Contact Homology, Capacity and Non-Squeezing in \mathbb{R}^{2n}\times S^{1} via Generating Functions

We study Liouville fillable contact manifolds (Σ, ξ) with non-zero Rabinowitz Floer homology and assign spectral numbers to paths of contactomorphisms. As a consequence we prove that Cont0(Σ, ξ) is orderable in the sense of Eliashberg and Polterovich. This provides a new class of orderable contact manifolds. If the contact manifold is in addition periodic or a prequantization space M ×S 1 for M a Liouville manifold, then we construct a contact capacity. This can be used to prove a general non-squeezing result, which amongst other examples in particular recovers the beautiful non-squeezing results from [EKP06].

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“…As before Z denotes a non-zero class in RFH * (Σ, W ). The basic definitions and results that follow are based on Sandon's article [San11].…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…Remark 1.16. The notion of contact capacity was introduced by Sandon in [San11]. She was the first to discover a connection between translated points and orderability and other contact rigidity phenomena.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Orderability, contact non-squeezing, and Rabinowitz Floer homology

Albers

Merry

2018

Journal of Symplectic Geometry

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“…In contrast to fixed points of contactomorphisms (that are completely flexible), translated and discriminant points turn out to be related to interesting contact rigidity phenomena. For example, translated and discriminant points for contactomorphisms of R 2n × S 1 play a crucial role in the proof of the contact non-squeezing theorem [EKP06,S11a], while translated and discriminant points of contactomorphisms of RP 2n+1 are intimately related to Givental's non-linear Maslov index [Giv90]. In view of these examples we believe that the discriminant metric should be an interesting object to study, since its properties on a given contact manifold might reflect the existence of contact rigidity phenomena such as contact non-squeezing, orderability, or the existence of quasimorphisms on the contactomorphism group.…”

Section: Introductionmentioning

confidence: 99%

The discriminant and oscillation lengths for contact and Legendrian isotopies

Colin¹,

Sandon²

2015

J. Eur. Math. Soc.

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We define an integer-valued non-degenerate bi-invariant metric (the discriminant metric) on the universal cover of the identity component of the contactomorphism group of any contact manifold. This metric has a very simple geometric definition, based on the notion of discriminant points of contactomorphisms. Using generating functions we prove that the discriminant metric is unbounded for the standard contact structures on R 2n × S 1 and RP 2n+1 . On the other hand we also show by elementary arguments that the discriminant metric is bounded for the standard contact structures on R 2n+1 and S 2n+1 . As an application of these results we get that the contact fragmentation norm is unbounded for R 2n × S 1 and RP 2n+1 . By elaborating on the construction of the discriminant metric we then define an integer-valued bi-invariant pseudo-metric, that we call the oscillation pseudo-metric, which is non-degenerate if and only if the contact manifold is orderable in the sense of Eliashberg and Polterovich and, in this case, it is compatible with the partial order. Finally we define the discriminant and oscillation lengths of a Legendrian isotopy, and prove that they are unbounded for T * B × S 1 for any closed manifold B, for RP 2n+1 and for some 3-dimensional circle bundles.1 Recall that any conjugation-invariant norm ν : G → [0, +∞) on a group G induces a bi-invariant metric dν on G by defining dν (f, g) = ν(f −1 g). 2 If M is not compact, we will consider the discriminant norm on the universal cover of the identity component of the group of compactly supported contactomorphisms.

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“…In these works the role played by discriminant and translated points is made more explicit, and appears to be similar to the one described in [Gi90]. Recall from [Sa11a,Sa12] that a point p of a contact manifold (V, ξ) is said to be a translated point of a contactomorphism φ with respect to a contact form α for ξ if p is a discriminant point of ϕ α −η • φ for some real number η (called the time-shift ), where ϕ α t denotes the Reeb flow. In terms of the Legendrian graph gr(φ) in the contact product V × V × R, discriminant and translated points correspond, respectively, to intersections and Reeb chords between gr(φ) and the diagonal ∆ × {0} = gr(id).…”

Section: Introductionmentioning

confidence: 87%

Givental’s Non-linear Maslov Index on Lens Spaces

Granja

Karshon

Pabiniak

et al. 2020

International Mathematics Research Notices

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Givental's non-linear Maslov index, constructed in 1990, is a quasimorphism on the universal cover of the identity component of the contactomorphism group of real projective space. This invariant was used by several authors to prove contact rigidity phenomena such as orderability, unboundedness of the discriminant and oscillation metrics, and a contact geometric version of the Arnold conjecture. In this article we give an analogue for lens spaces of Givental's construction and its applications.

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Contact Homology, Capacity and Non-Squeezing in \mathbb{R}^{2n}\times S^{1} via Generating Functions

Cited by 32 publications

References 18 publications

Orderability, contact non-squeezing, and Rabinowitz Floer homology

Orderability, contact non-squeezing, and Rabinowitz Floer homology

The discriminant and oscillation lengths for contact and Legendrian isotopies

Givental’s Non-linear Maslov Index on Lens Spaces

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