Quilted Floer cohomology

Abstract

We generalize Lagrangian Floer cohomology to sequences of Lagrangian correspondences. For sequences related by the geometric composition of Lagrangian correspondences we establish an isomorphism of the Floer cohomologies. This provides the foundation for the construction of a symplectic 2-category as well as for the definition of topological invariants via decomposition and representation in the symplectic category. Here we give some first direct symplectic applications: Calculations of Floer cohomology, displaceability of Lagrangian correspondences and transfer of displaceability under geometric composition. 53D40; 57R56

Quilted Floer trajectories with constant components: Corrigendum to the article “Quilted Floer cohomology”

“…Quilted Floer homology is defined in [10] for a cyclic generalized Lagrangian correspondence L, that is, a sequence of symplectic manifolds M 0 ; M 1 ; : : : ; M r ; M r C1 with M 0 D M r C1 for some r 0, and a sequence of compact Lagrangian correspondences…”

“…In [10] we moreover make monotonicity, grading, and Maslov index assumptions that guarantee compactification properties. These are not required for the results in this paper.…”

“…Generically this choice would not be of split form, ie induced by a tuple of Hamiltonian functions and almost complex structures on each symplectic manifold M j . The quilted definition of HF.L/ in [10] on the one hand generalizes this construction by allowing a choice of widths ı D .ı j > 0/ j D0;:::;r of the strips mapping to each M j . On the other hand, we claim in [10] that the quilted Floer complex can be constructed (in particular transversality of the quilted Cauchy-Riemann operator can be achieved) for Hamiltonians and almost complex structures of split type.…”

“…The quilted definition of HF.L/ in [10] on the one hand generalizes this construction by allowing a choice of widths ı D .ı j > 0/ j D0;:::;r of the strips mapping to each M j . On the other hand, we claim in [10] that the quilted Floer complex can be constructed (in particular transversality of the quilted Cauchy-Riemann operator can be achieved) for Hamiltonians and almost complex structures of split type. That is, we restrict our choice of perturbation data to a tuple of Hamiltonian functions and a tuple of almost complex structures in the complete metric spaces…”

“…

We fill a gap in the proof of the transversality result for quilted Floer trajectories in [10] by addressing trajectories for which some but not all components are constant. Namely we show that for generic sets of split Hamiltonian perturbations and split almost complex structures, the moduli spaces of parametrized quilted Floer trajectories of a given index are smooth of expected dimension.

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