Symplectic homology of Lefschetz fibrations and Floer homology of the monodromy map

McLean, Mark

doi:10.1007/s00029-011-0079-6

Cited by 10 publications

(12 citation statements)

References 18 publications

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“…One can partially describe (2.48) in topological terms, along the lines of [25]. Rather than encapsulating that into a spectral sequence, we break up the statement into pieces (the proofs of which will be described in Section 5b; we emphasize that this follows known ideas): Lemma 2.24.…”

Section: Lemma 223mentioning

confidence: 99%

“…This section reviews the groups HF * q,q −1 (p, r) from (2.48), following [42], and adds more concrete computational material along the lines of [25]. We only deal with fibrations which have trivial monodromy at infinity, which makes both the technical arguments and the outcome of the computations particularly simple.…”

Section: Hamiltonian Floer Cohomology For Lefschetz Fibrationsmentioning

confidence: 99%

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Fukaya A ∞-structures Associated to Lefschetz Fibrations. II

Seidel

2017

Algebra, Geometry, and Physics in the 21st Century

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Floer cohomology groups are usually defined over a field of formal functions (a Novikov field). Under certain assumptions, one can equip them with connections, which means operations of differentiation with respect to the Novikov variable. This allows one to write differential equations for Floer cohomology classes. Here, we apply that idea to symplectic cohomology groups associated to Lefschetz fibrations, and establish a relation with enumerative geometry.

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Section: Lemma 223mentioning

confidence: 99%

Section: Hamiltonian Floer Cohomology For Lefschetz Fibrationsmentioning

confidence: 99%

Fukaya A ∞-structures Associated to Lefschetz Fibrations. II

Seidel

2017

Algebra, Geometry, and Physics in the 21st Century

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“…For a > 0 small, the group HF * (φ, a) has already appeared in the work of Seidel [26] and McLean [17]. After the first version of this paper was written, the paper [23] by Seidel was posted on the ArXiv.…”

Section: Introductionmentioning

confidence: 99%

Floer homology of automorphisms of Liouville domains

Uljarevic

2017

Journal of Symplectic Geometry

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We introduce a combination of fixed point Floer homology and symplectic homology for Liouville domains. As an application, we detect non-trivial elements in the symplectic mapping class group of a Liouville domain. 2.Floer homology for symplectomorphism of a Liouville domain 2.1. Floer data and Admissible data.The exact symplectomorphisms form a group, denoted by Symp( W , λ/d). The functions associated to composition and inverse are given byWe define the subgroup Symp c (W, λ/d) of Symp( W , λ/d) byAn exact symplectomorphism is not required to be compactly supported. Every Hamiltonian symplectomorphism generated by a compactly supported Hamiltonian is exact. However, an exact symplectomorphism need not be Hamiltonian or even isotopic to the identity.Definition 2.2. A real number a is called admissible if it is not a period of a Reeb orbit of (M, β). Definition 2.3. Let φ ∈ Symp c (W, λ/d). Floer data for φ is a pair (H, J) of a (timedependent) Hamiltonian H t : W → R and a family J t of ω-compatible almost complex structures on W satisfying the following conditions. H and J are twisted by φ, i.e.In addition, the conditions below hold near infinity H t (x, r) = ar, (2.6) J t (x, r)ξ β = ξ β , (2.7) J t (x, r)∂ r = R β , (2.8)

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“…This idea can be generalized to other r > 0, leading to a version of the spectral sequence from [20], involving powers of µ (see [29,Lemma 6.14] for the statement). The situation for r < 0 is dual: there are nondegenerate pairings (2.7) HF * (E, −r) ⊗ HF 2n− * (E, r) −→ K.…”

Section: Main Constructionsmentioning

confidence: 99%

Fukaya 𝐴_{∞}-structures associated to Lefschetz fibrations. IV

Seidel¹

2019

Proceedings of Symposia in Pure Mathematics

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To a symplectic Lefschetz pencil on a monotone symplectic manifold, we associate an algebraic structure, which is a pencil of categories in the sense of noncommutative geometry. One fibre of this "noncommutative pencil" is related to the Fukaya category of the open (meaning, with the base locus removed, and hence exact symplectic) fibre of the original Lefschetz pencil; the other fibres are newly constructed kinds of Fukaya categories.

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Symplectic homology of Lefschetz fibrations and Floer homology of the monodromy map

Cited by 10 publications

References 18 publications

Fukaya A ∞-structures Associated to Lefschetz Fibrations. II

Fukaya A ∞-structures Associated to Lefschetz Fibrations. II

Floer homology of automorphisms of Liouville domains

Fukaya 𝐴_{∞}-structures associated to Lefschetz fibrations. IV

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