Floer homology of open subsets and a relative version of Arnold's conjecture

Kasturirangan, Rajesh; Oh, Yong─Geun

doi:10.1007/pl00004822

Cited by 23 publications

(64 citation statements)

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“…An important source of inspiration is Kashiwara's construction [22] of characteristic cycles for constructible sheaves on Z; and indeed, Nadler proves that, once derived, the resulting version of the Fukaya category is equivalent to the constructible derived category. (A similar point of view was taken in earlier papers of Kasturirangan and Oh [23,24,31]). Generally speaking, to get a finite resolution of the diagonal, this category has to be modified further, by restricting the behaviour at infinity; however, if one is only interested in applications to closed Lagrangian submanifolds, this step can be greatly simplified.…”

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confidence: 54%

The Symplectic Geometry of Cotangent Bundles from a Categorical Viewpoint

Fukaya

Seidel

Smith

2008

Homological Mirror Symmetry

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Abstract. We describe various approaches to understanding Fukaya categories of cotangent bundles. All of the approaches rely on introducing a suitable class of noncompact Lagrangian submanifolds. We review the work of Nadler-Zaslow [30,29] and the authors [15], before discussing a new approach using family Floer cohomology [11] and the "wrapped Fukaya category". The latter, inspired by Viterbo's symplectic homology, emphasises the connection to loop spaces, hence seems particularly suitable when trying to extend the existing theory beyond the simply-connected case.A classical problem in symplectic topology is to describe exact Lagrangian submanifolds inside a cotangent bundle. The main conjecture, usually attributed to Arnol'd [4], asserts that any (compact) submanifold of this kind should be Hamiltonian isotopic to the zero-section. In this sharp form, the result is known only for S 2 and RP 2 , and the proof uses methods which are specifically four-dimensional (both cases are due to Hind [17]; concerning the state of the art for surfaces of genus > 0, see [18]). In higher dimensions, work has concentrated on trying to establish topological restrictions on exact Lagrangian submanifolds. There are many results dealing with assorted partial aspects of this question ([26, 37, 38, 7, 35] and others), using a variety of techniques. Quite recently, more categorical methods have been added to the toolkit, and these have led to a result covering a fairly general situation. The basic statement (which we will later generalise somewhat) is as follows:Theorem 0.1 (Nadler, Fukaya-Seidel-Smith). Let Z be a closed, simply connected manifold which is spin, and M = T * Z its cotangent bundle. Suppose that L ⊂ M is an exact closed Lagrangian submanifold, which is also spin, and additionally has vanishing Maslov class. Then: (i) the projection L ֒→ M → Z has degree ±1; (ii) pullback by the projection is an isomorphism H * (Z; K) ∼ = H * (L; K) for any coefficient field K; and (iii) given any two Lagrangian submanifolds L 0 , L 1 ⊂ M with these properties, which meet transversally, one has |L 0 ∩ L 1 | ≥ dim H * (Z; K). Remarkably, three ways of arriving at this goal have emerged, which are essentially independent of each other, but share a basic philosophical outlook. One proof is due to Nadler [29], building on earlier work of Nadler and Zaslow [30] (the result in [29] is formulated for K = C, but it seems that the proof goes through for any K). Another one is given in [15], and involves, among other things, tools from [36] (for technical reasons, this actually works only for char(K) = 2). The third one, which is collaborative work of the three authors of this paper, is not complete at the time of writing, mostly because it relies on ongoing developments in general Floer homology theory. In spite of this, we included a description of it, to round off the overall picture.The best starting point may actually be the end of the proof, which can be taken to be roughly the same in all three cases. Let L be as in Theorem 0.1. The Floe...

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confidence: 54%

The Symplectic Geometry of Cotangent Bundles from a Categorical Viewpoint

Fukaya

Seidel

Smith

2008

Homological Mirror Symmetry

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“…In the context of Lagrangian graphs in the cotangent bundle of a compact manifold, Fukaya and Oh [9] extended this to an identification of the Morse and Fukaya A ∞ -categories by establishing an oriented diffeomorphism of the moduli spaces of gradient trees and holomorphic polygons involved in the definition of the higher composition maps. In the local setting of graphs over open sets with transverse boundaries, Kasturirangan and Oh [21,22] prove an equivalence of the Morse and Floer chain complexes. In this section, we adapt the approach of Fukaya and Oh to prove an A ∞ -equivalence of Morse and Fukaya A ∞ -categories which include all standard objects.…”

Section: Lemma 633mentioning

confidence: 99%

“…First, there is the index formula of Dubson [6] and Kashiwara [17]. This states that given a constructible complex of sheaves F, its Euler characteristic χ(X, F) is equal to the intersection of Lagrangian cycles CC(F) · [df ] where df is the graph of a sufficiently generic function f : X → R. More generally, given two constructible complexes of sheaves F 1 , F 2 , a formula of MacPherson (see the introduction of [11], the lecture notes [13], and a Floer-theoretic interpretation [21,22]) expresses the Euler characteristic of their tensor product in terms of the intersection of their characteristic cycles:…”

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confidence: 99%

Constructible sheaves and the Fukaya category

Nadler

Zaslow

2008

J. Amer. Math. Soc.

201

276

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Let X X be a compact real analytic manifold, and let T ∗ X T^*X be its cotangent bundle. Let S h ( X ) Sh(X) be the triangulated dg category of bounded, constructible complexes of sheaves on X X . In this paper, we develop a Fukaya A ∞ A_\infty -category F u k ( T ∗ X ) Fuk(T^*X) whose objects are exact, not necessarily compact Lagrangian branes in the cotangent bundle. We write T w F u k ( T ∗ X ) Tw Fuk(T^*X) for the A ∞ A_\infty -triangulated envelope of F u k ( T ∗ X ) Fuk(T^*X) consisting of twisted complexes of Lagrangian branes. Our main result is that S h ( X ) Sh(X) quasi-embeds into T w F u k ( T ∗ X ) Tw Fuk(T^*X) as an A ∞ A_\infty -category. Taking cohomology gives an embedding of the corresponding derived categories.

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“…According to [2], there is a "plumbing model" for the Fukaya category of a pair of exact graded Lagrangians which meet pairwise cleanly. We only require the somewhat simpler cohomological result, which goes back to Pozniak and follows fairly straightforwardly from Lemma 5.10; see for instance [24]. It may help to recall that given a diagram of cleanly intersecting submanifolds…”

Section: 4mentioning

confidence: 99%

Khovanov homology from Floer cohomology

Abouzaid

Smith

2018

J. Amer. Math. Soc.

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This paper realises the Khovanov homology of a link in S 3 as a Lagrangian Floer cohomology group, establishing a conjecture of Seidel and the second author. The starting point is the previously established formality theorem for the symplectic arc algebra over a field k of characteristic zero. Here we prove the symplectic cup and cap bimodules which relate different symplectic arc algebras are themselves formal over k, and construct a long exact triangle for symplectic Khovanov cohomology. We then prove the symplectic and combinatorial arc algebras are isomorphic over Z in a manner compatible with the cup bimodules. It follows that Khovanov cohomology and symplectic Khovanov cohomology co-incide in characteristic zero.

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Floer homology of open subsets and a relative version of Arnold's conjecture

Cited by 23 publications

References 13 publications

The Symplectic Geometry of Cotangent Bundles from a Categorical Viewpoint

The Symplectic Geometry of Cotangent Bundles from a Categorical Viewpoint

Constructible sheaves and the Fukaya category

Khovanov homology from Floer cohomology

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