arXiv research

This paper is concerned with the rational symplectic field theory in the Floer case. For this observe that in the general geometric setup for symplectic field theory the contact manifolds can be replaced by mapping tori of symplectic manifolds with symplectomorphisms. While the cylindrical contact homology is given by …

Study spectral invariants over integers, discovering unboundedness and field-dependence.

Area-preserving diffeomorphisms of a 2-disc can be regarded as time-1 maps of (non-autonomous) Hamiltonian flows on solid tori, periodic flow-lines of which define braid (conjugacy) classes, up to full twists. We examine the dynamics relative to such braid classes and define a braid Floer homology. This refinement of t…

Rabinowitz Floer homology is the semi-infinite dimensional Morse homology associated to the Rabinowitz action functional used in the pioneering work of Rabinowitz. Gradient flow lines are solutions of a vortex-like equation. In this survey article we describe the construction of Rabinowitz Floer homology and its applic…

We study the heat flow in the loop space of a closed Riemannian manifold $M$ as an adiabatic limit of the Floer equations in the cotangent bundle. Our main application is a proof that the Floer homology of the cotangent bundle, for the Hamiltonian function kinetic plus potential energy, is naturally isomorphic to the h…

A $g$-tuple of disjoint, linearly independent circles in a Riemann surface of genus $g$ determines a `Heegaard torus' in its $g$-fold symmetric product. Changing the circles by a handleslide produces a new torus. It is proved that, for symplectic forms with certain properties, these two tori are Hamiltonian-isotopic La…

Defines Floer homology with DG coefficients for symplectic manifolds.

Periodic Floer homology (PFH) is a Gromov-Floer type invariant for fibered three-manifolds with Hamiltonian structures. The cobordism maps on periodic Floer homology induced by symplectic cobordisms are currently only defined indirectly by using Seiberg-Witten theory. In this paper, we investigate the cobordism maps in…

In this article we prove existence of Reeb orbits for Bohr-Sommerfeld Legendrians in certain pre-quantization spaces. We give a quantitative estimate from below. These estimates are obtained by studying Floer homology for fibre-wise quadratic Hamiltonian functions on negative line bundles.

We show that the Hamiltonian Lagrangian monodromy group, in its homological version, is trivial for any weakly exact Lagrangian submanifold of a symplectic manifold. The proof relies on a sheaf approach to Floer homology given by a relative Seidel morphism.

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 …

Proves Arnold conjecture for singular symplectic manifolds using novel techniques.

This is the first part of an article in two parts, which builds the foundation of a Floer-theoretic invariant, (I_F). (See math.DG/0505013 for part II). The Floer homology can be trivial in many variants of the Floer theory; it is therefore interesting to consider more refined invariants of the Floer complex. We consid…

This is a sequel to the paper [Oh5] (or ArXiv:math.SG/0206092). The main purpose of the paper is to give the proof of an existence theorem, with energy bounds, of certain pseudo-holomorphic sections of the mapping cylinder that is needed for the proof of nondegeneracy of the homological invariant pseudo-norm which the …

In this paper we establish the existence of periodic orbits belonging to any $σ$-atoroidal free homotopy class for Hamiltonian systems in the twisted disc bundle, provided that the compactly supported time-dependent Hamiltonian function is sufficiently large over the zero section and the magnitude of the weakly exact $…

We show that if K: P \to R is an autonomous Hamiltonian on a symplectic manifold (P,Ω) which attains 0 as a Morse-Bott nondegenerate minimum along a symplectic submanifold M, and if c_1(TP)|_M vanishes in real cohomology, then the Hamiltonian flow of K has contractible periodic orbits with bounded period on all suffici…

In this paper we calculate the Lagrangian Floer homology $HF(L_0, L_1 : {\mathbb Z}_2)$ of a pair of real forms $(L_0,L_1)$ in a monotone Hermitian symmetric space $M$ of compact type in the case where $L_0$ is not necessarily congruent to $L_1$. In particular, we have a generalization of the Arnold-Givental inequality…

Floer invented his theory in the mid eighties in order to prove the Arnol'd conjectures on the number of fixed point of Hamiltonian diffeomorphisms and Lagrangian intersections. Over the last thirty years, many versions of Floer homology have been constructed. In symplectic and contact dynamics and geometry they have b…

Paper proves equivalence of two Floer theories using pearly trees and Hamiltonian flows.

We give a construction of the Floer homology of the pair of {\it non-compact} Lagrangian submanifolds, which satisfies natural continuity property under the Hamiltonian isotopy which moves the infinity but leaves the intersection set of the pair compact. This construction uses the concept of Lagrangian cobordism and ce…

The study connects ECH capacities to Anosov flows, proving infinite capacities and obstructions.

In this paper we use Floer theory to study topological restrictions on Lagrangian embeddings in closed symplectic manifolds. One of the phenomena arising from our results is ``homological rigidity'' of Lagrangian submanifolds. Namely, in certain symplectic manifolds, conditions on low dimensional topological invariants…

Extends Khovanov homology spectral sequence using Heegaard Floer homology.

Study links with annuli using sutured Floer homology.

Knot Floer homology matches fixed point Floer for fibred knots.

New link detection results using knot and link Floer homology.

In this paper we show how the rich algebraic formalism of Eliashberg-Givental-Hofer's symplectic field theory (SFT) can be used to define higher algebraic structures in Hamiltonian Floer theory. Using the SFT of Hamiltonian mapping tori we show how to define a homotopy extension of the well-known Lie bracket and discus…

Paper classifies Heegaard Floer minimal knots in sutured manifolds.

New colored knot Floer homology defined using infinite full twists.

New algebraic method for knot Floer homology computation.

Link Floer homology detects split links.

Lecture notes on Heegaard Floer homology for 3-manifolds and knots.

Maps from rational homology solid tori yield rank inequalities in Heegaard Floer homology.

Proves properties of instanton knot Floer homology and connected sum formula.

Study on knot concordance and homology cobordism using Heegaard Floer homology.

Instanton Floer homology matches Heegaard Floer for almost-rational plumbings.

Introduces linear K-systems for Hamiltonian Floer theory.

Develops a new method for equivariant Lagrangian Floer homology using symplectic homotopy quotients.

Lecture notes on Heegaard Floer homology for beginners.

Study shows rank of knot Floer homology detects Hopf links and classifies second smallest links.

Introduces Floer lasagna modules using link Floer homology.

Real Heegaard Floer Homology extends Li's real monopole Floer homology.

Study shows inequality in Floer homologies for 3-manifold covers.

New method for Lagrangian Floer homology groups using flow trees.

Study Brieskorn spheres using Floer homology, generating infinite rank summands in homology cobordism.

Contact gluing maps are shown to be equivalent in sutured Floer homology.

Formula for Heegaard Floer multicurves of double tangles from knot complements.

We define an extended field theory in dimensions $1+1+1$, that takes the form of a `quasi 2-functor' with values in a strict 2-category $\widehat{\mathcal{H}am}$, defined as the `completion of a partial 2-category' $\mathcal{H}am$, notions which we define. Our construction extends Wehrheim and Woodward's Floer Field th…