Polyfolds: A first and second look

Fabert, Oliver; Fish, Joel W.; Golovko, Roman; Wehrheim, Katrin

doi:10.4171/emss/16

Cited by 20 publications

(34 citation statements)

References 36 publications

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“…Rather than dealing with the general theory, which also allows for boundary with corners, we restrict ourselves to a special case and illustrate it with a discussion of stable maps, a topic closely related to Gromov-Witten theory. We also would like to mention the paper [9], where the ideas of polyfold theory are explained as well. The abstract theory has been applied in [28] as part of the general construction of SFT.…”

Section: Introductionmentioning

confidence: 99%

Polyfolds and Fredholm Theory

Hofer¹

2017

Oxford Scholarship Online

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Section: Introductionmentioning

confidence: 99%

Polyfolds and Fredholm Theory

Hofer¹

2017

Oxford Scholarship Online

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“…x,y,0 is the sc-Fredholm section on the Morse homology M-polyfold sketched in [7] for (H, g) and is in general position by the Sard-Smale assumption.…”

Section: Hamiltonian-floer Cohomologymentioning

confidence: 99%

“…We first give a brief description of the Morse homology M-polyfolds which contain compactified Morse trajectory spaces following [7].…”

Section: The Morse Homology M-polyfolds and Hamiltonian-floer Cohomolmentioning

confidence: 99%

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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“…Then there exists a neighborhood U ⊂ X of x 0 such that the map s : U → s(U ) is invertible with open image s(U ) ⊂ Y , and the inverse s −1 : s(U ) → U is continuously differentiable with differential ds −1 (s(x)) = ds(x) −1 . 1 A Banach space is a vector space with a norm X → [0, ∞), x → x that induces a complete topology. The spaces X = R n with any norm are Banach spaces, but the term usually denotes infinite dimensional Banach spaces such as the space of square integrable functions L 2 (R) = {f : R → R | f L 2 := |f (x)| 2 |dx < ∞ }.…”

Section: From Calculus To Scale Calculusmentioning

confidence: 99%

Counterexamples in scale calculus

Filippenko

Zhou

Wehrheim

2019

Proc. Natl. Acad. Sci. U.S.A.

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We construct counterexamples to classical calculus facts such as the Inverse and Implicit Function Theorems in Scale Calculus -a generalization of Multivariable Calculus to infinite dimensional vector spaces in which the reparameterization maps relevant to Symplectic Geometry are smooth. Scale Calculus is a cornerstone of Polyfold Theory, which was introduced by Hofer-Wysocki-Zehnder as a broadly applicable tool for regularizing moduli spaces of pseudoholomorphic curves. We show how the novel nonlinear scale-Fredholm notion in Polyfold Theory overcomes the lack of Implicit Function Theorems, by formally establishing an often implicitly used fact: The differentials of basic germs -the local models for scale-Fredholm maps -vary continuously in the space of bounded operators when the base point changes. We moreover demonstrate that this continuity holds only in specific coordinates, by constructing an example of a scale-diffeomorphism and scale-Fredholm map with discontinuous differentials. This justifies the high technical complexity in the foundations of Polyfold Theory.

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Polyfolds: A first and second look

Cited by 20 publications

References 36 publications

Polyfolds and Fredholm Theory

Polyfolds and Fredholm Theory

Quotient theorems in polyfold theory and S1‐equivariant transversality

Counterexamples in scale calculus

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