Motivated by the programmes initiated by Taubes and Perutz, we study the geometry of near-symplectic 4-manifolds, ie, manifolds equipped with a closed 2-form which is symplectic outside a union of embedded 1-dimensional submanifolds, and broken Lefschetz fibrations on them; see Auroux, Donaldson and Katzarkov [3] and Gay and Kirby [8]. We present a set of four moves which allow us to pass from any given broken fibration to any other which is deformation equivalent to it. Moreover, we study the change of the near-symplectic geometry under each of these moves. The arguments rely on the introduction of a more general class of maps, which we call wrinkled fibrations and which allow us to rely on classical singularity theory. Finally, we illustrate these constructions by showing how one can merge components of the zero-set of the near-symplectic form. We also disprove a conjecture of Gay and Kirby by showing that any achiral broken Lefschetz fibration can be turned into a broken Lefschetz fibration by applying a sequence of our moves.

This paper is a companion to the authors' forthcoming work extending Heegaard Floer theory from closed 3-manifolds to compact 3-manifolds with two boundary components via quilted Floer cohomology. We describe the first interesting case of this theory: the invariants of 3-manifolds bounding S 2 ⨿ T 2 , regarded as modules over the Fukaya category of the punctured 2-torus. We extract a short proof of exactness of the Dehn surgery triangle in Heegaard Floer homology. We show that A ∞ -structures on the graded algebra A formed by the cohomology of two basic objects in the Fukaya category of the punctured 2-torus are governed by just two parameters ðm 6 , m 8 Þ, extracted from the Hochschild cohomology of A. For the Fukaya category itself, m 6 ≠ 0.symplectic manifolds | 3-manifolds | Heegaard Floer theory | homological mirror symmetry | Hochschild cohomology T his article is an offshoot of the authors' forthcoming work on Lagrangian correspondences and invariants of three-manifolds with boundary. In that work we will combine a detailed geometric examination of the Lagrangian correspondences between symmetric products of Riemann surfaces, studied by the second author in refs. 1 and 2, with the A ∞ quilted Floer theory of Ma'uWehrheim-Woodward (3) and the functoriality principle of ref. 4 (see also ref. 5). By doing so, we will extend the package of Heegaard Floer cohomology invariants (6) from closed 3-manifolds to compact 3-manifolds with boundary. To be precise, we construct invariants for compact, oriented, connected 3-manifolds with precisely two boundary components, marked as 'incoming' and 'outgoing'. When these are both spherical, our invariants capture the Heegaard Floer cochains of the capped-off 3-manifold. We refer to Auroux's work (7) for the relationship of this theory to bordered Heegaard Floer theory (8).The format of our invariants is alarmingly abstract: they take the form of A ∞ -functors between A ∞ -categories associated with the boundary surfaces, satisfying a composition law under sewing of cobordisms. Enthusiasts for extended topological quantum field theory (TQFT) will approve of this formulation, but geometric topologists will want to know how to extract topological information from it.In this article, we examine the next-to-simplest case of the theory by applying it to manifolds Y 3 with incoming boundary of genus 0 (which we cap off to formȲ ) and outgoing boundary of genus 1. The relevant A ∞ -categories are certain versions of the Fukaya category of a symplectic 2-torus T with a distinguished point z. The simplest version of the invariant for Y is an A ∞ -moduleM Y over the Fukaya categorŷ F ðT 0 Þ of exact; embedded curves in T 0 ≔ T \fzg:This module evaluates on each object X (which is a circle X ⊂ T with an exactness constraint and certain decorations) to give a cochain complexM Y ðXÞ. This complex is quasi-isomorphic to the Heegaard Floer cochains d CF Ã ðȲ ∪ T U X Þ, where U X is the solid torus in which the circle X bounds a disc. The different objects X correspond to different Dehn ...

Following the approach of Haiden-Katzarkov-Kontsevich [15], to any homologically smooth Z-graded gentle algebra A we associate a triple (Σ A , Λ A ; η A ), where Σ A is an oriented smooth surface with non-empty boundary, Λ A is a set of stops on ∂Σ A and η A is a line field on Σ A , such that the derived category of perfect dg-modules of A is equivalent to the partially wrapped Fukaya category of (Σ A , Λ A ; η A ). Modifying arguments of Johnson and Kawazumi, we classify the orbit decomposition of the action of the (symplectic) mapping class group of Σ A on the homotopy classes of line fields. As a result we obtain a sufficient criterion for homologically smooth graded gentle algebras to be derived equivalent. Our criterion uses numerical invariants generalizing those given by Avella-Alaminos-Geiss in [7], as well as some other numerical invariants. As an application, we find many new cases when the AAG-invariants determine the derived Morita class. As another application, we establish some derived equivalences between the stacky nodal curves considered in [21].

This paper explores a refinement of homological mirror symmetry which relates exact symplectic topology to arithmetic algebraic geometry. We establish a derived equivalence of the Fukaya category of the 2-torus, relative to a basepoint, with the category of perfect complexes of coherent sheaves on the Tate curve over the formal disc Spec Z [[q]]. It specializes to a derived equivalence, over Z, of the Fukaya category of the punctured torus with perfect complexes on the curve y 2 + xy = x 3 over Spec Z, the central fibre of the Tate curve; and, over the 'punctured disc' Spec Z ((q)), to an integral refinement of the known statement of homological mirror symmetry for the 2-torus. We also prove that the wrapped Fukaya category of the punctured torus is derived-equivalent over Z to coherent sheaves on the central fiber of the Tate curve.1 Beware: the circumstances under which one expects to find such an X are more subtle than those claimed by our one-sentence précis of Kontsevich's conjecture.We shall need to say what we mean by a CY structure for a category over a commutative ring L. The categories in question are of form H 0 C, where C is an A ∞ -category, and this permits us to make an expedient (but not fully satisfactory) definition: Definition 2.1 A CY structure consists on the L-linear A ∞ -category C consists of cochain-level maps φ A,B : hom C (A, B) ≃ hom C (B, A[n]) ∨ such that the induced maps on cohomology [φ A,B ⊗ 1 F ] : Hom H 0 (C× L F) (A, B) ≃ Hom H 0 (C× L F) (B, A[n]) ∨

We study symplectic invariants of the open symplectic manifolds X Γ obtained by plumbing cotangent bundles of 2-spheres according to a plumbing tree Γ. For any tree Γ, we calculate (DG-)algebra models of the Fukaya category F(X Γ ) of closed exact Lagrangians in X Γ and the wrapped Fukaya category W(X Γ ). When Γ is a Dynkin tree of type A n or D n (and conjecturally also for E 6 , E 7 , E 8 ), we prove that these models for the Fukaya category F(X Γ ) and W(X Γ ) are related by (derived) Koszul duality. As an application, we give explicit computations of symplectic cohomology of X Γ for Γ = A n , D n , based on the Legendrian surgery formula of [14]. In the case that Γ is non-Dynkin, we obtain a spectral sequence that converges to symplectic cohomology whose E 2 -page is given by the Hochschild cohomology of the preprojective algebra associated to the corresponding Γ. It is conjectured that this spectral sequence is degenerate if the ground field has characteristic zero.

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It follows from the work of Burban and Drozd [Math. Ann. 351 (2011) 665-709] that for nodal curves C, the derived category of modules over the Auslander order AC provides a categorical (smooth and proper) resolution of the category of perfect complexes Perf(C). On the A-side, it follows from the work of Haiden-Katzarkov-Kontsevich [Publ. Math. Inst. HautesÉtudes Sci. 126 (2017) 247-318] that for punctured surfaces X with stops Λ at their boundary, the partially wrapped Fukaya category W(X, Λ) provides a categorical (smooth and proper) resolution of the compact Fukaya category F (X). Inspired by this analogy, we establish an equivalence between the derived category of modules over the Auslander orders over certain nodal stacky curves and partially wrapped Fukaya categories associated to punctured surfaces of arbitrary genus equipped with stops at their boundary. As an application, we deduce equivalences between derived categories of coherent sheaves (respectively perfect complexes) on such nodal stacky curves and the wrapped (respectively compact) Fukaya categories of punctured surfaces of arbitrary genus.

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