Simple homotopy equivalence of nearby Lagrangians

Abouzaid, Mohammed; Kragh, Thomas

doi:10.4310/acta.2018.v220.n2.a1

Cited by 32 publications

(33 citation statements)

References 15 publications

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“…As we have seen in §5, Arnol'd's nearby Lagrangian conjecture holds for Q = S 1 , and it is known also for Q = S 2 by a result of R. Hind [101]. For all other manifolds Q, the strongest result known so far is the following recent theorem due to Abouzaid and Kragh [9]. Theorem 7.3.…”

Section: 2mentioning

confidence: 85%

Floer Homologies, with Applications

Abbondandolo

Schlenk

2018

Jahresber. Dtsch. Math. Ver.

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Floer invented his theory in the mid eighties in order to prove the Arnol'd conjectures on the number of fixed point of Hamiltonian diffeomorphisms and Lagrangian intersections. Over the last thirty years, many versions of Floer homology have been constructed. In symplectic and contact dynamics and geometry they have become a principal tool, with applications that go far beyond the Arnol'd conjectures: The proof of the Conley conjecture and of many instances of the Weinstein conjecture, rigidity results on Lagrangian submanifolds and on the group of symplectomorphisms, lower bounds for the topological entropy of Reeb flows and obstructions to symplectic embeddings are just some of the applications of Floer's seminal ideas. Other Floer homologies are of topological nature. Among their applications are Property P for knots and the construction of compact topological manifolds of dimension greater than five that are not triangulisable. This is by no means a comprehensive survey on the presently known Floer homologies and their applications. Such a survey would take several hundred pages. We just describe some of the most classical versions and applications, together with the results that we know or like best. The text is written for non-specialists and the focus is on ideas rather than generality. Two intermediate sections recall basic notions and concepts from symplectic dynamics and geometry. 14 4.

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Section: 2mentioning

confidence: 85%

Floer Homologies, with Applications

Abbondandolo

Schlenk

2018

Jahresber. Dtsch. Math. Ver.

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“…The Floer-theoretic Wh 1 invariant was used by Abouzaid and Kragh in [AK18] to prove that the projection to the base of any nearby Lagrangian in a cotangent bundle is a simple homotopy equivalence. Another application was found by Suárez [Suá17] in her study of exact Lagrangian cobordisms.…”

Section: Appendix Overview Of Torsionmentioning

confidence: 99%

A Legendrian Turaev torsion via generating families

Álvarez-Gavela¹,

Igusa²

2020

Journal De L’École Polytechnique — Mathématiques

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“…The famous "nearby Lagrangian problem" asks whether there is a unique up to Hamiltonian isotopy exact closed Lagrangian submanifold in the standard T * M for a closed M . Though in this form the answer is unknown except for M = S 2 and T 2 , see [27,12], the answer is positive up to simple homotopy equivalence, [1], and hence according to Smale, Freedman and Perelman for M = S n up to homeomorphism, and for some dimensions, e.g. n = 3, 5, 6, 12, even up to diffeomorphism, [34].…”

Section: Symplectic Topology Of Weinstein Manifoldsmentioning

confidence: 99%

“…Problem 6.3. Can the uniquenes results from [1] be extended to a more general class of Weinstein structures on T * S n ?…”

Section: Symplectic Topology Of Weinstein Manifoldsmentioning

confidence: 99%

Weinstein manifolds revisited

Eliashberg¹

2018

Proceedings of Symposia in Pure Mathematics

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This is a very biased and incomplete survey of some basic notions, old and new results, as well as open problems concerning Weinstein symplectic manifolds. Weinstein manifolds, domains, cobordismsWe begin with a notion of a Liouville domain. Let (X, ω) be a 2n-dimensional compact symplectic manifold with boundary equipped with an exact symplectic form ω. A Liouville structure on (X, ω) is a choice of a primitive λ, dλ = ω, called Liouville form such that λ| ∂X is a contact form and the orientation of ∂X by the form λ ∧ dλ n−1 | ∂X coincides with its orientation as the boundary of symplectic manifold (X, ω). The vector field Z, that is ω-dual to λ, i.e. ι(Z)ω = λ, is also called Liouville. It satisfies the condition L Z ω = ω which means that its flow is conformally symplectically expanding. The contact boundary condition is equivalent to the outward transversality of Z to ∂X. A Liouville domain X can always be completed to a Liouville manifold X by attaching a cylindrical end:and extending λ to X as equal to e s (λ| ∂X ) on the attached end. We will be constantly going back and forth between these two tightly related notions of Liouville domains and Liouville manifolds.Given a Liouville structure L = (X, ω, Z) we say that a Liouville structure L = (X , ω, Z) is obtained by a radial deformation from L if there exists a function h : *

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Simple homotopy equivalence of nearby Lagrangians

Abstract: Given a closed exact Lagrangian in the cotangent bundle of a closed smooth manifold, we prove that the projection to the base is a simple homotopy equivalence.

Cited by 32 publications

References 15 publications

Floer Homologies, with Applications

Floer Homologies, with Applications

A Legendrian Turaev torsion via generating families

Weinstein manifolds revisited

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