Definition of cylindrical contact homology in dimension three

Abstract

In this paper we give a rigorous definition of cylindrical contact homology for contact 3-manifolds that admit nondegenerate contact forms with no contractible Reeb orbits, and show that the cylindrical contact homology is an invariant of the contact structure. ContentsThe curves v 0 and v 1 are also equipped with asymptotic markers at the positive and negative ends; this will be described more precisely later. Let M = M J+ be the moduli space of curves v 1 from γ to γ satisfying the above. We want to glue v 0 to M/R (or its compactification M/R); see Figure 1.By a slight modification of [17], there is an obstruction bundle, † More precisely, 'tame', which is defined in Section 3.1. ‡ This will be our usual convention. § By a curve from γ + to γ − we mean a curve which is asymptotic to γ + at the positive end and to γ − at the negative end.Remark 6.5. The proof is modeled on but is substantially easier than that of [17, Proposition 3.2]. This is due to the fact that the J-holomorphic equation is linear near each R × γ. This allows us to dispense with the quadratic estimates. Proof. Let J J be the subset of J <L,reg,δ0 consisting of J that are C 1 -close to J. Also let π : M → J J be the bundle with fiber π −1 (J ) = M cyl J (γ + , γ + )/R, π : M → M be the bundle with fiber π −1 ([u], J ) = π −1 R,J ([u]), where π R,J : M cyl J (γ + , γ + ) → M cyl J (γ + , γ + )/R We first consider the moduli spaces of the form M ind=−k,cyl J τ l (γ + , γ − ) for k 1. There are only finitely many τ l ∈ [0, 1] such that M ind=−k,cyl J τ l (γ + , γ − ) is nonempty since {J τ } is generic; and, for each such τ l , if (v, r) ∈ M ind=−k,cyl J τ l (γ + , γ − ), then v is a k-fold cover of a simple curve u, where (u, r ) ∈ M ind=−1,cyl J τ l ((γ + ) 1/k , γ 1/k − ), γ + is the k-fold cover of (γ + ) 1/k , and γ − is the k-fold cover of γ 1/k − . Remark 7.5. For each J τ l -holomorphic cylinder from (γ + ) 1/k to γ 1/k − of index −1, without markers but after quotienting by automorphisms, there are m((γ + ) 1/k ) · m(γ 1/k − ) elements of M ind=−1,cyl J τ l ((γ + ) 1/k , γ 1/k − ) and k · m((γ + ) 1/k ) · m(γ 1/k − ) elements of M ind=−k,cyl J τ l (γ + , γ − ).

Contact homology and virtual fundamental cycles

“…Embedded contact homology was introduced and constructed by Hutchings and Hutchings-Taubes [Hut02, Hut09, HT07, HT09]. Cylindrical contact homology was constructed by Hutchings-Nelson [HN16] for dynamically convex contact three-manifolds and by Bao-Honda [BH18] for hypertight contact threemanifolds. Soon after the present paper was released, work of Bao-Honda [BH16] appeared, as well as work of Ishikawa [Ish18].…”

Automatic transversality in contact homology II: filtrations and computations

“…Pardon has defined full contact homology via virtual fundamental cycles but this approach is not applicable to defining cylindrical contact homology in the presence of contractible Reeb orbits. In dimension three, in the absence of contractible Reeb orbits, and when paired with the action filtered versions of [, Theorems 1.6 and 1.9], the definition provided by Bao–Honda in can be shown to be isomorphic to the cylindrical contact homology. Using virtual techniques, Bao–Honda give a definition of the full contact homology differential graded algebra for any closed contact manifold in any dimension.…”

“…In dimension three, in the absence of contractible Reeb orbits, and when paired with the action filtered versions of [, Theorems 1.6 and 1.9], the definition provided by Bao–Honda in can be shown to be isomorphic to the cylindrical contact homology. Using virtual techniques, Bao–Honda give a definition of the full contact homology differential graded algebra for any closed contact manifold in any dimension. The approaches of Pardon and the latter of Bao–Honda use Kuranishi structures to construct contact and symplectic invariants and while they hold more generally, they are more difficult to work with in computations and applications.…”

Complex cobordism, Hamiltonian loops and global Kuranishi charts

“…Siebert [62] also considers Kuranishi charts involving sections of powers of a bundle which is fibrewise ample on the universal curve. Honda and Bao [7] introduce a thickening of a moduli space of punctured curves in which the auxiliary fields are controlled by finite-dimensional sums of eigenspaces of an asymptotic Laplace-type operator associated to periodic orbits at the punctures.…”