Counterexamples in scale calculus

Filippenko, Benjamin; Zhou, Zhengyi; Wehrheim, Katrin

doi:10.1073/pnas.1811701116

Cited by 5 publications

(21 citation statements)

References 7 publications

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“…The notation of an sc-Fredholm section was introduced in [21] to replace the naive definition of Fredholm section such that implicit function theorem holds. The novel extra condition in the sc-Fredholm notion is necessary as the naive definition does not imply implicit function theorem [10]. Therefore, preserving sc-Fredholm property is the most important property in any abstract constructions of polyfolds.…”

Section: Resultsmentioning

confidence: 99%

“…By Lemma A.2, there exists θ 1 ∈ Mor(F a (w a ), F a (w a )), θ 2 ∈ Mor(G a (w a ), G a (w a )), such that If we write 10) then h close to g and z close to x we have…”

Section: Suppose We Have Limmentioning

confidence: 99%

“…x0 , that is, 10) and the inverse to Id x0 is itself, we can require the neighborhoodW is small enough such that we can invert (3.10) to get…”

Section: Moreover For Every Open Neighborhoodwmentioning

confidence: 99%

“…By Theorem 5.10, there is a S 1 -equivariant sc + -perturbation γ supported in N such that s + γ is transverse to 0 on Z S 1 . By [10,Corollary 3.3], there is a S 1 -invariant neighborhood U ⊂ Z 3 of (Z S 1 ) 3 , such that s + τ is proper and transverse to 0 on U ⊂ Z 3 . Let C := U \Z S 1 , then C is a closed subset in Z 3 \Z S 1 .…”

Section: Polyfold Casementioning

confidence: 99%

“…Before proving Proposition B.5, we first state [10,Corollary 3.3], which will be useful for our proof. By Proposition B.3, we can find trivial sub-bundle F 1 extending F 1 hence covers coker D λ s over 1 .…”

Section: Suppose We Have Limmentioning

confidence: 99%

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Quotient theorems in polyfold theory and S1‐equivariant transversality

Zhou

2020

Proc. London Math. Soc.

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We introduce group actions on polyfolds and polyfold bundles. We prove quotient theorems for polyfolds, when the group action has finite isotropy. We prove that the sc-Fredholm property is preserved under quotient if the base polyfold is infinite dimensional. The quotient construction is the main technical tool in the construction of equivariant fundamental class in our future work. We also analyze the equivariant transversality near the fixed locus in the polyfold setting. In the case of S 1-action with fixed locus, we give a sufficient condition for the existence of equivariant transverse perturbations. We outline the application to Hamiltonian-Floer cohomology and a proof of the weak Arnold conjecture for general symplectic manifolds, assuming the existence of Hamiltonian-Floer cohomology polyfolds. Contents

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Section: Resultsmentioning

confidence: 99%

“…By Lemma A.2, there exists θ 1 ∈ Mor(F a (w a ), F a (w a )), θ 2 ∈ Mor(G a (w a ), G a (w a )), such that If we write 10) then h close to g and z close to x we have…”

Section: Suppose We Have Limmentioning

confidence: 99%

“…x0 , that is, 10) and the inverse to Id x0 is itself, we can require the neighborhoodW is small enough such that we can invert (3.10) to get…”

Section: Moreover For Every Open Neighborhoodwmentioning

confidence: 99%

Section: Polyfold Casementioning

confidence: 99%

Section: Suppose We Have Limmentioning

confidence: 99%

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Quotient theorems in polyfold theory and S1‐equivariant transversality

Zhou

2020

Proc. London Math. Soc.

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A polyfold proof of the Arnold conjecture

Filippenko

Wehrheim

2021

Sel. Math. New Ser.

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We give a detailed proof of the homological Arnold conjecture for nondegenerate periodic Hamiltonians on general closed symplectic manifolds M via a direct Piunikhin–Salamon–Schwarz morphism. Our constructions are based on a coherent polyfold description for moduli spaces of pseudoholomorphic curves in a family of symplectic manifolds degenerating from $${{\mathbb {C}}{\mathbb {P}}}^1\times M$$ C P 1 × M to $${{\mathbb {C}}}^+ \times M$$ C + × M and $${{\mathbb {C}}}^-\times M$$ C - × M , as developed by Fish–Hofer–Wysocki–Zehnder as part of the Symplectic Field Theory package. To make the paper self-contained we include all polyfold assumptions, describe the coherent perturbation iteration in detail, and prove an abstract regularization theorem for moduli spaces with evaluation maps relative to a countable collection of submanifolds. The 2011 sketch of this proof was joint work with Peter Albers, Joel Fish.

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Periodic delay orbits and the polyfold implicit function theorem

Albers

Seifert

2022

Comment. Math. Helv.

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We consider differential delay equations of the form \partial_tx(t) = X_{t}(x(t - \tau)) in \mathbb{R}^n , where (X_t)_{t\in S^1} is a time-dependent family of smooth vector fields on \mathbb{R}^n and \tau is a delay parameter. If there is a (suitably non-degenerate) periodic solution x_0 of this equation for \tau=0 , that is without delay, there are good reasons to expect existence of a family of periodic solutions for all sufficiently small delays, smoothly parametrized by \tau . However, it seems difficult to prove this using the classical implicit function theorem, since the equation above, considered as an operator, is not smooth in the delay parameter. In this paper, we show how to use the M-polyfold implicit function theorem by Hofer–Wysocki–Zehnder (2009, 2021) to overcome this problem in a natural setup.

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Counterexamples in scale calculus

Cited by 5 publications

References 7 publications

Quotient theorems in polyfold theory and S1‐equivariant transversality

Quotient theorems in polyfold theory and S1‐equivariant transversality

A polyfold proof of the Arnold conjecture

Periodic delay orbits and the polyfold implicit function theorem

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