A long exact sequence for symplectic Floer cohomology

By P. Seidel. 2001

235 citations recorded by the source corpus.

The long exact sequence describes how the Floer cohomology of two Lagrangian submanifolds changes if one of them is modified by applying a Dehn twist. We give a proof in the simplest case (no bubbling). The paper contains a certain amount of material that may be of interest independently of the exact sequence: in particular, chapter 1 covers "symplectic Picard-Lefschetz theory" in some detail, and chapter 2 contains a generalization of the usual TQFT setup for Floer cohomology.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f10e463135c94b4287a1db6d57bdf8ce81786866

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Citations in the corpus

Lecture notes on stabilization of contact open books (2010, background): “Seidel’s results on Dehn twists, see [21, 22, 23] for some of them, gave symplectic Dehn twists a prominent role in symplectic topology; for example, he showed that Dehn twists have infinite order in the compactly supported symplectic mapping class group of T S.”
The Symplectic Geometry of Cotangent Bundles from a Categorical Viewpoint (2007, background): “The rest of the argument essentially consists in applying the long exact sequence from [33].”
Quiver algebras as Fukaya categories (2013, methodology): “) Seidel’s theorem [34] yields a distinguished triangle in K.”
Sectorial descent for wrapped Fukaya categories (2018, result): “nuation map. Of course, Theorem 1.8 is very similar to existing results concerning the eﬀect in the Fukaya category of various other geometric operations, e.g. Seidel’s exact triangle of a Dehn twist [31], the exact triangle associated to Polterovich surgery, and Biran–Cornea’s work on exact triangl”
Floer cohomology and geometric composition of Lagrangian correspondences (2009, unknown): “13 68 v2 [ m at h. SG ] 1 3 A ug 2”
Symplectic homology as Hochschild homology (2006, background): “Choose a distinguished basis of paths (1,...,m) going from ∗ to the critical values of �. Each j gives rise to a Lefschetz thimble �j, which is a Lagrangian ball in E fibered over j. Its boundary, the vanishing cycle Lj = @�j, is a Lagrangian sphere in M [ 17, Section 1.3 ].”
Effect of Legendrian surgery (2009, result): “In this sense our results can be viewed as generalizations of P Seidel’s exact triangles for symplectic Dehn twists; see [38; 37; 39].”
Lagrangian matching invariants for fibred four-manifolds: I (2006, background): “Let Spinc(X) denote theH2(X;Z)–torsor of (isomorphism classes of) Spinc–structures on the oriented four–manifoldX.”
Homological mirror symmetry for the quartic surface (2003, methodology): “ the vanishing cycle, up to isotopy inside the group of exact symplectic automorphisms: (6.12) hγ≃τVc ∈Aut(Mc(0)). To be precise, Vc carries a small additional piece of structure, called a framing in [53, 55], and the deﬁnition of Dehn twist uses that too. However, framings contain no information if”
Uniqueness of symplectic structures (2012, methodology): “Seidel computed the Floer cohomology group HF(τΛ) with its module structure over the quantum cohomology ring, via his exact sequence [44, 45].”
A link invariant from the symplectic geometry of nilpotent slices (2004, background): “The monodromy is idX times the standard PicardLefschetz (Dehn twist) monodromy, see [38] for an explicit computation, and the Lemma is trivially true, since hγt(Lt) = Lt.”
Quilted Floer cohomology (2009, background): “One naturally conjectures the following relative version (for example, compare the Seidel triangle with the relative version in [32]).”
Braid cobordisms, triangulated categories, and flag varieties (2006, background): “Fuk(X p))). In particular, the action of the Dehn twist TL about the Lagrangian sphere L(the twist being independent of the grading on L) on the object [L′] ∈ Db(Fuk(X p)) can be described as follows [Se4]. Think of the identity in HF∗(L,L′)∗ ⊗ HF∗(L,L′) as a morphism from HF∗(L,L′) ⊗ [L] to [L′] in”