Functoriality for Lagrangian correspondences in Floer theory

Wehrheim, Katrin; Woodward, Chris

doi:10.4171/qt/4

Cited by 73 publications

(142 citation statements)

References 41 publications

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“…As a consequence, in F(M ) (or A(M ), the difference being irrelevant at this level) one cannot expect to have a finite resolution of a closed exact L ⊂ M in terms of cotangent fibres. However, using the Wehrheim-Woodward formalism of Lagrangian correspondences [40], Nadler proves a modified version of this statement, where the fibres are replaced by standard objects associated to certain contractible subsets of Z.…”

Section: Constructible Sheavesmentioning

confidence: 99%

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The Symplectic Geometry of Cotangent Bundles from a Categorical Viewpoint

Fukaya

Seidel

Smith

2008

Lecture Notes in Physics

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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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Section: Constructible Sheavesmentioning

confidence: 99%

“…However, the composition of all three is just the Yoneda embedding for A cpt , which is again full and faithful. In view of [40], and its chain-level analogue [27], each object C in A(M × M ) (and more generally, twisted complex built out of such objects) induces a convolution functor…”

Section: Constructible Sheavesmentioning

confidence: 99%

The Symplectic Geometry of Cotangent Bundles from a Categorical Viewpoint

Fukaya

Seidel

Smith

2008

Lecture Notes in Physics

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“…In sections 3.1 and 3.3 we put together necessary tools for construction of Floer homology of noncompact Lagrangians in Stein manifolds. From these plus Theorem 4.2.8 and the Functoriality Theorem of [19] we get the following.…”

Section: Theorem 428 Up To Isomorphism Of Generalized Correspondenmentioning

confidence: 97%

“…According to [19] Section 2.2, the symplectic category is the category whose objects are compact monotone symplectic manifolds (including exact ones) and whose morphisms are equivalence classes of compact generalized Lagrangian correspondences. The equivalence relation on morphisms is generated by the following two relations.…”

Section: Stein Symplectic Valued Field Theoriesmentioning

confidence: 99%

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Seidel–Smith cohomology for tangles

Rezazadegan

2009

Sel. Math. New Ser.

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Floer Homology on the Extended Moduli Space

Manolescu¹,

Woodward

2011

Progress in Mathematics

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Abstract. Starting from a Heegaard splitting of a three-manifold, we use Lagrangian Floer homology to construct a three-manifold invariant, in the form of a relatively Z/8Z-graded abelian group. Our motivation is to have a well-defined symplectic side of the Atiyah-Floer Conjecture, for arbitrary three-manifolds. The symplectic manifold used in the construction is the extended moduli space of flat SU (2)-connections on the Heegaard surface. An open subset of this moduli space carries a symplectic form, and each of the two handlebodies in the decomposition gives rise to a Lagrangian inside the open set. In order to define their Floer homology, we compactify the open subset by symplectic cutting; the resulting manifold is only semipositive, but we show that one can still develop a version of Floer homology in this setting.

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Functoriality for Lagrangian correspondences in Floer theory

Cited by 73 publications

References 41 publications

The Symplectic Geometry of Cotangent Bundles from a Categorical Viewpoint

The Symplectic Geometry of Cotangent Bundles from a Categorical Viewpoint

Seidel–Smith cohomology for tangles

Floer Homology on the Extended Moduli Space

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