The symplectic arc algebra is formal

Abouzaid, Mohammed; Smith, Ivan

doi:10.1215/00127094-3449459

Cited by 23 publications

(102 citation statements)

References 43 publications

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“…The cylindrical formulation. To visualize holomorphic disks in (Y n , K A , K B ) it will be convenient to use the following six-dimensional formulation; compare [2,Section 5.8]. First, an intersection point between K A and K B corresponds to an n-tuple of points…”

Section: A Brief Review Of Symplectic Khovanov Homologymentioning

confidence: 99%

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A flexible construction of equivariant Floer homology and applications

Hendricks

Lipshitz²,

Sarkar³

2016

J Topology

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Seidel–Smith and Hendricks used equivariant Floer cohomology to define some spectral sequences from symplectic Khovanov homology and Heegaard Floer homology. These spectral sequences give rise to Smith‐type inequalities. Similar‐looking spectral sequences have been defined by Lee, Bar–Natan, Ozsváth–Szabó, Lipshitz–Treumann, Szabó, Sarkar–Seed–Szabó, and others. In this paper, we give another construction of equivariant Floer cohomology with respect to a finite group action and use it to prove some invariance properties of these spectral sequences; prove that some of these spectral sequences agree; improve Hendricks's Smith‐type inequalities; give some theoretical and practical computability results for these spectral sequences; define some new spectral sequences conjecturally related to Sarkar–Seed–Szabó's; and introduce a new concordance homomorphism and concordance invariants. We also digress to prove invariance of Manolescu's reduced symplectic Khovanov homology.

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Section: A Brief Review Of Symplectic Khovanov Homologymentioning

confidence: 99%

“…where θ m = 2π/2 m , giving a subgroup inclusion D 2 m → O (2). Passing to group cohomologies (with coefficients in F 2 ), we get a map H * (D 2 m ) ← F 2 [w 1 , w 2 ], where w 1 ∈ H 1 (BO(2)) and w 2 ∈ H 2 (BO(2)) are the universal first and second Stiefel-Whitney classes.…”

Section: A Brief Review Of Cyclic and Dihedral Groupsmentioning

confidence: 99%

A flexible construction of equivariant Floer homology and applications

Hendricks

Lipshitz²,

Sarkar³

2016

J Topology

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“…where the connecting homomorphism (marked by [1] in our notation, because it has degree 1) is canonical in H * (Tw(A)). By applying this repeatedly, one can find a collection of exact triangles which decompose any twisted complex into objects of A (up to shifts).…”

Section: Twisted Complexes a ∞ -Modulesmentioning

confidence: 99%

“…The first example may have been the bigrading on the Floer cohomology of certain Lagrangian spheres in the Milnor fibres of type (A) hypersurface singularities in C n , described in [12] (it turned out later [21,Section 20] that this is compatible with the A ∞ -structure of the Fukaya category only if n 3). The geometric origin of such symmetries (on the infinitesimal level) has been studied in [25], with applications in [1,15,24]. A roadmap is provided (via mirror symmetry) by the theory of equivariant coherent sheaves, and its applications in algebraic geometry and geometric representation theory; the relevant literature is too vast to survey properly, but [8,18] have been influential for the developments presented here.…”

Section: Introductionmentioning

confidence: 99%

Picard–Lefschetz theory and dilating ℂ*-actions

Seidel¹

2015

J Topology

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Abstract. We consider C * -actions on Fukaya categories of exact symplectic manifolds. Such actions can be constructed by dimensional induction, going from the fibre of a Lefschetz fibration to its total space. We explore applications to the topology of Lagrangian submanifolds, with an emphasis on ease of computation. IntroductionIt has been gradually recognized that certain classes of non-closed symplectic manifolds admit symmetries of a new kind. These symmetries are not given by groups acting on the manifold, but instead appear as extra structure on Floer cohomology, or more properly on the Fukaya category. The first example may have been the bigrading on the Floer cohomology of certain Lagrangian spheres in the Milnor fibres of type (A) hypersurface singularities in C n , described in [12] (it turned out later [21, Section 20] that this is compatible with the A ∞ -structure of the Fukaya category only if n ≥ 3). The geometric origin of such symmetries (on the infinitesimal level) has been studied in [25], with applications in [24,15,1]. A roadmap is provided (via mirror symmetry) by the theory of equivariant coherent sheaves, and its applications in algebraic geometry and geometric representation theory; the relevant literature is too vast to survey properly, but [18,8] have been influential for the developments presented here.In [22], the example from [12] was used as a test case for talking about such symmetries in an algebraic language of A ∞ -categories with C * -actions. Here, we generalize that approach, and combine it with the symplectic version of Picard-Lefschetz theory [21]. Recall that classical Picard-Lefschetz theory provides (among other things) a way of computing the intersection pairing in the middle-dimensional homology of an affine algebraic variety, by induction on the dimension. As one consequence of our construction, one gets a similar machinery for defining and computing algebraic analogues of the q-intersection numbers from [25]. Aside from their intrinsic interest, these q-intersection numbers have implications for the topology of Lagrangian submanifolds. These are similar in spirit to those derived in [24], but benefit from the more rigid setup of C * -actions, as well as the greater ease of doing computations in a purely algebraic framework. In particular, Example 1.20 would be out of reach of the methods in [24], since those only involved the first derivative in the equivariant parameter q around q = 1, whereas (1.80) vanishes at least to order 2 at that point. Another noteworthy comparison is [14], which contains an example of a

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“…The curves ζ i generate all simple closed curves, in the sense that any homotopically essential curve must have non-trivial geometric intersection number with some ζ i , and the Dehn twists τ ζ j generate the mapping class group. 6 Over the ring A = i,j HF * (ζ i , ζ j ) the module Y σ defined by the curve σ is isomorphic to the sumŶ σ = S ζ 3 ⊕ S ζ 3 [1] of the simple module over ζ 3 and its shift, essentially because there are no non-constant holomorphic lunes (bigons) or triangles in the picture. However, the endomorphism algebra of…”

Section: Mapping Class Groupsmentioning

confidence: 99%