Related papers: Cluster Homology

We study the flux homomorphism for closed forms of arbitrary degree, with special emphasis on volume forms and on symplectic forms. The volume flux group is an invariant of the underlying manifold, whose non-vanishing implies that the…

Lagrangian Floer homology in a general case has been constructed by Fukaya, Oh, Ohta and Ono, where they construct an $\AI$-algebra or an $\AI$-bimodule from Lagrangian submanifolds, and studied the obstructions and deformation theories.…

We develop a general algebraic framework involving "Poincar\'e--Novikov structures" and "filtered matched pairs" to provide an abstract approach to the barcodes associated to the homologies of interlevel sets of $\mathbb{R}$- or…

We use closed geodesics to construct and compute Bott-type Morse homology groups for the energy functional on the loop space of flat $n$-dimensional tori, $n\ge 1$, and Bott-type Floer cohomology groups for their cotangent bundles equipped…

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…

We define Lagrangian Floer cohomology over $\mathbb Z_2$-coefficients by counting pearly trajectories for graded, exact Lagrangian immersions that satisfy certain positivity condition on the index of the non-embedded points, and show that…

Given a closed, orientable Lagrangian submanifold $L$ in a symplectic manifold $(X, \omega)$, we show that if $L$ is relatively exact then any Hamiltonian diffeomorphism preserving $L$ setwise must preserve its orientation. In contrast to…

We prove that Floer theory induces a filtration by ideals on equivariant quantum cohomology of symplectic manifolds equipped with a $\mathbb{C}^*$-action. In particular, this gives rise to Hilbert-Poincar\'e polynomials on ordinary…

We use Floer theory to describe invariants of symplectic $\mathbb{C}^*$-manifolds admitting several commuting $\mathbb{C}^*$-actions. The $\mathbb{C}^*$-actions induce filtrations by ideals on quantum cohomology, as well as filtrations on…

Let (M,\omega) be a compact symplectic manifold, and \phi be a symplectic diffeomorphism on M, we define a Floer-type homology FH_*(\phi) which is a gen- eralization of Floer homology for symplectic fixed points defined by Dostoglou and…

We introduce a homology theory whose Euler characteristics counts ASD bundles over four dimensional co-associative submanifolds in (almost) G_2 manifolds. As a TQFT, in relative situations, we have the Fukaya-Floer category of Lagrangians…

Positive loops of Legendrian embeddings are examined from the point of view of Floer homology of Lagrangian cobordisms. This leads to new obstructions to the existence of a positive loop containing a given Legendrian, expressed in terms of…

The goal of this paper is to set up the general framework of higher-dimensional Heegaard Floer homology, define the contact class, and use it to give an obstruction to the Liouville fillability of a contact manifold and a sufficient…

We apply Heegaard Floer homology to study deformations of singularities of plane algebraic curves. Our main result provides an obstruction to the existence of a deformation between two singularities. Generalizations include the case of…

In this note we give examples of Hamiltonian diffeomorphisms which are on one hand dynamically complicated, for instance with positive topological entropy, and on the other hand minimal from the perspective of Floer theory. The minimality…

In this note, we introduce a relative (or Lagrangian) version of the Seidel homomorphism that assigns to each homotopy class of paths in Ham(M), starting at the identity and ending on the subgroup that preserves a given Lagrangian…

We define a notion of Heegaard Floer homology for three dimensional orbifolds with arbitrary cyclic singularities, generalizing the recent work of Biji Wong where the singular locus is assumed to be connected.

We focus on L-spaces for which the boundary maps of the Heegaard Floer chain complexes vanish. In previous paper \cite{Usui}, we collect such manifolds systematically by using the smoothing order on links. In this paper, we classify such…

It follows implicitly from recent work in Heegaard Floer theory that lens spaces are homology cobordant exactly when they are oriented homeomorphic. We provide a new combinatorial proof using the Heegaard Floer d-invariants, which…

Bordered Floer homology is an invariant for 3-manifolds with boundary, defined by the authors in 2008. It extends the Heegaard Floer homology of closed 3-manifolds, defined in earlier work of Zolt\'an Szab\'o and the second author. In…