Yanki Lekili, Max Lipyanskiy
We prove that Floer cohomology of cyclic Lagrangian correspondences is invariant under transverse and embedded composition of Lagrangians under a general set of assumptions. In the Corrigendum, we introduce an additional assumption of monotonicity for cylinders which is needed to avoid bubbling at the Y-end.
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Quilted Floer trajectories with constant components
我们填补了arXiv：0905.1370中绗缝Floer轨迹的横截性结果证明中的空白，解决了一些但不是所有分量都是常数的轨迹。也就是说，我们表明，一般的分裂哈密顿扰动和分裂殆复结构的集合，模空间的参数化绗缝Floer轨迹的一个给定的索引是光滑的预期的尺寸。通用分裂哈密顿扰动的另一个好处是，它们扰动给定的循环拉格朗日对应，使得其因子的任何几何组成是横向的，因此是浸入的。
Given a closed, connected, oriented 3-manifold with positive first Betti number, one can define an instanton Floer group as well as a quilted Lagrangian Floer group. Each of these is equipped with a chain level grading. We show that the gradings agree.
Given a closed, connected, oriented 3-manifold with positive first Betti number, one can define an instanton Floer group as well as a quilted Lagrangian Floer group. The quilted Atiyah-Floer conjecture states that these cohomology groups are isomorphic. We initiate a program for proving this conjecture.
Quilted Floer Cohomology
我们推广拉格朗日弗洛尔上同调的拉格朗日对应序列。对于由拉格朗日对应的几何组成相关的序列，我们建立了Floer上同调的同构。我们给出应用程序的计算弗洛尔上同调，拉格朗日对应的可扩展性，和转移的几何组成下的可扩展性。
We use quilted Floer theory to construct functor-valued invariants of tangles arising from moduli spaces of flat bundles on punctured surfaces. As an application, we show the non-triviality of certain elements in the symplectic mapping class groups of moduli spaces of flat bundles on punctured spheres.
We use quilted Floer theory to generalize Seidel's long exact sequence in symplectic Floer theory to fibered Dehn twists. We then apply it to construct versions of the Floer and Khovanov-Rozansky exact triangles in Lagrangian Floer theory of moduli spaces of bundles.
We study Lagrangian correspondences between Liouville manifolds and construct functors between wrapped Fukaya categories. The study naturally brings up the question on comparing two versions of wrapped Fukaya categories of the product manifold, which we prove quasi-isomorphism on the level of wrapped Floer cohomology. To prove representability of these functors constructed from Lagrangian correspondences, we introduce the geometric compositions of Lagrangian correspondences under wrapping, as new classes of objects in the wrapped Fukaya category, which we prove to represent the functors by est…
Construction of holomorphic quilts in Cartesian product of closed surfaces
本文通过修改 Wehrheim-Woodward 的证明，在闭曲面辛流形上构造了一类浸入全纯拼缝，并给出例子支持 Bottman-Wehrheim 关于两类 Floer 同调经 bounding cochains 扭转后同构的猜想。
Functoriality for Lagrangian correspondences in Floer theory
利用绗缝Floer上同调和相对绗缝不变量，我们定义了一个复合函子类的拉格朗日对应的单调和精确辛Floer理论。我们表明，这个函子同意几何组成的情况下，组成是光滑的和嵌入的。因此，我们得到了拉格朗日对应的“分类与组成”。
We define relative Floer theoretic invariants arising from 'quilted pseudo-holomorphic surfaces': Collections of pseudoholomorphic maps to various target spaces with 'seam conditions' in Lagrangian correspondences. As application we construct a morphism on quantum homology associated to any monotone Lagrangian correspondence.
Categorical symplectic geometry is the study of a rich collection of invariants of symplectic manifolds, including the Fukaya $A_\infty$-category, Floer cohomology, and symplectic cohomology. Beginning with work of Wehrheim and Woodward in the late 2000s, several authors have developed techniques for functorial manipulation of these invariants. We survey these functorial structures, including Wehrheim-Woodward's quilted Floer cohomology and functors associated to Lagrangian correspondences, Fukaya's alternate approach to defining functors between Fukaya $A_\infty$-categories, and the second au…
We study wrapped Floer theory on product Liouville manifolds and prove that the wrapped Fukaya categories defined with respect to two different kinds of natural Hamiltonians and almost complex structures are equivalent. The implication is we can do quilted version of wrapped Floer theory, based on which we then construct functors between wrapped Fukaya categories of Liouville manifolds from certain classes of Lagrangian correspondences, by enlarging the wrapped Fukaya categories appropriately, allowing exact cylindrical Lagrangian immersions. For applications, we present a general Künneth form…
There have been a number of constructions of Lagrangian Floer homology invariants for $3$-manifolds defined in terms of symplectic character varieties arising from Heegaard splittings. With the aim of establishing an Atiyah-Floer counterpart of Kronheimer and Mrowka's singular instanton homology, we generalize one of these, due to H. Horton, to produce a Lagrangian Floer invariant of a knot or link $K \subset Y$ in a closed, oriented $3$-manifold, which we call symplectic instanton knot homology ($\mathrm{SIK}$). We use a multi-pointed Heegaard diagram to parametrize the gluing together of a p…
Fukaya categories of the torus and Dehn surgery
本文是作者即将开展的工作的同伴Heegaard Floer理论从封闭的3-流形到紧凑的3-流形与两个边界分量通过绗缝Floer上同调。我们描述了这个理论的第一个有趣的例子：包围S^2并T^2的3-流形的不变量，它们被看作是穿孔2-环面上的Rakaya范畴上的模。我们提取了一个简短的证明，在Heegaard弗洛尔同调的Dehn手术三角形的准确性。我们证明了由穿孔2-torus的Riskaya范畴中两个基本对象的上同调构成的分次代数A上的A-无穷结构仅由两个参数（m^6，m^8）控制，这两个参数是从A的Hochschild上同调中提取的.对于类本身，m^6是非零的。
Floer Cohomology and Geometric Composition of Lagrangian Correspondences
我们证明了在精确和单调的设置下的几何组成的拉格朗日对应的Floer上同调的同构。
A long exact sequence on the composition of Dehn twists
本文证明了关联一组 Lagrangian 球的 Floer 同调与其迭代 Dehn twist 不动点 Floer 同调的正合序列，推进了 Seidel 猜想。
Floer Cohomology of Certain pseudo-Anosov Maps
本文计算了亏格为$g>1$的曲面$Σ$的映射类群中某些元素的Floer上同调，这些元素是沿$Σ$中某些环的正、负dehn扭曲的合成.计算覆盖了一类伪Anasov映射。
HF=HM IV: The Seiberg-Witten Floer homology and ech correspondence
这是第四个五个文件，建设同构之间的塞伯格-威滕弗洛尔同调和Heegaard弗洛尔同调的一个给定的紧凑，面向3流形。的同构是作为一个组成的三个同构，其中第一个涉及到一个版本的嵌入式接触同源的辅助流形的Heegaard弗洛尔同调的原始。第二个同构将辅助流形上的嵌入接触同调的相关版本与同一流形上的Seiberg-Witten弗洛尔同调的版本联系起来。第三个同构将辅助流形上的Seiberg-Witten弗洛尔同调与原流形上的Seiberg-Witten弗洛尔同调的适当形式联系起来。本文描述了其中的第二个同构。
HF=HM III: Holomorphic curves and the differential for the ech/Heegaard Floer correspondence
这是第三个五个文件，建设同构之间的塞伯格，威滕弗洛尔同调和Heegaard弗洛尔同调的一个给定的紧凑，面向3流形。的同构是作为一个组成的三个同构，其中第一个涉及到一个版本的嵌入式接触同源的辅助流形的Heegaard弗洛尔同调的原始。本文描述了嵌入接触同调链复形上的微分与Heegaard Floer链复形上的微分之间的关系。本文还描述了这两个复形的同调群上的各种典型自同态之间的关系。
HF=HM II: Reeb orbits and holomorphic curves for the ech/Heegaard-Floer correspondence
这是第二个五个文件，建设之间的同构Seiberg-Witten弗洛尔同调和Heegaard弗洛尔同调的一个给定的紧凑，面向3流形。的同构是作为一个组成的三个同构，其中第一个涉及到一个版本的嵌入式接触同源的辅助流形的Heegaard弗洛尔同调的原始。本文描述了这一辅助流形，它的几何，嵌入接触同调链复形的生成元与Heegaard Floer链复形的生成元之间的关系.伪全纯曲线，定义嵌入式接触同调链复形上的微分，这里也被描述为第一步，后者复杂的微分与Heegaard Floer复杂。
Uniqueness of holomorphic quilts lifted from holomorphic bigons on surfaces
本文证明了从拉格朗日 Floer 链群双边形提升的全纯被子的唯一性，并据此给出闭曲面上浸入拉格朗日 Floer 链群边界映射的组合计算方法，构造了展示数字 8 气泡现象的例子，证实了 Cazassus–Herald–Kirk–Kotelskiy 猜想。
Symplectic cohomology and Viterbo's theorem
这是一本关于辛上同调的研究专著（伪装成高级研究生教科书），它提供了闭流形的余切丛的这个版本的Hamilton Floer上同调的构造。重点讨论了文献中忽略的几个方面：（1）不假定基是可定向的或Spin的;（2）利用自由闭路空间上的局部系统定义了Floer上同调的扭曲形式;（3）构造了（扭曲的）Batalin-Vilkovisky结构;（4）验证了BV关系.在此设置（即与所有适当的扭曲），证明维泰博定理有关的自由闭路空间的同调的辛上同调，我们表明，这张地图尊重BV结构。通过构造三个不同的映射，证明了Viterbo定理，并利用亏格为0的带边界Riemann曲面的模空间的退化，证明了其中两个复合是同构的.两个地图构建是新的，并使用灵感来自拉格朗日版本的家庭弗洛尔上同调的想法。
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