Related papers: Spectral invariants in Lagrangian Floer theory

Given a Lagrangian submanifold $L$ in a symplectic manifold $X$, the homological Lagrangian monodromy group $\mathcal{H}_L$ describes how Hamiltonian diffeomorphisms of $X$ preserving $L$ setwise act on $H_*(L)$. We begin a systematic study…

The variation operator in singularity theory maps relative homology cycles to compact cycles in the Milnor fiber using the monodromy. We construct its symplectic analogue for an isolated singularity. We define the monodromy Lagrangian Floer…

We define combinatorial Floer homology of a transverse pair of noncontractibe nonisotopic embedded loops in an oriented 2-manifold without boundary, prove that it is invariant under isotopy, and prove that it is isomorphic to the original…

Link Floer homology is an invariant for links defined using a suitable version of Lagrangian Floer homology. In an earlier paper, this invariant was given a combinatorial description with mod 2 coefficients. In the present paper, we give a…

Seiberg-Witten (Floer) theory, Ozsvath-Szabo's Heegaard Floer theory, Hutchings's embedded contact homology, in different stages of development, define (or are expected to define) packages of invariants for 3- and 4-manifolds (including…

We consider exact Lagrangian submanifolds in cotangent bundles. Under certain additional restrictions (triviality of the fundamental group of the cotangent bundle, and of the Maslov class and second Stiefel-Whitney class of the Lagrangian…

Inspired by the Ozsv\'ath-Szab\'o mixed invariant in ordinary Heegaard Floer theory, we define a mixed invariant $\Phi_{X, \mathfrak{s}}^{I}$ for closed, spin four-manifolds $(X, \mathfrak{s})$ using the cobordism maps on involutive…

For G a Lie group acting on a symplectic manifold $(M,\omega)$ preserving a pair of Lagrangians $L_0$, $L_1$, under certain hypotheses not including equivariant transversality we construct a G-equivariant Floer cohomology of $L_0$ and…

We consider Lagrangian Floer cohomology for a pair of Lagrangian submanifolds in a symplectic manifold M. Suppose that M carries a symplectic involution, which preserves both submanifolds. Under various topological hypotheses, we prove a…

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…

In this article, the authors review what the Floer homology is and what it does in symplectic geometry both in the closed string and in the open string context. In the first case, the authors will explain how the chain level Floer theory…

Manifold submetries of the round sphere are a class of partitions of the round sphere that generalizes both singular Riemannian foliations, and the orbit decompositions by the orthogonal representations of compact groups. We exhibit a…

The "zero in the spectrum conjecture" asserted (in its strongest form) that for any manifold M zero should be in the l2-spectrum of the Laplacian (on forms) of the universal covering of M, i.e. that at least one (unreduced) L2-cohomology…

We construct distinguished elements in the embedded contact homology (and monopole Floer homology) of a 3-torus, associated with Lagrangian tori in symplectic 4-manifolds and their isotopy classes. They turn out not to be new invariants,…

A new relation between homoclinic points and Lagrangian Floer homology is presented: In dimension two, we construct a Floer homology generated by primary homoclinic points. We compute two examples and prove an invariance theorem. Moreover,…

In this note we present a brief introduction to Lagrangian Floer homology and its relation with the solution of Arnol'd conjecture, on the minimal number of non-degenerate fixed points of a Hamiltonian diffeomorphism. We start with the…

We compare Hofer's geometries on two spaces associated with a closed symplectic manifold M. The first space is the group of Hamiltonian diffeomorphisms. The second space L consists of all Lagrangian submanifolds of $M \times M$ which are…

Let $L$ be a closed orientable Lagrangian submanifold of a closed symplectic six-manifold $(X, \omega)$. We assume that the first homology group $H_1 (L ; A)$ with coefficients in a commutative ring $A$ injects into the group $H_1 (X ; A)$…

Mirror symmetry gives predictions for the genus zero Gromov-Witten invariants of a closed Calabi--Yau variety in terms of period integrals on a mirror family of Calabi-Yau varieties. We deduce an analogous mirror theorem for the open…

Using the symplectic geometry of certain manifolds which appear naturally in Lie theory, we define an invariant which assigns a graded abelian group to an oriented link. The relevant manifolds are transverse slices to certain nilpotent…