A Lagrangian Piunikhin-Salamon-Schwarz Morphism and Two Comparison Homomorphisms in Floer Homology

By Peter Albers. 2005

85 citations recorded by the source corpus.

In this article we address two issues. First, we explore to what extent the techniques of Piunikhin, Salamon and Schwarz in [PSS96] can be carried over to Lagrangian Floer homology. In [PSS96] an isomorphism between Hamiltonian Floer homology and the singular homology is established. In contrast, Lagrangian Floer homology is not isomorphic to the singular homology of the Lagrangian submanifold, in general. Depending on the minimal Maslov number, we construct for certain degrees two homomorphisms between Lagrangian Floer homology and singular homology. In degrees where both maps are defined we prove them to be isomorphisms. Examples show that this statement is sharp. Second, we construct two comparison homomorphisms between Lagrangian and Hamiltonian Floer homology. They underly no degree restrictions and are proven to be the natural analogs to the homomorphisms in singular homology induced by the inclusion map of the Lagrangian submanifold into the ambient symplectic manifold.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ba9a517cf022c6fbbbe0c762129888343270b43a

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In Discrete Mathematics and Combinatorics

1. Nick Sheridan
2. P. Seidel
3. P. Biran
4. M. Abouzaid
5. O. Cornea
6. F. Zapolsky
7. E. Shelukhin
8. Alexandra Monzner

In Geometry and Topology

1. P. Seidel
2. I. Smith
3. M. Abouzaid
4. Nick Sheridan
5. D. Auroux
6. E. Shelukhin
7. Christopher Woodward
8. A. Kislev

In Mathematical Physics

1. P. Seidel
2. M. Abouzaid
3. Nick Sheridan
4. I. Smith
5. D. Auroux
6. O. Cornea
7. E. Shelukhin
8. A. Kislev

Citations in the corpus

• Gauged Floer Theory Of Toric Moment Fibers (2010, background): “The combined gradient trees/holomorphic disks have appeared in many places, for example the Piunikhin-Salamon-Schwarz [45], BiranCornea [8], Albers [3], and Seidel [51].”
• Symplectic Cohomology and q-Intersection Numbers (2010, background): “In the special case L0 = L1 = L, we have the PSS isomorphism [30, 4] (3.”
• The monotone wrapped Fukaya category and the open-closed string map (2012, unknown): “In the closed monotone case, Albers [11] defined maps τ : HF∗(H)→ HF (L,φ(1)H(L)) and χ : HF∗(L,φ 1 H(L))→ HF∗(H), identifiable with CO : QH(B) → HF (L,L) and OC : HF (L,L) → QH(B).”
• Bounds on spectral norms and barcodes (2018, background): “rl for L. There are well-dened ring isomorphisms PSS : QH(L;Dpearl) ! HF(L;H;D) which also commute with the Floer continuation maps. Let us brie y recall their construction (for the details see e.g. [5, 15, 16, 17, 62, 87, 89]). Given x2Crit(f), ( ;[ ]) 2Crit(A H), and A2ˇ 2(M;L). Consider the seque”
• Fukaya A∞-structures Associated to Lefschetz Fibrations. II (2014, background): “The appropriate version of the Lagrangian PSS isomorphism [6] says that”
• Rigidity and uniruling for Lagrangian submanifolds (2008, methodology): “In the case of H ;f ı f;H one of the internal vertices satisfies a perturbed equation but a function ˇ with support in an interval of type Œ r; r  is used instead of ˇ (see again Albers [1] and Biran and Cornea [12] for details).”
• On the Fukaya category of a Fano hypersurface in projective space (2013, methodology): “We recall the Lagrangian version [3] of the Piunikhin-Salamon-Schwarz morphism [29].”
• Partial quasi-morphisms and quasi-states on cotangent bundles, and symplectic homogenization (2011, background): “ as 1-periodic orbits. 23 This implies the following chain of inequalities: c−(φ) ≤ℓ−(φ) ≤ℓ+(φ) ≤c+(φ), where the rightmost inequality follows by duality. Proof. The proof is essentially contained in [Alb], the only point of difference being that there the theory is restricted to closed manifolds. Th”
• Symplectic cohomology and Viterbo's theorem (2013, background): “This idea has appeared in various incarnations so many times in the literature that we shall necessarily sin by omission in listing potential references: [8,12,39,75].”
• Fukaya A∞-structures associated to Lefschetz fibrations. I (2009, background): “3), one gets numbers n(x0) ∈ K for each x0 ∈ φH(Vj) ∩ Vj, which together form a distinguished cocycle Φ̄0 ∈ CFB(Vj , Vj) representing 1 ∈ H(Vj ;K) ∼= HF (Vj , Vj) (this is obvious if the isomorphism between Floer cohomology and classical cohomology is constructed as a PiunikhinSalamon-Schwarz type m”