This paper establishes the orderability of contact manifolds which are quotients of fillable contact manifolds under finite group actions compatible with the filling, the prototypical example being $\mathbb{R}P^{2n-1}$ as the quotient of $S^{2n-1}$. Our approach employs an equivariant formulation of the so-called contact Floer cohomology theory. This leads us to develop an analogue of Givental's nonlinear Maslov index using the $\mathbf{k}[[x]]$-module structure on an equivariant version of contact Floer cohomology. A key idea is that mapping cones of continuation maps detect crossings with the discriminant (recall that Givental's index is a continuous integer valued function on the complement of the discriminant). To properly handle the inherent non-canonicity in defining such mapping cones, we lift the structure of contact Floer cohomology to chain level by defining it as an $\infty$-functor on a suitable $\infty$-categorification of the Eliashberg-Polterovich orderability relation on the universal cover of the contactomorphism group.
A positive contactomorphism of a contact manifold $M$ is the end point of a contact isotopy on $M$ that is always positively transverse to the contact structure. Assume that $M$ contains a Legendrian sphere $Λ$, and that $(M,Λ)$ is fillable by a Liouville domain $(W,ω)$ with exact Lagrangian $L$ such that $ω|_{π_2(W,L)}=0$. We show that if the exponential growth of the action filtered wrapped Floer homology of $(W,L)$ is positive, then every positive contactomorphism of $M$ has positive topological entropy. This result generalizes the result of Alves and Meiwes from Reeb flows to positive cont…
We prove that if a contact manifold admits an exact filling then every local contactomorphism isotopic to the identity admits a translated point in the interior of its support, in the sense of Sandon [San11b]. In addition we prove that if the Rabinowitz Floer homology of the filling is non-zero then every contactomorphism isotopic to the identity admits a translated point, and if the Rabinowitz Floer homology of the filling is infinite dimensional then every contactmorphism isotopic to the identity has either infinitely many translated points, or a translated point on a closed leaf. Moreover i…
We prove a version of Sandon's conjecture on the number of translated points of contactomorphisms for the case of prequantization bundles over certain closed monotone symplectic toric manifolds. Namely we show that any contactomorphism of such a prequantization bundle lying in the identity component of the contactomorphism group possesses at least $N$ translated points, where $N$ is the minimal Chern number of the symplectic toric manifold. The proof relies on the theory of generating functions coupled with equivariant cohomology, whereby we adapt Givental's approach to the Arnold conjecture f…
Let $S^*Q$ be the spherization of a closed connected manifold of dimension at least two. Consider a contactomorphism $φ$ that can be reached by a contact isotopy that is everywhere positively transverse to the contact structure. In other words, $φ$ is the time-1-map of a time-dependent Reeb flow. We show that the volume growth of $φ$ is bounded from below by the topological complexity of the loop space of $Q$. Denote by $ΩQ_0(q)$ the component of the based loop space that contains the constant loop. We show that if the fundamental group or the homology of $ΩQ_0(q)$ grows exponentially, then th…
Using a simplified version of Kuranishi perturbation theory that we call semi-global Kuranishi structures, we give a definition of the equivariant Lagrangian Floer cohomology of a pair of Lagrangian submanifolds that are fixed under a finite symplectic group action and satisfy certain simplifying assumptions.
The Floer cohomology of a symplectic automorphism and that of its square are related by the pair-of-pants product. For exact symplectic automorphisms, we introduce an equivariant version of that product, and use it to prove a Smith-type inequality of ranks between Floer cohomology groups. Under additional topological assumptions, the same inequality was previously proved by Hendricks, using a different strategy.
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 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…
Inspired by Kronheimer and Mrowka's approach to monopole Floer homology, we develop a model for $\mathbb{Z}/2$-equivariant symplectic Floer theory using equivariant almost complex structures, which admits a localization map to a twisted version of Floer cohomology in the invariant set. We then present applications to Steenrod operations on Lagrangian Floer cohomology, and to the Heegaard Floer homology of double covers of three-manifolds.
We study Liouville fillable contact manifolds $(Σ,ξ)$ with non-zero Rabinowitz Floer homology and assign spectral numbers to paths of contactomorphisms. As a consequence we prove that $\widetilde{\mathrm{Cont}_0}(Σ,ξ)$ is orderable in the sense of Eliashberg and Polterovich. This provides a new class of orderable contact manifolds. If the contact manifold is in addition periodic or a prequantization space $M \times S^1$ for $M$ a Liouville manifold, then we construct a contact capacity. This can be used to prove a general non-squeezing result, which amongst other examples in particular recover…
In this article we consider a variant of Rabinowitz Floer homology in order to define a homological count of discriminant points for paths of contactomorphisms. The growth rate of this count can be seen as an analogue of Givental's nonlinear Maslov index. As an application we prove a Bott-Samelson type obstruction theorem for positive loops of contactomorphisms.
For G a Lie group acting on a symplectic manifold $(M,ω)$ 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 $L_1$.
A contact manifold admittting a supporting contact form without contractible Reeb orbits is called hypertight. In this paper we construct a Rabinowitz Floer homology associated to an arbitrary supporting contact form for a hypertight contact manifold, and use this to prove versions of conjectures of Sandon and Mazzucchelli on the existence of translated points and invariant Reeb orbits, and to show that positive loops of contactomorphisms give rise to non-contractible Reeb orbits.
The objective of this paper is to clarify the relationships between the quantum D-module and equivariant Floer theory. Equivariant Floer theory was introduced by Givental in his paper ``Homological Geometry''. He conjectured that the quantum D-module of a symplectic manifold is isomorphic to the equivariant Floer cohomology for the universal cover of the free loop space. First, motivated by the work of Guest, we formulate the notion of ``abstract quantum D-module''which generalizes the D-module defined by the small quantum cohomology algebra. Second, we define the equivariant Floer cohomology …
Let $M$ be an exact symplectic manifold equal to a symplectization near infinity and having stably trivializable tangent bundle, and $ϕ$ be an exact symplectomorphism of $M$ which, near infinity, is equal to either the identity or the symplectization of a contactomorphism $\hatϕ$ such that neither $\hatϕ$ nor $\hatϕ^2$ has fixed points. We give conditions under which Seidel and Smith's localization theorem for Lagrangian Floer cohomology implies the existence of a spectral sequence from $\mathit{HF}(ϕ^2)\otimes \mathbb Z_2((θ))$ to $\mathit{HF}(ϕ)\otimes \mathbb Z_2((θ))$.
The aim of this article is to apply a Floer theory to study symmetric periodic Reeb orbits. We define positive equivariant wrapped Floer homology using a (anti-)symplectic involution on a Liouville domain and investigate its algebraic properties. By a careful analysis of index iterations, we obtain a non-trivial lower bound on the minimal number of geometrically distinct symmetric periodic Reeb orbits on a certain class of real contact manifolds. This includes non-degenerate real dynamically convex starshaped hypersurfaces in $\mathbb{R}^{2n}$ which are invariant under complex conjugation. As …
Seidel-Smith and Hendricks used equivariant Floer cohomology to define some spectral sequences from symplectic Khovanov homology and Heegaard Floer homology. These spectral sequences give rise to Smith-type inequalities. Similar-looking spectral sequences have been defined by Lee, Bar-Natan, Ozsváth-Szabó, Lipshitz-Treumann, Szabó, Sarkar-Seed-Szabó, and others. In this paper we give another construction of equivariant Floer cohomology with respect to a finite group action and use it to prove some invariance properties of these spectral sequences; prove that some of these spectral sequences ag…
We explain how to adapt the methods of Abouzaid-McLean-Smith to the setting of Hamiltonian Floer theory. We develop a language around equivariant ``$\langle k \rangle$-manifolds'', which are a type of manifold-with-corners that suffices to capture the combinatorics of Floer-theoretic constructions. We describe some geometry which allows us to straightforwardly adapt Lashofs's stable equivariant smoothing theory and Bau-Xu's theory of FOP-perturbations to $\langle k \rangle$-manifolds. This allows us to compatibly smooth global Kuranishi charts on all Hamiltonian Floer trajectories at once, in …
We construct shift operators on equivariant symplectic cohomology which generalise the shift operators on equivariant quantum cohomology in algebraic geometry. That is, given a Hamiltonian action of the torus $T$, we assign to a cocharacter of $T$ an endomorphism of $(S^1 \times T)$-equivariant Floer cohomology based on the equivariant Floer Seidel map. We prove the shift operator commutes with a connection. This connection is a multivariate version of Seidel's $q$-connection on $S^1$-equivariant Floer cohomology and generalises the Dubrovin connection on equivariant quantum cohomology. We pro…
Let $p \geq 2$ be a prime, and $\mathbb{F}_p$ be the field with $p$ elements. Extending a result of Seidel for $p=2,$ we construct an isomorphism between the Floer cohomology of an exact or Hamiltonian symplectomorphism $ϕ,$ with $\mathbb{F}_p$ coefficients, and the $\mathbb{Z}/p \mathbb{Z}$-equivariant Tate Floer cohomology of its $p$-th power $ϕ^p.$ The construction involves a Kaledin-type quasi-Frobenius map, as well as a $\mathbb{Z}/p \mathbb{Z}$-equivariant pants product: an equivariant operation with $p$ inputs and $1$ output. Our method of proof involves a spectral sequence for the acti…
Building on Seidel-Solomon's fundamental work, we define the notion of a $\mathfrak{g}$-equivariant Lagrangian brane in an exact symplectic manifold $M$ where $\mathfrak{g} \subset SH^1(M)$ is a sub-Lie algebra of the symplectic cohomology of $M$. When $M$ is a (symplectic) mirror to an (algebraic) homogeneous space $G/P$, homological mirror symmetry predicts that there is an embedding of $\mathfrak{g}$ in $SH^1(M)$. This allows us to study a mirror theory to classical constructions of Borel-Weil and Bott. We give explicit computations recovering all finite dimensional irreducible representati…
We construct finite group actions on Lagrangian Floer theory when symplectic manifolds have finite group actions and Lagrangian submanifolds have induced group actions. We first define finite group actions on Novikov-Morse theory. We introduce the notion of a {\em spin profile} as an obstruction class of extending the group action on Lagrangian submanifold to the one on its spin structure, which is a group cohomology class in $H^2(G;\Z/2)$. For a class of Lagrangian submanifolds which have the same spin profiles, we define a finite group action on their Fukaya category. In consequence, we obta…
The Seidel maps are two maps associated to a Hamiltonian circle action on a convex symplectic manifold, one on Floer cohomology and one on quantum cohomology. We extend their definitions to $S^1$-equivariant Floer cohomology and $S^1$-equivariant quantum cohomology based on a construction of Maulik and Okounkov. The $S^1$-action used to construct $S^1$-equivariant Floer cohomology changes after applying the equivariant Seidel map (a similar phenomenon occurs for $S^1$-equivariant quantum cohomology). We show the equivariant Seidel map on $S^1$-equivariant quantum cohomology does not commute wi…
We introduce group actions on polyfolds and polyfold bundles. We prove quotient theorems for polyfolds, when the group action has finite isotropy. We prove that the sc-Fredholm property is preserved under quotient if the base polyfold is infinite dimensional. The quotient construction is the main technical tool in the construction of equivariant fundamental class in [42]. We also analyze the equivariant transversality near the fixed locus in the polyfold setting. In the case of $S^1$-action with fixed locus, we give a sufficient condition for the existence of equivariant transverse perturbatio…
We survey the different versions of Floer homology that can be associated to three-manifolds. We also discuss their applications, particularly to questions about surgery, homology cobordism, and four-manifolds with boundary. We then describe Floer stable homotopy types, the related Pin(2)-equivariant Seiberg-Witten Floer homology, and its application to the triangulation conjecture.
In this paper the 5-dimensional contact SO(3)-manifolds are classified up to equivariant contactomorphisms. The construction of such manifolds with singular orbits requires the use of generalized Dehn twists. We show as an application that all simply connected 5-manifoldswith singular orbits are realized by a Brieskorn manifold with exponents (k,2,2,2). The standard contact structure on such a manifold gives right-handed Dehn twists, and a second contact structure defined in the article gives left-handed twists.
In a previous paper, we showed that the original definition of cylindrical contact homology, with rational coefficients, is valid on a closed three-manifold with a dynamically convex contact form. However we did not show that this cylindrical contact homology is an invariant of the contact structure. In the present paper, we define "nonequivariant contact homology" and "S^1-equivariant contact homology", both with integer coefficients, for a contact form on a closed manifold in any dimension with no contractible Reeb orbits. We prove that these contact homologies depend only on the contact str…
Gromov's famous non-squeezing theorem (1985) states that the standard symplectic ball cannot be symplectically squeezed into any cylinder of smaller radius. Does there exist an analogue of this result in contact geometry? Our main finding is that the answer depends on the sizes of the domains in question: We establish contact non-squeezing on large scales, and show that it disappears on small scales. The algebraic counterpart of the (non)-squeezing problem for contact domains is the question of existence of a natural partial order on the universal cover of the contactomorphisms group of a cont…
We define $S^1$-equivariant symplectic homology for symplectically aspherical manifolds with contact boundary, using a Floer-type construction first proposed by Viterbo. We show that it is related to the usual symplectic homology by a Gysin exact sequence. As an important ingredient of the proof, we define a parametrized version of symplectic homology, corresponding to families of Hamiltonian functions indexed by a finite dimensional smooth parameter space.
Given a symplectic cohomology class of degree 1, we define the notion of an equivariant Lagrangian submanifold. The Floer cohomology of equivariant Lagrangian submanifolds has a natural endomorphism, which induces a grading by generalized eigenspaces. Taking Euler characteristics with respect to the induced grading yields a deformation of the intersection number. Dehn twists act naturally on equivariant Lagrangians. Cotangent bundles and Lefschetz fibrations give fully computable examples. A key step in computations is to impose the "dilation" condition stipulating that the BV operator applied…
We define a $\mathbb{Z}_k$-equivariant version of the cylindrical contact homology used by Eliashberg-Kim-Polterovich (2006) to prove contact non-squeezing for prequantized integer-capacity balls $B(R) \times S^1 \subset \mathbb{R}^{2n} \times S^1$, $R \in \mathbb{N}$ and we use it to extend their result to all $R \geq 1$. Specifically we prove if $R \geq 1$ there is no $ψ\in \text{Cont}(\mathbb{R}^{2n} \times S^1)$, the group of compactly supported contactomorphisms of $\mathbb{R}^{2n} \times S^1$ which squeezes $\hat{B}(R) = B(R) \times S^1$ into itself, i.e. maps the closure of $\hat{B}(R)$…
In the previous paper, the author defined equivariant Floer cohomology for a complete intersection in a toric variety and showed that it is isomorphic to the small quantum D-module after a mirror transformation when the first Chern class c_1(M) of the tangent bundle is nef. In this paper, even when c_1(M) is not nef, we show that the equivariant Floer cohomology reconstructs the big quantum D-module under certain conditions on the ambient toric variety. The proof is based on a mirror theorem by Coates and Givental. The reconstruction procedure here gives a generalized mirror transformation fir…
