The main purpose of the present paper is a study of orientations of the moduli spaces of pseudo-holomorphic discs with boundary lying on a \emph{real} Lagrangian submanifold, i.e., the fixed point set of an anti-symplectic involutions $τ$ on a symplectic manifold. We introduce the notion of $τ$-relatively spin structure for an anti-symplectic involution $τ$, and study how the orientations on the moduli space behave under the involution $τ$. We also apply this to the study of Lagrangian Floer theory of real Lagrangian submanifolds. In particular, we study unobstructedness of the $τ$-fixed point set of symplectic manifolds and in particular prove its unobstructedness in the case of Calabi-Yau manifolds. And we also do explicit calculation of Floer cohomology of $\R P^{2n+1}$ over $Λ_{0,nov}^{\Z}$ which provides an example whose Floer cohomology is not isomorphic to its classical cohomology. We study Floer cohomology of the diagonal of the square of a symplectic manifold, which leads to a rigorous construction of the quantum Massey product of symplectic manifold in complete generality.
We study the Floer cohomology of the Dehn twist along a real Lagrangian sphere in a symplectic manifold endowed with an anti-symplectic involution. We prove that there exists a distinguished element in the Floer group that is a fixed point of the automorphism induced by the involution. Our methods of proof are based on Mak-Wu's cobordism and Floer-theoretic considerations.
Floer theory relates the dynamics of Hamiltonian isotopies and the homology of the ambient manifold. It was extended to similarly relate the dynamics of symplectic isotopies and the Novikov homology associated to their flux. We discuss this picture regarding the fundamental group, and prove that when the flux is not too big, the associated Novikov fundamental group is generated by Floer moduli spaces associated to closed orbits of the symplectic isotopy.
This is an expanded version of the talk given be the first author at the conference "Topology, Geometry, and Dynamics: Rokhlin - 100". The purpose of this talk was to explain our current results on classification of rational symplectic 4-manifolds equipped with an anti-symplectic involution. Detailed exposition will appear elsewhere.
We show how to compute the Lagrangian Floer homology in the one-point blow up of the proper transform of Lagrangians submanifolds, solely in terms of information of the base manifold. As an example we present an alternative computation of the Lagrangian quantum homology in the one-point blow up of (CP^2,ω) of the proper transform of the Clifford torus.
We incorporate pearly Floer trajectories into the transversality scheme for pseudoholomorphic maps introduced by Cieliebak-Mohnke. By choosing generic domain-dependent almost complex structures we obtain zero and one-dimensional moduli spaces with the structure of cell complexes with rational fundamental classes. This gives a definition of Floer cohomology over Novikov rings via stabilizing divisors for compact symplectic manifolds with rational symplectic classes and Lagrangians that are fixed point sets of anti-symplectic involutions satisfying certain Maslov index conditions, in particular,…
We show that blow-ups or reverse flips (in the sense of the minimal model program) of rational symplectic manifolds with point centers create Floer-non-trivial Lagrangian tori. As applications, we demonstrate the existence of Hamiltonian non-displaceable Lagrangian tori in, for example, small symplectic blow-ups of compact symplectic manifolds and moduli spaces of polygons. These results are part of a conjectural description of generators for the Fukaya category of a compact symplectic manifold with a singularity-free running of the minimal model program.
We correct a mistake regarding almost complex structures on Hilbert schemes of points in surfaces in arXiv:1510.02449. The error does not affect the main results of the paper, and only affects the proofs of invariance of equivariant symplectic Khovanov homology and reduced symplectic Khovanov homology. We give an alternate proof of the invariance of equivariant symplectic Khovanov homology.
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.
We study the reduced symplectic cohomology of disk subbundles in negative symplectic line bundles. We show that this cohomology theory "sees" the spectrum of a quantum action on quantum cohomology. Precisely, quantum cohomology decomposes into generalized eigenspaces of the action of the first Chern class by quantum cup product. The reduced symplectic cohomology of a disk bundle of radius $R$ sees all eigenspaces whose eigenvalues have size less than $R$, up to rescaling by a fixed constant. Similarly, we show that the reduced symplectic cohomology of an annulus subbundle between radii $R_1$ a…
In this paper we first develop various enhancements of the theory of spectral invariants of Hamiltonian Floer homology and of Entovi-Polterovich theory of spectral symplectic quasi-states and quasimorphisms by incorporating \emph{bulk deformations}, i.e., deformations by ambient cycles of symplectic manifolds, of the Floer homology and quantum cohomology. Essentially the same kind of construction is independently carried out by Usher [Us4] in a slightly less general context. Then we explore various applications of these enhancements to the symplectic topology, especially new construction of sy…
The Z-graded symplectic Floer cohomology of monotone Lagrangian sub-manifolds
定义了一个整数分次辛Floer上同调和一个Fintushel-Stern型谱序列，它们是单调Lagrange子流形和正合同位素的新的不变量. Z-分次辛Floer上同调是通常的Z_Sigma（L）-分次Floer-Oh上同调的积分提升。证明了谱序列的Kunneth公式及其上的环结构，并通过谱序列由Lagrange子流形的上同调环结构导出了Z_Sigma（L）-分次Floer上同调环结构.利用Z-分次辛Floer上同调，我们给出了嵌入拉格朗日量的H_∞能量e_H（L）、拉格朗日量L的收敛谱序列的最小辛作用量σ（L）、最小Maslov指数σ（L）$ and the smallest integer k(L, phi$之间的一些相互交织的关系.
In this paper we connect algebraic properties of the pair-of-pants product in local Floer homology and Hamiltonian dynamics. We show that for an isolated periodic orbit the product is non-uniformly nilpotent and use this fact to give a simple proof of the Conley conjecture for closed manifolds with aspherical symplectic form. More precisely, we prove that on a closed symplectic manifold the mean action spectrum of a Hamiltonian diffeomorphism with isolated periodic orbits is infinite.
We show that the space of anti-symplectic involutions of a monotone $S^2\times S^2$ whose fixed points set is a Lagrangian sphere is connected. This follows from a stronger result, namely that any two anti-symplectic involutions in that space are Hamiltonian isotopic.
Since its inception, Floer homology has been an important tool in low-dimensional topology. Floer theoretic invariants of $3$-manifolds tend to be either gauge theoretic or symplecto-geometric in nature, and there is a general philosophy that each gauge theoretic Floer homology should have a corresponding symplectic Floer homology and vice-versa. In this article, we construct a Lagrangian Floer invariant for any closed, oriented $3$-manifold $Y$ (called the symplectic instanton homology of $Y$ and denoted $\mathrm{SI}(Y)$) which is conjecturally equivalent to a Floer homology defined using a c…
We use quilted Floer theory to construct functor-valued invariants of tangles arising from moduli spaces of flat bundles on punctured surfaces. As an application, we show the non-triviality of certain elements in the symplectic mapping class groups of moduli spaces of flat bundles on punctured spheres.
Categorical symplectic geometry is the study of a rich collection of invariants of symplectic manifolds, including the Fukaya $A_\infty$-category, Floer cohomology, and symplectic cohomology. Beginning with work of Wehrheim and Woodward in the late 2000s, several authors have developed techniques for functorial manipulation of these invariants. We survey these functorial structures, including Wehrheim-Woodward's quilted Floer cohomology and functors associated to Lagrangian correspondences, Fukaya's alternate approach to defining functors between Fukaya $A_\infty$-categories, and the second au…
We use quilted Floer theory to generalize Seidel's long exact sequence in symplectic Floer theory to fibered Dehn twists. We then apply it to construct versions of the Floer and Khovanov-Rozansky exact triangles in Lagrangian Floer theory of moduli spaces of bundles.
This paper realises the Khovanov homology of a link in the 3-sphere as a Lagrangian Floer cohomology group, establishing a conjecture of Seidel and the second author. The starting point is the previously established formality theorem for the symplectic arc algebra over a field k of characteristic zero. Here we prove the symplectic cup and cap bimodules which relate different symplectic arc algebras are themselves formal over k, and construct a long exact triangle for symplectic Khovanov cohomology. We then prove the symplectic and combinatorial arc algebras are isomorphic over the integers in …
