The augmentation category map induced by exact Lagrangian cobordisms

Pan, Yu

doi:10.2140/agt.2017.17.1813

Cited by 18 publications

(17 citation statements)

References 31 publications

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“…The next exact sequence is the following, analogous to results in [8, Theorem 1.1] and Pan [45,Theorem 1.2].…”

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confidence: 82%

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Lagrangian cobordism functor in microlocal sheaf theory

Li¹

2021

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Given a Lagrangian cobordism L of Legendrian submanifolds from Λ− to Λ+, we construct a functor Φ * L :) between sheaf categories of compact objects with singular support on Λ± and its adjoint on sheaf categories of proper objects, using Nadler-Shende's work. This gives a sheaf theory description analogous to the Lagrangian cobordism map on Legendrian contact homologies and the adjoint on their unital augmentation categories. We also deduce some long exact sequences and new obstructions to Lagrangian cobordisms between high dimensional Legendrian submanifolds.

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“…The next exact sequence is the following, analogous to results in [8, Theorem 1.1] and Pan [45,Theorem 1.2].…”

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confidence: 82%

“…tri is injective on objects as long as H 0 (L, Λ − ) = 0. The proof is the same as [45], where one uses the fact that…”

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confidence: 99%

Lagrangian cobordism functor in microlocal sheaf theory

Li¹

2021

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“…for any q that is a power of a prime number [Pan17]. (5) If Λ admits a Maslov 0 Lagrangian filling L, and if L denotes the augmentation of Λ induced by L, then…”

Section: 3mentioning

confidence: 99%

Constructions of Lagrangian cobordisms

Blackwell¹,

Legout²,

Leverson³

et al. 2021

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Lagrangian cobordisms between Legendrian knots arise in Symplectic Field Theory and impose an interesting and not well-understood relation on Legendrian knots. There are some known "elementary" building blocks for Lagrangian cobordisms that are smoothly the attachment of 0-and 1-handles. An important question is whether every pair of nonempty Legendrians that are related by a connected Lagrangian cobordism can be related by a ribbon Lagrangian cobordism, in particular one that is "decomposable" into a composition of these elementary building blocks. We will describe these and other combinatorial building blocks as well as some geometric methods, involving the theory of satellites, to construct Lagrangian cobordisms. We will then survey some known results, derived through Heegaard Floer Homology and contact surgery, that may provide a pathway to proving the existence of nondecomposable (nonribbon) Lagrangian cobordisms.

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“…In the case of the augmentation category Aug + (Λ) (defined in [NRS + ]), an exact Lagrangian cobordism from Λ − to Λ + also induces a functor F : Aug + (Λ − ) → Aug + (Λ + ). In particular, this functor was shown to be injective on equivalence classes of augmentations and cohomologically faithful by Yu Pan [Pan17].…”

Section: Legendrian Contact Homologymentioning

confidence: 99%

Product structures in Floer theory for Lagrangian cobordisms

Legout¹

2018

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We construct a product on the Floer complex associated to a pair of Lagrangian cobordisms. More precisely, given three exact transverse Lagrangian cobordisms in the symplectization of a contact manifold, we define a map m 2 by a count of rigid pseudo-holomorphic disks with boundary on the cobordisms and having punctures asymptotic to intersection points and Reeb chords of the negative Legendrian ends of the cobordisms. More generally, to a (d + 1)-tuple of exact transverse Lagrangian cobordisms we associate a map m d such that the family (m d ) d≥1 are A∞-maps. Finally, we extend the Ekholm-Seidel isomorphism to an A∞-morphism, giving in particular that it is a ring isomorphism. r∈Σ1∩Σ3 δ ′ 1

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The augmentation category map induced by exact Lagrangian cobordisms

Cited by 18 publications

References 31 publications

Lagrangian cobordism functor in microlocal sheaf theory

Lagrangian cobordism functor in microlocal sheaf theory

Constructions of Lagrangian cobordisms

Product structures in Floer theory for Lagrangian cobordisms

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