The Bott–Samelson theorem for positive Legendrian isotopies

Dahinden, Lucas

doi:10.1007/s12188-017-0180-7

Cited by 5 publications

(3 citation statements)

References 12 publications

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“…The definition of swappability makes sense for Lagrangians as well as for stops, where it turns out to have been studied before: Theorem 1.2 ( [13,8]). Let Λ = S * p Q ⊂ S * Q = ∂ ∞ T * Q be a swappable cosphere fiber.…”

Section: Introductionmentioning

confidence: 99%

On partially wrapped Fukaya categories

Sylvan

2019

Journal of Topology

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We define a new class of symplectic objects called 'stops', which, roughly speaking, are Liouville hypersurfaces in the boundary of a Liouville domain. Locally, these can be viewed as pages of a compatible open book. To a Liouville domain with a collection of disjoint stops, we assign an A∞-category called its partially wrapped Fukaya category. An exact Landau-Ginzburg model gives rise to a stop, and the corresponding partially wrapped Fukaya category is meant to agree with the Fukaya category one is supposed to assign to the Landau-Ginzburg model. As evidence, we prove a formula that relates these partially wrapped Fukaya categories to the wrapped Fukaya category of the underlying Liouville domain. This operation is mirror to removing a divisor. 373is the open-closed map. Here, the action filtration on Hochschild homology is induced from the action filtration on Floer cochain groups; for a precise definition, see Section 4.3.In particular, any punctured Riemann surface other than C and any cotangent bundle is strongly nondegenerate.Remark 1.1. A nondegenerate Liouville domain F is one for which some element e ∈ HH * (W(F )) satisfies OC(e) = 1. This condition is Abouzaid's generation criterion [2], which says that such an F admits a finite collection of Lagrangians which split-generate W(F ). Ganatra further proved that in this case, OC is an isomorphism [15], which in particular implies that e is unique.An approximate version of Theorem 4.21 can be stated as follows: Theorem 1.2. Let M be a Liouville domain, and let σ be a strongly nondegenerate stop in M . Let B ⊂ W σ (M ) be the full subcategory of objects supported near σ. Then the inclusion W σ (M ) → W(M ) induces a fully faithful functorwhere the quotient is a quotient of an A ∞ category by a full subcategory in the sense of .At the most basic level, Theorem 1.2 shows that the partially wrapped Fukaya category, together with B, really knows at least as much as the wrapped Fukaya category. That is, restricting to the zero-filtered part but remembering the stop does not lose information. At an intuitive level, this happens because the 'nice' presentation of W(M ) giving rise to the stop filtration uses a contact form with a large number of canceling Reeb chords. These chords live at different levels in the filtration, so passing to the zero-filtered part results in them no longer canceling. In a more minimal presentation, these chords could be eliminated geometrically.The key ingredient of the proof of Theorem 1.2 is an auxiliary filtration on W(M ), presented as the trivial quotient A = W(M )/B. The benefit of this quotient presentation is that it naturally contains the category A 0 = W σ (M )/B(σ) as the minimally filtered part, which makes it possible to build a homotopy which retracts A onto A 0 . The homotopy itself requires a filtered version of the annulus trick, which was introduced in [2] and extended in a forthcoming paper by Abouzaid and Auroux. Specifically, one factors the identity operation as a composition of a product and a coproduct, where ...

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Section: Introductionmentioning

confidence: 99%

On partially wrapped Fukaya categories

Sylvan

2019

Journal of Topology

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“…Finer versions are proven in [19,Chapter 7]. All these results hold true for Reeb flows on spherizations, even time-dependent ones, see [46,36]. Here, • ∞ denotes the supremum norm induced by the operator norm on endomorphisms of T M that is determined by any Riemannian metric on M. The limit defining Γ + exists because the sequence (log dφ n ∞ ) is subadittive.…”

Section: Sfrag Replacementsmentioning

confidence: 97%

Entropy collapse versus entropy rigidity for Reeb and Finsler flows

Abbondandolo¹,

Alves²,

Sağlam³

et al. 2021

Preprint

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“…Also, see the work [20] by Dahinden for obstructions in certain cases when the universal cover is not open.…”

Section: Introductionmentioning

confidence: 99%

Positive Legendrian isotopies and Floer theory

Chantraine¹,

Colin²,

Rizell³

2019

Annales De l'Institut Fourier

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Positive loops of Legendrian embeddings are examined from the point of view of Floer homology of Lagrangian cobordisms. This leads to new obstructions to the existence of a positive loop containing a given Legendrian, expressed in terms of the Legendrian contact homology of the Legendrian submanifold. As applications, old and new examples of orderable contact manifolds are obtained and discussed. We also show that contact manifolds filled by a Liouville domain with non-zero symplectic homology are strongly orderable in the sense of Liu.

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The Bott–Samelson theorem for positive Legendrian isotopies

Cited by 5 publications

References 12 publications

On partially wrapped Fukaya categories

On partially wrapped Fukaya categories

Entropy collapse versus entropy rigidity for Reeb and Finsler flows

Positive Legendrian isotopies and Floer theory

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