Viterbo conjecture for Zoll symmetric spaces

Shelukhin, Egor

doi:10.48550/arxiv.1811.05552

Cited by 10 publications

(27 citation statements)

References 92 publications

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“…Proof of Lemma 16. This is a direct consquence of [63,Theorem D], building on [40,Remark 62]. Alternatively, passing to the canonical Λ 0 -complex, and tensoring with Λ, this follows from the rank-nullity theorem: Proof of Proposition 9.…”

Section: Further Technical Proofsmentioning

confidence: 96%

“…Barcode of Hamiltonian diffeomorphism with isolated fixed points. Furthermore, it was shown in [63] following [40], that if φ has isolated fixed points, then the barcode B ′ (φ) consists of a finite number of bars, of them B(K) are infinite, and…”

Section: 33mentioning

confidence: 99%

“…We refer to [69,35,63] for more details of the identifications between the various descriptions of the bar-length spectrum, and notions of persistence.…”

Section: 34mentioning

confidence: 99%

“…In order to deal with this issue, we develop a version of the above spectral sequence by working with coefficients in Λ univ,K,0 , removing the recapping ambiguity, and describing a homotopy-canonical Λ 0 complex on the sum of the local homology groups that computes the bar-length spectrum in the barcode of φ. The latter is known to be finite: see [40] and [63].…”

Section: 37mentioning

confidence: 99%

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On the Hofer-Zehnder conjecture

Shelukhin¹

2019

Preprint

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We prove that if a Hamiltonian diffeomorphism on a closed monotone symplectic manifold with semisimple quantum homology has a finite number of contractible periodic points then the sum of the ranks of the local Floer homologies at its contractible fixed points is equal to the total dimension of the homology of the manifold. This constitutes a higher-dimensional homological generalization of a celebrated result of Franks from 1992, as conjectured by Hofer and Zehnder in 1994. Contents1. Setup 1 2. Outline 3 3. Preliminaries 5 4. Proof of Theorem B 21 5. Algebraic statements 22 6. The Z/(p)-equivariant Tate homology and natural isomorphisms 24 7. Proof of Theorem A 32 8. Further technical proofs 33 Acknowledgements 45 References 46 1. Setup Definition 1. We call a closed symplectic manifold (M, ω) monotone if the class

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Section: Further Technical Proofsmentioning

confidence: 96%

Section: 33mentioning

confidence: 99%

“…We refer to [69,35,63] for more details of the identifications between the various descriptions of the bar-length spectrum, and notions of persistence.…”

Section: 34mentioning

confidence: 99%

Section: 37mentioning

confidence: 99%

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On the Hofer-Zehnder conjecture

Shelukhin¹

2019

Preprint

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“…1 More generally, they defined the mapping µa for each a ∈ H 1 (M ), and proved that it has the properties anologous to those of a partial quasi-morphism (due to the later results of Shelukhin [20] and Kislev-Shelukhin [11], µa give rise to genuine quasi-morphisms for some M ).…”

Section: Introductionmentioning

confidence: 99%

A note on partial quasi-morphisms and products in Lagrangian Floer homology in cotangent bundles

Katic¹,

Milinković²,

Nikolić³

2021

Preprint

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An Arnold-type principle for non-smooth objects

Buhovsky

Humilière

Seyfaddini

2022

J. Fixed Point Theory Appl.

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In this article we study the Arnold conjecture in settings where objects under consideration are no longer smooth but only continuous. The example of a Hamiltonian homeomorphism, on any closed symplectic manifold of dimension greater than 2, having only one fixed point shows that the conjecture does not admit a direct generalization to continuous settings. However, it appears that the following Arnoldtype principle continues to hold in C 0 settings: Suppose that X is a non-smooth object for which one can define spectral invariants. If the number of spectral invariants associated to X is smaller than the number predicted by the (homological) Arnold conjecture, then the set of fixed/intersection points of X is homologically non-trivial, hence it is infinite.We recently proved that the above principle holds for Hamiltonian homeomorphisms of closed and aspherical symplectic manifolds. In this article, we verify this principle in two new settings: C 0 Lagrangians in cotangent bundles and Hausdorff limits of Legendrians in 1-jet bundles which are isotopic to 0-section.An unexpected consequence of the result on Legendrians is that the classical Arnold conjecture does hold for Hausdorff limits of Legendrians in 1-jet bundles. Contents 1 Introduction and main results2 Preliminaries from symplectic geometry 2.

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Viterbo conjecture for Zoll symmetric spaces

Cited by 10 publications

References 92 publications

On the Hofer-Zehnder conjecture

On the Hofer-Zehnder conjecture

A note on partial quasi-morphisms and products in Lagrangian Floer homology in cotangent bundles

An Arnold-type principle for non-smooth objects

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