Dehn twists and free subgroups of symplectic mapping class groups

Keating, Ailsa

doi:10.1112/jtopol/jtt033

Cited by 17 publications

(26 citation statements)

References 22 publications

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“…In the same paper he constructed symplectomorphisms that are smoothly, but not symplectically, isotopic. For generalisations of these results see [121,175].…”

Section: Psfrag Replacementsmentioning

confidence: 74%

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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“…In the same paper he constructed symplectomorphisms that are smoothly, but not symplectically, isotopic. For generalisations of these results see [121,175].…”

Section: Psfrag Replacementsmentioning

confidence: 74%

Floer Homologies, with Applications

Abbondandolo

Schlenk

2018

Jahresber. Dtsch. Math. Ver.

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“…The main theorems of this chapter have been proved; this last section is devoted to an additional observation on the relation between the Floer cohomology of a Lagrangian sphere and its associated Dehn twist. Keating [19] has recently obtained an exact sequence involving iterated Dehn twists in the Fukaya category of a symplectic manifold, extending Seidel's original exact sequence [36]. In this subsection we use it to prove Proposition 6.1, which is stated below.…”

Section: Proofs Of the Theorems About Lagrangian Spheres In Divisorsmentioning

confidence: 85%

Commuting symplectomorphisms and Dehn twists in divisors

Tonkonog¹

2015

Geom. Topol.

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Abstract. Two commuting symplectomorphisms of a symplectic manifold give rise to actions on Floer cohomologies of each other. We prove the elliptic relation saying that the supertraces of these two actions are equal.In the case when a symplectomorphism f commutes with a symplectic involution, the elliptic relation provides a lower bound on the dimension of HF * (f ) in terms of the Lefschetz number of f restricted to the fixed locus of the involution. We apply this bound to prove that Dehn twists around vanishing Lagrangian spheres inside most hypersurfaces in Grassmannians have infinite order in the symplectic mapping class group.

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“…Fix a compact subset B ⊂ B\∆ containing the point t. In this situation, [56] explained how to relate distinct fibres of χ by "rescaled symplectic parallel transport maps" which were defined on arbitrarily large compact subsets of the fibres. Here we expand on [56,Remark 30], defining symplectic parallel transport globally for paths which stay away from the discriminant locus; analogous arguments appear in [29,Section 6] and [25,Section 9], which we follow closely.…”

Section: The Isomorphismmentioning

confidence: 99%

Khovanov homology from Floer cohomology

Abouzaid

Smith

2018

J. Amer. Math. Soc.

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This paper realises the Khovanov homology of a link in S 3 as a Lagrangian Floer cohomology group, establishing a conjecture of Seidel and the second author. The starting point is the previously established formality theorem for the symplectic arc algebra over a field k of characteristic zero. Here we prove the symplectic cup and cap bimodules which relate different symplectic arc algebras are themselves formal over k, and construct a long exact triangle for symplectic Khovanov cohomology. We then prove the symplectic and combinatorial arc algebras are isomorphic over Z in a manner compatible with the cup bimodules. It follows that Khovanov cohomology and symplectic Khovanov cohomology co-incide in characteristic zero.

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Dehn twists and free subgroups of symplectic mapping class groups

Cited by 17 publications

References 22 publications

Floer Homologies, with Applications

Floer Homologies, with Applications

Commuting symplectomorphisms and Dehn twists in divisors

Khovanov homology from Floer cohomology

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