In this paper, we apply spectral invariants, constructed in [Oh5,8], to the study of Hamiltonian diffeomorphisms of closed symplectic manifolds $(M,ω)$. Using spectral invariants, we first construct an invariant norm called the {\it spectral norm} on the Hamiltonian diffeomorphism group and obtain several lower bounds for the spectral norm in terms of the $\e$-regularity theorem and the symplectic area of certain pseudo-holomorphic curves. We then apply spectral invariants to the study of length minimizing properties of certain Hamiltonian paths {\it among all paths}. In addition to the construction of spectral invariants, these applications rely heavily on the {\it chain level Floer theory} and on some existence theorems with energy bounds of pseudo-holomorphic sections of certain Hamiltonian fibrations with prescribed monodromy. The existence scheme that we develop in this paper in turn relies on some careful geometric analysis involving {\it adiabatic degeneration} and {\it thick-thin decomposition} of the Floer moduli spaces which has an independent interest of its own.
We assume that $(M,ω)$ is {\it strongly semi-positive} throughout, which will be removed in a sequel.
In this paper we first apply the chain level Floer theory to the study of Hofer's geometry of Hamiltonian diffeomorphism group in the cases without quantum contribution: we prove that any quasi-autonomous Hamiltonian path on weakly exact symplectic manifolds or any autonomous Hamiltonian path on arbitrary symplectic manifolds is length minimizing in its homotopy class with fixed ends, as long as it has a fixed maximum and a fixed minimum which are not over-twisted and all of its contractible periodic orbits of period less than one are sufficiently $C^1$-small. Next we give a construction of ne…
We prove that the Knot Floer homology group of a fibred knot of genus g in the Alexander grading 1-g is isomorphic to a version of the fixed point Floer homology of an area-preserving representative of the monodromy.
We define an index for the critical points of parametrized Hamiltonian action functionals. The expected dimension of moduli spaces of parametrized Floer trajectories equals the difference of indices of the asymptotes.
Let K \subset Y be a knot in a three manifold which admits a longitude-framed surgery such that the surgered manifold has first Betti number greater than that of Y. We find a formula which computes the twisted Floer homology of the surgered manifold, in terms of twisted knot Floer homology. Using this, we compute the twisted Heegaard Floer homology \underline{HF}^+ of the mapping torus of a diffeomorphism of a closed Riemann surface whose mapping class is periodic, giving an almost complete description of the structure of these groups. When the mapping class is nontrivial, we find in particula…
We prove a very general Weyl-type law for Periodic Floer Homology, estimating the action of twisted Periodic Floer Homology classes over essentially any coefficient ring in terms of the grading and the degree, and recovering the Calabi invariant of Hamiltonians in the limit. We also prove a strong non-vanishing result, showing that under a monotonicity assumption which holds for a dense set of maps, the Periodic Floer Homology has infinite rank. An application of these results yields that a $C^{\infty}$-generic area-preserving diffeomorphism of a closed surface has a dense set of periodic poin…
The symplectic Floer homology HF_*(f) of a symplectomorphism f:S->S encodes data about the fixed points of f using counts of holomorphic cylinders in R x M_f, where M_f is the mapping torus of f. We give an algorithm to compute HF_*(f) for f a surface symplectomorphism in a pseudo-Anosov or reducible mapping class, completing the computation of Seidel's HF_*(h) for h any orientation-preserving mapping class.
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.
Let (M,ω) be a compact symplectic manifold, and ϕbe a symplectic diffeomorphism on M, we define a Floer-type homology FH_*(ϕ) which is a gen- eralization of Floer homology for symplectic fixed points defined by Dostoglou and Salamon for monotone symplectic manifolds. These homology groups are modules over a suitable Novikov ring and depend only on ϕup to a Hamiltonian isotopy.
We define partial quasi-morphisms on the group of Hamiltonian diffeomorphisms of the cotangent bundle using the spectral invariants in Lagrangian Floer homology with conormal boundary conditions, where the product compatible with the PSS isomorphism and the homological intersection product is lacking.
We construct partial category-valued field theories in (2+1)-dimensions using Lagrangian Floer theory in moduli spaces of central-curvature unitary connections with fixed determinant of rank r and degree d where r,d are coprime positive integers. These theories associate to a closed, connected, oriented surface the Fukaya category of the moduli space, and to a connected bordism between two surfaces a functor between the Fukaya categories. We obtain the latter by combining Cerf theory with holomorphic quilt invariants.
Starting from a Heegaard splitting of a three-manifold, we use Lagrangian Floer homology to construct a three-manifold invariant, in the form of a relatively Z/8-graded abelian group. Our motivation is to have a well-defined symplectic side of the Atiyah-Floer Conjecture, for arbitrary three-manifolds. The symplectic manifold used in the construction is the extended moduli space of flat SU(2)-connections on the Heegaard surface. An open subset of this moduli space carries a symplectic form, and each of the two handlebodies in the decomposition gives rise to a Lagrangian inside the open set. In…
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 develop a new approach to Lagrangian-Floer gluing. The construction of the gluing map is based on the intersection theory in some Hilbert manifold of paths $\mathcal{P} $. We consider some moduli spaces of perturbed holomorphic curves whose domains are either strips or more general Riemann surfaces with strip-like ends. These moduli spaces can be injectively immersed into the Hilbert manifold $\mathcal{P} $ by taking the restriction to non-Lagrangian boundary. Some subsets $ \mathcal{M}^T(U) $ and $ \mathcal{M}^{\infty} (U) $ of the aforementioned moduli spaces of perturbed holomorphic stri…
On any closed symplectic manifold we construct a path-connected neighborhood of the identity in the Hamiltonian diffeomorphism group with the property that each Hamiltonian diffeomorphism in this neighborhood admits a Hofer and spectral length minimizing path to the identity. This neighborhood is open in the $C^1-$topology. The construction utilizes a continuation argument and chain level result in the Floer theory of Lagrangian intersections.
In this paper, we study the behavior of the local Floer homology of an isolated fixed point and the growth of the action gap under iterations. To be more specific, we prove that an isolated fixed point of a Hamiltonian diffeomorphism remains isolated for a certain class of iterations (the so-called admissible iterations) and that the local Floer homology groups for all such iterations are isomorphic to each other up to a shift of degree. Furthermore, we study the pair-of-pants product in local Floer homology, and characterize a particular class of isolated fixed points (the symplectically dege…
We define the $S^1$-equivariant Rabinowitz-Floer homology of a bounding contact hypersurface $Σ$ in an exact symplectic manifold, and show by a geometric argument that it vanishes if $Σ$ is displaceable. In the appendix we describe an approach to transversality for Floer homologies for which the moduli space $\widehat M_J$ of all gradient flow lines is compact for some almost complex structure $J$. This approach uses a large set of perturbations, namely vector fields on the loop space, and selects from the possibly non-compact perturbed moduli spaces a part near $\widehat M_J$ that turns out t…
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.
We consider symplectic Floer homology in the lowest nontrivial dimension, that is to say, for area-preserving diffeomorphisms of surfaces. Particular attention is paid to the quantum cap product; we show that it distinguishes the trivial element of the mapping class group from any nontrivial one.
We say that a fixed point of a diffeomorphism is non-degenerate if 1 is not an eigenvalue of the linearization at the fixed point. We use pseudo-holomorphic curves techniques to prove the following: the inclusion map $$i: \text{Diff} ^{1} (S ^{2} ) \to \text{Diff} (S^2)$$ vanishes on all homotopy groups, where $\text{Diff} ^{1} (S^{2} ) \subset \text {Diff} (S^{2} )$ denotes the space of orientation preserving diffeomorphisms of $S ^{2} $ with a prescribed non-degenerate fixed point. This complements the classical results of Smale and Eels and Earl.
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.
We construct absolute and relative versions of Hamiltonian Floer homology algebras for strongly semi-positive compact symplectic manifolds with convex boundary, where the ring structures are given by the appropriate versions of the pair-of-pants products. We establish the absolute and relative Piunikhin-Salamon-Schwarz isomorphisms between these Floer homology algebras and the corresponding absolute and relative quantum homology algebras. As a result, the absolute and relative analogues of the spectral invariants on the group of compactly supported Hamiltonian diffeomorphisms are defined.
This paper extends the definition of Rabinowitz Floer homology to non-compact hypersurfaces. We present a general framework for the construction of Rabinowitz Floer homology in the non-compact setting under suitable compactness assumptions on the periodic orbits and the moduli spaces of Floer trajectories. We introduce a class of hypersurfaces being the level sets of specific Hamiltonians: strongly tentacular Hamiltonians, for which the compactness conditions are satisfied, thus enabling us to define the Rabinowitz Floer homology for this class. Rabinowitz Floer homology in turn serves as a to…
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…
We construct the vortex Floer homology group $VHF (M,μ;H)$ for an aspherical Hamiltonian $G$-manifold $(M, ω)$ with moment map $μ$ and a class of $G$-invariant Hamiltonian loop $H_t$, following the proposal of [3]. This is a substitute for the ordinary Hamiltonian Floer homology of the symplectic quotient of $M$. We achieve the transversality of the moduli space by the classical perturbation argument instead of the virtual technique, so the homology can be defined over ${\mathbb Z}$ or ${\mathbb Z}_2$.
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 fill a gap in the proof of the transversality result for quilted Floer trajectories in arXiv:0905.1370 by addressing trajectories for which some but not all components are constant. Namely we show that for generic sets of split Hamiltonian perturbations and split almost complex structures, the moduli spaces of parametrized quilted Floer trajectories of a given index are smooth of expected dimension. An additional benefit of the generic split Hamiltonian perturbations is that they perturb the given cyclic Lagrangian correspondence such that any geometric composition of its factors is transve…
We calculate the Heegaard Floer homologies$HF^+(M,s) for mapping tori M associated to certain surface diffeomorphisms, where s is any Spin^c structure on M whose first Chern class is non-torsion. Let gamma and delta be a pair of geometrically dual nonseparating curves on a genus g Riemann surface Sigma_g, and let sigma be a curve separating Sigma_g into components of genus 1 and g-1. Write t-gamma, t_delta, and t_sigma for the right-handed Dehn twists about each of these curves. The examples we consider are the mapping tori of the diffeomorphisms t_gamma^m circ t_delta^n for m,n in Z and that …
The main purpose of this paper is to provide a description of the fundamental group of a symplectic manifold in terms of Floer theoretic objects. As an application, we show that when counted with a suitable notion of multiplicity, non degenerate Hamiltonian diffeomorphisms have enough fixed points to generate the fundamental group.
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 the Floer homology originally used for the Arnol'd Conjecture yields a Morse-type forcing theory for periodic points of area-preserving diffeomorphisms of the 2-disc based on braiding. Contributions of this paper include (1) a monotonicity lemma for the behavior of the nonlinear C…
We construct an embedding $Φ$ of $[0,1]^{\infty}$ into $Ham(M, ω)$, the group of Hamiltonian diffeomorphisms of a suitable closed symplectic manifold $(M, ω)$. We then prove that $Φ$ is in fact a quasi-isometry. After imposing further assumptions on $(M, ω)$, we adapt our methods to construct a similar embedding of $\mathbb{R} \oplus [0,1]^{\infty}$ into either $Ham(M, ω)$ or $\widetilde{Ham}(M, ω)$, the universal cover of $Ham(M, ω)$. Along the way, we prove results related to the filtered Floer chain complexes of radially symmetric Hamiltonians. Our proofs rely heavily on a continuity result…
We define Floer homology for a time-independent, or autonomous Hamiltonian on a symplectic manifold with contact type boundary, under the assumption that its 1-periodic orbits are transversally nondegenerate. Our construction is based on Morse-Bott techniques for Floer trajectories. Our main motivation is to understand the relationship between linearized contact homology of a fillable contact manifold and symplectic homology of its filling.
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.
In this paper, we use Floer theory to study the Hofer length functional for paths of Hamiltonian diffeomorphisms which are sufficiently short. In particular, the length minimizing properties of a short Hamiltonian path are related to the properties and number of its periodic orbits.
In this paper we study the Lagrangian Floer theory over $\Z$ or $\Z_2$. Under an appropriate assumption on ambient symplectic manifold, we show that the whole story of Lagrangian Floer theory in \cite{fooo-book} can be developed over $\Z_2$ coefficients, and over $\Z$ coefficients when Lagrangian submanifolds are relatively spin. The main technical tools used for the construction are the notion of the sheaf of groups, and stratification and compatibility of the normal cones applied to the Kuranishi structure of the moduli space of pseudo-holomorphic discs.
We introduce a new notion of persistence modules endowed with operators. It encapsulates the additional structure on Floer-type persistence modules coming from the intersection product with classes in the ambient (quantum) homology, along with a few other geometric situations. We provide sample applications to the $C^0$-geometry of Morse functions and to Hofer's geometry of Hamiltonian diffeomorphisms, that go beyond spectral invariants and traditional persistent homology.
Family Floer theory yields a functor from the Fukaya category of a symplectic manifold admitting a Lagrangian torus fibration to a (twisted) category of perfect complexes on the mirror rigid analytic space. This functor is shown to be faithful by a degeneration argument involving moduli spaces of annuli.
Lipshitz, Ozsváth and Thurston defined a bordered Heegaard Floer invariant CFDA for 3-manifolds with two boundary components, including mapping cylinders for surface diffeomorphisms. We define a related invariant for certain 4-dimensional cobordisms with corners, by associating a morphism F from CFDA(f) to CFDA(g) to each such cobordism between two mapping cylinders f and g. Like the Ozsváth-Szabó invariants of cobordisms between closed 3-manifolds, this morphism arises from counting holomorphic triangles on Heegaard triples. We demonstrate that the homotopy class of the morphism F only depend…
In the first part of the present series of papers, we studied the moduli spaces of holomorphic discs and strips into an open symplectic manifold, isomorphic to the complement of a smooth divisor in a closed symplectic manifold. In particular, we introduced a compactification of this moduli space, which is called the RGW compactification. The goal of this paper is to show that the RGW compactifications admit Kuranishi structures. This result provides the crucial ingredient for the main construction of this series of papers: Floer homology for monotone Lagrangians in a smooth divisor complement.
We present a set of notes on Morse Homology, which grew out of lectures the first named autor gave at Ludwig-Maximilian University in Munich, Seoul National University, and the University of Augsburg. Although we do not discuss Floer homology in these notes, they are written having Floer homology in mind. Therefore, we use concepts which can be generalized to the semi-infinite dimensional case. In the finite dimensional case, they might not always be the most efficient ones but on the other hand, the finite dimensional case has the advantage that these concepts can be visualized much more easi…
We use Liu-Tian's virtual moduli cycle methods to construct detailedly the explicit isomorphism between Floer homology and quantum homology for any closed symplectic manifold that was first outlined by Piunikhin, Salamon and Schwarz for the case of the semi-positive symplectic manifolds.
Bordered Heegaard Floer homology is an invariant for 3-manifolds, which associates to a surface F an algebra A(Z), and to a 3-manifold Y with boundary, together with an orientation-preserving diffeomorphism ϕfrom F to \bdy Y, a module over A(Z). We study the Grothendieck group of modules over A(Z), and define an invariant lying in this group for every bordered 3-manifold. We prove that this invariant recovers the kernel of the inclusion of H_1(\bdy Y; Z) into H_1(Y; Z) if H_1(Y, \bdy Y; Z) is finite, and is 0 otherwise. We also study the properties of this invariant corresponding to gluing. As…
We define a new homology theory we call symbol homology by using decorated moduli spaces of Whitney polygons. By decorating different types of moduli spaces we obtain different flavors of this homology theory together with morphisms between them. Each of these flavors encodes the properties of a different type of Heegaard Floer homology. The morphisms between the symbol homologies enable us to push properties from one Floer theory to a different one. Furthermore, we obtain a new presentation of Heegaard Floer theory in which maps correspond to multiplication from the right with suitable elemen…
Let M be a closed, oriented, n-dimensional manifold. In this paper we describe a spectrum in the sense of homotopy theory, Z(T^*M), whose homology is naturally isomorphic to the Floer homology of the cotangent bundle, T^*M. This Floer homology is taken with respect to a Hamiltonian H: S^1 x T^*M --> R, which is quadratic near infinity. Z(T^*M) is constructed assuming a basic smooth gluing result of J-holomorphic cylinders. This spectrum will have a C.W decomposition with one cell for every periodic solution of the equation defined by the Hamiltonian vector field X_H. Its induced cellular chain…
We develop techniques for defining and working with virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves which are not necessarily cut out transversally. Such techniques have the potential for applications as foundations for invariants in symplectic topology arising from "counting" pseudo-holomorphic curves. We introduce the notion of an *implicit atlas* on a moduli space, which is (roughly) a convenient system of local finite-dimensional reductions. We present a general intrinsic strategy for constructing a canonical implicit atlas on any moduli space of pseudo-holomorphic…
Bordered Heegaard Floer homology is a three-manifold invariant which associates to a surface F an algebra A(F) and to a three-manifold Y with boundary identified with F a module over A(F). In this paper, we establish naturality properties of this invariant. Changing the diffeomorphism between F and the boundary of Y tensors the bordered invariant with a suitable bimodule over A(F). These bimodules give an action of a suitably based mapping class group on the category of modules over A(F). The Hochschild homology of such a bimodule is identified with the knot Floer homology of the associated op…
Inspired by Hutchings' elementary alternative to ECH capacities, we introduce an elementary alternative to spectral invariants defined via periodic Floer homology (PFH). We use these spectral invariants to provide more elementary proofs of a number of results which have recently been obtained using PFH spectral invariants. Among these results are quantitative closing lemmas for area-preserving surface diffeomorphisms and the simplicity conjecture.
