A polyfold proof of the Arnold conjecture

Filippenko, Benjamin; Wehrheim, Katrin

doi:10.1007/s00029-021-00680-z

Cited by 4 publications

(2 citation statements)

References 24 publications

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“…Symplectic cohomology is a special case of Hamiltonian Floer cohomology, whose polyfold construction was sketched in [26]. An alternative approach is using the full SFT polyfolds [12] as in [11]. In those constructions, the linearization in the polyfold and the linearization of the Floer equation modulo an R-translation are the same.…”

Section: Symplectic Cohomology and Fiber Bundlesmentioning

confidence: 99%

On the cohomology ring of symplectic fillings

Zhou

2023

Algebr. Geom. Topol.

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We consider symplectic cohomology twisted by sphere bundles, which can be viewed as an analogue of symplectic cohomology with local systems. Using the associated Gysin exact sequence, we prove the uniqueness of part of the ring structure on cohomology of fillings for those asymptotically dynamically convex manifolds with vanishing property considered by Zhou (Int. Math. Res. Not. 2020 (2020) 9717-9729 and J. Topol. 14 (2021) 112-182). In particular, for any simply connected 4nC1dimensional flexibly fillable contact manifold Y , we show that the real cohomology H .W / is unique as a ring for any Liouville filling W of Y as long as c 1 .W / D 0. Uniqueness of real homotopy type of Liouville fillings is also obtained for a class of flexibly fillable contact manifolds.

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Section: Symplectic Cohomology and Fiber Bundlesmentioning

confidence: 99%

On the cohomology ring of symplectic fillings

Zhou

2023

Algebr. Geom. Topol.

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“…The most recent advancement towards the homological Arnold conjecture by Abouzaid-Blumberg [AB21], which relies more heavily on stable homotopy theory, bounds the number of periodic orbits from below by the sum of Betti numbers in any finite field. In addition to the aforementioned works, using different versions of the virtual technique, the weak Arnold conjecture over rational numbers is reproved in [Par16] and [FW22].…”

Section: Introductionmentioning

confidence: 99%

Arnold conjecture over integers

Bai¹,

Xu²

2022

Preprint

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For any closed symplectic manifold, we show that the number of 1-periodic orbits of a nondegenerate Hamiltonian thereon is bounded from below by a version of total Betti number over Z of the ambient space taking account of the total Betti number over Q and torsions of all characteristic. The proof is based on constructing a Hamiltonian Floer theory over the Novikov ring with integer coefficients, which generalizes our earlier work for constructing integervalued Gromov-Witten type invariants. In the course of the construction, we build a Hamiltonian Floer flow category with compatible smooth global Kuranishi charts. This generalizes a recent work of Abouzaid-McLean-Smith, which might be of independent interest. Contents 1. Introduction 1 2. Recap of the FOP natural transformation and multiplicativity 11 3. Abstract constructions of chain complexes and maps over the integers 22 4. Proof of the integral Arnold conjecture 52 5. Global Kuranishi charts on Floer moduli spaces 69 6. Smoothing 118 7. Constructions for PSS, SSP, and the homotopy 146 Appendix A. Product of canonical Whitney stratifications 155 Appendix B. Relative stable equivariant smoothing 160 Appendix C. A proof sketch of Proposition 5.64 163 References 166

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A local-to-global inequality for spectral invariants and an energy dichotomy for Floer trajectories

Buhovsky,

Tanny

2024

J. Fixed Point Theory Appl.

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We study a local-to-global inequality for spectral invariants of Hamiltonians whose supports have a “large enough” disjoint tubular neighborhood on semipositive symplectic manifolds. As a corollary, we deduce this inequality for disjointly supported Hamiltonians that are $$C^0$$ C 0 -small (when fixing the supports). In particular, we present the first examples of such an inequality when the Hamiltonians are not necessarily supported in domains with contact-type boundaries, or when the ambient manifold is irrational. This extends a series of previous works studying locality phenomena of spectral invariants [9, 13, 15, 20, 25, 27]. A main new tool is a lower bound, in the spirit of Sikorav, for the energy of Floer trajectories that cross the tubular neighborhood against the direction of the negative-gradient vector field.

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A polyfold proof of the Arnold conjecture

Cited by 4 publications

References 24 publications

On the cohomology ring of symplectic fillings

On the cohomology ring of symplectic fillings

Arnold conjecture over integers

A local-to-global inequality for spectral invariants and an energy dichotomy for Floer trajectories

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