Fibered Symplectic Cohomology and the Leray-Serre Spectral Sequence

A Leray-Serre spectral sequence for Lagrangian Floer theory

We consider symplectic fibrations as in Guillemin-Lerman-Sternberg, and derive a spectral sequence to compute the Floer cohomology of certain fibered Lagrangians sitting inside a compact symplectic fibration with small monotone fibers and a rational base. We show if the Floer cohomology with field coefficients of the fiber Lagrangian vanishes, then the Floer cohomology with field coefficients of the total Lagrangian also vanishes. We give an application to certain non-torus fibers of the Gelfand-Cetlin system in Flag manifolds, and show that their Floer cohomology vanishes.

李子代数胚的Serre谱序列

The Serre spectral sequence of a Lie subalgebroid

本文提出并研究李子代数胚的Serre谱序列，统一推广了纤维化、李代数及李代数胚扩张中的经典谱序列，并证明其在宽子代数胚和真子流形情形下分别收敛到相应上同调，同时应用于泊松几何。

Floer homology of families I

In principle, Floer theory can be extended to define homotopy invariants of families of equivalent objects (e.g. Hamiltonian isotopic symplectomorphisms, 3-manifolds, Legendrian knots, etc.) parametrized by a smooth manifold B. The invariant of a family consists of a filtered chain homotopy type, which gives rise to a spectral sequence whose E^2 term is the homology of B with twisted coefficients in the Floer homology of the fibers. The filtered chain homotopy type also gives rise to a "family Floer homology" to which the spectral sequence converges. For any particular version of Floer theory,…