The Seidel morphism of Cartesian products

Leclercq, Rémi

doi:10.2140/agt.2009.9.1951

Cited by 4 publications

(6 citation statements)

References 11 publications

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“…The PSS morphism as described in this particular case in Section 2.1.1 is compatible with the Künneth formula(20). This was the content of[13, Claim 3.4] in the more general case of monotone manifolds. More precisely, the Morse theoretic version of Künneth's formula is satisfied, that is HM (L) = HM (L ) ⊗ HM (L ) .…”

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confidence: 55%

“…Then define J σ as J σ t (x) = J σ(t) (x). Notice that (H σ , J σ ) is regular, and that there is a bijection between the moduli spaces M(γ − , γ + ; H, J) and M(γ σ − , γ σ + ; H σ , J σ ) so that (13) induces an action-preserving isomorphism of the differential complexes. Notice that geometrically the main objects (orbits and Floer's strips) remain the same.…”

Section: Comparison Between Lagrangian and Hamiltonian Spectral Invarmentioning

confidence: 99%

“…which finishes the proof of the triangle inequality. The pair (H, J) is regular for (L 0 , L 1 ): This is because the linearization of the operator u → ∂ s u + J t (u)(∂ t u − X t H (u)) splits into a product of the corresponding linearizations for (H , J ) and (H , J ); see, for example, [13] for further details. It follows that, for any two chords x − = (x − , x − ) and x + = (x + , x + ) of H, the moduli space M L 0 ,L 1 (x − , x + ; H, J), used in the definition of the Floer boundary map, coincides with the product…”

Section: 52mentioning

confidence: 99%

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Reduction of symplectic homeomorphisms

Humilière¹,

Leclercq²,

Seyfaddini³

2016

Ann. Sci. École Norm. Sup.

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In [9], we proved that symplectic homeomorphisms preserving a coisotropic submanifold C, preserve its characteristic foliation as well. As a consequence, such symplectic homeomorphisms descend to the reduction of the coisotropic C.In this article we show that these reduced homeomorphisms continue to exhibit certain symplectic properties. In particular, in the specific setting where the symplectic manifold is a torus and the coisotropic is a standard subtorus, we prove that the reduced homeomorphism preserves spectral invariants and hence the spectral capacity.To prove our main result, we use Lagrangian Floer theory to construct a new class of spectral invariants which satisfy a non-standard triangle inequality.Résumé. Nous avons démontré dans [9], qu'un homéomorphisme symplectique qui laisse invariante une sous-variété coisotrope C, préserve également son feuilletage caractéristique. Il induit donc un homéomorphisme sur la réduction symplectique de C.Dans cet article, nous démontrons que l'homéomorphisme ainsi obtenu exhibe certaines propriétés symplectiques. En particulier, dans le cas où la variété symplectique ambiante est un tore, et la sous-variété coisotrope est un sous-tore standard, nous démontrons que l'homéomorphisme réduit préserve les invariants spectraux et donc aussi la capacité spectrale.Pour démontrer notre résultat principal, nous construisons, à l'aide de l'homologie de Floer Lagrangienne, une nouvelle famille d'invariants spectraux qui satisfont un nouveau type d'inégalité triangulaire.

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mentioning

confidence: 55%

Section: Comparison Between Lagrangian and Hamiltonian Spectral Invarmentioning

confidence: 99%

Section: 52mentioning

confidence: 99%

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Reduction of symplectic homeomorphisms

Humilière¹,

Leclercq²,

Seyfaddini³

2016

Ann. Sci. École Norm. Sup.

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“…By studying these constraints on the Seidel element and properties of the quantum homology, we can show that I vanishes for products of projective spaces and the Grassmannian G.2; 4/. Theorem 1.1 is related to results of Pedroza [10] and Leclercq [5]. They showed that, for X 0 and X 00 monotone symplectic manifolds, 0 2 1 .Ham.X 0 //, 00 2 1 .Ham.X 00 //, then S. 0 00 / D S. 0 /˝S.…”

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confidence: 86%

Symplectic manifolds with vanishing action–Maslov homomorphism

Branson

2011

Algebr. Geom. Topol.

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The action-Maslov homomorphism I W 1 .Ham.X; !// ! R is an important tool for understanding the topology of the Hamiltonian group of monotone symplectic manifolds. We explore conditions for the vanishing of this homomorphism, and show that it is identically zero when the Seidel element has finite order and the homology satisfies property D (a generalization of having homology generated by divisor classes). We use these results to show that I D 0 for products of projective spaces and the Grassmannian of 2 planes in C 4 .53D45; 53D35, 53D40, 20F69

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“…Recall that π 1 (Ham(S 2 , ω)) ≃ Z 2 and Ham(Σ g , η) is simply connected for g ≥ 1. By [8] and [13] we can say that π 1 (Ham(S 2 × Σ g , ω ⊕ η)) has an element of order two. We can say more by using Theorem 1.3.…”

Section: Examplesmentioning

confidence: 99%