 2-form ω4 is relatively cohomologous to the concatenation ωX ˝v ωX . Indeed, their 1 difference is dpf ^ λq for a compactly supported smooth function f :Ω ˝v Ω Ñ R. Thus we 1 can use ω4 to compute the cobordism map of X ˝v X. 1 As the underlying 4-manifold of X ˝v X is completely stretched along k ˆ Σ in Figure 6, 3 we need the following result concerning the monopole Floer homology of the 3-torus T : 2 3 2 1 2 3 Lemma 3.8. Let T be the 2-torus and T “ T ˆ S . Let d P H pT , Zq be the Poincar´e 1 2 3 dual of tptu ˆ S and set rωs “ iδ ¨ d P H pT , iRq for some δ P R. Following the notations 3 from [KM07, Section 30], we write HM ˚pT , s, c; Rωq for the monopole Floer homology of 3 c T associated to the period class c “ ´2πirωs “ 2πδ ¨ d and the spin structure s, which is defined using the Seiberg-Witten equations (3.2) for some imaginary valued 2-form ω in the class rωs. If in addition δ ‰ 0 and |δ| ă 2π, then this group can be computed as follows: R if c psq “ 0, HM p 3, s, c; R q “ 1 ˚ T ω t0u otherwise. " 2 3 Proof of Lemma 3.8. Pick a flat metric of T and equip T with the product metric. Take ω to be a multiple of the volume form dvolT2 . In this case, the 3-dimensional Seiberg-Witten equations (3.2) can be solved explicitly. If |δ| ă 2π and c1psq ‰ 0, (3.2) has no solutions at all. If δ ‰ 0 and c1psq “ 0, (3.2) has a unique solution γ˚, which is irreducible; see [Tau01, 22 DONGHAO WANG

Lemma 3.1]. Since we have worked with a non-balanced perturbation, the perturbed Chern- Simons-Dirac functional Lω is not full gauge invariant. By [Tau01, Proposition 4.4], the moduli space of down-ward gradient flowlines of Lω connecting γ˚ to itself is not empty, although the formal dimension predicted by the index theory is always zero. This issue can be circumvented using an admissible perturbation q of Lω supported away from γ˚, as explained in [KM07, Section 15]. Moreover, q can be made small so that γ˚ is still the unique critical point of the perturbed functional, giving rise to the unique generator 3 of HM ˚pT , s, c; Rωq.  To complete the proof of Theorem 3.6, we need another result regarding the monopole 2 2 c invariants of M :“ D ˆT , which we recall below. Let sstd be the standard spin structure 3 c on T with c1psstdq “ 0. A relative spin structure sM on M is a pair psM , ϕq where sM is a c spin structure and ϕ : sM |BM Ñ sstd is a fixed isomorphism. In particular, one may define 2 2 2 1 its relative Chern class c1psM q P H pM, BM; Zq – Hp pD ,S ; Zq. We shall work with a flat 2 2 metric of T and make D into a surface with a cylindrical end: p s Ñ ¨ ¨ ¨

Figure 8. The disk with a cylindrical end.

2 2 1 Let d P H pM, Zq be the dual of tptu ˆ pD ,S q and rωM s :“ iδ ¨ d for some δ P R. The monopole invariant of M is defined as a generating function ωM ´E psM q mpM, rωM sq :“ mpM, sM , rωM sq ¨ q top P R s ÿM p with m M, s , ω and p p M r M sq P F2 p

ωM 1 (3.5) E psM q :“ pFAt ´ 2ωM q ^ pFAt ´ 2ωM q “ ´2πδ ¨ pc1psq Y dqrM, BMs. topp 4 0 0 żM Here A is a reference spinc connection on M, s . The coefficient m M, s , ω is 0 p p M q p p M r M sq defined by counting finite energy solutions to the Seiberg-Witten equations (3.1) with ωM “ 2 c iδ ¨ dvolT2 { VolpT q for the relative spin manifoldppM, sM q. In practice, one hasp to perturb ωM by a compactly supported closed 2-form (see [Tau01]) or add a tame perturbation to Lω as in the proof of Lemma 3.8, to ensure that the modulip space is transversely cut out. 3 More invariantly, one should regard mpM, rωM sq as an element in HM ˚pT , sstd, c; Rωq. 3 Lemma 3.9. Using the canonical identification HM ˚pT , sstd, c; Rωq – R in the proof of Lemma 3.8, the monopole invariant mpM, rωM sq can be computed as pt ´ t´1q´1 “ t´1 ` t´3 ` t´5 ` ¨ ¨ ¨ P R with t “ q2π|δ|. Proof of Lemma 3.9. Although we have used non-exact perturbations, this computation is not different from the one in [KM07, Section 38.2], in which case exact perturbations and a non-trivial local coefficient system are used. This formula can be found in [KM07, P.719].  MONOPOLES AND LANDAU-GINZBURG MODELS III: A GLUING THEOREM 23

Back to the proof of Theorem 3.6. Once the neck is completely stretched along k ˆ Σ 2 in Figure 6, we glue two copies of D ˆ Σ to obtain the product cobordism from Y1 Y2 to itself, which induces the identity map on monopole Floer homology groups. Since our n piq š surface Σ “ i“1 Σ is disconnected, we have p p n n š 2 2 piq ´1 ´1 mpD ˆ Σ, rµsq “ mpD ˆ Σ , rµisq “ pti ´ ti q “ η, i“1 i“1 ź ź |2πixµ,rΣpiqsy| with µi :“ µ|Σpiq and ti “ q . As a result, β α η2 HM 1 Id . ˝ “ ˚pX ˝v Xq “ HM ˚pY1qbHM ˚pY2q The computation for α ˝ β is similar and is omitted here. This completes the proof of Theorem 3.6.  c c Proof of Theorem 3.7. Fix relative spin structures si P SpinRpYiq for i “ 1, 2. The state- ment about relative spinc gradings is obvious. Indeed, the 4-manifold cobordism in Figure c 3 can be upgraded into a relative spin cobordism: p pX, sX q : pY1, s1q pY2, s2q Ñ pY1 ˝h Y2, sq pRs ˆ Σ, sstdq only if s “ s1 ˝h s2. To deal withž the canonical grading,ž we make use of an equivalent π p p p p p p p p definition of Ξ pYi, siq following [Wan20b, (18.2)]. Instead of oriented relative 2-plane fields, we investigatep p thep space of unit-length relative spinors on Y , denoted by ΞpY, sq. A spinor Ψ P ΓpY,Sq isp calledp relative, if Ψ “ Ψ˚{|Ψ˚| on the cylindrical end r0, 8qs ˆ Σ; see [Wan20b, Definition 18.2]. Finally, we have p p p π 2 Ξ pY, sq :“ π0pΞpY, sqq{H pY, BY ; Zq 2 where H pY, BY ; Zq “ π0pGpY qq is the component group of the gauge group GpY q. p p Let ai P CpYi, siq be a critical point of the perturbed Chern-Simons-Dirac functional on Yi for i “ 1, 2. In fact, beforep passing to the quotient configuration space, we canp assign an element p p p grpaiq P π0pΞpY, sqq π whose image in Ξ pY, sq is the chain level grading of rais. Let Ψi P ΓpYi,Siq be a unit-length relative spinor representing grpaiq. p p Since the cobordismp map HM ˚pXq is defined by counting 0-dimensionalp moduli spaces on the 4-manifold X , the chain level map exists between

pa1, a2q and pa3, a˚q only if an index condition holds, where a˚ “ pB˚, Ψ˚q is the canonical solution on Rs ˆ Σ and a3 is a critical point on Y1 ˝h Y2. The relative spinor representing the grading of a˚ is Ψ˚{|Ψ˚|, by the Normalization Axiom [Wan20b, Section 18]. Let Ψ3 be the one for a3. By the Index Axiom from [Wan20b, Section 18], this index condition can be stated in c terms of the quadruple pΨ1, Ψ2;Ψ3, Ψ˚{|Ψ˚|q. For a fixed relative spin cobordism sX , we construct a non-vanishing spinor ΦX on X zX as follows:

‚ ΦX ” Ψi on p´8, 1st ˆ Yi for i “ 1, 2; p ‚ ΦX ” Ψ3 on r1, 8qt ˆ pY1 ˝h Y2q; p 24 DONGHAO WANG

‚ ΦX ” Ψ˚{|Ψ˚| on r1, 8qt ˆ Rs ˆ Σ and ‚ ΦX ” Ψ˚{|Ψ˚| also on Rt ˆ r1, 8qs ˆ Σ and Rt ˆ p´8, ´1ss ˆ Σ. The index condition is then equivalent to saying that the relative Euler class of ΦX vanishes ` (3.6) epSX ;ΦX qrX, BXs “ 0, which determines the class rΨ3s P π0pΞpY1 ˝h Y2, s1 ˝h s2qq in terms of rΨ1s and rΨ2s. c There is an obvious spin cobordism sX on X (see Figure 3): we pick the product relative c spin structures on r´1, 1st ˆ Y1,´ and r´1, 1st ˆ pY2,´ respectively,p and choose any relative c spin structure on Ω ˆ Σ. In this case,p the characteristic condition (3.6) holds trivially, if we take Ψ3 to be the concatenation of Ψ1 and Ψ2. c 1 In general, any other relative spin cobordism sX differs from sX by taking the tensor 2 product with a relative line bundle L P H pX, BX; Zq, an action that leaves the relative Euler number unaffected. Thus the image of rΨ3s pin p 2 π0pΞpY1 ˝h Y2, s1 ˝h s2qq{H pY1 ˝h Y2; Zq 1 is independent of sX . This completes the proof of Theorem 3.7.  p p 3.7. Proof of (G2). The proof of Theorem 2.7 will dominate the rest of Section 3. In this p subsection, we focus on (G2). For any 1-cells Y12 P TpΣ1, Σ2q and Y23 P TpΣ2, Σ3q, the isomorphism α : HM ˚pY12q bR HM ˚pY23q Ñ HM ˚pY12 ˝h Y23q, is constructed in the same way as in the special case discussed in Subsection 3.6. It remains 1 1 to verify that α is a functor. Let X12 : Y12 Ñ Y12 and X23 : Y23 Ñ Y23 be 2-cell morphisms in TpΣ1, Σ2q and TpΣ2, Σ3q respectively. We have to show that

HM ˚pX12 ˝h X23q ˝ α “ α ˝ pHM ˚pX12q b HM ˚pX23qq 1 1 as maps from HM ˚pY12q bR HM ˚pY23q to HM ˚pY12 ˝h Y23q. Indeed, both of them agree with the map induced by the 4-manifold cobordism in Figure 9. This completes the proof of (G2). 

Y12 Y23

p´8, ´1st ˆ Y12 Ñ Ð p´8, 1st ˆ Y23 p X12 X23 p

1 1 r1, 8qt ˆ Y12 Ñ Ð r1, 8qt ˆ Y23

1 1 Y12 ˝h Y23 Figure 9 MONOPOLES AND LANDAU-GINZBURG MODELS III: A GLUING THEOREM 25

3.8. Proof of (G3). For any triple pY12, Y23, Y34q P TpΣ1, Σ2q ˆ TpΣ2, Σ3q ˆ TpΣ3, Σ4q, the composition α ˝ pα b Idq in the diagram (2.1) is induced from the cobordism in Figure 10 below, with the red dot line indicating a copy of the completion Y12 ˝h Y23.

{

Y12 ñ ð Y23

Y12 ˝h Y23

{

pY12 ˝h Y23q ˝h Y34 ð

Y34 Figure 10

On the other hand, the map α ˝ pId bαq is identical to the one induced from Figure 11, which can be obtained from Figure 10 by continuously varying the metric and the closed 2-form, which implies that the diagram (2.1) is commutative and completes the proof of (G3).  Y23

Y12 ˝h Y23

{

Y12 ñ

Y12 ˝h pY23 ˝h Y34q Y34 Figure 11

3.9. Proof of (G4). We have to show that the gluing map

α : HM ˚pY12q b HM ˚peΣ2 q Ñ HM ˚pY12 ˝h eΣ2 q “ HM ˚pY12q agrees with Id bRιΣ2 . Equivalently, we prove that

Id bι´1 p1q Σ2 α α˜ : HM ˚pY12q ÝÝÝÝÝÝÑ HM ˚pY12q b HM ˚peΣ2 q ÝÑ HM ˚pY12q. 26 DONGHAO WANG is the identity map. We start with a few reductions: Step 1.α ˜ is an isomorphism. This is by Theorem 3.6.

Step 2. It suffices to verify the special case when Σ1 “ Σ2 and Y12 “ eΣ2 . Indeed, if the statement holds for this special case, consider the diagram:

αpId bαq

Id bι´1 p1qbι´1 p1q Σ2 Σ2 HM ˚pY12q HM ˚pY12q b HM ˚peΣ2 q b HM ˚peΣ2 q HM ˚pY12q.

αpαbIdq

2 Applying Theorem 2.7 (G3) to the triple pY12, eΣ2 , eΣ2 q, we obtain thatα ˜ “ α˜; soα ˜ “ Id.

Step 3. In the case when Σ1 “ Σ2 and Y12 “ eΣ2 , the group HM ˚peΣ2 q has rank 1; soα ˜ : HM ˚peΣ2 q Ñ HM ˚peΣ2 q is a multiplication map. Let X be the cobordism inducing the gluing map α. In this special case, the opposite cobordism X1 of X is identical to X. Theorem 3.6 then implies thatα ˜2 “ Id; soα ˜ “ Id. This completes the proof of Theorem 2.7 (G4). 

4. Generalized Cobordism Maps 2-cell morphisms in the category TpH, Σq are given by strict cobordisms: the induced cobordisms between boundaries are necessarily standard products, as required by Property (Q2). The primary goal of this section is to remove this constraint and define the generalized cobordism maps. For any 1-cells Y1 P TpH, Σ1q and Y2 P TpH, Σ2q, a general cobordism pX,W q : pY1, Σ1q Ñ pY2, Σ2q is a 4-manifold with corners. To better package the data of closed 2-forms, however, we shall adopt a different point of view and introduce a new category T1.

Definition 4.1. Consider the category T1 with objects

Ob T1 “ Ob TpH, Σq. Σ ž For any Y1 P TpH, Σ1q and Y2 P TpH, Σ2q, a morphism in T1pY1, Y2q is a pair pX12, W12q where

W12 P TpΣ1, Σ2q, X12 P HomTpH,Σ2qpY1 ˝h W12, Y2q.

For morphisms pX12, W12q P T1pY1, Y2q and pX23, W23q P T1pY2, Y3q, their composition pX13, W13q P T1pY1, Y3q is defined as

W13 “ W12 ˝h W23, X13 “ pX12 ˝h IdW23 q ˝ X23, MONOPOLES AND LANDAU-GINZBURG MODELS III: A GLUING THEOREM 27

The associativity can be easily checked using the digram that represents X13:

Y1 W12 W23 H Σ1 Σ2 Σ3

X12 IdW23 Y2 W23 (4.1) H Σ2 Σ3 ♦

X23 Y3 H Σ3.

Corollary 4.2. We define a fake functor HM ˚ : T1 Ñ Rp‹q as follows. For any object Yi P TpH, Σiq, we assign its monopole Floer homology group HM ˚pYiq. For any morphism pX12, W12q : Y1 Ñ Y2, we assign the map HM ˚pX12, W12q :“ HM ˚pX12q ˝ α :

α HM ˚pX12q HM ˚pY1q b HM ˚pW12q ÝÑ HM ˚pY1 ˝h W12q ÝÝÝÝÝÝÝÑ HM ˚pY2q.

This assignment HM ˚ fails to be a functor, since the ordinary composition law is violated. The replacement is a commutative diagram relating HM ˚pX12, W12q, HM ˚pX23, W23q and the map of their composition pX13, W13q “ pX12, W12q ˝ pX23, W23q: Id bα HM ˚pY1q b HM ˚pW12q b HM ˚pW23q HM ˚pY1q b HM ˚pW13q

(4.2) HM pX12,W12qbId HM ˚pX13,W13q HM ˚pX23,W23q HM ˚pY2q b HM ˚pW23q HM ˚pY3q. Proof. The commutativity of (4.2) is obtained by applying the monopole Floer 2-functor HM ˚ in Theorem 2.7 to the digram (4.1). 

In order to obtain a genuine functor, we have to enlarge the category T1 to incorporate an element of HM ˚pW12q for each morphism pX12, W12q.

Definition 4.3. The category T2 has the set of same objects as T1. For any objects Y1 P TpH, Σ1q and Y2 P TpH, Σ2q, a morphism in T2pY1, Y2q is now a triple pX12, W12, a12q where pX12, W12q P T1pY1, Y2q and a12 P HM ˚pW12q. For pX12, W12, a12q P T2pY1, Y2q and pX23, W23, a23q P T2pY2, Y3q, their composition pX13, W13, a13q P T2pY1, Y3q is defined as

W13 “ W12 ˝h W23, X13 “ pX12 ˝h IdW23 q ˝ X23, a23 “ αpa12 b a23q. ♦

Corollary 4.4. We define a functor HM ˚ : T2 Ñ Rp‹q as follows. For any object Yi P TpH, Σiq, we assign its monopole Floer homology group HM ˚pYiq. For any morphism pX12, W12, a12q : Y1 Ñ Y2, define the map HM ˚pX12, W12, a12q to be the composition

αp¨,a12q HM ˚pX12q HM ˚pY1q ÝÝÝÝÝÑ HM ˚pY1 ˝h W12q ÝÝÝÝÝÝÝÑ HM ˚pY2q.

Then HM ˚ : T2 Ñ Rp‹q is a functor in the classical sense.

Proof. The functoriality of HM ˚ follows from the commutative digram (4.2).  28 DONGHAO WANG

5. Invariance of Boundary Metrics For any 1-cell Y P TpH, Σq, we defined its monopole Floer homology group in [Wan20b]; but its invariance is only proved in a weak sense: Theorem 5.1 ([Wan20b, Remark 1.8 & Corollary 19.10]). For any T -surface Σ and any 1-cell Y P TpH, Σq, the monopole Floer functor from Theorem 3.2 implies the invariance of HM ˚pY, sq when ‚ we change the cylindrical metric gY in (P3); 1 ‚ we replacep ω by ω ` dY b for a compactly supported 1-form b P Ωc pY, iRq in (P4); ‚ we apply an isotopy to the diffeomorphism ψ : BY Ñ Σ. p In this section, we strengthen this invariance result by showing that HM ˚pYq is in- 2 dependent of the flat metric of Σ and depends only on the class rωs P H pY ; iRq and 1 r˚Σλs P H pΣ, iRq. In particular, the Floer homology HM ˚pYq is a topological invariant for the triple pY, rωs, r˚Σλsq. The main result is as follows:

Theorem 5.2. Suppose that Σi “ pΣ, gi, λi, µiq, i “ 1, 2 have the same underlying oriented surface Σ and 1 2 r˚1λ1s “ r˚2λ2s P H pΣ, iRq, rµ1s “ rµ2s P H pΣ, iRq. If 1-cells Yi P TpH, Σiq, i “ 1, 2 have the same underlying 3-manifold Y1 – Y2 (this diffeo- morphism is compatible with the identification maps ψi : BYi Ñ Σ, i “ 1, 2) and 2 rω1s “ rω2s P H pY, iRq, c c then HM ˚pY1, sq – HM ˚pY2, sq for any relative spin structure s P SpinRpY q. The proof of Theorem 5.2 is based on the Gluing Theorem 2.7 and a special property concerning thep product manifoldp r´3, 3ss ˆ Σ: p

Lemma 5.3. Under the assumptions of Theorem 5.2, consider the 1-cell Y0 P TpΣ1, Σ2q with Y0 “ r´3, 3ss ˆ Σ and ψ “ Id : BY0 Ñ p´Σq Y Σ. Then for any cylindrical metric of 2 Y0 and any compatible 2-form ω0 P Ω pY0, iRq, we have R if s “ s , HM p , sq – std ˚ Y0 t0u otherwise. " p p Proof of Theorem 5.2. By applying thep gluing functor from Theorem 2.7 α : TpH, Σ1q ˆ TpΣ1, Σ2q Ñ TpH, Σ2q. to the pair pY1, Y0q, we deduce from Theorem 3.7 and Lemma 5.3 that

HM ˚pY1, sq – HM ˚pY1 ˝h Y0, sq. c for any s P SpinRpY q. Now Y1 ˝h Y0 and Y2 are 1-cells in the same strict cobordism 2 category TpH, Σ2q. The difference ω1p˝h ω0 ´ ω2 determinesp a class β P H pY, BY ; iRq. Our goalp here is to choose ω0 properly so that β “ 0. This can be always done, because 1 rω2s “ rω1s “ rω1 ˝h ω0s; so β lies in the image of H pBY ; iRq: 1 2 2 ¨ ¨ ¨ Ñ H pBY ; iRq Ñ H pY, BY ; iRq Ñ H pY ; iRq Ñ ¨ ¨ ¨ β ÞÑ 0. MONOPOLES AND LANDAU-GINZBURG MODELS III: A GLUING THEOREM 29

When β “ 0, we can then identify HM ˚pY1 ˝h Y0, sq with HM ˚pY2, sq using Theorem 5.1. This completes the proof of Theorem 5.2.  3 The proof of Lemma 5.3 relies on a computationp for the 3-torus pT , which generalizes Lemma 3.8. 2 3 Lemma 5.4. For any closed 2-form ω P Ω pT , iRq, suppose that the period class c :“ c 3 ´2πirωs is neither negative monotone nor balanced for any spin structures on T , then R if c psq “ 0, HM p 3, s, c; R q “ 1 ˚ T ω t0u otherwise. " Proof of Lemma 5.4. The meaning of Lemma 5.4 will become more transparent when we review the theory of closed 3-manifolds in Section 6. When c1psq ‰ 0, the statement can be verified as in the proof of Lemma 3.8, by working with a product metric and a harmonic 2-form ω. When c1psq “ 0, the statement follows from [Tau01, Lemma 3.1]. Alternatively, we may apply Proposition 6.2 to reduce the problem to the case of exact perturbations.  Proof of Lemma 5.3. With loss of generality, we assume that Σ is connected. Since the 1 roles of Σ1 and Σ2 are symmetric, consider a similar 1-cell Y0 P TpΣ2, Σ1q. Now regard 1 Y0 and Y0 as 1-cells in TpH, Σ2 Y p´Σ1qq and TpΣ2 Y p´Σ1q, Hq respectively. We apply Theorem 3.6 to obtain that 1 3 (5.1) HM ˚pY0q bR HM ˚pY0q – HM ˚pT , s, c; Rωq 1 3 1 where ω “ ω0 ˝h ω0 is a closed 2-form on the 3-torus T “ S ˆ Σ. The condition of Lemma 5.4 can be verified, since |xω, rΣsy| ă 2π and ‰ 0 by properties (T3) and (T4). We conclude from (5.1) and Lemma 5.4 that 1 rankR HM ˚pY0q “ rankR HM ˚pY0q “ 1. c In particular, the group HM ˚pY0, sq vanishes except for one particular relative spin structure. By Theorem 3.4, the Euler characteristic χpHM ˚pY0, sqq is independent of the metric and the 2-form ω0. We conclude fromp the computation of HM ˚peΣ1 , sstdq that χpHM ˚pY0, sstdqq “p1. p Thus HM ˚pY0, sstdq ‰ t0u. This completes the proof of Lemma 5.3.  p p 30 DONGHAO WANG

Part 3. Monopoles and Thurston Norms n Let F “ i“1 Fi be a compact oriented surface with Fi connected. Recall that the norm of F is defined to be Ť xpF q :“ ´ mintχpFiq, 0u. i ÿ For any 3-manifold Y with toroidal boundary, Thurston [Thu86] introduced a semi-norm xp¨q on H2pY, BY ; Rq such that for any integral class κ P H2pY, BY ; Zq, we have xpκq :“ mintxpF q : pF, BF q Ă pY, BY q properly embedded and rF s “ κu.

In this part, we show that the monopole Floer homology HM ˚pYq defined in [Wan20b] detects the Thurston norm and fiberness of Y , if the underlying 3-manifold Y is connected and irreducible, generalizing the previous results for closed 3-manifolds [KM07, Ni08, Ni09a]. The same detection theorems for link Floer homology have been obtained in [OS08, Ni09b, Juh08, GL19] and [Ni07]. The main results are Theorem 7.2 and 7.3. For the proof of Theorem 7.2, we use the Gluing Theorem 2.7 to reduce the problem to the double of Y , and apply the non-vanishing result [KM07, Corollary 41.4.3] for closed 3-manifold. However, we have to adapt this corollary first to the case of non-exact pertur- bations, which is done in Section 6. The proof of Theorem 7.3 is accomplished in Section 9, which exploits the relation of HM ˚pYq with sutured Floer homology, as discussed in Section 8. Any 3-manifold with toroidal boundary is a sutured manifold in the sense of Gabai [Gab83], but it is not an example of balanced sutured manifolds in the sense of [Juh06, Definition 2.2]. One may think of our construction as a natural extension of the sutured Floer homology. Although the author was not able to prove a general sutured manifold decomposition theorem, a preliminary result, Theorem 8.1, will be supplied in Section 8 to justify this heuristic.

6. Closed 3-Manifolds Revisited Throughout this section, we will take Y to be a closed connected oriented 3-manifold. In [KM07], Kronheimer and Mrowka introduced three flavors of monopole Floer homology groups for any spinc structure s on Y , which fit into a long exact sequence:

i˚ j˚ p˚ i˚ (6.1) ¨ ¨ ¨ ÝÑ HM ˚pY, s;Γq ÝÑ HM ˚pY, s;Γq ÝÑ HM ˚pY, s;Γq ÝѨ ¨ ¨ Here Γ is any local coefficient system on the blown-up configuration space BσpY, sq. For } y instance, one may take Γ to be the trivial system with Z coefficient. For any real 1-cycle ξ, we can also define the local coefficient system Γξ as in [KM07, Section 3.7], whose fiber is always R. We write HM ˚pY, s;Γq :“ Im j˚ for the reduced Floer homology. The third group HM ˚pY, s;Γq in (6.1) is trivial if 2 ‚ c1psq P H pY ; Zq is non-torsion, or ‚ Γ “ Γξ and rξs ‰ 0 P H1pY ; Rq; see [KM07, Proposition 3.9.1].

In either case, the map j˚ is an isomorphism, so HM ˚pY, s;Γq – HM ˚pY, s;Γq. Monopole Floer homology detects the Thurston norm xp¨q on H2pY ; Rq. The next theo- rem expand on what lies behind this slogan: } MONOPOLES AND LANDAU-GINZBURG MODELS III: A GLUING THEOREM 31

Theorem 6.1 ([KM07, Corollary 40.1.2 & 41.4.3]). Let Y be a closed oriented 3-manifold with b1pY q ą 0 and κ P H2pY, Zq be any integral class. (1) pThe adjunction inequalitiesq For any local coefficient system Γ and any spinc structure s with |xc1psq, κy| ą }κ}T h, the monopole Floer homology group HM ˚pY, s;Γq is triv- ial. (2) If in addition Y is irreducible, then there is a spinc structure s on Y such that

xc1psq, κy “ }κ}T h and HM ˚pY, s;Γξq ‰ t0u for any 1-cycle ξ with rξs ‰ 0 P H1pY ; Rq.

In particular, when rξs ‰ 0 P H1pY, Rq, the subset 2 tc1psq : HM ˚pY, s;Γξq ‰ t0uu Ă H pY, Rq determines the Thurston norm on H2pY, Rq in the same way that the Newton polytope of the Alexander polynomial determines the Alexander norm; see [McM02]. However, Theorem 6.1 is stated only for the monopole Floer homology defined using exact perturbations. In [KM07, Section 30], these groups are extended for non-exact per- turbations:

i˚ j˚ p˚ i˚ (6.2) ¨ ¨ ¨ ÝÑ HM ˚pY, s, c;Γq ÝÑ HM ˚pY, s, c;Γq ÝÑ HM ˚pY, s, c;Γq ÝѨ ¨ ¨ 2 where c P H pY ; Rq is the period class and Γ is any c-complete local coefficient system in the sense of [KM07,} Definition 30.2.2].y These groups are defined using the Seiberg-Witten i 2 equations (3.2) on Y with the closed 2-form ω belonging to the class rωs “ 2π ¨c P H pY, iRq.

The purpose of this section is to understand the extent to which Theorem 6.1 generalizes to these groups defined using non-exact perturbations. With that said, the results of this section follow almost trivially from the general theory in [KM07]; no originality is claimed here. 6.1. Statements. Fix a spinc structure s on Y . Recall from [KM07, Definition 29.1.1] that 2 the non-exact perturbation associated to a closed 2-form ω P Ω pY, iRq is called monotone if 2 2 2π c1psq ` c “ 2π c1psq ¨ t for some t P R, where c :“ ´2πirωs is the period class of this perturbation. Furthermore, it is called balanced, positively or negatively monotone if t “ 0, t ą 0 or ă 0 respectively. We write HM ˚pY, s, c;Γq :“ Im j˚ Ă HM ˚pY, s, c;Γq for the reduced monopole Floer homology. The third group HM ˚pY, s, c;Γq in (6.2) is always 2 trivial unless the period class c “ cb :“ ´2π c1psqyis balanced. Thus HM ˚pY, s, c;Γq “ HM ˚pY, s, c;Γq if c ‰ cb. σ We focus on the local coefficient systems Rω on B pY, sq, whose fiber at every point σ 1 isy always the mod 2 Novikov ring R. The fundamental group of B pY, sq is H pY, Zq “ 1 fpzq π0pGpY qq. The monodromy of Rω is then defined by sending z P H pY, Zq to q with 2 (6.3) fpzq :“ xp2π c1psq ` cq Y z, rY sy P R. 32 DONGHAO WANG

For the adjunction inequality, it suffices to consider spinc structures with non-torsion 3 2 1 c1psq. The problem already occurs for the 3-torus T “ T ˆ S in Lemma 3.8. Let 2 3 1 d P H pT , Zq be the Poincar´edual of tptu ˆ S . Suppose ω “ iδ ¨ d and consider the c 3 spin structure s on T with c1psq “ 2 ¨ d. If δ ă ´2π, then this perturbation is negatively monotone. In this case, the moduli space of the Seiberg-Witten equations (3.2) is non- 2 empty, and is diffeomorphic to T ˆ tptu. As one may verify, for either Γ “ Z or Rω, 3 the monopole Floer homology group HM ˚pT , s, c;Γq is non-trivial, and so the adjunction inequality in Theorem 6.1 is violated in this case. However, by Proposition 6.2, negatively monotone perturbations are the only exceptions.

Proposition 6.2. Let Y be any closed oriented 3-manifold with b1pY q ě 2 and s be any c 2 non-torsion spin structure. Suppose that rωs P H pY, iRq is neither negatively monotone nor balanced with respect to s, then there is an isomorphism

HM ˚pY, s, c; Rωq – HM ˚pY, s; Rωq where the second group is defined using an exact perturbation with local system Rω. In par- ticular, the adjunction inequality from Theorem 6.1 holds also for the group HM ˚pY, s, c; Rωq: it is trivial, whenever xc1psq, κy ą }κ}T h for some integral class κ P H2pY ; Zq. The non-vanishing result is more robust: it suffices to rule out exact perturbations:

Proposition 6.3. Let Y be any irreducible closed oriented 3-manifold with b1pY q ě 1. If the 2 2 period class c “ ´2πirωs P H pY ; Rq is non-exact, then for any integral class κ P H pY ; Zq, there is a spinc structure s such that

xc1psq, κy “ }κ}T h and HM ˚pY, s, c; Rωq ‰ t0u. To ease our notation, we introduce the following shorthand: 2 Definition 6.4. For any closed 2-form ω P Ω pY ; iRq and any integral class κ P H2pY ; Zq, we write HM pY, rωsq for the direct sum

HM ˚pY, s, c; Rωq s à and HM pY, rωs|κq for the subgroup

HM ˚pY, s, c; Rωq. xc1psq,κy“xpκq à For any connected oriented subsurface F Ă Y , define similarly

HM pY, rωs|F q “ HM ˚pY, s, c; Rωq. ♦ xc1psq,κy“xpF q à 2 Remark 6.5. When rωs “ 0 P H pY ; iRq and c1psq ‰ 0, the local system Rω is not trivial. Nevertheless, by [KM07, P.288 (16.5)], we still have an isomorphism:

HM ˚pY, s, c; Rωq – HM ˚pY, s; Rq. The latter group is defined using the trivial system with coefficients R. Set

HM ˚pY |κq :“ HM ˚pY, s; Rq. xc1psq,κy“xpκq à MONOPOLES AND LANDAU-GINZBURG MODELS III: A GLUING THEOREM 33

Then HM ˚pY |κq – HM ˚pY, r0s|κq. We define similarly the group HM ˚pY |F q. ♦ 2 Corollary 6.6. If Y is irreducible with b1pY q ą 0 and rωs ‰ 0 P H pY ; iRq, then HM pY, rωs|κq ‰ 2 t0u for any integral class κ P H pY ; Zq. The proof of Proposition 6.2 and 6.3 will dominate the rest of this section.

6.2. Completions of Chain Complexes. The proof of Proposition 6.2 is achieved in two steps: the first step is to relate the group HM ˚pY, s, c; Rωq with the Floer homology 2 associated to a balanced class cb “ ´2π c1psq. For monotone perturbations, this is already done in [KM07, Theorem 31.1.1 & 31.5.1]. We shall describe a slightly different setup that allows generalization for any non-balanced perturbations. Consider the monopole Floer chain complexes of pY, s, cbq associated to the local system Rω:

C˚pY, s, cb; Rωq, C˚pY, s, cb; Rωq, C˚pY, s, cb; Rωq. As modules, they are free over the Novikov ring R, but none of them is finitely generated. q p Since the period class cb is balanced, we have to blow up the configuration space in order to define the Floer homology. Suppose an admissible perturbation of the Chern-Simons-Dirac functional Lω is chosen, then each reducible solution rαs of the perturbed 3-dimensional Seiberg-Witten equations will contribute infinitely many generators to each of C˚, C˚, C˚, corresponding to the eigenvectors of the Dirac operator at rαs. As noted in [KM07, Section 30.1], we can form a chain level completion using theq filtrationp of eigenvalues. Label the reducible solutions in the quotient configuration space BpY, sq as 1 p r rα s, ¨ ¨ ¨ , rα s and label the corresponding critical points in the blown-up space as rαi s with r r i P Z and 1 ď r ď p, so that rαi s corresponds to the eigenvalue λi of the perturbed Dirac operator at rαrs and r r ¨ ¨ ¨ ă λi ă λi`1 ă ¨ ¨ ¨ , r with λ0 being the first positive one. For any m ě 1, let

(6.4) C˚pY, s, cb; Rωqm Ă C˚pY, s, cb; Rωq, be the subgroup generated by rαrs with i ď ´m. This defines a filtration on C , called the p i p ˚ λ-filtration. We form the completion p C‚pY, s, cb; Rωq Ą C˚pY, s, cb; Rωq. The same construction applies also to the bar-version; so we obtain p p C‚pY, s, cb; Rωq Ą C˚pY, s, cb; Rωq.

On the other hand, C˚ carries an additional filtration arising from the base ring R, called the R-filtration. One may formally write p C˚pY, s, cb; Rωq “ Ren, ně0 à by identifying the fibers of Rω atp different critical points using a collection of paths in the blown-down space BpY, sq. We may start with the group ring F2rRs and obtain R by taking 34 DONGHAO WANG

´ the completion in the negative direction. This filtration on R “ F2rRs then induces the R-filtration on the free module C˚pY, s, cb; Rωq; let C pY, s, c ; R q Ă C pY, s, c ; R q, ˛ p b ω ‚ b ω be the resulting completion. Any element a ¨ e P C pY, s, c ; R q may have infinitely p ně0p n n ˛ b ω many non-zero coefficients an P R, but under the topology of R, we must have ř p (6.5) lim an “ 0. nÑ8 The bar-version analogue is constructed in a slightly different way. First, take the R- completion of its m-th filtered subgroup C˚pY, s, cb; Rωqm Ă C˚pY, s, cb; Rωq for each m P Z, denoted by

C˛pY, s, cb; Rωqm. Next, we form the union:

C˛pY, s, cb; Rωq :“ C˛pY, s, cb; Rωqm Ă C‚pY, s, cb; Rωq. mP ďZ An element in C˛pY, s, cb; Rωq has only finitely many non-zero coefficients for critical points r with λi ą 0, while a infinite sum may occur in the negative direction satisfying the conver- gence condition (6.5). The upshot is that the homology groups of

C˚pY, s, cb; Rωq, C˛pY, s, cb; Rωq, C˛pY, s, cb; Rωq form a long exact sequence (by the proof of [KM07, Proposition 22.2.1]), which fits into a diagram below: p p

i˚ j˚ p˚ i˚ ¨ ¨ ¨ HM ˚pY, s, cb; Rωq HM ˚pY, s, cb; Rωq HM ˚pY, s, cb; Rωq ¨ ¨ ¨

x˚ } y (6.6) i˛ j˛ p˛ i˛ ¨ ¨ ¨ HM ˚pY, s, cb; Rωq HM ˛pY, s, cb; Rωq HM ˛pY, s, cb; Rωq ¨ ¨ ¨

y˛ } y i‚ j‚ p‚ i‚ ¨ ¨ ¨ HM ˚pY, s, cb; Rωq HM ‚pY, s, cb; Rωq HM ‚pY, s, cb; Rωq ¨ ¨ ¨

The next proposition} says that HM ˚ypY, s, c; Rωq can be computed in terms of the group 2 HM ˛ associated the balanced period class cb :“ ´2π c1psq. Proposition 6.7. For any pY, sq and any non-balanced perturbation with period class c “ y ´2πirωs, we have an isomorphism

HM ˚pY, s, c; Rωq – HM ˛pY, s, cb; Rωq, Proof. This result follows from the proof of [KM07, Theorem 31.1.1]. See [KM07, Section y y 31.2]. Since we have used a c-complete local coefficient system Rω here, there is no need top to estimate the topological energy E1 in [KM07, (31.5) P.611]. The argument is even simpler.  MONOPOLES AND LANDAU-GINZBURG MODELS III: A GLUING THEOREM 35

The second step in the proof of Proposition 6.2 is to relate HM ˛pY, s, cb; Rωq with the bullet-version HM ‚pY, s, cb; Rωq using the vertical map y˛ in (6.6). In fact, more is true when the class rωs is not monotone with respect to c1psq: y y Proposition 6.8. Let Y be any closed oriented 3-manifold with b1pY q ě 2 and s be any non-torsion spinc structure. Suppose that the period class c “ ´2πirωs is not monotone, then the vertical maps x˚ and y˛ in digram (6.6) are isomorphisms

x˚ y˛ HM ˚pY, s, cb; Rωq ÝÑ HM ˛pY, s, cb; Rωq ÝÑ HM ‚pY, s, cb; Rωq. – – Now we readyy to prove Propositiony 6.2. y Proof of Proposition 6.2. When c is not monotone, we can combine Proposition 6.7 & 6.8 with the following isomorphism

(6.7) HM ‚pY, s, cb; Rωq – HM ˚pY, s; Rωq from [KM07, Theorem 31.1.1] to conclude. The case when c is positively monotone is y already addressed in [KM07, Theorem 31.1.2].  Proof of Proposition 6.8. Since the left vertical maps in digram (6.6) are identity maps, the statement then follows from the fact that for any ˝ P t˚, ˛, ‚u,

HM ˝pY, s, cb; Rωq “ t0u. The case when ˝ “ ‚ is addressed already in [KM07, Theorem 31.1.1]; in fact, the group HM ‚pY, s, cb;Γq always vanishes, no matter which local system Γ we use. In general, for any ˝ P t˚, ˛, ‚u, the group HM ˝pY, s, cb; Rωq is an instance of coupled Morse homology (see [KM07, Section 34]) associated to the Picard torus of Y

b 1 1 T :“ H pY ; Rq{H pY ; Zq, with b :“ b1pY q ě 2. In particular, we may use [KM07, Theorem 35.1.6] to compute HM ˝pY, s, cb; Rωq using a standard chain complex, which we recall below. Consider the b ´1 local coefficient system Γω on T with fiber RrT,T s and with monodromy given by b 1 ´1 ˆ (6.8) π1pT q – H pY, Zq Ñ RrT,T s z ÞÑ qfpzqT gpzq

1 where fpzq is as in (6.3) and gpzq “ x 2 c1psq Y z, rY sy. Then the group HM ˚pY, s, cb; Rωq is isomorphic to the homology of pC, B“B1 ` B3q, where

N b ´1 C “ CpT , h;Γωq :“ RrT,T sxn ně0 à b is the Morse complex of T with local coefficients Γω, defined using a suitable Morse function b h : T Ñ R and B1 is the Morse differential. As a result, C is a finite-rank free module over ´1 the ring RrT,T s, generated by critical points txnu of h. The Morse index then induces 36 DONGHAO WANG an additional grading on C such that deg Bi “ ´i for i “ 1, 3. The m-th λ-filtered subgroup of C is given by N ´m ´1 Cm :“ T RrT sxn, m P Z. ně0 à The upshot is that te differential B is RrT,T ´1s-linear. The bar-version chain complex ¯ pC˚pY, s, cb; Rωq, Bq ¯ which defines HM ˚pY, s, c; Rωq admits a very similar structure to pC, Bq, but B is only linear in R and it may have higher components with respect to the Morse grading: B“¯ B¯1 ` B¯3 ` ¯ ¯ B5 ` ¨ ¨ ¨ . By [KM07, Propositionp 34.4.1 & 33.3.8, Theorem 35.1.6], pC˚pY, s, cb; Rωq, Bq is homotopic to such a standard complex pC, Bq, preserving the topology induced by tpC˚qmu and tCmu respectively. To compute the homology of pC, Bq, we exploit the spectral sequence induced from the Morse grading, which abuts to HpC, Bq and whose E1-page is HpC, B1q. Since the period class c “ ´2πirωs is not monotone, one verifies immediately that HpC, B1q “ t0u. Indeed, b 1 b 1 b 1 we may write T “ S ˆ T ´ such that a integral multiple of tptu ˆ T ´ is dual to the 1 b b b´1 map g P H pT , Zq “ HompH1pT , Zq, Zq. One may further decompose T into a product of circles and let h be the sum of Morse functions on circles. It follows that the homology b´1 1 of T with local coefficients Γω is trivial, since the holonomy of Γω along some S -factor is non-trivial and lies in R. Finally, we deduce that HpC, B1q “ t0u using the K¨unneth formula. This finishes the proof when ˝ “ ˚. When ˝ “ ˛ or ‚, it suffices to replace the base ring RrT,T ´1s in (6.8) by ´1 ´1 RrT,T s˛ or RrT,T ss ´1 ´1 respectively. The ring RrT,T s˛ is obtained by taking the completion of RrT s with ´1 ´1 respect to the topology of R and then inverting T . Any element in RrT,T s˛ is then of 8 ´n the form něm anT for some m P Z and an P R such that limnÑ8 an “ 0 in R; so ´1 ´1 ´1 ř RrT,T s Ă RrT,T s˛ Ă RrT,T ss. One may now apply the universal coefficient theorem to conclude.  6.3. Proof of Proposition 6.3. To deal with the non-vanishing result, we first look at a non-torsion spinc structure s on Y . If the period class c is non-balanced with respect to s, then the map j‚ “ y˛ ˝ j˛ in diagram (6.6) is an isomorphism, since the group HM ‚pY, s, cb; Rωq vanishes by [KM07, Theorem 31.1.1]. This shows that the vertical map

y˛ : HM ˛pY, s, cb; Rωq Ñ HM ‚pY, s, cb; Rωq. is always surjective. By (6.7) and Proposition 6.7, we conclude that y y (6.9) rankR HM ˚pY, s, c; Rωq ě rankR HM ˚pY, s; Rωq. ˚ If the period class c is balanced with respect to s, then HM ˚pY, s, c; Rωq :“ Im j in the digram (6.6). We apply the same argument to y˛ ˝ x˚; so the inequality (6.9) still holds. MONOPOLES AND LANDAU-GINZBURG MODELS III: A GLUING THEOREM 37

When c1psq is torsion and c is non-exact, the inequality (6.9) follows from [KM07, The- orem 31.1.3]. It remains to verify that for the special spinc structure s given by Theorem 6.1 p2q, we have HM ˚pY, s; Rωq ‰ t0u. Combined with (6.9), this will complete the proof of Proposition 6.3. The rest of the argument is completely formal and is inspired by [KM07, Section 32.3]. The group HM ˚pY, s; Rωq can be understood using a smaller local coefficient system Πω, whose fiber at any point in BpY, sq is the group ring R :“ ZrZs and whose monodromy is ˆ again (6.3) (note that the image of f lies in a Z-subgroup in R). Let pC˚, Bq be the resulting chain complex defined over R. There are a few other base rings that we will look at:

gα p R “ ZrZs RrZs R

F2rZs K R. where K is the rational field of F2rZs and for any α P R, gα is the ring homomorphism α sending the generator t P Z to e P R. Note that both F2rZs and RrZs are principal ideal ˆ domains. Moreover, K Ñ R is a field extension. The goal is to show that HpC˚ bR F2rZs, Bq ˆ contains at least one free summand, so that HM ˚pY, s; Rωq “ HpC˚ bR R, Bq ‰ t0u. Consider a finite presentation of HpC˚q over the Noetherian ring R: p p n A k (6.10) R ÝÑpR Ñ HpC˚q Ñ 0, and let I Ă R be the ideal generated by all k ˆ k minors of A, called the Fitting ideal of p HpC˚q. By [Eis95, Corollary 20.4], I is independent of the resolution that we choose. ˆ Suppose on the contrary that the finitely generated F2rZs-module HpC˚ bR F2rZs, Bq is torsion.p Then we claim that I ‰ t0u; otherwise, the extension IF2 of I in F2rZs is trivial.

IF2 is also the Fitting ideal of HpC˚q bZ F2, which can be seen by taking thep tensor product of (6.10) with F2 (over Z). As a result, HpC˚q bZ F2 contains a free F2rZs-summand. The universal coefficient theorem [Hat02,p Theorem 3A.3] then provides an injection: p ˆ 0 Ñ HpC˚q bZ F2 Ñ HpC˚ bZ F2, Bq, which contradicts the assumption that we start with. p p ˆ Now consider the tensor product of (6.10) with R. The Fitting ideal IR of HpC˚q bZ R is then the extension of I in RrZs. Since I ‰ t0u, IR ‰ t0u. Using the universal coefficient theorem, we conclude that ˆ ˆ HpC˚q bZ R – HpC˚ bZ Rq is a torsion RrZs-module and is therefore annihilated by some polynomial uptq P RrZs. α On the one hand, one may pick α P R such that gαpuptqq “ upe q ‰ 0 P R; as a result, ˆ the group HpC˚ bgα R, Bq is trivial for this particular ring homomorphism gα : RrZs Ñ R. On the other hand, the local coefficient system Πω bgα R agrees with Γξ for some 1-cycle ξ with rξs ‰ 0p P H1pY ; Zq, since rωs is non-balanced. Theorem 6.1 (2) then asserts that HM ˚pY, s;Γξq is non-trivial. A contradiction.  38 DONGHAO WANG

6.4. Computation for Product 3-manifolds. For future reference, we recall a classical 1 result for the 3-manifold Σg ˆ S , where Σg is a closed surface of genus g ě 2. Lemma 6.9. Following the shorthands from Definition 6.4, for any closed 2-form ω P 2 1 Ω pΣg ˆ S , iRq with

(6.11) ixrωs, rΣgsy ă 2πpg ´ 1q, 1 we have HM pΣg ˆ S , rωs|Σgq – R. c Proof of Lemma 6.9. Let s be any spin structures contributing to the group HM pΣg ˆ 1 S , rωs|Σgq. Under the assumption (6.11), the period class c “ ´2πirωs is neither negatively monotone nor balanced with respect to s; indeed, 2 2 x2π c1psq ` c, rΣgsy “ 4π pg ´ 1q ´ 2πixrωs, rΣgsy ą 0. Using Proposition 6.2, we reduce the computation to the case when the perturbation is exact, but the coefficient system is still Rω. Now we use [KM10, Lemma 2.2] to conclude the proof.  7. Thurston Norm Detection From now on, we will always take Y to be an oriented 3-manifold with toroidal boundary.

Definition 7.1. For any T -surface Σ “ pΣ, gΣ, λ, µq and any 1-cell Y P TpH, Σq, consider the set of monopole classes: 2 MpYq :“ tc1psq : HM ˚pY, sq ‰ t0uu Ă H pY, BY ; Zq, and for any κ P H2pY, BY ; Zq, define p p ϕYpκq :“ max xc1psq, κy, κ P H2pY, BY ; Zq. c1psqPMpYq

Our convention here is that ϕY ”p ´8, if MppY q “ H. ♦ When Y is connected and irreducible, it is tempting to generalize Theorem 6.1 and relate p MpYq with the Thurston norm on H2pY, BY ; Rq. However, the author was not able to show that MpYq is symmetric about the origin; so only a weaker statement is obtained in this paper.

Theorem 7.2. For any T -surface Σ “ pΣ, gΣ, λ, µq, let Y P TpH, Σq be any 1-cell with Y connected and irreducible. Then the set of monopole classes MpYq is non-empty and determines the Thurston norm on H2pY, BY ; Rq in the following sense: 1 xpκq “ pϕ pκq ` ϕ p´κqq, @κ P H pY, BY ; q. 2 Y Y 2 Z

In general, we only have an inequality ϕYpκq ` ϕYp´κq ď 2xpκq. One may approach the more desirable statement

xpκq “ ϕYpκq “ ϕYp´κq. by either proving the symmetry of MpYq or the adjunction inequality:

ϕYpκq ď xpκq. MONOPOLES AND LANDAU-GINZBURG MODELS III: A GLUING THEOREM 39

But the author was unable to verify either of them directly. This Thurston norm detection result is accompanied with a fiberness detection result:

Theorem 7.3. For any T -surface Σ “ pΣ, gΣ, λ, µq, let Y P TpH, Σq be any 1-cell with Y connected and irreducible, and κ P H2pY, BY ; Zq be any integral class. Consider the subgroup

HM ˚pY|κq :“ HM ˚pY, sq. xc1psq,κy“ϕ pκq à Y If rankR HM ˚pY|κq “ rankR HM ˚pY| ´ κqp “ 1, then κ can bep represented by a Thurston norm minimizing surface F and Y fibers over S1 with F as fiber. The rest of this section is devoted to the proof of Theorem 7.2, while the proof of Theorem 7.3 is deferred to Section 9, after a digression into sutured monopole Floer homology in Section 8. Proof of Theorem 7.2. We focus on the case when Y is irreducible. The strategy is to apply the Gluing Theorem 2.7 and reduce the problem to the double of Y : Y˜ :“ Y p´Y q. Σ ď Since Y is irreducible, so is Y˜ . Then we can conclude using Proposition 6.2 and 6.3. For any 1-cell Y “ pY, gY , ω, ¨ ¨ ¨ q, consider its orientation reversal: 1 p´Yq “ p´Y, gY , ω, ¨ ¨ ¨ q P TpΣ , Hq. 1 The problem here is that the T -surface Σ “ pΣ, gΣ, ´λ, µq is different from Σ in that the sign of λ is reversed. We can not form the horizontal composition Y ˝h p´Yq. This problem is circumvented by changing the 2-form ω on p´Y q. Since HM ˚pY, sq is independent of the metric gY , we assume that it is cylindrical on p´2, 1ss ˆ Σ Ă Y . Pick a cut-off function χ1 : p´2, 1ss Ñ R such that χ1 ” 0 if s ď ´3{2 and ” 1 if s ě ´1. Writep ω “ ω ` χ1psqds ^ λ and set 1 1 1 1 Y :“ pY, gY , ω , ¨ ¨ ¨ q P TpH, Σ q with ω :“ ω ´ χ1psqds ^ λ. We shall work instead with the orientation reversal p´Y1q P TpΣ, Hq. At this point, we need a lemma relating the Floer homology of Y1 with that of Y: c Lemma 7.4. For any T -surface Σ, let Y P TpH, Σq be any 1-cell and s P SpinRpY q be any relative spinc structure. Then we have the following isomorphisms: ˚ (1) (Poincar´eDuality) HM ˚pp´Yq, sq – HM pY, sq. p 1 (2) (Reversing the sign of λ) HM ˚pY, sq – HM ˚pY , sq. Let us finish the proof of Theorem 7.2p assuming Lemmap 7.4. Consider the horizontal 1 p p composition of Y and ´Y : ˜ 1 pY, ω,˜ ¨ ¨ ¨ q :“ Y ˝h p´Y q. We verify that the closed 2-formω ˜ is neither negatively monotone nor balanced with respect to any spinc structures on the double Y˜ . Indeed, by (T3) and (T4), |xω,˜ rΣpiqsy| ă 2π and ‰ 0 for any component Σpiq Ă Σ. 40 DONGHAO WANG

Any integral class κ P H2pY, BY ; Zq can be represented by a properly embedded oriented surface F Ă Y minimizing the Thurston norm. Combined with its orientation reversal p´F q Ă p´Y q, they form a closed surface F˜ :“ F Y p´F q Ă Y.˜ Since Y is irreducible and Y ‰ S1 ˆ D2, the surface F has no sphere or disk components. By [Gab83, Lemma 6.15], the double F˜ Ă Y˜ is also norm-minimizing. By Theorem 3.7, we have ˜ 1 (7.1) HM ˚pY, s, c˜; Rω˜ q – HM ˚pY, s1q bR HM ˚p´Y , s2q. s1˝hs2“s à c c where the sum is over all pairs ps1, sp2qp P SpinRpY q ˆp SpinRp´Y q withps1 ˝h s2 “ s. Note that

(7.2) xc1ps1 ˝h s2q,prF˜psy “ xc1ps1q, rF sy ` xc1ps2q, r´F sy, p p By the adjunction inequality from Proposition 6.2, the left hand side of (7.1) vanishes whenever p p p p

xc1psq, rF˜sy ą xpF˜q “ 2xpF q. Combined with Corollary 6.6 and (7.1)(7.2), this implies that the group ˜ ˜ 1 (7.3) HM pY, rω˜s|rF sq – HM ˚pY|κq bR HM ˚p´Y | ´ κq, is non-vanishing. By Lemma 7.4, we have 1 (7.4) rankR HM ˚p´Y | ´ κq “ rankR HM ˚pY| ´ κq, so ˜ 2xpκq “ xpF q “ ϕYpκq ` ϕp´Y1qp´κq “ ϕYpκq ` ϕYp´κq. This completes the proof of Theorem 7.2 when Y is irreducible. In the general case, the inequality

ϕYpκq ` ϕYp´κq ď 2xpκq, follows from the vanishing result that we used earlier.  Proof of Lemma 7.4. The first isomorphism (1) is due to Poincar´eduality. Changing the orientation of Y has the same effect as changing the sign of the perturbed Chern-Simons- Dirac functional Lω in (3.3), while keeping the orientation fixed; see [KM07, Section 22.5] for more details.p Since we have worked with a ring R over F2, there is no need to deal with orientations. Our Floer homology HM ˚pYq is defined using a local coefficient system with fibers R. Thus the most relevant analogue for closed 3-manifolds is [KM07, P.624 (32.2)]. The second isomorphism (2) is induced from a concrete strict cobordism

1 X : Y eΣ Ñ Y , as we describe now. Similar to the constructionž in Section 3, we start with a hexagon Ω1 with boundary consisting of geodesic segments of length 2 and whose internal angles are always π{2. Also, the metric of Ω1 is flat near the boundary. MONOPOLES AND LANDAU-GINZBURG MODELS III: A GLUING THEOREM 41

s2 s2

Rs2 ˆ Σ

s1 t12 Ω1 t32 s3 γ12 γ23

t13 Y Y γ13 p p

Figure 12

The desired cobordism X is then obtained from Ω1 ˆ Σ by attaching r´1, 1st ˆ Y´ to γ13 ˆ Σ in Figure 12, where Y´ :“ ts ď ´1u Ă Y . In this case, we use s1, s2, s3 for spatial coordinates and t12, t13, t32 for time coordinates; their relations are indicated as in Figure 12. p We have to specify the closed 2-form ωX on X in order to construct the cobordism map. Recall that the 2-form ω on Y is decomposed as

ω “ ω ` χ1psqds ^ λ.

First, pull back ω to r´1, 1st ˆ Y´ and extend it on Ω1 ˆ Σ constantly as the harmonic 1 2-form µ P ΩhpΣ, iRq. Second, consider a function f : p´2, 8qs Ñ R such that fpsq ” s 1 when s ě ´1 and f psq “ χ1psq when s P p´2, 1s. We shall regard f as a function on Y by extending f constantly when s ď ´3{2. Now consider a function h : X Ñ R with the following properties: p p ‚ h ” f on a neighborhood r´1, ´1 ` qt13 ˆ Y and ” ´f on p1 ´ , 1st13 ˆ Y ;

‚ h ” s1 ” s2 on γ12 ˆ r1, 8qs1 ˆ Σ and ” ´s3 ” s2 on γ23 ˆ r1, 8qs3 ˆ Σ;

‚ h ” s2 on a neighborhood p1 ´ , 1st12 ˆ Rs2pˆ Σ. p The extension of h over the interior of X can be arbitrary. Finally, set

ωX “ ω ` dh ^ λ on X.

The strict cobordism X “ pX, ωX q then induces a map: p Id b1 HM ˚pXq 1 α1 : HM ˚pY, sq ÝÝÝÑ HM ˚pY, sq b HM ˚peΣ, sstdq ÝÝÝÝÝÑ HM ˚pY , sq. 1 1 1 Switching the roles of Y and Y produces a strict cobordism X : Y eΣ Ñ Y and a map in the opposite direction:p p p p 1 š 1 1 Id b1 1 HM ˚pX q α1 : HM ˚pY , sq ÝÝÝÑ HM ˚pY , sq b HM ˚peΣ1 , sstdq ÝÝÝÝÝÝÑ HM ˚pY, sq. 1 It remains to verify that α1 and α1 are mutual inverses to each other (up to a non-zero scalar). To see this, composep X with Xp1 along the commonp boundary Y1. Thep resulting cobordism Y eΣ eΣ1 Ñ Y ž ž 42 DONGHAO WANG induces the gluing map α in Subsection 3.6, which is an isomorphism by Theorem 3.6. 1 Indeed, one can change the 2-form ωX ˝ ωX into the standard one in (3.4) by adding an 1 exact 2-form. Thus the map α1 ˝ α1 : HM ˚pY, sq Ñ HM ˚pY, sq is invertible. The other 1 composition α1 ˝ α1 is dealt with in a similar manner. This completes the proof of Lemma 7.4. p p  8. Relations with Sutured Floer Homology Before we prove the fiberness detection result, Theorem 7.3, we add a digression to explain the relations of HM ˚pYq with the sutured Floer homology SHM [KM10] and SFH [Juh06, Juh08]. In the original definition of Gabai [Gab83, Definition 2.6], any 3-manifolds with toroidal boundary are sutured manifolds with only toral components. However, they are not examples of balanced sutured manifolds in the sense of Juh´asz[Juh06, Definition 2.2]; thus the sutured Floer homology, either SHM or SFH, is never defined for this class of sutured manifolds. One may regard our construction as a natural extension of the existing sutured Floer theory, and ask if the suture manifold decomposition theorem, e.g. [Juh08, Theorem 1.3] and [KM10, Proposition 6.9], continue to hold in our case. In this section, we will only prove a preliminary result towards this direction. Since our Floer homology HM ˚pYq also relies on the closed 2-form ω, it is not clear to the author whether a general result is available. It is worth mentioning that the sutured monopole Floer homology SHM introduced by Kronheimer-Mrowka [KM10] is defined over Z, but the construction extends naturally to the mod 2 Novikov ring R, as explained in [Siv12, Section 2.2]. We shall work with the latter case.

8.1. Cutting along a surface. For any T -surface Σ “ pΣ, gΣ, λ, µq and any 1-cell Y P TpH, Σq, we listed a few cohomological conditions on ω in (P4) (P5). In this section, we shall think of them geometrically and work with a more restrictive setup: (P5) there exists a properly embedded oriented surface F Ă Y such that BF intersects each component of Σ in parallel circles, and ir˚2λs is dual to zrBF s P H1pΣ, Zq for some z P R; moreover, F has no closed component or disk components. In particular, χpF q ď 0. 2 (P6) the Poincar´edual of ´2πirωs P H pY ; Rq can be represented by a real 1-cycle η that lies on the surface F . If Y satisfies the additional properties (P5)(P6), then we can cut Y along the surface F to obtain a balanced sutured manifold, denoted by MpY,F q. Let SHM pMpY,F qq be the sutured monopole Floer homology of MpY,F q defined over R with trivial coefficient system; cf. [Siv12, Section 2.2].

Theorem 8.1. For any T -surface Σ and any 1-cell Y P TpH, Σq satisfying (P5)(P6), we have SHM pMpY,F qq – HM ˚pY, sq. xc1psq,rF sy“xpF q à Moreover, ϕ prF sq ď xpF q with ϕ defined as in Definition 7.1. Y Y p p MONOPOLES AND LANDAU-GINZBURG MODELS III: A GLUING THEOREM 43

For the proof of Theorem 8.1, we have to first understand the special case when F is connected and Y fibers over S1 with F as a fiber. Thus F is a genus g surface with n boundary circles, and Y is the mapping torus of a self-diffeomorphism φ : F Ñ F such that

φ|π0pBF q has at least two orbits. Lemma 8.2. For any T -surface Σ and any 1-cell Y P TpH, Σq, if F is connected and Y is a mapping torus over F , then for κ “ ˘rF s, ϕYpκq “ ´χpF q and HM pY|κq – R. 2 Proof of Lemma 8.2. If χpF q “ 0, then F is an annulus and Y “ r´3, 3ss ˆ T is a product. This case is addressed already in Lemma 5.3. We focus on the case when χpF q ă 0. With loss of generality, let κ “ rF s. Note that Y is irreducible and F minimizes the Thurston norm. We follow the notations and arguments in the proof of Theorem 7.2. Let pY,˜ F˜q denote the double of pY,F q, then Y˜ is a mapping torus over F˜ with gpF˜q ě 2. By (7.3), we have ˜ ˜ 1 (8.1) HM pY, rω˜s|F q “ HM pY|κq bR HM p´Y | ´ κq. The rest of the proof is divided into four steps. Step 1. We can arrange the 2-formω ˜ so that xrω˜s, rF˜sy “ 0. 1 Instead of Y ˝h p´Y q, we consider the horizontal composition ˜ 1 pY, ω,˜ ¨ ¨ ¨ q “ Y ˝h Y1 ˝h p´Y q, 2 where Y1 “ pr´3, 3ss ˆΣ, ω1, ¨ ¨ ¨ q P TpΣ, Σq. Here ω1 “ µ`ds^λ`ω2 and ω2 P Ωc pY1, iRq is compactly supported. By Lemma 5.3, inserting this extra 1-cell Y1 does not affect the 2 identity (8.1), but changing the class rω2s P H pY1, BY1; iRq can effectively alter the pairing xrω˜s, rF˜sy. From now on, we shall always assume that xrω˜s, rF˜sy “ 0. Step 2. The group HM pY,˜ rω˜s|F˜q in (8.1) has rank 1. Since gpF˜q ě 2, this computation for mapping tori is [KM10, Lemma 4.7], ifω ˜ “ 0. The general case is not really different. Since xrω˜s, rF˜sy “ 0, we can still apply Floer’s excision theorem [KM10, Theorem 3.1] to reduce the problem to the case when Y˜ “ F˜ ˆ S1 is a product. The same trick is used also in the proof of Theorem 8.1 below. Now the statement follows from Lemma 6.9. Step 3. We conclude from (8.1), Lemma 7.4 and Step 2 that

rankR HM pY|κq “ rankR HM pY| ´ κq “ 1.

Step 4. By Theorem 7.2, ϕYpκq ` ϕYp´κq “ 2xpκq. We have to verify that ϕYpκq “ ϕYp´κq. This equality now follows from the symmetry of the graded Euler characteristics:

SWpY q : s ÞÑ χpHM ˚pY, sqq; see Theorem 3.4. This function is invariant under the conjugacy of relative spinc structures: ˚ p p s Ø s by [MT96, Tau01]. The computation of HM pY|κq shows that χpHM ˚pY, sqq ‰ 0 for exactly one s with xc1psq, κy ě ϕYpκq. This proves the equality ϕYpκq “ ϕYp´κq and pcompletesp the proof of Lemma 8.2. p  p p 44 DONGHAO WANG

Proof of Theorem 8.1. In [KM10], the group SHM pMpY,F qq is defined as the monopole Floer homology of a suitable closure of MpY,F q, which can be described as follows. Consider an 1-cell Y1 P TpΣ, Hq such that Y1 is a mapping torus over a connected surface F1. We require that ‚ gpF1q ě 2; ‚ rBF1s “ ´rBF s P H1pΣ, Zq, and ‚ the 1-cell Y1 satisfies property (P5)(P6) for the surface F1. Consider the closed 3-manifold obtained by gluing Y and Y1:

pY2, ω2, ¨ ¨ ¨ q :“ Y ˝h Y1,

We can arrange so that BF is identical to BF1 on Σ; so Y2 contains a closed oriented surface F2 :“ F Y F1 with gpF2q ě 2. Moreover, F2 is connected. Following the shorthands from Definition 6.4, the sutured monopole Floer homology of MpY,Kq is then defined as

SHM pMpY,Kqq :“ HM ˚pY2|F2q; see [KM10, Definition 4.3] and [Siv12, Section 2.2]. For the latter group, we have

HM ˚pY2|F2q – HM ˚pY2, r0s|F2q, 2 by Remark 6.5. Let c2 “ ´2πirω2s be the period class of ω2 P Ω pY2, iRq. Using Property (P6), we can arrange so that the Poincar´edual of c2 is represented by a real 1-cycle η2 lying over F2 Ă Y2. Now consider the subgroup

Gy :“ HM ˚pY, sq Ă HM ˚pYq. xc1psq,rF sy“y à Theorem 3.7 and Lemma 8.2 thenp imply that p

(8.2) HM ˚pY2, s, c2; Rω2 q “ Gb bR HM pY1|rF1sq – Gb, xc1psq,rF2sy“y`xpF1q à for any y ě ϕYprF sq. By the adjunction inequality from Proposition 6.2, the left hand side of (8.2) vanishes whenever b ` xpF1q ą xpF2q. As a result,

ϕYprF sq ď xpF q “ ´χpF q. Let y “ xpF q in (8.2), then HM pY2, rω2s|F2q – GxpF q. To complete the proof of Theorem 8.1, it remains to verify that

(8.3) HM pY2, r0s|F2q – HM pY2, rω2s|F2q.

This isomorphism, which involves only the closed 3-manifold Y2, is similar to the one in [KM10, Corollary 3.4], except that ω2 is used for non-exact perturbations here. The proof of (8.3) relies on the property that the real 1-cycle η2 that represents c2 lies on the surface 1 1 F2, so we can pick ω2 P rω2s such that ω2 is supported on a tubular neighborhood of F2:

r´1, 1s ˆ F2 Ă Y2. 1 1 By identifying t˘1uˆF2, ω2 becomes a closed 2-form on F2 ˆS . The same process applied to Y2zr´1, 1s ˆ F2 yields another copy of Y2, but the 2-form is zero now. MONOPOLES AND LANDAU-GINZBURG MODELS III: A GLUING THEOREM 45

As in the proof of [KM10, Corollary 3.4], we apply Floer’s excision theorem [KM10, Theorem 3.1] to obtain that 1 1 HM pY2, rω2s|F2q “ HM pY2, r0s|F2q bR HM pF2 ˆ S , rω2s|F2q. 1 1 By Lemma 6.9, HM pF2 ˆ S , rω2s|F2q – R. This completes the proof of Theorem 8.1.  The proof of Theorem 8.1 has an immediate corollary:

Corollary 8.3. For any T -surface Σ and any 1-cell Y P TpH, Σq satisfying (P5), ϕYprF sq ď xpF q. Remark 8.4. The property (P5) is crucial to the proof of Theorem 8.1 for the following reason: for the gluing argument to work, we have to make sure that (i) rBF s “ ´rBF1s P H1pΣ; Zq; (ii) there exists classes a P H2pY, BY ; Rq and a1 P H2pY1, BY1; Rq such that

rBas “ ´rBa1s “ the Poincar´edual of ir˚Σλs P H1pΣ; Rq;

In general, it is not clear to the author whether Y and Y1 can be always glued. However, Property (P5) reduces (i)(ii) into a single condition, which is easier to verify in practice. ♦ 8.2. Relations with Link Floer Homology. As a special case of Theorem 8.1, our construction recovers the monopole knot Floer homology KHM introduced by Kronheimer and Mrowka [KM10]. This statement is also true for the link Floer homology; let us expand on its meaning. n For any link L “ tLiui“1 inside a closed oriented 3-manifold Z0, consider the balanced sutured manifold Z0pLq :“ pZ0zNpLq, γq where spγq X NpLiq are two meridional sutures on BNpLiq oriented in opposite ways. The link Floer homology of pZ0,Lq is defined as the sutured monopole Floer homology of Z0pLq:

LHM pZ0,Lq :“ SHM pZ0pLqq, and we shall work with the mod 2 Novikov ring R. On the other hand, pick a meridian mi for each link component Li and consider the link complement Y pZ0,Lq :“ Z0zNpL Y m1 Y m2 Y ¨ ¨ ¨ Y mnq.

Each mi bounds a disk in Z0, which becomes an annulus Ai in Y pZ0,Lq. Let F be the n union i“1 Ai (with any fixed orientation). The balanced sutured manifold Z0pLq is then obtained from Y pZ0,Lq by cutting along F . To applyŤ Theorem 8.1, we have to specify the choice of ω: 1 ‚ let Σ “BY pZ0,Lq. Pick a flat metric gΣ and λ P ΩhpΣ, iRq such that ir˚Σλs is dual to zrBF s P H1pΣ, Rq for some z ‰ 0 P R; ‚ η is a real 1-cycle on F such that η X Ai is a segment joining two components of 2 BAi. Let ´2πirωs P H pY pZ0,Lq, Rq be the dual of η. Finally, let YpZ0,Lq “ pY pZ0,Lq, ω, ¨ ¨ ¨ q P TpH, Σq be any 1-cell satisfying these prop- erties. As a corollary of Theorem 8.1, we have 46 DONGHAO WANG

Corollary 8.5. For any 1-cell YpZ0,Lq constructed above, we have an isomorphism

HM ˚pYpZ0,Lqq – SHM pZ0pLqq “ LHM pZ0,Lq.

Moreover, if HM ˚pYpZ0,Lq, sq ‰ t0u, then xc1psq, rAisy “ 0 for any 1 ď i ď n.

Proof of Corollary 8.5. If HM ˚pYpZ0,Lq, sq ‰ t0u, then Theorem 8.1 implies that p p xc1psq, rF sy ď ϕYprF sq ď xpF q “ 0. p Applying the same argument for r´F s, we conclude that xc1psq, rF sy “ 0. The desired isomorphism then follows from thep first part of Theorem 8.1. It remains to verify the stronger statement: xc1psq, rAisy “ 0.p In the proof of Theorem 8.1, we composed Y with a mapping torus over a connected surface F1. We now make F1 1 disconnected: take F1 “ p´F q and Y1 “ p´F q ˆ S .p When Y pZ0,Lq are glued with Y1, we require that 1 ‚B NpLi Y miq Ă BY pZ0,Lq is identified with Bp´Aiq ˆ S ĂBY1; ‚B Ai ĂBY pZ0,Lq is identified with Bp´Aiq ˆ tptu Ă BY1. 2 The closed 3-manifold Y ˝h Y1 contains a collection of 2-tori Ti , which are doubles of Ai’s. Now we use the adjunction inequality in Proposition 6.2 and Lemma 5.3 to conclude.  9. Fiberness Detection 9.1. Some Preparations. In this section, we complete the proof of Theorem 7.3. We start with a preliminary result:

Lemma 9.1. Under the assumption of Theorem 7.3, if κ P H2pY, BY ; Zq is primitive, then κ can be represented by a connected Thurston norm minimizing surface F that intersects each component of Σ non-trivially. Moreover, ϕYpκq “ ϕYp´κq. Proof. This lemma follows from [McM02, Theorem 4.1 & Proposition 6.1]. Let 1 φ P H pY ; Zq “ Hompπ1pY q, Zq be the Poincar´edual of κ and b1pker φq be the first Betti number of the subgroup ker φ Ă π1pY q. Then [McM02, Proposition 6.1] states that κ can be represented by such a norm- minimizing surface F if

(9.1) b1pker φq ă 8. To verify that F intersects each component of Σ non-trivially, one has to go through the proof of [McM02, Proposition 6.1]. The condition (9.1) will follow from [McM02, Theorem 4.1], if we can verify its assumptions. Consider the set of Alexander classes 2 ∆pY q :“ tc1psq : χpHM ˚pY, sqq ‰ 0u Ă H pY, BY ; Zq. By Theorem 3.4, ∆pY q is precisely the support of the Alexander polynomial of Y and is symmetric about the origin. Sincep p

rankR HM ˚pY|κq “ rankR HM ˚pY| ´ κq “ 1, we conclude that ∆pY q ‰ H and ϕYpκq “ ϕYp´κq. Moreover, the maximum max φpa ´ bq a,bP∆pY q MONOPOLES AND LANDAU-GINZBURG MODELS III: A GLUING THEOREM 47 is achieved for exactly one pair of elements pa, bq. The other assumption of [McM02, Theo- rem 4.1] is certified by [McM02, Theorem 5.1]. This completes the proof of Lemma 9.1.  9.2. Proof of Theorem 7.3. The proof of Theorem 7.3 is based on the fiberness detection result for balanced sutured manifolds, adapted to the case of mod 2 Novikov ring R: Theorem 9.2 ([KM10, Theorem 6.1]). Suppose that a balanced sutured manifold pM, γq is taut and a homology product. Then pM, γq is a product sutured manifold if and only if SHM pM, γq – R. For the proof of Theorem 7.3, it suffices to deal with the case when κ is primitive. Let F be the surface given by Lemma 9.1. We will address the cases when χpF q ă 0 and when χpF q “ 0 separately.

Proof of Theorem 7.3 when χpF q ă 0. Since Y is irreducible, by Theorem 7.3 and Lemma 9.1, we have

ϕYpκq “ ϕYp´κq “ xpF q. Let pY,˜ F˜q be the double of pY,F q, then gpF˜q ě 2. Following the notations in the proof of Theorem 7.3, we have ˜ ˜ 1 (9.2) HM pY, rω˜s|F q “ HM ˚pY|κq bR HM ˚p´Y | ´ κq – R. The rest of the proof is divided into six steps. Step 1. We can effectively change the 2-formω ˜ so that xrω˜s, rF˜sy “ 0. This is Step 1 in the proof of Lemma 8.2. Step 2. Let M˜ be the 3-manifold with boundary obtained by cutting Y˜ along F˜. Write ˜ ˜ ˜ ˜ ˜ ˜ ˜ BM “ F` Y F´. Then pM, F`q is a homology product, i.e. H˚pF`; Zq Ñ H˚pM; Zq is an isomorphism. This follows from the fact [Tur98, Theorem 1] that for the graded Euler ˜ ˜ characteristic of HM ˚pY, rω˜sq recovers the Minor-Turaev torsion invariant T pY q. Then we apply [Ni09a, Proposition 3.1] and (9.2). Step 3. HM pY˜ |F˜q – R. Let N :“ r´1, 1s ˆ F˜ Ă Y˜ be a tubular neighborhood of F˜. Then Y˜ zN – M˜ . We claim that rω˜s is represented by a 2-formω ˜1 supported in r´1, 1s ˆ F˜. This follows from Step 1, Step 2 and a diagram-chasing:

rω˜sÞÑ0 H2pY,˜ t1u ˆ F˜q H2pY˜ q H2pF˜q

–

H2pY,˜ M˜ q – H2pN, BNq.

Given such a 2-formω ˜1, we use Floer’s excision theorem as in the proof of Theorem 8.1 to deduce that HM pY˜ |F˜q – HM pY,˜ rω˜s|F˜q – R. Step 4. Let pMpY,F q, γq be the balanced sutured manifold obtained by cutting Y along F . Then MpY,F q is a homology product. 48 DONGHAO WANG

Note that M˜ is the double of MpY,F q along the annuli Apγq. Write M˜ “ M1 Y M2 ˜ and Fi :“ Mi X F` for i “ 1, 2. Then the statement follows by examining the long exact sequences: ˜ ¨ ¨ ¨ H˚pF1 X F2q H˚pF1q ‘ H˚pF2q H˚pF`q ¨ ¨ ¨

– – ˜ ¨ ¨ ¨ H˚pM1 X M2q H˚pM1q ‘ H˚pM2q H˚pMq ¨ ¨ ¨ By Step 3 and the five lemma, the middle vertical map is also an isomorphism. Step 5. If properties (P5)(P6) hold for pΣ, Y,F q, then Theorem 7.3 holds. In this special case, we can use Theorem 8.1 and Lemma 9.1 to obtain that SHM pMpY,F q, γq – HM pY|rF sq – R. Since Y is irreducible and F is norm-minimizing, pMpY,F q, γq is taut. By Step 4 and Theorem 9.2, MpY,F q is a product sutured manifold. Step 6. Reduce the general case to Step 5. Let Σ2 be another T -surface with the same underlying oriented surface as Σ and Y2 P TpH, Σ2q has the same underlying 3-manifold as Y. We require that properties (P5)(P6) hold for pΣ2, Y2,F q. The goal is to show that HM pY2|rF sq – R. Let ˜ 1 pY, ω˜2, ¨ ¨ ¨ q “ Y2 ˝h p´Y2q.

By Step 1, we may assume that xrω˜2s, rF sy “ 0. By Step 3, we have

R – HM pY,˜ rω˜s|F˜q – HM pY˜ |F˜q – HM pY,˜ rω˜2s|F˜q.

By (9.2), HM pY2|rF sq – R. Now we use Step 5 to complete the proof of Theorem 7.3 when χpF q ă 0.  Proof of Theorem 7.3 when χpF q “ 0. In this case, F is an annulus, Σ has 2-components ˜ and the double F is a 2-torus. By Lemma 9.1, ϕYpκq “ ϕYp´κq “ 0. Our assumptions then imply that HM ˚pYq – R. The proof when χpF q ă 0 carries over with no essential changes. Let us explain where the differences arise: ‚ In Step 1, we require instead that xirω˜s, rF sy “ a for some fixed a P R with a ‰ 0 and |a| ă 2π; ‚ In Step 2, Ni’s result [Ni09a, Proposition 3.1b] is stated for a closed surface with genus ě 2; but its proof in [Ni09a, Section 3.3] relies only on the property of the Milnor-Turaev torsion invariant T pY˜ q (note also that b1pY˜ q ě 3). Thus we can still ˜ ˜ conclude from (9.2) that pM, F`q is a homology product; ‚ Step 3 is replaced by the isomorphism

HM pY,˜ rω˜s|F˜q – HM pY,˜ rω˜2s|F˜q 2 ˜ for any classes rω˜s “ rω˜2s P H pY , iRq with xirω˜s, rF sy “ xirω˜2s, rF sy “ a. Then the difference rω˜s ´ rω˜2s is represented by a 2-formω ˜1 supported in the neighborhood N “ r´1, 1s ˆ F˜. As in the proof of Gluing Theorem 3.6, one may adapt Floer’s MONOPOLES AND LANDAU-GINZBURG MODELS III: A GLUING THEOREM 49

excision theorem [KM10, Theorem 3.1] to the case of a 2-torus using non-exact perturbations, The rest of proof can now proceed with no difficulty. The conclusion says that Y is a 2 mapping torus over an annulus; so Y “ r´3, 3ss ˆ T is in fact a cylinder.  As an immediate corollary of Theorem 7.3, we have

Corollary 9.3. For any T -surface Σ “ pΣ, gΣ, λ, µq and any 1-cell Y P TpH, Σq, if Y is 2 connected, irreducible and HM ˚pY, sq – R, then Y “ r´1, 1ss ˆ T .

Proof of Corollary 9.3. Let κ P H2pY, BY ; Zq be any primitive class, i.e., κ is not divisible by any other integral classes non-trivially.p By the symmetry of the graded Euler characteristic SWpY q in Theorem 3.4, the conditions of Theorem 7.3 are verified with ϕYpκq “ ϕYp´κq “ 2 0. By Theorem 7.3, Y is a mapping torus over an annulus; so Y “ r´1, 1ss ˆ T is a product.  10. Connected Sum Formulae Having discussed irreducible 3-manifolds with toroidal boundary, we focus on reducible 3-manifolds in this section and derive the connected sum formulae. 10.1. Connected Sums with 3-Manifolds with Toroidal Boundary. For i “ 1, 2, let Σi be any T -surface and Yi P TpH, Σiq be any 1-cell. Then we can form an 1-cell Y3 “ pY3, ω3, ¨ ¨ ¨ q P TpH, Σ1 Y Σ2q with the following properties: 2 (C1) the underlying 3-manifold Y3 is Y1#Y2; let S Ă Y3 be the 2-sphere separating Y1 and Y2; 2 (C2) rω3s P H pY3; iRq is determined by the relations: kiprω3sq “ liprωisq for i “ 1, 2 in 2 the digram below. As a result, xrω3s, rS sy “ 0.

2 pk1,k2q 2 3 2 3 2 2 0 H pY1#Y2; iRq H pY1zB1 ; iRq ‘ H pY2zB2 ; iRq H pS ; iRq

l1‘l2

2 2 prω1s, rω2sq P H pY1; iRq ‘ H pY2; iRq.

Although Y3 is not uniquely determined by these properties, we say that Y3 is a connected sum of Y1 and Y2. By Theorem 5.2, the isomorphism type of HM ˚pY3q is determined by (C1)(C2).

Proposition 10.1. The monopole Floer homology of Y3 can be computed as follows: HM ˚pY3q – HM ˚pY1q bR HM ˚pY2q bR V, where V is a 2-dimensional vector space over R.

Proof. By Theorem 5.2, we can compute the group HM ˚pY3q using a convenient metric 1 on Y1#Y2 and a 2-form ω3. Take a component Σi Ă Σi for each i “ 1, 2. Consider the 3-manifold 1 pr´3, 3s ˆ Σ , µ | 1 ` ds ^ λ | 1 q, i “ 1, 2. s i i Σi i Σi 50 DONGHAO WANG

1 1 1 1 Let Y4 “ pY4, ω4, ¨ ¨ ¨ q P TpΣ1 Y Σ2, Σ1 Y Σ2q be their connected sum. The 1-cell Y3 can be then obtained as the horizontal composition

pY1 Y2q ˝h Y4. Using the Gluing Theorem 3.6, we obtainž that

HM ˚pY3q – HM ˚pY1q bR HM ˚pY2q bR HM ˚pY4q. 1 It remains to verify that rankR HM ˚pY4q “ 2. Regard Y4 as a 1-cell in TpH, p´Σ1q Y 1 1 1 p´Σ2q Y Σ1 Y Σ2q and consider the horizontal composition with 1 1 1 1 e 1 P Tpp´Σ q Y Σ , Hq and e 1 P Tpp´Σ q Y Σ , Hq. Σ1 1 1 Σ2 2 2 Another application of Theorem 3.6 implies that 1 1 1 1 1 1 HM ˚pY4q – HM ˚ppΣ1 ˆ S q#pΣ2 ˆ S q, rω1s#rω2sq, 1 1 1 1 where ω “ µ | 1 ` dθ ^ λ | 1 and θ denotes the coordinate of S . rω s#rω s is obtained i i Σi i Σi 1 2 1 1 from rω1s and rω2s using (C2). Here we have used the shorthands from Definition 6.4. By Lemma 5.4, we have already known that 1 1 1 HM ˚pΣi ˆ S , rωisq – R for i “ 1, 2. To conclude, we apply the connected sum formula [Lin17, Theorem 5] for closed 3-manifolds. c As a result, the group HM ˚pY4q is concentrated in a single spin grading and has rank 2.  10.2. Connected Sums with Closed 3-Manifolds. Let us first review the definition of HM ˚pMq for any closed 3-manifold M. Definition 10.2. For any closed 3-manifold Z, we obtain a balanced sutured manifold pĄZp1q, δq by taking Zp1q “ ZzB3 and the suture spδq Ă BZp1q to be the equator. Define HM pZq :“ SHM pZp1q, δq. ♦ Let Σ be any T -surface, Y P TpH, Σq be any 1-cell and Z be any closed 3-manifold. Ą Consider an 1-cell Y5 “ pY5, ω5, ¨ ¨ ¨ q P TpH, Σq that satisfies the following properties: ‚ the underlying 3-manifold Y5 is Y #Z; 2 ‚r ω5s P H pY5; iRq is obtained from rωs using (C2) and the zero form on Z. Proposition 10.3. The monopole Floer homology of Y5 can be computed as follows:

HM ˚pY5q – HM ˚pYq bR HM pZq. Proof. Following the proof of Proposition 10.1, it suffices to verify that 1 1Ą 1 (10.1) HM pMq – HM ˚pM#pΣ ˆ S q, r0s#rω sq 1 1 where Σ is a connected component of Σ and ω :“ µ|Σ1 ` dθ ^ λ|Σ1 . One can argue as in Step 1 of the proof of LemmaĄ 8.2 and work instead with the 2-form 2 1 1 ω ” µ|Σ1 on S ˆ Σ . Then the Poincar´edual of rω2s is proportional to the 1-cycle tptu ˆ S1. Since r0s#rω2s is neither balanced nor negatively monotone for any spinc structure on M#pS1 ˆΣ1q, one may apply Proposition 6.2 to compute the right hand side of (10.1) using exact perturbations. Now the isomorphism (10.1) follows from [KM10, Proposition 4.6].  MONOPOLES AND LANDAU-GINZBURG MODELS III: A GLUING THEOREM 51

References [BD95] P. J. Braam and S. K. Donaldson. Floer’s work on instanton homology, knots and surgery. In The Floer memorial volume, volume 133 of Progr. Math., pages 195–256. Birkh¨auser,Basel, 1995. [Ben67] Jean Benabou. Introduction to bicategories. In Reports of the Midwest Category Seminar, pages 1–77. Springer, Berlin, 1967. [EE67] C. J. Earle and J. Eells. The diffeomorphism group of a compact Riemann surface. Bull. Amer. Math. Soc., 73:557–559, 1967. [Eis95] David Eisenbud. Commutative algebra, volume 150 of Graduate Texts in Mathematics. Springer- Verlag, New York, 1995. With a view toward algebraic geometry. [Gab83] David Gabai. Foliations and the topology of 3-manifolds. J. Differential Geom., 18(3):445–503, 1983. [Ghi08] Paolo Ghiggini. Knot Floer homology detects genus-one fibred knots. Amer. J. Math., 130(5):1151–1169, 2008. [GL19] Sudipta Ghosh and Zhenkun Li. Decomposing sutured monopole and instanton floer homologies, 2019. [Hat02] Allen Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002. [Juh06] Andr´asJuh´asz.Holomorphic discs and sutured manifolds. Algebr. Geom. Topol., 6:1429–1457, 2006. [Juh08] Andr´asJuh´asz.Floer homology and surface decompositions. Geom. Topol., 12(1):299–350, 2008. [KM97] P. B. Kronheimer and T. S. Mrowka. Scalar curvature and the Thurston norm. Math. Res. Lett., 4(6):931–937, 1997. [KM07] Peter Kronheimer and Tomasz Mrowka. Monopoles and three-manifolds, volume 10 of New Math- ematical Monographs. Cambridge University Press, Cambridge, 2007. [KM10] Peter Kronheimer and Tomasz Mrowka. Knots, sutures, and excision. J. Differential Geom., 84(2):301–364, 2010. [Li18] Zhenkun Li. Contact structures, excisions, and sutured monopole floer homology, 2018. [Lin17] Francesco Lin. Pinp2q-monopole Floer homology, higher compositions and connected sums. J. Topol., 10(4):921–969, 2017. [Man16] Ciprian Manolescu. An introduction to knot Floer homology. In Physics and mathematics of link homology, volume 680 of Contemp. Math., pages 99–135. Amer. Math. Soc., Providence, RI, 2016. [McM02] Curtis T. McMullen. The Alexander polynomial of a 3-manifold and the Thurston norm on coho- mology. Ann. Sci. Ecole´ Norm. Sup. (4), 35(2):153–171, 2002. [MT96] Guowu Meng and Clifford Henry Taubes. SW “ Milnor torsion. Math. Res. Lett., 3(5):661–674, 1996. [Ni07] Yi Ni. Knot Floer homology detects fibred knots. Invent. Math., 170(3):577–608, 2007. [Ni08] Yi Ni. Addendum to: ”knots, sutures and excision”, 2008. [Ni09a] Yi Ni. Heegaard Floer homology and fibred 3-manifolds. Amer. J. Math., 131(4):1047–1063, 2009. [Ni09b] Yi Ni. Link Floer homology detects the Thurston norm. Geom. Topol., 13(5):2991–3019, 2009. [OS04] Peter Ozsv´athand Zolt´anSzab´o.Holomorphic disks and knot invariants. Adv. Math., 186(1):58– 116, 2004. [OS08] Peter Ozsv´athand Zolt´anSzab´o.Link Floer homology and the Thurston norm. J. Amer. Math. Soc., 21(3):671–709, 2008. [Ras03] Jacob Rasmussen. Floer homology and knot complements. arXiv:math/0306378, 2003. [Sei08] Paul Seidel. Fukaya categories and Picard-Lefschetz theory. Zurich Lectures in Advanced Mathe- matics. European Mathematical Society (EMS), Z¨urich, 2008. [Siv12] Steven Sivek. Monopole Floer homology and Legendrian knots. Geom. Topol., 16(2):751–779, 2012. [Tau01] Clifford Henry Taubes. The Seiberg-Witten invariants and 4-manifolds with essential tori. Geom. Topol., 5:441–519, 2001. [Thu86] William P. Thurston. A norm for the homology of 3-manifolds. Mem. Amer. Math. Soc., 59(339):i– vi and 99–130, 1986. 52 DONGHAO WANG

[Tur98] Vladimir Turaev. A combinatorial formulation for the Seiberg-Witten invariants of 3-manifolds. Math. Res. Lett., 5(5):583–598, 1998. [Wan20a] Donghao Wang. Monopoles and landau-ginzburg models i. arXiv:2004.06227, 2020. [Wan20b] Donghao Wang. Monopoles and landau-ginzburg models ii: Floer homology. arXiv:2005.04333, 2020.

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Email address: [email protected]