We show that any smooth, closed, oriented, connected 4‐manifold can be trisected into three copies of \ k .S 1 B 3 /, intersecting pairwise in 3‐dimensional handlebodies, with triple intersection a closed 2‐dimensional surface. Such a trisection is unique up to a natural stabilization operation. This is analogous to the existence, and uniqueness up to stabilization, of Heegaard splittings of 3‐manifolds. A trisection of a 4‐manifold X arises from a Morse 2‐function GW X! B 2 and the obvious trisection of B 2 , in much the same way that a Heegaard splitting of a 3‐manifold Y arises from a Morse function gW Y ! B 1 and the obvious bisection of B 1 .

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Trisecting 4–manifolds

We consider mirror symmetry for (essentially arbitrary) hypersurfaces in (possibly noncompact) toric varieties from the perspective of the Strominger-Yau-Zaslow (SYZ) conjecture. Given a hypersurface $H$
 in a toric variety $V$
 we construct a Landau-Ginzburg model which is SYZ mirror to the blowup of $V\times \mathbf {C}$
 along $H\times0$
 , under a positivity assumption. This construction also yields SYZ mirrors to affine conic bundles, as well as a Landau-Ginzburg model which can be naturally viewed as a mirror to 
 $H$
 . The main applications concern affine hypersurfaces of general type, for which our results provide a geometric basis for various mirror symmetry statements that appear in the recent literature. We also obtain analogous results for complete intersections.

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Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces

This paper shows that there are symplectic four-manifolds M with the following property: a single isotopy class of smooth embedded two-spheres in M contains infinitely many Lagrangian submanifolds, no two of which are isotopic as Lagrangian submanifolds. The examples are constructed using a special class of symplectic automorphisms ("generalized Dehn twists"). The proof uses Floer homology. 
Revised version: one footnote removed, one reference added

Lagrangian two-spheres can be symplectically knotted

We develop a set of tools for doing computations in and of (partially) wrapped Fukaya categories. In particular, we prove (1) a descent (cosheaf) property for the wrapped Fukaya category with respect to so-called Weinstein sectorial coverings and (2) that the partially wrapped Fukaya category of a Weinstein manifold with respect to a mostly Legendrian stop is generated by the cocores of the critical handles and the linking disks to the stop. We also prove (3) a `stop removal equals localization' result, and (4) that the Fukaya--Seidel category of a Lefschetz fibration with Weinstein fiber is generated by the Lefschetz thimbles. These results are derived from three main ingredients, also of independent use: (5) a K\"unneth formula (6) an exact triangle in the Fukaya category associated to wrapping a Lagrangian through a Legendrian stop at infinity and (7) a geometric criterion for when a pushforward functor between wrapped Fukaya categories of Liouville sectors is fully faithful.

Sectorial descent for wrapped Fukaya categories

The Nadler-Zaslow correspondence famously identifies the finite-dimensional Floer homology groups between Lagrangians in cotangent bundles with the finite-dimensional Hom spaces between corresponding constructible sheaves. We generalize this correspondence to incorporate the infinite-dimensional spaces of morphisms "at infinity", given on the Floer side by Reeb trajectories (also known as "wrapping") and on the sheaf side by allowing unbounded infinite rank sheaves which are categorically compact. When combined with existing sheaf theoretic computations, our results confirm many new instances of homological mirror symmetry. 
More precisely, given a real analytic manifold $M$ and a subanalytic isotropic subset $\Lambda$ of its co-sphere bundle $S^*M$, we show that the partially wrapped Fukaya category of $T^*M$ stopped at $\Lambda$ is equivalent to the category of compact objects in the unbounded derived category of sheaves on $M$ with microsupport inside $\Lambda$. By an embedding trick, we also deduce a sheaf theoretic description of the wrapped Fukaya category of any Weinstein sector admitting a stable polarization.

Microlocal Morse theory of wrapped Fukaya categories

Following the approach of Haiden–Katzarkov–Kontsevich (Publ Math Inst Hautes Etudes Sci 126:247–318, 2017), to any homologically smooth $$\mathbb {Z}$$-graded gentle algebra A we associate a triple $$(\Sigma _A, \Lambda _A; \eta _A)$$, where $$\Sigma _A$$ is an oriented smooth surface with non-empty boundary, $$\Lambda _A$$ is a set of stops on $$\partial \Sigma _A$$ and $$\eta _A$$ is a line field on $$\Sigma _A$$, such that the derived category of perfect dg-modules of A is equivalent to the partially wrapped Fukaya category of $$(\Sigma _A, \Lambda _A ;\eta _A)$$. Modifying arguments of Johnson and Kawazumi, we classify the orbit decomposition of the action of the (symplectic) mapping class group of $$\Sigma _A$$ 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 Avella et al. (J Pure Appl Algebra 212(1):228–243, 2008), 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 Lekili and Polishchuk (J Topology 11:615–444, 2018)

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Derived equivalences of gentle algebras via Fukaya categories

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.

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Wrinkled fibrations on near-symplectic manifolds

Consider a pair $(X,L)$, of a Weinstein manifold $X$ with an exact Lagrangian submanifold $L$, with ideal contact boundary $(Y,\Lambda)$, where $Y$ is a contact manifold and $\Lambda\subset Y$ is a Legendrian submanifold. We introduce the Chekanov-Eliashberg DG-algebra, $CE^{\ast}(\Lambda)$, with coefficients in chains of the based loop space of $\Lambda$ and study its relation to the Floer cohomology $CF^{\ast}(L)$ of $L$. Using the augmentation induced by $L$, $CE^{\ast}(\Lambda)$ can be expressed as the Adams cobar construction $\Omega$ applied to a Legendrian coalgebra, $LC_{\ast}(\Lambda)$. We define a twisting cochain:\[\mathfrak{t} \colon LC_{\ast}(\Lambda) \to \mathrm{B} (CF^*(L))^\#\]via holomorphic curve counts, where $\mathrm{B}$ denotes the bar construction and $\#$ the graded linear dual. We show under simply-connectedness assumptions that the corresponding Koszul complex is acyclic which then implies that $CE^*(\Lambda)$ and $CF^{\ast}(L)$ are Koszul dual. In particular, $\mathfrak{t}$ induces a quasi-isomorphism between $CE^*(\Lambda)$ and the cobar of the Floer homology of $L$, $\Omega CF_*(L)$. We use the duality result to show that under certain connectivity and locally finiteness assumptions, $CE^*(\Lambda)$ is quasi-isomorphic to $C_{-*}(\Omega L)$ for any Lagrangian filling $L$ of $\Lambda$. Our constructions have interpretations in terms of wrapped Floer cohomology after versions of Lagrangian handle attachments. In particular, we outline a proof that $CE^{\ast}(\Lambda)$ is quasi-isomorphic to the wrapped Floer cohomology of a fiber disk $C$ in the Weinstein domain obtained by attaching $T^{\ast}(\Lambda\times[0,\infty))$ to $X$ along $\Lambda$ (or, in the terminology of arXiv:1604.02540 the wrapped Floer cohomology of $C$ in $X$ with wrapping stopped by $\Lambda$). Along the way, we give a definition of wrapped Floer cohomology without Hamiltonian perturbations.

Duality between Lagrangian and Legendrian invariants

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 , 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 (m6, m8), extracted from the Hochschild cohomology of A. For the Fukaya category itself, m6  ≠ 0.

https://www.pnas.org/content/pnas/108/20/8106.full.pdf

Fukaya categories of the torus and Dehn surgery

http://homepages.math.uic.edu/~lekili/quiltsLL.pdf

Yanki Lekili

Papers

Derived equivalences of gentle algebras via Fukaya categories

Wrinkled fibrations on near-symplectic manifolds

Duality between Lagrangian and Legendrian invariants

Fukaya categories of the torus and Dehn surgery

Geometric composition in quilted Floer theory