A $$C^0$$ C 0 counterexample to the Arnold conjecture

Buhovsky, Lev; Humilière, Vincent; Seyfaddini, Sobhan

doi:10.1007/s00222-018-0797-x

Cited by 21 publications

(31 citation statements)

References 37 publications

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“…Buhovsky-Humilière-Seyfaddini [BHS18] constructed a counterexample of the Arnold conjecture for a Hamiltonian homeomorphism of a closed symplectic manifold with dimension at least 4. However, one still obtains an Arnold-type theorem for a Hamiltonian homeomorphism if one reformulates the conjecture with the notion of spectral invariants, as in [BHS21,Kaw19,BHS19].…”

Section: Related Workmentioning

confidence: 99%

Completeness of derived interleaving distances and sheaf quantization of non-smooth objects

Asano¹,

Ike²

2022

Preprint

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We investigate sheaf-theoretic methods to deal with non-smooth objects in symplectic geometry. We show the completeness of a derived category of sheaves with respect to the interleaving-like distance and construct a sheaf quantization of a hameomorphism. We also develop Lusternik-Schnirelmann theory in the microlocal theory of sheaves. With these new tools, we prove an Arnold-type theorem for the image of the zero-section under a hameomorphism by a purely sheaf-theoretic method.

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Section: Related Workmentioning

confidence: 99%

Completeness of derived interleaving distances and sheaf quantization of non-smooth objects

Asano¹,

Ike²

2022

Preprint

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“…Recently, the notion of barcodes appears as a great tool to study C 0 symplectic geometry, let us cite for example the work of Buhovski, Humilière and Seyfaddini [5], Jannaud [12] and Le Roux Seyfaddini and Viterbo [25].…”

Section: Barcodes In Symplectic Geometrymentioning

confidence: 99%

Barcodes for Hamiltonian homeomorphisms of surfaces

Joly¹

2021

Preprint

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In this article, the main goal is to give a dynamical point of view of Floer homology barcodes for Hamiltonian homeomorphisms of surfaces. More specifically, we describe a way to construct barcodes for Hamiltonian homeomorphisms of surfaces from graphs. We will define graphes associated to maximal isotopies of a Hamiltonian homeomorphism using Le Calvez's positively transverse foliation theory and to those graphs we will associate barcodes. In particular, we will prove that for the simplest cases, our constructions coincide with the Floer Homology barcodes.

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“…There is a priori no reason for this group to be non trivial. Indeed, the flexibility results such as the C 0 -counter example to the Arnold conjecture ( [9]) show that sometimes symplectic homeomorphisms behave very differently than their smooth counter parts. This led Ivan Smith to ask 1 the following question.…”

Section: Dehn-seidel Twist and C 0 Symplectic Mapping Class Groupmentioning

confidence: 99%

Dehn-Seidel twist, $C^0$ symplectic topology and barcodes

Alexandre¹

2021

Preprint

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We initiate the study of the C 0 symplectic mapping class group, i.e. the group of isotopy classes of symplectic homeomorphisms. We prove that the square of the Dehn-Seidel twist does not belong to the connected component of the identity of the group of symplectic homeomorphisms of some Liouville domain. This generalizes to C 0 settings a celebrated result of Seidel. As a consequence, we obtain the non-triviality of the C 0 symplectic mapping class group in these domains.For that purpose, we develop a method coming from Floer theory and barcodes theory. This builds on the recent developments of C 0 -symplectic topology. In particular, we adapt and generalize to our context results by Buhovsky-Humilière-Seyfaddini and Kislev-Shelukhin. ContentsIntroduction 1 1 Preliminaries 13 2 Barcodes and action selectors in symplectic topology 18 3 Continuity of the barcode 35 4 The Dehn-Seidel twist in C 0 -symplectic geometry 52

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A $$C^0$$ C 0 counterexample to the Arnold conjecture

Cited by 21 publications

References 37 publications

Completeness of derived interleaving distances and sheaf quantization of non-smooth objects

Completeness of derived interleaving distances and sheaf quantization of non-smooth objects

Barcodes for Hamiltonian homeomorphisms of surfaces

Dehn-Seidel twist, $C^0$ symplectic topology and barcodes

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