Barcode of a pair of compact exact Lagrangians in a punctured exact two-dimensional symplectic manifold

2022/08/20 by Pasquer, Tangi
#FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2208.09704

Abstract

In this article, we modify the classical Floer complex CF(L₀,L₁) of a pair of two compact exact Lagrangian submanifolds L₀,L₁ of an exact symplectic 2-manifold M into a ℤ₂[T]-complex CF_h(L₀,L₁), whose differential keeps track of how many times a pseudo-holomorphic strip passes through a distinguished point h∈ M. We show that this complex is invariant under Hamiltonian isotopy, and we prove that its barcode, if it exists, is the same as both barcodes B(CF(L₀,L₁;M)) and B(CF(L₀,L₁;M∖ \h \)). This allows us to extend a conjecture of Viterbo, which states that for every Hamiltonian isotopy ϕ¹_H(L) in D^* L, the spectral norm γ(L,ϕ¹_H(L)) remains bounded independently of H, to the case of D^* L with a point removed.

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