Givental’s Non-linear Maslov Index on Lens Spaces

Granja, Gustavo; Karshon, Yael; Pabiniak, Milena; Sandon, Sheila

doi:10.1093/imrn/rnz350

Cited by 9 publications

(4 citation statements)

References 35 publications

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“…The actions on the skeleta Lk p are the easiest examples of actions with index k + 1. Using the associativity and commutativity of the join up to natural homeomorphism (see [GKPS,Section B.2]) it is easy to see that the index is additive on skeleta of lens spaces as…”

Section: We Write Lkmentioning

confidence: 99%

The cohomological index of free $\Z/p$-actions is not additive with respect to join

Gomes¹,

Granja²

2020

Preprint

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Let p denote an odd prime. We show by example that the inequalities obtained in [GKPS] for the behaviour of the cohomological index of a join of free Z/pactions are sharp. Namely, for all odd integers k, l at least one of which is greater than one, we give examples of finite free Z/p-CW complexes of cohomological indices k and l whose join has index k + l + 1 and also examples where the join has index k + l − 1.

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Section: We Write Lkmentioning

confidence: 99%

The cohomological index of free $\Z/p$-actions is not additive with respect to join

Gomes¹,

Granja²

2020

Preprint

Self Cite

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“…Just to mention a few, there are constructions by Barge-Ghys [4], Borman [10], Entov [20], Entov-Polterovich [22], Gambaudo-Ghys [32], Givental [33], McDuff [40], Ostrover [47], Py [48] and Shelukhin [50]. Contact counterparts are also considered by Givental [33], Borman-Zapolsky [11] and Granja-Karshon-Pabiniak-Sandon [34]. In particular, Entov-Polterovich [22] where .M; !/ is a closed monotone symplectic manifold which satisfies some property.…”

Section: Introductionmentioning

confidence: 99%

Homogeneous quasimorphisms, $C^0$-topology and Lagrangian intersection

Kawamoto

2022

Comment. Math. Helv.

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We construct an example of a non-trivial homogeneous quasimorphism on the group of Hamiltonian diffeomorphisms of the two and four dimensional quadric hypersurfaces which is continuous with respect to both the C^0 -metric and the Hofer metric. This answers a variant of a question of Entov–Polterovich–Py which is one of the open problems listed in the monograph of McDuff–Salamon. Throughout the proof, we make extensive use of the idea of working with different coefficient fields in quantum cohomology rings. As a by-product of the arguments in the paper, we answer a question of Polterovich–Wu regarding homogeneous quasimorphisms on the group of Hamiltonian diffeomorphisms of the complex projective plane and prove some intersection results about Lagrangians in the four dimensional quadric hypersurface.

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“…Seemingly independently of this algebraic structure studies, one can seek an interesting geometry on these groups. Inspired by the Hofer and Viterbo distances in symplectic geometry, there have been several recent papers on invariant norms on contact transformation groups, see Sandon 2010;Fraser, Polterovich, and Rosen 2017;Colin and Sandon 2015;Borman and Zapolsky 2015;Granja, Karshon, Pabiniak, and Sandon 2017, and the survey Sandon 2015. A conjugation invariant norm on a group G is a function ν : G → [0, ∞) satisfying the following properties:…”

Section: Introductionmentioning

confidence: 99%

Contactomorphism groups and Legendrian flexibility

Courte¹,

Massot²

2018

Preprint

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We explain a connection between the algebraic and geometric properties of groups of contact transformations, open book decompositions, and flexible Legendrian embeddings. The main result is that, if a closed contact manifold (V, ξ) has a supporting open book whose pages are flexible Weinstein manifolds, then the connected component G of the identity in its automorphism group is a uniformly simple group: for every non-trivial element g, every other element is a product of at most 128(dim V + 1) conjugates of g ±1 . In particular any conjugation invariant norm on this group is bounded. We also prove the later statement still holds for the universal cover of G.

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Givental’s Non-linear Maslov Index on Lens Spaces

Cited by 9 publications

References 35 publications

The cohomological index of free $\Z/p$-actions is not additive with respect to join

The cohomological index of free $\Z/p$-actions is not additive with respect to join

Homogeneous quasimorphisms, $C^0$-topology and Lagrangian intersection

Contactomorphism groups and Legendrian flexibility

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