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Monopoles and Landau-Ginzburg Models Iii: a Gluing Theorem

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Monopoles and Landau-Ginzburg Models Iii: a Gluing Theorem 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; !q, where Y is an oriented 3-manifold with toroidal boundary and ! 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 ! 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; !q, where Y is a compact oriented 3-manifold with toroidal boundary and ! 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 !|BY is small and yet non-vanishing. Under this assumption, we will prove that the monopole Floer homology of pY; !q is a topological invariant: it depends only on the 3-manifold Y , the cohomology class r!s 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;!1q and pY2;!2q are glued suitably along their common boundary; see Theorem 2.7. The gluing map α : HM ˚pY1;!1q bR HM ˚pY2;!2q Ñ HM ˚pY1 ˝h Y2;!1 ˝h !2q: 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; !q 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 :“ 1¤j¤n 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 disconnected² 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 ¤ j ¤ 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 ¤ j ¤ n. Such a quadruple Σ “ pΣ; gΣ; λ, µq is called a T -surface. We choose a closed 2-form 2 ! P Ω pY; iRq on Y such that ! “ µ ` ds ^ λ in a collar neighborhood p´1; 0ss ˆ Σ Ă Y . Denote such a pair pY; !q, 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 ! 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 r!s 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 'Ypκq :“ maxaPMpYqxa; κy, κ P H2pY; BY ; Rq. We set 'Y ” ´8 if MpYq “ H. Then p p 1 p' pκq ` ' p´κqq ¤ 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 'Ypκq “ 'Yp´κq whenever MpYq is non-empty; but the author is unable to prove this symmetry directly, due to the presence of the 2-form !. However, the ideal adjunction inequality 'Ypκq ¤ 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).
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